Download - NN31545.1077 X - WUR E-depot home
1 NN315451077
X laquoOTA 1077 augustus 1978
Instituut voor Cultuurtechniek en Waterhuishouding Wageningen
[
A NUMERICAL MODEL FOR NON-STATIONARY SATURATED
GROUNDWATER FLOW IN A MULTI-LAYERED SYSTEM
i r P J T van Bakel
SpoundIcircIcirc r
amp
I Notas van het Instituut zijn in principe interne communicatiemiddeshylen dus geen officieumlle publikaties Hun inhoud varieert sterk en kan zowel betrekking hebben op een eenshyvoudige weergave van cijferreeksen als op een concluderende discussie van onderzoeksresultaten In de meeste gevallen zullen de conclusies van voorlopige aard zijn omdat het onderzoek nog niet is afgesloten Bepaalde notas komen niet voor verspreiding buiten het Instituut in aanmerking
E LANDBOUWCATALOGUS
0000 0044 7090
C O N T E N T S
b i z
I INTRODUCTION 1
II GENERAL DESCRIPTION OF THE PROBLEM 2
21 Physical background 2
211 The law of linear resistance (Darcys law) 2
212 The law of continuity 3
213 Combination of Darcys law and the law of
continuity 3
22 Hydraulic properties and types of aquifers 3
23 Schematization of flow 5
24 Water balance terms 6
25 Solution of groundwater flow problems 9
III NUMERICAL SOLUTIONS 10
31 Finite element method 10
32 Application of the finite element method on the
flow in groundwater basins 13
33 Computer program 17
IV INPUT EXECUTION AND OUTPUT 19
41 Input 19
411 Endogenous variables 19
412 Exogenous variables 21
413 Data input to Program FEMSAT 21
42 Running the program on the Cyber 7200 25
43 Output 26
44 Computing time 27
mdash 1
biz
V NUMERICAL EXPERIMENTS 27
51 Flow to a well in an unconfined circular-
shaped aquifer 27
511 Steady state flow in an unconfined aquifer 28
512 Unsteady-state flow in an unconfined aquifer 30
52 Flow to a well in a 3-layered system 33
VI SUMMARY AND CONCLUSIONS 37
VII REFERENCES 39
i
I INTRODUCTION
The groundwater system is part of the hydrological cycle and
changes in this system affects phenomena such as plant production
river flow etc Generally one tries to use groundwater resources
in such a way that the utility for the community as a whole is
maximal
Numerical flow models can be valuable tools in the management
of water resources In many situations these models are the only
possible way to indicate effects of certain interferences in the
groundwater system
In this paper a numerical model is presented which simulates
non-stationary saturated groundwater flow in a certain region
Purpose of the model is to predict effects of certain operations
upon the various terms of the water balance of the groundwater basin
and on the hydraulic head
The basic idea of the model is that groundwater is flowing
horizontally in waterbearing layers and vertically in less-permeable
layers Each layer is discretized into a number of elements To
each element both Darcys law and the law of continuity are applied
(chapter II)
In this way a set of equations is obtained which are solved
using a finite element method and the Gauss-Seidel iteration method
(chapter III)
In chapter IV the input and output of the model are treated
while in chapter V some numerical experiments are presented
Finally in chapter VI an evaluation of the model is given
II GENERAL DESCRIPTION OF THE PROBLEM
21 P h y s i c a l b a c k g r o u n d
Throughout this report water potential is expressed on a unit
weight basis and is being called hydraulic head with symbol h
h = 2- + z [L] Pg
where
mdash = piezometer head |L | Pg u -1
z = gravitational head |_Lj
The hydraulic head is an expression for the capacity of a unit
weight of water to do work as compared to the work capacity of the
same weight of water with the same chemical composition at a certain
reference level and under atmosferic pressure
The problem is restricted to saturated flow of groundwater with
one constant density
211 The law of linear resistance (Darcys law)
According to Darcys law the rate of flow through a porous medium
is proportional to the gradient in hydraulic head It may be written
as
ocirc h ltSh ocirch N q = _k Tmdash q = -k vmdash q = -k -rmdash ( 1 ) x y ocircx y y lt5y nz z ocircz
where
q = volume flux of water passing through a unit area per x y z
per unit time perpendicular to x y z direction
respectively |_LT J
k = hydraulic conductivity in x y and z direction x gt y gt z _ _ i _
respectively | LT J
The coefficient k in the Darcy flow equation is taken to be
a constant depending on both the properties of the porous medium
and the fluid
212 The law of continuity
The other important physical law states that no matter can be
lost or created
The conservation equation can be found by applying the principle
of continuity to an infinite small volume
6v 6v lt5v 60 x y z TT = T + x + ~T7 + 1 (2)
Ot OX OcircX oz
where
0 = volume of water per unit bulk volume of soil |_-J
q = sink or source term T J
213 Combination of Darcys law and the law of continuity
Darcys law combined with the law of continuity gives
60 lt5 oh lt5 ocirch 6 oh _ ocirct ocircx x ocircx ocircy y oy ocircz z ocircz
or
c 5h 6 ocirch ocirc Ocirch 6 ocirch c ot ox x ocircx ocircy y oy laquoz z 6z
where
S = volume of water released or stored per unit bulk c I
volume of soil per unit change in hydraulic head |L I
22 H y d r a u l i c p r o p e r t i e s a n d t y p e s o f
a q u i f e r s
The flow in a natural groundwater basin takes place in layers
with different hydraulic properties Usually the layers are divided
into good permeable layers or waterbearing layers (aquifers) and
less permeable layers (aquitards)
The most important hydraulic properties are
a Transmissiviy - the product of the average hydraulic conductivity
k and the saturated vertical thickness d of the waterbearing
layer
It is a measure for the transport capacity of aquifers
Notation T = kd |_ L T ~
b Hydraulic resistance - the ratio of vertical thickness of
aquitards d and hydraulic conductivity k It is a measure for
the resistance capacity of aquitards denoted as c = d k _ T J
c Specific yield - the volume of water released or stored in the
phreatic zone per unit surface area of an unconfined aquifer per
unit change in hydraulic head
It is also called effective porosity
The symbol used for specific storage will be S j_-J
d Specific storage - the volume of water released or stored per
unit volume of the saturated parts of an aquifer or aquitard per
unit change in hydraulic head It is a measure for the elasticity
of the soil material and the fluid Excluded are irreversible
processes Notation S _ L J
e Storage coefficient - the volume of water released or stored
pe7 nit surface of a layer per unit change in hydraulic head
It is the sum of specific yield and the product of specific
storage and thickness of the aquifer or aquitard
S = S + S d r - 1 c y s - -1
Note in confined layers S = 0 y
In fig 1 a 3-layered configuration is given of two aquifers
separated by an aquitard Aquifer I has a free water surface and is
an unconfined aquifer Aquifer II is a completely saturated aquifer
Dependent on the hydraulic resistance of the aquitard KRUSEMAN and
DE RIDDER (1976) distinguish between confined semi-confined and
semi-unconfined aquifers For our purpose it is not useful to make
this distinction We restrict the aquifer types to unconfined and
confined aquifers
WIHtUtttWttllttlt(ltmtttltHlttltlil(lltlittlfllnl ) bull (
d V U i l i l i ii i Hi il i M M bull f i m
i tMi| i i | J i j j i _ I I bull
aq ui tard
aquifer II
Fig 1 Schematic representatipn of flow direction and variation
of hydraulic head h in a 3-rlayered system
In an unconfined aquifers ( I ) poundhe vertical thickness pf the
waterbearing layer d is a variable while in a confined aquifer II
this is a constant (denoted as D ) Other important properties of
unconfined aquifers are the hydraulic head being the same as the
phreatic surface and the specific yield being different from zero
23 S c h eacute m a t i s a t i o n o f f l o w
The groundwater flow in a groundwater basin is schematized
into a horizontal flow in aquifers and into a vertical flow in
aquitards Consequently the vertical variation of the hydraulic
head is as drawn in figl
The validity of this assumption has )een investigated by
NEUMAN and WITHERSPOON (1969) The found that if the contrast
in hydraulic conductivity between two adjacent layers is bigger
than a factor two the relative error caused by this assumption
is usually smaller than 5
This schematization gives a simplification of the general flow
equation (eq 4 ) In an aquifer the vertical is zero and eq (4)
then reduces to
-Sbdquo -r- 7 - (k d -r-) + 7 mdash (k d 7-) + qd c lt5t ox x ox oy y oy
or
bull(S + S d) g y s ot
-r- (k d -=-) + -T- (k d -=-) + Q ocircx x ox oy y ocircy
(5)
where
Q = volume of water substracted from or added to unit surface
of aquifer per unit time |_ L T J
In an aquitard eq (4) reduces to
-sQ |pound = (k Q) s 6t oz Z OcircZ
(6)
As will be discussed in the next paragraph the term q in an
aquitard is set equal to zero
24 W a t e r b a l a n c e t e r m s
In eq (4) the term q is mentioned With this sink or source
term different terms of the water balance are taken into account
Fig 2 depicts schematically the situation for an unconfined aquifer
(for other layers some terms dont exist)
recharge- - discharge
laquo-b
Fig 2 Schematic representation of the various terms involved in the
water balance
Description of the different terms
a The flow through the phreatic surface q f
If this term is negative (water goes out of the saturated system)
it is called capillary rise If positive it is called percolation
(or drainage)
This flow can be calculated as a rest term of the water balance
of the unsaturated zone Very often this term is expressed as a
flow per unit surface (flux) and is called effective precipitation
For the time being this flux is a given input
b Flow to adjacent aquitards q
The flux to adjacent aquitards can be calculated by applying eq
(6) to the aquitard
c Flow to the primary surface water system q
The primary surface water system penetrates fully the groundwater
basin So it acts as a boundary with given values for the
hydraulic head (is equal to the phreatic level of the primary
system)
d Flow to the secondary surface water system q
The secondary surface water system does not penetrate fully the
groundwater basin which causes a resistance against flow between
groundwater and surface water The flow q can be calculated as
follows
h - h
- mdash f i i gt
where
h = hydraulic head of the groundwater at the place of
the secondary system |_ L |
h = phreatic level of secondary system |_ L J i s -
R = radial and entrance resistance of the
secondary system per unit surface |_ T J
According to ERNST (1962) the radial resistance H per unit
surface can be caKulated with
Uuml = 4 i n 4 e8) TTk B
where
1 = area drained by 1 m conduit L ^ J
d = thickness of the layer near the conduit _ L J
B = wet perimeter of the conduit |_ L J
The entrance resistance is caused by a layer which shows a
lower conductivity around the conduit than its surroundings
In practise radial and entrance resistancies are difficult to
deduct from hydraulic properties They can be found indirectly
by measuring simultaneously q h and h
t s
e Flow to the tertiary surface water system q
In smaller conduits usually the phreatic level is not known
So eq (7) cant be applied The flow to these conduits however
is sometimes very important (eg in areas with shallow groundwater
tables) One may a-sume (ERNST 1978) that a relationship exists
between the hydraulic head of an aquifor and the flow q
According to Ernst the tertiary system can be divided into a
number of categories with respect to their mutual distance and
bottom height For each category the flow is h - h (b)
q ( h b ) - mdash (9)
where
h = average hydraulic head | L J
h (b) = phreatic level in the category under
consideration a function of bottom height |_L J
a = geometry factor to convert h into h _ - J
Y = drainage resistance _ T J
hm = hydraulic head midway between two
conduits |_ L J m
Summation of the q(hb)-relations of the different categories
yields the overall relation between hydraulic head and flow to
the tertiary system we are looking for
f Artificial recharge or discharge q and prescribed boundary
flux qfo
The magnitudes of these flows are not influenced by the conditions
inside the groundwater basin but remain on a prescribed level
Thus q is an internal boundary condition q an external boundary cl D
condition
For confined aquifers some of the above mentioned terms dont
exist These terms are the flow through the phreatic surface qf
and the flow to the tertiary surface water system q
Aquitards are considered only as transmitters of water from one
aquifer to another with possibilities of storage (eq 6 ) This means
that for aquitards the water balance terms ar through f are set
equal to zero
25 S o l u t i o n o f g r o u n d w a t e r f l o w p r o b l e m s
The basic equations for saturated groundwater flow are eq (5)
for aquifers and eq (6) for aquitards The solution of these
equations require knowledge of the physical system and the auxiliary
conditions describing the system constraints
These auxiliary conditions are (REMSON etal 1971)
a Geometry of the system
Eg the diviation of the system into a restricted number of
layers and the horizontal extension of the system under
consideration
b The matrix of hydraulic properties including in-homogeneity
and an-isotropy
c Initial conditions Because we are dealing with non-stationary
flow the values of pertinent system variables (such as hydraulic
head at the initial time) must be known
d Boundary conditions They describe the conditions at the boundaries
pf the system as a function of time
One can distinguish between
- hydraulic head boundary
- flow (including zero flow) boundary
- both head and flow boundary
After specifying the auxiliary conditions an unique solution of
the variation of the hydraulic head in space and time can be
obtained
For special shapes of flow and assumptions exact analytical
solutions exist In nature however one is usually confronted with
problems like
- spacial variation in the hydraulic properties of the soil
- irregular shape of the groundwater basin
- empirical relations between q and h
- d not being a constant
For such situations by numerical and analogy models only an
approximation of the actual situation can be obtained In the next
chapter a numerical solution called the finite element method is
treated
III NUMERICAL SOLUTIONS
In this chapter the numerical solution of the aquifer equation
(eq 5) and the aquitard equation (eq 6) will be treated The
method of solving is the finite element method In section 31 only
a short mathematical description of this method will be given For
more mathematical background the reader is referred to publications
on this subject (eg ZIENKIEWICZ 1971 NEUMAN etal 1974 and
WESSELING 1976)
In section 32 the application of the finite element method on
the flow in groundwater basins is treated while in section 33 the
translation into a computer program is given
3 1 F i n i t e e l e m e n t m e t h o d
For reasons of simplicity the finite element method will be
explained for the stationary version of eq 5 ie
10
(k d^) + (k d ^ ) - Q = O (10) ocircx x 6x 6y y Sy
The basic idea of each numerical solution of eq (10) is to
apply this equation on a small but finite volume The flow region
of interest R is divided into a finite number of volumes called
elements The elements can have different shapes but for reasons
of conveniency we only use triangles- Quadrilaterals can also be
used because they can be divided into two triangles The sizes of
the triangles may vary They can be adapted according to the
geometry of the region and the accuracy desired (fig 11 gives an
example of the diviation of a region into triangles and quadrilaterals)
The corner points of the elements represent the nodal points
If there are N nodal points then the continuous variation of h
in region R is approximated by the expression
h (x_v) ~ h f (xy) (11) a n n
where
h (AV) = approximation of u (xy) a
h = value of h in nodal point n n
f ltxgty) = global coordinate function
In nodal point n a must be equal to h so f (x y ) = 1 and a n n n n
f (xy) = 0 i n Between f (x v ) and f (xy) of the nodal n j ] n n n n j j
points surrounding nodal point n f (xy) varies linearly from
1 to 0
In fig 3 a picture of f (xy) is given n
If one knows the values h in the nodal points n we know the approximation h of h We can find these values of b bv using for
a n
example the Galerkin method
Galerkins principle states that an approximate solution of h in
eq (10) can be obtained from the following system of equations
N ~~ (k d -A + -^- (k d T-y E h f (xy) f(xy)dxdy +
-ox x ox ocircy y oy - _ n n i R n~
J J Q f(xy) dx dy = 0 i = 1 N (12)
1 1
Fig 3 The global coordinate function f (xy)
The integration can be performed element by element with the
aid of the coordinate functions f (NEUMAN etal 1974)
For non-stationary problems it is not possible to apply the
Galerkin principle on the time derivative But it is possible
to discretize this derivative and to set up the system of equation
(eq 12) for each finite time step
One obtains in this way a series of sets of laquocriations tha s
an approximation of the non-stationary flow problem
Treatment of boundary conditions
a Initial conditions
In non-stationary flow problern gtne must start with given
values of a at time t = t ri c
b Boundary conditions
- Prescribed head boundary
Nodal points which lie on a prescribed head boundary have a
prescribed value for h bull r Hgte r umber of unknown h s is n n
reduced and accordingly no the number of equations
- Prescribed flux through z-- boundary
If one knows the flux (iuumlr jw or outflow per unit area) and
the area of the boundary oi-r element the inflow or outflow
through the boundary per flemlt=nt can be calculated
One may assume this flow to J bull part of the sink term Q in
eq (12)
12
32 A p p l i c a t i o n o f t h e f i n i t e e l e m e n t
m e t h o d o n t h e f l o w i n g r o u n d w a t e r
b a s i n s
From hydrogeological investigations the geometry and the matrix
of the hydraulic properties are known Knowing the geometry the
flow can be split up into a finite number of layers This holds
also for the nature of the boundaries
Now a nodal grid is superimposed on each layer in such a way
that the nodal points of the different layers are situated right
above each other (see fig 4) and in each layer the nodal point
lies in the middle of this layer
-nodal point
Fig 4 Schematic representation of the nodal points in the
xyz-space
Each layer has N nodal points of which P nodal points have a
prescribed head In each layer remain N-P = N nodal points with
unknown head Eq (12) can be applied to each point with unknown
head in each separate layer
As was stated in the last paragraph for each time step one
can write
13
Ah bdquo ~ -r1 + Q = O in = 12N in n m At yi A h + B (13)
where
A = in
ocircf ocircf ocircf of (k d f mdashri- ~ + k df - ~ -jS) dxdy
J x n ox ox y n oy oy
in (S d + S ) ff dx dy O if i icirc n
s y i n
Q f dx dy
The matrices A B and the vector Q have been defined in in l
elsewhere (NEUMAN etal 1974) A is the conductivity matrix in
B the storage matrix and Q the source vector in l
In order to clarify these rather complicate expressions apply
equation (13) poundo point 4 of fig 5
6 7
Fig 5 Top view of part of the nodal grid
lhl + A42h2 + A43h3 + A44h4 + A45h5 + A46h6 + A47h7 +
h - h
+ B _iuumli i ^ + Q = o 44 t - t x4
n n-1
(14)
where h and h are values of h at time t and t 4n 4n-1 4 n n-1
respectively
14
If the h-values belonging to the A-values are taken at time
t = t we get an equation with one unknown h for time t = t n-1 4n n
The equation can then be solved directly
However this is an explicit formulation which puts strong limitations
on the magnitude of the time step t - t (REMSON etal 1971) n n-1
Values larger than a certain value cause instability With an implicit method the values of h are taken at time
t = t This method is unconditionally stable Now one gets an equation n
with 7 unknowns h through h If one has 7 equations for every 1 n n
point 1 through 7 we can solve the values h i = 17
in
Of all the methods possible the iterative method of Gauss-Seidel
has been chosen The Gauss-Seidel method starts by choosing a first estimate of h at time t = t (usually equal to h ) At each in n j -i in-l iteration step a new value of h is calculated using the last
in calculated values of h If one passes the nodal points from 1 to
i n 7 eq (14) becomes
h-j + 1 = A h^ + 1 + A bdquo h | + 1 + A _ h | + 1 + A h j + A r h j + 4 n 4 1 1 n 4 2 2 n 43 3 n 45 5 n 4 6 6 n
B h B + A h J + - 4 4 4 n ~ + 0 J (A + mdash) A 4 7 h 7 n At n U 4 4 At
(15)
where
h-i = in
j = iteration step or number of iteration
i = number of nodal point n = time = t
n
The iteration process converges if the successive differences k+1 k
between h and h become continuously smaller in in
The iteration process is stopped if at each nodal point the k+1 k
difference between h and h is smaller than a certain prescribed in in
value
15
To accelerate the convergence an overrelaxation factor W is used
(for more information about W see FORSYTHE and WASOW 1960)
k+1 k T7 k+1 k h = h + W (h - h )
in in in in (16)
If W = 1 there is no overrelaxation Usually W varies between 1 and
2 and the optimum value very often is found by trial and error
Eq (15) is valid for points in aquifers The connection with
adjacent layers is made by the term Q because in this term the
flow from adjacent aquitards q is included (two aquifers are always
separated by an aquitard)
Aquitards
We dont apply the finite element method on points in aquitards
As stated before there is no horizontal flow in aquitards So it has
no sense to replace the continuous variation of h in the aquitard
by equation (11)
The nodal points in aquitards represent a certain volume of an
aquitard hich transmit water from one aquifer to another and which
has storage capacity The nodal points are included in the Gauss-
Seidel scheme -s follows (see fig 6 )
1 aauifer
a -_vS laquo 2 [~^-t^yM^^^-^r~ aqutarci l-Z
aquifer
Fig 6 Schematic situation for points in aquitards
Point 2 is situated in the middle of the aquitard with thickness
d storage coefficient S and vertical hvdraulic conductivity k c J v
Suppose the Gauss-Seidel iteration starts in the upper layer and
ends in the bottom-layer Then the flux from point 1 in aquifer 1 to
16
point 2 in the aquitard at time t = t and in iteration step j + 1
is
k q = (hf -hf )-iumlmdash f (17)
1 n 2 n OSxd1
and the flux from point 3 in aquifer 2 to point 2 in the aquitard
is
1 c 1
qbdquo = (h - hf) mdashHmdashr- (18) 2 3 n 2 n OSxd
The net flux ltl-q9 must be equal to the change in storage per unit
area per unit time This change in storage can numerically be
approximated by
te - h-
n n-1
Combina t ion of ( 1 7 ) (18) ana (19) g i v e s
l tr1 - h (hJ + 1 + hj - 2 hj+1) - - - - - - S1 - ilaquoS L u l l = o (20)
] i 3 n 2 n 1 c t - t j j x d n n -1
T + 1
From t h i s e q u a t i o n h^ kan be s o l v e d
3 3 C o m p u t e r o g r a m
The problem outl^ugrave so far has been translated into a computer
program written in FORTRAN IV
The flow chart ot Uis program is given in fig 7 whereas in
appendix A the complete listing of the program can be found
In the flow chart the 5 main parts of the program are
indicated
A Reading and printing of input data
B Determination of the matrices A and B and the vector Q per layer
17
M
1 A
FLOH to unsaturated zono
FLOW to tertiary syster
FLOW to secondnrv system
Leakage to adjacent aquitards
SOLVE equation (14)
yes
CALCULATE flux through constant head boundnr
CALCULATE waterbalance n p r 1 OAro r r J j - ~
PRINT of output
Ier node per layer
k = k+1
t = t + At
7 DUMP variables on tape
-gt-
( STOP )
Fig 7 Flow chart of the program FEMSAT with diviation in 5 main
parts A B C D and E
C START PROGRAM FEMSAT
f READ general information
PRINT general information
f READ element information
PRINT element information 7 READ stationary nodal point information
PRINT stationary nodal point information 7
READ non-stationary nodal point information for t = t
FRINT non-stationary nodal point information for t = t
L 2 (
^ yes R E A D non-stat input for t gt t
J PRINT non-stat input for t gt t (- 1 -- M degY
GENERATE A B and AREA-matrix for confined aquifers
_L
GENERATE A B and AREA-matrix for unconfined aouifer
bull
C Solving h with Gauss-Seidel
D Determination of the water balance terms per layer
E Printing of results
The program exists of a main program with several subroutines
The main program calls the subroutines In the subroutines most
calculations are performed
In this way an easy adaptation to different problems is
obtained (see also section 44) Comments in the program will
help the reader to understand the program
In the next chapter the preparation of the input data executing
of the program and the output will be treated
Acknowledgement
Some parts of the computer program were taken from the program
UNSAT2 of NEUMAN etal (1974)
IV INPUT EXECUTION AND OUTPUT
41 I n p u t
The input is divided into endogenous variables (chosen by the
user or calculated by the program during execution) and exogenous
variables (based on physical data)
411 Endogenous variables
Translation of groundwater basin into a nodal point network
Probably the most important but also the most difficult stage
in the process of simulating groundwater flow is the translation
of the natural groundwater basin into a nodal point network Questions
which arise are
a In how many layer the geo-hydrological system mustcan be
schematized
b The locations of the boundaries and the nature of these boundaries
(impervious prescribed head prescribed flux)
c The configuration of the nodal points
19
The answers of these questions depend on the problem under
consideration
- Schematization into layers
Usually our knowledge about the geo-hydrological situation is
such that no more than 5 different layers can be distinguished
(remember contrast rule)
- Boundary conditions
The top boundary condition of a groundwater basin usually is
a prescribed flux (effective precipitation)
The bottom boundary condition usually is a prescribed flux
boundary with flux zero (impervious hydrological base)
For the boundaries at the other sides of the system one sees
either prescribed head boundaries (eg a fully penetrating
river) or prescribed flux boundaries (including zero flux)
- Nodal point configuration
Nodal points are situated
on the boundaries
along internal boundaries which indicate changes in
hydraulic properties within the layer
along the tertiary surface water system
on places where artificial recharge or discharge wiii tgtke
place
elsewhere
Construction of the finite element mesh
Assume we have schemetized the groundwater basin iatu a finite
number of layers with different direction of flow (horizontal and
vertical) Within one layer the hydrological properties may vary
from place to place
Now a nodal grid is superimposed on each layer in a way already
described in section 32 Thus each layer has the same number of
nodal points situated on the same places in the horizontal plane
but with different z-coordinates
We are free to construct quadrilaterals or triangles because
the program automatically divides a quadrilateral into two triangles
having identical material properties The dimensions of the elements
20
should be small at places where large changes in hydrological
properties occur or where large gradients in hydraulic head are
expected to occur
Coding of nodal points
Each node in the finite element network is assigned an integer
code which indicates the type of calculation for this node
The coding is as follows
- Code = 1 nodal points with described head as function of time
- Code 2 internal nodal points
- Code = 3 internal nodal points which represent secondary surface
water system
- Code = 5 nodal points with prescribed artificial recharge or
discharge and points with prescribed boundary flow
(both as functions of time)
Other important endogenous variables are the magnitude of the
time step the value of the variable which stops the iteration
process and the maximum number of iterations allowed per time step
(see also group A and B of section 413)
412 Exogenous variables
After constructing the nodal grid each node represents part of
the groundwater basin Geo-hydrological properties can be assigned
to each mode based on physical data (group C D and E of section
413) In group F of section 413 time-dependent variables can
be assigned to each node
413 Data input to Program FEMSAT
The data input must be punched on cards according to the
following instructions
GROUP COLUMNS FORMAT SYMBOL DESCRIPTION
A 1-80 20A4 HEAD Desired heading to be printed at
the beginning of the execution
1-5 15 NUMNP number of nodal points
21
GROUP COLUMNS FORMAT SYMBOL DESCRIPTION
6-10 15
11-15 15
16-20 15
21-25 15
26-30 15
31-35 15
NUMEL
NUMLAY
MBAND
NTSPI
NTSPO
MAXIT
1-10 F103 ST
11-20 F103 DELT
21-30 F103 TMAX
31-40 F105 TOL
number of elements
number of layers
maximum difference in node number
between two adjacent nodes
number of time steps between a
change in non-stationary input data
number of time steps between two
successive outputs
maximum number of iterations per
time step
time to start with the execution
initial time interval
maximum time to be reached before
execution is stopped
maximum allowed absolute change in
the values of h between any two
successive iterations in a given
time step
Iterations continue until all changes
are less than or equal to TOL or
until MAXIT is exceeded
1- 5 15
6-10 15
11-15 15
16-20 15
KX(N1)
KX(N2)
KX(N3)
KX(N4)
sequential number of i-th corner of
element N
sequential number of j-th corner
sequential number of k-th corner
sequential number of 1-th corner
(see fig 8)
Fig 8 Numbering of element corner nodes
22
GROUP COLUMNS FORMAcircT SYMBOL DESCRIPTION
One card must be provided for each element in sequential order
starting with N = 1 and ending with N = NVJMEL
I-K 1103 SANG(NLY)
1i-iO MO3 C1(NLY)
21-30 F 103 C2(NLY)
21-4- 1-104 THI(NLY)
angle in degrees between CI and
the laquo-coordinate to he used in
nodal print N in layer LY (see
poundig 9)
First principal conductivity of
material in node N and layer LY
idem second principal conductivity
thickness of layer LY at nodal
point N
ig V Definition of SANGCI and C2
One card icircuiPt be provided for each model point ctarting wih M = 1
and ending with N = NUMNP If there is norlt3 than 1 layer for each
layer a set must be provided starting with the upper layer LY - 1
and ending with the bottom layer LY = NIJMLAY The variables SANG
M and 2 are not relevant tor aquitards but for reasons of
conveniency this procedure is maintained (making SANG CI and C2
dummy variables)
1-10 110 KODE(N)
11-20 II0 NTERR(N)
coding of nodal points (see
paragraph 42)
number of tertiary relation in node N
The q-h relationship of the flow to
23
GROUP COLUMNS FORMAT SYMBOL DESCRIPTION
21-30 FlOl X(N)
31-40 FlOl Y(N)
41-50 FlOl HGL(N)
51-60 FlOl HBSC(N)
61-70 FlOl LESC(N)
the tertiary system as discussed
in par 24 are schematized in a small
number of different relations
x-coardinate of node N
y-coordinate of node N
height ground level in node N
height bottom secondary conduit in
node N If node N is not a secondary
conduit this variable is a dummy
variable
length of secondary conduit in area
represented by node N
One card must be provided for each node starting with N = 1 and
ending with N = NUMNP
E 1-10 FlOl P(NLY1) initial value of h at node N in
layer LY
11-20 FlOl VERRES(NLY) vertical resistance at node N and
layer LY If the layer is an aquifer
this value is a dummy variable
21-30 F101 RADRES(NLY) radial resistance for flow to
secondary system at node N in layer
LY
31-40 FlOl ENRES(NLY) entrance resistance for flow to
secondary system at node N in layer
LY
41-50 FlOl WP(NLY) wet perimeter of secondary system at
node N in layer LY
51-60 FlOl P0R(NLY) specific yield of the material at
node N in layer LY
61-70 FlOl SS(NLY) specific storage of the material
at node N in layer LY
24
GROUP COLUMNS FORMAT SYMBOL DESCRIPTION
One card must be provided for each node starting with N = 1 and
ending with N = NUMNP If there is more than 1 layer for each layer
a set of cards must be provided starting with LY = 1 and ending
with LY = NUMLAY
F 1-10 F101 P(NLY1) value of h at different times at
node N in layer LY Node N lies
on a prescribed head boundary
11-20 F101 WTSC(N) water table in secondary system at
node N at different times
21-30 F 101 ARRED(NLY) artificial recharge to or discharge
from node N in layer LY or
prescribed boundary flow at
different times
31-40 F 101 QPHR(N) flux through phreatic surface at
node N at different times
The data in group F are non-stationary input data One card must be
provided for each node starting with N = 1 and ending with N = NUMNP
In case of more than 1 layer for each layer a set must be provided
Not relevant variables are taken as dummy variables
After a certain number of time steps (NTSPI) the data of group E
are read So the set cards E must contain NxLYxC cards where C =
(TMAXNTSPIXDELT) + 1 if DELT is a constant
42 R u n n i n g t h e p r o g r a m o n t h e C y b e r 7200
The program has been executed on a Cyber 7200 computer of
IWIS-TNO in the Hague
- The demanded input is punched on cards according to the instructions
of the last paragraph and is put on a permanent file
- For each specific problem inthe program only the dimension statements
in the main program have to be adjusted according to the instructions
in the lines of comment
- After compilation the program together with the input file is
executed
25
- Intermediate data necessary for continuation of the calculations
are saved on another permanent file This gives the opportunity
to restart the program at a time the calculation was stopped
The only change in the input file is to adjust the starting time
Remark
With the aid of a terminal small changes in the program and in the
input file can be made easily and the output immediately appears
on the terminal In this way it is possible to stop erroneous
calculations This is especially useful when being in the testing
phase of the program
43 O u t p u t
Immediately after reading the input data they are printed so
that possible errors can be detected During execution after each
certain number of time steps (NTSPO) calculated values of h and
water balance terms are printed In appendix B an example is given
The output data can be devided into two groups
A Data per node per layer
- Coding of the nodal points KODE
- Hydraulic head HEAD
- Flow through phreatic surface at the time indicated in head
of table FLOWUN
- Flow to tertiary surface water system FLOWTS
- Flow to secondary surface water system FLOWSS
- Flow to adjacent layers LEAKAGE
- Artificial recharge or discharge and prescribed boundary
flow ARRED
- Change in storage during last time step STORAGE
- Flow through prescribed head boundary LATFLOW
B Data per layer
Summation of the terms already mentioned under A gives the
magnitude of the water balance terms per layer at the time
indicated (instanteneous water balance terms) Summation of
these instanteneous terms multiplied by their respective time
intervals since time is zero gives the cumulative water balance
terms per layer
26
After each print out we are able to check the calculations
because the sum of the water balance terms per layer must be
approximately zero
44 C o m p u t i n g t i m e
The computing time depends on the number of nodal points and
the number of layers But it depends also on the magnitude of the
changes in hydraulic head and on the value which terminates the
Gauss-Seidel iteration (TOL) To give an impression TOL = 001 m
Costs per day simulation (in hlfl) on the Cyber 7200
number of nodes number of layer bull bdquo bull bull _ 1mdash x a x b (21)
where
a = costs in Dutch guilders per system second (ƒ 027)
b = empirical factor with magnitude about 001
V NUMERICAL EXPERIMENTS
In general there are two possibilities to test the validity of
numerical models The first and most commonly used method is to
compare numerical results with analytical solutions The second
possibility is to compare numerical results with measured field
data Since the last alternative requires many data that are
difficult to obtain we restrict ourselves for the time being to
comparison with the analytical solutions
5 1 F l o w t o a w e l l i n a n u n c o n f i n e d
c i r c u l a r - s h a p e d a q u i f e r
Flow to a well can be divided into steady-state flow and
unsteady-state flow (KRUSEMAN and DE RIDDER 1976)
27
511 Steady state flow in an unconfined aquifer
For this case the formula of Thiem-Dupuit is valid (see fig 10)
Q bull k lth2-hicircgt ln(r2r])
(22)
where
hhbdquo = hydraulic head in 1 and 2 respectively
Q = discharge from well
rbdquo = distance from well
CL31
hmdash^ri --r2
Fig 10 Schematic cross-section of a pumped unconfined aquifer
Fig 11 gives the nodal grid superimposed on a well in a
circular shaped aquifer Near the well the nodal point distance
is made smaller Quadrilaterals as well as triangles are used
Numerical calculations with this finite element network and 3 -1
an extraction rate of 200 m day were performed for 10 days
At that moment a steady-state situation was almost achieved
28
2tmdashnodal point number
160 180 X-axesim)
Fig 11 Nodal point configuration for flow to a well in a circular-
shaped aquifer The outer boundary is taken to be constant
The well is situated in nodal point 42
29
In fig 12 the results of the simulation are compared with the
analytical solution The agreement between numerical results and
the analytical solution is excellent
draw- 15irS (m) down
analytical solution imdashx calculated values
distance from well r (m)
Fig 12 Comparison between analyt ical and calculated s teady-state
drawdown for the problem of f ig 11 with 1
-1 Q bull - 200 m 3 d a y _ 1
k = 10 mday
d = 100 m
S = 2 y
S = 0 s
512 Unsteady-state flow in an unconfined aquifer
Analytical solutions for unsteady-state flow in an unconfined
aquifer only exist if the aquifer extends infinitely For the
aquifer given in fig 11 this condition is not fulfilled However
during the first one or two days after starting the extraction the
influence of a constant hydraulic head at the outer boundary on the
30
drawdown near the well can be neglected (see fig 13) After one
day the lateral inflow is only 25 of the extraction the remainder
being the change in storage due to drawdown
2 0 0
m 3 day 1
Q=200m 3 day
100-
vraquomdash Lateral inflow
N amp storage
I I ^ V - W 10
t (days)
Fig 13 Values of Q lateral inflow and change in storage as
functions of time
The nonsteady-state or Theis equation can be written as
i _ Q s 4iTkD
dy = 4irkD
W(u) (23)
where
u = r2S
4irkDt
r = distance from well
S = storage coefficient
Q = constant discharge from well
D = thickness of the waterbearing layer
at the beginning of pumping
t = time after starting the extraction
s = corrected drawdown
= s - 322D
s = calculated drawdown
W(u) = Theiss well function
[J
CIAT1
[T]
]
31
In fig 14 the analytical unsteady-state solution for the case
described in section 511 is given together with the numerical
results at times 05 and 10 days The agreement between numerical
results and the analytical solution is rather good
Drawdown s1(m)
10
10
10
Theiss solution
x caculated values t=Q5
bdquo t= lO
10 10- 10 10 t r2(daym2)
Fig 14 Comparison between analytical and calculated steady-state
drawdown for the problem of fig 11 with
Q
k
d
S y
s s
=
=
=
=
=
200
1
10
3 m
0
0
2
0
-1 day
m day
m
These results indicate that the assumption of a linear variation
of h between two nodal points (see fig 3 ) even with the rather
rough element network of fig 11 is acceptable So there is no
need to improve for example the finite element network by introducing
higher order coordinate functions
32
52 F l o w t o w e l l i n 3-1 a y e r e d s y s t e m
In fig 15 a schematic cross-section of a 3-layered system of
two aquifers separated by an aquitard is given A well of
infinitisemal radius is completed in the lower aquifer and discharges
at a constant rate QTT Each layer is homogeneous isotropic and
extending radially infinite
bdquo bull - bull bull ~ i
mdash Q II
aquifer I
lugrave^i~IcircIuml7i-i~~ aquitard --r-rl~-
-II--I i - i
aquifer
Fig 15 Schematic cross-section of a 3-layered system
For this situation NEUMAN and WITHERSPOON (1969a) give rather
complicated analytical solutions In the special case where the
hydraulic properties of the two aquifers are identical they give
graphical illustrations of their solutions for three different
cases
These graphical solutions give the opportunity to verify
roughly numerical results of a 3-layered system Therefore a nodal
grid is superimposed on a circular shaped groundwater flow system
with the extraction in the centre of the lower aquifer and with
a constant head on the outer boundary Fig 16 gives the nodal
point situation in the x - y plane of 1 layer The nodes in the
other two layers are situated in the middle of the layers at the
same places in the x - y plane only their z-coordinate is different
33
A
-Jy
bull33
V 5
iumlff
Fig 16 Nodal point configuration for the 3-layered problem (top view)
34
In fig 17 numerical results are compared with the analytical
solution The agreement is rather bad especially for the two
aquifers But in terms of the water balance the difference between
aquifers and aquitard is not so big because the aquitard has
about 16 times bigger storage capacity than the aquifers The
small value of the specific storage of the aquifers is the reason
that small errors in the components of the water balance have a
big influence on the calculated values of the hydraulic head
Eig 17 Dimensionless drawdown vs dimensionless time in a
3-layered system with
II = 50 m day
3 m (distance from well)
aquifer I
aquitard
aquifer II
Thi (m)
100
25
100
S s
001
0642
001
k(mday )
1
0277
1
The discrepancies are partly caused by errors in discretization
In the z-direction these errors can be eliminated by taking the
35
vertical resistance of the aquitard to be infinite Then the
calculated drawdown in the pumped aquifer can be compared with
the analytical Theiss solution
This is done in fig 18 for different distances from the well
One sees that the agreement is better when the distance is bigger
Apparently the linearization of the logarithmic drawdown curve
near the well is responsible for the differences The differences
for r = 3 m are of the same magnitude as for the pumped aquifer in
the 3-layered case
joo gt A 50
20
Theiss solution
bull calculated values r=3m (distance f rom well) x bdquo bdquo r=7m
1 L
I l_ 001 003 0 0 5 Ol Q2
- I I 5 10 t - K t
Fig 18 Dimensionless drawdown vs dimensionless time in a
confined aquifer with
Q =500 m day
k = 10 mday
D = 100 m
S = 001 s
The differences between numerical results and analytical solutions
are not caused by taking a finite area because changing the outer
boundary into an impermeable boundary has only minor effects
36
VI SUMMARY AND CONCLUSIONS
In this report a numerical model for non-stationary saturated
groundwater flow in a multi-layered system has been treated
Groundwater flow can be described in different ways
a Physical-mathematical by combining Darcys law and the law of
continuity (section 21)
b Geo-hydrological the flow in a natural groundwater basin takes
place in layers with different hydraulic properties Definitions
of the most important hydraulic properties are given in section
22
c Schematical groundwater flows horizontally in aquifers and
vertically in aquitards (section 23)
d Approximating by dividing the region of flow into a finite
number of elements and applying Darcys law and the law of
continuity on each of these elements In these ways one gets
a set of equations which can be solved given (initial and)
boundary conditions (section 25)
e Numerical the set of equations has been solved using a Galerkin
type finite element method and the implicit Gauss-Seidel method
with overrelaxation factor (section 31 and 32)
f Computer-solvable the numerical solution has been translated
in a FORTRAN IV program called FEMSAT (section 33)
In chapter IV the input execution and output of the program
FEMSAT are treated
Finally in chapter V some numerical applications are given
simulation of flow to a well in unconfined aquifer and simulation
of flow to a well in a 3-layered configuration
Comparison of hydraulic heads obtained by numerical calculations
and analytical true solutions showed an excellent agreement in case
of stationary flow and a good to moderate agreement in case of non-
stationary flow However the diviations caused by the discretization
process must be seen in relation to the errors made in the
establishment of the hydraulic properties
37
As was stated in the introduction numerical models can be
valuable tools in the management of water resources Therefore
numerical models must be suitable to simulate effects of certain
operations upon the spreading of water over saturated groundwater
unsaturated groundwater and surface water The model described in
this report is a model for saturated groundwater flow The relation
with the surface water can be taken into account via radial and
entrance resistances or via a relation between flow to the surface
water system and the hydraulic head The relation with the unsaturated
zone has been restricted to the input factor effective precipitation
In practise however the unsaturated and saturated system are
interconnected Therefore the model should be improved by taking
into account the unsaturated zone This can be done in at least
two ways
a Replacing the unsaturated zone by a parametrical model
b Coupling the unsaturated zone in an iterative way with the
saturated system
In order to find out the usefulness of the numerical model for
water resources management it will be applied to actual regional
groundwater systemslaquoAlso sensitivity analyses will be carried out
The computer model itself can be improved by applying standard
programs eg automatical generation of a nodal grid and graphical
plotting of the results
38
VII REFERENCES
ERNST LF 1962 Grondwaterstromingen in de verzadigde zone en hun
berekening bij aanwezigheid van horizontale evenwijdige open
leidingen Thesis University Utrecht pp 189
1978 Drainage of undulating soils with high groundwater
tables (in press)
FORSYTHE GE and WR WASON 1960 Finite Difference Methods for
Partial Differential Equations John Wiley and Sons New York
pp 444
KRUSEMAN GP and NA DE RIDDER 1976 Analysis and evaluation of
pumping test data ILRI-bulletin No 11 pp 200
LAAT PJM DE and C VAN DEN AKKER 1976 Simulatiemodel voor grond-
waterstroming en verdamping in Modelonderzoek 1971-1974 ten
behoeve van de waterhuishouding in Gelderland deel II
Grondslagen pp 145-197
NEUMAN SP RA FEDDES and E BRESLER 1974 Finite element
simulation of flow in saturated -unsaturated soils considering
water uptake by plants Hydraulic Engineering Laboratory
Technion Israel pp 104
and PA WITHERSPOON 1969a Theory of Flow in a confined
two Aquifer System Water Resources Research Vol 5 No 4
pp 803-816
^_ and PA WITHERSPOON 1969b Applicability of Current Theories
of Flow in leaky aquifers Water Resources Research Vol 5
No 4 pp 817-829
PINDER GF and EO FRIND 1972 Application of Galerkins Procedure
to Aquifer Analysis Water Resources Research Vol 8 no 1
pp 108-120
REMSON I GE HORNBERGER and FJ MOLZ 1971 Numerical Methods
in Subsurface Hydrology John Wiley and Sons New York
pp 389
WESSELING JG 1977 Toepassing van Finite Difference- en Finite
Element Methoden voor het oplossen van een aantal grondwatershy
stromingsproblemen (met computerprogrammas) ICW-nota 944
pp 63
39
ZIENKIEWICZ OC 1971 Finite Element Method in Engineering Science
McGraw-Hill London pp 521
40
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X - e - o
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LUX
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APPENDIX 8 EXAMPLE OF OUTPUT
HEAD AND TERMS OF UATERBALANCE PER NODE PER LAYER
AT TIME 260 IN II AND I13DAYS
T
LAYER NODE KODE HEAD FLOUUN FLOUTS FLOUSS LEAKAGE STORAGE LAT FLOU
13 19 25 31 37 43
125008 125000 12S000 12S000 12S000 12S000 12S000 12S000
0000 0000 0000 0000 0000 0000 0000 0000
0000 0000 0000 0000 0000 0000 0000 0000
0000 0000 0000 0000 0000 0000 0000 0000
- 000 - 000 -000 - 000 - 000 - 000 - 000 0000
0000 0000 0000 0000 0000 0000 0000 0000
0000 000 000 000 000 000 000 000
000 0000 0000 0P100 0000 0000 0000 0000
1 7
13 19 25 31 37
12S000 125000 125000 125000 125000 12S000 125000 125000
0000 0000 0000 0000 0000 0000 0000 0000
- 001 -007 - 023 - 020 - 009 - 003 0000
0000 0000 0000 0000 0000 0000 0000 0000
000
001
007
023
020
009
003
003
0000 0000 0000 0000 0000 0000 0000 0000
1 7
13 19 25 31 37 43
125000 125000 124998 124972 124895 124787 124647 124358
0000 0000 0000 0000 0000 0000 0000 0000
- 000 001 007 023 020 009 003
0000
0000 0000 0000 0000 0000 0000 0000
50000
0000 182
1982 3862 1674 419 078 041
0000 0000 0000 0000 0000 0000 0000 0000
TERMS OF WATERBALANCE PER LAYER DURING LAST TIMESTEP AND
AND SINCE BEGINNING IN M3DAY M3
FLOU TO UNSATURATED ZONE
FLOU TO TERTIARY SYSTEM
0000
0000
0000
0000
LAYER NO 1
FLOU TO SECONDARY SYSTEM
FLOU TO AD3ACENT LAYERCS)
ARTIFICIAL FLOU AND PRESCRIBED BOUNDARY FLOU
ACCUMULATION TERMS
LATERAL FLOU THROUGH PRESCR HEAD BOUNDARY
0000
-000
0000
000
000
0000
-000
0000
000
000
LAYER NO 2
FLOU TO SECONDARY SYSTEM
FLOU TO AD3ACENT LAYERCS1
ARTIFICIAL FLOU AND PRESCRIBED BOUNDARY FLOU
ACCUMULATION TERMS
LATERAL FLOU THROUGH PRESCR HEAD BOUNDARY
0000
-378
0000
381
0000
0000
-055
0000
056
0000
LAYER NO 3
FLOU TO SECONDARY SYSTEM
FLOU TO AD3ACENT LAYERCS5
ARTIFICIAL FLOU AND PRESCRIBED BOUNDARY FLOU
ACCUMULATION TERMS
LATERAL FLOU THROUGH PRESCR HEAD BOUNDARY
0000
370
50000
49376
0000
0000
055
12979
12909
0000
C O N T E N T S
b i z
I INTRODUCTION 1
II GENERAL DESCRIPTION OF THE PROBLEM 2
21 Physical background 2
211 The law of linear resistance (Darcys law) 2
212 The law of continuity 3
213 Combination of Darcys law and the law of
continuity 3
22 Hydraulic properties and types of aquifers 3
23 Schematization of flow 5
24 Water balance terms 6
25 Solution of groundwater flow problems 9
III NUMERICAL SOLUTIONS 10
31 Finite element method 10
32 Application of the finite element method on the
flow in groundwater basins 13
33 Computer program 17
IV INPUT EXECUTION AND OUTPUT 19
41 Input 19
411 Endogenous variables 19
412 Exogenous variables 21
413 Data input to Program FEMSAT 21
42 Running the program on the Cyber 7200 25
43 Output 26
44 Computing time 27
mdash 1
biz
V NUMERICAL EXPERIMENTS 27
51 Flow to a well in an unconfined circular-
shaped aquifer 27
511 Steady state flow in an unconfined aquifer 28
512 Unsteady-state flow in an unconfined aquifer 30
52 Flow to a well in a 3-layered system 33
VI SUMMARY AND CONCLUSIONS 37
VII REFERENCES 39
i
I INTRODUCTION
The groundwater system is part of the hydrological cycle and
changes in this system affects phenomena such as plant production
river flow etc Generally one tries to use groundwater resources
in such a way that the utility for the community as a whole is
maximal
Numerical flow models can be valuable tools in the management
of water resources In many situations these models are the only
possible way to indicate effects of certain interferences in the
groundwater system
In this paper a numerical model is presented which simulates
non-stationary saturated groundwater flow in a certain region
Purpose of the model is to predict effects of certain operations
upon the various terms of the water balance of the groundwater basin
and on the hydraulic head
The basic idea of the model is that groundwater is flowing
horizontally in waterbearing layers and vertically in less-permeable
layers Each layer is discretized into a number of elements To
each element both Darcys law and the law of continuity are applied
(chapter II)
In this way a set of equations is obtained which are solved
using a finite element method and the Gauss-Seidel iteration method
(chapter III)
In chapter IV the input and output of the model are treated
while in chapter V some numerical experiments are presented
Finally in chapter VI an evaluation of the model is given
II GENERAL DESCRIPTION OF THE PROBLEM
21 P h y s i c a l b a c k g r o u n d
Throughout this report water potential is expressed on a unit
weight basis and is being called hydraulic head with symbol h
h = 2- + z [L] Pg
where
mdash = piezometer head |L | Pg u -1
z = gravitational head |_Lj
The hydraulic head is an expression for the capacity of a unit
weight of water to do work as compared to the work capacity of the
same weight of water with the same chemical composition at a certain
reference level and under atmosferic pressure
The problem is restricted to saturated flow of groundwater with
one constant density
211 The law of linear resistance (Darcys law)
According to Darcys law the rate of flow through a porous medium
is proportional to the gradient in hydraulic head It may be written
as
ocirc h ltSh ocirch N q = _k Tmdash q = -k vmdash q = -k -rmdash ( 1 ) x y ocircx y y lt5y nz z ocircz
where
q = volume flux of water passing through a unit area per x y z
per unit time perpendicular to x y z direction
respectively |_LT J
k = hydraulic conductivity in x y and z direction x gt y gt z _ _ i _
respectively | LT J
The coefficient k in the Darcy flow equation is taken to be
a constant depending on both the properties of the porous medium
and the fluid
212 The law of continuity
The other important physical law states that no matter can be
lost or created
The conservation equation can be found by applying the principle
of continuity to an infinite small volume
6v 6v lt5v 60 x y z TT = T + x + ~T7 + 1 (2)
Ot OX OcircX oz
where
0 = volume of water per unit bulk volume of soil |_-J
q = sink or source term T J
213 Combination of Darcys law and the law of continuity
Darcys law combined with the law of continuity gives
60 lt5 oh lt5 ocirch 6 oh _ ocirct ocircx x ocircx ocircy y oy ocircz z ocircz
or
c 5h 6 ocirch ocirc Ocirch 6 ocirch c ot ox x ocircx ocircy y oy laquoz z 6z
where
S = volume of water released or stored per unit bulk c I
volume of soil per unit change in hydraulic head |L I
22 H y d r a u l i c p r o p e r t i e s a n d t y p e s o f
a q u i f e r s
The flow in a natural groundwater basin takes place in layers
with different hydraulic properties Usually the layers are divided
into good permeable layers or waterbearing layers (aquifers) and
less permeable layers (aquitards)
The most important hydraulic properties are
a Transmissiviy - the product of the average hydraulic conductivity
k and the saturated vertical thickness d of the waterbearing
layer
It is a measure for the transport capacity of aquifers
Notation T = kd |_ L T ~
b Hydraulic resistance - the ratio of vertical thickness of
aquitards d and hydraulic conductivity k It is a measure for
the resistance capacity of aquitards denoted as c = d k _ T J
c Specific yield - the volume of water released or stored in the
phreatic zone per unit surface area of an unconfined aquifer per
unit change in hydraulic head
It is also called effective porosity
The symbol used for specific storage will be S j_-J
d Specific storage - the volume of water released or stored per
unit volume of the saturated parts of an aquifer or aquitard per
unit change in hydraulic head It is a measure for the elasticity
of the soil material and the fluid Excluded are irreversible
processes Notation S _ L J
e Storage coefficient - the volume of water released or stored
pe7 nit surface of a layer per unit change in hydraulic head
It is the sum of specific yield and the product of specific
storage and thickness of the aquifer or aquitard
S = S + S d r - 1 c y s - -1
Note in confined layers S = 0 y
In fig 1 a 3-layered configuration is given of two aquifers
separated by an aquitard Aquifer I has a free water surface and is
an unconfined aquifer Aquifer II is a completely saturated aquifer
Dependent on the hydraulic resistance of the aquitard KRUSEMAN and
DE RIDDER (1976) distinguish between confined semi-confined and
semi-unconfined aquifers For our purpose it is not useful to make
this distinction We restrict the aquifer types to unconfined and
confined aquifers
WIHtUtttWttllttlt(ltmtttltHlttltlil(lltlittlfllnl ) bull (
d V U i l i l i ii i Hi il i M M bull f i m
i tMi| i i | J i j j i _ I I bull
aq ui tard
aquifer II
Fig 1 Schematic representatipn of flow direction and variation
of hydraulic head h in a 3-rlayered system
In an unconfined aquifers ( I ) poundhe vertical thickness pf the
waterbearing layer d is a variable while in a confined aquifer II
this is a constant (denoted as D ) Other important properties of
unconfined aquifers are the hydraulic head being the same as the
phreatic surface and the specific yield being different from zero
23 S c h eacute m a t i s a t i o n o f f l o w
The groundwater flow in a groundwater basin is schematized
into a horizontal flow in aquifers and into a vertical flow in
aquitards Consequently the vertical variation of the hydraulic
head is as drawn in figl
The validity of this assumption has )een investigated by
NEUMAN and WITHERSPOON (1969) The found that if the contrast
in hydraulic conductivity between two adjacent layers is bigger
than a factor two the relative error caused by this assumption
is usually smaller than 5
This schematization gives a simplification of the general flow
equation (eq 4 ) In an aquifer the vertical is zero and eq (4)
then reduces to
-Sbdquo -r- 7 - (k d -r-) + 7 mdash (k d 7-) + qd c lt5t ox x ox oy y oy
or
bull(S + S d) g y s ot
-r- (k d -=-) + -T- (k d -=-) + Q ocircx x ox oy y ocircy
(5)
where
Q = volume of water substracted from or added to unit surface
of aquifer per unit time |_ L T J
In an aquitard eq (4) reduces to
-sQ |pound = (k Q) s 6t oz Z OcircZ
(6)
As will be discussed in the next paragraph the term q in an
aquitard is set equal to zero
24 W a t e r b a l a n c e t e r m s
In eq (4) the term q is mentioned With this sink or source
term different terms of the water balance are taken into account
Fig 2 depicts schematically the situation for an unconfined aquifer
(for other layers some terms dont exist)
recharge- - discharge
laquo-b
Fig 2 Schematic representation of the various terms involved in the
water balance
Description of the different terms
a The flow through the phreatic surface q f
If this term is negative (water goes out of the saturated system)
it is called capillary rise If positive it is called percolation
(or drainage)
This flow can be calculated as a rest term of the water balance
of the unsaturated zone Very often this term is expressed as a
flow per unit surface (flux) and is called effective precipitation
For the time being this flux is a given input
b Flow to adjacent aquitards q
The flux to adjacent aquitards can be calculated by applying eq
(6) to the aquitard
c Flow to the primary surface water system q
The primary surface water system penetrates fully the groundwater
basin So it acts as a boundary with given values for the
hydraulic head (is equal to the phreatic level of the primary
system)
d Flow to the secondary surface water system q
The secondary surface water system does not penetrate fully the
groundwater basin which causes a resistance against flow between
groundwater and surface water The flow q can be calculated as
follows
h - h
- mdash f i i gt
where
h = hydraulic head of the groundwater at the place of
the secondary system |_ L |
h = phreatic level of secondary system |_ L J i s -
R = radial and entrance resistance of the
secondary system per unit surface |_ T J
According to ERNST (1962) the radial resistance H per unit
surface can be caKulated with
Uuml = 4 i n 4 e8) TTk B
where
1 = area drained by 1 m conduit L ^ J
d = thickness of the layer near the conduit _ L J
B = wet perimeter of the conduit |_ L J
The entrance resistance is caused by a layer which shows a
lower conductivity around the conduit than its surroundings
In practise radial and entrance resistancies are difficult to
deduct from hydraulic properties They can be found indirectly
by measuring simultaneously q h and h
t s
e Flow to the tertiary surface water system q
In smaller conduits usually the phreatic level is not known
So eq (7) cant be applied The flow to these conduits however
is sometimes very important (eg in areas with shallow groundwater
tables) One may a-sume (ERNST 1978) that a relationship exists
between the hydraulic head of an aquifor and the flow q
According to Ernst the tertiary system can be divided into a
number of categories with respect to their mutual distance and
bottom height For each category the flow is h - h (b)
q ( h b ) - mdash (9)
where
h = average hydraulic head | L J
h (b) = phreatic level in the category under
consideration a function of bottom height |_L J
a = geometry factor to convert h into h _ - J
Y = drainage resistance _ T J
hm = hydraulic head midway between two
conduits |_ L J m
Summation of the q(hb)-relations of the different categories
yields the overall relation between hydraulic head and flow to
the tertiary system we are looking for
f Artificial recharge or discharge q and prescribed boundary
flux qfo
The magnitudes of these flows are not influenced by the conditions
inside the groundwater basin but remain on a prescribed level
Thus q is an internal boundary condition q an external boundary cl D
condition
For confined aquifers some of the above mentioned terms dont
exist These terms are the flow through the phreatic surface qf
and the flow to the tertiary surface water system q
Aquitards are considered only as transmitters of water from one
aquifer to another with possibilities of storage (eq 6 ) This means
that for aquitards the water balance terms ar through f are set
equal to zero
25 S o l u t i o n o f g r o u n d w a t e r f l o w p r o b l e m s
The basic equations for saturated groundwater flow are eq (5)
for aquifers and eq (6) for aquitards The solution of these
equations require knowledge of the physical system and the auxiliary
conditions describing the system constraints
These auxiliary conditions are (REMSON etal 1971)
a Geometry of the system
Eg the diviation of the system into a restricted number of
layers and the horizontal extension of the system under
consideration
b The matrix of hydraulic properties including in-homogeneity
and an-isotropy
c Initial conditions Because we are dealing with non-stationary
flow the values of pertinent system variables (such as hydraulic
head at the initial time) must be known
d Boundary conditions They describe the conditions at the boundaries
pf the system as a function of time
One can distinguish between
- hydraulic head boundary
- flow (including zero flow) boundary
- both head and flow boundary
After specifying the auxiliary conditions an unique solution of
the variation of the hydraulic head in space and time can be
obtained
For special shapes of flow and assumptions exact analytical
solutions exist In nature however one is usually confronted with
problems like
- spacial variation in the hydraulic properties of the soil
- irregular shape of the groundwater basin
- empirical relations between q and h
- d not being a constant
For such situations by numerical and analogy models only an
approximation of the actual situation can be obtained In the next
chapter a numerical solution called the finite element method is
treated
III NUMERICAL SOLUTIONS
In this chapter the numerical solution of the aquifer equation
(eq 5) and the aquitard equation (eq 6) will be treated The
method of solving is the finite element method In section 31 only
a short mathematical description of this method will be given For
more mathematical background the reader is referred to publications
on this subject (eg ZIENKIEWICZ 1971 NEUMAN etal 1974 and
WESSELING 1976)
In section 32 the application of the finite element method on
the flow in groundwater basins is treated while in section 33 the
translation into a computer program is given
3 1 F i n i t e e l e m e n t m e t h o d
For reasons of simplicity the finite element method will be
explained for the stationary version of eq 5 ie
10
(k d^) + (k d ^ ) - Q = O (10) ocircx x 6x 6y y Sy
The basic idea of each numerical solution of eq (10) is to
apply this equation on a small but finite volume The flow region
of interest R is divided into a finite number of volumes called
elements The elements can have different shapes but for reasons
of conveniency we only use triangles- Quadrilaterals can also be
used because they can be divided into two triangles The sizes of
the triangles may vary They can be adapted according to the
geometry of the region and the accuracy desired (fig 11 gives an
example of the diviation of a region into triangles and quadrilaterals)
The corner points of the elements represent the nodal points
If there are N nodal points then the continuous variation of h
in region R is approximated by the expression
h (x_v) ~ h f (xy) (11) a n n
where
h (AV) = approximation of u (xy) a
h = value of h in nodal point n n
f ltxgty) = global coordinate function
In nodal point n a must be equal to h so f (x y ) = 1 and a n n n n
f (xy) = 0 i n Between f (x v ) and f (xy) of the nodal n j ] n n n n j j
points surrounding nodal point n f (xy) varies linearly from
1 to 0
In fig 3 a picture of f (xy) is given n
If one knows the values h in the nodal points n we know the approximation h of h We can find these values of b bv using for
a n
example the Galerkin method
Galerkins principle states that an approximate solution of h in
eq (10) can be obtained from the following system of equations
N ~~ (k d -A + -^- (k d T-y E h f (xy) f(xy)dxdy +
-ox x ox ocircy y oy - _ n n i R n~
J J Q f(xy) dx dy = 0 i = 1 N (12)
1 1
Fig 3 The global coordinate function f (xy)
The integration can be performed element by element with the
aid of the coordinate functions f (NEUMAN etal 1974)
For non-stationary problems it is not possible to apply the
Galerkin principle on the time derivative But it is possible
to discretize this derivative and to set up the system of equation
(eq 12) for each finite time step
One obtains in this way a series of sets of laquocriations tha s
an approximation of the non-stationary flow problem
Treatment of boundary conditions
a Initial conditions
In non-stationary flow problern gtne must start with given
values of a at time t = t ri c
b Boundary conditions
- Prescribed head boundary
Nodal points which lie on a prescribed head boundary have a
prescribed value for h bull r Hgte r umber of unknown h s is n n
reduced and accordingly no the number of equations
- Prescribed flux through z-- boundary
If one knows the flux (iuumlr jw or outflow per unit area) and
the area of the boundary oi-r element the inflow or outflow
through the boundary per flemlt=nt can be calculated
One may assume this flow to J bull part of the sink term Q in
eq (12)
12
32 A p p l i c a t i o n o f t h e f i n i t e e l e m e n t
m e t h o d o n t h e f l o w i n g r o u n d w a t e r
b a s i n s
From hydrogeological investigations the geometry and the matrix
of the hydraulic properties are known Knowing the geometry the
flow can be split up into a finite number of layers This holds
also for the nature of the boundaries
Now a nodal grid is superimposed on each layer in such a way
that the nodal points of the different layers are situated right
above each other (see fig 4) and in each layer the nodal point
lies in the middle of this layer
-nodal point
Fig 4 Schematic representation of the nodal points in the
xyz-space
Each layer has N nodal points of which P nodal points have a
prescribed head In each layer remain N-P = N nodal points with
unknown head Eq (12) can be applied to each point with unknown
head in each separate layer
As was stated in the last paragraph for each time step one
can write
13
Ah bdquo ~ -r1 + Q = O in = 12N in n m At yi A h + B (13)
where
A = in
ocircf ocircf ocircf of (k d f mdashri- ~ + k df - ~ -jS) dxdy
J x n ox ox y n oy oy
in (S d + S ) ff dx dy O if i icirc n
s y i n
Q f dx dy
The matrices A B and the vector Q have been defined in in l
elsewhere (NEUMAN etal 1974) A is the conductivity matrix in
B the storage matrix and Q the source vector in l
In order to clarify these rather complicate expressions apply
equation (13) poundo point 4 of fig 5
6 7
Fig 5 Top view of part of the nodal grid
lhl + A42h2 + A43h3 + A44h4 + A45h5 + A46h6 + A47h7 +
h - h
+ B _iuumli i ^ + Q = o 44 t - t x4
n n-1
(14)
where h and h are values of h at time t and t 4n 4n-1 4 n n-1
respectively
14
If the h-values belonging to the A-values are taken at time
t = t we get an equation with one unknown h for time t = t n-1 4n n
The equation can then be solved directly
However this is an explicit formulation which puts strong limitations
on the magnitude of the time step t - t (REMSON etal 1971) n n-1
Values larger than a certain value cause instability With an implicit method the values of h are taken at time
t = t This method is unconditionally stable Now one gets an equation n
with 7 unknowns h through h If one has 7 equations for every 1 n n
point 1 through 7 we can solve the values h i = 17
in
Of all the methods possible the iterative method of Gauss-Seidel
has been chosen The Gauss-Seidel method starts by choosing a first estimate of h at time t = t (usually equal to h ) At each in n j -i in-l iteration step a new value of h is calculated using the last
in calculated values of h If one passes the nodal points from 1 to
i n 7 eq (14) becomes
h-j + 1 = A h^ + 1 + A bdquo h | + 1 + A _ h | + 1 + A h j + A r h j + 4 n 4 1 1 n 4 2 2 n 43 3 n 45 5 n 4 6 6 n
B h B + A h J + - 4 4 4 n ~ + 0 J (A + mdash) A 4 7 h 7 n At n U 4 4 At
(15)
where
h-i = in
j = iteration step or number of iteration
i = number of nodal point n = time = t
n
The iteration process converges if the successive differences k+1 k
between h and h become continuously smaller in in
The iteration process is stopped if at each nodal point the k+1 k
difference between h and h is smaller than a certain prescribed in in
value
15
To accelerate the convergence an overrelaxation factor W is used
(for more information about W see FORSYTHE and WASOW 1960)
k+1 k T7 k+1 k h = h + W (h - h )
in in in in (16)
If W = 1 there is no overrelaxation Usually W varies between 1 and
2 and the optimum value very often is found by trial and error
Eq (15) is valid for points in aquifers The connection with
adjacent layers is made by the term Q because in this term the
flow from adjacent aquitards q is included (two aquifers are always
separated by an aquitard)
Aquitards
We dont apply the finite element method on points in aquitards
As stated before there is no horizontal flow in aquitards So it has
no sense to replace the continuous variation of h in the aquitard
by equation (11)
The nodal points in aquitards represent a certain volume of an
aquitard hich transmit water from one aquifer to another and which
has storage capacity The nodal points are included in the Gauss-
Seidel scheme -s follows (see fig 6 )
1 aauifer
a -_vS laquo 2 [~^-t^yM^^^-^r~ aqutarci l-Z
aquifer
Fig 6 Schematic situation for points in aquitards
Point 2 is situated in the middle of the aquitard with thickness
d storage coefficient S and vertical hvdraulic conductivity k c J v
Suppose the Gauss-Seidel iteration starts in the upper layer and
ends in the bottom-layer Then the flux from point 1 in aquifer 1 to
16
point 2 in the aquitard at time t = t and in iteration step j + 1
is
k q = (hf -hf )-iumlmdash f (17)
1 n 2 n OSxd1
and the flux from point 3 in aquifer 2 to point 2 in the aquitard
is
1 c 1
qbdquo = (h - hf) mdashHmdashr- (18) 2 3 n 2 n OSxd
The net flux ltl-q9 must be equal to the change in storage per unit
area per unit time This change in storage can numerically be
approximated by
te - h-
n n-1
Combina t ion of ( 1 7 ) (18) ana (19) g i v e s
l tr1 - h (hJ + 1 + hj - 2 hj+1) - - - - - - S1 - ilaquoS L u l l = o (20)
] i 3 n 2 n 1 c t - t j j x d n n -1
T + 1
From t h i s e q u a t i o n h^ kan be s o l v e d
3 3 C o m p u t e r o g r a m
The problem outl^ugrave so far has been translated into a computer
program written in FORTRAN IV
The flow chart ot Uis program is given in fig 7 whereas in
appendix A the complete listing of the program can be found
In the flow chart the 5 main parts of the program are
indicated
A Reading and printing of input data
B Determination of the matrices A and B and the vector Q per layer
17
M
1 A
FLOH to unsaturated zono
FLOW to tertiary syster
FLOW to secondnrv system
Leakage to adjacent aquitards
SOLVE equation (14)
yes
CALCULATE flux through constant head boundnr
CALCULATE waterbalance n p r 1 OAro r r J j - ~
PRINT of output
Ier node per layer
k = k+1
t = t + At
7 DUMP variables on tape
-gt-
( STOP )
Fig 7 Flow chart of the program FEMSAT with diviation in 5 main
parts A B C D and E
C START PROGRAM FEMSAT
f READ general information
PRINT general information
f READ element information
PRINT element information 7 READ stationary nodal point information
PRINT stationary nodal point information 7
READ non-stationary nodal point information for t = t
FRINT non-stationary nodal point information for t = t
L 2 (
^ yes R E A D non-stat input for t gt t
J PRINT non-stat input for t gt t (- 1 -- M degY
GENERATE A B and AREA-matrix for confined aquifers
_L
GENERATE A B and AREA-matrix for unconfined aouifer
bull
C Solving h with Gauss-Seidel
D Determination of the water balance terms per layer
E Printing of results
The program exists of a main program with several subroutines
The main program calls the subroutines In the subroutines most
calculations are performed
In this way an easy adaptation to different problems is
obtained (see also section 44) Comments in the program will
help the reader to understand the program
In the next chapter the preparation of the input data executing
of the program and the output will be treated
Acknowledgement
Some parts of the computer program were taken from the program
UNSAT2 of NEUMAN etal (1974)
IV INPUT EXECUTION AND OUTPUT
41 I n p u t
The input is divided into endogenous variables (chosen by the
user or calculated by the program during execution) and exogenous
variables (based on physical data)
411 Endogenous variables
Translation of groundwater basin into a nodal point network
Probably the most important but also the most difficult stage
in the process of simulating groundwater flow is the translation
of the natural groundwater basin into a nodal point network Questions
which arise are
a In how many layer the geo-hydrological system mustcan be
schematized
b The locations of the boundaries and the nature of these boundaries
(impervious prescribed head prescribed flux)
c The configuration of the nodal points
19
The answers of these questions depend on the problem under
consideration
- Schematization into layers
Usually our knowledge about the geo-hydrological situation is
such that no more than 5 different layers can be distinguished
(remember contrast rule)
- Boundary conditions
The top boundary condition of a groundwater basin usually is
a prescribed flux (effective precipitation)
The bottom boundary condition usually is a prescribed flux
boundary with flux zero (impervious hydrological base)
For the boundaries at the other sides of the system one sees
either prescribed head boundaries (eg a fully penetrating
river) or prescribed flux boundaries (including zero flux)
- Nodal point configuration
Nodal points are situated
on the boundaries
along internal boundaries which indicate changes in
hydraulic properties within the layer
along the tertiary surface water system
on places where artificial recharge or discharge wiii tgtke
place
elsewhere
Construction of the finite element mesh
Assume we have schemetized the groundwater basin iatu a finite
number of layers with different direction of flow (horizontal and
vertical) Within one layer the hydrological properties may vary
from place to place
Now a nodal grid is superimposed on each layer in a way already
described in section 32 Thus each layer has the same number of
nodal points situated on the same places in the horizontal plane
but with different z-coordinates
We are free to construct quadrilaterals or triangles because
the program automatically divides a quadrilateral into two triangles
having identical material properties The dimensions of the elements
20
should be small at places where large changes in hydrological
properties occur or where large gradients in hydraulic head are
expected to occur
Coding of nodal points
Each node in the finite element network is assigned an integer
code which indicates the type of calculation for this node
The coding is as follows
- Code = 1 nodal points with described head as function of time
- Code 2 internal nodal points
- Code = 3 internal nodal points which represent secondary surface
water system
- Code = 5 nodal points with prescribed artificial recharge or
discharge and points with prescribed boundary flow
(both as functions of time)
Other important endogenous variables are the magnitude of the
time step the value of the variable which stops the iteration
process and the maximum number of iterations allowed per time step
(see also group A and B of section 413)
412 Exogenous variables
After constructing the nodal grid each node represents part of
the groundwater basin Geo-hydrological properties can be assigned
to each mode based on physical data (group C D and E of section
413) In group F of section 413 time-dependent variables can
be assigned to each node
413 Data input to Program FEMSAT
The data input must be punched on cards according to the
following instructions
GROUP COLUMNS FORMAT SYMBOL DESCRIPTION
A 1-80 20A4 HEAD Desired heading to be printed at
the beginning of the execution
1-5 15 NUMNP number of nodal points
21
GROUP COLUMNS FORMAT SYMBOL DESCRIPTION
6-10 15
11-15 15
16-20 15
21-25 15
26-30 15
31-35 15
NUMEL
NUMLAY
MBAND
NTSPI
NTSPO
MAXIT
1-10 F103 ST
11-20 F103 DELT
21-30 F103 TMAX
31-40 F105 TOL
number of elements
number of layers
maximum difference in node number
between two adjacent nodes
number of time steps between a
change in non-stationary input data
number of time steps between two
successive outputs
maximum number of iterations per
time step
time to start with the execution
initial time interval
maximum time to be reached before
execution is stopped
maximum allowed absolute change in
the values of h between any two
successive iterations in a given
time step
Iterations continue until all changes
are less than or equal to TOL or
until MAXIT is exceeded
1- 5 15
6-10 15
11-15 15
16-20 15
KX(N1)
KX(N2)
KX(N3)
KX(N4)
sequential number of i-th corner of
element N
sequential number of j-th corner
sequential number of k-th corner
sequential number of 1-th corner
(see fig 8)
Fig 8 Numbering of element corner nodes
22
GROUP COLUMNS FORMAcircT SYMBOL DESCRIPTION
One card must be provided for each element in sequential order
starting with N = 1 and ending with N = NVJMEL
I-K 1103 SANG(NLY)
1i-iO MO3 C1(NLY)
21-30 F 103 C2(NLY)
21-4- 1-104 THI(NLY)
angle in degrees between CI and
the laquo-coordinate to he used in
nodal print N in layer LY (see
poundig 9)
First principal conductivity of
material in node N and layer LY
idem second principal conductivity
thickness of layer LY at nodal
point N
ig V Definition of SANGCI and C2
One card icircuiPt be provided for each model point ctarting wih M = 1
and ending with N = NUMNP If there is norlt3 than 1 layer for each
layer a set must be provided starting with the upper layer LY - 1
and ending with the bottom layer LY = NIJMLAY The variables SANG
M and 2 are not relevant tor aquitards but for reasons of
conveniency this procedure is maintained (making SANG CI and C2
dummy variables)
1-10 110 KODE(N)
11-20 II0 NTERR(N)
coding of nodal points (see
paragraph 42)
number of tertiary relation in node N
The q-h relationship of the flow to
23
GROUP COLUMNS FORMAT SYMBOL DESCRIPTION
21-30 FlOl X(N)
31-40 FlOl Y(N)
41-50 FlOl HGL(N)
51-60 FlOl HBSC(N)
61-70 FlOl LESC(N)
the tertiary system as discussed
in par 24 are schematized in a small
number of different relations
x-coardinate of node N
y-coordinate of node N
height ground level in node N
height bottom secondary conduit in
node N If node N is not a secondary
conduit this variable is a dummy
variable
length of secondary conduit in area
represented by node N
One card must be provided for each node starting with N = 1 and
ending with N = NUMNP
E 1-10 FlOl P(NLY1) initial value of h at node N in
layer LY
11-20 FlOl VERRES(NLY) vertical resistance at node N and
layer LY If the layer is an aquifer
this value is a dummy variable
21-30 F101 RADRES(NLY) radial resistance for flow to
secondary system at node N in layer
LY
31-40 FlOl ENRES(NLY) entrance resistance for flow to
secondary system at node N in layer
LY
41-50 FlOl WP(NLY) wet perimeter of secondary system at
node N in layer LY
51-60 FlOl P0R(NLY) specific yield of the material at
node N in layer LY
61-70 FlOl SS(NLY) specific storage of the material
at node N in layer LY
24
GROUP COLUMNS FORMAT SYMBOL DESCRIPTION
One card must be provided for each node starting with N = 1 and
ending with N = NUMNP If there is more than 1 layer for each layer
a set of cards must be provided starting with LY = 1 and ending
with LY = NUMLAY
F 1-10 F101 P(NLY1) value of h at different times at
node N in layer LY Node N lies
on a prescribed head boundary
11-20 F101 WTSC(N) water table in secondary system at
node N at different times
21-30 F 101 ARRED(NLY) artificial recharge to or discharge
from node N in layer LY or
prescribed boundary flow at
different times
31-40 F 101 QPHR(N) flux through phreatic surface at
node N at different times
The data in group F are non-stationary input data One card must be
provided for each node starting with N = 1 and ending with N = NUMNP
In case of more than 1 layer for each layer a set must be provided
Not relevant variables are taken as dummy variables
After a certain number of time steps (NTSPI) the data of group E
are read So the set cards E must contain NxLYxC cards where C =
(TMAXNTSPIXDELT) + 1 if DELT is a constant
42 R u n n i n g t h e p r o g r a m o n t h e C y b e r 7200
The program has been executed on a Cyber 7200 computer of
IWIS-TNO in the Hague
- The demanded input is punched on cards according to the instructions
of the last paragraph and is put on a permanent file
- For each specific problem inthe program only the dimension statements
in the main program have to be adjusted according to the instructions
in the lines of comment
- After compilation the program together with the input file is
executed
25
- Intermediate data necessary for continuation of the calculations
are saved on another permanent file This gives the opportunity
to restart the program at a time the calculation was stopped
The only change in the input file is to adjust the starting time
Remark
With the aid of a terminal small changes in the program and in the
input file can be made easily and the output immediately appears
on the terminal In this way it is possible to stop erroneous
calculations This is especially useful when being in the testing
phase of the program
43 O u t p u t
Immediately after reading the input data they are printed so
that possible errors can be detected During execution after each
certain number of time steps (NTSPO) calculated values of h and
water balance terms are printed In appendix B an example is given
The output data can be devided into two groups
A Data per node per layer
- Coding of the nodal points KODE
- Hydraulic head HEAD
- Flow through phreatic surface at the time indicated in head
of table FLOWUN
- Flow to tertiary surface water system FLOWTS
- Flow to secondary surface water system FLOWSS
- Flow to adjacent layers LEAKAGE
- Artificial recharge or discharge and prescribed boundary
flow ARRED
- Change in storage during last time step STORAGE
- Flow through prescribed head boundary LATFLOW
B Data per layer
Summation of the terms already mentioned under A gives the
magnitude of the water balance terms per layer at the time
indicated (instanteneous water balance terms) Summation of
these instanteneous terms multiplied by their respective time
intervals since time is zero gives the cumulative water balance
terms per layer
26
After each print out we are able to check the calculations
because the sum of the water balance terms per layer must be
approximately zero
44 C o m p u t i n g t i m e
The computing time depends on the number of nodal points and
the number of layers But it depends also on the magnitude of the
changes in hydraulic head and on the value which terminates the
Gauss-Seidel iteration (TOL) To give an impression TOL = 001 m
Costs per day simulation (in hlfl) on the Cyber 7200
number of nodes number of layer bull bdquo bull bull _ 1mdash x a x b (21)
where
a = costs in Dutch guilders per system second (ƒ 027)
b = empirical factor with magnitude about 001
V NUMERICAL EXPERIMENTS
In general there are two possibilities to test the validity of
numerical models The first and most commonly used method is to
compare numerical results with analytical solutions The second
possibility is to compare numerical results with measured field
data Since the last alternative requires many data that are
difficult to obtain we restrict ourselves for the time being to
comparison with the analytical solutions
5 1 F l o w t o a w e l l i n a n u n c o n f i n e d
c i r c u l a r - s h a p e d a q u i f e r
Flow to a well can be divided into steady-state flow and
unsteady-state flow (KRUSEMAN and DE RIDDER 1976)
27
511 Steady state flow in an unconfined aquifer
For this case the formula of Thiem-Dupuit is valid (see fig 10)
Q bull k lth2-hicircgt ln(r2r])
(22)
where
hhbdquo = hydraulic head in 1 and 2 respectively
Q = discharge from well
rbdquo = distance from well
CL31
hmdash^ri --r2
Fig 10 Schematic cross-section of a pumped unconfined aquifer
Fig 11 gives the nodal grid superimposed on a well in a
circular shaped aquifer Near the well the nodal point distance
is made smaller Quadrilaterals as well as triangles are used
Numerical calculations with this finite element network and 3 -1
an extraction rate of 200 m day were performed for 10 days
At that moment a steady-state situation was almost achieved
28
2tmdashnodal point number
160 180 X-axesim)
Fig 11 Nodal point configuration for flow to a well in a circular-
shaped aquifer The outer boundary is taken to be constant
The well is situated in nodal point 42
29
In fig 12 the results of the simulation are compared with the
analytical solution The agreement between numerical results and
the analytical solution is excellent
draw- 15irS (m) down
analytical solution imdashx calculated values
distance from well r (m)
Fig 12 Comparison between analyt ical and calculated s teady-state
drawdown for the problem of f ig 11 with 1
-1 Q bull - 200 m 3 d a y _ 1
k = 10 mday
d = 100 m
S = 2 y
S = 0 s
512 Unsteady-state flow in an unconfined aquifer
Analytical solutions for unsteady-state flow in an unconfined
aquifer only exist if the aquifer extends infinitely For the
aquifer given in fig 11 this condition is not fulfilled However
during the first one or two days after starting the extraction the
influence of a constant hydraulic head at the outer boundary on the
30
drawdown near the well can be neglected (see fig 13) After one
day the lateral inflow is only 25 of the extraction the remainder
being the change in storage due to drawdown
2 0 0
m 3 day 1
Q=200m 3 day
100-
vraquomdash Lateral inflow
N amp storage
I I ^ V - W 10
t (days)
Fig 13 Values of Q lateral inflow and change in storage as
functions of time
The nonsteady-state or Theis equation can be written as
i _ Q s 4iTkD
dy = 4irkD
W(u) (23)
where
u = r2S
4irkDt
r = distance from well
S = storage coefficient
Q = constant discharge from well
D = thickness of the waterbearing layer
at the beginning of pumping
t = time after starting the extraction
s = corrected drawdown
= s - 322D
s = calculated drawdown
W(u) = Theiss well function
[J
CIAT1
[T]
]
31
In fig 14 the analytical unsteady-state solution for the case
described in section 511 is given together with the numerical
results at times 05 and 10 days The agreement between numerical
results and the analytical solution is rather good
Drawdown s1(m)
10
10
10
Theiss solution
x caculated values t=Q5
bdquo t= lO
10 10- 10 10 t r2(daym2)
Fig 14 Comparison between analytical and calculated steady-state
drawdown for the problem of fig 11 with
Q
k
d
S y
s s
=
=
=
=
=
200
1
10
3 m
0
0
2
0
-1 day
m day
m
These results indicate that the assumption of a linear variation
of h between two nodal points (see fig 3 ) even with the rather
rough element network of fig 11 is acceptable So there is no
need to improve for example the finite element network by introducing
higher order coordinate functions
32
52 F l o w t o w e l l i n 3-1 a y e r e d s y s t e m
In fig 15 a schematic cross-section of a 3-layered system of
two aquifers separated by an aquitard is given A well of
infinitisemal radius is completed in the lower aquifer and discharges
at a constant rate QTT Each layer is homogeneous isotropic and
extending radially infinite
bdquo bull - bull bull ~ i
mdash Q II
aquifer I
lugrave^i~IcircIuml7i-i~~ aquitard --r-rl~-
-II--I i - i
aquifer
Fig 15 Schematic cross-section of a 3-layered system
For this situation NEUMAN and WITHERSPOON (1969a) give rather
complicated analytical solutions In the special case where the
hydraulic properties of the two aquifers are identical they give
graphical illustrations of their solutions for three different
cases
These graphical solutions give the opportunity to verify
roughly numerical results of a 3-layered system Therefore a nodal
grid is superimposed on a circular shaped groundwater flow system
with the extraction in the centre of the lower aquifer and with
a constant head on the outer boundary Fig 16 gives the nodal
point situation in the x - y plane of 1 layer The nodes in the
other two layers are situated in the middle of the layers at the
same places in the x - y plane only their z-coordinate is different
33
A
-Jy
bull33
V 5
iumlff
Fig 16 Nodal point configuration for the 3-layered problem (top view)
34
In fig 17 numerical results are compared with the analytical
solution The agreement is rather bad especially for the two
aquifers But in terms of the water balance the difference between
aquifers and aquitard is not so big because the aquitard has
about 16 times bigger storage capacity than the aquifers The
small value of the specific storage of the aquifers is the reason
that small errors in the components of the water balance have a
big influence on the calculated values of the hydraulic head
Eig 17 Dimensionless drawdown vs dimensionless time in a
3-layered system with
II = 50 m day
3 m (distance from well)
aquifer I
aquitard
aquifer II
Thi (m)
100
25
100
S s
001
0642
001
k(mday )
1
0277
1
The discrepancies are partly caused by errors in discretization
In the z-direction these errors can be eliminated by taking the
35
vertical resistance of the aquitard to be infinite Then the
calculated drawdown in the pumped aquifer can be compared with
the analytical Theiss solution
This is done in fig 18 for different distances from the well
One sees that the agreement is better when the distance is bigger
Apparently the linearization of the logarithmic drawdown curve
near the well is responsible for the differences The differences
for r = 3 m are of the same magnitude as for the pumped aquifer in
the 3-layered case
joo gt A 50
20
Theiss solution
bull calculated values r=3m (distance f rom well) x bdquo bdquo r=7m
1 L
I l_ 001 003 0 0 5 Ol Q2
- I I 5 10 t - K t
Fig 18 Dimensionless drawdown vs dimensionless time in a
confined aquifer with
Q =500 m day
k = 10 mday
D = 100 m
S = 001 s
The differences between numerical results and analytical solutions
are not caused by taking a finite area because changing the outer
boundary into an impermeable boundary has only minor effects
36
VI SUMMARY AND CONCLUSIONS
In this report a numerical model for non-stationary saturated
groundwater flow in a multi-layered system has been treated
Groundwater flow can be described in different ways
a Physical-mathematical by combining Darcys law and the law of
continuity (section 21)
b Geo-hydrological the flow in a natural groundwater basin takes
place in layers with different hydraulic properties Definitions
of the most important hydraulic properties are given in section
22
c Schematical groundwater flows horizontally in aquifers and
vertically in aquitards (section 23)
d Approximating by dividing the region of flow into a finite
number of elements and applying Darcys law and the law of
continuity on each of these elements In these ways one gets
a set of equations which can be solved given (initial and)
boundary conditions (section 25)
e Numerical the set of equations has been solved using a Galerkin
type finite element method and the implicit Gauss-Seidel method
with overrelaxation factor (section 31 and 32)
f Computer-solvable the numerical solution has been translated
in a FORTRAN IV program called FEMSAT (section 33)
In chapter IV the input execution and output of the program
FEMSAT are treated
Finally in chapter V some numerical applications are given
simulation of flow to a well in unconfined aquifer and simulation
of flow to a well in a 3-layered configuration
Comparison of hydraulic heads obtained by numerical calculations
and analytical true solutions showed an excellent agreement in case
of stationary flow and a good to moderate agreement in case of non-
stationary flow However the diviations caused by the discretization
process must be seen in relation to the errors made in the
establishment of the hydraulic properties
37
As was stated in the introduction numerical models can be
valuable tools in the management of water resources Therefore
numerical models must be suitable to simulate effects of certain
operations upon the spreading of water over saturated groundwater
unsaturated groundwater and surface water The model described in
this report is a model for saturated groundwater flow The relation
with the surface water can be taken into account via radial and
entrance resistances or via a relation between flow to the surface
water system and the hydraulic head The relation with the unsaturated
zone has been restricted to the input factor effective precipitation
In practise however the unsaturated and saturated system are
interconnected Therefore the model should be improved by taking
into account the unsaturated zone This can be done in at least
two ways
a Replacing the unsaturated zone by a parametrical model
b Coupling the unsaturated zone in an iterative way with the
saturated system
In order to find out the usefulness of the numerical model for
water resources management it will be applied to actual regional
groundwater systemslaquoAlso sensitivity analyses will be carried out
The computer model itself can be improved by applying standard
programs eg automatical generation of a nodal grid and graphical
plotting of the results
38
VII REFERENCES
ERNST LF 1962 Grondwaterstromingen in de verzadigde zone en hun
berekening bij aanwezigheid van horizontale evenwijdige open
leidingen Thesis University Utrecht pp 189
1978 Drainage of undulating soils with high groundwater
tables (in press)
FORSYTHE GE and WR WASON 1960 Finite Difference Methods for
Partial Differential Equations John Wiley and Sons New York
pp 444
KRUSEMAN GP and NA DE RIDDER 1976 Analysis and evaluation of
pumping test data ILRI-bulletin No 11 pp 200
LAAT PJM DE and C VAN DEN AKKER 1976 Simulatiemodel voor grond-
waterstroming en verdamping in Modelonderzoek 1971-1974 ten
behoeve van de waterhuishouding in Gelderland deel II
Grondslagen pp 145-197
NEUMAN SP RA FEDDES and E BRESLER 1974 Finite element
simulation of flow in saturated -unsaturated soils considering
water uptake by plants Hydraulic Engineering Laboratory
Technion Israel pp 104
and PA WITHERSPOON 1969a Theory of Flow in a confined
two Aquifer System Water Resources Research Vol 5 No 4
pp 803-816
^_ and PA WITHERSPOON 1969b Applicability of Current Theories
of Flow in leaky aquifers Water Resources Research Vol 5
No 4 pp 817-829
PINDER GF and EO FRIND 1972 Application of Galerkins Procedure
to Aquifer Analysis Water Resources Research Vol 8 no 1
pp 108-120
REMSON I GE HORNBERGER and FJ MOLZ 1971 Numerical Methods
in Subsurface Hydrology John Wiley and Sons New York
pp 389
WESSELING JG 1977 Toepassing van Finite Difference- en Finite
Element Methoden voor het oplossen van een aantal grondwatershy
stromingsproblemen (met computerprogrammas) ICW-nota 944
pp 63
39
ZIENKIEWICZ OC 1971 Finite Element Method in Engineering Science
McGraw-Hill London pp 521
40
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X - e - o
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LUX
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APPENDIX 8 EXAMPLE OF OUTPUT
HEAD AND TERMS OF UATERBALANCE PER NODE PER LAYER
AT TIME 260 IN II AND I13DAYS
T
LAYER NODE KODE HEAD FLOUUN FLOUTS FLOUSS LEAKAGE STORAGE LAT FLOU
13 19 25 31 37 43
125008 125000 12S000 12S000 12S000 12S000 12S000 12S000
0000 0000 0000 0000 0000 0000 0000 0000
0000 0000 0000 0000 0000 0000 0000 0000
0000 0000 0000 0000 0000 0000 0000 0000
- 000 - 000 -000 - 000 - 000 - 000 - 000 0000
0000 0000 0000 0000 0000 0000 0000 0000
0000 000 000 000 000 000 000 000
000 0000 0000 0P100 0000 0000 0000 0000
1 7
13 19 25 31 37
12S000 125000 125000 125000 125000 12S000 125000 125000
0000 0000 0000 0000 0000 0000 0000 0000
- 001 -007 - 023 - 020 - 009 - 003 0000
0000 0000 0000 0000 0000 0000 0000 0000
000
001
007
023
020
009
003
003
0000 0000 0000 0000 0000 0000 0000 0000
1 7
13 19 25 31 37 43
125000 125000 124998 124972 124895 124787 124647 124358
0000 0000 0000 0000 0000 0000 0000 0000
- 000 001 007 023 020 009 003
0000
0000 0000 0000 0000 0000 0000 0000
50000
0000 182
1982 3862 1674 419 078 041
0000 0000 0000 0000 0000 0000 0000 0000
TERMS OF WATERBALANCE PER LAYER DURING LAST TIMESTEP AND
AND SINCE BEGINNING IN M3DAY M3
FLOU TO UNSATURATED ZONE
FLOU TO TERTIARY SYSTEM
0000
0000
0000
0000
LAYER NO 1
FLOU TO SECONDARY SYSTEM
FLOU TO AD3ACENT LAYERCS)
ARTIFICIAL FLOU AND PRESCRIBED BOUNDARY FLOU
ACCUMULATION TERMS
LATERAL FLOU THROUGH PRESCR HEAD BOUNDARY
0000
-000
0000
000
000
0000
-000
0000
000
000
LAYER NO 2
FLOU TO SECONDARY SYSTEM
FLOU TO AD3ACENT LAYERCS1
ARTIFICIAL FLOU AND PRESCRIBED BOUNDARY FLOU
ACCUMULATION TERMS
LATERAL FLOU THROUGH PRESCR HEAD BOUNDARY
0000
-378
0000
381
0000
0000
-055
0000
056
0000
LAYER NO 3
FLOU TO SECONDARY SYSTEM
FLOU TO AD3ACENT LAYERCS5
ARTIFICIAL FLOU AND PRESCRIBED BOUNDARY FLOU
ACCUMULATION TERMS
LATERAL FLOU THROUGH PRESCR HEAD BOUNDARY
0000
370
50000
49376
0000
0000
055
12979
12909
0000
mdash 1
biz
V NUMERICAL EXPERIMENTS 27
51 Flow to a well in an unconfined circular-
shaped aquifer 27
511 Steady state flow in an unconfined aquifer 28
512 Unsteady-state flow in an unconfined aquifer 30
52 Flow to a well in a 3-layered system 33
VI SUMMARY AND CONCLUSIONS 37
VII REFERENCES 39
i
I INTRODUCTION
The groundwater system is part of the hydrological cycle and
changes in this system affects phenomena such as plant production
river flow etc Generally one tries to use groundwater resources
in such a way that the utility for the community as a whole is
maximal
Numerical flow models can be valuable tools in the management
of water resources In many situations these models are the only
possible way to indicate effects of certain interferences in the
groundwater system
In this paper a numerical model is presented which simulates
non-stationary saturated groundwater flow in a certain region
Purpose of the model is to predict effects of certain operations
upon the various terms of the water balance of the groundwater basin
and on the hydraulic head
The basic idea of the model is that groundwater is flowing
horizontally in waterbearing layers and vertically in less-permeable
layers Each layer is discretized into a number of elements To
each element both Darcys law and the law of continuity are applied
(chapter II)
In this way a set of equations is obtained which are solved
using a finite element method and the Gauss-Seidel iteration method
(chapter III)
In chapter IV the input and output of the model are treated
while in chapter V some numerical experiments are presented
Finally in chapter VI an evaluation of the model is given
II GENERAL DESCRIPTION OF THE PROBLEM
21 P h y s i c a l b a c k g r o u n d
Throughout this report water potential is expressed on a unit
weight basis and is being called hydraulic head with symbol h
h = 2- + z [L] Pg
where
mdash = piezometer head |L | Pg u -1
z = gravitational head |_Lj
The hydraulic head is an expression for the capacity of a unit
weight of water to do work as compared to the work capacity of the
same weight of water with the same chemical composition at a certain
reference level and under atmosferic pressure
The problem is restricted to saturated flow of groundwater with
one constant density
211 The law of linear resistance (Darcys law)
According to Darcys law the rate of flow through a porous medium
is proportional to the gradient in hydraulic head It may be written
as
ocirc h ltSh ocirch N q = _k Tmdash q = -k vmdash q = -k -rmdash ( 1 ) x y ocircx y y lt5y nz z ocircz
where
q = volume flux of water passing through a unit area per x y z
per unit time perpendicular to x y z direction
respectively |_LT J
k = hydraulic conductivity in x y and z direction x gt y gt z _ _ i _
respectively | LT J
The coefficient k in the Darcy flow equation is taken to be
a constant depending on both the properties of the porous medium
and the fluid
212 The law of continuity
The other important physical law states that no matter can be
lost or created
The conservation equation can be found by applying the principle
of continuity to an infinite small volume
6v 6v lt5v 60 x y z TT = T + x + ~T7 + 1 (2)
Ot OX OcircX oz
where
0 = volume of water per unit bulk volume of soil |_-J
q = sink or source term T J
213 Combination of Darcys law and the law of continuity
Darcys law combined with the law of continuity gives
60 lt5 oh lt5 ocirch 6 oh _ ocirct ocircx x ocircx ocircy y oy ocircz z ocircz
or
c 5h 6 ocirch ocirc Ocirch 6 ocirch c ot ox x ocircx ocircy y oy laquoz z 6z
where
S = volume of water released or stored per unit bulk c I
volume of soil per unit change in hydraulic head |L I
22 H y d r a u l i c p r o p e r t i e s a n d t y p e s o f
a q u i f e r s
The flow in a natural groundwater basin takes place in layers
with different hydraulic properties Usually the layers are divided
into good permeable layers or waterbearing layers (aquifers) and
less permeable layers (aquitards)
The most important hydraulic properties are
a Transmissiviy - the product of the average hydraulic conductivity
k and the saturated vertical thickness d of the waterbearing
layer
It is a measure for the transport capacity of aquifers
Notation T = kd |_ L T ~
b Hydraulic resistance - the ratio of vertical thickness of
aquitards d and hydraulic conductivity k It is a measure for
the resistance capacity of aquitards denoted as c = d k _ T J
c Specific yield - the volume of water released or stored in the
phreatic zone per unit surface area of an unconfined aquifer per
unit change in hydraulic head
It is also called effective porosity
The symbol used for specific storage will be S j_-J
d Specific storage - the volume of water released or stored per
unit volume of the saturated parts of an aquifer or aquitard per
unit change in hydraulic head It is a measure for the elasticity
of the soil material and the fluid Excluded are irreversible
processes Notation S _ L J
e Storage coefficient - the volume of water released or stored
pe7 nit surface of a layer per unit change in hydraulic head
It is the sum of specific yield and the product of specific
storage and thickness of the aquifer or aquitard
S = S + S d r - 1 c y s - -1
Note in confined layers S = 0 y
In fig 1 a 3-layered configuration is given of two aquifers
separated by an aquitard Aquifer I has a free water surface and is
an unconfined aquifer Aquifer II is a completely saturated aquifer
Dependent on the hydraulic resistance of the aquitard KRUSEMAN and
DE RIDDER (1976) distinguish between confined semi-confined and
semi-unconfined aquifers For our purpose it is not useful to make
this distinction We restrict the aquifer types to unconfined and
confined aquifers
WIHtUtttWttllttlt(ltmtttltHlttltlil(lltlittlfllnl ) bull (
d V U i l i l i ii i Hi il i M M bull f i m
i tMi| i i | J i j j i _ I I bull
aq ui tard
aquifer II
Fig 1 Schematic representatipn of flow direction and variation
of hydraulic head h in a 3-rlayered system
In an unconfined aquifers ( I ) poundhe vertical thickness pf the
waterbearing layer d is a variable while in a confined aquifer II
this is a constant (denoted as D ) Other important properties of
unconfined aquifers are the hydraulic head being the same as the
phreatic surface and the specific yield being different from zero
23 S c h eacute m a t i s a t i o n o f f l o w
The groundwater flow in a groundwater basin is schematized
into a horizontal flow in aquifers and into a vertical flow in
aquitards Consequently the vertical variation of the hydraulic
head is as drawn in figl
The validity of this assumption has )een investigated by
NEUMAN and WITHERSPOON (1969) The found that if the contrast
in hydraulic conductivity between two adjacent layers is bigger
than a factor two the relative error caused by this assumption
is usually smaller than 5
This schematization gives a simplification of the general flow
equation (eq 4 ) In an aquifer the vertical is zero and eq (4)
then reduces to
-Sbdquo -r- 7 - (k d -r-) + 7 mdash (k d 7-) + qd c lt5t ox x ox oy y oy
or
bull(S + S d) g y s ot
-r- (k d -=-) + -T- (k d -=-) + Q ocircx x ox oy y ocircy
(5)
where
Q = volume of water substracted from or added to unit surface
of aquifer per unit time |_ L T J
In an aquitard eq (4) reduces to
-sQ |pound = (k Q) s 6t oz Z OcircZ
(6)
As will be discussed in the next paragraph the term q in an
aquitard is set equal to zero
24 W a t e r b a l a n c e t e r m s
In eq (4) the term q is mentioned With this sink or source
term different terms of the water balance are taken into account
Fig 2 depicts schematically the situation for an unconfined aquifer
(for other layers some terms dont exist)
recharge- - discharge
laquo-b
Fig 2 Schematic representation of the various terms involved in the
water balance
Description of the different terms
a The flow through the phreatic surface q f
If this term is negative (water goes out of the saturated system)
it is called capillary rise If positive it is called percolation
(or drainage)
This flow can be calculated as a rest term of the water balance
of the unsaturated zone Very often this term is expressed as a
flow per unit surface (flux) and is called effective precipitation
For the time being this flux is a given input
b Flow to adjacent aquitards q
The flux to adjacent aquitards can be calculated by applying eq
(6) to the aquitard
c Flow to the primary surface water system q
The primary surface water system penetrates fully the groundwater
basin So it acts as a boundary with given values for the
hydraulic head (is equal to the phreatic level of the primary
system)
d Flow to the secondary surface water system q
The secondary surface water system does not penetrate fully the
groundwater basin which causes a resistance against flow between
groundwater and surface water The flow q can be calculated as
follows
h - h
- mdash f i i gt
where
h = hydraulic head of the groundwater at the place of
the secondary system |_ L |
h = phreatic level of secondary system |_ L J i s -
R = radial and entrance resistance of the
secondary system per unit surface |_ T J
According to ERNST (1962) the radial resistance H per unit
surface can be caKulated with
Uuml = 4 i n 4 e8) TTk B
where
1 = area drained by 1 m conduit L ^ J
d = thickness of the layer near the conduit _ L J
B = wet perimeter of the conduit |_ L J
The entrance resistance is caused by a layer which shows a
lower conductivity around the conduit than its surroundings
In practise radial and entrance resistancies are difficult to
deduct from hydraulic properties They can be found indirectly
by measuring simultaneously q h and h
t s
e Flow to the tertiary surface water system q
In smaller conduits usually the phreatic level is not known
So eq (7) cant be applied The flow to these conduits however
is sometimes very important (eg in areas with shallow groundwater
tables) One may a-sume (ERNST 1978) that a relationship exists
between the hydraulic head of an aquifor and the flow q
According to Ernst the tertiary system can be divided into a
number of categories with respect to their mutual distance and
bottom height For each category the flow is h - h (b)
q ( h b ) - mdash (9)
where
h = average hydraulic head | L J
h (b) = phreatic level in the category under
consideration a function of bottom height |_L J
a = geometry factor to convert h into h _ - J
Y = drainage resistance _ T J
hm = hydraulic head midway between two
conduits |_ L J m
Summation of the q(hb)-relations of the different categories
yields the overall relation between hydraulic head and flow to
the tertiary system we are looking for
f Artificial recharge or discharge q and prescribed boundary
flux qfo
The magnitudes of these flows are not influenced by the conditions
inside the groundwater basin but remain on a prescribed level
Thus q is an internal boundary condition q an external boundary cl D
condition
For confined aquifers some of the above mentioned terms dont
exist These terms are the flow through the phreatic surface qf
and the flow to the tertiary surface water system q
Aquitards are considered only as transmitters of water from one
aquifer to another with possibilities of storage (eq 6 ) This means
that for aquitards the water balance terms ar through f are set
equal to zero
25 S o l u t i o n o f g r o u n d w a t e r f l o w p r o b l e m s
The basic equations for saturated groundwater flow are eq (5)
for aquifers and eq (6) for aquitards The solution of these
equations require knowledge of the physical system and the auxiliary
conditions describing the system constraints
These auxiliary conditions are (REMSON etal 1971)
a Geometry of the system
Eg the diviation of the system into a restricted number of
layers and the horizontal extension of the system under
consideration
b The matrix of hydraulic properties including in-homogeneity
and an-isotropy
c Initial conditions Because we are dealing with non-stationary
flow the values of pertinent system variables (such as hydraulic
head at the initial time) must be known
d Boundary conditions They describe the conditions at the boundaries
pf the system as a function of time
One can distinguish between
- hydraulic head boundary
- flow (including zero flow) boundary
- both head and flow boundary
After specifying the auxiliary conditions an unique solution of
the variation of the hydraulic head in space and time can be
obtained
For special shapes of flow and assumptions exact analytical
solutions exist In nature however one is usually confronted with
problems like
- spacial variation in the hydraulic properties of the soil
- irregular shape of the groundwater basin
- empirical relations between q and h
- d not being a constant
For such situations by numerical and analogy models only an
approximation of the actual situation can be obtained In the next
chapter a numerical solution called the finite element method is
treated
III NUMERICAL SOLUTIONS
In this chapter the numerical solution of the aquifer equation
(eq 5) and the aquitard equation (eq 6) will be treated The
method of solving is the finite element method In section 31 only
a short mathematical description of this method will be given For
more mathematical background the reader is referred to publications
on this subject (eg ZIENKIEWICZ 1971 NEUMAN etal 1974 and
WESSELING 1976)
In section 32 the application of the finite element method on
the flow in groundwater basins is treated while in section 33 the
translation into a computer program is given
3 1 F i n i t e e l e m e n t m e t h o d
For reasons of simplicity the finite element method will be
explained for the stationary version of eq 5 ie
10
(k d^) + (k d ^ ) - Q = O (10) ocircx x 6x 6y y Sy
The basic idea of each numerical solution of eq (10) is to
apply this equation on a small but finite volume The flow region
of interest R is divided into a finite number of volumes called
elements The elements can have different shapes but for reasons
of conveniency we only use triangles- Quadrilaterals can also be
used because they can be divided into two triangles The sizes of
the triangles may vary They can be adapted according to the
geometry of the region and the accuracy desired (fig 11 gives an
example of the diviation of a region into triangles and quadrilaterals)
The corner points of the elements represent the nodal points
If there are N nodal points then the continuous variation of h
in region R is approximated by the expression
h (x_v) ~ h f (xy) (11) a n n
where
h (AV) = approximation of u (xy) a
h = value of h in nodal point n n
f ltxgty) = global coordinate function
In nodal point n a must be equal to h so f (x y ) = 1 and a n n n n
f (xy) = 0 i n Between f (x v ) and f (xy) of the nodal n j ] n n n n j j
points surrounding nodal point n f (xy) varies linearly from
1 to 0
In fig 3 a picture of f (xy) is given n
If one knows the values h in the nodal points n we know the approximation h of h We can find these values of b bv using for
a n
example the Galerkin method
Galerkins principle states that an approximate solution of h in
eq (10) can be obtained from the following system of equations
N ~~ (k d -A + -^- (k d T-y E h f (xy) f(xy)dxdy +
-ox x ox ocircy y oy - _ n n i R n~
J J Q f(xy) dx dy = 0 i = 1 N (12)
1 1
Fig 3 The global coordinate function f (xy)
The integration can be performed element by element with the
aid of the coordinate functions f (NEUMAN etal 1974)
For non-stationary problems it is not possible to apply the
Galerkin principle on the time derivative But it is possible
to discretize this derivative and to set up the system of equation
(eq 12) for each finite time step
One obtains in this way a series of sets of laquocriations tha s
an approximation of the non-stationary flow problem
Treatment of boundary conditions
a Initial conditions
In non-stationary flow problern gtne must start with given
values of a at time t = t ri c
b Boundary conditions
- Prescribed head boundary
Nodal points which lie on a prescribed head boundary have a
prescribed value for h bull r Hgte r umber of unknown h s is n n
reduced and accordingly no the number of equations
- Prescribed flux through z-- boundary
If one knows the flux (iuumlr jw or outflow per unit area) and
the area of the boundary oi-r element the inflow or outflow
through the boundary per flemlt=nt can be calculated
One may assume this flow to J bull part of the sink term Q in
eq (12)
12
32 A p p l i c a t i o n o f t h e f i n i t e e l e m e n t
m e t h o d o n t h e f l o w i n g r o u n d w a t e r
b a s i n s
From hydrogeological investigations the geometry and the matrix
of the hydraulic properties are known Knowing the geometry the
flow can be split up into a finite number of layers This holds
also for the nature of the boundaries
Now a nodal grid is superimposed on each layer in such a way
that the nodal points of the different layers are situated right
above each other (see fig 4) and in each layer the nodal point
lies in the middle of this layer
-nodal point
Fig 4 Schematic representation of the nodal points in the
xyz-space
Each layer has N nodal points of which P nodal points have a
prescribed head In each layer remain N-P = N nodal points with
unknown head Eq (12) can be applied to each point with unknown
head in each separate layer
As was stated in the last paragraph for each time step one
can write
13
Ah bdquo ~ -r1 + Q = O in = 12N in n m At yi A h + B (13)
where
A = in
ocircf ocircf ocircf of (k d f mdashri- ~ + k df - ~ -jS) dxdy
J x n ox ox y n oy oy
in (S d + S ) ff dx dy O if i icirc n
s y i n
Q f dx dy
The matrices A B and the vector Q have been defined in in l
elsewhere (NEUMAN etal 1974) A is the conductivity matrix in
B the storage matrix and Q the source vector in l
In order to clarify these rather complicate expressions apply
equation (13) poundo point 4 of fig 5
6 7
Fig 5 Top view of part of the nodal grid
lhl + A42h2 + A43h3 + A44h4 + A45h5 + A46h6 + A47h7 +
h - h
+ B _iuumli i ^ + Q = o 44 t - t x4
n n-1
(14)
where h and h are values of h at time t and t 4n 4n-1 4 n n-1
respectively
14
If the h-values belonging to the A-values are taken at time
t = t we get an equation with one unknown h for time t = t n-1 4n n
The equation can then be solved directly
However this is an explicit formulation which puts strong limitations
on the magnitude of the time step t - t (REMSON etal 1971) n n-1
Values larger than a certain value cause instability With an implicit method the values of h are taken at time
t = t This method is unconditionally stable Now one gets an equation n
with 7 unknowns h through h If one has 7 equations for every 1 n n
point 1 through 7 we can solve the values h i = 17
in
Of all the methods possible the iterative method of Gauss-Seidel
has been chosen The Gauss-Seidel method starts by choosing a first estimate of h at time t = t (usually equal to h ) At each in n j -i in-l iteration step a new value of h is calculated using the last
in calculated values of h If one passes the nodal points from 1 to
i n 7 eq (14) becomes
h-j + 1 = A h^ + 1 + A bdquo h | + 1 + A _ h | + 1 + A h j + A r h j + 4 n 4 1 1 n 4 2 2 n 43 3 n 45 5 n 4 6 6 n
B h B + A h J + - 4 4 4 n ~ + 0 J (A + mdash) A 4 7 h 7 n At n U 4 4 At
(15)
where
h-i = in
j = iteration step or number of iteration
i = number of nodal point n = time = t
n
The iteration process converges if the successive differences k+1 k
between h and h become continuously smaller in in
The iteration process is stopped if at each nodal point the k+1 k
difference between h and h is smaller than a certain prescribed in in
value
15
To accelerate the convergence an overrelaxation factor W is used
(for more information about W see FORSYTHE and WASOW 1960)
k+1 k T7 k+1 k h = h + W (h - h )
in in in in (16)
If W = 1 there is no overrelaxation Usually W varies between 1 and
2 and the optimum value very often is found by trial and error
Eq (15) is valid for points in aquifers The connection with
adjacent layers is made by the term Q because in this term the
flow from adjacent aquitards q is included (two aquifers are always
separated by an aquitard)
Aquitards
We dont apply the finite element method on points in aquitards
As stated before there is no horizontal flow in aquitards So it has
no sense to replace the continuous variation of h in the aquitard
by equation (11)
The nodal points in aquitards represent a certain volume of an
aquitard hich transmit water from one aquifer to another and which
has storage capacity The nodal points are included in the Gauss-
Seidel scheme -s follows (see fig 6 )
1 aauifer
a -_vS laquo 2 [~^-t^yM^^^-^r~ aqutarci l-Z
aquifer
Fig 6 Schematic situation for points in aquitards
Point 2 is situated in the middle of the aquitard with thickness
d storage coefficient S and vertical hvdraulic conductivity k c J v
Suppose the Gauss-Seidel iteration starts in the upper layer and
ends in the bottom-layer Then the flux from point 1 in aquifer 1 to
16
point 2 in the aquitard at time t = t and in iteration step j + 1
is
k q = (hf -hf )-iumlmdash f (17)
1 n 2 n OSxd1
and the flux from point 3 in aquifer 2 to point 2 in the aquitard
is
1 c 1
qbdquo = (h - hf) mdashHmdashr- (18) 2 3 n 2 n OSxd
The net flux ltl-q9 must be equal to the change in storage per unit
area per unit time This change in storage can numerically be
approximated by
te - h-
n n-1
Combina t ion of ( 1 7 ) (18) ana (19) g i v e s
l tr1 - h (hJ + 1 + hj - 2 hj+1) - - - - - - S1 - ilaquoS L u l l = o (20)
] i 3 n 2 n 1 c t - t j j x d n n -1
T + 1
From t h i s e q u a t i o n h^ kan be s o l v e d
3 3 C o m p u t e r o g r a m
The problem outl^ugrave so far has been translated into a computer
program written in FORTRAN IV
The flow chart ot Uis program is given in fig 7 whereas in
appendix A the complete listing of the program can be found
In the flow chart the 5 main parts of the program are
indicated
A Reading and printing of input data
B Determination of the matrices A and B and the vector Q per layer
17
M
1 A
FLOH to unsaturated zono
FLOW to tertiary syster
FLOW to secondnrv system
Leakage to adjacent aquitards
SOLVE equation (14)
yes
CALCULATE flux through constant head boundnr
CALCULATE waterbalance n p r 1 OAro r r J j - ~
PRINT of output
Ier node per layer
k = k+1
t = t + At
7 DUMP variables on tape
-gt-
( STOP )
Fig 7 Flow chart of the program FEMSAT with diviation in 5 main
parts A B C D and E
C START PROGRAM FEMSAT
f READ general information
PRINT general information
f READ element information
PRINT element information 7 READ stationary nodal point information
PRINT stationary nodal point information 7
READ non-stationary nodal point information for t = t
FRINT non-stationary nodal point information for t = t
L 2 (
^ yes R E A D non-stat input for t gt t
J PRINT non-stat input for t gt t (- 1 -- M degY
GENERATE A B and AREA-matrix for confined aquifers
_L
GENERATE A B and AREA-matrix for unconfined aouifer
bull
C Solving h with Gauss-Seidel
D Determination of the water balance terms per layer
E Printing of results
The program exists of a main program with several subroutines
The main program calls the subroutines In the subroutines most
calculations are performed
In this way an easy adaptation to different problems is
obtained (see also section 44) Comments in the program will
help the reader to understand the program
In the next chapter the preparation of the input data executing
of the program and the output will be treated
Acknowledgement
Some parts of the computer program were taken from the program
UNSAT2 of NEUMAN etal (1974)
IV INPUT EXECUTION AND OUTPUT
41 I n p u t
The input is divided into endogenous variables (chosen by the
user or calculated by the program during execution) and exogenous
variables (based on physical data)
411 Endogenous variables
Translation of groundwater basin into a nodal point network
Probably the most important but also the most difficult stage
in the process of simulating groundwater flow is the translation
of the natural groundwater basin into a nodal point network Questions
which arise are
a In how many layer the geo-hydrological system mustcan be
schematized
b The locations of the boundaries and the nature of these boundaries
(impervious prescribed head prescribed flux)
c The configuration of the nodal points
19
The answers of these questions depend on the problem under
consideration
- Schematization into layers
Usually our knowledge about the geo-hydrological situation is
such that no more than 5 different layers can be distinguished
(remember contrast rule)
- Boundary conditions
The top boundary condition of a groundwater basin usually is
a prescribed flux (effective precipitation)
The bottom boundary condition usually is a prescribed flux
boundary with flux zero (impervious hydrological base)
For the boundaries at the other sides of the system one sees
either prescribed head boundaries (eg a fully penetrating
river) or prescribed flux boundaries (including zero flux)
- Nodal point configuration
Nodal points are situated
on the boundaries
along internal boundaries which indicate changes in
hydraulic properties within the layer
along the tertiary surface water system
on places where artificial recharge or discharge wiii tgtke
place
elsewhere
Construction of the finite element mesh
Assume we have schemetized the groundwater basin iatu a finite
number of layers with different direction of flow (horizontal and
vertical) Within one layer the hydrological properties may vary
from place to place
Now a nodal grid is superimposed on each layer in a way already
described in section 32 Thus each layer has the same number of
nodal points situated on the same places in the horizontal plane
but with different z-coordinates
We are free to construct quadrilaterals or triangles because
the program automatically divides a quadrilateral into two triangles
having identical material properties The dimensions of the elements
20
should be small at places where large changes in hydrological
properties occur or where large gradients in hydraulic head are
expected to occur
Coding of nodal points
Each node in the finite element network is assigned an integer
code which indicates the type of calculation for this node
The coding is as follows
- Code = 1 nodal points with described head as function of time
- Code 2 internal nodal points
- Code = 3 internal nodal points which represent secondary surface
water system
- Code = 5 nodal points with prescribed artificial recharge or
discharge and points with prescribed boundary flow
(both as functions of time)
Other important endogenous variables are the magnitude of the
time step the value of the variable which stops the iteration
process and the maximum number of iterations allowed per time step
(see also group A and B of section 413)
412 Exogenous variables
After constructing the nodal grid each node represents part of
the groundwater basin Geo-hydrological properties can be assigned
to each mode based on physical data (group C D and E of section
413) In group F of section 413 time-dependent variables can
be assigned to each node
413 Data input to Program FEMSAT
The data input must be punched on cards according to the
following instructions
GROUP COLUMNS FORMAT SYMBOL DESCRIPTION
A 1-80 20A4 HEAD Desired heading to be printed at
the beginning of the execution
1-5 15 NUMNP number of nodal points
21
GROUP COLUMNS FORMAT SYMBOL DESCRIPTION
6-10 15
11-15 15
16-20 15
21-25 15
26-30 15
31-35 15
NUMEL
NUMLAY
MBAND
NTSPI
NTSPO
MAXIT
1-10 F103 ST
11-20 F103 DELT
21-30 F103 TMAX
31-40 F105 TOL
number of elements
number of layers
maximum difference in node number
between two adjacent nodes
number of time steps between a
change in non-stationary input data
number of time steps between two
successive outputs
maximum number of iterations per
time step
time to start with the execution
initial time interval
maximum time to be reached before
execution is stopped
maximum allowed absolute change in
the values of h between any two
successive iterations in a given
time step
Iterations continue until all changes
are less than or equal to TOL or
until MAXIT is exceeded
1- 5 15
6-10 15
11-15 15
16-20 15
KX(N1)
KX(N2)
KX(N3)
KX(N4)
sequential number of i-th corner of
element N
sequential number of j-th corner
sequential number of k-th corner
sequential number of 1-th corner
(see fig 8)
Fig 8 Numbering of element corner nodes
22
GROUP COLUMNS FORMAcircT SYMBOL DESCRIPTION
One card must be provided for each element in sequential order
starting with N = 1 and ending with N = NVJMEL
I-K 1103 SANG(NLY)
1i-iO MO3 C1(NLY)
21-30 F 103 C2(NLY)
21-4- 1-104 THI(NLY)
angle in degrees between CI and
the laquo-coordinate to he used in
nodal print N in layer LY (see
poundig 9)
First principal conductivity of
material in node N and layer LY
idem second principal conductivity
thickness of layer LY at nodal
point N
ig V Definition of SANGCI and C2
One card icircuiPt be provided for each model point ctarting wih M = 1
and ending with N = NUMNP If there is norlt3 than 1 layer for each
layer a set must be provided starting with the upper layer LY - 1
and ending with the bottom layer LY = NIJMLAY The variables SANG
M and 2 are not relevant tor aquitards but for reasons of
conveniency this procedure is maintained (making SANG CI and C2
dummy variables)
1-10 110 KODE(N)
11-20 II0 NTERR(N)
coding of nodal points (see
paragraph 42)
number of tertiary relation in node N
The q-h relationship of the flow to
23
GROUP COLUMNS FORMAT SYMBOL DESCRIPTION
21-30 FlOl X(N)
31-40 FlOl Y(N)
41-50 FlOl HGL(N)
51-60 FlOl HBSC(N)
61-70 FlOl LESC(N)
the tertiary system as discussed
in par 24 are schematized in a small
number of different relations
x-coardinate of node N
y-coordinate of node N
height ground level in node N
height bottom secondary conduit in
node N If node N is not a secondary
conduit this variable is a dummy
variable
length of secondary conduit in area
represented by node N
One card must be provided for each node starting with N = 1 and
ending with N = NUMNP
E 1-10 FlOl P(NLY1) initial value of h at node N in
layer LY
11-20 FlOl VERRES(NLY) vertical resistance at node N and
layer LY If the layer is an aquifer
this value is a dummy variable
21-30 F101 RADRES(NLY) radial resistance for flow to
secondary system at node N in layer
LY
31-40 FlOl ENRES(NLY) entrance resistance for flow to
secondary system at node N in layer
LY
41-50 FlOl WP(NLY) wet perimeter of secondary system at
node N in layer LY
51-60 FlOl P0R(NLY) specific yield of the material at
node N in layer LY
61-70 FlOl SS(NLY) specific storage of the material
at node N in layer LY
24
GROUP COLUMNS FORMAT SYMBOL DESCRIPTION
One card must be provided for each node starting with N = 1 and
ending with N = NUMNP If there is more than 1 layer for each layer
a set of cards must be provided starting with LY = 1 and ending
with LY = NUMLAY
F 1-10 F101 P(NLY1) value of h at different times at
node N in layer LY Node N lies
on a prescribed head boundary
11-20 F101 WTSC(N) water table in secondary system at
node N at different times
21-30 F 101 ARRED(NLY) artificial recharge to or discharge
from node N in layer LY or
prescribed boundary flow at
different times
31-40 F 101 QPHR(N) flux through phreatic surface at
node N at different times
The data in group F are non-stationary input data One card must be
provided for each node starting with N = 1 and ending with N = NUMNP
In case of more than 1 layer for each layer a set must be provided
Not relevant variables are taken as dummy variables
After a certain number of time steps (NTSPI) the data of group E
are read So the set cards E must contain NxLYxC cards where C =
(TMAXNTSPIXDELT) + 1 if DELT is a constant
42 R u n n i n g t h e p r o g r a m o n t h e C y b e r 7200
The program has been executed on a Cyber 7200 computer of
IWIS-TNO in the Hague
- The demanded input is punched on cards according to the instructions
of the last paragraph and is put on a permanent file
- For each specific problem inthe program only the dimension statements
in the main program have to be adjusted according to the instructions
in the lines of comment
- After compilation the program together with the input file is
executed
25
- Intermediate data necessary for continuation of the calculations
are saved on another permanent file This gives the opportunity
to restart the program at a time the calculation was stopped
The only change in the input file is to adjust the starting time
Remark
With the aid of a terminal small changes in the program and in the
input file can be made easily and the output immediately appears
on the terminal In this way it is possible to stop erroneous
calculations This is especially useful when being in the testing
phase of the program
43 O u t p u t
Immediately after reading the input data they are printed so
that possible errors can be detected During execution after each
certain number of time steps (NTSPO) calculated values of h and
water balance terms are printed In appendix B an example is given
The output data can be devided into two groups
A Data per node per layer
- Coding of the nodal points KODE
- Hydraulic head HEAD
- Flow through phreatic surface at the time indicated in head
of table FLOWUN
- Flow to tertiary surface water system FLOWTS
- Flow to secondary surface water system FLOWSS
- Flow to adjacent layers LEAKAGE
- Artificial recharge or discharge and prescribed boundary
flow ARRED
- Change in storage during last time step STORAGE
- Flow through prescribed head boundary LATFLOW
B Data per layer
Summation of the terms already mentioned under A gives the
magnitude of the water balance terms per layer at the time
indicated (instanteneous water balance terms) Summation of
these instanteneous terms multiplied by their respective time
intervals since time is zero gives the cumulative water balance
terms per layer
26
After each print out we are able to check the calculations
because the sum of the water balance terms per layer must be
approximately zero
44 C o m p u t i n g t i m e
The computing time depends on the number of nodal points and
the number of layers But it depends also on the magnitude of the
changes in hydraulic head and on the value which terminates the
Gauss-Seidel iteration (TOL) To give an impression TOL = 001 m
Costs per day simulation (in hlfl) on the Cyber 7200
number of nodes number of layer bull bdquo bull bull _ 1mdash x a x b (21)
where
a = costs in Dutch guilders per system second (ƒ 027)
b = empirical factor with magnitude about 001
V NUMERICAL EXPERIMENTS
In general there are two possibilities to test the validity of
numerical models The first and most commonly used method is to
compare numerical results with analytical solutions The second
possibility is to compare numerical results with measured field
data Since the last alternative requires many data that are
difficult to obtain we restrict ourselves for the time being to
comparison with the analytical solutions
5 1 F l o w t o a w e l l i n a n u n c o n f i n e d
c i r c u l a r - s h a p e d a q u i f e r
Flow to a well can be divided into steady-state flow and
unsteady-state flow (KRUSEMAN and DE RIDDER 1976)
27
511 Steady state flow in an unconfined aquifer
For this case the formula of Thiem-Dupuit is valid (see fig 10)
Q bull k lth2-hicircgt ln(r2r])
(22)
where
hhbdquo = hydraulic head in 1 and 2 respectively
Q = discharge from well
rbdquo = distance from well
CL31
hmdash^ri --r2
Fig 10 Schematic cross-section of a pumped unconfined aquifer
Fig 11 gives the nodal grid superimposed on a well in a
circular shaped aquifer Near the well the nodal point distance
is made smaller Quadrilaterals as well as triangles are used
Numerical calculations with this finite element network and 3 -1
an extraction rate of 200 m day were performed for 10 days
At that moment a steady-state situation was almost achieved
28
2tmdashnodal point number
160 180 X-axesim)
Fig 11 Nodal point configuration for flow to a well in a circular-
shaped aquifer The outer boundary is taken to be constant
The well is situated in nodal point 42
29
In fig 12 the results of the simulation are compared with the
analytical solution The agreement between numerical results and
the analytical solution is excellent
draw- 15irS (m) down
analytical solution imdashx calculated values
distance from well r (m)
Fig 12 Comparison between analyt ical and calculated s teady-state
drawdown for the problem of f ig 11 with 1
-1 Q bull - 200 m 3 d a y _ 1
k = 10 mday
d = 100 m
S = 2 y
S = 0 s
512 Unsteady-state flow in an unconfined aquifer
Analytical solutions for unsteady-state flow in an unconfined
aquifer only exist if the aquifer extends infinitely For the
aquifer given in fig 11 this condition is not fulfilled However
during the first one or two days after starting the extraction the
influence of a constant hydraulic head at the outer boundary on the
30
drawdown near the well can be neglected (see fig 13) After one
day the lateral inflow is only 25 of the extraction the remainder
being the change in storage due to drawdown
2 0 0
m 3 day 1
Q=200m 3 day
100-
vraquomdash Lateral inflow
N amp storage
I I ^ V - W 10
t (days)
Fig 13 Values of Q lateral inflow and change in storage as
functions of time
The nonsteady-state or Theis equation can be written as
i _ Q s 4iTkD
dy = 4irkD
W(u) (23)
where
u = r2S
4irkDt
r = distance from well
S = storage coefficient
Q = constant discharge from well
D = thickness of the waterbearing layer
at the beginning of pumping
t = time after starting the extraction
s = corrected drawdown
= s - 322D
s = calculated drawdown
W(u) = Theiss well function
[J
CIAT1
[T]
]
31
In fig 14 the analytical unsteady-state solution for the case
described in section 511 is given together with the numerical
results at times 05 and 10 days The agreement between numerical
results and the analytical solution is rather good
Drawdown s1(m)
10
10
10
Theiss solution
x caculated values t=Q5
bdquo t= lO
10 10- 10 10 t r2(daym2)
Fig 14 Comparison between analytical and calculated steady-state
drawdown for the problem of fig 11 with
Q
k
d
S y
s s
=
=
=
=
=
200
1
10
3 m
0
0
2
0
-1 day
m day
m
These results indicate that the assumption of a linear variation
of h between two nodal points (see fig 3 ) even with the rather
rough element network of fig 11 is acceptable So there is no
need to improve for example the finite element network by introducing
higher order coordinate functions
32
52 F l o w t o w e l l i n 3-1 a y e r e d s y s t e m
In fig 15 a schematic cross-section of a 3-layered system of
two aquifers separated by an aquitard is given A well of
infinitisemal radius is completed in the lower aquifer and discharges
at a constant rate QTT Each layer is homogeneous isotropic and
extending radially infinite
bdquo bull - bull bull ~ i
mdash Q II
aquifer I
lugrave^i~IcircIuml7i-i~~ aquitard --r-rl~-
-II--I i - i
aquifer
Fig 15 Schematic cross-section of a 3-layered system
For this situation NEUMAN and WITHERSPOON (1969a) give rather
complicated analytical solutions In the special case where the
hydraulic properties of the two aquifers are identical they give
graphical illustrations of their solutions for three different
cases
These graphical solutions give the opportunity to verify
roughly numerical results of a 3-layered system Therefore a nodal
grid is superimposed on a circular shaped groundwater flow system
with the extraction in the centre of the lower aquifer and with
a constant head on the outer boundary Fig 16 gives the nodal
point situation in the x - y plane of 1 layer The nodes in the
other two layers are situated in the middle of the layers at the
same places in the x - y plane only their z-coordinate is different
33
A
-Jy
bull33
V 5
iumlff
Fig 16 Nodal point configuration for the 3-layered problem (top view)
34
In fig 17 numerical results are compared with the analytical
solution The agreement is rather bad especially for the two
aquifers But in terms of the water balance the difference between
aquifers and aquitard is not so big because the aquitard has
about 16 times bigger storage capacity than the aquifers The
small value of the specific storage of the aquifers is the reason
that small errors in the components of the water balance have a
big influence on the calculated values of the hydraulic head
Eig 17 Dimensionless drawdown vs dimensionless time in a
3-layered system with
II = 50 m day
3 m (distance from well)
aquifer I
aquitard
aquifer II
Thi (m)
100
25
100
S s
001
0642
001
k(mday )
1
0277
1
The discrepancies are partly caused by errors in discretization
In the z-direction these errors can be eliminated by taking the
35
vertical resistance of the aquitard to be infinite Then the
calculated drawdown in the pumped aquifer can be compared with
the analytical Theiss solution
This is done in fig 18 for different distances from the well
One sees that the agreement is better when the distance is bigger
Apparently the linearization of the logarithmic drawdown curve
near the well is responsible for the differences The differences
for r = 3 m are of the same magnitude as for the pumped aquifer in
the 3-layered case
joo gt A 50
20
Theiss solution
bull calculated values r=3m (distance f rom well) x bdquo bdquo r=7m
1 L
I l_ 001 003 0 0 5 Ol Q2
- I I 5 10 t - K t
Fig 18 Dimensionless drawdown vs dimensionless time in a
confined aquifer with
Q =500 m day
k = 10 mday
D = 100 m
S = 001 s
The differences between numerical results and analytical solutions
are not caused by taking a finite area because changing the outer
boundary into an impermeable boundary has only minor effects
36
VI SUMMARY AND CONCLUSIONS
In this report a numerical model for non-stationary saturated
groundwater flow in a multi-layered system has been treated
Groundwater flow can be described in different ways
a Physical-mathematical by combining Darcys law and the law of
continuity (section 21)
b Geo-hydrological the flow in a natural groundwater basin takes
place in layers with different hydraulic properties Definitions
of the most important hydraulic properties are given in section
22
c Schematical groundwater flows horizontally in aquifers and
vertically in aquitards (section 23)
d Approximating by dividing the region of flow into a finite
number of elements and applying Darcys law and the law of
continuity on each of these elements In these ways one gets
a set of equations which can be solved given (initial and)
boundary conditions (section 25)
e Numerical the set of equations has been solved using a Galerkin
type finite element method and the implicit Gauss-Seidel method
with overrelaxation factor (section 31 and 32)
f Computer-solvable the numerical solution has been translated
in a FORTRAN IV program called FEMSAT (section 33)
In chapter IV the input execution and output of the program
FEMSAT are treated
Finally in chapter V some numerical applications are given
simulation of flow to a well in unconfined aquifer and simulation
of flow to a well in a 3-layered configuration
Comparison of hydraulic heads obtained by numerical calculations
and analytical true solutions showed an excellent agreement in case
of stationary flow and a good to moderate agreement in case of non-
stationary flow However the diviations caused by the discretization
process must be seen in relation to the errors made in the
establishment of the hydraulic properties
37
As was stated in the introduction numerical models can be
valuable tools in the management of water resources Therefore
numerical models must be suitable to simulate effects of certain
operations upon the spreading of water over saturated groundwater
unsaturated groundwater and surface water The model described in
this report is a model for saturated groundwater flow The relation
with the surface water can be taken into account via radial and
entrance resistances or via a relation between flow to the surface
water system and the hydraulic head The relation with the unsaturated
zone has been restricted to the input factor effective precipitation
In practise however the unsaturated and saturated system are
interconnected Therefore the model should be improved by taking
into account the unsaturated zone This can be done in at least
two ways
a Replacing the unsaturated zone by a parametrical model
b Coupling the unsaturated zone in an iterative way with the
saturated system
In order to find out the usefulness of the numerical model for
water resources management it will be applied to actual regional
groundwater systemslaquoAlso sensitivity analyses will be carried out
The computer model itself can be improved by applying standard
programs eg automatical generation of a nodal grid and graphical
plotting of the results
38
VII REFERENCES
ERNST LF 1962 Grondwaterstromingen in de verzadigde zone en hun
berekening bij aanwezigheid van horizontale evenwijdige open
leidingen Thesis University Utrecht pp 189
1978 Drainage of undulating soils with high groundwater
tables (in press)
FORSYTHE GE and WR WASON 1960 Finite Difference Methods for
Partial Differential Equations John Wiley and Sons New York
pp 444
KRUSEMAN GP and NA DE RIDDER 1976 Analysis and evaluation of
pumping test data ILRI-bulletin No 11 pp 200
LAAT PJM DE and C VAN DEN AKKER 1976 Simulatiemodel voor grond-
waterstroming en verdamping in Modelonderzoek 1971-1974 ten
behoeve van de waterhuishouding in Gelderland deel II
Grondslagen pp 145-197
NEUMAN SP RA FEDDES and E BRESLER 1974 Finite element
simulation of flow in saturated -unsaturated soils considering
water uptake by plants Hydraulic Engineering Laboratory
Technion Israel pp 104
and PA WITHERSPOON 1969a Theory of Flow in a confined
two Aquifer System Water Resources Research Vol 5 No 4
pp 803-816
^_ and PA WITHERSPOON 1969b Applicability of Current Theories
of Flow in leaky aquifers Water Resources Research Vol 5
No 4 pp 817-829
PINDER GF and EO FRIND 1972 Application of Galerkins Procedure
to Aquifer Analysis Water Resources Research Vol 8 no 1
pp 108-120
REMSON I GE HORNBERGER and FJ MOLZ 1971 Numerical Methods
in Subsurface Hydrology John Wiley and Sons New York
pp 389
WESSELING JG 1977 Toepassing van Finite Difference- en Finite
Element Methoden voor het oplossen van een aantal grondwatershy
stromingsproblemen (met computerprogrammas) ICW-nota 944
pp 63
39
ZIENKIEWICZ OC 1971 Finite Element Method in Engineering Science
McGraw-Hill London pp 521
40
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X - e - o
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LUX
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APPENDIX 8 EXAMPLE OF OUTPUT
HEAD AND TERMS OF UATERBALANCE PER NODE PER LAYER
AT TIME 260 IN II AND I13DAYS
T
LAYER NODE KODE HEAD FLOUUN FLOUTS FLOUSS LEAKAGE STORAGE LAT FLOU
13 19 25 31 37 43
125008 125000 12S000 12S000 12S000 12S000 12S000 12S000
0000 0000 0000 0000 0000 0000 0000 0000
0000 0000 0000 0000 0000 0000 0000 0000
0000 0000 0000 0000 0000 0000 0000 0000
- 000 - 000 -000 - 000 - 000 - 000 - 000 0000
0000 0000 0000 0000 0000 0000 0000 0000
0000 000 000 000 000 000 000 000
000 0000 0000 0P100 0000 0000 0000 0000
1 7
13 19 25 31 37
12S000 125000 125000 125000 125000 12S000 125000 125000
0000 0000 0000 0000 0000 0000 0000 0000
- 001 -007 - 023 - 020 - 009 - 003 0000
0000 0000 0000 0000 0000 0000 0000 0000
000
001
007
023
020
009
003
003
0000 0000 0000 0000 0000 0000 0000 0000
1 7
13 19 25 31 37 43
125000 125000 124998 124972 124895 124787 124647 124358
0000 0000 0000 0000 0000 0000 0000 0000
- 000 001 007 023 020 009 003
0000
0000 0000 0000 0000 0000 0000 0000
50000
0000 182
1982 3862 1674 419 078 041
0000 0000 0000 0000 0000 0000 0000 0000
TERMS OF WATERBALANCE PER LAYER DURING LAST TIMESTEP AND
AND SINCE BEGINNING IN M3DAY M3
FLOU TO UNSATURATED ZONE
FLOU TO TERTIARY SYSTEM
0000
0000
0000
0000
LAYER NO 1
FLOU TO SECONDARY SYSTEM
FLOU TO AD3ACENT LAYERCS)
ARTIFICIAL FLOU AND PRESCRIBED BOUNDARY FLOU
ACCUMULATION TERMS
LATERAL FLOU THROUGH PRESCR HEAD BOUNDARY
0000
-000
0000
000
000
0000
-000
0000
000
000
LAYER NO 2
FLOU TO SECONDARY SYSTEM
FLOU TO AD3ACENT LAYERCS1
ARTIFICIAL FLOU AND PRESCRIBED BOUNDARY FLOU
ACCUMULATION TERMS
LATERAL FLOU THROUGH PRESCR HEAD BOUNDARY
0000
-378
0000
381
0000
0000
-055
0000
056
0000
LAYER NO 3
FLOU TO SECONDARY SYSTEM
FLOU TO AD3ACENT LAYERCS5
ARTIFICIAL FLOU AND PRESCRIBED BOUNDARY FLOU
ACCUMULATION TERMS
LATERAL FLOU THROUGH PRESCR HEAD BOUNDARY
0000
370
50000
49376
0000
0000
055
12979
12909
0000
I INTRODUCTION
The groundwater system is part of the hydrological cycle and
changes in this system affects phenomena such as plant production
river flow etc Generally one tries to use groundwater resources
in such a way that the utility for the community as a whole is
maximal
Numerical flow models can be valuable tools in the management
of water resources In many situations these models are the only
possible way to indicate effects of certain interferences in the
groundwater system
In this paper a numerical model is presented which simulates
non-stationary saturated groundwater flow in a certain region
Purpose of the model is to predict effects of certain operations
upon the various terms of the water balance of the groundwater basin
and on the hydraulic head
The basic idea of the model is that groundwater is flowing
horizontally in waterbearing layers and vertically in less-permeable
layers Each layer is discretized into a number of elements To
each element both Darcys law and the law of continuity are applied
(chapter II)
In this way a set of equations is obtained which are solved
using a finite element method and the Gauss-Seidel iteration method
(chapter III)
In chapter IV the input and output of the model are treated
while in chapter V some numerical experiments are presented
Finally in chapter VI an evaluation of the model is given
II GENERAL DESCRIPTION OF THE PROBLEM
21 P h y s i c a l b a c k g r o u n d
Throughout this report water potential is expressed on a unit
weight basis and is being called hydraulic head with symbol h
h = 2- + z [L] Pg
where
mdash = piezometer head |L | Pg u -1
z = gravitational head |_Lj
The hydraulic head is an expression for the capacity of a unit
weight of water to do work as compared to the work capacity of the
same weight of water with the same chemical composition at a certain
reference level and under atmosferic pressure
The problem is restricted to saturated flow of groundwater with
one constant density
211 The law of linear resistance (Darcys law)
According to Darcys law the rate of flow through a porous medium
is proportional to the gradient in hydraulic head It may be written
as
ocirc h ltSh ocirch N q = _k Tmdash q = -k vmdash q = -k -rmdash ( 1 ) x y ocircx y y lt5y nz z ocircz
where
q = volume flux of water passing through a unit area per x y z
per unit time perpendicular to x y z direction
respectively |_LT J
k = hydraulic conductivity in x y and z direction x gt y gt z _ _ i _
respectively | LT J
The coefficient k in the Darcy flow equation is taken to be
a constant depending on both the properties of the porous medium
and the fluid
212 The law of continuity
The other important physical law states that no matter can be
lost or created
The conservation equation can be found by applying the principle
of continuity to an infinite small volume
6v 6v lt5v 60 x y z TT = T + x + ~T7 + 1 (2)
Ot OX OcircX oz
where
0 = volume of water per unit bulk volume of soil |_-J
q = sink or source term T J
213 Combination of Darcys law and the law of continuity
Darcys law combined with the law of continuity gives
60 lt5 oh lt5 ocirch 6 oh _ ocirct ocircx x ocircx ocircy y oy ocircz z ocircz
or
c 5h 6 ocirch ocirc Ocirch 6 ocirch c ot ox x ocircx ocircy y oy laquoz z 6z
where
S = volume of water released or stored per unit bulk c I
volume of soil per unit change in hydraulic head |L I
22 H y d r a u l i c p r o p e r t i e s a n d t y p e s o f
a q u i f e r s
The flow in a natural groundwater basin takes place in layers
with different hydraulic properties Usually the layers are divided
into good permeable layers or waterbearing layers (aquifers) and
less permeable layers (aquitards)
The most important hydraulic properties are
a Transmissiviy - the product of the average hydraulic conductivity
k and the saturated vertical thickness d of the waterbearing
layer
It is a measure for the transport capacity of aquifers
Notation T = kd |_ L T ~
b Hydraulic resistance - the ratio of vertical thickness of
aquitards d and hydraulic conductivity k It is a measure for
the resistance capacity of aquitards denoted as c = d k _ T J
c Specific yield - the volume of water released or stored in the
phreatic zone per unit surface area of an unconfined aquifer per
unit change in hydraulic head
It is also called effective porosity
The symbol used for specific storage will be S j_-J
d Specific storage - the volume of water released or stored per
unit volume of the saturated parts of an aquifer or aquitard per
unit change in hydraulic head It is a measure for the elasticity
of the soil material and the fluid Excluded are irreversible
processes Notation S _ L J
e Storage coefficient - the volume of water released or stored
pe7 nit surface of a layer per unit change in hydraulic head
It is the sum of specific yield and the product of specific
storage and thickness of the aquifer or aquitard
S = S + S d r - 1 c y s - -1
Note in confined layers S = 0 y
In fig 1 a 3-layered configuration is given of two aquifers
separated by an aquitard Aquifer I has a free water surface and is
an unconfined aquifer Aquifer II is a completely saturated aquifer
Dependent on the hydraulic resistance of the aquitard KRUSEMAN and
DE RIDDER (1976) distinguish between confined semi-confined and
semi-unconfined aquifers For our purpose it is not useful to make
this distinction We restrict the aquifer types to unconfined and
confined aquifers
WIHtUtttWttllttlt(ltmtttltHlttltlil(lltlittlfllnl ) bull (
d V U i l i l i ii i Hi il i M M bull f i m
i tMi| i i | J i j j i _ I I bull
aq ui tard
aquifer II
Fig 1 Schematic representatipn of flow direction and variation
of hydraulic head h in a 3-rlayered system
In an unconfined aquifers ( I ) poundhe vertical thickness pf the
waterbearing layer d is a variable while in a confined aquifer II
this is a constant (denoted as D ) Other important properties of
unconfined aquifers are the hydraulic head being the same as the
phreatic surface and the specific yield being different from zero
23 S c h eacute m a t i s a t i o n o f f l o w
The groundwater flow in a groundwater basin is schematized
into a horizontal flow in aquifers and into a vertical flow in
aquitards Consequently the vertical variation of the hydraulic
head is as drawn in figl
The validity of this assumption has )een investigated by
NEUMAN and WITHERSPOON (1969) The found that if the contrast
in hydraulic conductivity between two adjacent layers is bigger
than a factor two the relative error caused by this assumption
is usually smaller than 5
This schematization gives a simplification of the general flow
equation (eq 4 ) In an aquifer the vertical is zero and eq (4)
then reduces to
-Sbdquo -r- 7 - (k d -r-) + 7 mdash (k d 7-) + qd c lt5t ox x ox oy y oy
or
bull(S + S d) g y s ot
-r- (k d -=-) + -T- (k d -=-) + Q ocircx x ox oy y ocircy
(5)
where
Q = volume of water substracted from or added to unit surface
of aquifer per unit time |_ L T J
In an aquitard eq (4) reduces to
-sQ |pound = (k Q) s 6t oz Z OcircZ
(6)
As will be discussed in the next paragraph the term q in an
aquitard is set equal to zero
24 W a t e r b a l a n c e t e r m s
In eq (4) the term q is mentioned With this sink or source
term different terms of the water balance are taken into account
Fig 2 depicts schematically the situation for an unconfined aquifer
(for other layers some terms dont exist)
recharge- - discharge
laquo-b
Fig 2 Schematic representation of the various terms involved in the
water balance
Description of the different terms
a The flow through the phreatic surface q f
If this term is negative (water goes out of the saturated system)
it is called capillary rise If positive it is called percolation
(or drainage)
This flow can be calculated as a rest term of the water balance
of the unsaturated zone Very often this term is expressed as a
flow per unit surface (flux) and is called effective precipitation
For the time being this flux is a given input
b Flow to adjacent aquitards q
The flux to adjacent aquitards can be calculated by applying eq
(6) to the aquitard
c Flow to the primary surface water system q
The primary surface water system penetrates fully the groundwater
basin So it acts as a boundary with given values for the
hydraulic head (is equal to the phreatic level of the primary
system)
d Flow to the secondary surface water system q
The secondary surface water system does not penetrate fully the
groundwater basin which causes a resistance against flow between
groundwater and surface water The flow q can be calculated as
follows
h - h
- mdash f i i gt
where
h = hydraulic head of the groundwater at the place of
the secondary system |_ L |
h = phreatic level of secondary system |_ L J i s -
R = radial and entrance resistance of the
secondary system per unit surface |_ T J
According to ERNST (1962) the radial resistance H per unit
surface can be caKulated with
Uuml = 4 i n 4 e8) TTk B
where
1 = area drained by 1 m conduit L ^ J
d = thickness of the layer near the conduit _ L J
B = wet perimeter of the conduit |_ L J
The entrance resistance is caused by a layer which shows a
lower conductivity around the conduit than its surroundings
In practise radial and entrance resistancies are difficult to
deduct from hydraulic properties They can be found indirectly
by measuring simultaneously q h and h
t s
e Flow to the tertiary surface water system q
In smaller conduits usually the phreatic level is not known
So eq (7) cant be applied The flow to these conduits however
is sometimes very important (eg in areas with shallow groundwater
tables) One may a-sume (ERNST 1978) that a relationship exists
between the hydraulic head of an aquifor and the flow q
According to Ernst the tertiary system can be divided into a
number of categories with respect to their mutual distance and
bottom height For each category the flow is h - h (b)
q ( h b ) - mdash (9)
where
h = average hydraulic head | L J
h (b) = phreatic level in the category under
consideration a function of bottom height |_L J
a = geometry factor to convert h into h _ - J
Y = drainage resistance _ T J
hm = hydraulic head midway between two
conduits |_ L J m
Summation of the q(hb)-relations of the different categories
yields the overall relation between hydraulic head and flow to
the tertiary system we are looking for
f Artificial recharge or discharge q and prescribed boundary
flux qfo
The magnitudes of these flows are not influenced by the conditions
inside the groundwater basin but remain on a prescribed level
Thus q is an internal boundary condition q an external boundary cl D
condition
For confined aquifers some of the above mentioned terms dont
exist These terms are the flow through the phreatic surface qf
and the flow to the tertiary surface water system q
Aquitards are considered only as transmitters of water from one
aquifer to another with possibilities of storage (eq 6 ) This means
that for aquitards the water balance terms ar through f are set
equal to zero
25 S o l u t i o n o f g r o u n d w a t e r f l o w p r o b l e m s
The basic equations for saturated groundwater flow are eq (5)
for aquifers and eq (6) for aquitards The solution of these
equations require knowledge of the physical system and the auxiliary
conditions describing the system constraints
These auxiliary conditions are (REMSON etal 1971)
a Geometry of the system
Eg the diviation of the system into a restricted number of
layers and the horizontal extension of the system under
consideration
b The matrix of hydraulic properties including in-homogeneity
and an-isotropy
c Initial conditions Because we are dealing with non-stationary
flow the values of pertinent system variables (such as hydraulic
head at the initial time) must be known
d Boundary conditions They describe the conditions at the boundaries
pf the system as a function of time
One can distinguish between
- hydraulic head boundary
- flow (including zero flow) boundary
- both head and flow boundary
After specifying the auxiliary conditions an unique solution of
the variation of the hydraulic head in space and time can be
obtained
For special shapes of flow and assumptions exact analytical
solutions exist In nature however one is usually confronted with
problems like
- spacial variation in the hydraulic properties of the soil
- irregular shape of the groundwater basin
- empirical relations between q and h
- d not being a constant
For such situations by numerical and analogy models only an
approximation of the actual situation can be obtained In the next
chapter a numerical solution called the finite element method is
treated
III NUMERICAL SOLUTIONS
In this chapter the numerical solution of the aquifer equation
(eq 5) and the aquitard equation (eq 6) will be treated The
method of solving is the finite element method In section 31 only
a short mathematical description of this method will be given For
more mathematical background the reader is referred to publications
on this subject (eg ZIENKIEWICZ 1971 NEUMAN etal 1974 and
WESSELING 1976)
In section 32 the application of the finite element method on
the flow in groundwater basins is treated while in section 33 the
translation into a computer program is given
3 1 F i n i t e e l e m e n t m e t h o d
For reasons of simplicity the finite element method will be
explained for the stationary version of eq 5 ie
10
(k d^) + (k d ^ ) - Q = O (10) ocircx x 6x 6y y Sy
The basic idea of each numerical solution of eq (10) is to
apply this equation on a small but finite volume The flow region
of interest R is divided into a finite number of volumes called
elements The elements can have different shapes but for reasons
of conveniency we only use triangles- Quadrilaterals can also be
used because they can be divided into two triangles The sizes of
the triangles may vary They can be adapted according to the
geometry of the region and the accuracy desired (fig 11 gives an
example of the diviation of a region into triangles and quadrilaterals)
The corner points of the elements represent the nodal points
If there are N nodal points then the continuous variation of h
in region R is approximated by the expression
h (x_v) ~ h f (xy) (11) a n n
where
h (AV) = approximation of u (xy) a
h = value of h in nodal point n n
f ltxgty) = global coordinate function
In nodal point n a must be equal to h so f (x y ) = 1 and a n n n n
f (xy) = 0 i n Between f (x v ) and f (xy) of the nodal n j ] n n n n j j
points surrounding nodal point n f (xy) varies linearly from
1 to 0
In fig 3 a picture of f (xy) is given n
If one knows the values h in the nodal points n we know the approximation h of h We can find these values of b bv using for
a n
example the Galerkin method
Galerkins principle states that an approximate solution of h in
eq (10) can be obtained from the following system of equations
N ~~ (k d -A + -^- (k d T-y E h f (xy) f(xy)dxdy +
-ox x ox ocircy y oy - _ n n i R n~
J J Q f(xy) dx dy = 0 i = 1 N (12)
1 1
Fig 3 The global coordinate function f (xy)
The integration can be performed element by element with the
aid of the coordinate functions f (NEUMAN etal 1974)
For non-stationary problems it is not possible to apply the
Galerkin principle on the time derivative But it is possible
to discretize this derivative and to set up the system of equation
(eq 12) for each finite time step
One obtains in this way a series of sets of laquocriations tha s
an approximation of the non-stationary flow problem
Treatment of boundary conditions
a Initial conditions
In non-stationary flow problern gtne must start with given
values of a at time t = t ri c
b Boundary conditions
- Prescribed head boundary
Nodal points which lie on a prescribed head boundary have a
prescribed value for h bull r Hgte r umber of unknown h s is n n
reduced and accordingly no the number of equations
- Prescribed flux through z-- boundary
If one knows the flux (iuumlr jw or outflow per unit area) and
the area of the boundary oi-r element the inflow or outflow
through the boundary per flemlt=nt can be calculated
One may assume this flow to J bull part of the sink term Q in
eq (12)
12
32 A p p l i c a t i o n o f t h e f i n i t e e l e m e n t
m e t h o d o n t h e f l o w i n g r o u n d w a t e r
b a s i n s
From hydrogeological investigations the geometry and the matrix
of the hydraulic properties are known Knowing the geometry the
flow can be split up into a finite number of layers This holds
also for the nature of the boundaries
Now a nodal grid is superimposed on each layer in such a way
that the nodal points of the different layers are situated right
above each other (see fig 4) and in each layer the nodal point
lies in the middle of this layer
-nodal point
Fig 4 Schematic representation of the nodal points in the
xyz-space
Each layer has N nodal points of which P nodal points have a
prescribed head In each layer remain N-P = N nodal points with
unknown head Eq (12) can be applied to each point with unknown
head in each separate layer
As was stated in the last paragraph for each time step one
can write
13
Ah bdquo ~ -r1 + Q = O in = 12N in n m At yi A h + B (13)
where
A = in
ocircf ocircf ocircf of (k d f mdashri- ~ + k df - ~ -jS) dxdy
J x n ox ox y n oy oy
in (S d + S ) ff dx dy O if i icirc n
s y i n
Q f dx dy
The matrices A B and the vector Q have been defined in in l
elsewhere (NEUMAN etal 1974) A is the conductivity matrix in
B the storage matrix and Q the source vector in l
In order to clarify these rather complicate expressions apply
equation (13) poundo point 4 of fig 5
6 7
Fig 5 Top view of part of the nodal grid
lhl + A42h2 + A43h3 + A44h4 + A45h5 + A46h6 + A47h7 +
h - h
+ B _iuumli i ^ + Q = o 44 t - t x4
n n-1
(14)
where h and h are values of h at time t and t 4n 4n-1 4 n n-1
respectively
14
If the h-values belonging to the A-values are taken at time
t = t we get an equation with one unknown h for time t = t n-1 4n n
The equation can then be solved directly
However this is an explicit formulation which puts strong limitations
on the magnitude of the time step t - t (REMSON etal 1971) n n-1
Values larger than a certain value cause instability With an implicit method the values of h are taken at time
t = t This method is unconditionally stable Now one gets an equation n
with 7 unknowns h through h If one has 7 equations for every 1 n n
point 1 through 7 we can solve the values h i = 17
in
Of all the methods possible the iterative method of Gauss-Seidel
has been chosen The Gauss-Seidel method starts by choosing a first estimate of h at time t = t (usually equal to h ) At each in n j -i in-l iteration step a new value of h is calculated using the last
in calculated values of h If one passes the nodal points from 1 to
i n 7 eq (14) becomes
h-j + 1 = A h^ + 1 + A bdquo h | + 1 + A _ h | + 1 + A h j + A r h j + 4 n 4 1 1 n 4 2 2 n 43 3 n 45 5 n 4 6 6 n
B h B + A h J + - 4 4 4 n ~ + 0 J (A + mdash) A 4 7 h 7 n At n U 4 4 At
(15)
where
h-i = in
j = iteration step or number of iteration
i = number of nodal point n = time = t
n
The iteration process converges if the successive differences k+1 k
between h and h become continuously smaller in in
The iteration process is stopped if at each nodal point the k+1 k
difference between h and h is smaller than a certain prescribed in in
value
15
To accelerate the convergence an overrelaxation factor W is used
(for more information about W see FORSYTHE and WASOW 1960)
k+1 k T7 k+1 k h = h + W (h - h )
in in in in (16)
If W = 1 there is no overrelaxation Usually W varies between 1 and
2 and the optimum value very often is found by trial and error
Eq (15) is valid for points in aquifers The connection with
adjacent layers is made by the term Q because in this term the
flow from adjacent aquitards q is included (two aquifers are always
separated by an aquitard)
Aquitards
We dont apply the finite element method on points in aquitards
As stated before there is no horizontal flow in aquitards So it has
no sense to replace the continuous variation of h in the aquitard
by equation (11)
The nodal points in aquitards represent a certain volume of an
aquitard hich transmit water from one aquifer to another and which
has storage capacity The nodal points are included in the Gauss-
Seidel scheme -s follows (see fig 6 )
1 aauifer
a -_vS laquo 2 [~^-t^yM^^^-^r~ aqutarci l-Z
aquifer
Fig 6 Schematic situation for points in aquitards
Point 2 is situated in the middle of the aquitard with thickness
d storage coefficient S and vertical hvdraulic conductivity k c J v
Suppose the Gauss-Seidel iteration starts in the upper layer and
ends in the bottom-layer Then the flux from point 1 in aquifer 1 to
16
point 2 in the aquitard at time t = t and in iteration step j + 1
is
k q = (hf -hf )-iumlmdash f (17)
1 n 2 n OSxd1
and the flux from point 3 in aquifer 2 to point 2 in the aquitard
is
1 c 1
qbdquo = (h - hf) mdashHmdashr- (18) 2 3 n 2 n OSxd
The net flux ltl-q9 must be equal to the change in storage per unit
area per unit time This change in storage can numerically be
approximated by
te - h-
n n-1
Combina t ion of ( 1 7 ) (18) ana (19) g i v e s
l tr1 - h (hJ + 1 + hj - 2 hj+1) - - - - - - S1 - ilaquoS L u l l = o (20)
] i 3 n 2 n 1 c t - t j j x d n n -1
T + 1
From t h i s e q u a t i o n h^ kan be s o l v e d
3 3 C o m p u t e r o g r a m
The problem outl^ugrave so far has been translated into a computer
program written in FORTRAN IV
The flow chart ot Uis program is given in fig 7 whereas in
appendix A the complete listing of the program can be found
In the flow chart the 5 main parts of the program are
indicated
A Reading and printing of input data
B Determination of the matrices A and B and the vector Q per layer
17
M
1 A
FLOH to unsaturated zono
FLOW to tertiary syster
FLOW to secondnrv system
Leakage to adjacent aquitards
SOLVE equation (14)
yes
CALCULATE flux through constant head boundnr
CALCULATE waterbalance n p r 1 OAro r r J j - ~
PRINT of output
Ier node per layer
k = k+1
t = t + At
7 DUMP variables on tape
-gt-
( STOP )
Fig 7 Flow chart of the program FEMSAT with diviation in 5 main
parts A B C D and E
C START PROGRAM FEMSAT
f READ general information
PRINT general information
f READ element information
PRINT element information 7 READ stationary nodal point information
PRINT stationary nodal point information 7
READ non-stationary nodal point information for t = t
FRINT non-stationary nodal point information for t = t
L 2 (
^ yes R E A D non-stat input for t gt t
J PRINT non-stat input for t gt t (- 1 -- M degY
GENERATE A B and AREA-matrix for confined aquifers
_L
GENERATE A B and AREA-matrix for unconfined aouifer
bull
C Solving h with Gauss-Seidel
D Determination of the water balance terms per layer
E Printing of results
The program exists of a main program with several subroutines
The main program calls the subroutines In the subroutines most
calculations are performed
In this way an easy adaptation to different problems is
obtained (see also section 44) Comments in the program will
help the reader to understand the program
In the next chapter the preparation of the input data executing
of the program and the output will be treated
Acknowledgement
Some parts of the computer program were taken from the program
UNSAT2 of NEUMAN etal (1974)
IV INPUT EXECUTION AND OUTPUT
41 I n p u t
The input is divided into endogenous variables (chosen by the
user or calculated by the program during execution) and exogenous
variables (based on physical data)
411 Endogenous variables
Translation of groundwater basin into a nodal point network
Probably the most important but also the most difficult stage
in the process of simulating groundwater flow is the translation
of the natural groundwater basin into a nodal point network Questions
which arise are
a In how many layer the geo-hydrological system mustcan be
schematized
b The locations of the boundaries and the nature of these boundaries
(impervious prescribed head prescribed flux)
c The configuration of the nodal points
19
The answers of these questions depend on the problem under
consideration
- Schematization into layers
Usually our knowledge about the geo-hydrological situation is
such that no more than 5 different layers can be distinguished
(remember contrast rule)
- Boundary conditions
The top boundary condition of a groundwater basin usually is
a prescribed flux (effective precipitation)
The bottom boundary condition usually is a prescribed flux
boundary with flux zero (impervious hydrological base)
For the boundaries at the other sides of the system one sees
either prescribed head boundaries (eg a fully penetrating
river) or prescribed flux boundaries (including zero flux)
- Nodal point configuration
Nodal points are situated
on the boundaries
along internal boundaries which indicate changes in
hydraulic properties within the layer
along the tertiary surface water system
on places where artificial recharge or discharge wiii tgtke
place
elsewhere
Construction of the finite element mesh
Assume we have schemetized the groundwater basin iatu a finite
number of layers with different direction of flow (horizontal and
vertical) Within one layer the hydrological properties may vary
from place to place
Now a nodal grid is superimposed on each layer in a way already
described in section 32 Thus each layer has the same number of
nodal points situated on the same places in the horizontal plane
but with different z-coordinates
We are free to construct quadrilaterals or triangles because
the program automatically divides a quadrilateral into two triangles
having identical material properties The dimensions of the elements
20
should be small at places where large changes in hydrological
properties occur or where large gradients in hydraulic head are
expected to occur
Coding of nodal points
Each node in the finite element network is assigned an integer
code which indicates the type of calculation for this node
The coding is as follows
- Code = 1 nodal points with described head as function of time
- Code 2 internal nodal points
- Code = 3 internal nodal points which represent secondary surface
water system
- Code = 5 nodal points with prescribed artificial recharge or
discharge and points with prescribed boundary flow
(both as functions of time)
Other important endogenous variables are the magnitude of the
time step the value of the variable which stops the iteration
process and the maximum number of iterations allowed per time step
(see also group A and B of section 413)
412 Exogenous variables
After constructing the nodal grid each node represents part of
the groundwater basin Geo-hydrological properties can be assigned
to each mode based on physical data (group C D and E of section
413) In group F of section 413 time-dependent variables can
be assigned to each node
413 Data input to Program FEMSAT
The data input must be punched on cards according to the
following instructions
GROUP COLUMNS FORMAT SYMBOL DESCRIPTION
A 1-80 20A4 HEAD Desired heading to be printed at
the beginning of the execution
1-5 15 NUMNP number of nodal points
21
GROUP COLUMNS FORMAT SYMBOL DESCRIPTION
6-10 15
11-15 15
16-20 15
21-25 15
26-30 15
31-35 15
NUMEL
NUMLAY
MBAND
NTSPI
NTSPO
MAXIT
1-10 F103 ST
11-20 F103 DELT
21-30 F103 TMAX
31-40 F105 TOL
number of elements
number of layers
maximum difference in node number
between two adjacent nodes
number of time steps between a
change in non-stationary input data
number of time steps between two
successive outputs
maximum number of iterations per
time step
time to start with the execution
initial time interval
maximum time to be reached before
execution is stopped
maximum allowed absolute change in
the values of h between any two
successive iterations in a given
time step
Iterations continue until all changes
are less than or equal to TOL or
until MAXIT is exceeded
1- 5 15
6-10 15
11-15 15
16-20 15
KX(N1)
KX(N2)
KX(N3)
KX(N4)
sequential number of i-th corner of
element N
sequential number of j-th corner
sequential number of k-th corner
sequential number of 1-th corner
(see fig 8)
Fig 8 Numbering of element corner nodes
22
GROUP COLUMNS FORMAcircT SYMBOL DESCRIPTION
One card must be provided for each element in sequential order
starting with N = 1 and ending with N = NVJMEL
I-K 1103 SANG(NLY)
1i-iO MO3 C1(NLY)
21-30 F 103 C2(NLY)
21-4- 1-104 THI(NLY)
angle in degrees between CI and
the laquo-coordinate to he used in
nodal print N in layer LY (see
poundig 9)
First principal conductivity of
material in node N and layer LY
idem second principal conductivity
thickness of layer LY at nodal
point N
ig V Definition of SANGCI and C2
One card icircuiPt be provided for each model point ctarting wih M = 1
and ending with N = NUMNP If there is norlt3 than 1 layer for each
layer a set must be provided starting with the upper layer LY - 1
and ending with the bottom layer LY = NIJMLAY The variables SANG
M and 2 are not relevant tor aquitards but for reasons of
conveniency this procedure is maintained (making SANG CI and C2
dummy variables)
1-10 110 KODE(N)
11-20 II0 NTERR(N)
coding of nodal points (see
paragraph 42)
number of tertiary relation in node N
The q-h relationship of the flow to
23
GROUP COLUMNS FORMAT SYMBOL DESCRIPTION
21-30 FlOl X(N)
31-40 FlOl Y(N)
41-50 FlOl HGL(N)
51-60 FlOl HBSC(N)
61-70 FlOl LESC(N)
the tertiary system as discussed
in par 24 are schematized in a small
number of different relations
x-coardinate of node N
y-coordinate of node N
height ground level in node N
height bottom secondary conduit in
node N If node N is not a secondary
conduit this variable is a dummy
variable
length of secondary conduit in area
represented by node N
One card must be provided for each node starting with N = 1 and
ending with N = NUMNP
E 1-10 FlOl P(NLY1) initial value of h at node N in
layer LY
11-20 FlOl VERRES(NLY) vertical resistance at node N and
layer LY If the layer is an aquifer
this value is a dummy variable
21-30 F101 RADRES(NLY) radial resistance for flow to
secondary system at node N in layer
LY
31-40 FlOl ENRES(NLY) entrance resistance for flow to
secondary system at node N in layer
LY
41-50 FlOl WP(NLY) wet perimeter of secondary system at
node N in layer LY
51-60 FlOl P0R(NLY) specific yield of the material at
node N in layer LY
61-70 FlOl SS(NLY) specific storage of the material
at node N in layer LY
24
GROUP COLUMNS FORMAT SYMBOL DESCRIPTION
One card must be provided for each node starting with N = 1 and
ending with N = NUMNP If there is more than 1 layer for each layer
a set of cards must be provided starting with LY = 1 and ending
with LY = NUMLAY
F 1-10 F101 P(NLY1) value of h at different times at
node N in layer LY Node N lies
on a prescribed head boundary
11-20 F101 WTSC(N) water table in secondary system at
node N at different times
21-30 F 101 ARRED(NLY) artificial recharge to or discharge
from node N in layer LY or
prescribed boundary flow at
different times
31-40 F 101 QPHR(N) flux through phreatic surface at
node N at different times
The data in group F are non-stationary input data One card must be
provided for each node starting with N = 1 and ending with N = NUMNP
In case of more than 1 layer for each layer a set must be provided
Not relevant variables are taken as dummy variables
After a certain number of time steps (NTSPI) the data of group E
are read So the set cards E must contain NxLYxC cards where C =
(TMAXNTSPIXDELT) + 1 if DELT is a constant
42 R u n n i n g t h e p r o g r a m o n t h e C y b e r 7200
The program has been executed on a Cyber 7200 computer of
IWIS-TNO in the Hague
- The demanded input is punched on cards according to the instructions
of the last paragraph and is put on a permanent file
- For each specific problem inthe program only the dimension statements
in the main program have to be adjusted according to the instructions
in the lines of comment
- After compilation the program together with the input file is
executed
25
- Intermediate data necessary for continuation of the calculations
are saved on another permanent file This gives the opportunity
to restart the program at a time the calculation was stopped
The only change in the input file is to adjust the starting time
Remark
With the aid of a terminal small changes in the program and in the
input file can be made easily and the output immediately appears
on the terminal In this way it is possible to stop erroneous
calculations This is especially useful when being in the testing
phase of the program
43 O u t p u t
Immediately after reading the input data they are printed so
that possible errors can be detected During execution after each
certain number of time steps (NTSPO) calculated values of h and
water balance terms are printed In appendix B an example is given
The output data can be devided into two groups
A Data per node per layer
- Coding of the nodal points KODE
- Hydraulic head HEAD
- Flow through phreatic surface at the time indicated in head
of table FLOWUN
- Flow to tertiary surface water system FLOWTS
- Flow to secondary surface water system FLOWSS
- Flow to adjacent layers LEAKAGE
- Artificial recharge or discharge and prescribed boundary
flow ARRED
- Change in storage during last time step STORAGE
- Flow through prescribed head boundary LATFLOW
B Data per layer
Summation of the terms already mentioned under A gives the
magnitude of the water balance terms per layer at the time
indicated (instanteneous water balance terms) Summation of
these instanteneous terms multiplied by their respective time
intervals since time is zero gives the cumulative water balance
terms per layer
26
After each print out we are able to check the calculations
because the sum of the water balance terms per layer must be
approximately zero
44 C o m p u t i n g t i m e
The computing time depends on the number of nodal points and
the number of layers But it depends also on the magnitude of the
changes in hydraulic head and on the value which terminates the
Gauss-Seidel iteration (TOL) To give an impression TOL = 001 m
Costs per day simulation (in hlfl) on the Cyber 7200
number of nodes number of layer bull bdquo bull bull _ 1mdash x a x b (21)
where
a = costs in Dutch guilders per system second (ƒ 027)
b = empirical factor with magnitude about 001
V NUMERICAL EXPERIMENTS
In general there are two possibilities to test the validity of
numerical models The first and most commonly used method is to
compare numerical results with analytical solutions The second
possibility is to compare numerical results with measured field
data Since the last alternative requires many data that are
difficult to obtain we restrict ourselves for the time being to
comparison with the analytical solutions
5 1 F l o w t o a w e l l i n a n u n c o n f i n e d
c i r c u l a r - s h a p e d a q u i f e r
Flow to a well can be divided into steady-state flow and
unsteady-state flow (KRUSEMAN and DE RIDDER 1976)
27
511 Steady state flow in an unconfined aquifer
For this case the formula of Thiem-Dupuit is valid (see fig 10)
Q bull k lth2-hicircgt ln(r2r])
(22)
where
hhbdquo = hydraulic head in 1 and 2 respectively
Q = discharge from well
rbdquo = distance from well
CL31
hmdash^ri --r2
Fig 10 Schematic cross-section of a pumped unconfined aquifer
Fig 11 gives the nodal grid superimposed on a well in a
circular shaped aquifer Near the well the nodal point distance
is made smaller Quadrilaterals as well as triangles are used
Numerical calculations with this finite element network and 3 -1
an extraction rate of 200 m day were performed for 10 days
At that moment a steady-state situation was almost achieved
28
2tmdashnodal point number
160 180 X-axesim)
Fig 11 Nodal point configuration for flow to a well in a circular-
shaped aquifer The outer boundary is taken to be constant
The well is situated in nodal point 42
29
In fig 12 the results of the simulation are compared with the
analytical solution The agreement between numerical results and
the analytical solution is excellent
draw- 15irS (m) down
analytical solution imdashx calculated values
distance from well r (m)
Fig 12 Comparison between analyt ical and calculated s teady-state
drawdown for the problem of f ig 11 with 1
-1 Q bull - 200 m 3 d a y _ 1
k = 10 mday
d = 100 m
S = 2 y
S = 0 s
512 Unsteady-state flow in an unconfined aquifer
Analytical solutions for unsteady-state flow in an unconfined
aquifer only exist if the aquifer extends infinitely For the
aquifer given in fig 11 this condition is not fulfilled However
during the first one or two days after starting the extraction the
influence of a constant hydraulic head at the outer boundary on the
30
drawdown near the well can be neglected (see fig 13) After one
day the lateral inflow is only 25 of the extraction the remainder
being the change in storage due to drawdown
2 0 0
m 3 day 1
Q=200m 3 day
100-
vraquomdash Lateral inflow
N amp storage
I I ^ V - W 10
t (days)
Fig 13 Values of Q lateral inflow and change in storage as
functions of time
The nonsteady-state or Theis equation can be written as
i _ Q s 4iTkD
dy = 4irkD
W(u) (23)
where
u = r2S
4irkDt
r = distance from well
S = storage coefficient
Q = constant discharge from well
D = thickness of the waterbearing layer
at the beginning of pumping
t = time after starting the extraction
s = corrected drawdown
= s - 322D
s = calculated drawdown
W(u) = Theiss well function
[J
CIAT1
[T]
]
31
In fig 14 the analytical unsteady-state solution for the case
described in section 511 is given together with the numerical
results at times 05 and 10 days The agreement between numerical
results and the analytical solution is rather good
Drawdown s1(m)
10
10
10
Theiss solution
x caculated values t=Q5
bdquo t= lO
10 10- 10 10 t r2(daym2)
Fig 14 Comparison between analytical and calculated steady-state
drawdown for the problem of fig 11 with
Q
k
d
S y
s s
=
=
=
=
=
200
1
10
3 m
0
0
2
0
-1 day
m day
m
These results indicate that the assumption of a linear variation
of h between two nodal points (see fig 3 ) even with the rather
rough element network of fig 11 is acceptable So there is no
need to improve for example the finite element network by introducing
higher order coordinate functions
32
52 F l o w t o w e l l i n 3-1 a y e r e d s y s t e m
In fig 15 a schematic cross-section of a 3-layered system of
two aquifers separated by an aquitard is given A well of
infinitisemal radius is completed in the lower aquifer and discharges
at a constant rate QTT Each layer is homogeneous isotropic and
extending radially infinite
bdquo bull - bull bull ~ i
mdash Q II
aquifer I
lugrave^i~IcircIuml7i-i~~ aquitard --r-rl~-
-II--I i - i
aquifer
Fig 15 Schematic cross-section of a 3-layered system
For this situation NEUMAN and WITHERSPOON (1969a) give rather
complicated analytical solutions In the special case where the
hydraulic properties of the two aquifers are identical they give
graphical illustrations of their solutions for three different
cases
These graphical solutions give the opportunity to verify
roughly numerical results of a 3-layered system Therefore a nodal
grid is superimposed on a circular shaped groundwater flow system
with the extraction in the centre of the lower aquifer and with
a constant head on the outer boundary Fig 16 gives the nodal
point situation in the x - y plane of 1 layer The nodes in the
other two layers are situated in the middle of the layers at the
same places in the x - y plane only their z-coordinate is different
33
A
-Jy
bull33
V 5
iumlff
Fig 16 Nodal point configuration for the 3-layered problem (top view)
34
In fig 17 numerical results are compared with the analytical
solution The agreement is rather bad especially for the two
aquifers But in terms of the water balance the difference between
aquifers and aquitard is not so big because the aquitard has
about 16 times bigger storage capacity than the aquifers The
small value of the specific storage of the aquifers is the reason
that small errors in the components of the water balance have a
big influence on the calculated values of the hydraulic head
Eig 17 Dimensionless drawdown vs dimensionless time in a
3-layered system with
II = 50 m day
3 m (distance from well)
aquifer I
aquitard
aquifer II
Thi (m)
100
25
100
S s
001
0642
001
k(mday )
1
0277
1
The discrepancies are partly caused by errors in discretization
In the z-direction these errors can be eliminated by taking the
35
vertical resistance of the aquitard to be infinite Then the
calculated drawdown in the pumped aquifer can be compared with
the analytical Theiss solution
This is done in fig 18 for different distances from the well
One sees that the agreement is better when the distance is bigger
Apparently the linearization of the logarithmic drawdown curve
near the well is responsible for the differences The differences
for r = 3 m are of the same magnitude as for the pumped aquifer in
the 3-layered case
joo gt A 50
20
Theiss solution
bull calculated values r=3m (distance f rom well) x bdquo bdquo r=7m
1 L
I l_ 001 003 0 0 5 Ol Q2
- I I 5 10 t - K t
Fig 18 Dimensionless drawdown vs dimensionless time in a
confined aquifer with
Q =500 m day
k = 10 mday
D = 100 m
S = 001 s
The differences between numerical results and analytical solutions
are not caused by taking a finite area because changing the outer
boundary into an impermeable boundary has only minor effects
36
VI SUMMARY AND CONCLUSIONS
In this report a numerical model for non-stationary saturated
groundwater flow in a multi-layered system has been treated
Groundwater flow can be described in different ways
a Physical-mathematical by combining Darcys law and the law of
continuity (section 21)
b Geo-hydrological the flow in a natural groundwater basin takes
place in layers with different hydraulic properties Definitions
of the most important hydraulic properties are given in section
22
c Schematical groundwater flows horizontally in aquifers and
vertically in aquitards (section 23)
d Approximating by dividing the region of flow into a finite
number of elements and applying Darcys law and the law of
continuity on each of these elements In these ways one gets
a set of equations which can be solved given (initial and)
boundary conditions (section 25)
e Numerical the set of equations has been solved using a Galerkin
type finite element method and the implicit Gauss-Seidel method
with overrelaxation factor (section 31 and 32)
f Computer-solvable the numerical solution has been translated
in a FORTRAN IV program called FEMSAT (section 33)
In chapter IV the input execution and output of the program
FEMSAT are treated
Finally in chapter V some numerical applications are given
simulation of flow to a well in unconfined aquifer and simulation
of flow to a well in a 3-layered configuration
Comparison of hydraulic heads obtained by numerical calculations
and analytical true solutions showed an excellent agreement in case
of stationary flow and a good to moderate agreement in case of non-
stationary flow However the diviations caused by the discretization
process must be seen in relation to the errors made in the
establishment of the hydraulic properties
37
As was stated in the introduction numerical models can be
valuable tools in the management of water resources Therefore
numerical models must be suitable to simulate effects of certain
operations upon the spreading of water over saturated groundwater
unsaturated groundwater and surface water The model described in
this report is a model for saturated groundwater flow The relation
with the surface water can be taken into account via radial and
entrance resistances or via a relation between flow to the surface
water system and the hydraulic head The relation with the unsaturated
zone has been restricted to the input factor effective precipitation
In practise however the unsaturated and saturated system are
interconnected Therefore the model should be improved by taking
into account the unsaturated zone This can be done in at least
two ways
a Replacing the unsaturated zone by a parametrical model
b Coupling the unsaturated zone in an iterative way with the
saturated system
In order to find out the usefulness of the numerical model for
water resources management it will be applied to actual regional
groundwater systemslaquoAlso sensitivity analyses will be carried out
The computer model itself can be improved by applying standard
programs eg automatical generation of a nodal grid and graphical
plotting of the results
38
VII REFERENCES
ERNST LF 1962 Grondwaterstromingen in de verzadigde zone en hun
berekening bij aanwezigheid van horizontale evenwijdige open
leidingen Thesis University Utrecht pp 189
1978 Drainage of undulating soils with high groundwater
tables (in press)
FORSYTHE GE and WR WASON 1960 Finite Difference Methods for
Partial Differential Equations John Wiley and Sons New York
pp 444
KRUSEMAN GP and NA DE RIDDER 1976 Analysis and evaluation of
pumping test data ILRI-bulletin No 11 pp 200
LAAT PJM DE and C VAN DEN AKKER 1976 Simulatiemodel voor grond-
waterstroming en verdamping in Modelonderzoek 1971-1974 ten
behoeve van de waterhuishouding in Gelderland deel II
Grondslagen pp 145-197
NEUMAN SP RA FEDDES and E BRESLER 1974 Finite element
simulation of flow in saturated -unsaturated soils considering
water uptake by plants Hydraulic Engineering Laboratory
Technion Israel pp 104
and PA WITHERSPOON 1969a Theory of Flow in a confined
two Aquifer System Water Resources Research Vol 5 No 4
pp 803-816
^_ and PA WITHERSPOON 1969b Applicability of Current Theories
of Flow in leaky aquifers Water Resources Research Vol 5
No 4 pp 817-829
PINDER GF and EO FRIND 1972 Application of Galerkins Procedure
to Aquifer Analysis Water Resources Research Vol 8 no 1
pp 108-120
REMSON I GE HORNBERGER and FJ MOLZ 1971 Numerical Methods
in Subsurface Hydrology John Wiley and Sons New York
pp 389
WESSELING JG 1977 Toepassing van Finite Difference- en Finite
Element Methoden voor het oplossen van een aantal grondwatershy
stromingsproblemen (met computerprogrammas) ICW-nota 944
pp 63
39
ZIENKIEWICZ OC 1971 Finite Element Method in Engineering Science
McGraw-Hill London pp 521
40
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X - e - o
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laquo ltE laquo h C Z Z E O t l l O E Q E O E Z E OC laquo a c laquo Q C laquo a c gt - lt o c x x x x x a c L U O U I O L U O a C O lt C lt laquo 0 0 0 L U acuacuacLi ouUcorococoroz
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APPENDIX 8 EXAMPLE OF OUTPUT
HEAD AND TERMS OF UATERBALANCE PER NODE PER LAYER
AT TIME 260 IN II AND I13DAYS
T
LAYER NODE KODE HEAD FLOUUN FLOUTS FLOUSS LEAKAGE STORAGE LAT FLOU
13 19 25 31 37 43
125008 125000 12S000 12S000 12S000 12S000 12S000 12S000
0000 0000 0000 0000 0000 0000 0000 0000
0000 0000 0000 0000 0000 0000 0000 0000
0000 0000 0000 0000 0000 0000 0000 0000
- 000 - 000 -000 - 000 - 000 - 000 - 000 0000
0000 0000 0000 0000 0000 0000 0000 0000
0000 000 000 000 000 000 000 000
000 0000 0000 0P100 0000 0000 0000 0000
1 7
13 19 25 31 37
12S000 125000 125000 125000 125000 12S000 125000 125000
0000 0000 0000 0000 0000 0000 0000 0000
- 001 -007 - 023 - 020 - 009 - 003 0000
0000 0000 0000 0000 0000 0000 0000 0000
000
001
007
023
020
009
003
003
0000 0000 0000 0000 0000 0000 0000 0000
1 7
13 19 25 31 37 43
125000 125000 124998 124972 124895 124787 124647 124358
0000 0000 0000 0000 0000 0000 0000 0000
- 000 001 007 023 020 009 003
0000
0000 0000 0000 0000 0000 0000 0000
50000
0000 182
1982 3862 1674 419 078 041
0000 0000 0000 0000 0000 0000 0000 0000
TERMS OF WATERBALANCE PER LAYER DURING LAST TIMESTEP AND
AND SINCE BEGINNING IN M3DAY M3
FLOU TO UNSATURATED ZONE
FLOU TO TERTIARY SYSTEM
0000
0000
0000
0000
LAYER NO 1
FLOU TO SECONDARY SYSTEM
FLOU TO AD3ACENT LAYERCS)
ARTIFICIAL FLOU AND PRESCRIBED BOUNDARY FLOU
ACCUMULATION TERMS
LATERAL FLOU THROUGH PRESCR HEAD BOUNDARY
0000
-000
0000
000
000
0000
-000
0000
000
000
LAYER NO 2
FLOU TO SECONDARY SYSTEM
FLOU TO AD3ACENT LAYERCS1
ARTIFICIAL FLOU AND PRESCRIBED BOUNDARY FLOU
ACCUMULATION TERMS
LATERAL FLOU THROUGH PRESCR HEAD BOUNDARY
0000
-378
0000
381
0000
0000
-055
0000
056
0000
LAYER NO 3
FLOU TO SECONDARY SYSTEM
FLOU TO AD3ACENT LAYERCS5
ARTIFICIAL FLOU AND PRESCRIBED BOUNDARY FLOU
ACCUMULATION TERMS
LATERAL FLOU THROUGH PRESCR HEAD BOUNDARY
0000
370
50000
49376
0000
0000
055
12979
12909
0000
II GENERAL DESCRIPTION OF THE PROBLEM
21 P h y s i c a l b a c k g r o u n d
Throughout this report water potential is expressed on a unit
weight basis and is being called hydraulic head with symbol h
h = 2- + z [L] Pg
where
mdash = piezometer head |L | Pg u -1
z = gravitational head |_Lj
The hydraulic head is an expression for the capacity of a unit
weight of water to do work as compared to the work capacity of the
same weight of water with the same chemical composition at a certain
reference level and under atmosferic pressure
The problem is restricted to saturated flow of groundwater with
one constant density
211 The law of linear resistance (Darcys law)
According to Darcys law the rate of flow through a porous medium
is proportional to the gradient in hydraulic head It may be written
as
ocirc h ltSh ocirch N q = _k Tmdash q = -k vmdash q = -k -rmdash ( 1 ) x y ocircx y y lt5y nz z ocircz
where
q = volume flux of water passing through a unit area per x y z
per unit time perpendicular to x y z direction
respectively |_LT J
k = hydraulic conductivity in x y and z direction x gt y gt z _ _ i _
respectively | LT J
The coefficient k in the Darcy flow equation is taken to be
a constant depending on both the properties of the porous medium
and the fluid
212 The law of continuity
The other important physical law states that no matter can be
lost or created
The conservation equation can be found by applying the principle
of continuity to an infinite small volume
6v 6v lt5v 60 x y z TT = T + x + ~T7 + 1 (2)
Ot OX OcircX oz
where
0 = volume of water per unit bulk volume of soil |_-J
q = sink or source term T J
213 Combination of Darcys law and the law of continuity
Darcys law combined with the law of continuity gives
60 lt5 oh lt5 ocirch 6 oh _ ocirct ocircx x ocircx ocircy y oy ocircz z ocircz
or
c 5h 6 ocirch ocirc Ocirch 6 ocirch c ot ox x ocircx ocircy y oy laquoz z 6z
where
S = volume of water released or stored per unit bulk c I
volume of soil per unit change in hydraulic head |L I
22 H y d r a u l i c p r o p e r t i e s a n d t y p e s o f
a q u i f e r s
The flow in a natural groundwater basin takes place in layers
with different hydraulic properties Usually the layers are divided
into good permeable layers or waterbearing layers (aquifers) and
less permeable layers (aquitards)
The most important hydraulic properties are
a Transmissiviy - the product of the average hydraulic conductivity
k and the saturated vertical thickness d of the waterbearing
layer
It is a measure for the transport capacity of aquifers
Notation T = kd |_ L T ~
b Hydraulic resistance - the ratio of vertical thickness of
aquitards d and hydraulic conductivity k It is a measure for
the resistance capacity of aquitards denoted as c = d k _ T J
c Specific yield - the volume of water released or stored in the
phreatic zone per unit surface area of an unconfined aquifer per
unit change in hydraulic head
It is also called effective porosity
The symbol used for specific storage will be S j_-J
d Specific storage - the volume of water released or stored per
unit volume of the saturated parts of an aquifer or aquitard per
unit change in hydraulic head It is a measure for the elasticity
of the soil material and the fluid Excluded are irreversible
processes Notation S _ L J
e Storage coefficient - the volume of water released or stored
pe7 nit surface of a layer per unit change in hydraulic head
It is the sum of specific yield and the product of specific
storage and thickness of the aquifer or aquitard
S = S + S d r - 1 c y s - -1
Note in confined layers S = 0 y
In fig 1 a 3-layered configuration is given of two aquifers
separated by an aquitard Aquifer I has a free water surface and is
an unconfined aquifer Aquifer II is a completely saturated aquifer
Dependent on the hydraulic resistance of the aquitard KRUSEMAN and
DE RIDDER (1976) distinguish between confined semi-confined and
semi-unconfined aquifers For our purpose it is not useful to make
this distinction We restrict the aquifer types to unconfined and
confined aquifers
WIHtUtttWttllttlt(ltmtttltHlttltlil(lltlittlfllnl ) bull (
d V U i l i l i ii i Hi il i M M bull f i m
i tMi| i i | J i j j i _ I I bull
aq ui tard
aquifer II
Fig 1 Schematic representatipn of flow direction and variation
of hydraulic head h in a 3-rlayered system
In an unconfined aquifers ( I ) poundhe vertical thickness pf the
waterbearing layer d is a variable while in a confined aquifer II
this is a constant (denoted as D ) Other important properties of
unconfined aquifers are the hydraulic head being the same as the
phreatic surface and the specific yield being different from zero
23 S c h eacute m a t i s a t i o n o f f l o w
The groundwater flow in a groundwater basin is schematized
into a horizontal flow in aquifers and into a vertical flow in
aquitards Consequently the vertical variation of the hydraulic
head is as drawn in figl
The validity of this assumption has )een investigated by
NEUMAN and WITHERSPOON (1969) The found that if the contrast
in hydraulic conductivity between two adjacent layers is bigger
than a factor two the relative error caused by this assumption
is usually smaller than 5
This schematization gives a simplification of the general flow
equation (eq 4 ) In an aquifer the vertical is zero and eq (4)
then reduces to
-Sbdquo -r- 7 - (k d -r-) + 7 mdash (k d 7-) + qd c lt5t ox x ox oy y oy
or
bull(S + S d) g y s ot
-r- (k d -=-) + -T- (k d -=-) + Q ocircx x ox oy y ocircy
(5)
where
Q = volume of water substracted from or added to unit surface
of aquifer per unit time |_ L T J
In an aquitard eq (4) reduces to
-sQ |pound = (k Q) s 6t oz Z OcircZ
(6)
As will be discussed in the next paragraph the term q in an
aquitard is set equal to zero
24 W a t e r b a l a n c e t e r m s
In eq (4) the term q is mentioned With this sink or source
term different terms of the water balance are taken into account
Fig 2 depicts schematically the situation for an unconfined aquifer
(for other layers some terms dont exist)
recharge- - discharge
laquo-b
Fig 2 Schematic representation of the various terms involved in the
water balance
Description of the different terms
a The flow through the phreatic surface q f
If this term is negative (water goes out of the saturated system)
it is called capillary rise If positive it is called percolation
(or drainage)
This flow can be calculated as a rest term of the water balance
of the unsaturated zone Very often this term is expressed as a
flow per unit surface (flux) and is called effective precipitation
For the time being this flux is a given input
b Flow to adjacent aquitards q
The flux to adjacent aquitards can be calculated by applying eq
(6) to the aquitard
c Flow to the primary surface water system q
The primary surface water system penetrates fully the groundwater
basin So it acts as a boundary with given values for the
hydraulic head (is equal to the phreatic level of the primary
system)
d Flow to the secondary surface water system q
The secondary surface water system does not penetrate fully the
groundwater basin which causes a resistance against flow between
groundwater and surface water The flow q can be calculated as
follows
h - h
- mdash f i i gt
where
h = hydraulic head of the groundwater at the place of
the secondary system |_ L |
h = phreatic level of secondary system |_ L J i s -
R = radial and entrance resistance of the
secondary system per unit surface |_ T J
According to ERNST (1962) the radial resistance H per unit
surface can be caKulated with
Uuml = 4 i n 4 e8) TTk B
where
1 = area drained by 1 m conduit L ^ J
d = thickness of the layer near the conduit _ L J
B = wet perimeter of the conduit |_ L J
The entrance resistance is caused by a layer which shows a
lower conductivity around the conduit than its surroundings
In practise radial and entrance resistancies are difficult to
deduct from hydraulic properties They can be found indirectly
by measuring simultaneously q h and h
t s
e Flow to the tertiary surface water system q
In smaller conduits usually the phreatic level is not known
So eq (7) cant be applied The flow to these conduits however
is sometimes very important (eg in areas with shallow groundwater
tables) One may a-sume (ERNST 1978) that a relationship exists
between the hydraulic head of an aquifor and the flow q
According to Ernst the tertiary system can be divided into a
number of categories with respect to their mutual distance and
bottom height For each category the flow is h - h (b)
q ( h b ) - mdash (9)
where
h = average hydraulic head | L J
h (b) = phreatic level in the category under
consideration a function of bottom height |_L J
a = geometry factor to convert h into h _ - J
Y = drainage resistance _ T J
hm = hydraulic head midway between two
conduits |_ L J m
Summation of the q(hb)-relations of the different categories
yields the overall relation between hydraulic head and flow to
the tertiary system we are looking for
f Artificial recharge or discharge q and prescribed boundary
flux qfo
The magnitudes of these flows are not influenced by the conditions
inside the groundwater basin but remain on a prescribed level
Thus q is an internal boundary condition q an external boundary cl D
condition
For confined aquifers some of the above mentioned terms dont
exist These terms are the flow through the phreatic surface qf
and the flow to the tertiary surface water system q
Aquitards are considered only as transmitters of water from one
aquifer to another with possibilities of storage (eq 6 ) This means
that for aquitards the water balance terms ar through f are set
equal to zero
25 S o l u t i o n o f g r o u n d w a t e r f l o w p r o b l e m s
The basic equations for saturated groundwater flow are eq (5)
for aquifers and eq (6) for aquitards The solution of these
equations require knowledge of the physical system and the auxiliary
conditions describing the system constraints
These auxiliary conditions are (REMSON etal 1971)
a Geometry of the system
Eg the diviation of the system into a restricted number of
layers and the horizontal extension of the system under
consideration
b The matrix of hydraulic properties including in-homogeneity
and an-isotropy
c Initial conditions Because we are dealing with non-stationary
flow the values of pertinent system variables (such as hydraulic
head at the initial time) must be known
d Boundary conditions They describe the conditions at the boundaries
pf the system as a function of time
One can distinguish between
- hydraulic head boundary
- flow (including zero flow) boundary
- both head and flow boundary
After specifying the auxiliary conditions an unique solution of
the variation of the hydraulic head in space and time can be
obtained
For special shapes of flow and assumptions exact analytical
solutions exist In nature however one is usually confronted with
problems like
- spacial variation in the hydraulic properties of the soil
- irregular shape of the groundwater basin
- empirical relations between q and h
- d not being a constant
For such situations by numerical and analogy models only an
approximation of the actual situation can be obtained In the next
chapter a numerical solution called the finite element method is
treated
III NUMERICAL SOLUTIONS
In this chapter the numerical solution of the aquifer equation
(eq 5) and the aquitard equation (eq 6) will be treated The
method of solving is the finite element method In section 31 only
a short mathematical description of this method will be given For
more mathematical background the reader is referred to publications
on this subject (eg ZIENKIEWICZ 1971 NEUMAN etal 1974 and
WESSELING 1976)
In section 32 the application of the finite element method on
the flow in groundwater basins is treated while in section 33 the
translation into a computer program is given
3 1 F i n i t e e l e m e n t m e t h o d
For reasons of simplicity the finite element method will be
explained for the stationary version of eq 5 ie
10
(k d^) + (k d ^ ) - Q = O (10) ocircx x 6x 6y y Sy
The basic idea of each numerical solution of eq (10) is to
apply this equation on a small but finite volume The flow region
of interest R is divided into a finite number of volumes called
elements The elements can have different shapes but for reasons
of conveniency we only use triangles- Quadrilaterals can also be
used because they can be divided into two triangles The sizes of
the triangles may vary They can be adapted according to the
geometry of the region and the accuracy desired (fig 11 gives an
example of the diviation of a region into triangles and quadrilaterals)
The corner points of the elements represent the nodal points
If there are N nodal points then the continuous variation of h
in region R is approximated by the expression
h (x_v) ~ h f (xy) (11) a n n
where
h (AV) = approximation of u (xy) a
h = value of h in nodal point n n
f ltxgty) = global coordinate function
In nodal point n a must be equal to h so f (x y ) = 1 and a n n n n
f (xy) = 0 i n Between f (x v ) and f (xy) of the nodal n j ] n n n n j j
points surrounding nodal point n f (xy) varies linearly from
1 to 0
In fig 3 a picture of f (xy) is given n
If one knows the values h in the nodal points n we know the approximation h of h We can find these values of b bv using for
a n
example the Galerkin method
Galerkins principle states that an approximate solution of h in
eq (10) can be obtained from the following system of equations
N ~~ (k d -A + -^- (k d T-y E h f (xy) f(xy)dxdy +
-ox x ox ocircy y oy - _ n n i R n~
J J Q f(xy) dx dy = 0 i = 1 N (12)
1 1
Fig 3 The global coordinate function f (xy)
The integration can be performed element by element with the
aid of the coordinate functions f (NEUMAN etal 1974)
For non-stationary problems it is not possible to apply the
Galerkin principle on the time derivative But it is possible
to discretize this derivative and to set up the system of equation
(eq 12) for each finite time step
One obtains in this way a series of sets of laquocriations tha s
an approximation of the non-stationary flow problem
Treatment of boundary conditions
a Initial conditions
In non-stationary flow problern gtne must start with given
values of a at time t = t ri c
b Boundary conditions
- Prescribed head boundary
Nodal points which lie on a prescribed head boundary have a
prescribed value for h bull r Hgte r umber of unknown h s is n n
reduced and accordingly no the number of equations
- Prescribed flux through z-- boundary
If one knows the flux (iuumlr jw or outflow per unit area) and
the area of the boundary oi-r element the inflow or outflow
through the boundary per flemlt=nt can be calculated
One may assume this flow to J bull part of the sink term Q in
eq (12)
12
32 A p p l i c a t i o n o f t h e f i n i t e e l e m e n t
m e t h o d o n t h e f l o w i n g r o u n d w a t e r
b a s i n s
From hydrogeological investigations the geometry and the matrix
of the hydraulic properties are known Knowing the geometry the
flow can be split up into a finite number of layers This holds
also for the nature of the boundaries
Now a nodal grid is superimposed on each layer in such a way
that the nodal points of the different layers are situated right
above each other (see fig 4) and in each layer the nodal point
lies in the middle of this layer
-nodal point
Fig 4 Schematic representation of the nodal points in the
xyz-space
Each layer has N nodal points of which P nodal points have a
prescribed head In each layer remain N-P = N nodal points with
unknown head Eq (12) can be applied to each point with unknown
head in each separate layer
As was stated in the last paragraph for each time step one
can write
13
Ah bdquo ~ -r1 + Q = O in = 12N in n m At yi A h + B (13)
where
A = in
ocircf ocircf ocircf of (k d f mdashri- ~ + k df - ~ -jS) dxdy
J x n ox ox y n oy oy
in (S d + S ) ff dx dy O if i icirc n
s y i n
Q f dx dy
The matrices A B and the vector Q have been defined in in l
elsewhere (NEUMAN etal 1974) A is the conductivity matrix in
B the storage matrix and Q the source vector in l
In order to clarify these rather complicate expressions apply
equation (13) poundo point 4 of fig 5
6 7
Fig 5 Top view of part of the nodal grid
lhl + A42h2 + A43h3 + A44h4 + A45h5 + A46h6 + A47h7 +
h - h
+ B _iuumli i ^ + Q = o 44 t - t x4
n n-1
(14)
where h and h are values of h at time t and t 4n 4n-1 4 n n-1
respectively
14
If the h-values belonging to the A-values are taken at time
t = t we get an equation with one unknown h for time t = t n-1 4n n
The equation can then be solved directly
However this is an explicit formulation which puts strong limitations
on the magnitude of the time step t - t (REMSON etal 1971) n n-1
Values larger than a certain value cause instability With an implicit method the values of h are taken at time
t = t This method is unconditionally stable Now one gets an equation n
with 7 unknowns h through h If one has 7 equations for every 1 n n
point 1 through 7 we can solve the values h i = 17
in
Of all the methods possible the iterative method of Gauss-Seidel
has been chosen The Gauss-Seidel method starts by choosing a first estimate of h at time t = t (usually equal to h ) At each in n j -i in-l iteration step a new value of h is calculated using the last
in calculated values of h If one passes the nodal points from 1 to
i n 7 eq (14) becomes
h-j + 1 = A h^ + 1 + A bdquo h | + 1 + A _ h | + 1 + A h j + A r h j + 4 n 4 1 1 n 4 2 2 n 43 3 n 45 5 n 4 6 6 n
B h B + A h J + - 4 4 4 n ~ + 0 J (A + mdash) A 4 7 h 7 n At n U 4 4 At
(15)
where
h-i = in
j = iteration step or number of iteration
i = number of nodal point n = time = t
n
The iteration process converges if the successive differences k+1 k
between h and h become continuously smaller in in
The iteration process is stopped if at each nodal point the k+1 k
difference between h and h is smaller than a certain prescribed in in
value
15
To accelerate the convergence an overrelaxation factor W is used
(for more information about W see FORSYTHE and WASOW 1960)
k+1 k T7 k+1 k h = h + W (h - h )
in in in in (16)
If W = 1 there is no overrelaxation Usually W varies between 1 and
2 and the optimum value very often is found by trial and error
Eq (15) is valid for points in aquifers The connection with
adjacent layers is made by the term Q because in this term the
flow from adjacent aquitards q is included (two aquifers are always
separated by an aquitard)
Aquitards
We dont apply the finite element method on points in aquitards
As stated before there is no horizontal flow in aquitards So it has
no sense to replace the continuous variation of h in the aquitard
by equation (11)
The nodal points in aquitards represent a certain volume of an
aquitard hich transmit water from one aquifer to another and which
has storage capacity The nodal points are included in the Gauss-
Seidel scheme -s follows (see fig 6 )
1 aauifer
a -_vS laquo 2 [~^-t^yM^^^-^r~ aqutarci l-Z
aquifer
Fig 6 Schematic situation for points in aquitards
Point 2 is situated in the middle of the aquitard with thickness
d storage coefficient S and vertical hvdraulic conductivity k c J v
Suppose the Gauss-Seidel iteration starts in the upper layer and
ends in the bottom-layer Then the flux from point 1 in aquifer 1 to
16
point 2 in the aquitard at time t = t and in iteration step j + 1
is
k q = (hf -hf )-iumlmdash f (17)
1 n 2 n OSxd1
and the flux from point 3 in aquifer 2 to point 2 in the aquitard
is
1 c 1
qbdquo = (h - hf) mdashHmdashr- (18) 2 3 n 2 n OSxd
The net flux ltl-q9 must be equal to the change in storage per unit
area per unit time This change in storage can numerically be
approximated by
te - h-
n n-1
Combina t ion of ( 1 7 ) (18) ana (19) g i v e s
l tr1 - h (hJ + 1 + hj - 2 hj+1) - - - - - - S1 - ilaquoS L u l l = o (20)
] i 3 n 2 n 1 c t - t j j x d n n -1
T + 1
From t h i s e q u a t i o n h^ kan be s o l v e d
3 3 C o m p u t e r o g r a m
The problem outl^ugrave so far has been translated into a computer
program written in FORTRAN IV
The flow chart ot Uis program is given in fig 7 whereas in
appendix A the complete listing of the program can be found
In the flow chart the 5 main parts of the program are
indicated
A Reading and printing of input data
B Determination of the matrices A and B and the vector Q per layer
17
M
1 A
FLOH to unsaturated zono
FLOW to tertiary syster
FLOW to secondnrv system
Leakage to adjacent aquitards
SOLVE equation (14)
yes
CALCULATE flux through constant head boundnr
CALCULATE waterbalance n p r 1 OAro r r J j - ~
PRINT of output
Ier node per layer
k = k+1
t = t + At
7 DUMP variables on tape
-gt-
( STOP )
Fig 7 Flow chart of the program FEMSAT with diviation in 5 main
parts A B C D and E
C START PROGRAM FEMSAT
f READ general information
PRINT general information
f READ element information
PRINT element information 7 READ stationary nodal point information
PRINT stationary nodal point information 7
READ non-stationary nodal point information for t = t
FRINT non-stationary nodal point information for t = t
L 2 (
^ yes R E A D non-stat input for t gt t
J PRINT non-stat input for t gt t (- 1 -- M degY
GENERATE A B and AREA-matrix for confined aquifers
_L
GENERATE A B and AREA-matrix for unconfined aouifer
bull
C Solving h with Gauss-Seidel
D Determination of the water balance terms per layer
E Printing of results
The program exists of a main program with several subroutines
The main program calls the subroutines In the subroutines most
calculations are performed
In this way an easy adaptation to different problems is
obtained (see also section 44) Comments in the program will
help the reader to understand the program
In the next chapter the preparation of the input data executing
of the program and the output will be treated
Acknowledgement
Some parts of the computer program were taken from the program
UNSAT2 of NEUMAN etal (1974)
IV INPUT EXECUTION AND OUTPUT
41 I n p u t
The input is divided into endogenous variables (chosen by the
user or calculated by the program during execution) and exogenous
variables (based on physical data)
411 Endogenous variables
Translation of groundwater basin into a nodal point network
Probably the most important but also the most difficult stage
in the process of simulating groundwater flow is the translation
of the natural groundwater basin into a nodal point network Questions
which arise are
a In how many layer the geo-hydrological system mustcan be
schematized
b The locations of the boundaries and the nature of these boundaries
(impervious prescribed head prescribed flux)
c The configuration of the nodal points
19
The answers of these questions depend on the problem under
consideration
- Schematization into layers
Usually our knowledge about the geo-hydrological situation is
such that no more than 5 different layers can be distinguished
(remember contrast rule)
- Boundary conditions
The top boundary condition of a groundwater basin usually is
a prescribed flux (effective precipitation)
The bottom boundary condition usually is a prescribed flux
boundary with flux zero (impervious hydrological base)
For the boundaries at the other sides of the system one sees
either prescribed head boundaries (eg a fully penetrating
river) or prescribed flux boundaries (including zero flux)
- Nodal point configuration
Nodal points are situated
on the boundaries
along internal boundaries which indicate changes in
hydraulic properties within the layer
along the tertiary surface water system
on places where artificial recharge or discharge wiii tgtke
place
elsewhere
Construction of the finite element mesh
Assume we have schemetized the groundwater basin iatu a finite
number of layers with different direction of flow (horizontal and
vertical) Within one layer the hydrological properties may vary
from place to place
Now a nodal grid is superimposed on each layer in a way already
described in section 32 Thus each layer has the same number of
nodal points situated on the same places in the horizontal plane
but with different z-coordinates
We are free to construct quadrilaterals or triangles because
the program automatically divides a quadrilateral into two triangles
having identical material properties The dimensions of the elements
20
should be small at places where large changes in hydrological
properties occur or where large gradients in hydraulic head are
expected to occur
Coding of nodal points
Each node in the finite element network is assigned an integer
code which indicates the type of calculation for this node
The coding is as follows
- Code = 1 nodal points with described head as function of time
- Code 2 internal nodal points
- Code = 3 internal nodal points which represent secondary surface
water system
- Code = 5 nodal points with prescribed artificial recharge or
discharge and points with prescribed boundary flow
(both as functions of time)
Other important endogenous variables are the magnitude of the
time step the value of the variable which stops the iteration
process and the maximum number of iterations allowed per time step
(see also group A and B of section 413)
412 Exogenous variables
After constructing the nodal grid each node represents part of
the groundwater basin Geo-hydrological properties can be assigned
to each mode based on physical data (group C D and E of section
413) In group F of section 413 time-dependent variables can
be assigned to each node
413 Data input to Program FEMSAT
The data input must be punched on cards according to the
following instructions
GROUP COLUMNS FORMAT SYMBOL DESCRIPTION
A 1-80 20A4 HEAD Desired heading to be printed at
the beginning of the execution
1-5 15 NUMNP number of nodal points
21
GROUP COLUMNS FORMAT SYMBOL DESCRIPTION
6-10 15
11-15 15
16-20 15
21-25 15
26-30 15
31-35 15
NUMEL
NUMLAY
MBAND
NTSPI
NTSPO
MAXIT
1-10 F103 ST
11-20 F103 DELT
21-30 F103 TMAX
31-40 F105 TOL
number of elements
number of layers
maximum difference in node number
between two adjacent nodes
number of time steps between a
change in non-stationary input data
number of time steps between two
successive outputs
maximum number of iterations per
time step
time to start with the execution
initial time interval
maximum time to be reached before
execution is stopped
maximum allowed absolute change in
the values of h between any two
successive iterations in a given
time step
Iterations continue until all changes
are less than or equal to TOL or
until MAXIT is exceeded
1- 5 15
6-10 15
11-15 15
16-20 15
KX(N1)
KX(N2)
KX(N3)
KX(N4)
sequential number of i-th corner of
element N
sequential number of j-th corner
sequential number of k-th corner
sequential number of 1-th corner
(see fig 8)
Fig 8 Numbering of element corner nodes
22
GROUP COLUMNS FORMAcircT SYMBOL DESCRIPTION
One card must be provided for each element in sequential order
starting with N = 1 and ending with N = NVJMEL
I-K 1103 SANG(NLY)
1i-iO MO3 C1(NLY)
21-30 F 103 C2(NLY)
21-4- 1-104 THI(NLY)
angle in degrees between CI and
the laquo-coordinate to he used in
nodal print N in layer LY (see
poundig 9)
First principal conductivity of
material in node N and layer LY
idem second principal conductivity
thickness of layer LY at nodal
point N
ig V Definition of SANGCI and C2
One card icircuiPt be provided for each model point ctarting wih M = 1
and ending with N = NUMNP If there is norlt3 than 1 layer for each
layer a set must be provided starting with the upper layer LY - 1
and ending with the bottom layer LY = NIJMLAY The variables SANG
M and 2 are not relevant tor aquitards but for reasons of
conveniency this procedure is maintained (making SANG CI and C2
dummy variables)
1-10 110 KODE(N)
11-20 II0 NTERR(N)
coding of nodal points (see
paragraph 42)
number of tertiary relation in node N
The q-h relationship of the flow to
23
GROUP COLUMNS FORMAT SYMBOL DESCRIPTION
21-30 FlOl X(N)
31-40 FlOl Y(N)
41-50 FlOl HGL(N)
51-60 FlOl HBSC(N)
61-70 FlOl LESC(N)
the tertiary system as discussed
in par 24 are schematized in a small
number of different relations
x-coardinate of node N
y-coordinate of node N
height ground level in node N
height bottom secondary conduit in
node N If node N is not a secondary
conduit this variable is a dummy
variable
length of secondary conduit in area
represented by node N
One card must be provided for each node starting with N = 1 and
ending with N = NUMNP
E 1-10 FlOl P(NLY1) initial value of h at node N in
layer LY
11-20 FlOl VERRES(NLY) vertical resistance at node N and
layer LY If the layer is an aquifer
this value is a dummy variable
21-30 F101 RADRES(NLY) radial resistance for flow to
secondary system at node N in layer
LY
31-40 FlOl ENRES(NLY) entrance resistance for flow to
secondary system at node N in layer
LY
41-50 FlOl WP(NLY) wet perimeter of secondary system at
node N in layer LY
51-60 FlOl P0R(NLY) specific yield of the material at
node N in layer LY
61-70 FlOl SS(NLY) specific storage of the material
at node N in layer LY
24
GROUP COLUMNS FORMAT SYMBOL DESCRIPTION
One card must be provided for each node starting with N = 1 and
ending with N = NUMNP If there is more than 1 layer for each layer
a set of cards must be provided starting with LY = 1 and ending
with LY = NUMLAY
F 1-10 F101 P(NLY1) value of h at different times at
node N in layer LY Node N lies
on a prescribed head boundary
11-20 F101 WTSC(N) water table in secondary system at
node N at different times
21-30 F 101 ARRED(NLY) artificial recharge to or discharge
from node N in layer LY or
prescribed boundary flow at
different times
31-40 F 101 QPHR(N) flux through phreatic surface at
node N at different times
The data in group F are non-stationary input data One card must be
provided for each node starting with N = 1 and ending with N = NUMNP
In case of more than 1 layer for each layer a set must be provided
Not relevant variables are taken as dummy variables
After a certain number of time steps (NTSPI) the data of group E
are read So the set cards E must contain NxLYxC cards where C =
(TMAXNTSPIXDELT) + 1 if DELT is a constant
42 R u n n i n g t h e p r o g r a m o n t h e C y b e r 7200
The program has been executed on a Cyber 7200 computer of
IWIS-TNO in the Hague
- The demanded input is punched on cards according to the instructions
of the last paragraph and is put on a permanent file
- For each specific problem inthe program only the dimension statements
in the main program have to be adjusted according to the instructions
in the lines of comment
- After compilation the program together with the input file is
executed
25
- Intermediate data necessary for continuation of the calculations
are saved on another permanent file This gives the opportunity
to restart the program at a time the calculation was stopped
The only change in the input file is to adjust the starting time
Remark
With the aid of a terminal small changes in the program and in the
input file can be made easily and the output immediately appears
on the terminal In this way it is possible to stop erroneous
calculations This is especially useful when being in the testing
phase of the program
43 O u t p u t
Immediately after reading the input data they are printed so
that possible errors can be detected During execution after each
certain number of time steps (NTSPO) calculated values of h and
water balance terms are printed In appendix B an example is given
The output data can be devided into two groups
A Data per node per layer
- Coding of the nodal points KODE
- Hydraulic head HEAD
- Flow through phreatic surface at the time indicated in head
of table FLOWUN
- Flow to tertiary surface water system FLOWTS
- Flow to secondary surface water system FLOWSS
- Flow to adjacent layers LEAKAGE
- Artificial recharge or discharge and prescribed boundary
flow ARRED
- Change in storage during last time step STORAGE
- Flow through prescribed head boundary LATFLOW
B Data per layer
Summation of the terms already mentioned under A gives the
magnitude of the water balance terms per layer at the time
indicated (instanteneous water balance terms) Summation of
these instanteneous terms multiplied by their respective time
intervals since time is zero gives the cumulative water balance
terms per layer
26
After each print out we are able to check the calculations
because the sum of the water balance terms per layer must be
approximately zero
44 C o m p u t i n g t i m e
The computing time depends on the number of nodal points and
the number of layers But it depends also on the magnitude of the
changes in hydraulic head and on the value which terminates the
Gauss-Seidel iteration (TOL) To give an impression TOL = 001 m
Costs per day simulation (in hlfl) on the Cyber 7200
number of nodes number of layer bull bdquo bull bull _ 1mdash x a x b (21)
where
a = costs in Dutch guilders per system second (ƒ 027)
b = empirical factor with magnitude about 001
V NUMERICAL EXPERIMENTS
In general there are two possibilities to test the validity of
numerical models The first and most commonly used method is to
compare numerical results with analytical solutions The second
possibility is to compare numerical results with measured field
data Since the last alternative requires many data that are
difficult to obtain we restrict ourselves for the time being to
comparison with the analytical solutions
5 1 F l o w t o a w e l l i n a n u n c o n f i n e d
c i r c u l a r - s h a p e d a q u i f e r
Flow to a well can be divided into steady-state flow and
unsteady-state flow (KRUSEMAN and DE RIDDER 1976)
27
511 Steady state flow in an unconfined aquifer
For this case the formula of Thiem-Dupuit is valid (see fig 10)
Q bull k lth2-hicircgt ln(r2r])
(22)
where
hhbdquo = hydraulic head in 1 and 2 respectively
Q = discharge from well
rbdquo = distance from well
CL31
hmdash^ri --r2
Fig 10 Schematic cross-section of a pumped unconfined aquifer
Fig 11 gives the nodal grid superimposed on a well in a
circular shaped aquifer Near the well the nodal point distance
is made smaller Quadrilaterals as well as triangles are used
Numerical calculations with this finite element network and 3 -1
an extraction rate of 200 m day were performed for 10 days
At that moment a steady-state situation was almost achieved
28
2tmdashnodal point number
160 180 X-axesim)
Fig 11 Nodal point configuration for flow to a well in a circular-
shaped aquifer The outer boundary is taken to be constant
The well is situated in nodal point 42
29
In fig 12 the results of the simulation are compared with the
analytical solution The agreement between numerical results and
the analytical solution is excellent
draw- 15irS (m) down
analytical solution imdashx calculated values
distance from well r (m)
Fig 12 Comparison between analyt ical and calculated s teady-state
drawdown for the problem of f ig 11 with 1
-1 Q bull - 200 m 3 d a y _ 1
k = 10 mday
d = 100 m
S = 2 y
S = 0 s
512 Unsteady-state flow in an unconfined aquifer
Analytical solutions for unsteady-state flow in an unconfined
aquifer only exist if the aquifer extends infinitely For the
aquifer given in fig 11 this condition is not fulfilled However
during the first one or two days after starting the extraction the
influence of a constant hydraulic head at the outer boundary on the
30
drawdown near the well can be neglected (see fig 13) After one
day the lateral inflow is only 25 of the extraction the remainder
being the change in storage due to drawdown
2 0 0
m 3 day 1
Q=200m 3 day
100-
vraquomdash Lateral inflow
N amp storage
I I ^ V - W 10
t (days)
Fig 13 Values of Q lateral inflow and change in storage as
functions of time
The nonsteady-state or Theis equation can be written as
i _ Q s 4iTkD
dy = 4irkD
W(u) (23)
where
u = r2S
4irkDt
r = distance from well
S = storage coefficient
Q = constant discharge from well
D = thickness of the waterbearing layer
at the beginning of pumping
t = time after starting the extraction
s = corrected drawdown
= s - 322D
s = calculated drawdown
W(u) = Theiss well function
[J
CIAT1
[T]
]
31
In fig 14 the analytical unsteady-state solution for the case
described in section 511 is given together with the numerical
results at times 05 and 10 days The agreement between numerical
results and the analytical solution is rather good
Drawdown s1(m)
10
10
10
Theiss solution
x caculated values t=Q5
bdquo t= lO
10 10- 10 10 t r2(daym2)
Fig 14 Comparison between analytical and calculated steady-state
drawdown for the problem of fig 11 with
Q
k
d
S y
s s
=
=
=
=
=
200
1
10
3 m
0
0
2
0
-1 day
m day
m
These results indicate that the assumption of a linear variation
of h between two nodal points (see fig 3 ) even with the rather
rough element network of fig 11 is acceptable So there is no
need to improve for example the finite element network by introducing
higher order coordinate functions
32
52 F l o w t o w e l l i n 3-1 a y e r e d s y s t e m
In fig 15 a schematic cross-section of a 3-layered system of
two aquifers separated by an aquitard is given A well of
infinitisemal radius is completed in the lower aquifer and discharges
at a constant rate QTT Each layer is homogeneous isotropic and
extending radially infinite
bdquo bull - bull bull ~ i
mdash Q II
aquifer I
lugrave^i~IcircIuml7i-i~~ aquitard --r-rl~-
-II--I i - i
aquifer
Fig 15 Schematic cross-section of a 3-layered system
For this situation NEUMAN and WITHERSPOON (1969a) give rather
complicated analytical solutions In the special case where the
hydraulic properties of the two aquifers are identical they give
graphical illustrations of their solutions for three different
cases
These graphical solutions give the opportunity to verify
roughly numerical results of a 3-layered system Therefore a nodal
grid is superimposed on a circular shaped groundwater flow system
with the extraction in the centre of the lower aquifer and with
a constant head on the outer boundary Fig 16 gives the nodal
point situation in the x - y plane of 1 layer The nodes in the
other two layers are situated in the middle of the layers at the
same places in the x - y plane only their z-coordinate is different
33
A
-Jy
bull33
V 5
iumlff
Fig 16 Nodal point configuration for the 3-layered problem (top view)
34
In fig 17 numerical results are compared with the analytical
solution The agreement is rather bad especially for the two
aquifers But in terms of the water balance the difference between
aquifers and aquitard is not so big because the aquitard has
about 16 times bigger storage capacity than the aquifers The
small value of the specific storage of the aquifers is the reason
that small errors in the components of the water balance have a
big influence on the calculated values of the hydraulic head
Eig 17 Dimensionless drawdown vs dimensionless time in a
3-layered system with
II = 50 m day
3 m (distance from well)
aquifer I
aquitard
aquifer II
Thi (m)
100
25
100
S s
001
0642
001
k(mday )
1
0277
1
The discrepancies are partly caused by errors in discretization
In the z-direction these errors can be eliminated by taking the
35
vertical resistance of the aquitard to be infinite Then the
calculated drawdown in the pumped aquifer can be compared with
the analytical Theiss solution
This is done in fig 18 for different distances from the well
One sees that the agreement is better when the distance is bigger
Apparently the linearization of the logarithmic drawdown curve
near the well is responsible for the differences The differences
for r = 3 m are of the same magnitude as for the pumped aquifer in
the 3-layered case
joo gt A 50
20
Theiss solution
bull calculated values r=3m (distance f rom well) x bdquo bdquo r=7m
1 L
I l_ 001 003 0 0 5 Ol Q2
- I I 5 10 t - K t
Fig 18 Dimensionless drawdown vs dimensionless time in a
confined aquifer with
Q =500 m day
k = 10 mday
D = 100 m
S = 001 s
The differences between numerical results and analytical solutions
are not caused by taking a finite area because changing the outer
boundary into an impermeable boundary has only minor effects
36
VI SUMMARY AND CONCLUSIONS
In this report a numerical model for non-stationary saturated
groundwater flow in a multi-layered system has been treated
Groundwater flow can be described in different ways
a Physical-mathematical by combining Darcys law and the law of
continuity (section 21)
b Geo-hydrological the flow in a natural groundwater basin takes
place in layers with different hydraulic properties Definitions
of the most important hydraulic properties are given in section
22
c Schematical groundwater flows horizontally in aquifers and
vertically in aquitards (section 23)
d Approximating by dividing the region of flow into a finite
number of elements and applying Darcys law and the law of
continuity on each of these elements In these ways one gets
a set of equations which can be solved given (initial and)
boundary conditions (section 25)
e Numerical the set of equations has been solved using a Galerkin
type finite element method and the implicit Gauss-Seidel method
with overrelaxation factor (section 31 and 32)
f Computer-solvable the numerical solution has been translated
in a FORTRAN IV program called FEMSAT (section 33)
In chapter IV the input execution and output of the program
FEMSAT are treated
Finally in chapter V some numerical applications are given
simulation of flow to a well in unconfined aquifer and simulation
of flow to a well in a 3-layered configuration
Comparison of hydraulic heads obtained by numerical calculations
and analytical true solutions showed an excellent agreement in case
of stationary flow and a good to moderate agreement in case of non-
stationary flow However the diviations caused by the discretization
process must be seen in relation to the errors made in the
establishment of the hydraulic properties
37
As was stated in the introduction numerical models can be
valuable tools in the management of water resources Therefore
numerical models must be suitable to simulate effects of certain
operations upon the spreading of water over saturated groundwater
unsaturated groundwater and surface water The model described in
this report is a model for saturated groundwater flow The relation
with the surface water can be taken into account via radial and
entrance resistances or via a relation between flow to the surface
water system and the hydraulic head The relation with the unsaturated
zone has been restricted to the input factor effective precipitation
In practise however the unsaturated and saturated system are
interconnected Therefore the model should be improved by taking
into account the unsaturated zone This can be done in at least
two ways
a Replacing the unsaturated zone by a parametrical model
b Coupling the unsaturated zone in an iterative way with the
saturated system
In order to find out the usefulness of the numerical model for
water resources management it will be applied to actual regional
groundwater systemslaquoAlso sensitivity analyses will be carried out
The computer model itself can be improved by applying standard
programs eg automatical generation of a nodal grid and graphical
plotting of the results
38
VII REFERENCES
ERNST LF 1962 Grondwaterstromingen in de verzadigde zone en hun
berekening bij aanwezigheid van horizontale evenwijdige open
leidingen Thesis University Utrecht pp 189
1978 Drainage of undulating soils with high groundwater
tables (in press)
FORSYTHE GE and WR WASON 1960 Finite Difference Methods for
Partial Differential Equations John Wiley and Sons New York
pp 444
KRUSEMAN GP and NA DE RIDDER 1976 Analysis and evaluation of
pumping test data ILRI-bulletin No 11 pp 200
LAAT PJM DE and C VAN DEN AKKER 1976 Simulatiemodel voor grond-
waterstroming en verdamping in Modelonderzoek 1971-1974 ten
behoeve van de waterhuishouding in Gelderland deel II
Grondslagen pp 145-197
NEUMAN SP RA FEDDES and E BRESLER 1974 Finite element
simulation of flow in saturated -unsaturated soils considering
water uptake by plants Hydraulic Engineering Laboratory
Technion Israel pp 104
and PA WITHERSPOON 1969a Theory of Flow in a confined
two Aquifer System Water Resources Research Vol 5 No 4
pp 803-816
^_ and PA WITHERSPOON 1969b Applicability of Current Theories
of Flow in leaky aquifers Water Resources Research Vol 5
No 4 pp 817-829
PINDER GF and EO FRIND 1972 Application of Galerkins Procedure
to Aquifer Analysis Water Resources Research Vol 8 no 1
pp 108-120
REMSON I GE HORNBERGER and FJ MOLZ 1971 Numerical Methods
in Subsurface Hydrology John Wiley and Sons New York
pp 389
WESSELING JG 1977 Toepassing van Finite Difference- en Finite
Element Methoden voor het oplossen van een aantal grondwatershy
stromingsproblemen (met computerprogrammas) ICW-nota 944
pp 63
39
ZIENKIEWICZ OC 1971 Finite Element Method in Engineering Science
McGraw-Hill London pp 521
40
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APPENDIX 8 EXAMPLE OF OUTPUT
HEAD AND TERMS OF UATERBALANCE PER NODE PER LAYER
AT TIME 260 IN II AND I13DAYS
T
LAYER NODE KODE HEAD FLOUUN FLOUTS FLOUSS LEAKAGE STORAGE LAT FLOU
13 19 25 31 37 43
125008 125000 12S000 12S000 12S000 12S000 12S000 12S000
0000 0000 0000 0000 0000 0000 0000 0000
0000 0000 0000 0000 0000 0000 0000 0000
0000 0000 0000 0000 0000 0000 0000 0000
- 000 - 000 -000 - 000 - 000 - 000 - 000 0000
0000 0000 0000 0000 0000 0000 0000 0000
0000 000 000 000 000 000 000 000
000 0000 0000 0P100 0000 0000 0000 0000
1 7
13 19 25 31 37
12S000 125000 125000 125000 125000 12S000 125000 125000
0000 0000 0000 0000 0000 0000 0000 0000
- 001 -007 - 023 - 020 - 009 - 003 0000
0000 0000 0000 0000 0000 0000 0000 0000
000
001
007
023
020
009
003
003
0000 0000 0000 0000 0000 0000 0000 0000
1 7
13 19 25 31 37 43
125000 125000 124998 124972 124895 124787 124647 124358
0000 0000 0000 0000 0000 0000 0000 0000
- 000 001 007 023 020 009 003
0000
0000 0000 0000 0000 0000 0000 0000
50000
0000 182
1982 3862 1674 419 078 041
0000 0000 0000 0000 0000 0000 0000 0000
TERMS OF WATERBALANCE PER LAYER DURING LAST TIMESTEP AND
AND SINCE BEGINNING IN M3DAY M3
FLOU TO UNSATURATED ZONE
FLOU TO TERTIARY SYSTEM
0000
0000
0000
0000
LAYER NO 1
FLOU TO SECONDARY SYSTEM
FLOU TO AD3ACENT LAYERCS)
ARTIFICIAL FLOU AND PRESCRIBED BOUNDARY FLOU
ACCUMULATION TERMS
LATERAL FLOU THROUGH PRESCR HEAD BOUNDARY
0000
-000
0000
000
000
0000
-000
0000
000
000
LAYER NO 2
FLOU TO SECONDARY SYSTEM
FLOU TO AD3ACENT LAYERCS1
ARTIFICIAL FLOU AND PRESCRIBED BOUNDARY FLOU
ACCUMULATION TERMS
LATERAL FLOU THROUGH PRESCR HEAD BOUNDARY
0000
-378
0000
381
0000
0000
-055
0000
056
0000
LAYER NO 3
FLOU TO SECONDARY SYSTEM
FLOU TO AD3ACENT LAYERCS5
ARTIFICIAL FLOU AND PRESCRIBED BOUNDARY FLOU
ACCUMULATION TERMS
LATERAL FLOU THROUGH PRESCR HEAD BOUNDARY
0000
370
50000
49376
0000
0000
055
12979
12909
0000
The coefficient k in the Darcy flow equation is taken to be
a constant depending on both the properties of the porous medium
and the fluid
212 The law of continuity
The other important physical law states that no matter can be
lost or created
The conservation equation can be found by applying the principle
of continuity to an infinite small volume
6v 6v lt5v 60 x y z TT = T + x + ~T7 + 1 (2)
Ot OX OcircX oz
where
0 = volume of water per unit bulk volume of soil |_-J
q = sink or source term T J
213 Combination of Darcys law and the law of continuity
Darcys law combined with the law of continuity gives
60 lt5 oh lt5 ocirch 6 oh _ ocirct ocircx x ocircx ocircy y oy ocircz z ocircz
or
c 5h 6 ocirch ocirc Ocirch 6 ocirch c ot ox x ocircx ocircy y oy laquoz z 6z
where
S = volume of water released or stored per unit bulk c I
volume of soil per unit change in hydraulic head |L I
22 H y d r a u l i c p r o p e r t i e s a n d t y p e s o f
a q u i f e r s
The flow in a natural groundwater basin takes place in layers
with different hydraulic properties Usually the layers are divided
into good permeable layers or waterbearing layers (aquifers) and
less permeable layers (aquitards)
The most important hydraulic properties are
a Transmissiviy - the product of the average hydraulic conductivity
k and the saturated vertical thickness d of the waterbearing
layer
It is a measure for the transport capacity of aquifers
Notation T = kd |_ L T ~
b Hydraulic resistance - the ratio of vertical thickness of
aquitards d and hydraulic conductivity k It is a measure for
the resistance capacity of aquitards denoted as c = d k _ T J
c Specific yield - the volume of water released or stored in the
phreatic zone per unit surface area of an unconfined aquifer per
unit change in hydraulic head
It is also called effective porosity
The symbol used for specific storage will be S j_-J
d Specific storage - the volume of water released or stored per
unit volume of the saturated parts of an aquifer or aquitard per
unit change in hydraulic head It is a measure for the elasticity
of the soil material and the fluid Excluded are irreversible
processes Notation S _ L J
e Storage coefficient - the volume of water released or stored
pe7 nit surface of a layer per unit change in hydraulic head
It is the sum of specific yield and the product of specific
storage and thickness of the aquifer or aquitard
S = S + S d r - 1 c y s - -1
Note in confined layers S = 0 y
In fig 1 a 3-layered configuration is given of two aquifers
separated by an aquitard Aquifer I has a free water surface and is
an unconfined aquifer Aquifer II is a completely saturated aquifer
Dependent on the hydraulic resistance of the aquitard KRUSEMAN and
DE RIDDER (1976) distinguish between confined semi-confined and
semi-unconfined aquifers For our purpose it is not useful to make
this distinction We restrict the aquifer types to unconfined and
confined aquifers
WIHtUtttWttllttlt(ltmtttltHlttltlil(lltlittlfllnl ) bull (
d V U i l i l i ii i Hi il i M M bull f i m
i tMi| i i | J i j j i _ I I bull
aq ui tard
aquifer II
Fig 1 Schematic representatipn of flow direction and variation
of hydraulic head h in a 3-rlayered system
In an unconfined aquifers ( I ) poundhe vertical thickness pf the
waterbearing layer d is a variable while in a confined aquifer II
this is a constant (denoted as D ) Other important properties of
unconfined aquifers are the hydraulic head being the same as the
phreatic surface and the specific yield being different from zero
23 S c h eacute m a t i s a t i o n o f f l o w
The groundwater flow in a groundwater basin is schematized
into a horizontal flow in aquifers and into a vertical flow in
aquitards Consequently the vertical variation of the hydraulic
head is as drawn in figl
The validity of this assumption has )een investigated by
NEUMAN and WITHERSPOON (1969) The found that if the contrast
in hydraulic conductivity between two adjacent layers is bigger
than a factor two the relative error caused by this assumption
is usually smaller than 5
This schematization gives a simplification of the general flow
equation (eq 4 ) In an aquifer the vertical is zero and eq (4)
then reduces to
-Sbdquo -r- 7 - (k d -r-) + 7 mdash (k d 7-) + qd c lt5t ox x ox oy y oy
or
bull(S + S d) g y s ot
-r- (k d -=-) + -T- (k d -=-) + Q ocircx x ox oy y ocircy
(5)
where
Q = volume of water substracted from or added to unit surface
of aquifer per unit time |_ L T J
In an aquitard eq (4) reduces to
-sQ |pound = (k Q) s 6t oz Z OcircZ
(6)
As will be discussed in the next paragraph the term q in an
aquitard is set equal to zero
24 W a t e r b a l a n c e t e r m s
In eq (4) the term q is mentioned With this sink or source
term different terms of the water balance are taken into account
Fig 2 depicts schematically the situation for an unconfined aquifer
(for other layers some terms dont exist)
recharge- - discharge
laquo-b
Fig 2 Schematic representation of the various terms involved in the
water balance
Description of the different terms
a The flow through the phreatic surface q f
If this term is negative (water goes out of the saturated system)
it is called capillary rise If positive it is called percolation
(or drainage)
This flow can be calculated as a rest term of the water balance
of the unsaturated zone Very often this term is expressed as a
flow per unit surface (flux) and is called effective precipitation
For the time being this flux is a given input
b Flow to adjacent aquitards q
The flux to adjacent aquitards can be calculated by applying eq
(6) to the aquitard
c Flow to the primary surface water system q
The primary surface water system penetrates fully the groundwater
basin So it acts as a boundary with given values for the
hydraulic head (is equal to the phreatic level of the primary
system)
d Flow to the secondary surface water system q
The secondary surface water system does not penetrate fully the
groundwater basin which causes a resistance against flow between
groundwater and surface water The flow q can be calculated as
follows
h - h
- mdash f i i gt
where
h = hydraulic head of the groundwater at the place of
the secondary system |_ L |
h = phreatic level of secondary system |_ L J i s -
R = radial and entrance resistance of the
secondary system per unit surface |_ T J
According to ERNST (1962) the radial resistance H per unit
surface can be caKulated with
Uuml = 4 i n 4 e8) TTk B
where
1 = area drained by 1 m conduit L ^ J
d = thickness of the layer near the conduit _ L J
B = wet perimeter of the conduit |_ L J
The entrance resistance is caused by a layer which shows a
lower conductivity around the conduit than its surroundings
In practise radial and entrance resistancies are difficult to
deduct from hydraulic properties They can be found indirectly
by measuring simultaneously q h and h
t s
e Flow to the tertiary surface water system q
In smaller conduits usually the phreatic level is not known
So eq (7) cant be applied The flow to these conduits however
is sometimes very important (eg in areas with shallow groundwater
tables) One may a-sume (ERNST 1978) that a relationship exists
between the hydraulic head of an aquifor and the flow q
According to Ernst the tertiary system can be divided into a
number of categories with respect to their mutual distance and
bottom height For each category the flow is h - h (b)
q ( h b ) - mdash (9)
where
h = average hydraulic head | L J
h (b) = phreatic level in the category under
consideration a function of bottom height |_L J
a = geometry factor to convert h into h _ - J
Y = drainage resistance _ T J
hm = hydraulic head midway between two
conduits |_ L J m
Summation of the q(hb)-relations of the different categories
yields the overall relation between hydraulic head and flow to
the tertiary system we are looking for
f Artificial recharge or discharge q and prescribed boundary
flux qfo
The magnitudes of these flows are not influenced by the conditions
inside the groundwater basin but remain on a prescribed level
Thus q is an internal boundary condition q an external boundary cl D
condition
For confined aquifers some of the above mentioned terms dont
exist These terms are the flow through the phreatic surface qf
and the flow to the tertiary surface water system q
Aquitards are considered only as transmitters of water from one
aquifer to another with possibilities of storage (eq 6 ) This means
that for aquitards the water balance terms ar through f are set
equal to zero
25 S o l u t i o n o f g r o u n d w a t e r f l o w p r o b l e m s
The basic equations for saturated groundwater flow are eq (5)
for aquifers and eq (6) for aquitards The solution of these
equations require knowledge of the physical system and the auxiliary
conditions describing the system constraints
These auxiliary conditions are (REMSON etal 1971)
a Geometry of the system
Eg the diviation of the system into a restricted number of
layers and the horizontal extension of the system under
consideration
b The matrix of hydraulic properties including in-homogeneity
and an-isotropy
c Initial conditions Because we are dealing with non-stationary
flow the values of pertinent system variables (such as hydraulic
head at the initial time) must be known
d Boundary conditions They describe the conditions at the boundaries
pf the system as a function of time
One can distinguish between
- hydraulic head boundary
- flow (including zero flow) boundary
- both head and flow boundary
After specifying the auxiliary conditions an unique solution of
the variation of the hydraulic head in space and time can be
obtained
For special shapes of flow and assumptions exact analytical
solutions exist In nature however one is usually confronted with
problems like
- spacial variation in the hydraulic properties of the soil
- irregular shape of the groundwater basin
- empirical relations between q and h
- d not being a constant
For such situations by numerical and analogy models only an
approximation of the actual situation can be obtained In the next
chapter a numerical solution called the finite element method is
treated
III NUMERICAL SOLUTIONS
In this chapter the numerical solution of the aquifer equation
(eq 5) and the aquitard equation (eq 6) will be treated The
method of solving is the finite element method In section 31 only
a short mathematical description of this method will be given For
more mathematical background the reader is referred to publications
on this subject (eg ZIENKIEWICZ 1971 NEUMAN etal 1974 and
WESSELING 1976)
In section 32 the application of the finite element method on
the flow in groundwater basins is treated while in section 33 the
translation into a computer program is given
3 1 F i n i t e e l e m e n t m e t h o d
For reasons of simplicity the finite element method will be
explained for the stationary version of eq 5 ie
10
(k d^) + (k d ^ ) - Q = O (10) ocircx x 6x 6y y Sy
The basic idea of each numerical solution of eq (10) is to
apply this equation on a small but finite volume The flow region
of interest R is divided into a finite number of volumes called
elements The elements can have different shapes but for reasons
of conveniency we only use triangles- Quadrilaterals can also be
used because they can be divided into two triangles The sizes of
the triangles may vary They can be adapted according to the
geometry of the region and the accuracy desired (fig 11 gives an
example of the diviation of a region into triangles and quadrilaterals)
The corner points of the elements represent the nodal points
If there are N nodal points then the continuous variation of h
in region R is approximated by the expression
h (x_v) ~ h f (xy) (11) a n n
where
h (AV) = approximation of u (xy) a
h = value of h in nodal point n n
f ltxgty) = global coordinate function
In nodal point n a must be equal to h so f (x y ) = 1 and a n n n n
f (xy) = 0 i n Between f (x v ) and f (xy) of the nodal n j ] n n n n j j
points surrounding nodal point n f (xy) varies linearly from
1 to 0
In fig 3 a picture of f (xy) is given n
If one knows the values h in the nodal points n we know the approximation h of h We can find these values of b bv using for
a n
example the Galerkin method
Galerkins principle states that an approximate solution of h in
eq (10) can be obtained from the following system of equations
N ~~ (k d -A + -^- (k d T-y E h f (xy) f(xy)dxdy +
-ox x ox ocircy y oy - _ n n i R n~
J J Q f(xy) dx dy = 0 i = 1 N (12)
1 1
Fig 3 The global coordinate function f (xy)
The integration can be performed element by element with the
aid of the coordinate functions f (NEUMAN etal 1974)
For non-stationary problems it is not possible to apply the
Galerkin principle on the time derivative But it is possible
to discretize this derivative and to set up the system of equation
(eq 12) for each finite time step
One obtains in this way a series of sets of laquocriations tha s
an approximation of the non-stationary flow problem
Treatment of boundary conditions
a Initial conditions
In non-stationary flow problern gtne must start with given
values of a at time t = t ri c
b Boundary conditions
- Prescribed head boundary
Nodal points which lie on a prescribed head boundary have a
prescribed value for h bull r Hgte r umber of unknown h s is n n
reduced and accordingly no the number of equations
- Prescribed flux through z-- boundary
If one knows the flux (iuumlr jw or outflow per unit area) and
the area of the boundary oi-r element the inflow or outflow
through the boundary per flemlt=nt can be calculated
One may assume this flow to J bull part of the sink term Q in
eq (12)
12
32 A p p l i c a t i o n o f t h e f i n i t e e l e m e n t
m e t h o d o n t h e f l o w i n g r o u n d w a t e r
b a s i n s
From hydrogeological investigations the geometry and the matrix
of the hydraulic properties are known Knowing the geometry the
flow can be split up into a finite number of layers This holds
also for the nature of the boundaries
Now a nodal grid is superimposed on each layer in such a way
that the nodal points of the different layers are situated right
above each other (see fig 4) and in each layer the nodal point
lies in the middle of this layer
-nodal point
Fig 4 Schematic representation of the nodal points in the
xyz-space
Each layer has N nodal points of which P nodal points have a
prescribed head In each layer remain N-P = N nodal points with
unknown head Eq (12) can be applied to each point with unknown
head in each separate layer
As was stated in the last paragraph for each time step one
can write
13
Ah bdquo ~ -r1 + Q = O in = 12N in n m At yi A h + B (13)
where
A = in
ocircf ocircf ocircf of (k d f mdashri- ~ + k df - ~ -jS) dxdy
J x n ox ox y n oy oy
in (S d + S ) ff dx dy O if i icirc n
s y i n
Q f dx dy
The matrices A B and the vector Q have been defined in in l
elsewhere (NEUMAN etal 1974) A is the conductivity matrix in
B the storage matrix and Q the source vector in l
In order to clarify these rather complicate expressions apply
equation (13) poundo point 4 of fig 5
6 7
Fig 5 Top view of part of the nodal grid
lhl + A42h2 + A43h3 + A44h4 + A45h5 + A46h6 + A47h7 +
h - h
+ B _iuumli i ^ + Q = o 44 t - t x4
n n-1
(14)
where h and h are values of h at time t and t 4n 4n-1 4 n n-1
respectively
14
If the h-values belonging to the A-values are taken at time
t = t we get an equation with one unknown h for time t = t n-1 4n n
The equation can then be solved directly
However this is an explicit formulation which puts strong limitations
on the magnitude of the time step t - t (REMSON etal 1971) n n-1
Values larger than a certain value cause instability With an implicit method the values of h are taken at time
t = t This method is unconditionally stable Now one gets an equation n
with 7 unknowns h through h If one has 7 equations for every 1 n n
point 1 through 7 we can solve the values h i = 17
in
Of all the methods possible the iterative method of Gauss-Seidel
has been chosen The Gauss-Seidel method starts by choosing a first estimate of h at time t = t (usually equal to h ) At each in n j -i in-l iteration step a new value of h is calculated using the last
in calculated values of h If one passes the nodal points from 1 to
i n 7 eq (14) becomes
h-j + 1 = A h^ + 1 + A bdquo h | + 1 + A _ h | + 1 + A h j + A r h j + 4 n 4 1 1 n 4 2 2 n 43 3 n 45 5 n 4 6 6 n
B h B + A h J + - 4 4 4 n ~ + 0 J (A + mdash) A 4 7 h 7 n At n U 4 4 At
(15)
where
h-i = in
j = iteration step or number of iteration
i = number of nodal point n = time = t
n
The iteration process converges if the successive differences k+1 k
between h and h become continuously smaller in in
The iteration process is stopped if at each nodal point the k+1 k
difference between h and h is smaller than a certain prescribed in in
value
15
To accelerate the convergence an overrelaxation factor W is used
(for more information about W see FORSYTHE and WASOW 1960)
k+1 k T7 k+1 k h = h + W (h - h )
in in in in (16)
If W = 1 there is no overrelaxation Usually W varies between 1 and
2 and the optimum value very often is found by trial and error
Eq (15) is valid for points in aquifers The connection with
adjacent layers is made by the term Q because in this term the
flow from adjacent aquitards q is included (two aquifers are always
separated by an aquitard)
Aquitards
We dont apply the finite element method on points in aquitards
As stated before there is no horizontal flow in aquitards So it has
no sense to replace the continuous variation of h in the aquitard
by equation (11)
The nodal points in aquitards represent a certain volume of an
aquitard hich transmit water from one aquifer to another and which
has storage capacity The nodal points are included in the Gauss-
Seidel scheme -s follows (see fig 6 )
1 aauifer
a -_vS laquo 2 [~^-t^yM^^^-^r~ aqutarci l-Z
aquifer
Fig 6 Schematic situation for points in aquitards
Point 2 is situated in the middle of the aquitard with thickness
d storage coefficient S and vertical hvdraulic conductivity k c J v
Suppose the Gauss-Seidel iteration starts in the upper layer and
ends in the bottom-layer Then the flux from point 1 in aquifer 1 to
16
point 2 in the aquitard at time t = t and in iteration step j + 1
is
k q = (hf -hf )-iumlmdash f (17)
1 n 2 n OSxd1
and the flux from point 3 in aquifer 2 to point 2 in the aquitard
is
1 c 1
qbdquo = (h - hf) mdashHmdashr- (18) 2 3 n 2 n OSxd
The net flux ltl-q9 must be equal to the change in storage per unit
area per unit time This change in storage can numerically be
approximated by
te - h-
n n-1
Combina t ion of ( 1 7 ) (18) ana (19) g i v e s
l tr1 - h (hJ + 1 + hj - 2 hj+1) - - - - - - S1 - ilaquoS L u l l = o (20)
] i 3 n 2 n 1 c t - t j j x d n n -1
T + 1
From t h i s e q u a t i o n h^ kan be s o l v e d
3 3 C o m p u t e r o g r a m
The problem outl^ugrave so far has been translated into a computer
program written in FORTRAN IV
The flow chart ot Uis program is given in fig 7 whereas in
appendix A the complete listing of the program can be found
In the flow chart the 5 main parts of the program are
indicated
A Reading and printing of input data
B Determination of the matrices A and B and the vector Q per layer
17
M
1 A
FLOH to unsaturated zono
FLOW to tertiary syster
FLOW to secondnrv system
Leakage to adjacent aquitards
SOLVE equation (14)
yes
CALCULATE flux through constant head boundnr
CALCULATE waterbalance n p r 1 OAro r r J j - ~
PRINT of output
Ier node per layer
k = k+1
t = t + At
7 DUMP variables on tape
-gt-
( STOP )
Fig 7 Flow chart of the program FEMSAT with diviation in 5 main
parts A B C D and E
C START PROGRAM FEMSAT
f READ general information
PRINT general information
f READ element information
PRINT element information 7 READ stationary nodal point information
PRINT stationary nodal point information 7
READ non-stationary nodal point information for t = t
FRINT non-stationary nodal point information for t = t
L 2 (
^ yes R E A D non-stat input for t gt t
J PRINT non-stat input for t gt t (- 1 -- M degY
GENERATE A B and AREA-matrix for confined aquifers
_L
GENERATE A B and AREA-matrix for unconfined aouifer
bull
C Solving h with Gauss-Seidel
D Determination of the water balance terms per layer
E Printing of results
The program exists of a main program with several subroutines
The main program calls the subroutines In the subroutines most
calculations are performed
In this way an easy adaptation to different problems is
obtained (see also section 44) Comments in the program will
help the reader to understand the program
In the next chapter the preparation of the input data executing
of the program and the output will be treated
Acknowledgement
Some parts of the computer program were taken from the program
UNSAT2 of NEUMAN etal (1974)
IV INPUT EXECUTION AND OUTPUT
41 I n p u t
The input is divided into endogenous variables (chosen by the
user or calculated by the program during execution) and exogenous
variables (based on physical data)
411 Endogenous variables
Translation of groundwater basin into a nodal point network
Probably the most important but also the most difficult stage
in the process of simulating groundwater flow is the translation
of the natural groundwater basin into a nodal point network Questions
which arise are
a In how many layer the geo-hydrological system mustcan be
schematized
b The locations of the boundaries and the nature of these boundaries
(impervious prescribed head prescribed flux)
c The configuration of the nodal points
19
The answers of these questions depend on the problem under
consideration
- Schematization into layers
Usually our knowledge about the geo-hydrological situation is
such that no more than 5 different layers can be distinguished
(remember contrast rule)
- Boundary conditions
The top boundary condition of a groundwater basin usually is
a prescribed flux (effective precipitation)
The bottom boundary condition usually is a prescribed flux
boundary with flux zero (impervious hydrological base)
For the boundaries at the other sides of the system one sees
either prescribed head boundaries (eg a fully penetrating
river) or prescribed flux boundaries (including zero flux)
- Nodal point configuration
Nodal points are situated
on the boundaries
along internal boundaries which indicate changes in
hydraulic properties within the layer
along the tertiary surface water system
on places where artificial recharge or discharge wiii tgtke
place
elsewhere
Construction of the finite element mesh
Assume we have schemetized the groundwater basin iatu a finite
number of layers with different direction of flow (horizontal and
vertical) Within one layer the hydrological properties may vary
from place to place
Now a nodal grid is superimposed on each layer in a way already
described in section 32 Thus each layer has the same number of
nodal points situated on the same places in the horizontal plane
but with different z-coordinates
We are free to construct quadrilaterals or triangles because
the program automatically divides a quadrilateral into two triangles
having identical material properties The dimensions of the elements
20
should be small at places where large changes in hydrological
properties occur or where large gradients in hydraulic head are
expected to occur
Coding of nodal points
Each node in the finite element network is assigned an integer
code which indicates the type of calculation for this node
The coding is as follows
- Code = 1 nodal points with described head as function of time
- Code 2 internal nodal points
- Code = 3 internal nodal points which represent secondary surface
water system
- Code = 5 nodal points with prescribed artificial recharge or
discharge and points with prescribed boundary flow
(both as functions of time)
Other important endogenous variables are the magnitude of the
time step the value of the variable which stops the iteration
process and the maximum number of iterations allowed per time step
(see also group A and B of section 413)
412 Exogenous variables
After constructing the nodal grid each node represents part of
the groundwater basin Geo-hydrological properties can be assigned
to each mode based on physical data (group C D and E of section
413) In group F of section 413 time-dependent variables can
be assigned to each node
413 Data input to Program FEMSAT
The data input must be punched on cards according to the
following instructions
GROUP COLUMNS FORMAT SYMBOL DESCRIPTION
A 1-80 20A4 HEAD Desired heading to be printed at
the beginning of the execution
1-5 15 NUMNP number of nodal points
21
GROUP COLUMNS FORMAT SYMBOL DESCRIPTION
6-10 15
11-15 15
16-20 15
21-25 15
26-30 15
31-35 15
NUMEL
NUMLAY
MBAND
NTSPI
NTSPO
MAXIT
1-10 F103 ST
11-20 F103 DELT
21-30 F103 TMAX
31-40 F105 TOL
number of elements
number of layers
maximum difference in node number
between two adjacent nodes
number of time steps between a
change in non-stationary input data
number of time steps between two
successive outputs
maximum number of iterations per
time step
time to start with the execution
initial time interval
maximum time to be reached before
execution is stopped
maximum allowed absolute change in
the values of h between any two
successive iterations in a given
time step
Iterations continue until all changes
are less than or equal to TOL or
until MAXIT is exceeded
1- 5 15
6-10 15
11-15 15
16-20 15
KX(N1)
KX(N2)
KX(N3)
KX(N4)
sequential number of i-th corner of
element N
sequential number of j-th corner
sequential number of k-th corner
sequential number of 1-th corner
(see fig 8)
Fig 8 Numbering of element corner nodes
22
GROUP COLUMNS FORMAcircT SYMBOL DESCRIPTION
One card must be provided for each element in sequential order
starting with N = 1 and ending with N = NVJMEL
I-K 1103 SANG(NLY)
1i-iO MO3 C1(NLY)
21-30 F 103 C2(NLY)
21-4- 1-104 THI(NLY)
angle in degrees between CI and
the laquo-coordinate to he used in
nodal print N in layer LY (see
poundig 9)
First principal conductivity of
material in node N and layer LY
idem second principal conductivity
thickness of layer LY at nodal
point N
ig V Definition of SANGCI and C2
One card icircuiPt be provided for each model point ctarting wih M = 1
and ending with N = NUMNP If there is norlt3 than 1 layer for each
layer a set must be provided starting with the upper layer LY - 1
and ending with the bottom layer LY = NIJMLAY The variables SANG
M and 2 are not relevant tor aquitards but for reasons of
conveniency this procedure is maintained (making SANG CI and C2
dummy variables)
1-10 110 KODE(N)
11-20 II0 NTERR(N)
coding of nodal points (see
paragraph 42)
number of tertiary relation in node N
The q-h relationship of the flow to
23
GROUP COLUMNS FORMAT SYMBOL DESCRIPTION
21-30 FlOl X(N)
31-40 FlOl Y(N)
41-50 FlOl HGL(N)
51-60 FlOl HBSC(N)
61-70 FlOl LESC(N)
the tertiary system as discussed
in par 24 are schematized in a small
number of different relations
x-coardinate of node N
y-coordinate of node N
height ground level in node N
height bottom secondary conduit in
node N If node N is not a secondary
conduit this variable is a dummy
variable
length of secondary conduit in area
represented by node N
One card must be provided for each node starting with N = 1 and
ending with N = NUMNP
E 1-10 FlOl P(NLY1) initial value of h at node N in
layer LY
11-20 FlOl VERRES(NLY) vertical resistance at node N and
layer LY If the layer is an aquifer
this value is a dummy variable
21-30 F101 RADRES(NLY) radial resistance for flow to
secondary system at node N in layer
LY
31-40 FlOl ENRES(NLY) entrance resistance for flow to
secondary system at node N in layer
LY
41-50 FlOl WP(NLY) wet perimeter of secondary system at
node N in layer LY
51-60 FlOl P0R(NLY) specific yield of the material at
node N in layer LY
61-70 FlOl SS(NLY) specific storage of the material
at node N in layer LY
24
GROUP COLUMNS FORMAT SYMBOL DESCRIPTION
One card must be provided for each node starting with N = 1 and
ending with N = NUMNP If there is more than 1 layer for each layer
a set of cards must be provided starting with LY = 1 and ending
with LY = NUMLAY
F 1-10 F101 P(NLY1) value of h at different times at
node N in layer LY Node N lies
on a prescribed head boundary
11-20 F101 WTSC(N) water table in secondary system at
node N at different times
21-30 F 101 ARRED(NLY) artificial recharge to or discharge
from node N in layer LY or
prescribed boundary flow at
different times
31-40 F 101 QPHR(N) flux through phreatic surface at
node N at different times
The data in group F are non-stationary input data One card must be
provided for each node starting with N = 1 and ending with N = NUMNP
In case of more than 1 layer for each layer a set must be provided
Not relevant variables are taken as dummy variables
After a certain number of time steps (NTSPI) the data of group E
are read So the set cards E must contain NxLYxC cards where C =
(TMAXNTSPIXDELT) + 1 if DELT is a constant
42 R u n n i n g t h e p r o g r a m o n t h e C y b e r 7200
The program has been executed on a Cyber 7200 computer of
IWIS-TNO in the Hague
- The demanded input is punched on cards according to the instructions
of the last paragraph and is put on a permanent file
- For each specific problem inthe program only the dimension statements
in the main program have to be adjusted according to the instructions
in the lines of comment
- After compilation the program together with the input file is
executed
25
- Intermediate data necessary for continuation of the calculations
are saved on another permanent file This gives the opportunity
to restart the program at a time the calculation was stopped
The only change in the input file is to adjust the starting time
Remark
With the aid of a terminal small changes in the program and in the
input file can be made easily and the output immediately appears
on the terminal In this way it is possible to stop erroneous
calculations This is especially useful when being in the testing
phase of the program
43 O u t p u t
Immediately after reading the input data they are printed so
that possible errors can be detected During execution after each
certain number of time steps (NTSPO) calculated values of h and
water balance terms are printed In appendix B an example is given
The output data can be devided into two groups
A Data per node per layer
- Coding of the nodal points KODE
- Hydraulic head HEAD
- Flow through phreatic surface at the time indicated in head
of table FLOWUN
- Flow to tertiary surface water system FLOWTS
- Flow to secondary surface water system FLOWSS
- Flow to adjacent layers LEAKAGE
- Artificial recharge or discharge and prescribed boundary
flow ARRED
- Change in storage during last time step STORAGE
- Flow through prescribed head boundary LATFLOW
B Data per layer
Summation of the terms already mentioned under A gives the
magnitude of the water balance terms per layer at the time
indicated (instanteneous water balance terms) Summation of
these instanteneous terms multiplied by their respective time
intervals since time is zero gives the cumulative water balance
terms per layer
26
After each print out we are able to check the calculations
because the sum of the water balance terms per layer must be
approximately zero
44 C o m p u t i n g t i m e
The computing time depends on the number of nodal points and
the number of layers But it depends also on the magnitude of the
changes in hydraulic head and on the value which terminates the
Gauss-Seidel iteration (TOL) To give an impression TOL = 001 m
Costs per day simulation (in hlfl) on the Cyber 7200
number of nodes number of layer bull bdquo bull bull _ 1mdash x a x b (21)
where
a = costs in Dutch guilders per system second (ƒ 027)
b = empirical factor with magnitude about 001
V NUMERICAL EXPERIMENTS
In general there are two possibilities to test the validity of
numerical models The first and most commonly used method is to
compare numerical results with analytical solutions The second
possibility is to compare numerical results with measured field
data Since the last alternative requires many data that are
difficult to obtain we restrict ourselves for the time being to
comparison with the analytical solutions
5 1 F l o w t o a w e l l i n a n u n c o n f i n e d
c i r c u l a r - s h a p e d a q u i f e r
Flow to a well can be divided into steady-state flow and
unsteady-state flow (KRUSEMAN and DE RIDDER 1976)
27
511 Steady state flow in an unconfined aquifer
For this case the formula of Thiem-Dupuit is valid (see fig 10)
Q bull k lth2-hicircgt ln(r2r])
(22)
where
hhbdquo = hydraulic head in 1 and 2 respectively
Q = discharge from well
rbdquo = distance from well
CL31
hmdash^ri --r2
Fig 10 Schematic cross-section of a pumped unconfined aquifer
Fig 11 gives the nodal grid superimposed on a well in a
circular shaped aquifer Near the well the nodal point distance
is made smaller Quadrilaterals as well as triangles are used
Numerical calculations with this finite element network and 3 -1
an extraction rate of 200 m day were performed for 10 days
At that moment a steady-state situation was almost achieved
28
2tmdashnodal point number
160 180 X-axesim)
Fig 11 Nodal point configuration for flow to a well in a circular-
shaped aquifer The outer boundary is taken to be constant
The well is situated in nodal point 42
29
In fig 12 the results of the simulation are compared with the
analytical solution The agreement between numerical results and
the analytical solution is excellent
draw- 15irS (m) down
analytical solution imdashx calculated values
distance from well r (m)
Fig 12 Comparison between analyt ical and calculated s teady-state
drawdown for the problem of f ig 11 with 1
-1 Q bull - 200 m 3 d a y _ 1
k = 10 mday
d = 100 m
S = 2 y
S = 0 s
512 Unsteady-state flow in an unconfined aquifer
Analytical solutions for unsteady-state flow in an unconfined
aquifer only exist if the aquifer extends infinitely For the
aquifer given in fig 11 this condition is not fulfilled However
during the first one or two days after starting the extraction the
influence of a constant hydraulic head at the outer boundary on the
30
drawdown near the well can be neglected (see fig 13) After one
day the lateral inflow is only 25 of the extraction the remainder
being the change in storage due to drawdown
2 0 0
m 3 day 1
Q=200m 3 day
100-
vraquomdash Lateral inflow
N amp storage
I I ^ V - W 10
t (days)
Fig 13 Values of Q lateral inflow and change in storage as
functions of time
The nonsteady-state or Theis equation can be written as
i _ Q s 4iTkD
dy = 4irkD
W(u) (23)
where
u = r2S
4irkDt
r = distance from well
S = storage coefficient
Q = constant discharge from well
D = thickness of the waterbearing layer
at the beginning of pumping
t = time after starting the extraction
s = corrected drawdown
= s - 322D
s = calculated drawdown
W(u) = Theiss well function
[J
CIAT1
[T]
]
31
In fig 14 the analytical unsteady-state solution for the case
described in section 511 is given together with the numerical
results at times 05 and 10 days The agreement between numerical
results and the analytical solution is rather good
Drawdown s1(m)
10
10
10
Theiss solution
x caculated values t=Q5
bdquo t= lO
10 10- 10 10 t r2(daym2)
Fig 14 Comparison between analytical and calculated steady-state
drawdown for the problem of fig 11 with
Q
k
d
S y
s s
=
=
=
=
=
200
1
10
3 m
0
0
2
0
-1 day
m day
m
These results indicate that the assumption of a linear variation
of h between two nodal points (see fig 3 ) even with the rather
rough element network of fig 11 is acceptable So there is no
need to improve for example the finite element network by introducing
higher order coordinate functions
32
52 F l o w t o w e l l i n 3-1 a y e r e d s y s t e m
In fig 15 a schematic cross-section of a 3-layered system of
two aquifers separated by an aquitard is given A well of
infinitisemal radius is completed in the lower aquifer and discharges
at a constant rate QTT Each layer is homogeneous isotropic and
extending radially infinite
bdquo bull - bull bull ~ i
mdash Q II
aquifer I
lugrave^i~IcircIuml7i-i~~ aquitard --r-rl~-
-II--I i - i
aquifer
Fig 15 Schematic cross-section of a 3-layered system
For this situation NEUMAN and WITHERSPOON (1969a) give rather
complicated analytical solutions In the special case where the
hydraulic properties of the two aquifers are identical they give
graphical illustrations of their solutions for three different
cases
These graphical solutions give the opportunity to verify
roughly numerical results of a 3-layered system Therefore a nodal
grid is superimposed on a circular shaped groundwater flow system
with the extraction in the centre of the lower aquifer and with
a constant head on the outer boundary Fig 16 gives the nodal
point situation in the x - y plane of 1 layer The nodes in the
other two layers are situated in the middle of the layers at the
same places in the x - y plane only their z-coordinate is different
33
A
-Jy
bull33
V 5
iumlff
Fig 16 Nodal point configuration for the 3-layered problem (top view)
34
In fig 17 numerical results are compared with the analytical
solution The agreement is rather bad especially for the two
aquifers But in terms of the water balance the difference between
aquifers and aquitard is not so big because the aquitard has
about 16 times bigger storage capacity than the aquifers The
small value of the specific storage of the aquifers is the reason
that small errors in the components of the water balance have a
big influence on the calculated values of the hydraulic head
Eig 17 Dimensionless drawdown vs dimensionless time in a
3-layered system with
II = 50 m day
3 m (distance from well)
aquifer I
aquitard
aquifer II
Thi (m)
100
25
100
S s
001
0642
001
k(mday )
1
0277
1
The discrepancies are partly caused by errors in discretization
In the z-direction these errors can be eliminated by taking the
35
vertical resistance of the aquitard to be infinite Then the
calculated drawdown in the pumped aquifer can be compared with
the analytical Theiss solution
This is done in fig 18 for different distances from the well
One sees that the agreement is better when the distance is bigger
Apparently the linearization of the logarithmic drawdown curve
near the well is responsible for the differences The differences
for r = 3 m are of the same magnitude as for the pumped aquifer in
the 3-layered case
joo gt A 50
20
Theiss solution
bull calculated values r=3m (distance f rom well) x bdquo bdquo r=7m
1 L
I l_ 001 003 0 0 5 Ol Q2
- I I 5 10 t - K t
Fig 18 Dimensionless drawdown vs dimensionless time in a
confined aquifer with
Q =500 m day
k = 10 mday
D = 100 m
S = 001 s
The differences between numerical results and analytical solutions
are not caused by taking a finite area because changing the outer
boundary into an impermeable boundary has only minor effects
36
VI SUMMARY AND CONCLUSIONS
In this report a numerical model for non-stationary saturated
groundwater flow in a multi-layered system has been treated
Groundwater flow can be described in different ways
a Physical-mathematical by combining Darcys law and the law of
continuity (section 21)
b Geo-hydrological the flow in a natural groundwater basin takes
place in layers with different hydraulic properties Definitions
of the most important hydraulic properties are given in section
22
c Schematical groundwater flows horizontally in aquifers and
vertically in aquitards (section 23)
d Approximating by dividing the region of flow into a finite
number of elements and applying Darcys law and the law of
continuity on each of these elements In these ways one gets
a set of equations which can be solved given (initial and)
boundary conditions (section 25)
e Numerical the set of equations has been solved using a Galerkin
type finite element method and the implicit Gauss-Seidel method
with overrelaxation factor (section 31 and 32)
f Computer-solvable the numerical solution has been translated
in a FORTRAN IV program called FEMSAT (section 33)
In chapter IV the input execution and output of the program
FEMSAT are treated
Finally in chapter V some numerical applications are given
simulation of flow to a well in unconfined aquifer and simulation
of flow to a well in a 3-layered configuration
Comparison of hydraulic heads obtained by numerical calculations
and analytical true solutions showed an excellent agreement in case
of stationary flow and a good to moderate agreement in case of non-
stationary flow However the diviations caused by the discretization
process must be seen in relation to the errors made in the
establishment of the hydraulic properties
37
As was stated in the introduction numerical models can be
valuable tools in the management of water resources Therefore
numerical models must be suitable to simulate effects of certain
operations upon the spreading of water over saturated groundwater
unsaturated groundwater and surface water The model described in
this report is a model for saturated groundwater flow The relation
with the surface water can be taken into account via radial and
entrance resistances or via a relation between flow to the surface
water system and the hydraulic head The relation with the unsaturated
zone has been restricted to the input factor effective precipitation
In practise however the unsaturated and saturated system are
interconnected Therefore the model should be improved by taking
into account the unsaturated zone This can be done in at least
two ways
a Replacing the unsaturated zone by a parametrical model
b Coupling the unsaturated zone in an iterative way with the
saturated system
In order to find out the usefulness of the numerical model for
water resources management it will be applied to actual regional
groundwater systemslaquoAlso sensitivity analyses will be carried out
The computer model itself can be improved by applying standard
programs eg automatical generation of a nodal grid and graphical
plotting of the results
38
VII REFERENCES
ERNST LF 1962 Grondwaterstromingen in de verzadigde zone en hun
berekening bij aanwezigheid van horizontale evenwijdige open
leidingen Thesis University Utrecht pp 189
1978 Drainage of undulating soils with high groundwater
tables (in press)
FORSYTHE GE and WR WASON 1960 Finite Difference Methods for
Partial Differential Equations John Wiley and Sons New York
pp 444
KRUSEMAN GP and NA DE RIDDER 1976 Analysis and evaluation of
pumping test data ILRI-bulletin No 11 pp 200
LAAT PJM DE and C VAN DEN AKKER 1976 Simulatiemodel voor grond-
waterstroming en verdamping in Modelonderzoek 1971-1974 ten
behoeve van de waterhuishouding in Gelderland deel II
Grondslagen pp 145-197
NEUMAN SP RA FEDDES and E BRESLER 1974 Finite element
simulation of flow in saturated -unsaturated soils considering
water uptake by plants Hydraulic Engineering Laboratory
Technion Israel pp 104
and PA WITHERSPOON 1969a Theory of Flow in a confined
two Aquifer System Water Resources Research Vol 5 No 4
pp 803-816
^_ and PA WITHERSPOON 1969b Applicability of Current Theories
of Flow in leaky aquifers Water Resources Research Vol 5
No 4 pp 817-829
PINDER GF and EO FRIND 1972 Application of Galerkins Procedure
to Aquifer Analysis Water Resources Research Vol 8 no 1
pp 108-120
REMSON I GE HORNBERGER and FJ MOLZ 1971 Numerical Methods
in Subsurface Hydrology John Wiley and Sons New York
pp 389
WESSELING JG 1977 Toepassing van Finite Difference- en Finite
Element Methoden voor het oplossen van een aantal grondwatershy
stromingsproblemen (met computerprogrammas) ICW-nota 944
pp 63
39
ZIENKIEWICZ OC 1971 Finite Element Method in Engineering Science
McGraw-Hill London pp 521
40
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APPENDIX 8 EXAMPLE OF OUTPUT
HEAD AND TERMS OF UATERBALANCE PER NODE PER LAYER
AT TIME 260 IN II AND I13DAYS
T
LAYER NODE KODE HEAD FLOUUN FLOUTS FLOUSS LEAKAGE STORAGE LAT FLOU
13 19 25 31 37 43
125008 125000 12S000 12S000 12S000 12S000 12S000 12S000
0000 0000 0000 0000 0000 0000 0000 0000
0000 0000 0000 0000 0000 0000 0000 0000
0000 0000 0000 0000 0000 0000 0000 0000
- 000 - 000 -000 - 000 - 000 - 000 - 000 0000
0000 0000 0000 0000 0000 0000 0000 0000
0000 000 000 000 000 000 000 000
000 0000 0000 0P100 0000 0000 0000 0000
1 7
13 19 25 31 37
12S000 125000 125000 125000 125000 12S000 125000 125000
0000 0000 0000 0000 0000 0000 0000 0000
- 001 -007 - 023 - 020 - 009 - 003 0000
0000 0000 0000 0000 0000 0000 0000 0000
000
001
007
023
020
009
003
003
0000 0000 0000 0000 0000 0000 0000 0000
1 7
13 19 25 31 37 43
125000 125000 124998 124972 124895 124787 124647 124358
0000 0000 0000 0000 0000 0000 0000 0000
- 000 001 007 023 020 009 003
0000
0000 0000 0000 0000 0000 0000 0000
50000
0000 182
1982 3862 1674 419 078 041
0000 0000 0000 0000 0000 0000 0000 0000
TERMS OF WATERBALANCE PER LAYER DURING LAST TIMESTEP AND
AND SINCE BEGINNING IN M3DAY M3
FLOU TO UNSATURATED ZONE
FLOU TO TERTIARY SYSTEM
0000
0000
0000
0000
LAYER NO 1
FLOU TO SECONDARY SYSTEM
FLOU TO AD3ACENT LAYERCS)
ARTIFICIAL FLOU AND PRESCRIBED BOUNDARY FLOU
ACCUMULATION TERMS
LATERAL FLOU THROUGH PRESCR HEAD BOUNDARY
0000
-000
0000
000
000
0000
-000
0000
000
000
LAYER NO 2
FLOU TO SECONDARY SYSTEM
FLOU TO AD3ACENT LAYERCS1
ARTIFICIAL FLOU AND PRESCRIBED BOUNDARY FLOU
ACCUMULATION TERMS
LATERAL FLOU THROUGH PRESCR HEAD BOUNDARY
0000
-378
0000
381
0000
0000
-055
0000
056
0000
LAYER NO 3
FLOU TO SECONDARY SYSTEM
FLOU TO AD3ACENT LAYERCS5
ARTIFICIAL FLOU AND PRESCRIBED BOUNDARY FLOU
ACCUMULATION TERMS
LATERAL FLOU THROUGH PRESCR HEAD BOUNDARY
0000
370
50000
49376
0000
0000
055
12979
12909
0000
The most important hydraulic properties are
a Transmissiviy - the product of the average hydraulic conductivity
k and the saturated vertical thickness d of the waterbearing
layer
It is a measure for the transport capacity of aquifers
Notation T = kd |_ L T ~
b Hydraulic resistance - the ratio of vertical thickness of
aquitards d and hydraulic conductivity k It is a measure for
the resistance capacity of aquitards denoted as c = d k _ T J
c Specific yield - the volume of water released or stored in the
phreatic zone per unit surface area of an unconfined aquifer per
unit change in hydraulic head
It is also called effective porosity
The symbol used for specific storage will be S j_-J
d Specific storage - the volume of water released or stored per
unit volume of the saturated parts of an aquifer or aquitard per
unit change in hydraulic head It is a measure for the elasticity
of the soil material and the fluid Excluded are irreversible
processes Notation S _ L J
e Storage coefficient - the volume of water released or stored
pe7 nit surface of a layer per unit change in hydraulic head
It is the sum of specific yield and the product of specific
storage and thickness of the aquifer or aquitard
S = S + S d r - 1 c y s - -1
Note in confined layers S = 0 y
In fig 1 a 3-layered configuration is given of two aquifers
separated by an aquitard Aquifer I has a free water surface and is
an unconfined aquifer Aquifer II is a completely saturated aquifer
Dependent on the hydraulic resistance of the aquitard KRUSEMAN and
DE RIDDER (1976) distinguish between confined semi-confined and
semi-unconfined aquifers For our purpose it is not useful to make
this distinction We restrict the aquifer types to unconfined and
confined aquifers
WIHtUtttWttllttlt(ltmtttltHlttltlil(lltlittlfllnl ) bull (
d V U i l i l i ii i Hi il i M M bull f i m
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aq ui tard
aquifer II
Fig 1 Schematic representatipn of flow direction and variation
of hydraulic head h in a 3-rlayered system
In an unconfined aquifers ( I ) poundhe vertical thickness pf the
waterbearing layer d is a variable while in a confined aquifer II
this is a constant (denoted as D ) Other important properties of
unconfined aquifers are the hydraulic head being the same as the
phreatic surface and the specific yield being different from zero
23 S c h eacute m a t i s a t i o n o f f l o w
The groundwater flow in a groundwater basin is schematized
into a horizontal flow in aquifers and into a vertical flow in
aquitards Consequently the vertical variation of the hydraulic
head is as drawn in figl
The validity of this assumption has )een investigated by
NEUMAN and WITHERSPOON (1969) The found that if the contrast
in hydraulic conductivity between two adjacent layers is bigger
than a factor two the relative error caused by this assumption
is usually smaller than 5
This schematization gives a simplification of the general flow
equation (eq 4 ) In an aquifer the vertical is zero and eq (4)
then reduces to
-Sbdquo -r- 7 - (k d -r-) + 7 mdash (k d 7-) + qd c lt5t ox x ox oy y oy
or
bull(S + S d) g y s ot
-r- (k d -=-) + -T- (k d -=-) + Q ocircx x ox oy y ocircy
(5)
where
Q = volume of water substracted from or added to unit surface
of aquifer per unit time |_ L T J
In an aquitard eq (4) reduces to
-sQ |pound = (k Q) s 6t oz Z OcircZ
(6)
As will be discussed in the next paragraph the term q in an
aquitard is set equal to zero
24 W a t e r b a l a n c e t e r m s
In eq (4) the term q is mentioned With this sink or source
term different terms of the water balance are taken into account
Fig 2 depicts schematically the situation for an unconfined aquifer
(for other layers some terms dont exist)
recharge- - discharge
laquo-b
Fig 2 Schematic representation of the various terms involved in the
water balance
Description of the different terms
a The flow through the phreatic surface q f
If this term is negative (water goes out of the saturated system)
it is called capillary rise If positive it is called percolation
(or drainage)
This flow can be calculated as a rest term of the water balance
of the unsaturated zone Very often this term is expressed as a
flow per unit surface (flux) and is called effective precipitation
For the time being this flux is a given input
b Flow to adjacent aquitards q
The flux to adjacent aquitards can be calculated by applying eq
(6) to the aquitard
c Flow to the primary surface water system q
The primary surface water system penetrates fully the groundwater
basin So it acts as a boundary with given values for the
hydraulic head (is equal to the phreatic level of the primary
system)
d Flow to the secondary surface water system q
The secondary surface water system does not penetrate fully the
groundwater basin which causes a resistance against flow between
groundwater and surface water The flow q can be calculated as
follows
h - h
- mdash f i i gt
where
h = hydraulic head of the groundwater at the place of
the secondary system |_ L |
h = phreatic level of secondary system |_ L J i s -
R = radial and entrance resistance of the
secondary system per unit surface |_ T J
According to ERNST (1962) the radial resistance H per unit
surface can be caKulated with
Uuml = 4 i n 4 e8) TTk B
where
1 = area drained by 1 m conduit L ^ J
d = thickness of the layer near the conduit _ L J
B = wet perimeter of the conduit |_ L J
The entrance resistance is caused by a layer which shows a
lower conductivity around the conduit than its surroundings
In practise radial and entrance resistancies are difficult to
deduct from hydraulic properties They can be found indirectly
by measuring simultaneously q h and h
t s
e Flow to the tertiary surface water system q
In smaller conduits usually the phreatic level is not known
So eq (7) cant be applied The flow to these conduits however
is sometimes very important (eg in areas with shallow groundwater
tables) One may a-sume (ERNST 1978) that a relationship exists
between the hydraulic head of an aquifor and the flow q
According to Ernst the tertiary system can be divided into a
number of categories with respect to their mutual distance and
bottom height For each category the flow is h - h (b)
q ( h b ) - mdash (9)
where
h = average hydraulic head | L J
h (b) = phreatic level in the category under
consideration a function of bottom height |_L J
a = geometry factor to convert h into h _ - J
Y = drainage resistance _ T J
hm = hydraulic head midway between two
conduits |_ L J m
Summation of the q(hb)-relations of the different categories
yields the overall relation between hydraulic head and flow to
the tertiary system we are looking for
f Artificial recharge or discharge q and prescribed boundary
flux qfo
The magnitudes of these flows are not influenced by the conditions
inside the groundwater basin but remain on a prescribed level
Thus q is an internal boundary condition q an external boundary cl D
condition
For confined aquifers some of the above mentioned terms dont
exist These terms are the flow through the phreatic surface qf
and the flow to the tertiary surface water system q
Aquitards are considered only as transmitters of water from one
aquifer to another with possibilities of storage (eq 6 ) This means
that for aquitards the water balance terms ar through f are set
equal to zero
25 S o l u t i o n o f g r o u n d w a t e r f l o w p r o b l e m s
The basic equations for saturated groundwater flow are eq (5)
for aquifers and eq (6) for aquitards The solution of these
equations require knowledge of the physical system and the auxiliary
conditions describing the system constraints
These auxiliary conditions are (REMSON etal 1971)
a Geometry of the system
Eg the diviation of the system into a restricted number of
layers and the horizontal extension of the system under
consideration
b The matrix of hydraulic properties including in-homogeneity
and an-isotropy
c Initial conditions Because we are dealing with non-stationary
flow the values of pertinent system variables (such as hydraulic
head at the initial time) must be known
d Boundary conditions They describe the conditions at the boundaries
pf the system as a function of time
One can distinguish between
- hydraulic head boundary
- flow (including zero flow) boundary
- both head and flow boundary
After specifying the auxiliary conditions an unique solution of
the variation of the hydraulic head in space and time can be
obtained
For special shapes of flow and assumptions exact analytical
solutions exist In nature however one is usually confronted with
problems like
- spacial variation in the hydraulic properties of the soil
- irregular shape of the groundwater basin
- empirical relations between q and h
- d not being a constant
For such situations by numerical and analogy models only an
approximation of the actual situation can be obtained In the next
chapter a numerical solution called the finite element method is
treated
III NUMERICAL SOLUTIONS
In this chapter the numerical solution of the aquifer equation
(eq 5) and the aquitard equation (eq 6) will be treated The
method of solving is the finite element method In section 31 only
a short mathematical description of this method will be given For
more mathematical background the reader is referred to publications
on this subject (eg ZIENKIEWICZ 1971 NEUMAN etal 1974 and
WESSELING 1976)
In section 32 the application of the finite element method on
the flow in groundwater basins is treated while in section 33 the
translation into a computer program is given
3 1 F i n i t e e l e m e n t m e t h o d
For reasons of simplicity the finite element method will be
explained for the stationary version of eq 5 ie
10
(k d^) + (k d ^ ) - Q = O (10) ocircx x 6x 6y y Sy
The basic idea of each numerical solution of eq (10) is to
apply this equation on a small but finite volume The flow region
of interest R is divided into a finite number of volumes called
elements The elements can have different shapes but for reasons
of conveniency we only use triangles- Quadrilaterals can also be
used because they can be divided into two triangles The sizes of
the triangles may vary They can be adapted according to the
geometry of the region and the accuracy desired (fig 11 gives an
example of the diviation of a region into triangles and quadrilaterals)
The corner points of the elements represent the nodal points
If there are N nodal points then the continuous variation of h
in region R is approximated by the expression
h (x_v) ~ h f (xy) (11) a n n
where
h (AV) = approximation of u (xy) a
h = value of h in nodal point n n
f ltxgty) = global coordinate function
In nodal point n a must be equal to h so f (x y ) = 1 and a n n n n
f (xy) = 0 i n Between f (x v ) and f (xy) of the nodal n j ] n n n n j j
points surrounding nodal point n f (xy) varies linearly from
1 to 0
In fig 3 a picture of f (xy) is given n
If one knows the values h in the nodal points n we know the approximation h of h We can find these values of b bv using for
a n
example the Galerkin method
Galerkins principle states that an approximate solution of h in
eq (10) can be obtained from the following system of equations
N ~~ (k d -A + -^- (k d T-y E h f (xy) f(xy)dxdy +
-ox x ox ocircy y oy - _ n n i R n~
J J Q f(xy) dx dy = 0 i = 1 N (12)
1 1
Fig 3 The global coordinate function f (xy)
The integration can be performed element by element with the
aid of the coordinate functions f (NEUMAN etal 1974)
For non-stationary problems it is not possible to apply the
Galerkin principle on the time derivative But it is possible
to discretize this derivative and to set up the system of equation
(eq 12) for each finite time step
One obtains in this way a series of sets of laquocriations tha s
an approximation of the non-stationary flow problem
Treatment of boundary conditions
a Initial conditions
In non-stationary flow problern gtne must start with given
values of a at time t = t ri c
b Boundary conditions
- Prescribed head boundary
Nodal points which lie on a prescribed head boundary have a
prescribed value for h bull r Hgte r umber of unknown h s is n n
reduced and accordingly no the number of equations
- Prescribed flux through z-- boundary
If one knows the flux (iuumlr jw or outflow per unit area) and
the area of the boundary oi-r element the inflow or outflow
through the boundary per flemlt=nt can be calculated
One may assume this flow to J bull part of the sink term Q in
eq (12)
12
32 A p p l i c a t i o n o f t h e f i n i t e e l e m e n t
m e t h o d o n t h e f l o w i n g r o u n d w a t e r
b a s i n s
From hydrogeological investigations the geometry and the matrix
of the hydraulic properties are known Knowing the geometry the
flow can be split up into a finite number of layers This holds
also for the nature of the boundaries
Now a nodal grid is superimposed on each layer in such a way
that the nodal points of the different layers are situated right
above each other (see fig 4) and in each layer the nodal point
lies in the middle of this layer
-nodal point
Fig 4 Schematic representation of the nodal points in the
xyz-space
Each layer has N nodal points of which P nodal points have a
prescribed head In each layer remain N-P = N nodal points with
unknown head Eq (12) can be applied to each point with unknown
head in each separate layer
As was stated in the last paragraph for each time step one
can write
13
Ah bdquo ~ -r1 + Q = O in = 12N in n m At yi A h + B (13)
where
A = in
ocircf ocircf ocircf of (k d f mdashri- ~ + k df - ~ -jS) dxdy
J x n ox ox y n oy oy
in (S d + S ) ff dx dy O if i icirc n
s y i n
Q f dx dy
The matrices A B and the vector Q have been defined in in l
elsewhere (NEUMAN etal 1974) A is the conductivity matrix in
B the storage matrix and Q the source vector in l
In order to clarify these rather complicate expressions apply
equation (13) poundo point 4 of fig 5
6 7
Fig 5 Top view of part of the nodal grid
lhl + A42h2 + A43h3 + A44h4 + A45h5 + A46h6 + A47h7 +
h - h
+ B _iuumli i ^ + Q = o 44 t - t x4
n n-1
(14)
where h and h are values of h at time t and t 4n 4n-1 4 n n-1
respectively
14
If the h-values belonging to the A-values are taken at time
t = t we get an equation with one unknown h for time t = t n-1 4n n
The equation can then be solved directly
However this is an explicit formulation which puts strong limitations
on the magnitude of the time step t - t (REMSON etal 1971) n n-1
Values larger than a certain value cause instability With an implicit method the values of h are taken at time
t = t This method is unconditionally stable Now one gets an equation n
with 7 unknowns h through h If one has 7 equations for every 1 n n
point 1 through 7 we can solve the values h i = 17
in
Of all the methods possible the iterative method of Gauss-Seidel
has been chosen The Gauss-Seidel method starts by choosing a first estimate of h at time t = t (usually equal to h ) At each in n j -i in-l iteration step a new value of h is calculated using the last
in calculated values of h If one passes the nodal points from 1 to
i n 7 eq (14) becomes
h-j + 1 = A h^ + 1 + A bdquo h | + 1 + A _ h | + 1 + A h j + A r h j + 4 n 4 1 1 n 4 2 2 n 43 3 n 45 5 n 4 6 6 n
B h B + A h J + - 4 4 4 n ~ + 0 J (A + mdash) A 4 7 h 7 n At n U 4 4 At
(15)
where
h-i = in
j = iteration step or number of iteration
i = number of nodal point n = time = t
n
The iteration process converges if the successive differences k+1 k
between h and h become continuously smaller in in
The iteration process is stopped if at each nodal point the k+1 k
difference between h and h is smaller than a certain prescribed in in
value
15
To accelerate the convergence an overrelaxation factor W is used
(for more information about W see FORSYTHE and WASOW 1960)
k+1 k T7 k+1 k h = h + W (h - h )
in in in in (16)
If W = 1 there is no overrelaxation Usually W varies between 1 and
2 and the optimum value very often is found by trial and error
Eq (15) is valid for points in aquifers The connection with
adjacent layers is made by the term Q because in this term the
flow from adjacent aquitards q is included (two aquifers are always
separated by an aquitard)
Aquitards
We dont apply the finite element method on points in aquitards
As stated before there is no horizontal flow in aquitards So it has
no sense to replace the continuous variation of h in the aquitard
by equation (11)
The nodal points in aquitards represent a certain volume of an
aquitard hich transmit water from one aquifer to another and which
has storage capacity The nodal points are included in the Gauss-
Seidel scheme -s follows (see fig 6 )
1 aauifer
a -_vS laquo 2 [~^-t^yM^^^-^r~ aqutarci l-Z
aquifer
Fig 6 Schematic situation for points in aquitards
Point 2 is situated in the middle of the aquitard with thickness
d storage coefficient S and vertical hvdraulic conductivity k c J v
Suppose the Gauss-Seidel iteration starts in the upper layer and
ends in the bottom-layer Then the flux from point 1 in aquifer 1 to
16
point 2 in the aquitard at time t = t and in iteration step j + 1
is
k q = (hf -hf )-iumlmdash f (17)
1 n 2 n OSxd1
and the flux from point 3 in aquifer 2 to point 2 in the aquitard
is
1 c 1
qbdquo = (h - hf) mdashHmdashr- (18) 2 3 n 2 n OSxd
The net flux ltl-q9 must be equal to the change in storage per unit
area per unit time This change in storage can numerically be
approximated by
te - h-
n n-1
Combina t ion of ( 1 7 ) (18) ana (19) g i v e s
l tr1 - h (hJ + 1 + hj - 2 hj+1) - - - - - - S1 - ilaquoS L u l l = o (20)
] i 3 n 2 n 1 c t - t j j x d n n -1
T + 1
From t h i s e q u a t i o n h^ kan be s o l v e d
3 3 C o m p u t e r o g r a m
The problem outl^ugrave so far has been translated into a computer
program written in FORTRAN IV
The flow chart ot Uis program is given in fig 7 whereas in
appendix A the complete listing of the program can be found
In the flow chart the 5 main parts of the program are
indicated
A Reading and printing of input data
B Determination of the matrices A and B and the vector Q per layer
17
M
1 A
FLOH to unsaturated zono
FLOW to tertiary syster
FLOW to secondnrv system
Leakage to adjacent aquitards
SOLVE equation (14)
yes
CALCULATE flux through constant head boundnr
CALCULATE waterbalance n p r 1 OAro r r J j - ~
PRINT of output
Ier node per layer
k = k+1
t = t + At
7 DUMP variables on tape
-gt-
( STOP )
Fig 7 Flow chart of the program FEMSAT with diviation in 5 main
parts A B C D and E
C START PROGRAM FEMSAT
f READ general information
PRINT general information
f READ element information
PRINT element information 7 READ stationary nodal point information
PRINT stationary nodal point information 7
READ non-stationary nodal point information for t = t
FRINT non-stationary nodal point information for t = t
L 2 (
^ yes R E A D non-stat input for t gt t
J PRINT non-stat input for t gt t (- 1 -- M degY
GENERATE A B and AREA-matrix for confined aquifers
_L
GENERATE A B and AREA-matrix for unconfined aouifer
bull
C Solving h with Gauss-Seidel
D Determination of the water balance terms per layer
E Printing of results
The program exists of a main program with several subroutines
The main program calls the subroutines In the subroutines most
calculations are performed
In this way an easy adaptation to different problems is
obtained (see also section 44) Comments in the program will
help the reader to understand the program
In the next chapter the preparation of the input data executing
of the program and the output will be treated
Acknowledgement
Some parts of the computer program were taken from the program
UNSAT2 of NEUMAN etal (1974)
IV INPUT EXECUTION AND OUTPUT
41 I n p u t
The input is divided into endogenous variables (chosen by the
user or calculated by the program during execution) and exogenous
variables (based on physical data)
411 Endogenous variables
Translation of groundwater basin into a nodal point network
Probably the most important but also the most difficult stage
in the process of simulating groundwater flow is the translation
of the natural groundwater basin into a nodal point network Questions
which arise are
a In how many layer the geo-hydrological system mustcan be
schematized
b The locations of the boundaries and the nature of these boundaries
(impervious prescribed head prescribed flux)
c The configuration of the nodal points
19
The answers of these questions depend on the problem under
consideration
- Schematization into layers
Usually our knowledge about the geo-hydrological situation is
such that no more than 5 different layers can be distinguished
(remember contrast rule)
- Boundary conditions
The top boundary condition of a groundwater basin usually is
a prescribed flux (effective precipitation)
The bottom boundary condition usually is a prescribed flux
boundary with flux zero (impervious hydrological base)
For the boundaries at the other sides of the system one sees
either prescribed head boundaries (eg a fully penetrating
river) or prescribed flux boundaries (including zero flux)
- Nodal point configuration
Nodal points are situated
on the boundaries
along internal boundaries which indicate changes in
hydraulic properties within the layer
along the tertiary surface water system
on places where artificial recharge or discharge wiii tgtke
place
elsewhere
Construction of the finite element mesh
Assume we have schemetized the groundwater basin iatu a finite
number of layers with different direction of flow (horizontal and
vertical) Within one layer the hydrological properties may vary
from place to place
Now a nodal grid is superimposed on each layer in a way already
described in section 32 Thus each layer has the same number of
nodal points situated on the same places in the horizontal plane
but with different z-coordinates
We are free to construct quadrilaterals or triangles because
the program automatically divides a quadrilateral into two triangles
having identical material properties The dimensions of the elements
20
should be small at places where large changes in hydrological
properties occur or where large gradients in hydraulic head are
expected to occur
Coding of nodal points
Each node in the finite element network is assigned an integer
code which indicates the type of calculation for this node
The coding is as follows
- Code = 1 nodal points with described head as function of time
- Code 2 internal nodal points
- Code = 3 internal nodal points which represent secondary surface
water system
- Code = 5 nodal points with prescribed artificial recharge or
discharge and points with prescribed boundary flow
(both as functions of time)
Other important endogenous variables are the magnitude of the
time step the value of the variable which stops the iteration
process and the maximum number of iterations allowed per time step
(see also group A and B of section 413)
412 Exogenous variables
After constructing the nodal grid each node represents part of
the groundwater basin Geo-hydrological properties can be assigned
to each mode based on physical data (group C D and E of section
413) In group F of section 413 time-dependent variables can
be assigned to each node
413 Data input to Program FEMSAT
The data input must be punched on cards according to the
following instructions
GROUP COLUMNS FORMAT SYMBOL DESCRIPTION
A 1-80 20A4 HEAD Desired heading to be printed at
the beginning of the execution
1-5 15 NUMNP number of nodal points
21
GROUP COLUMNS FORMAT SYMBOL DESCRIPTION
6-10 15
11-15 15
16-20 15
21-25 15
26-30 15
31-35 15
NUMEL
NUMLAY
MBAND
NTSPI
NTSPO
MAXIT
1-10 F103 ST
11-20 F103 DELT
21-30 F103 TMAX
31-40 F105 TOL
number of elements
number of layers
maximum difference in node number
between two adjacent nodes
number of time steps between a
change in non-stationary input data
number of time steps between two
successive outputs
maximum number of iterations per
time step
time to start with the execution
initial time interval
maximum time to be reached before
execution is stopped
maximum allowed absolute change in
the values of h between any two
successive iterations in a given
time step
Iterations continue until all changes
are less than or equal to TOL or
until MAXIT is exceeded
1- 5 15
6-10 15
11-15 15
16-20 15
KX(N1)
KX(N2)
KX(N3)
KX(N4)
sequential number of i-th corner of
element N
sequential number of j-th corner
sequential number of k-th corner
sequential number of 1-th corner
(see fig 8)
Fig 8 Numbering of element corner nodes
22
GROUP COLUMNS FORMAcircT SYMBOL DESCRIPTION
One card must be provided for each element in sequential order
starting with N = 1 and ending with N = NVJMEL
I-K 1103 SANG(NLY)
1i-iO MO3 C1(NLY)
21-30 F 103 C2(NLY)
21-4- 1-104 THI(NLY)
angle in degrees between CI and
the laquo-coordinate to he used in
nodal print N in layer LY (see
poundig 9)
First principal conductivity of
material in node N and layer LY
idem second principal conductivity
thickness of layer LY at nodal
point N
ig V Definition of SANGCI and C2
One card icircuiPt be provided for each model point ctarting wih M = 1
and ending with N = NUMNP If there is norlt3 than 1 layer for each
layer a set must be provided starting with the upper layer LY - 1
and ending with the bottom layer LY = NIJMLAY The variables SANG
M and 2 are not relevant tor aquitards but for reasons of
conveniency this procedure is maintained (making SANG CI and C2
dummy variables)
1-10 110 KODE(N)
11-20 II0 NTERR(N)
coding of nodal points (see
paragraph 42)
number of tertiary relation in node N
The q-h relationship of the flow to
23
GROUP COLUMNS FORMAT SYMBOL DESCRIPTION
21-30 FlOl X(N)
31-40 FlOl Y(N)
41-50 FlOl HGL(N)
51-60 FlOl HBSC(N)
61-70 FlOl LESC(N)
the tertiary system as discussed
in par 24 are schematized in a small
number of different relations
x-coardinate of node N
y-coordinate of node N
height ground level in node N
height bottom secondary conduit in
node N If node N is not a secondary
conduit this variable is a dummy
variable
length of secondary conduit in area
represented by node N
One card must be provided for each node starting with N = 1 and
ending with N = NUMNP
E 1-10 FlOl P(NLY1) initial value of h at node N in
layer LY
11-20 FlOl VERRES(NLY) vertical resistance at node N and
layer LY If the layer is an aquifer
this value is a dummy variable
21-30 F101 RADRES(NLY) radial resistance for flow to
secondary system at node N in layer
LY
31-40 FlOl ENRES(NLY) entrance resistance for flow to
secondary system at node N in layer
LY
41-50 FlOl WP(NLY) wet perimeter of secondary system at
node N in layer LY
51-60 FlOl P0R(NLY) specific yield of the material at
node N in layer LY
61-70 FlOl SS(NLY) specific storage of the material
at node N in layer LY
24
GROUP COLUMNS FORMAT SYMBOL DESCRIPTION
One card must be provided for each node starting with N = 1 and
ending with N = NUMNP If there is more than 1 layer for each layer
a set of cards must be provided starting with LY = 1 and ending
with LY = NUMLAY
F 1-10 F101 P(NLY1) value of h at different times at
node N in layer LY Node N lies
on a prescribed head boundary
11-20 F101 WTSC(N) water table in secondary system at
node N at different times
21-30 F 101 ARRED(NLY) artificial recharge to or discharge
from node N in layer LY or
prescribed boundary flow at
different times
31-40 F 101 QPHR(N) flux through phreatic surface at
node N at different times
The data in group F are non-stationary input data One card must be
provided for each node starting with N = 1 and ending with N = NUMNP
In case of more than 1 layer for each layer a set must be provided
Not relevant variables are taken as dummy variables
After a certain number of time steps (NTSPI) the data of group E
are read So the set cards E must contain NxLYxC cards where C =
(TMAXNTSPIXDELT) + 1 if DELT is a constant
42 R u n n i n g t h e p r o g r a m o n t h e C y b e r 7200
The program has been executed on a Cyber 7200 computer of
IWIS-TNO in the Hague
- The demanded input is punched on cards according to the instructions
of the last paragraph and is put on a permanent file
- For each specific problem inthe program only the dimension statements
in the main program have to be adjusted according to the instructions
in the lines of comment
- After compilation the program together with the input file is
executed
25
- Intermediate data necessary for continuation of the calculations
are saved on another permanent file This gives the opportunity
to restart the program at a time the calculation was stopped
The only change in the input file is to adjust the starting time
Remark
With the aid of a terminal small changes in the program and in the
input file can be made easily and the output immediately appears
on the terminal In this way it is possible to stop erroneous
calculations This is especially useful when being in the testing
phase of the program
43 O u t p u t
Immediately after reading the input data they are printed so
that possible errors can be detected During execution after each
certain number of time steps (NTSPO) calculated values of h and
water balance terms are printed In appendix B an example is given
The output data can be devided into two groups
A Data per node per layer
- Coding of the nodal points KODE
- Hydraulic head HEAD
- Flow through phreatic surface at the time indicated in head
of table FLOWUN
- Flow to tertiary surface water system FLOWTS
- Flow to secondary surface water system FLOWSS
- Flow to adjacent layers LEAKAGE
- Artificial recharge or discharge and prescribed boundary
flow ARRED
- Change in storage during last time step STORAGE
- Flow through prescribed head boundary LATFLOW
B Data per layer
Summation of the terms already mentioned under A gives the
magnitude of the water balance terms per layer at the time
indicated (instanteneous water balance terms) Summation of
these instanteneous terms multiplied by their respective time
intervals since time is zero gives the cumulative water balance
terms per layer
26
After each print out we are able to check the calculations
because the sum of the water balance terms per layer must be
approximately zero
44 C o m p u t i n g t i m e
The computing time depends on the number of nodal points and
the number of layers But it depends also on the magnitude of the
changes in hydraulic head and on the value which terminates the
Gauss-Seidel iteration (TOL) To give an impression TOL = 001 m
Costs per day simulation (in hlfl) on the Cyber 7200
number of nodes number of layer bull bdquo bull bull _ 1mdash x a x b (21)
where
a = costs in Dutch guilders per system second (ƒ 027)
b = empirical factor with magnitude about 001
V NUMERICAL EXPERIMENTS
In general there are two possibilities to test the validity of
numerical models The first and most commonly used method is to
compare numerical results with analytical solutions The second
possibility is to compare numerical results with measured field
data Since the last alternative requires many data that are
difficult to obtain we restrict ourselves for the time being to
comparison with the analytical solutions
5 1 F l o w t o a w e l l i n a n u n c o n f i n e d
c i r c u l a r - s h a p e d a q u i f e r
Flow to a well can be divided into steady-state flow and
unsteady-state flow (KRUSEMAN and DE RIDDER 1976)
27
511 Steady state flow in an unconfined aquifer
For this case the formula of Thiem-Dupuit is valid (see fig 10)
Q bull k lth2-hicircgt ln(r2r])
(22)
where
hhbdquo = hydraulic head in 1 and 2 respectively
Q = discharge from well
rbdquo = distance from well
CL31
hmdash^ri --r2
Fig 10 Schematic cross-section of a pumped unconfined aquifer
Fig 11 gives the nodal grid superimposed on a well in a
circular shaped aquifer Near the well the nodal point distance
is made smaller Quadrilaterals as well as triangles are used
Numerical calculations with this finite element network and 3 -1
an extraction rate of 200 m day were performed for 10 days
At that moment a steady-state situation was almost achieved
28
2tmdashnodal point number
160 180 X-axesim)
Fig 11 Nodal point configuration for flow to a well in a circular-
shaped aquifer The outer boundary is taken to be constant
The well is situated in nodal point 42
29
In fig 12 the results of the simulation are compared with the
analytical solution The agreement between numerical results and
the analytical solution is excellent
draw- 15irS (m) down
analytical solution imdashx calculated values
distance from well r (m)
Fig 12 Comparison between analyt ical and calculated s teady-state
drawdown for the problem of f ig 11 with 1
-1 Q bull - 200 m 3 d a y _ 1
k = 10 mday
d = 100 m
S = 2 y
S = 0 s
512 Unsteady-state flow in an unconfined aquifer
Analytical solutions for unsteady-state flow in an unconfined
aquifer only exist if the aquifer extends infinitely For the
aquifer given in fig 11 this condition is not fulfilled However
during the first one or two days after starting the extraction the
influence of a constant hydraulic head at the outer boundary on the
30
drawdown near the well can be neglected (see fig 13) After one
day the lateral inflow is only 25 of the extraction the remainder
being the change in storage due to drawdown
2 0 0
m 3 day 1
Q=200m 3 day
100-
vraquomdash Lateral inflow
N amp storage
I I ^ V - W 10
t (days)
Fig 13 Values of Q lateral inflow and change in storage as
functions of time
The nonsteady-state or Theis equation can be written as
i _ Q s 4iTkD
dy = 4irkD
W(u) (23)
where
u = r2S
4irkDt
r = distance from well
S = storage coefficient
Q = constant discharge from well
D = thickness of the waterbearing layer
at the beginning of pumping
t = time after starting the extraction
s = corrected drawdown
= s - 322D
s = calculated drawdown
W(u) = Theiss well function
[J
CIAT1
[T]
]
31
In fig 14 the analytical unsteady-state solution for the case
described in section 511 is given together with the numerical
results at times 05 and 10 days The agreement between numerical
results and the analytical solution is rather good
Drawdown s1(m)
10
10
10
Theiss solution
x caculated values t=Q5
bdquo t= lO
10 10- 10 10 t r2(daym2)
Fig 14 Comparison between analytical and calculated steady-state
drawdown for the problem of fig 11 with
Q
k
d
S y
s s
=
=
=
=
=
200
1
10
3 m
0
0
2
0
-1 day
m day
m
These results indicate that the assumption of a linear variation
of h between two nodal points (see fig 3 ) even with the rather
rough element network of fig 11 is acceptable So there is no
need to improve for example the finite element network by introducing
higher order coordinate functions
32
52 F l o w t o w e l l i n 3-1 a y e r e d s y s t e m
In fig 15 a schematic cross-section of a 3-layered system of
two aquifers separated by an aquitard is given A well of
infinitisemal radius is completed in the lower aquifer and discharges
at a constant rate QTT Each layer is homogeneous isotropic and
extending radially infinite
bdquo bull - bull bull ~ i
mdash Q II
aquifer I
lugrave^i~IcircIuml7i-i~~ aquitard --r-rl~-
-II--I i - i
aquifer
Fig 15 Schematic cross-section of a 3-layered system
For this situation NEUMAN and WITHERSPOON (1969a) give rather
complicated analytical solutions In the special case where the
hydraulic properties of the two aquifers are identical they give
graphical illustrations of their solutions for three different
cases
These graphical solutions give the opportunity to verify
roughly numerical results of a 3-layered system Therefore a nodal
grid is superimposed on a circular shaped groundwater flow system
with the extraction in the centre of the lower aquifer and with
a constant head on the outer boundary Fig 16 gives the nodal
point situation in the x - y plane of 1 layer The nodes in the
other two layers are situated in the middle of the layers at the
same places in the x - y plane only their z-coordinate is different
33
A
-Jy
bull33
V 5
iumlff
Fig 16 Nodal point configuration for the 3-layered problem (top view)
34
In fig 17 numerical results are compared with the analytical
solution The agreement is rather bad especially for the two
aquifers But in terms of the water balance the difference between
aquifers and aquitard is not so big because the aquitard has
about 16 times bigger storage capacity than the aquifers The
small value of the specific storage of the aquifers is the reason
that small errors in the components of the water balance have a
big influence on the calculated values of the hydraulic head
Eig 17 Dimensionless drawdown vs dimensionless time in a
3-layered system with
II = 50 m day
3 m (distance from well)
aquifer I
aquitard
aquifer II
Thi (m)
100
25
100
S s
001
0642
001
k(mday )
1
0277
1
The discrepancies are partly caused by errors in discretization
In the z-direction these errors can be eliminated by taking the
35
vertical resistance of the aquitard to be infinite Then the
calculated drawdown in the pumped aquifer can be compared with
the analytical Theiss solution
This is done in fig 18 for different distances from the well
One sees that the agreement is better when the distance is bigger
Apparently the linearization of the logarithmic drawdown curve
near the well is responsible for the differences The differences
for r = 3 m are of the same magnitude as for the pumped aquifer in
the 3-layered case
joo gt A 50
20
Theiss solution
bull calculated values r=3m (distance f rom well) x bdquo bdquo r=7m
1 L
I l_ 001 003 0 0 5 Ol Q2
- I I 5 10 t - K t
Fig 18 Dimensionless drawdown vs dimensionless time in a
confined aquifer with
Q =500 m day
k = 10 mday
D = 100 m
S = 001 s
The differences between numerical results and analytical solutions
are not caused by taking a finite area because changing the outer
boundary into an impermeable boundary has only minor effects
36
VI SUMMARY AND CONCLUSIONS
In this report a numerical model for non-stationary saturated
groundwater flow in a multi-layered system has been treated
Groundwater flow can be described in different ways
a Physical-mathematical by combining Darcys law and the law of
continuity (section 21)
b Geo-hydrological the flow in a natural groundwater basin takes
place in layers with different hydraulic properties Definitions
of the most important hydraulic properties are given in section
22
c Schematical groundwater flows horizontally in aquifers and
vertically in aquitards (section 23)
d Approximating by dividing the region of flow into a finite
number of elements and applying Darcys law and the law of
continuity on each of these elements In these ways one gets
a set of equations which can be solved given (initial and)
boundary conditions (section 25)
e Numerical the set of equations has been solved using a Galerkin
type finite element method and the implicit Gauss-Seidel method
with overrelaxation factor (section 31 and 32)
f Computer-solvable the numerical solution has been translated
in a FORTRAN IV program called FEMSAT (section 33)
In chapter IV the input execution and output of the program
FEMSAT are treated
Finally in chapter V some numerical applications are given
simulation of flow to a well in unconfined aquifer and simulation
of flow to a well in a 3-layered configuration
Comparison of hydraulic heads obtained by numerical calculations
and analytical true solutions showed an excellent agreement in case
of stationary flow and a good to moderate agreement in case of non-
stationary flow However the diviations caused by the discretization
process must be seen in relation to the errors made in the
establishment of the hydraulic properties
37
As was stated in the introduction numerical models can be
valuable tools in the management of water resources Therefore
numerical models must be suitable to simulate effects of certain
operations upon the spreading of water over saturated groundwater
unsaturated groundwater and surface water The model described in
this report is a model for saturated groundwater flow The relation
with the surface water can be taken into account via radial and
entrance resistances or via a relation between flow to the surface
water system and the hydraulic head The relation with the unsaturated
zone has been restricted to the input factor effective precipitation
In practise however the unsaturated and saturated system are
interconnected Therefore the model should be improved by taking
into account the unsaturated zone This can be done in at least
two ways
a Replacing the unsaturated zone by a parametrical model
b Coupling the unsaturated zone in an iterative way with the
saturated system
In order to find out the usefulness of the numerical model for
water resources management it will be applied to actual regional
groundwater systemslaquoAlso sensitivity analyses will be carried out
The computer model itself can be improved by applying standard
programs eg automatical generation of a nodal grid and graphical
plotting of the results
38
VII REFERENCES
ERNST LF 1962 Grondwaterstromingen in de verzadigde zone en hun
berekening bij aanwezigheid van horizontale evenwijdige open
leidingen Thesis University Utrecht pp 189
1978 Drainage of undulating soils with high groundwater
tables (in press)
FORSYTHE GE and WR WASON 1960 Finite Difference Methods for
Partial Differential Equations John Wiley and Sons New York
pp 444
KRUSEMAN GP and NA DE RIDDER 1976 Analysis and evaluation of
pumping test data ILRI-bulletin No 11 pp 200
LAAT PJM DE and C VAN DEN AKKER 1976 Simulatiemodel voor grond-
waterstroming en verdamping in Modelonderzoek 1971-1974 ten
behoeve van de waterhuishouding in Gelderland deel II
Grondslagen pp 145-197
NEUMAN SP RA FEDDES and E BRESLER 1974 Finite element
simulation of flow in saturated -unsaturated soils considering
water uptake by plants Hydraulic Engineering Laboratory
Technion Israel pp 104
and PA WITHERSPOON 1969a Theory of Flow in a confined
two Aquifer System Water Resources Research Vol 5 No 4
pp 803-816
^_ and PA WITHERSPOON 1969b Applicability of Current Theories
of Flow in leaky aquifers Water Resources Research Vol 5
No 4 pp 817-829
PINDER GF and EO FRIND 1972 Application of Galerkins Procedure
to Aquifer Analysis Water Resources Research Vol 8 no 1
pp 108-120
REMSON I GE HORNBERGER and FJ MOLZ 1971 Numerical Methods
in Subsurface Hydrology John Wiley and Sons New York
pp 389
WESSELING JG 1977 Toepassing van Finite Difference- en Finite
Element Methoden voor het oplossen van een aantal grondwatershy
stromingsproblemen (met computerprogrammas) ICW-nota 944
pp 63
39
ZIENKIEWICZ OC 1971 Finite Element Method in Engineering Science
McGraw-Hill London pp 521
40
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APPENDIX 8 EXAMPLE OF OUTPUT
HEAD AND TERMS OF UATERBALANCE PER NODE PER LAYER
AT TIME 260 IN II AND I13DAYS
T
LAYER NODE KODE HEAD FLOUUN FLOUTS FLOUSS LEAKAGE STORAGE LAT FLOU
13 19 25 31 37 43
125008 125000 12S000 12S000 12S000 12S000 12S000 12S000
0000 0000 0000 0000 0000 0000 0000 0000
0000 0000 0000 0000 0000 0000 0000 0000
0000 0000 0000 0000 0000 0000 0000 0000
- 000 - 000 -000 - 000 - 000 - 000 - 000 0000
0000 0000 0000 0000 0000 0000 0000 0000
0000 000 000 000 000 000 000 000
000 0000 0000 0P100 0000 0000 0000 0000
1 7
13 19 25 31 37
12S000 125000 125000 125000 125000 12S000 125000 125000
0000 0000 0000 0000 0000 0000 0000 0000
- 001 -007 - 023 - 020 - 009 - 003 0000
0000 0000 0000 0000 0000 0000 0000 0000
000
001
007
023
020
009
003
003
0000 0000 0000 0000 0000 0000 0000 0000
1 7
13 19 25 31 37 43
125000 125000 124998 124972 124895 124787 124647 124358
0000 0000 0000 0000 0000 0000 0000 0000
- 000 001 007 023 020 009 003
0000
0000 0000 0000 0000 0000 0000 0000
50000
0000 182
1982 3862 1674 419 078 041
0000 0000 0000 0000 0000 0000 0000 0000
TERMS OF WATERBALANCE PER LAYER DURING LAST TIMESTEP AND
AND SINCE BEGINNING IN M3DAY M3
FLOU TO UNSATURATED ZONE
FLOU TO TERTIARY SYSTEM
0000
0000
0000
0000
LAYER NO 1
FLOU TO SECONDARY SYSTEM
FLOU TO AD3ACENT LAYERCS)
ARTIFICIAL FLOU AND PRESCRIBED BOUNDARY FLOU
ACCUMULATION TERMS
LATERAL FLOU THROUGH PRESCR HEAD BOUNDARY
0000
-000
0000
000
000
0000
-000
0000
000
000
LAYER NO 2
FLOU TO SECONDARY SYSTEM
FLOU TO AD3ACENT LAYERCS1
ARTIFICIAL FLOU AND PRESCRIBED BOUNDARY FLOU
ACCUMULATION TERMS
LATERAL FLOU THROUGH PRESCR HEAD BOUNDARY
0000
-378
0000
381
0000
0000
-055
0000
056
0000
LAYER NO 3
FLOU TO SECONDARY SYSTEM
FLOU TO AD3ACENT LAYERCS5
ARTIFICIAL FLOU AND PRESCRIBED BOUNDARY FLOU
ACCUMULATION TERMS
LATERAL FLOU THROUGH PRESCR HEAD BOUNDARY
0000
370
50000
49376
0000
0000
055
12979
12909
0000
WIHtUtttWttllttlt(ltmtttltHlttltlil(lltlittlfllnl ) bull (
d V U i l i l i ii i Hi il i M M bull f i m
i tMi| i i | J i j j i _ I I bull
aq ui tard
aquifer II
Fig 1 Schematic representatipn of flow direction and variation
of hydraulic head h in a 3-rlayered system
In an unconfined aquifers ( I ) poundhe vertical thickness pf the
waterbearing layer d is a variable while in a confined aquifer II
this is a constant (denoted as D ) Other important properties of
unconfined aquifers are the hydraulic head being the same as the
phreatic surface and the specific yield being different from zero
23 S c h eacute m a t i s a t i o n o f f l o w
The groundwater flow in a groundwater basin is schematized
into a horizontal flow in aquifers and into a vertical flow in
aquitards Consequently the vertical variation of the hydraulic
head is as drawn in figl
The validity of this assumption has )een investigated by
NEUMAN and WITHERSPOON (1969) The found that if the contrast
in hydraulic conductivity between two adjacent layers is bigger
than a factor two the relative error caused by this assumption
is usually smaller than 5
This schematization gives a simplification of the general flow
equation (eq 4 ) In an aquifer the vertical is zero and eq (4)
then reduces to
-Sbdquo -r- 7 - (k d -r-) + 7 mdash (k d 7-) + qd c lt5t ox x ox oy y oy
or
bull(S + S d) g y s ot
-r- (k d -=-) + -T- (k d -=-) + Q ocircx x ox oy y ocircy
(5)
where
Q = volume of water substracted from or added to unit surface
of aquifer per unit time |_ L T J
In an aquitard eq (4) reduces to
-sQ |pound = (k Q) s 6t oz Z OcircZ
(6)
As will be discussed in the next paragraph the term q in an
aquitard is set equal to zero
24 W a t e r b a l a n c e t e r m s
In eq (4) the term q is mentioned With this sink or source
term different terms of the water balance are taken into account
Fig 2 depicts schematically the situation for an unconfined aquifer
(for other layers some terms dont exist)
recharge- - discharge
laquo-b
Fig 2 Schematic representation of the various terms involved in the
water balance
Description of the different terms
a The flow through the phreatic surface q f
If this term is negative (water goes out of the saturated system)
it is called capillary rise If positive it is called percolation
(or drainage)
This flow can be calculated as a rest term of the water balance
of the unsaturated zone Very often this term is expressed as a
flow per unit surface (flux) and is called effective precipitation
For the time being this flux is a given input
b Flow to adjacent aquitards q
The flux to adjacent aquitards can be calculated by applying eq
(6) to the aquitard
c Flow to the primary surface water system q
The primary surface water system penetrates fully the groundwater
basin So it acts as a boundary with given values for the
hydraulic head (is equal to the phreatic level of the primary
system)
d Flow to the secondary surface water system q
The secondary surface water system does not penetrate fully the
groundwater basin which causes a resistance against flow between
groundwater and surface water The flow q can be calculated as
follows
h - h
- mdash f i i gt
where
h = hydraulic head of the groundwater at the place of
the secondary system |_ L |
h = phreatic level of secondary system |_ L J i s -
R = radial and entrance resistance of the
secondary system per unit surface |_ T J
According to ERNST (1962) the radial resistance H per unit
surface can be caKulated with
Uuml = 4 i n 4 e8) TTk B
where
1 = area drained by 1 m conduit L ^ J
d = thickness of the layer near the conduit _ L J
B = wet perimeter of the conduit |_ L J
The entrance resistance is caused by a layer which shows a
lower conductivity around the conduit than its surroundings
In practise radial and entrance resistancies are difficult to
deduct from hydraulic properties They can be found indirectly
by measuring simultaneously q h and h
t s
e Flow to the tertiary surface water system q
In smaller conduits usually the phreatic level is not known
So eq (7) cant be applied The flow to these conduits however
is sometimes very important (eg in areas with shallow groundwater
tables) One may a-sume (ERNST 1978) that a relationship exists
between the hydraulic head of an aquifor and the flow q
According to Ernst the tertiary system can be divided into a
number of categories with respect to their mutual distance and
bottom height For each category the flow is h - h (b)
q ( h b ) - mdash (9)
where
h = average hydraulic head | L J
h (b) = phreatic level in the category under
consideration a function of bottom height |_L J
a = geometry factor to convert h into h _ - J
Y = drainage resistance _ T J
hm = hydraulic head midway between two
conduits |_ L J m
Summation of the q(hb)-relations of the different categories
yields the overall relation between hydraulic head and flow to
the tertiary system we are looking for
f Artificial recharge or discharge q and prescribed boundary
flux qfo
The magnitudes of these flows are not influenced by the conditions
inside the groundwater basin but remain on a prescribed level
Thus q is an internal boundary condition q an external boundary cl D
condition
For confined aquifers some of the above mentioned terms dont
exist These terms are the flow through the phreatic surface qf
and the flow to the tertiary surface water system q
Aquitards are considered only as transmitters of water from one
aquifer to another with possibilities of storage (eq 6 ) This means
that for aquitards the water balance terms ar through f are set
equal to zero
25 S o l u t i o n o f g r o u n d w a t e r f l o w p r o b l e m s
The basic equations for saturated groundwater flow are eq (5)
for aquifers and eq (6) for aquitards The solution of these
equations require knowledge of the physical system and the auxiliary
conditions describing the system constraints
These auxiliary conditions are (REMSON etal 1971)
a Geometry of the system
Eg the diviation of the system into a restricted number of
layers and the horizontal extension of the system under
consideration
b The matrix of hydraulic properties including in-homogeneity
and an-isotropy
c Initial conditions Because we are dealing with non-stationary
flow the values of pertinent system variables (such as hydraulic
head at the initial time) must be known
d Boundary conditions They describe the conditions at the boundaries
pf the system as a function of time
One can distinguish between
- hydraulic head boundary
- flow (including zero flow) boundary
- both head and flow boundary
After specifying the auxiliary conditions an unique solution of
the variation of the hydraulic head in space and time can be
obtained
For special shapes of flow and assumptions exact analytical
solutions exist In nature however one is usually confronted with
problems like
- spacial variation in the hydraulic properties of the soil
- irregular shape of the groundwater basin
- empirical relations between q and h
- d not being a constant
For such situations by numerical and analogy models only an
approximation of the actual situation can be obtained In the next
chapter a numerical solution called the finite element method is
treated
III NUMERICAL SOLUTIONS
In this chapter the numerical solution of the aquifer equation
(eq 5) and the aquitard equation (eq 6) will be treated The
method of solving is the finite element method In section 31 only
a short mathematical description of this method will be given For
more mathematical background the reader is referred to publications
on this subject (eg ZIENKIEWICZ 1971 NEUMAN etal 1974 and
WESSELING 1976)
In section 32 the application of the finite element method on
the flow in groundwater basins is treated while in section 33 the
translation into a computer program is given
3 1 F i n i t e e l e m e n t m e t h o d
For reasons of simplicity the finite element method will be
explained for the stationary version of eq 5 ie
10
(k d^) + (k d ^ ) - Q = O (10) ocircx x 6x 6y y Sy
The basic idea of each numerical solution of eq (10) is to
apply this equation on a small but finite volume The flow region
of interest R is divided into a finite number of volumes called
elements The elements can have different shapes but for reasons
of conveniency we only use triangles- Quadrilaterals can also be
used because they can be divided into two triangles The sizes of
the triangles may vary They can be adapted according to the
geometry of the region and the accuracy desired (fig 11 gives an
example of the diviation of a region into triangles and quadrilaterals)
The corner points of the elements represent the nodal points
If there are N nodal points then the continuous variation of h
in region R is approximated by the expression
h (x_v) ~ h f (xy) (11) a n n
where
h (AV) = approximation of u (xy) a
h = value of h in nodal point n n
f ltxgty) = global coordinate function
In nodal point n a must be equal to h so f (x y ) = 1 and a n n n n
f (xy) = 0 i n Between f (x v ) and f (xy) of the nodal n j ] n n n n j j
points surrounding nodal point n f (xy) varies linearly from
1 to 0
In fig 3 a picture of f (xy) is given n
If one knows the values h in the nodal points n we know the approximation h of h We can find these values of b bv using for
a n
example the Galerkin method
Galerkins principle states that an approximate solution of h in
eq (10) can be obtained from the following system of equations
N ~~ (k d -A + -^- (k d T-y E h f (xy) f(xy)dxdy +
-ox x ox ocircy y oy - _ n n i R n~
J J Q f(xy) dx dy = 0 i = 1 N (12)
1 1
Fig 3 The global coordinate function f (xy)
The integration can be performed element by element with the
aid of the coordinate functions f (NEUMAN etal 1974)
For non-stationary problems it is not possible to apply the
Galerkin principle on the time derivative But it is possible
to discretize this derivative and to set up the system of equation
(eq 12) for each finite time step
One obtains in this way a series of sets of laquocriations tha s
an approximation of the non-stationary flow problem
Treatment of boundary conditions
a Initial conditions
In non-stationary flow problern gtne must start with given
values of a at time t = t ri c
b Boundary conditions
- Prescribed head boundary
Nodal points which lie on a prescribed head boundary have a
prescribed value for h bull r Hgte r umber of unknown h s is n n
reduced and accordingly no the number of equations
- Prescribed flux through z-- boundary
If one knows the flux (iuumlr jw or outflow per unit area) and
the area of the boundary oi-r element the inflow or outflow
through the boundary per flemlt=nt can be calculated
One may assume this flow to J bull part of the sink term Q in
eq (12)
12
32 A p p l i c a t i o n o f t h e f i n i t e e l e m e n t
m e t h o d o n t h e f l o w i n g r o u n d w a t e r
b a s i n s
From hydrogeological investigations the geometry and the matrix
of the hydraulic properties are known Knowing the geometry the
flow can be split up into a finite number of layers This holds
also for the nature of the boundaries
Now a nodal grid is superimposed on each layer in such a way
that the nodal points of the different layers are situated right
above each other (see fig 4) and in each layer the nodal point
lies in the middle of this layer
-nodal point
Fig 4 Schematic representation of the nodal points in the
xyz-space
Each layer has N nodal points of which P nodal points have a
prescribed head In each layer remain N-P = N nodal points with
unknown head Eq (12) can be applied to each point with unknown
head in each separate layer
As was stated in the last paragraph for each time step one
can write
13
Ah bdquo ~ -r1 + Q = O in = 12N in n m At yi A h + B (13)
where
A = in
ocircf ocircf ocircf of (k d f mdashri- ~ + k df - ~ -jS) dxdy
J x n ox ox y n oy oy
in (S d + S ) ff dx dy O if i icirc n
s y i n
Q f dx dy
The matrices A B and the vector Q have been defined in in l
elsewhere (NEUMAN etal 1974) A is the conductivity matrix in
B the storage matrix and Q the source vector in l
In order to clarify these rather complicate expressions apply
equation (13) poundo point 4 of fig 5
6 7
Fig 5 Top view of part of the nodal grid
lhl + A42h2 + A43h3 + A44h4 + A45h5 + A46h6 + A47h7 +
h - h
+ B _iuumli i ^ + Q = o 44 t - t x4
n n-1
(14)
where h and h are values of h at time t and t 4n 4n-1 4 n n-1
respectively
14
If the h-values belonging to the A-values are taken at time
t = t we get an equation with one unknown h for time t = t n-1 4n n
The equation can then be solved directly
However this is an explicit formulation which puts strong limitations
on the magnitude of the time step t - t (REMSON etal 1971) n n-1
Values larger than a certain value cause instability With an implicit method the values of h are taken at time
t = t This method is unconditionally stable Now one gets an equation n
with 7 unknowns h through h If one has 7 equations for every 1 n n
point 1 through 7 we can solve the values h i = 17
in
Of all the methods possible the iterative method of Gauss-Seidel
has been chosen The Gauss-Seidel method starts by choosing a first estimate of h at time t = t (usually equal to h ) At each in n j -i in-l iteration step a new value of h is calculated using the last
in calculated values of h If one passes the nodal points from 1 to
i n 7 eq (14) becomes
h-j + 1 = A h^ + 1 + A bdquo h | + 1 + A _ h | + 1 + A h j + A r h j + 4 n 4 1 1 n 4 2 2 n 43 3 n 45 5 n 4 6 6 n
B h B + A h J + - 4 4 4 n ~ + 0 J (A + mdash) A 4 7 h 7 n At n U 4 4 At
(15)
where
h-i = in
j = iteration step or number of iteration
i = number of nodal point n = time = t
n
The iteration process converges if the successive differences k+1 k
between h and h become continuously smaller in in
The iteration process is stopped if at each nodal point the k+1 k
difference between h and h is smaller than a certain prescribed in in
value
15
To accelerate the convergence an overrelaxation factor W is used
(for more information about W see FORSYTHE and WASOW 1960)
k+1 k T7 k+1 k h = h + W (h - h )
in in in in (16)
If W = 1 there is no overrelaxation Usually W varies between 1 and
2 and the optimum value very often is found by trial and error
Eq (15) is valid for points in aquifers The connection with
adjacent layers is made by the term Q because in this term the
flow from adjacent aquitards q is included (two aquifers are always
separated by an aquitard)
Aquitards
We dont apply the finite element method on points in aquitards
As stated before there is no horizontal flow in aquitards So it has
no sense to replace the continuous variation of h in the aquitard
by equation (11)
The nodal points in aquitards represent a certain volume of an
aquitard hich transmit water from one aquifer to another and which
has storage capacity The nodal points are included in the Gauss-
Seidel scheme -s follows (see fig 6 )
1 aauifer
a -_vS laquo 2 [~^-t^yM^^^-^r~ aqutarci l-Z
aquifer
Fig 6 Schematic situation for points in aquitards
Point 2 is situated in the middle of the aquitard with thickness
d storage coefficient S and vertical hvdraulic conductivity k c J v
Suppose the Gauss-Seidel iteration starts in the upper layer and
ends in the bottom-layer Then the flux from point 1 in aquifer 1 to
16
point 2 in the aquitard at time t = t and in iteration step j + 1
is
k q = (hf -hf )-iumlmdash f (17)
1 n 2 n OSxd1
and the flux from point 3 in aquifer 2 to point 2 in the aquitard
is
1 c 1
qbdquo = (h - hf) mdashHmdashr- (18) 2 3 n 2 n OSxd
The net flux ltl-q9 must be equal to the change in storage per unit
area per unit time This change in storage can numerically be
approximated by
te - h-
n n-1
Combina t ion of ( 1 7 ) (18) ana (19) g i v e s
l tr1 - h (hJ + 1 + hj - 2 hj+1) - - - - - - S1 - ilaquoS L u l l = o (20)
] i 3 n 2 n 1 c t - t j j x d n n -1
T + 1
From t h i s e q u a t i o n h^ kan be s o l v e d
3 3 C o m p u t e r o g r a m
The problem outl^ugrave so far has been translated into a computer
program written in FORTRAN IV
The flow chart ot Uis program is given in fig 7 whereas in
appendix A the complete listing of the program can be found
In the flow chart the 5 main parts of the program are
indicated
A Reading and printing of input data
B Determination of the matrices A and B and the vector Q per layer
17
M
1 A
FLOH to unsaturated zono
FLOW to tertiary syster
FLOW to secondnrv system
Leakage to adjacent aquitards
SOLVE equation (14)
yes
CALCULATE flux through constant head boundnr
CALCULATE waterbalance n p r 1 OAro r r J j - ~
PRINT of output
Ier node per layer
k = k+1
t = t + At
7 DUMP variables on tape
-gt-
( STOP )
Fig 7 Flow chart of the program FEMSAT with diviation in 5 main
parts A B C D and E
C START PROGRAM FEMSAT
f READ general information
PRINT general information
f READ element information
PRINT element information 7 READ stationary nodal point information
PRINT stationary nodal point information 7
READ non-stationary nodal point information for t = t
FRINT non-stationary nodal point information for t = t
L 2 (
^ yes R E A D non-stat input for t gt t
J PRINT non-stat input for t gt t (- 1 -- M degY
GENERATE A B and AREA-matrix for confined aquifers
_L
GENERATE A B and AREA-matrix for unconfined aouifer
bull
C Solving h with Gauss-Seidel
D Determination of the water balance terms per layer
E Printing of results
The program exists of a main program with several subroutines
The main program calls the subroutines In the subroutines most
calculations are performed
In this way an easy adaptation to different problems is
obtained (see also section 44) Comments in the program will
help the reader to understand the program
In the next chapter the preparation of the input data executing
of the program and the output will be treated
Acknowledgement
Some parts of the computer program were taken from the program
UNSAT2 of NEUMAN etal (1974)
IV INPUT EXECUTION AND OUTPUT
41 I n p u t
The input is divided into endogenous variables (chosen by the
user or calculated by the program during execution) and exogenous
variables (based on physical data)
411 Endogenous variables
Translation of groundwater basin into a nodal point network
Probably the most important but also the most difficult stage
in the process of simulating groundwater flow is the translation
of the natural groundwater basin into a nodal point network Questions
which arise are
a In how many layer the geo-hydrological system mustcan be
schematized
b The locations of the boundaries and the nature of these boundaries
(impervious prescribed head prescribed flux)
c The configuration of the nodal points
19
The answers of these questions depend on the problem under
consideration
- Schematization into layers
Usually our knowledge about the geo-hydrological situation is
such that no more than 5 different layers can be distinguished
(remember contrast rule)
- Boundary conditions
The top boundary condition of a groundwater basin usually is
a prescribed flux (effective precipitation)
The bottom boundary condition usually is a prescribed flux
boundary with flux zero (impervious hydrological base)
For the boundaries at the other sides of the system one sees
either prescribed head boundaries (eg a fully penetrating
river) or prescribed flux boundaries (including zero flux)
- Nodal point configuration
Nodal points are situated
on the boundaries
along internal boundaries which indicate changes in
hydraulic properties within the layer
along the tertiary surface water system
on places where artificial recharge or discharge wiii tgtke
place
elsewhere
Construction of the finite element mesh
Assume we have schemetized the groundwater basin iatu a finite
number of layers with different direction of flow (horizontal and
vertical) Within one layer the hydrological properties may vary
from place to place
Now a nodal grid is superimposed on each layer in a way already
described in section 32 Thus each layer has the same number of
nodal points situated on the same places in the horizontal plane
but with different z-coordinates
We are free to construct quadrilaterals or triangles because
the program automatically divides a quadrilateral into two triangles
having identical material properties The dimensions of the elements
20
should be small at places where large changes in hydrological
properties occur or where large gradients in hydraulic head are
expected to occur
Coding of nodal points
Each node in the finite element network is assigned an integer
code which indicates the type of calculation for this node
The coding is as follows
- Code = 1 nodal points with described head as function of time
- Code 2 internal nodal points
- Code = 3 internal nodal points which represent secondary surface
water system
- Code = 5 nodal points with prescribed artificial recharge or
discharge and points with prescribed boundary flow
(both as functions of time)
Other important endogenous variables are the magnitude of the
time step the value of the variable which stops the iteration
process and the maximum number of iterations allowed per time step
(see also group A and B of section 413)
412 Exogenous variables
After constructing the nodal grid each node represents part of
the groundwater basin Geo-hydrological properties can be assigned
to each mode based on physical data (group C D and E of section
413) In group F of section 413 time-dependent variables can
be assigned to each node
413 Data input to Program FEMSAT
The data input must be punched on cards according to the
following instructions
GROUP COLUMNS FORMAT SYMBOL DESCRIPTION
A 1-80 20A4 HEAD Desired heading to be printed at
the beginning of the execution
1-5 15 NUMNP number of nodal points
21
GROUP COLUMNS FORMAT SYMBOL DESCRIPTION
6-10 15
11-15 15
16-20 15
21-25 15
26-30 15
31-35 15
NUMEL
NUMLAY
MBAND
NTSPI
NTSPO
MAXIT
1-10 F103 ST
11-20 F103 DELT
21-30 F103 TMAX
31-40 F105 TOL
number of elements
number of layers
maximum difference in node number
between two adjacent nodes
number of time steps between a
change in non-stationary input data
number of time steps between two
successive outputs
maximum number of iterations per
time step
time to start with the execution
initial time interval
maximum time to be reached before
execution is stopped
maximum allowed absolute change in
the values of h between any two
successive iterations in a given
time step
Iterations continue until all changes
are less than or equal to TOL or
until MAXIT is exceeded
1- 5 15
6-10 15
11-15 15
16-20 15
KX(N1)
KX(N2)
KX(N3)
KX(N4)
sequential number of i-th corner of
element N
sequential number of j-th corner
sequential number of k-th corner
sequential number of 1-th corner
(see fig 8)
Fig 8 Numbering of element corner nodes
22
GROUP COLUMNS FORMAcircT SYMBOL DESCRIPTION
One card must be provided for each element in sequential order
starting with N = 1 and ending with N = NVJMEL
I-K 1103 SANG(NLY)
1i-iO MO3 C1(NLY)
21-30 F 103 C2(NLY)
21-4- 1-104 THI(NLY)
angle in degrees between CI and
the laquo-coordinate to he used in
nodal print N in layer LY (see
poundig 9)
First principal conductivity of
material in node N and layer LY
idem second principal conductivity
thickness of layer LY at nodal
point N
ig V Definition of SANGCI and C2
One card icircuiPt be provided for each model point ctarting wih M = 1
and ending with N = NUMNP If there is norlt3 than 1 layer for each
layer a set must be provided starting with the upper layer LY - 1
and ending with the bottom layer LY = NIJMLAY The variables SANG
M and 2 are not relevant tor aquitards but for reasons of
conveniency this procedure is maintained (making SANG CI and C2
dummy variables)
1-10 110 KODE(N)
11-20 II0 NTERR(N)
coding of nodal points (see
paragraph 42)
number of tertiary relation in node N
The q-h relationship of the flow to
23
GROUP COLUMNS FORMAT SYMBOL DESCRIPTION
21-30 FlOl X(N)
31-40 FlOl Y(N)
41-50 FlOl HGL(N)
51-60 FlOl HBSC(N)
61-70 FlOl LESC(N)
the tertiary system as discussed
in par 24 are schematized in a small
number of different relations
x-coardinate of node N
y-coordinate of node N
height ground level in node N
height bottom secondary conduit in
node N If node N is not a secondary
conduit this variable is a dummy
variable
length of secondary conduit in area
represented by node N
One card must be provided for each node starting with N = 1 and
ending with N = NUMNP
E 1-10 FlOl P(NLY1) initial value of h at node N in
layer LY
11-20 FlOl VERRES(NLY) vertical resistance at node N and
layer LY If the layer is an aquifer
this value is a dummy variable
21-30 F101 RADRES(NLY) radial resistance for flow to
secondary system at node N in layer
LY
31-40 FlOl ENRES(NLY) entrance resistance for flow to
secondary system at node N in layer
LY
41-50 FlOl WP(NLY) wet perimeter of secondary system at
node N in layer LY
51-60 FlOl P0R(NLY) specific yield of the material at
node N in layer LY
61-70 FlOl SS(NLY) specific storage of the material
at node N in layer LY
24
GROUP COLUMNS FORMAT SYMBOL DESCRIPTION
One card must be provided for each node starting with N = 1 and
ending with N = NUMNP If there is more than 1 layer for each layer
a set of cards must be provided starting with LY = 1 and ending
with LY = NUMLAY
F 1-10 F101 P(NLY1) value of h at different times at
node N in layer LY Node N lies
on a prescribed head boundary
11-20 F101 WTSC(N) water table in secondary system at
node N at different times
21-30 F 101 ARRED(NLY) artificial recharge to or discharge
from node N in layer LY or
prescribed boundary flow at
different times
31-40 F 101 QPHR(N) flux through phreatic surface at
node N at different times
The data in group F are non-stationary input data One card must be
provided for each node starting with N = 1 and ending with N = NUMNP
In case of more than 1 layer for each layer a set must be provided
Not relevant variables are taken as dummy variables
After a certain number of time steps (NTSPI) the data of group E
are read So the set cards E must contain NxLYxC cards where C =
(TMAXNTSPIXDELT) + 1 if DELT is a constant
42 R u n n i n g t h e p r o g r a m o n t h e C y b e r 7200
The program has been executed on a Cyber 7200 computer of
IWIS-TNO in the Hague
- The demanded input is punched on cards according to the instructions
of the last paragraph and is put on a permanent file
- For each specific problem inthe program only the dimension statements
in the main program have to be adjusted according to the instructions
in the lines of comment
- After compilation the program together with the input file is
executed
25
- Intermediate data necessary for continuation of the calculations
are saved on another permanent file This gives the opportunity
to restart the program at a time the calculation was stopped
The only change in the input file is to adjust the starting time
Remark
With the aid of a terminal small changes in the program and in the
input file can be made easily and the output immediately appears
on the terminal In this way it is possible to stop erroneous
calculations This is especially useful when being in the testing
phase of the program
43 O u t p u t
Immediately after reading the input data they are printed so
that possible errors can be detected During execution after each
certain number of time steps (NTSPO) calculated values of h and
water balance terms are printed In appendix B an example is given
The output data can be devided into two groups
A Data per node per layer
- Coding of the nodal points KODE
- Hydraulic head HEAD
- Flow through phreatic surface at the time indicated in head
of table FLOWUN
- Flow to tertiary surface water system FLOWTS
- Flow to secondary surface water system FLOWSS
- Flow to adjacent layers LEAKAGE
- Artificial recharge or discharge and prescribed boundary
flow ARRED
- Change in storage during last time step STORAGE
- Flow through prescribed head boundary LATFLOW
B Data per layer
Summation of the terms already mentioned under A gives the
magnitude of the water balance terms per layer at the time
indicated (instanteneous water balance terms) Summation of
these instanteneous terms multiplied by their respective time
intervals since time is zero gives the cumulative water balance
terms per layer
26
After each print out we are able to check the calculations
because the sum of the water balance terms per layer must be
approximately zero
44 C o m p u t i n g t i m e
The computing time depends on the number of nodal points and
the number of layers But it depends also on the magnitude of the
changes in hydraulic head and on the value which terminates the
Gauss-Seidel iteration (TOL) To give an impression TOL = 001 m
Costs per day simulation (in hlfl) on the Cyber 7200
number of nodes number of layer bull bdquo bull bull _ 1mdash x a x b (21)
where
a = costs in Dutch guilders per system second (ƒ 027)
b = empirical factor with magnitude about 001
V NUMERICAL EXPERIMENTS
In general there are two possibilities to test the validity of
numerical models The first and most commonly used method is to
compare numerical results with analytical solutions The second
possibility is to compare numerical results with measured field
data Since the last alternative requires many data that are
difficult to obtain we restrict ourselves for the time being to
comparison with the analytical solutions
5 1 F l o w t o a w e l l i n a n u n c o n f i n e d
c i r c u l a r - s h a p e d a q u i f e r
Flow to a well can be divided into steady-state flow and
unsteady-state flow (KRUSEMAN and DE RIDDER 1976)
27
511 Steady state flow in an unconfined aquifer
For this case the formula of Thiem-Dupuit is valid (see fig 10)
Q bull k lth2-hicircgt ln(r2r])
(22)
where
hhbdquo = hydraulic head in 1 and 2 respectively
Q = discharge from well
rbdquo = distance from well
CL31
hmdash^ri --r2
Fig 10 Schematic cross-section of a pumped unconfined aquifer
Fig 11 gives the nodal grid superimposed on a well in a
circular shaped aquifer Near the well the nodal point distance
is made smaller Quadrilaterals as well as triangles are used
Numerical calculations with this finite element network and 3 -1
an extraction rate of 200 m day were performed for 10 days
At that moment a steady-state situation was almost achieved
28
2tmdashnodal point number
160 180 X-axesim)
Fig 11 Nodal point configuration for flow to a well in a circular-
shaped aquifer The outer boundary is taken to be constant
The well is situated in nodal point 42
29
In fig 12 the results of the simulation are compared with the
analytical solution The agreement between numerical results and
the analytical solution is excellent
draw- 15irS (m) down
analytical solution imdashx calculated values
distance from well r (m)
Fig 12 Comparison between analyt ical and calculated s teady-state
drawdown for the problem of f ig 11 with 1
-1 Q bull - 200 m 3 d a y _ 1
k = 10 mday
d = 100 m
S = 2 y
S = 0 s
512 Unsteady-state flow in an unconfined aquifer
Analytical solutions for unsteady-state flow in an unconfined
aquifer only exist if the aquifer extends infinitely For the
aquifer given in fig 11 this condition is not fulfilled However
during the first one or two days after starting the extraction the
influence of a constant hydraulic head at the outer boundary on the
30
drawdown near the well can be neglected (see fig 13) After one
day the lateral inflow is only 25 of the extraction the remainder
being the change in storage due to drawdown
2 0 0
m 3 day 1
Q=200m 3 day
100-
vraquomdash Lateral inflow
N amp storage
I I ^ V - W 10
t (days)
Fig 13 Values of Q lateral inflow and change in storage as
functions of time
The nonsteady-state or Theis equation can be written as
i _ Q s 4iTkD
dy = 4irkD
W(u) (23)
where
u = r2S
4irkDt
r = distance from well
S = storage coefficient
Q = constant discharge from well
D = thickness of the waterbearing layer
at the beginning of pumping
t = time after starting the extraction
s = corrected drawdown
= s - 322D
s = calculated drawdown
W(u) = Theiss well function
[J
CIAT1
[T]
]
31
In fig 14 the analytical unsteady-state solution for the case
described in section 511 is given together with the numerical
results at times 05 and 10 days The agreement between numerical
results and the analytical solution is rather good
Drawdown s1(m)
10
10
10
Theiss solution
x caculated values t=Q5
bdquo t= lO
10 10- 10 10 t r2(daym2)
Fig 14 Comparison between analytical and calculated steady-state
drawdown for the problem of fig 11 with
Q
k
d
S y
s s
=
=
=
=
=
200
1
10
3 m
0
0
2
0
-1 day
m day
m
These results indicate that the assumption of a linear variation
of h between two nodal points (see fig 3 ) even with the rather
rough element network of fig 11 is acceptable So there is no
need to improve for example the finite element network by introducing
higher order coordinate functions
32
52 F l o w t o w e l l i n 3-1 a y e r e d s y s t e m
In fig 15 a schematic cross-section of a 3-layered system of
two aquifers separated by an aquitard is given A well of
infinitisemal radius is completed in the lower aquifer and discharges
at a constant rate QTT Each layer is homogeneous isotropic and
extending radially infinite
bdquo bull - bull bull ~ i
mdash Q II
aquifer I
lugrave^i~IcircIuml7i-i~~ aquitard --r-rl~-
-II--I i - i
aquifer
Fig 15 Schematic cross-section of a 3-layered system
For this situation NEUMAN and WITHERSPOON (1969a) give rather
complicated analytical solutions In the special case where the
hydraulic properties of the two aquifers are identical they give
graphical illustrations of their solutions for three different
cases
These graphical solutions give the opportunity to verify
roughly numerical results of a 3-layered system Therefore a nodal
grid is superimposed on a circular shaped groundwater flow system
with the extraction in the centre of the lower aquifer and with
a constant head on the outer boundary Fig 16 gives the nodal
point situation in the x - y plane of 1 layer The nodes in the
other two layers are situated in the middle of the layers at the
same places in the x - y plane only their z-coordinate is different
33
A
-Jy
bull33
V 5
iumlff
Fig 16 Nodal point configuration for the 3-layered problem (top view)
34
In fig 17 numerical results are compared with the analytical
solution The agreement is rather bad especially for the two
aquifers But in terms of the water balance the difference between
aquifers and aquitard is not so big because the aquitard has
about 16 times bigger storage capacity than the aquifers The
small value of the specific storage of the aquifers is the reason
that small errors in the components of the water balance have a
big influence on the calculated values of the hydraulic head
Eig 17 Dimensionless drawdown vs dimensionless time in a
3-layered system with
II = 50 m day
3 m (distance from well)
aquifer I
aquitard
aquifer II
Thi (m)
100
25
100
S s
001
0642
001
k(mday )
1
0277
1
The discrepancies are partly caused by errors in discretization
In the z-direction these errors can be eliminated by taking the
35
vertical resistance of the aquitard to be infinite Then the
calculated drawdown in the pumped aquifer can be compared with
the analytical Theiss solution
This is done in fig 18 for different distances from the well
One sees that the agreement is better when the distance is bigger
Apparently the linearization of the logarithmic drawdown curve
near the well is responsible for the differences The differences
for r = 3 m are of the same magnitude as for the pumped aquifer in
the 3-layered case
joo gt A 50
20
Theiss solution
bull calculated values r=3m (distance f rom well) x bdquo bdquo r=7m
1 L
I l_ 001 003 0 0 5 Ol Q2
- I I 5 10 t - K t
Fig 18 Dimensionless drawdown vs dimensionless time in a
confined aquifer with
Q =500 m day
k = 10 mday
D = 100 m
S = 001 s
The differences between numerical results and analytical solutions
are not caused by taking a finite area because changing the outer
boundary into an impermeable boundary has only minor effects
36
VI SUMMARY AND CONCLUSIONS
In this report a numerical model for non-stationary saturated
groundwater flow in a multi-layered system has been treated
Groundwater flow can be described in different ways
a Physical-mathematical by combining Darcys law and the law of
continuity (section 21)
b Geo-hydrological the flow in a natural groundwater basin takes
place in layers with different hydraulic properties Definitions
of the most important hydraulic properties are given in section
22
c Schematical groundwater flows horizontally in aquifers and
vertically in aquitards (section 23)
d Approximating by dividing the region of flow into a finite
number of elements and applying Darcys law and the law of
continuity on each of these elements In these ways one gets
a set of equations which can be solved given (initial and)
boundary conditions (section 25)
e Numerical the set of equations has been solved using a Galerkin
type finite element method and the implicit Gauss-Seidel method
with overrelaxation factor (section 31 and 32)
f Computer-solvable the numerical solution has been translated
in a FORTRAN IV program called FEMSAT (section 33)
In chapter IV the input execution and output of the program
FEMSAT are treated
Finally in chapter V some numerical applications are given
simulation of flow to a well in unconfined aquifer and simulation
of flow to a well in a 3-layered configuration
Comparison of hydraulic heads obtained by numerical calculations
and analytical true solutions showed an excellent agreement in case
of stationary flow and a good to moderate agreement in case of non-
stationary flow However the diviations caused by the discretization
process must be seen in relation to the errors made in the
establishment of the hydraulic properties
37
As was stated in the introduction numerical models can be
valuable tools in the management of water resources Therefore
numerical models must be suitable to simulate effects of certain
operations upon the spreading of water over saturated groundwater
unsaturated groundwater and surface water The model described in
this report is a model for saturated groundwater flow The relation
with the surface water can be taken into account via radial and
entrance resistances or via a relation between flow to the surface
water system and the hydraulic head The relation with the unsaturated
zone has been restricted to the input factor effective precipitation
In practise however the unsaturated and saturated system are
interconnected Therefore the model should be improved by taking
into account the unsaturated zone This can be done in at least
two ways
a Replacing the unsaturated zone by a parametrical model
b Coupling the unsaturated zone in an iterative way with the
saturated system
In order to find out the usefulness of the numerical model for
water resources management it will be applied to actual regional
groundwater systemslaquoAlso sensitivity analyses will be carried out
The computer model itself can be improved by applying standard
programs eg automatical generation of a nodal grid and graphical
plotting of the results
38
VII REFERENCES
ERNST LF 1962 Grondwaterstromingen in de verzadigde zone en hun
berekening bij aanwezigheid van horizontale evenwijdige open
leidingen Thesis University Utrecht pp 189
1978 Drainage of undulating soils with high groundwater
tables (in press)
FORSYTHE GE and WR WASON 1960 Finite Difference Methods for
Partial Differential Equations John Wiley and Sons New York
pp 444
KRUSEMAN GP and NA DE RIDDER 1976 Analysis and evaluation of
pumping test data ILRI-bulletin No 11 pp 200
LAAT PJM DE and C VAN DEN AKKER 1976 Simulatiemodel voor grond-
waterstroming en verdamping in Modelonderzoek 1971-1974 ten
behoeve van de waterhuishouding in Gelderland deel II
Grondslagen pp 145-197
NEUMAN SP RA FEDDES and E BRESLER 1974 Finite element
simulation of flow in saturated -unsaturated soils considering
water uptake by plants Hydraulic Engineering Laboratory
Technion Israel pp 104
and PA WITHERSPOON 1969a Theory of Flow in a confined
two Aquifer System Water Resources Research Vol 5 No 4
pp 803-816
^_ and PA WITHERSPOON 1969b Applicability of Current Theories
of Flow in leaky aquifers Water Resources Research Vol 5
No 4 pp 817-829
PINDER GF and EO FRIND 1972 Application of Galerkins Procedure
to Aquifer Analysis Water Resources Research Vol 8 no 1
pp 108-120
REMSON I GE HORNBERGER and FJ MOLZ 1971 Numerical Methods
in Subsurface Hydrology John Wiley and Sons New York
pp 389
WESSELING JG 1977 Toepassing van Finite Difference- en Finite
Element Methoden voor het oplossen van een aantal grondwatershy
stromingsproblemen (met computerprogrammas) ICW-nota 944
pp 63
39
ZIENKIEWICZ OC 1971 Finite Element Method in Engineering Science
McGraw-Hill London pp 521
40
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C 3 C 3 - r 3 r - 3 m H i -z agt z z en m 0 bulllt -a -j-j lt i raquo
n z -bulln C
- t n c r 3 m w o n n r i gt
-gt r - en xraquo 3 2 Tgt C -lt Z 3 bull-lt C
3 K n r -lt Tgt Jgt bull~J 33 bulllt - 30 n m c n o - i -H - o z r raquo c m ^ 3 W Z c a r 3gt
-lt t n
Z C 3 r xraquo
-lt n
- lt -gt w Z - c
3 Z TJ
Z C 3 r Xlaquo
-lt agt
raquo 33 m o
z c 3
i-i en bull o - o z en i gt
Cn 33 TJ -lt 33 w m z - c c r -3 m - i Z Xgt 3gt -o T 3 - - 33 z xraquo m C 33 O 3 33 -i - m n x o w -lt raquo H - 3gt O hJ Ci 33 w n -- - mdashi x c t o O mdash H O O m 33
z n C v 3 Z
-n en r c ( - 00 m 3 iumlgt 0 - C
mdash1
z m
~a 33
z H O
O
Z
3 Z Tgt
C 3
X
-lt en
-
TJ m i n n 33 z m o 0
z H O
mdash1 0 0 t-t w 0 lt- - n
o - t z z z z z i - -C -H -H O H 33 t-t raquo1 c w
- 7
Z Z Z - 1 Z -lt Cl C C m m
bull-gt c r - m
-lt 11
z 0 c bullH
r~
-lt + a
z
r -
-lt
w -H Il bull
poundgt Z
Z 3 - Z r - T)
- lt laquo-raquo lt-bull O I C
Xgt mdash1 mdash( CJ 0
Z NJ - s
- -c
C
7C
r -c
h j
x 0 11 -bull
z z + -u r
M bull n
7
i bulllt r
n 0
- t
Z CT
1mdash
-lt 1
gt U
7
fmdash bull lt
-0
-lt ro
- 1 0
s
T D O H H
z 0
ro
O M O ^ O (Si - lt - n o - n C T gt - c z laquo-raquo r- 3 03 - X tO C gt Z m 33
u c o r o c D m c o c z o
+ u n
ro
3 œ Xraquo
- c 0 33 3 to c u f f l z i r z M H 11 -i 1 z
1gt bull-bull
11 m bulllt -0 0 w j - i n
z - j z c -- m 3 r
z z gt 0
r laquo_ -lt 0 lt-raquo 0
TJ -H -bull O Z - t o r CD - c
ro
Il - z z bullbull z m - C X Z 3 laquo -n C T 3 H
Z -C 3 3gt 03 X TJ w r~ -lt x a
2gt - z gt-raquo Cl -lt JO 0 -H
CD
Cl
-H O
S
LJ - 0 - z 0 o c m lt-bull 3 -z z -0 c -0 -3 - Z Z Z C O C 3 - 3 Z z r -0 c xgt -3 -lt Z ~ -J C 3gt - 3 -lt -H ~ s-gt -H 3gt - Z -lt c 0 -
laquo bull
T 1 T 1 3 H H T 1 H z m n - n 0 - n
X O H lt-bull r- 33 r- O C C TJ 3 TJ
33 m gt-raquo i gt - Z 33 Z - 1 Z
33 - -gt -m raquo v j t o - -raquo -laquo-bull W CD ro Z u x laquo_
r - c i -c r --lt H - 1
bull X Z 3 ) Z C l m r - z u bull 1-1 0 -o (S Z O 33 bull u f t l M w w bull z C t H H 3 m m z - w c i ro t o Iuml gt j j -11 - X z
a ro 3 U N m ii t n Cl X xgt c i t n 1 - 3 r o - 3gt bull z I -
s t 33 r- Z 33 i-t
rt 3gt m r- X - t Z - 33 -lt -1
w - z -0 -bull ro - r w f~ f Ti r - xraquo -n -c
O Uuml I H o i-t - n Z C l H 3 P H-l Xgt bull z u r C tO H
C l - 3 w 3gt Cl
r o n c f l H H c ouml lt-laquo - n
z z a - 1 u i - H 3 T
+ Z II C l ^ I C l f l H
11 m w bull ro c i z
3 C 3 3gt 3
T C O II Z
+ 3 l
O
r o
3
r o
3
) - 03
o z
o o gt c ^ c ^ o ^ o ^ t n t n o i i 0 t n t n y i t n t n i K ^ ^ ^ ^ ^ ^ gt ^ t j c o u c o o J W u x t o w c n t c ^ ^ S s 3 X N ^ o T gt 4 ^ w ^ œ N O O N j ( gt c n t J w ^ a D s 3 œ N i o ^ c n gt c ugrave r o ^ c a s 3 œ N j O W I S l B S Iuml S S iuml I B l S f f l C B l S O O O S I S Œ l S l S i Œ J I S S l C D l S S l S l S icirc S l B i S l I B C C l B C D S i S C D l S l B C D i S i i S Il I Il I raquo Il II li II M li II II II II li H II 11 II II II il II II II h 11 l II II II II II II t |t jj raquo Il M il
t n iS
m -x r -Z T- O c - t z
ja- w ro j s es S (S CD C3 IS
- n - 0 - r j T j - n x T i - O T icirc - o T i - D
an CD
rf 0 n -o - n -0 x C X O 3 3 O 3 3 O 3 3 O 3 0 O 3 3 3 O 3 3 O 3 3 V 3 3 ^ - I 3 3 - 3 3 raquo - I 3 3 gt - I 3 3 ) - ( 3 3 I - I
C - I 3 Z 3 Z 3 Z 3 Z 3 Z 3 Z 33 1-1 Z Z
ff
I gt - f X - - l 3 gt - H X raquo - I gt - H 3 gt - i bull-i mdashl mdash1 mdash1 mdashi mdash1 laquo | J gt - j gt - j ^ - f c gt 0 4 N ( r o - gt j gt i S i N C D r n Œ c i reg t n s t n reg t n s ^o -x - x - x - x - x - x r
r - w 2 gt uuml l X gt ] gt T i - n T i - r j r -raquo z n n icirc s j r r r r raquo H o n c - i iuml o r o i - lt m c c s w m c n c c n m raquo bull H j ^ - n o xgt ^ 3 3 Tgt lt-r tz mdash w iuml- i - t m H r r - r - r - - lt r r c r - o - lt z
bulllt x _bull s- -lt -lt _ C3 -n _ - i - _ rgt _ bull t n - -i - - M rlt r - c - m n -gt o n o œ n u n n - n ^ c - z - i ^ 3 gt 3 gt - n o r - c a
z o r s n r z u icirc N H O - 3 3 0 3 3 m f - O f J l w i c ^ ^ c m z ^ raquo f x mdash 33 r- 0 - 1 p s | -O bull-raquo 3 bulllt 3gt - - - raquo - lt - lt
3 J M raquo H _ - 3 Z 3 Z t n 3gt H Xraquo- -H S - 1 -H
I 03 f - NI r -fc copy bull 3D -lt t n t n -il X ~i X ~i -raquo - n - n - T U mdash n r z r - - t r c c 0 cn a z 3 c - e - - n a i gt -s - n - - n -c 0 r 0 r
mdashf c - i cn c z m 2 33 en H xgt i - H 3gt C 33 33 -lt X
_ 3
-k
33 s c e s t n
j t i T T i - i i n n A Z O S O C O O X d 33
U) 3gt V Z N O bull
t o
Z n m
00 m 0
z z
z C l
z
raquo J J
- -n bullbull-gt r w 33 J Z 2 1gt n bullbull r n
3 Z 3 3 - 1 -H 1 gt - H 1 gt B H H H
V t n Ni
mdash1 m 33 3 t n
- n
c xgt
m x 55
f j
M zgt -n
O)
_ D t 3 - T) C bulln c 0 3= 0 t o 0 x
3= w gt0 3 raquo-i 3 3 r x raquo i gt r - r 0 3 Z u 3 X 3 Z 33 -lt 33
- 1 - i Z Z m bull 33 lt-raquo -lt t n iuml gt
o ^ r - r - c c o z m z m z CD t o t o F I H
t n t n
- n as - T] S) -raquobull bull œ w raquo - t o ro bull-gt CD X
en - n
es
u bull-bull
bull-1 m c -z bull ^ ^ - z r w --lt - 0 - raquo w yO 1mdash1
3gt Z Xgt n - f n 0 n 1-1 [Tl r-i Z 03 Z
-lt -lt icirc-f
- z o 0 - f-1 bull-- z z
J
0 11
w z -0 c 33 3 w Z Z O H -
t gt bull t o -
CS
r
-lt z
7Z O O
H - t S) 3gt - 1
- 1 r o r - r - lt-raquo r o
h ) ^ f m n f i V n i gt t n
J X I - 3gt z c i r c m i gt 3 ~lt r m bulljgt raquo
-lt rgt z raquo 0 3 3 0 m m 0
3 0 0 m
t o H
^ 00 X
iuml gt - 4
-H
3 m
- n
t n
t o
bullbull0 x
z
3
3gt
C -lt Igt - t Ci
APPENDIX 8 EXAMPLE OF OUTPUT
HEAD AND TERMS OF UATERBALANCE PER NODE PER LAYER
AT TIME 260 IN II AND I13DAYS
T
LAYER NODE KODE HEAD FLOUUN FLOUTS FLOUSS LEAKAGE STORAGE LAT FLOU
13 19 25 31 37 43
125008 125000 12S000 12S000 12S000 12S000 12S000 12S000
0000 0000 0000 0000 0000 0000 0000 0000
0000 0000 0000 0000 0000 0000 0000 0000
0000 0000 0000 0000 0000 0000 0000 0000
- 000 - 000 -000 - 000 - 000 - 000 - 000 0000
0000 0000 0000 0000 0000 0000 0000 0000
0000 000 000 000 000 000 000 000
000 0000 0000 0P100 0000 0000 0000 0000
1 7
13 19 25 31 37
12S000 125000 125000 125000 125000 12S000 125000 125000
0000 0000 0000 0000 0000 0000 0000 0000
- 001 -007 - 023 - 020 - 009 - 003 0000
0000 0000 0000 0000 0000 0000 0000 0000
000
001
007
023
020
009
003
003
0000 0000 0000 0000 0000 0000 0000 0000
1 7
13 19 25 31 37 43
125000 125000 124998 124972 124895 124787 124647 124358
0000 0000 0000 0000 0000 0000 0000 0000
- 000 001 007 023 020 009 003
0000
0000 0000 0000 0000 0000 0000 0000
50000
0000 182
1982 3862 1674 419 078 041
0000 0000 0000 0000 0000 0000 0000 0000
TERMS OF WATERBALANCE PER LAYER DURING LAST TIMESTEP AND
AND SINCE BEGINNING IN M3DAY M3
FLOU TO UNSATURATED ZONE
FLOU TO TERTIARY SYSTEM
0000
0000
0000
0000
LAYER NO 1
FLOU TO SECONDARY SYSTEM
FLOU TO AD3ACENT LAYERCS)
ARTIFICIAL FLOU AND PRESCRIBED BOUNDARY FLOU
ACCUMULATION TERMS
LATERAL FLOU THROUGH PRESCR HEAD BOUNDARY
0000
-000
0000
000
000
0000
-000
0000
000
000
LAYER NO 2
FLOU TO SECONDARY SYSTEM
FLOU TO AD3ACENT LAYERCS1
ARTIFICIAL FLOU AND PRESCRIBED BOUNDARY FLOU
ACCUMULATION TERMS
LATERAL FLOU THROUGH PRESCR HEAD BOUNDARY
0000
-378
0000
381
0000
0000
-055
0000
056
0000
LAYER NO 3
FLOU TO SECONDARY SYSTEM
FLOU TO AD3ACENT LAYERCS5
ARTIFICIAL FLOU AND PRESCRIBED BOUNDARY FLOU
ACCUMULATION TERMS
LATERAL FLOU THROUGH PRESCR HEAD BOUNDARY
0000
370
50000
49376
0000
0000
055
12979
12909
0000
bull(S + S d) g y s ot
-r- (k d -=-) + -T- (k d -=-) + Q ocircx x ox oy y ocircy
(5)
where
Q = volume of water substracted from or added to unit surface
of aquifer per unit time |_ L T J
In an aquitard eq (4) reduces to
-sQ |pound = (k Q) s 6t oz Z OcircZ
(6)
As will be discussed in the next paragraph the term q in an
aquitard is set equal to zero
24 W a t e r b a l a n c e t e r m s
In eq (4) the term q is mentioned With this sink or source
term different terms of the water balance are taken into account
Fig 2 depicts schematically the situation for an unconfined aquifer
(for other layers some terms dont exist)
recharge- - discharge
laquo-b
Fig 2 Schematic representation of the various terms involved in the
water balance
Description of the different terms
a The flow through the phreatic surface q f
If this term is negative (water goes out of the saturated system)
it is called capillary rise If positive it is called percolation
(or drainage)
This flow can be calculated as a rest term of the water balance
of the unsaturated zone Very often this term is expressed as a
flow per unit surface (flux) and is called effective precipitation
For the time being this flux is a given input
b Flow to adjacent aquitards q
The flux to adjacent aquitards can be calculated by applying eq
(6) to the aquitard
c Flow to the primary surface water system q
The primary surface water system penetrates fully the groundwater
basin So it acts as a boundary with given values for the
hydraulic head (is equal to the phreatic level of the primary
system)
d Flow to the secondary surface water system q
The secondary surface water system does not penetrate fully the
groundwater basin which causes a resistance against flow between
groundwater and surface water The flow q can be calculated as
follows
h - h
- mdash f i i gt
where
h = hydraulic head of the groundwater at the place of
the secondary system |_ L |
h = phreatic level of secondary system |_ L J i s -
R = radial and entrance resistance of the
secondary system per unit surface |_ T J
According to ERNST (1962) the radial resistance H per unit
surface can be caKulated with
Uuml = 4 i n 4 e8) TTk B
where
1 = area drained by 1 m conduit L ^ J
d = thickness of the layer near the conduit _ L J
B = wet perimeter of the conduit |_ L J
The entrance resistance is caused by a layer which shows a
lower conductivity around the conduit than its surroundings
In practise radial and entrance resistancies are difficult to
deduct from hydraulic properties They can be found indirectly
by measuring simultaneously q h and h
t s
e Flow to the tertiary surface water system q
In smaller conduits usually the phreatic level is not known
So eq (7) cant be applied The flow to these conduits however
is sometimes very important (eg in areas with shallow groundwater
tables) One may a-sume (ERNST 1978) that a relationship exists
between the hydraulic head of an aquifor and the flow q
According to Ernst the tertiary system can be divided into a
number of categories with respect to their mutual distance and
bottom height For each category the flow is h - h (b)
q ( h b ) - mdash (9)
where
h = average hydraulic head | L J
h (b) = phreatic level in the category under
consideration a function of bottom height |_L J
a = geometry factor to convert h into h _ - J
Y = drainage resistance _ T J
hm = hydraulic head midway between two
conduits |_ L J m
Summation of the q(hb)-relations of the different categories
yields the overall relation between hydraulic head and flow to
the tertiary system we are looking for
f Artificial recharge or discharge q and prescribed boundary
flux qfo
The magnitudes of these flows are not influenced by the conditions
inside the groundwater basin but remain on a prescribed level
Thus q is an internal boundary condition q an external boundary cl D
condition
For confined aquifers some of the above mentioned terms dont
exist These terms are the flow through the phreatic surface qf
and the flow to the tertiary surface water system q
Aquitards are considered only as transmitters of water from one
aquifer to another with possibilities of storage (eq 6 ) This means
that for aquitards the water balance terms ar through f are set
equal to zero
25 S o l u t i o n o f g r o u n d w a t e r f l o w p r o b l e m s
The basic equations for saturated groundwater flow are eq (5)
for aquifers and eq (6) for aquitards The solution of these
equations require knowledge of the physical system and the auxiliary
conditions describing the system constraints
These auxiliary conditions are (REMSON etal 1971)
a Geometry of the system
Eg the diviation of the system into a restricted number of
layers and the horizontal extension of the system under
consideration
b The matrix of hydraulic properties including in-homogeneity
and an-isotropy
c Initial conditions Because we are dealing with non-stationary
flow the values of pertinent system variables (such as hydraulic
head at the initial time) must be known
d Boundary conditions They describe the conditions at the boundaries
pf the system as a function of time
One can distinguish between
- hydraulic head boundary
- flow (including zero flow) boundary
- both head and flow boundary
After specifying the auxiliary conditions an unique solution of
the variation of the hydraulic head in space and time can be
obtained
For special shapes of flow and assumptions exact analytical
solutions exist In nature however one is usually confronted with
problems like
- spacial variation in the hydraulic properties of the soil
- irregular shape of the groundwater basin
- empirical relations between q and h
- d not being a constant
For such situations by numerical and analogy models only an
approximation of the actual situation can be obtained In the next
chapter a numerical solution called the finite element method is
treated
III NUMERICAL SOLUTIONS
In this chapter the numerical solution of the aquifer equation
(eq 5) and the aquitard equation (eq 6) will be treated The
method of solving is the finite element method In section 31 only
a short mathematical description of this method will be given For
more mathematical background the reader is referred to publications
on this subject (eg ZIENKIEWICZ 1971 NEUMAN etal 1974 and
WESSELING 1976)
In section 32 the application of the finite element method on
the flow in groundwater basins is treated while in section 33 the
translation into a computer program is given
3 1 F i n i t e e l e m e n t m e t h o d
For reasons of simplicity the finite element method will be
explained for the stationary version of eq 5 ie
10
(k d^) + (k d ^ ) - Q = O (10) ocircx x 6x 6y y Sy
The basic idea of each numerical solution of eq (10) is to
apply this equation on a small but finite volume The flow region
of interest R is divided into a finite number of volumes called
elements The elements can have different shapes but for reasons
of conveniency we only use triangles- Quadrilaterals can also be
used because they can be divided into two triangles The sizes of
the triangles may vary They can be adapted according to the
geometry of the region and the accuracy desired (fig 11 gives an
example of the diviation of a region into triangles and quadrilaterals)
The corner points of the elements represent the nodal points
If there are N nodal points then the continuous variation of h
in region R is approximated by the expression
h (x_v) ~ h f (xy) (11) a n n
where
h (AV) = approximation of u (xy) a
h = value of h in nodal point n n
f ltxgty) = global coordinate function
In nodal point n a must be equal to h so f (x y ) = 1 and a n n n n
f (xy) = 0 i n Between f (x v ) and f (xy) of the nodal n j ] n n n n j j
points surrounding nodal point n f (xy) varies linearly from
1 to 0
In fig 3 a picture of f (xy) is given n
If one knows the values h in the nodal points n we know the approximation h of h We can find these values of b bv using for
a n
example the Galerkin method
Galerkins principle states that an approximate solution of h in
eq (10) can be obtained from the following system of equations
N ~~ (k d -A + -^- (k d T-y E h f (xy) f(xy)dxdy +
-ox x ox ocircy y oy - _ n n i R n~
J J Q f(xy) dx dy = 0 i = 1 N (12)
1 1
Fig 3 The global coordinate function f (xy)
The integration can be performed element by element with the
aid of the coordinate functions f (NEUMAN etal 1974)
For non-stationary problems it is not possible to apply the
Galerkin principle on the time derivative But it is possible
to discretize this derivative and to set up the system of equation
(eq 12) for each finite time step
One obtains in this way a series of sets of laquocriations tha s
an approximation of the non-stationary flow problem
Treatment of boundary conditions
a Initial conditions
In non-stationary flow problern gtne must start with given
values of a at time t = t ri c
b Boundary conditions
- Prescribed head boundary
Nodal points which lie on a prescribed head boundary have a
prescribed value for h bull r Hgte r umber of unknown h s is n n
reduced and accordingly no the number of equations
- Prescribed flux through z-- boundary
If one knows the flux (iuumlr jw or outflow per unit area) and
the area of the boundary oi-r element the inflow or outflow
through the boundary per flemlt=nt can be calculated
One may assume this flow to J bull part of the sink term Q in
eq (12)
12
32 A p p l i c a t i o n o f t h e f i n i t e e l e m e n t
m e t h o d o n t h e f l o w i n g r o u n d w a t e r
b a s i n s
From hydrogeological investigations the geometry and the matrix
of the hydraulic properties are known Knowing the geometry the
flow can be split up into a finite number of layers This holds
also for the nature of the boundaries
Now a nodal grid is superimposed on each layer in such a way
that the nodal points of the different layers are situated right
above each other (see fig 4) and in each layer the nodal point
lies in the middle of this layer
-nodal point
Fig 4 Schematic representation of the nodal points in the
xyz-space
Each layer has N nodal points of which P nodal points have a
prescribed head In each layer remain N-P = N nodal points with
unknown head Eq (12) can be applied to each point with unknown
head in each separate layer
As was stated in the last paragraph for each time step one
can write
13
Ah bdquo ~ -r1 + Q = O in = 12N in n m At yi A h + B (13)
where
A = in
ocircf ocircf ocircf of (k d f mdashri- ~ + k df - ~ -jS) dxdy
J x n ox ox y n oy oy
in (S d + S ) ff dx dy O if i icirc n
s y i n
Q f dx dy
The matrices A B and the vector Q have been defined in in l
elsewhere (NEUMAN etal 1974) A is the conductivity matrix in
B the storage matrix and Q the source vector in l
In order to clarify these rather complicate expressions apply
equation (13) poundo point 4 of fig 5
6 7
Fig 5 Top view of part of the nodal grid
lhl + A42h2 + A43h3 + A44h4 + A45h5 + A46h6 + A47h7 +
h - h
+ B _iuumli i ^ + Q = o 44 t - t x4
n n-1
(14)
where h and h are values of h at time t and t 4n 4n-1 4 n n-1
respectively
14
If the h-values belonging to the A-values are taken at time
t = t we get an equation with one unknown h for time t = t n-1 4n n
The equation can then be solved directly
However this is an explicit formulation which puts strong limitations
on the magnitude of the time step t - t (REMSON etal 1971) n n-1
Values larger than a certain value cause instability With an implicit method the values of h are taken at time
t = t This method is unconditionally stable Now one gets an equation n
with 7 unknowns h through h If one has 7 equations for every 1 n n
point 1 through 7 we can solve the values h i = 17
in
Of all the methods possible the iterative method of Gauss-Seidel
has been chosen The Gauss-Seidel method starts by choosing a first estimate of h at time t = t (usually equal to h ) At each in n j -i in-l iteration step a new value of h is calculated using the last
in calculated values of h If one passes the nodal points from 1 to
i n 7 eq (14) becomes
h-j + 1 = A h^ + 1 + A bdquo h | + 1 + A _ h | + 1 + A h j + A r h j + 4 n 4 1 1 n 4 2 2 n 43 3 n 45 5 n 4 6 6 n
B h B + A h J + - 4 4 4 n ~ + 0 J (A + mdash) A 4 7 h 7 n At n U 4 4 At
(15)
where
h-i = in
j = iteration step or number of iteration
i = number of nodal point n = time = t
n
The iteration process converges if the successive differences k+1 k
between h and h become continuously smaller in in
The iteration process is stopped if at each nodal point the k+1 k
difference between h and h is smaller than a certain prescribed in in
value
15
To accelerate the convergence an overrelaxation factor W is used
(for more information about W see FORSYTHE and WASOW 1960)
k+1 k T7 k+1 k h = h + W (h - h )
in in in in (16)
If W = 1 there is no overrelaxation Usually W varies between 1 and
2 and the optimum value very often is found by trial and error
Eq (15) is valid for points in aquifers The connection with
adjacent layers is made by the term Q because in this term the
flow from adjacent aquitards q is included (two aquifers are always
separated by an aquitard)
Aquitards
We dont apply the finite element method on points in aquitards
As stated before there is no horizontal flow in aquitards So it has
no sense to replace the continuous variation of h in the aquitard
by equation (11)
The nodal points in aquitards represent a certain volume of an
aquitard hich transmit water from one aquifer to another and which
has storage capacity The nodal points are included in the Gauss-
Seidel scheme -s follows (see fig 6 )
1 aauifer
a -_vS laquo 2 [~^-t^yM^^^-^r~ aqutarci l-Z
aquifer
Fig 6 Schematic situation for points in aquitards
Point 2 is situated in the middle of the aquitard with thickness
d storage coefficient S and vertical hvdraulic conductivity k c J v
Suppose the Gauss-Seidel iteration starts in the upper layer and
ends in the bottom-layer Then the flux from point 1 in aquifer 1 to
16
point 2 in the aquitard at time t = t and in iteration step j + 1
is
k q = (hf -hf )-iumlmdash f (17)
1 n 2 n OSxd1
and the flux from point 3 in aquifer 2 to point 2 in the aquitard
is
1 c 1
qbdquo = (h - hf) mdashHmdashr- (18) 2 3 n 2 n OSxd
The net flux ltl-q9 must be equal to the change in storage per unit
area per unit time This change in storage can numerically be
approximated by
te - h-
n n-1
Combina t ion of ( 1 7 ) (18) ana (19) g i v e s
l tr1 - h (hJ + 1 + hj - 2 hj+1) - - - - - - S1 - ilaquoS L u l l = o (20)
] i 3 n 2 n 1 c t - t j j x d n n -1
T + 1
From t h i s e q u a t i o n h^ kan be s o l v e d
3 3 C o m p u t e r o g r a m
The problem outl^ugrave so far has been translated into a computer
program written in FORTRAN IV
The flow chart ot Uis program is given in fig 7 whereas in
appendix A the complete listing of the program can be found
In the flow chart the 5 main parts of the program are
indicated
A Reading and printing of input data
B Determination of the matrices A and B and the vector Q per layer
17
M
1 A
FLOH to unsaturated zono
FLOW to tertiary syster
FLOW to secondnrv system
Leakage to adjacent aquitards
SOLVE equation (14)
yes
CALCULATE flux through constant head boundnr
CALCULATE waterbalance n p r 1 OAro r r J j - ~
PRINT of output
Ier node per layer
k = k+1
t = t + At
7 DUMP variables on tape
-gt-
( STOP )
Fig 7 Flow chart of the program FEMSAT with diviation in 5 main
parts A B C D and E
C START PROGRAM FEMSAT
f READ general information
PRINT general information
f READ element information
PRINT element information 7 READ stationary nodal point information
PRINT stationary nodal point information 7
READ non-stationary nodal point information for t = t
FRINT non-stationary nodal point information for t = t
L 2 (
^ yes R E A D non-stat input for t gt t
J PRINT non-stat input for t gt t (- 1 -- M degY
GENERATE A B and AREA-matrix for confined aquifers
_L
GENERATE A B and AREA-matrix for unconfined aouifer
bull
C Solving h with Gauss-Seidel
D Determination of the water balance terms per layer
E Printing of results
The program exists of a main program with several subroutines
The main program calls the subroutines In the subroutines most
calculations are performed
In this way an easy adaptation to different problems is
obtained (see also section 44) Comments in the program will
help the reader to understand the program
In the next chapter the preparation of the input data executing
of the program and the output will be treated
Acknowledgement
Some parts of the computer program were taken from the program
UNSAT2 of NEUMAN etal (1974)
IV INPUT EXECUTION AND OUTPUT
41 I n p u t
The input is divided into endogenous variables (chosen by the
user or calculated by the program during execution) and exogenous
variables (based on physical data)
411 Endogenous variables
Translation of groundwater basin into a nodal point network
Probably the most important but also the most difficult stage
in the process of simulating groundwater flow is the translation
of the natural groundwater basin into a nodal point network Questions
which arise are
a In how many layer the geo-hydrological system mustcan be
schematized
b The locations of the boundaries and the nature of these boundaries
(impervious prescribed head prescribed flux)
c The configuration of the nodal points
19
The answers of these questions depend on the problem under
consideration
- Schematization into layers
Usually our knowledge about the geo-hydrological situation is
such that no more than 5 different layers can be distinguished
(remember contrast rule)
- Boundary conditions
The top boundary condition of a groundwater basin usually is
a prescribed flux (effective precipitation)
The bottom boundary condition usually is a prescribed flux
boundary with flux zero (impervious hydrological base)
For the boundaries at the other sides of the system one sees
either prescribed head boundaries (eg a fully penetrating
river) or prescribed flux boundaries (including zero flux)
- Nodal point configuration
Nodal points are situated
on the boundaries
along internal boundaries which indicate changes in
hydraulic properties within the layer
along the tertiary surface water system
on places where artificial recharge or discharge wiii tgtke
place
elsewhere
Construction of the finite element mesh
Assume we have schemetized the groundwater basin iatu a finite
number of layers with different direction of flow (horizontal and
vertical) Within one layer the hydrological properties may vary
from place to place
Now a nodal grid is superimposed on each layer in a way already
described in section 32 Thus each layer has the same number of
nodal points situated on the same places in the horizontal plane
but with different z-coordinates
We are free to construct quadrilaterals or triangles because
the program automatically divides a quadrilateral into two triangles
having identical material properties The dimensions of the elements
20
should be small at places where large changes in hydrological
properties occur or where large gradients in hydraulic head are
expected to occur
Coding of nodal points
Each node in the finite element network is assigned an integer
code which indicates the type of calculation for this node
The coding is as follows
- Code = 1 nodal points with described head as function of time
- Code 2 internal nodal points
- Code = 3 internal nodal points which represent secondary surface
water system
- Code = 5 nodal points with prescribed artificial recharge or
discharge and points with prescribed boundary flow
(both as functions of time)
Other important endogenous variables are the magnitude of the
time step the value of the variable which stops the iteration
process and the maximum number of iterations allowed per time step
(see also group A and B of section 413)
412 Exogenous variables
After constructing the nodal grid each node represents part of
the groundwater basin Geo-hydrological properties can be assigned
to each mode based on physical data (group C D and E of section
413) In group F of section 413 time-dependent variables can
be assigned to each node
413 Data input to Program FEMSAT
The data input must be punched on cards according to the
following instructions
GROUP COLUMNS FORMAT SYMBOL DESCRIPTION
A 1-80 20A4 HEAD Desired heading to be printed at
the beginning of the execution
1-5 15 NUMNP number of nodal points
21
GROUP COLUMNS FORMAT SYMBOL DESCRIPTION
6-10 15
11-15 15
16-20 15
21-25 15
26-30 15
31-35 15
NUMEL
NUMLAY
MBAND
NTSPI
NTSPO
MAXIT
1-10 F103 ST
11-20 F103 DELT
21-30 F103 TMAX
31-40 F105 TOL
number of elements
number of layers
maximum difference in node number
between two adjacent nodes
number of time steps between a
change in non-stationary input data
number of time steps between two
successive outputs
maximum number of iterations per
time step
time to start with the execution
initial time interval
maximum time to be reached before
execution is stopped
maximum allowed absolute change in
the values of h between any two
successive iterations in a given
time step
Iterations continue until all changes
are less than or equal to TOL or
until MAXIT is exceeded
1- 5 15
6-10 15
11-15 15
16-20 15
KX(N1)
KX(N2)
KX(N3)
KX(N4)
sequential number of i-th corner of
element N
sequential number of j-th corner
sequential number of k-th corner
sequential number of 1-th corner
(see fig 8)
Fig 8 Numbering of element corner nodes
22
GROUP COLUMNS FORMAcircT SYMBOL DESCRIPTION
One card must be provided for each element in sequential order
starting with N = 1 and ending with N = NVJMEL
I-K 1103 SANG(NLY)
1i-iO MO3 C1(NLY)
21-30 F 103 C2(NLY)
21-4- 1-104 THI(NLY)
angle in degrees between CI and
the laquo-coordinate to he used in
nodal print N in layer LY (see
poundig 9)
First principal conductivity of
material in node N and layer LY
idem second principal conductivity
thickness of layer LY at nodal
point N
ig V Definition of SANGCI and C2
One card icircuiPt be provided for each model point ctarting wih M = 1
and ending with N = NUMNP If there is norlt3 than 1 layer for each
layer a set must be provided starting with the upper layer LY - 1
and ending with the bottom layer LY = NIJMLAY The variables SANG
M and 2 are not relevant tor aquitards but for reasons of
conveniency this procedure is maintained (making SANG CI and C2
dummy variables)
1-10 110 KODE(N)
11-20 II0 NTERR(N)
coding of nodal points (see
paragraph 42)
number of tertiary relation in node N
The q-h relationship of the flow to
23
GROUP COLUMNS FORMAT SYMBOL DESCRIPTION
21-30 FlOl X(N)
31-40 FlOl Y(N)
41-50 FlOl HGL(N)
51-60 FlOl HBSC(N)
61-70 FlOl LESC(N)
the tertiary system as discussed
in par 24 are schematized in a small
number of different relations
x-coardinate of node N
y-coordinate of node N
height ground level in node N
height bottom secondary conduit in
node N If node N is not a secondary
conduit this variable is a dummy
variable
length of secondary conduit in area
represented by node N
One card must be provided for each node starting with N = 1 and
ending with N = NUMNP
E 1-10 FlOl P(NLY1) initial value of h at node N in
layer LY
11-20 FlOl VERRES(NLY) vertical resistance at node N and
layer LY If the layer is an aquifer
this value is a dummy variable
21-30 F101 RADRES(NLY) radial resistance for flow to
secondary system at node N in layer
LY
31-40 FlOl ENRES(NLY) entrance resistance for flow to
secondary system at node N in layer
LY
41-50 FlOl WP(NLY) wet perimeter of secondary system at
node N in layer LY
51-60 FlOl P0R(NLY) specific yield of the material at
node N in layer LY
61-70 FlOl SS(NLY) specific storage of the material
at node N in layer LY
24
GROUP COLUMNS FORMAT SYMBOL DESCRIPTION
One card must be provided for each node starting with N = 1 and
ending with N = NUMNP If there is more than 1 layer for each layer
a set of cards must be provided starting with LY = 1 and ending
with LY = NUMLAY
F 1-10 F101 P(NLY1) value of h at different times at
node N in layer LY Node N lies
on a prescribed head boundary
11-20 F101 WTSC(N) water table in secondary system at
node N at different times
21-30 F 101 ARRED(NLY) artificial recharge to or discharge
from node N in layer LY or
prescribed boundary flow at
different times
31-40 F 101 QPHR(N) flux through phreatic surface at
node N at different times
The data in group F are non-stationary input data One card must be
provided for each node starting with N = 1 and ending with N = NUMNP
In case of more than 1 layer for each layer a set must be provided
Not relevant variables are taken as dummy variables
After a certain number of time steps (NTSPI) the data of group E
are read So the set cards E must contain NxLYxC cards where C =
(TMAXNTSPIXDELT) + 1 if DELT is a constant
42 R u n n i n g t h e p r o g r a m o n t h e C y b e r 7200
The program has been executed on a Cyber 7200 computer of
IWIS-TNO in the Hague
- The demanded input is punched on cards according to the instructions
of the last paragraph and is put on a permanent file
- For each specific problem inthe program only the dimension statements
in the main program have to be adjusted according to the instructions
in the lines of comment
- After compilation the program together with the input file is
executed
25
- Intermediate data necessary for continuation of the calculations
are saved on another permanent file This gives the opportunity
to restart the program at a time the calculation was stopped
The only change in the input file is to adjust the starting time
Remark
With the aid of a terminal small changes in the program and in the
input file can be made easily and the output immediately appears
on the terminal In this way it is possible to stop erroneous
calculations This is especially useful when being in the testing
phase of the program
43 O u t p u t
Immediately after reading the input data they are printed so
that possible errors can be detected During execution after each
certain number of time steps (NTSPO) calculated values of h and
water balance terms are printed In appendix B an example is given
The output data can be devided into two groups
A Data per node per layer
- Coding of the nodal points KODE
- Hydraulic head HEAD
- Flow through phreatic surface at the time indicated in head
of table FLOWUN
- Flow to tertiary surface water system FLOWTS
- Flow to secondary surface water system FLOWSS
- Flow to adjacent layers LEAKAGE
- Artificial recharge or discharge and prescribed boundary
flow ARRED
- Change in storage during last time step STORAGE
- Flow through prescribed head boundary LATFLOW
B Data per layer
Summation of the terms already mentioned under A gives the
magnitude of the water balance terms per layer at the time
indicated (instanteneous water balance terms) Summation of
these instanteneous terms multiplied by their respective time
intervals since time is zero gives the cumulative water balance
terms per layer
26
After each print out we are able to check the calculations
because the sum of the water balance terms per layer must be
approximately zero
44 C o m p u t i n g t i m e
The computing time depends on the number of nodal points and
the number of layers But it depends also on the magnitude of the
changes in hydraulic head and on the value which terminates the
Gauss-Seidel iteration (TOL) To give an impression TOL = 001 m
Costs per day simulation (in hlfl) on the Cyber 7200
number of nodes number of layer bull bdquo bull bull _ 1mdash x a x b (21)
where
a = costs in Dutch guilders per system second (ƒ 027)
b = empirical factor with magnitude about 001
V NUMERICAL EXPERIMENTS
In general there are two possibilities to test the validity of
numerical models The first and most commonly used method is to
compare numerical results with analytical solutions The second
possibility is to compare numerical results with measured field
data Since the last alternative requires many data that are
difficult to obtain we restrict ourselves for the time being to
comparison with the analytical solutions
5 1 F l o w t o a w e l l i n a n u n c o n f i n e d
c i r c u l a r - s h a p e d a q u i f e r
Flow to a well can be divided into steady-state flow and
unsteady-state flow (KRUSEMAN and DE RIDDER 1976)
27
511 Steady state flow in an unconfined aquifer
For this case the formula of Thiem-Dupuit is valid (see fig 10)
Q bull k lth2-hicircgt ln(r2r])
(22)
where
hhbdquo = hydraulic head in 1 and 2 respectively
Q = discharge from well
rbdquo = distance from well
CL31
hmdash^ri --r2
Fig 10 Schematic cross-section of a pumped unconfined aquifer
Fig 11 gives the nodal grid superimposed on a well in a
circular shaped aquifer Near the well the nodal point distance
is made smaller Quadrilaterals as well as triangles are used
Numerical calculations with this finite element network and 3 -1
an extraction rate of 200 m day were performed for 10 days
At that moment a steady-state situation was almost achieved
28
2tmdashnodal point number
160 180 X-axesim)
Fig 11 Nodal point configuration for flow to a well in a circular-
shaped aquifer The outer boundary is taken to be constant
The well is situated in nodal point 42
29
In fig 12 the results of the simulation are compared with the
analytical solution The agreement between numerical results and
the analytical solution is excellent
draw- 15irS (m) down
analytical solution imdashx calculated values
distance from well r (m)
Fig 12 Comparison between analyt ical and calculated s teady-state
drawdown for the problem of f ig 11 with 1
-1 Q bull - 200 m 3 d a y _ 1
k = 10 mday
d = 100 m
S = 2 y
S = 0 s
512 Unsteady-state flow in an unconfined aquifer
Analytical solutions for unsteady-state flow in an unconfined
aquifer only exist if the aquifer extends infinitely For the
aquifer given in fig 11 this condition is not fulfilled However
during the first one or two days after starting the extraction the
influence of a constant hydraulic head at the outer boundary on the
30
drawdown near the well can be neglected (see fig 13) After one
day the lateral inflow is only 25 of the extraction the remainder
being the change in storage due to drawdown
2 0 0
m 3 day 1
Q=200m 3 day
100-
vraquomdash Lateral inflow
N amp storage
I I ^ V - W 10
t (days)
Fig 13 Values of Q lateral inflow and change in storage as
functions of time
The nonsteady-state or Theis equation can be written as
i _ Q s 4iTkD
dy = 4irkD
W(u) (23)
where
u = r2S
4irkDt
r = distance from well
S = storage coefficient
Q = constant discharge from well
D = thickness of the waterbearing layer
at the beginning of pumping
t = time after starting the extraction
s = corrected drawdown
= s - 322D
s = calculated drawdown
W(u) = Theiss well function
[J
CIAT1
[T]
]
31
In fig 14 the analytical unsteady-state solution for the case
described in section 511 is given together with the numerical
results at times 05 and 10 days The agreement between numerical
results and the analytical solution is rather good
Drawdown s1(m)
10
10
10
Theiss solution
x caculated values t=Q5
bdquo t= lO
10 10- 10 10 t r2(daym2)
Fig 14 Comparison between analytical and calculated steady-state
drawdown for the problem of fig 11 with
Q
k
d
S y
s s
=
=
=
=
=
200
1
10
3 m
0
0
2
0
-1 day
m day
m
These results indicate that the assumption of a linear variation
of h between two nodal points (see fig 3 ) even with the rather
rough element network of fig 11 is acceptable So there is no
need to improve for example the finite element network by introducing
higher order coordinate functions
32
52 F l o w t o w e l l i n 3-1 a y e r e d s y s t e m
In fig 15 a schematic cross-section of a 3-layered system of
two aquifers separated by an aquitard is given A well of
infinitisemal radius is completed in the lower aquifer and discharges
at a constant rate QTT Each layer is homogeneous isotropic and
extending radially infinite
bdquo bull - bull bull ~ i
mdash Q II
aquifer I
lugrave^i~IcircIuml7i-i~~ aquitard --r-rl~-
-II--I i - i
aquifer
Fig 15 Schematic cross-section of a 3-layered system
For this situation NEUMAN and WITHERSPOON (1969a) give rather
complicated analytical solutions In the special case where the
hydraulic properties of the two aquifers are identical they give
graphical illustrations of their solutions for three different
cases
These graphical solutions give the opportunity to verify
roughly numerical results of a 3-layered system Therefore a nodal
grid is superimposed on a circular shaped groundwater flow system
with the extraction in the centre of the lower aquifer and with
a constant head on the outer boundary Fig 16 gives the nodal
point situation in the x - y plane of 1 layer The nodes in the
other two layers are situated in the middle of the layers at the
same places in the x - y plane only their z-coordinate is different
33
A
-Jy
bull33
V 5
iumlff
Fig 16 Nodal point configuration for the 3-layered problem (top view)
34
In fig 17 numerical results are compared with the analytical
solution The agreement is rather bad especially for the two
aquifers But in terms of the water balance the difference between
aquifers and aquitard is not so big because the aquitard has
about 16 times bigger storage capacity than the aquifers The
small value of the specific storage of the aquifers is the reason
that small errors in the components of the water balance have a
big influence on the calculated values of the hydraulic head
Eig 17 Dimensionless drawdown vs dimensionless time in a
3-layered system with
II = 50 m day
3 m (distance from well)
aquifer I
aquitard
aquifer II
Thi (m)
100
25
100
S s
001
0642
001
k(mday )
1
0277
1
The discrepancies are partly caused by errors in discretization
In the z-direction these errors can be eliminated by taking the
35
vertical resistance of the aquitard to be infinite Then the
calculated drawdown in the pumped aquifer can be compared with
the analytical Theiss solution
This is done in fig 18 for different distances from the well
One sees that the agreement is better when the distance is bigger
Apparently the linearization of the logarithmic drawdown curve
near the well is responsible for the differences The differences
for r = 3 m are of the same magnitude as for the pumped aquifer in
the 3-layered case
joo gt A 50
20
Theiss solution
bull calculated values r=3m (distance f rom well) x bdquo bdquo r=7m
1 L
I l_ 001 003 0 0 5 Ol Q2
- I I 5 10 t - K t
Fig 18 Dimensionless drawdown vs dimensionless time in a
confined aquifer with
Q =500 m day
k = 10 mday
D = 100 m
S = 001 s
The differences between numerical results and analytical solutions
are not caused by taking a finite area because changing the outer
boundary into an impermeable boundary has only minor effects
36
VI SUMMARY AND CONCLUSIONS
In this report a numerical model for non-stationary saturated
groundwater flow in a multi-layered system has been treated
Groundwater flow can be described in different ways
a Physical-mathematical by combining Darcys law and the law of
continuity (section 21)
b Geo-hydrological the flow in a natural groundwater basin takes
place in layers with different hydraulic properties Definitions
of the most important hydraulic properties are given in section
22
c Schematical groundwater flows horizontally in aquifers and
vertically in aquitards (section 23)
d Approximating by dividing the region of flow into a finite
number of elements and applying Darcys law and the law of
continuity on each of these elements In these ways one gets
a set of equations which can be solved given (initial and)
boundary conditions (section 25)
e Numerical the set of equations has been solved using a Galerkin
type finite element method and the implicit Gauss-Seidel method
with overrelaxation factor (section 31 and 32)
f Computer-solvable the numerical solution has been translated
in a FORTRAN IV program called FEMSAT (section 33)
In chapter IV the input execution and output of the program
FEMSAT are treated
Finally in chapter V some numerical applications are given
simulation of flow to a well in unconfined aquifer and simulation
of flow to a well in a 3-layered configuration
Comparison of hydraulic heads obtained by numerical calculations
and analytical true solutions showed an excellent agreement in case
of stationary flow and a good to moderate agreement in case of non-
stationary flow However the diviations caused by the discretization
process must be seen in relation to the errors made in the
establishment of the hydraulic properties
37
As was stated in the introduction numerical models can be
valuable tools in the management of water resources Therefore
numerical models must be suitable to simulate effects of certain
operations upon the spreading of water over saturated groundwater
unsaturated groundwater and surface water The model described in
this report is a model for saturated groundwater flow The relation
with the surface water can be taken into account via radial and
entrance resistances or via a relation between flow to the surface
water system and the hydraulic head The relation with the unsaturated
zone has been restricted to the input factor effective precipitation
In practise however the unsaturated and saturated system are
interconnected Therefore the model should be improved by taking
into account the unsaturated zone This can be done in at least
two ways
a Replacing the unsaturated zone by a parametrical model
b Coupling the unsaturated zone in an iterative way with the
saturated system
In order to find out the usefulness of the numerical model for
water resources management it will be applied to actual regional
groundwater systemslaquoAlso sensitivity analyses will be carried out
The computer model itself can be improved by applying standard
programs eg automatical generation of a nodal grid and graphical
plotting of the results
38
VII REFERENCES
ERNST LF 1962 Grondwaterstromingen in de verzadigde zone en hun
berekening bij aanwezigheid van horizontale evenwijdige open
leidingen Thesis University Utrecht pp 189
1978 Drainage of undulating soils with high groundwater
tables (in press)
FORSYTHE GE and WR WASON 1960 Finite Difference Methods for
Partial Differential Equations John Wiley and Sons New York
pp 444
KRUSEMAN GP and NA DE RIDDER 1976 Analysis and evaluation of
pumping test data ILRI-bulletin No 11 pp 200
LAAT PJM DE and C VAN DEN AKKER 1976 Simulatiemodel voor grond-
waterstroming en verdamping in Modelonderzoek 1971-1974 ten
behoeve van de waterhuishouding in Gelderland deel II
Grondslagen pp 145-197
NEUMAN SP RA FEDDES and E BRESLER 1974 Finite element
simulation of flow in saturated -unsaturated soils considering
water uptake by plants Hydraulic Engineering Laboratory
Technion Israel pp 104
and PA WITHERSPOON 1969a Theory of Flow in a confined
two Aquifer System Water Resources Research Vol 5 No 4
pp 803-816
^_ and PA WITHERSPOON 1969b Applicability of Current Theories
of Flow in leaky aquifers Water Resources Research Vol 5
No 4 pp 817-829
PINDER GF and EO FRIND 1972 Application of Galerkins Procedure
to Aquifer Analysis Water Resources Research Vol 8 no 1
pp 108-120
REMSON I GE HORNBERGER and FJ MOLZ 1971 Numerical Methods
in Subsurface Hydrology John Wiley and Sons New York
pp 389
WESSELING JG 1977 Toepassing van Finite Difference- en Finite
Element Methoden voor het oplossen van een aantal grondwatershy
stromingsproblemen (met computerprogrammas) ICW-nota 944
pp 63
39
ZIENKIEWICZ OC 1971 Finite Element Method in Engineering Science
McGraw-Hill London pp 521
40
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X - e - o
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LUX
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M II II II II II II
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z o r s n r z u icirc N H O - 3 3 0 3 3 m f - O f J l w i c ^ ^ c m z ^ raquo f x mdash 33 r- 0 - 1 p s | -O bull-raquo 3 bulllt 3gt - - - raquo - lt - lt
3 J M raquo H _ - 3 Z 3 Z t n 3gt H Xraquo- -H S - 1 -H
I 03 f - NI r -fc copy bull 3D -lt t n t n -il X ~i X ~i -raquo - n - n - T U mdash n r z r - - t r c c 0 cn a z 3 c - e - - n a i gt -s - n - - n -c 0 r 0 r
mdashf c - i cn c z m 2 33 en H xgt i - H 3gt C 33 33 -lt X
_ 3
-k
33 s c e s t n
j t i T T i - i i n n A Z O S O C O O X d 33
U) 3gt V Z N O bull
t o
Z n m
00 m 0
z z
z C l
z
raquo J J
- -n bullbull-gt r w 33 J Z 2 1gt n bullbull r n
3 Z 3 3 - 1 -H 1 gt - H 1 gt B H H H
V t n Ni
mdash1 m 33 3 t n
- n
c xgt
m x 55
f j
M zgt -n
O)
_ D t 3 - T) C bulln c 0 3= 0 t o 0 x
3= w gt0 3 raquo-i 3 3 r x raquo i gt r - r 0 3 Z u 3 X 3 Z 33 -lt 33
- 1 - i Z Z m bull 33 lt-raquo -lt t n iuml gt
o ^ r - r - c c o z m z m z CD t o t o F I H
t n t n
- n as - T] S) -raquobull bull œ w raquo - t o ro bull-gt CD X
en - n
es
u bull-bull
bull-1 m c -z bull ^ ^ - z r w --lt - 0 - raquo w yO 1mdash1
3gt Z Xgt n - f n 0 n 1-1 [Tl r-i Z 03 Z
-lt -lt icirc-f
- z o 0 - f-1 bull-- z z
J
0 11
w z -0 c 33 3 w Z Z O H -
t gt bull t o -
CS
r
-lt z
7Z O O
H - t S) 3gt - 1
- 1 r o r - r - lt-raquo r o
h ) ^ f m n f i V n i gt t n
J X I - 3gt z c i r c m i gt 3 ~lt r m bulljgt raquo
-lt rgt z raquo 0 3 3 0 m m 0
3 0 0 m
t o H
^ 00 X
iuml gt - 4
-H
3 m
- n
t n
t o
bullbull0 x
z
3
3gt
C -lt Igt - t Ci
APPENDIX 8 EXAMPLE OF OUTPUT
HEAD AND TERMS OF UATERBALANCE PER NODE PER LAYER
AT TIME 260 IN II AND I13DAYS
T
LAYER NODE KODE HEAD FLOUUN FLOUTS FLOUSS LEAKAGE STORAGE LAT FLOU
13 19 25 31 37 43
125008 125000 12S000 12S000 12S000 12S000 12S000 12S000
0000 0000 0000 0000 0000 0000 0000 0000
0000 0000 0000 0000 0000 0000 0000 0000
0000 0000 0000 0000 0000 0000 0000 0000
- 000 - 000 -000 - 000 - 000 - 000 - 000 0000
0000 0000 0000 0000 0000 0000 0000 0000
0000 000 000 000 000 000 000 000
000 0000 0000 0P100 0000 0000 0000 0000
1 7
13 19 25 31 37
12S000 125000 125000 125000 125000 12S000 125000 125000
0000 0000 0000 0000 0000 0000 0000 0000
- 001 -007 - 023 - 020 - 009 - 003 0000
0000 0000 0000 0000 0000 0000 0000 0000
000
001
007
023
020
009
003
003
0000 0000 0000 0000 0000 0000 0000 0000
1 7
13 19 25 31 37 43
125000 125000 124998 124972 124895 124787 124647 124358
0000 0000 0000 0000 0000 0000 0000 0000
- 000 001 007 023 020 009 003
0000
0000 0000 0000 0000 0000 0000 0000
50000
0000 182
1982 3862 1674 419 078 041
0000 0000 0000 0000 0000 0000 0000 0000
TERMS OF WATERBALANCE PER LAYER DURING LAST TIMESTEP AND
AND SINCE BEGINNING IN M3DAY M3
FLOU TO UNSATURATED ZONE
FLOU TO TERTIARY SYSTEM
0000
0000
0000
0000
LAYER NO 1
FLOU TO SECONDARY SYSTEM
FLOU TO AD3ACENT LAYERCS)
ARTIFICIAL FLOU AND PRESCRIBED BOUNDARY FLOU
ACCUMULATION TERMS
LATERAL FLOU THROUGH PRESCR HEAD BOUNDARY
0000
-000
0000
000
000
0000
-000
0000
000
000
LAYER NO 2
FLOU TO SECONDARY SYSTEM
FLOU TO AD3ACENT LAYERCS1
ARTIFICIAL FLOU AND PRESCRIBED BOUNDARY FLOU
ACCUMULATION TERMS
LATERAL FLOU THROUGH PRESCR HEAD BOUNDARY
0000
-378
0000
381
0000
0000
-055
0000
056
0000
LAYER NO 3
FLOU TO SECONDARY SYSTEM
FLOU TO AD3ACENT LAYERCS5
ARTIFICIAL FLOU AND PRESCRIBED BOUNDARY FLOU
ACCUMULATION TERMS
LATERAL FLOU THROUGH PRESCR HEAD BOUNDARY
0000
370
50000
49376
0000
0000
055
12979
12909
0000
Description of the different terms
a The flow through the phreatic surface q f
If this term is negative (water goes out of the saturated system)
it is called capillary rise If positive it is called percolation
(or drainage)
This flow can be calculated as a rest term of the water balance
of the unsaturated zone Very often this term is expressed as a
flow per unit surface (flux) and is called effective precipitation
For the time being this flux is a given input
b Flow to adjacent aquitards q
The flux to adjacent aquitards can be calculated by applying eq
(6) to the aquitard
c Flow to the primary surface water system q
The primary surface water system penetrates fully the groundwater
basin So it acts as a boundary with given values for the
hydraulic head (is equal to the phreatic level of the primary
system)
d Flow to the secondary surface water system q
The secondary surface water system does not penetrate fully the
groundwater basin which causes a resistance against flow between
groundwater and surface water The flow q can be calculated as
follows
h - h
- mdash f i i gt
where
h = hydraulic head of the groundwater at the place of
the secondary system |_ L |
h = phreatic level of secondary system |_ L J i s -
R = radial and entrance resistance of the
secondary system per unit surface |_ T J
According to ERNST (1962) the radial resistance H per unit
surface can be caKulated with
Uuml = 4 i n 4 e8) TTk B
where
1 = area drained by 1 m conduit L ^ J
d = thickness of the layer near the conduit _ L J
B = wet perimeter of the conduit |_ L J
The entrance resistance is caused by a layer which shows a
lower conductivity around the conduit than its surroundings
In practise radial and entrance resistancies are difficult to
deduct from hydraulic properties They can be found indirectly
by measuring simultaneously q h and h
t s
e Flow to the tertiary surface water system q
In smaller conduits usually the phreatic level is not known
So eq (7) cant be applied The flow to these conduits however
is sometimes very important (eg in areas with shallow groundwater
tables) One may a-sume (ERNST 1978) that a relationship exists
between the hydraulic head of an aquifor and the flow q
According to Ernst the tertiary system can be divided into a
number of categories with respect to their mutual distance and
bottom height For each category the flow is h - h (b)
q ( h b ) - mdash (9)
where
h = average hydraulic head | L J
h (b) = phreatic level in the category under
consideration a function of bottom height |_L J
a = geometry factor to convert h into h _ - J
Y = drainage resistance _ T J
hm = hydraulic head midway between two
conduits |_ L J m
Summation of the q(hb)-relations of the different categories
yields the overall relation between hydraulic head and flow to
the tertiary system we are looking for
f Artificial recharge or discharge q and prescribed boundary
flux qfo
The magnitudes of these flows are not influenced by the conditions
inside the groundwater basin but remain on a prescribed level
Thus q is an internal boundary condition q an external boundary cl D
condition
For confined aquifers some of the above mentioned terms dont
exist These terms are the flow through the phreatic surface qf
and the flow to the tertiary surface water system q
Aquitards are considered only as transmitters of water from one
aquifer to another with possibilities of storage (eq 6 ) This means
that for aquitards the water balance terms ar through f are set
equal to zero
25 S o l u t i o n o f g r o u n d w a t e r f l o w p r o b l e m s
The basic equations for saturated groundwater flow are eq (5)
for aquifers and eq (6) for aquitards The solution of these
equations require knowledge of the physical system and the auxiliary
conditions describing the system constraints
These auxiliary conditions are (REMSON etal 1971)
a Geometry of the system
Eg the diviation of the system into a restricted number of
layers and the horizontal extension of the system under
consideration
b The matrix of hydraulic properties including in-homogeneity
and an-isotropy
c Initial conditions Because we are dealing with non-stationary
flow the values of pertinent system variables (such as hydraulic
head at the initial time) must be known
d Boundary conditions They describe the conditions at the boundaries
pf the system as a function of time
One can distinguish between
- hydraulic head boundary
- flow (including zero flow) boundary
- both head and flow boundary
After specifying the auxiliary conditions an unique solution of
the variation of the hydraulic head in space and time can be
obtained
For special shapes of flow and assumptions exact analytical
solutions exist In nature however one is usually confronted with
problems like
- spacial variation in the hydraulic properties of the soil
- irregular shape of the groundwater basin
- empirical relations between q and h
- d not being a constant
For such situations by numerical and analogy models only an
approximation of the actual situation can be obtained In the next
chapter a numerical solution called the finite element method is
treated
III NUMERICAL SOLUTIONS
In this chapter the numerical solution of the aquifer equation
(eq 5) and the aquitard equation (eq 6) will be treated The
method of solving is the finite element method In section 31 only
a short mathematical description of this method will be given For
more mathematical background the reader is referred to publications
on this subject (eg ZIENKIEWICZ 1971 NEUMAN etal 1974 and
WESSELING 1976)
In section 32 the application of the finite element method on
the flow in groundwater basins is treated while in section 33 the
translation into a computer program is given
3 1 F i n i t e e l e m e n t m e t h o d
For reasons of simplicity the finite element method will be
explained for the stationary version of eq 5 ie
10
(k d^) + (k d ^ ) - Q = O (10) ocircx x 6x 6y y Sy
The basic idea of each numerical solution of eq (10) is to
apply this equation on a small but finite volume The flow region
of interest R is divided into a finite number of volumes called
elements The elements can have different shapes but for reasons
of conveniency we only use triangles- Quadrilaterals can also be
used because they can be divided into two triangles The sizes of
the triangles may vary They can be adapted according to the
geometry of the region and the accuracy desired (fig 11 gives an
example of the diviation of a region into triangles and quadrilaterals)
The corner points of the elements represent the nodal points
If there are N nodal points then the continuous variation of h
in region R is approximated by the expression
h (x_v) ~ h f (xy) (11) a n n
where
h (AV) = approximation of u (xy) a
h = value of h in nodal point n n
f ltxgty) = global coordinate function
In nodal point n a must be equal to h so f (x y ) = 1 and a n n n n
f (xy) = 0 i n Between f (x v ) and f (xy) of the nodal n j ] n n n n j j
points surrounding nodal point n f (xy) varies linearly from
1 to 0
In fig 3 a picture of f (xy) is given n
If one knows the values h in the nodal points n we know the approximation h of h We can find these values of b bv using for
a n
example the Galerkin method
Galerkins principle states that an approximate solution of h in
eq (10) can be obtained from the following system of equations
N ~~ (k d -A + -^- (k d T-y E h f (xy) f(xy)dxdy +
-ox x ox ocircy y oy - _ n n i R n~
J J Q f(xy) dx dy = 0 i = 1 N (12)
1 1
Fig 3 The global coordinate function f (xy)
The integration can be performed element by element with the
aid of the coordinate functions f (NEUMAN etal 1974)
For non-stationary problems it is not possible to apply the
Galerkin principle on the time derivative But it is possible
to discretize this derivative and to set up the system of equation
(eq 12) for each finite time step
One obtains in this way a series of sets of laquocriations tha s
an approximation of the non-stationary flow problem
Treatment of boundary conditions
a Initial conditions
In non-stationary flow problern gtne must start with given
values of a at time t = t ri c
b Boundary conditions
- Prescribed head boundary
Nodal points which lie on a prescribed head boundary have a
prescribed value for h bull r Hgte r umber of unknown h s is n n
reduced and accordingly no the number of equations
- Prescribed flux through z-- boundary
If one knows the flux (iuumlr jw or outflow per unit area) and
the area of the boundary oi-r element the inflow or outflow
through the boundary per flemlt=nt can be calculated
One may assume this flow to J bull part of the sink term Q in
eq (12)
12
32 A p p l i c a t i o n o f t h e f i n i t e e l e m e n t
m e t h o d o n t h e f l o w i n g r o u n d w a t e r
b a s i n s
From hydrogeological investigations the geometry and the matrix
of the hydraulic properties are known Knowing the geometry the
flow can be split up into a finite number of layers This holds
also for the nature of the boundaries
Now a nodal grid is superimposed on each layer in such a way
that the nodal points of the different layers are situated right
above each other (see fig 4) and in each layer the nodal point
lies in the middle of this layer
-nodal point
Fig 4 Schematic representation of the nodal points in the
xyz-space
Each layer has N nodal points of which P nodal points have a
prescribed head In each layer remain N-P = N nodal points with
unknown head Eq (12) can be applied to each point with unknown
head in each separate layer
As was stated in the last paragraph for each time step one
can write
13
Ah bdquo ~ -r1 + Q = O in = 12N in n m At yi A h + B (13)
where
A = in
ocircf ocircf ocircf of (k d f mdashri- ~ + k df - ~ -jS) dxdy
J x n ox ox y n oy oy
in (S d + S ) ff dx dy O if i icirc n
s y i n
Q f dx dy
The matrices A B and the vector Q have been defined in in l
elsewhere (NEUMAN etal 1974) A is the conductivity matrix in
B the storage matrix and Q the source vector in l
In order to clarify these rather complicate expressions apply
equation (13) poundo point 4 of fig 5
6 7
Fig 5 Top view of part of the nodal grid
lhl + A42h2 + A43h3 + A44h4 + A45h5 + A46h6 + A47h7 +
h - h
+ B _iuumli i ^ + Q = o 44 t - t x4
n n-1
(14)
where h and h are values of h at time t and t 4n 4n-1 4 n n-1
respectively
14
If the h-values belonging to the A-values are taken at time
t = t we get an equation with one unknown h for time t = t n-1 4n n
The equation can then be solved directly
However this is an explicit formulation which puts strong limitations
on the magnitude of the time step t - t (REMSON etal 1971) n n-1
Values larger than a certain value cause instability With an implicit method the values of h are taken at time
t = t This method is unconditionally stable Now one gets an equation n
with 7 unknowns h through h If one has 7 equations for every 1 n n
point 1 through 7 we can solve the values h i = 17
in
Of all the methods possible the iterative method of Gauss-Seidel
has been chosen The Gauss-Seidel method starts by choosing a first estimate of h at time t = t (usually equal to h ) At each in n j -i in-l iteration step a new value of h is calculated using the last
in calculated values of h If one passes the nodal points from 1 to
i n 7 eq (14) becomes
h-j + 1 = A h^ + 1 + A bdquo h | + 1 + A _ h | + 1 + A h j + A r h j + 4 n 4 1 1 n 4 2 2 n 43 3 n 45 5 n 4 6 6 n
B h B + A h J + - 4 4 4 n ~ + 0 J (A + mdash) A 4 7 h 7 n At n U 4 4 At
(15)
where
h-i = in
j = iteration step or number of iteration
i = number of nodal point n = time = t
n
The iteration process converges if the successive differences k+1 k
between h and h become continuously smaller in in
The iteration process is stopped if at each nodal point the k+1 k
difference between h and h is smaller than a certain prescribed in in
value
15
To accelerate the convergence an overrelaxation factor W is used
(for more information about W see FORSYTHE and WASOW 1960)
k+1 k T7 k+1 k h = h + W (h - h )
in in in in (16)
If W = 1 there is no overrelaxation Usually W varies between 1 and
2 and the optimum value very often is found by trial and error
Eq (15) is valid for points in aquifers The connection with
adjacent layers is made by the term Q because in this term the
flow from adjacent aquitards q is included (two aquifers are always
separated by an aquitard)
Aquitards
We dont apply the finite element method on points in aquitards
As stated before there is no horizontal flow in aquitards So it has
no sense to replace the continuous variation of h in the aquitard
by equation (11)
The nodal points in aquitards represent a certain volume of an
aquitard hich transmit water from one aquifer to another and which
has storage capacity The nodal points are included in the Gauss-
Seidel scheme -s follows (see fig 6 )
1 aauifer
a -_vS laquo 2 [~^-t^yM^^^-^r~ aqutarci l-Z
aquifer
Fig 6 Schematic situation for points in aquitards
Point 2 is situated in the middle of the aquitard with thickness
d storage coefficient S and vertical hvdraulic conductivity k c J v
Suppose the Gauss-Seidel iteration starts in the upper layer and
ends in the bottom-layer Then the flux from point 1 in aquifer 1 to
16
point 2 in the aquitard at time t = t and in iteration step j + 1
is
k q = (hf -hf )-iumlmdash f (17)
1 n 2 n OSxd1
and the flux from point 3 in aquifer 2 to point 2 in the aquitard
is
1 c 1
qbdquo = (h - hf) mdashHmdashr- (18) 2 3 n 2 n OSxd
The net flux ltl-q9 must be equal to the change in storage per unit
area per unit time This change in storage can numerically be
approximated by
te - h-
n n-1
Combina t ion of ( 1 7 ) (18) ana (19) g i v e s
l tr1 - h (hJ + 1 + hj - 2 hj+1) - - - - - - S1 - ilaquoS L u l l = o (20)
] i 3 n 2 n 1 c t - t j j x d n n -1
T + 1
From t h i s e q u a t i o n h^ kan be s o l v e d
3 3 C o m p u t e r o g r a m
The problem outl^ugrave so far has been translated into a computer
program written in FORTRAN IV
The flow chart ot Uis program is given in fig 7 whereas in
appendix A the complete listing of the program can be found
In the flow chart the 5 main parts of the program are
indicated
A Reading and printing of input data
B Determination of the matrices A and B and the vector Q per layer
17
M
1 A
FLOH to unsaturated zono
FLOW to tertiary syster
FLOW to secondnrv system
Leakage to adjacent aquitards
SOLVE equation (14)
yes
CALCULATE flux through constant head boundnr
CALCULATE waterbalance n p r 1 OAro r r J j - ~
PRINT of output
Ier node per layer
k = k+1
t = t + At
7 DUMP variables on tape
-gt-
( STOP )
Fig 7 Flow chart of the program FEMSAT with diviation in 5 main
parts A B C D and E
C START PROGRAM FEMSAT
f READ general information
PRINT general information
f READ element information
PRINT element information 7 READ stationary nodal point information
PRINT stationary nodal point information 7
READ non-stationary nodal point information for t = t
FRINT non-stationary nodal point information for t = t
L 2 (
^ yes R E A D non-stat input for t gt t
J PRINT non-stat input for t gt t (- 1 -- M degY
GENERATE A B and AREA-matrix for confined aquifers
_L
GENERATE A B and AREA-matrix for unconfined aouifer
bull
C Solving h with Gauss-Seidel
D Determination of the water balance terms per layer
E Printing of results
The program exists of a main program with several subroutines
The main program calls the subroutines In the subroutines most
calculations are performed
In this way an easy adaptation to different problems is
obtained (see also section 44) Comments in the program will
help the reader to understand the program
In the next chapter the preparation of the input data executing
of the program and the output will be treated
Acknowledgement
Some parts of the computer program were taken from the program
UNSAT2 of NEUMAN etal (1974)
IV INPUT EXECUTION AND OUTPUT
41 I n p u t
The input is divided into endogenous variables (chosen by the
user or calculated by the program during execution) and exogenous
variables (based on physical data)
411 Endogenous variables
Translation of groundwater basin into a nodal point network
Probably the most important but also the most difficult stage
in the process of simulating groundwater flow is the translation
of the natural groundwater basin into a nodal point network Questions
which arise are
a In how many layer the geo-hydrological system mustcan be
schematized
b The locations of the boundaries and the nature of these boundaries
(impervious prescribed head prescribed flux)
c The configuration of the nodal points
19
The answers of these questions depend on the problem under
consideration
- Schematization into layers
Usually our knowledge about the geo-hydrological situation is
such that no more than 5 different layers can be distinguished
(remember contrast rule)
- Boundary conditions
The top boundary condition of a groundwater basin usually is
a prescribed flux (effective precipitation)
The bottom boundary condition usually is a prescribed flux
boundary with flux zero (impervious hydrological base)
For the boundaries at the other sides of the system one sees
either prescribed head boundaries (eg a fully penetrating
river) or prescribed flux boundaries (including zero flux)
- Nodal point configuration
Nodal points are situated
on the boundaries
along internal boundaries which indicate changes in
hydraulic properties within the layer
along the tertiary surface water system
on places where artificial recharge or discharge wiii tgtke
place
elsewhere
Construction of the finite element mesh
Assume we have schemetized the groundwater basin iatu a finite
number of layers with different direction of flow (horizontal and
vertical) Within one layer the hydrological properties may vary
from place to place
Now a nodal grid is superimposed on each layer in a way already
described in section 32 Thus each layer has the same number of
nodal points situated on the same places in the horizontal plane
but with different z-coordinates
We are free to construct quadrilaterals or triangles because
the program automatically divides a quadrilateral into two triangles
having identical material properties The dimensions of the elements
20
should be small at places where large changes in hydrological
properties occur or where large gradients in hydraulic head are
expected to occur
Coding of nodal points
Each node in the finite element network is assigned an integer
code which indicates the type of calculation for this node
The coding is as follows
- Code = 1 nodal points with described head as function of time
- Code 2 internal nodal points
- Code = 3 internal nodal points which represent secondary surface
water system
- Code = 5 nodal points with prescribed artificial recharge or
discharge and points with prescribed boundary flow
(both as functions of time)
Other important endogenous variables are the magnitude of the
time step the value of the variable which stops the iteration
process and the maximum number of iterations allowed per time step
(see also group A and B of section 413)
412 Exogenous variables
After constructing the nodal grid each node represents part of
the groundwater basin Geo-hydrological properties can be assigned
to each mode based on physical data (group C D and E of section
413) In group F of section 413 time-dependent variables can
be assigned to each node
413 Data input to Program FEMSAT
The data input must be punched on cards according to the
following instructions
GROUP COLUMNS FORMAT SYMBOL DESCRIPTION
A 1-80 20A4 HEAD Desired heading to be printed at
the beginning of the execution
1-5 15 NUMNP number of nodal points
21
GROUP COLUMNS FORMAT SYMBOL DESCRIPTION
6-10 15
11-15 15
16-20 15
21-25 15
26-30 15
31-35 15
NUMEL
NUMLAY
MBAND
NTSPI
NTSPO
MAXIT
1-10 F103 ST
11-20 F103 DELT
21-30 F103 TMAX
31-40 F105 TOL
number of elements
number of layers
maximum difference in node number
between two adjacent nodes
number of time steps between a
change in non-stationary input data
number of time steps between two
successive outputs
maximum number of iterations per
time step
time to start with the execution
initial time interval
maximum time to be reached before
execution is stopped
maximum allowed absolute change in
the values of h between any two
successive iterations in a given
time step
Iterations continue until all changes
are less than or equal to TOL or
until MAXIT is exceeded
1- 5 15
6-10 15
11-15 15
16-20 15
KX(N1)
KX(N2)
KX(N3)
KX(N4)
sequential number of i-th corner of
element N
sequential number of j-th corner
sequential number of k-th corner
sequential number of 1-th corner
(see fig 8)
Fig 8 Numbering of element corner nodes
22
GROUP COLUMNS FORMAcircT SYMBOL DESCRIPTION
One card must be provided for each element in sequential order
starting with N = 1 and ending with N = NVJMEL
I-K 1103 SANG(NLY)
1i-iO MO3 C1(NLY)
21-30 F 103 C2(NLY)
21-4- 1-104 THI(NLY)
angle in degrees between CI and
the laquo-coordinate to he used in
nodal print N in layer LY (see
poundig 9)
First principal conductivity of
material in node N and layer LY
idem second principal conductivity
thickness of layer LY at nodal
point N
ig V Definition of SANGCI and C2
One card icircuiPt be provided for each model point ctarting wih M = 1
and ending with N = NUMNP If there is norlt3 than 1 layer for each
layer a set must be provided starting with the upper layer LY - 1
and ending with the bottom layer LY = NIJMLAY The variables SANG
M and 2 are not relevant tor aquitards but for reasons of
conveniency this procedure is maintained (making SANG CI and C2
dummy variables)
1-10 110 KODE(N)
11-20 II0 NTERR(N)
coding of nodal points (see
paragraph 42)
number of tertiary relation in node N
The q-h relationship of the flow to
23
GROUP COLUMNS FORMAT SYMBOL DESCRIPTION
21-30 FlOl X(N)
31-40 FlOl Y(N)
41-50 FlOl HGL(N)
51-60 FlOl HBSC(N)
61-70 FlOl LESC(N)
the tertiary system as discussed
in par 24 are schematized in a small
number of different relations
x-coardinate of node N
y-coordinate of node N
height ground level in node N
height bottom secondary conduit in
node N If node N is not a secondary
conduit this variable is a dummy
variable
length of secondary conduit in area
represented by node N
One card must be provided for each node starting with N = 1 and
ending with N = NUMNP
E 1-10 FlOl P(NLY1) initial value of h at node N in
layer LY
11-20 FlOl VERRES(NLY) vertical resistance at node N and
layer LY If the layer is an aquifer
this value is a dummy variable
21-30 F101 RADRES(NLY) radial resistance for flow to
secondary system at node N in layer
LY
31-40 FlOl ENRES(NLY) entrance resistance for flow to
secondary system at node N in layer
LY
41-50 FlOl WP(NLY) wet perimeter of secondary system at
node N in layer LY
51-60 FlOl P0R(NLY) specific yield of the material at
node N in layer LY
61-70 FlOl SS(NLY) specific storage of the material
at node N in layer LY
24
GROUP COLUMNS FORMAT SYMBOL DESCRIPTION
One card must be provided for each node starting with N = 1 and
ending with N = NUMNP If there is more than 1 layer for each layer
a set of cards must be provided starting with LY = 1 and ending
with LY = NUMLAY
F 1-10 F101 P(NLY1) value of h at different times at
node N in layer LY Node N lies
on a prescribed head boundary
11-20 F101 WTSC(N) water table in secondary system at
node N at different times
21-30 F 101 ARRED(NLY) artificial recharge to or discharge
from node N in layer LY or
prescribed boundary flow at
different times
31-40 F 101 QPHR(N) flux through phreatic surface at
node N at different times
The data in group F are non-stationary input data One card must be
provided for each node starting with N = 1 and ending with N = NUMNP
In case of more than 1 layer for each layer a set must be provided
Not relevant variables are taken as dummy variables
After a certain number of time steps (NTSPI) the data of group E
are read So the set cards E must contain NxLYxC cards where C =
(TMAXNTSPIXDELT) + 1 if DELT is a constant
42 R u n n i n g t h e p r o g r a m o n t h e C y b e r 7200
The program has been executed on a Cyber 7200 computer of
IWIS-TNO in the Hague
- The demanded input is punched on cards according to the instructions
of the last paragraph and is put on a permanent file
- For each specific problem inthe program only the dimension statements
in the main program have to be adjusted according to the instructions
in the lines of comment
- After compilation the program together with the input file is
executed
25
- Intermediate data necessary for continuation of the calculations
are saved on another permanent file This gives the opportunity
to restart the program at a time the calculation was stopped
The only change in the input file is to adjust the starting time
Remark
With the aid of a terminal small changes in the program and in the
input file can be made easily and the output immediately appears
on the terminal In this way it is possible to stop erroneous
calculations This is especially useful when being in the testing
phase of the program
43 O u t p u t
Immediately after reading the input data they are printed so
that possible errors can be detected During execution after each
certain number of time steps (NTSPO) calculated values of h and
water balance terms are printed In appendix B an example is given
The output data can be devided into two groups
A Data per node per layer
- Coding of the nodal points KODE
- Hydraulic head HEAD
- Flow through phreatic surface at the time indicated in head
of table FLOWUN
- Flow to tertiary surface water system FLOWTS
- Flow to secondary surface water system FLOWSS
- Flow to adjacent layers LEAKAGE
- Artificial recharge or discharge and prescribed boundary
flow ARRED
- Change in storage during last time step STORAGE
- Flow through prescribed head boundary LATFLOW
B Data per layer
Summation of the terms already mentioned under A gives the
magnitude of the water balance terms per layer at the time
indicated (instanteneous water balance terms) Summation of
these instanteneous terms multiplied by their respective time
intervals since time is zero gives the cumulative water balance
terms per layer
26
After each print out we are able to check the calculations
because the sum of the water balance terms per layer must be
approximately zero
44 C o m p u t i n g t i m e
The computing time depends on the number of nodal points and
the number of layers But it depends also on the magnitude of the
changes in hydraulic head and on the value which terminates the
Gauss-Seidel iteration (TOL) To give an impression TOL = 001 m
Costs per day simulation (in hlfl) on the Cyber 7200
number of nodes number of layer bull bdquo bull bull _ 1mdash x a x b (21)
where
a = costs in Dutch guilders per system second (ƒ 027)
b = empirical factor with magnitude about 001
V NUMERICAL EXPERIMENTS
In general there are two possibilities to test the validity of
numerical models The first and most commonly used method is to
compare numerical results with analytical solutions The second
possibility is to compare numerical results with measured field
data Since the last alternative requires many data that are
difficult to obtain we restrict ourselves for the time being to
comparison with the analytical solutions
5 1 F l o w t o a w e l l i n a n u n c o n f i n e d
c i r c u l a r - s h a p e d a q u i f e r
Flow to a well can be divided into steady-state flow and
unsteady-state flow (KRUSEMAN and DE RIDDER 1976)
27
511 Steady state flow in an unconfined aquifer
For this case the formula of Thiem-Dupuit is valid (see fig 10)
Q bull k lth2-hicircgt ln(r2r])
(22)
where
hhbdquo = hydraulic head in 1 and 2 respectively
Q = discharge from well
rbdquo = distance from well
CL31
hmdash^ri --r2
Fig 10 Schematic cross-section of a pumped unconfined aquifer
Fig 11 gives the nodal grid superimposed on a well in a
circular shaped aquifer Near the well the nodal point distance
is made smaller Quadrilaterals as well as triangles are used
Numerical calculations with this finite element network and 3 -1
an extraction rate of 200 m day were performed for 10 days
At that moment a steady-state situation was almost achieved
28
2tmdashnodal point number
160 180 X-axesim)
Fig 11 Nodal point configuration for flow to a well in a circular-
shaped aquifer The outer boundary is taken to be constant
The well is situated in nodal point 42
29
In fig 12 the results of the simulation are compared with the
analytical solution The agreement between numerical results and
the analytical solution is excellent
draw- 15irS (m) down
analytical solution imdashx calculated values
distance from well r (m)
Fig 12 Comparison between analyt ical and calculated s teady-state
drawdown for the problem of f ig 11 with 1
-1 Q bull - 200 m 3 d a y _ 1
k = 10 mday
d = 100 m
S = 2 y
S = 0 s
512 Unsteady-state flow in an unconfined aquifer
Analytical solutions for unsteady-state flow in an unconfined
aquifer only exist if the aquifer extends infinitely For the
aquifer given in fig 11 this condition is not fulfilled However
during the first one or two days after starting the extraction the
influence of a constant hydraulic head at the outer boundary on the
30
drawdown near the well can be neglected (see fig 13) After one
day the lateral inflow is only 25 of the extraction the remainder
being the change in storage due to drawdown
2 0 0
m 3 day 1
Q=200m 3 day
100-
vraquomdash Lateral inflow
N amp storage
I I ^ V - W 10
t (days)
Fig 13 Values of Q lateral inflow and change in storage as
functions of time
The nonsteady-state or Theis equation can be written as
i _ Q s 4iTkD
dy = 4irkD
W(u) (23)
where
u = r2S
4irkDt
r = distance from well
S = storage coefficient
Q = constant discharge from well
D = thickness of the waterbearing layer
at the beginning of pumping
t = time after starting the extraction
s = corrected drawdown
= s - 322D
s = calculated drawdown
W(u) = Theiss well function
[J
CIAT1
[T]
]
31
In fig 14 the analytical unsteady-state solution for the case
described in section 511 is given together with the numerical
results at times 05 and 10 days The agreement between numerical
results and the analytical solution is rather good
Drawdown s1(m)
10
10
10
Theiss solution
x caculated values t=Q5
bdquo t= lO
10 10- 10 10 t r2(daym2)
Fig 14 Comparison between analytical and calculated steady-state
drawdown for the problem of fig 11 with
Q
k
d
S y
s s
=
=
=
=
=
200
1
10
3 m
0
0
2
0
-1 day
m day
m
These results indicate that the assumption of a linear variation
of h between two nodal points (see fig 3 ) even with the rather
rough element network of fig 11 is acceptable So there is no
need to improve for example the finite element network by introducing
higher order coordinate functions
32
52 F l o w t o w e l l i n 3-1 a y e r e d s y s t e m
In fig 15 a schematic cross-section of a 3-layered system of
two aquifers separated by an aquitard is given A well of
infinitisemal radius is completed in the lower aquifer and discharges
at a constant rate QTT Each layer is homogeneous isotropic and
extending radially infinite
bdquo bull - bull bull ~ i
mdash Q II
aquifer I
lugrave^i~IcircIuml7i-i~~ aquitard --r-rl~-
-II--I i - i
aquifer
Fig 15 Schematic cross-section of a 3-layered system
For this situation NEUMAN and WITHERSPOON (1969a) give rather
complicated analytical solutions In the special case where the
hydraulic properties of the two aquifers are identical they give
graphical illustrations of their solutions for three different
cases
These graphical solutions give the opportunity to verify
roughly numerical results of a 3-layered system Therefore a nodal
grid is superimposed on a circular shaped groundwater flow system
with the extraction in the centre of the lower aquifer and with
a constant head on the outer boundary Fig 16 gives the nodal
point situation in the x - y plane of 1 layer The nodes in the
other two layers are situated in the middle of the layers at the
same places in the x - y plane only their z-coordinate is different
33
A
-Jy
bull33
V 5
iumlff
Fig 16 Nodal point configuration for the 3-layered problem (top view)
34
In fig 17 numerical results are compared with the analytical
solution The agreement is rather bad especially for the two
aquifers But in terms of the water balance the difference between
aquifers and aquitard is not so big because the aquitard has
about 16 times bigger storage capacity than the aquifers The
small value of the specific storage of the aquifers is the reason
that small errors in the components of the water balance have a
big influence on the calculated values of the hydraulic head
Eig 17 Dimensionless drawdown vs dimensionless time in a
3-layered system with
II = 50 m day
3 m (distance from well)
aquifer I
aquitard
aquifer II
Thi (m)
100
25
100
S s
001
0642
001
k(mday )
1
0277
1
The discrepancies are partly caused by errors in discretization
In the z-direction these errors can be eliminated by taking the
35
vertical resistance of the aquitard to be infinite Then the
calculated drawdown in the pumped aquifer can be compared with
the analytical Theiss solution
This is done in fig 18 for different distances from the well
One sees that the agreement is better when the distance is bigger
Apparently the linearization of the logarithmic drawdown curve
near the well is responsible for the differences The differences
for r = 3 m are of the same magnitude as for the pumped aquifer in
the 3-layered case
joo gt A 50
20
Theiss solution
bull calculated values r=3m (distance f rom well) x bdquo bdquo r=7m
1 L
I l_ 001 003 0 0 5 Ol Q2
- I I 5 10 t - K t
Fig 18 Dimensionless drawdown vs dimensionless time in a
confined aquifer with
Q =500 m day
k = 10 mday
D = 100 m
S = 001 s
The differences between numerical results and analytical solutions
are not caused by taking a finite area because changing the outer
boundary into an impermeable boundary has only minor effects
36
VI SUMMARY AND CONCLUSIONS
In this report a numerical model for non-stationary saturated
groundwater flow in a multi-layered system has been treated
Groundwater flow can be described in different ways
a Physical-mathematical by combining Darcys law and the law of
continuity (section 21)
b Geo-hydrological the flow in a natural groundwater basin takes
place in layers with different hydraulic properties Definitions
of the most important hydraulic properties are given in section
22
c Schematical groundwater flows horizontally in aquifers and
vertically in aquitards (section 23)
d Approximating by dividing the region of flow into a finite
number of elements and applying Darcys law and the law of
continuity on each of these elements In these ways one gets
a set of equations which can be solved given (initial and)
boundary conditions (section 25)
e Numerical the set of equations has been solved using a Galerkin
type finite element method and the implicit Gauss-Seidel method
with overrelaxation factor (section 31 and 32)
f Computer-solvable the numerical solution has been translated
in a FORTRAN IV program called FEMSAT (section 33)
In chapter IV the input execution and output of the program
FEMSAT are treated
Finally in chapter V some numerical applications are given
simulation of flow to a well in unconfined aquifer and simulation
of flow to a well in a 3-layered configuration
Comparison of hydraulic heads obtained by numerical calculations
and analytical true solutions showed an excellent agreement in case
of stationary flow and a good to moderate agreement in case of non-
stationary flow However the diviations caused by the discretization
process must be seen in relation to the errors made in the
establishment of the hydraulic properties
37
As was stated in the introduction numerical models can be
valuable tools in the management of water resources Therefore
numerical models must be suitable to simulate effects of certain
operations upon the spreading of water over saturated groundwater
unsaturated groundwater and surface water The model described in
this report is a model for saturated groundwater flow The relation
with the surface water can be taken into account via radial and
entrance resistances or via a relation between flow to the surface
water system and the hydraulic head The relation with the unsaturated
zone has been restricted to the input factor effective precipitation
In practise however the unsaturated and saturated system are
interconnected Therefore the model should be improved by taking
into account the unsaturated zone This can be done in at least
two ways
a Replacing the unsaturated zone by a parametrical model
b Coupling the unsaturated zone in an iterative way with the
saturated system
In order to find out the usefulness of the numerical model for
water resources management it will be applied to actual regional
groundwater systemslaquoAlso sensitivity analyses will be carried out
The computer model itself can be improved by applying standard
programs eg automatical generation of a nodal grid and graphical
plotting of the results
38
VII REFERENCES
ERNST LF 1962 Grondwaterstromingen in de verzadigde zone en hun
berekening bij aanwezigheid van horizontale evenwijdige open
leidingen Thesis University Utrecht pp 189
1978 Drainage of undulating soils with high groundwater
tables (in press)
FORSYTHE GE and WR WASON 1960 Finite Difference Methods for
Partial Differential Equations John Wiley and Sons New York
pp 444
KRUSEMAN GP and NA DE RIDDER 1976 Analysis and evaluation of
pumping test data ILRI-bulletin No 11 pp 200
LAAT PJM DE and C VAN DEN AKKER 1976 Simulatiemodel voor grond-
waterstroming en verdamping in Modelonderzoek 1971-1974 ten
behoeve van de waterhuishouding in Gelderland deel II
Grondslagen pp 145-197
NEUMAN SP RA FEDDES and E BRESLER 1974 Finite element
simulation of flow in saturated -unsaturated soils considering
water uptake by plants Hydraulic Engineering Laboratory
Technion Israel pp 104
and PA WITHERSPOON 1969a Theory of Flow in a confined
two Aquifer System Water Resources Research Vol 5 No 4
pp 803-816
^_ and PA WITHERSPOON 1969b Applicability of Current Theories
of Flow in leaky aquifers Water Resources Research Vol 5
No 4 pp 817-829
PINDER GF and EO FRIND 1972 Application of Galerkins Procedure
to Aquifer Analysis Water Resources Research Vol 8 no 1
pp 108-120
REMSON I GE HORNBERGER and FJ MOLZ 1971 Numerical Methods
in Subsurface Hydrology John Wiley and Sons New York
pp 389
WESSELING JG 1977 Toepassing van Finite Difference- en Finite
Element Methoden voor het oplossen van een aantal grondwatershy
stromingsproblemen (met computerprogrammas) ICW-nota 944
pp 63
39
ZIENKIEWICZ OC 1971 Finite Element Method in Engineering Science
McGraw-Hill London pp 521
40
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APPENDIX 8 EXAMPLE OF OUTPUT
HEAD AND TERMS OF UATERBALANCE PER NODE PER LAYER
AT TIME 260 IN II AND I13DAYS
T
LAYER NODE KODE HEAD FLOUUN FLOUTS FLOUSS LEAKAGE STORAGE LAT FLOU
13 19 25 31 37 43
125008 125000 12S000 12S000 12S000 12S000 12S000 12S000
0000 0000 0000 0000 0000 0000 0000 0000
0000 0000 0000 0000 0000 0000 0000 0000
0000 0000 0000 0000 0000 0000 0000 0000
- 000 - 000 -000 - 000 - 000 - 000 - 000 0000
0000 0000 0000 0000 0000 0000 0000 0000
0000 000 000 000 000 000 000 000
000 0000 0000 0P100 0000 0000 0000 0000
1 7
13 19 25 31 37
12S000 125000 125000 125000 125000 12S000 125000 125000
0000 0000 0000 0000 0000 0000 0000 0000
- 001 -007 - 023 - 020 - 009 - 003 0000
0000 0000 0000 0000 0000 0000 0000 0000
000
001
007
023
020
009
003
003
0000 0000 0000 0000 0000 0000 0000 0000
1 7
13 19 25 31 37 43
125000 125000 124998 124972 124895 124787 124647 124358
0000 0000 0000 0000 0000 0000 0000 0000
- 000 001 007 023 020 009 003
0000
0000 0000 0000 0000 0000 0000 0000
50000
0000 182
1982 3862 1674 419 078 041
0000 0000 0000 0000 0000 0000 0000 0000
TERMS OF WATERBALANCE PER LAYER DURING LAST TIMESTEP AND
AND SINCE BEGINNING IN M3DAY M3
FLOU TO UNSATURATED ZONE
FLOU TO TERTIARY SYSTEM
0000
0000
0000
0000
LAYER NO 1
FLOU TO SECONDARY SYSTEM
FLOU TO AD3ACENT LAYERCS)
ARTIFICIAL FLOU AND PRESCRIBED BOUNDARY FLOU
ACCUMULATION TERMS
LATERAL FLOU THROUGH PRESCR HEAD BOUNDARY
0000
-000
0000
000
000
0000
-000
0000
000
000
LAYER NO 2
FLOU TO SECONDARY SYSTEM
FLOU TO AD3ACENT LAYERCS1
ARTIFICIAL FLOU AND PRESCRIBED BOUNDARY FLOU
ACCUMULATION TERMS
LATERAL FLOU THROUGH PRESCR HEAD BOUNDARY
0000
-378
0000
381
0000
0000
-055
0000
056
0000
LAYER NO 3
FLOU TO SECONDARY SYSTEM
FLOU TO AD3ACENT LAYERCS5
ARTIFICIAL FLOU AND PRESCRIBED BOUNDARY FLOU
ACCUMULATION TERMS
LATERAL FLOU THROUGH PRESCR HEAD BOUNDARY
0000
370
50000
49376
0000
0000
055
12979
12909
0000
Uuml = 4 i n 4 e8) TTk B
where
1 = area drained by 1 m conduit L ^ J
d = thickness of the layer near the conduit _ L J
B = wet perimeter of the conduit |_ L J
The entrance resistance is caused by a layer which shows a
lower conductivity around the conduit than its surroundings
In practise radial and entrance resistancies are difficult to
deduct from hydraulic properties They can be found indirectly
by measuring simultaneously q h and h
t s
e Flow to the tertiary surface water system q
In smaller conduits usually the phreatic level is not known
So eq (7) cant be applied The flow to these conduits however
is sometimes very important (eg in areas with shallow groundwater
tables) One may a-sume (ERNST 1978) that a relationship exists
between the hydraulic head of an aquifor and the flow q
According to Ernst the tertiary system can be divided into a
number of categories with respect to their mutual distance and
bottom height For each category the flow is h - h (b)
q ( h b ) - mdash (9)
where
h = average hydraulic head | L J
h (b) = phreatic level in the category under
consideration a function of bottom height |_L J
a = geometry factor to convert h into h _ - J
Y = drainage resistance _ T J
hm = hydraulic head midway between two
conduits |_ L J m
Summation of the q(hb)-relations of the different categories
yields the overall relation between hydraulic head and flow to
the tertiary system we are looking for
f Artificial recharge or discharge q and prescribed boundary
flux qfo
The magnitudes of these flows are not influenced by the conditions
inside the groundwater basin but remain on a prescribed level
Thus q is an internal boundary condition q an external boundary cl D
condition
For confined aquifers some of the above mentioned terms dont
exist These terms are the flow through the phreatic surface qf
and the flow to the tertiary surface water system q
Aquitards are considered only as transmitters of water from one
aquifer to another with possibilities of storage (eq 6 ) This means
that for aquitards the water balance terms ar through f are set
equal to zero
25 S o l u t i o n o f g r o u n d w a t e r f l o w p r o b l e m s
The basic equations for saturated groundwater flow are eq (5)
for aquifers and eq (6) for aquitards The solution of these
equations require knowledge of the physical system and the auxiliary
conditions describing the system constraints
These auxiliary conditions are (REMSON etal 1971)
a Geometry of the system
Eg the diviation of the system into a restricted number of
layers and the horizontal extension of the system under
consideration
b The matrix of hydraulic properties including in-homogeneity
and an-isotropy
c Initial conditions Because we are dealing with non-stationary
flow the values of pertinent system variables (such as hydraulic
head at the initial time) must be known
d Boundary conditions They describe the conditions at the boundaries
pf the system as a function of time
One can distinguish between
- hydraulic head boundary
- flow (including zero flow) boundary
- both head and flow boundary
After specifying the auxiliary conditions an unique solution of
the variation of the hydraulic head in space and time can be
obtained
For special shapes of flow and assumptions exact analytical
solutions exist In nature however one is usually confronted with
problems like
- spacial variation in the hydraulic properties of the soil
- irregular shape of the groundwater basin
- empirical relations between q and h
- d not being a constant
For such situations by numerical and analogy models only an
approximation of the actual situation can be obtained In the next
chapter a numerical solution called the finite element method is
treated
III NUMERICAL SOLUTIONS
In this chapter the numerical solution of the aquifer equation
(eq 5) and the aquitard equation (eq 6) will be treated The
method of solving is the finite element method In section 31 only
a short mathematical description of this method will be given For
more mathematical background the reader is referred to publications
on this subject (eg ZIENKIEWICZ 1971 NEUMAN etal 1974 and
WESSELING 1976)
In section 32 the application of the finite element method on
the flow in groundwater basins is treated while in section 33 the
translation into a computer program is given
3 1 F i n i t e e l e m e n t m e t h o d
For reasons of simplicity the finite element method will be
explained for the stationary version of eq 5 ie
10
(k d^) + (k d ^ ) - Q = O (10) ocircx x 6x 6y y Sy
The basic idea of each numerical solution of eq (10) is to
apply this equation on a small but finite volume The flow region
of interest R is divided into a finite number of volumes called
elements The elements can have different shapes but for reasons
of conveniency we only use triangles- Quadrilaterals can also be
used because they can be divided into two triangles The sizes of
the triangles may vary They can be adapted according to the
geometry of the region and the accuracy desired (fig 11 gives an
example of the diviation of a region into triangles and quadrilaterals)
The corner points of the elements represent the nodal points
If there are N nodal points then the continuous variation of h
in region R is approximated by the expression
h (x_v) ~ h f (xy) (11) a n n
where
h (AV) = approximation of u (xy) a
h = value of h in nodal point n n
f ltxgty) = global coordinate function
In nodal point n a must be equal to h so f (x y ) = 1 and a n n n n
f (xy) = 0 i n Between f (x v ) and f (xy) of the nodal n j ] n n n n j j
points surrounding nodal point n f (xy) varies linearly from
1 to 0
In fig 3 a picture of f (xy) is given n
If one knows the values h in the nodal points n we know the approximation h of h We can find these values of b bv using for
a n
example the Galerkin method
Galerkins principle states that an approximate solution of h in
eq (10) can be obtained from the following system of equations
N ~~ (k d -A + -^- (k d T-y E h f (xy) f(xy)dxdy +
-ox x ox ocircy y oy - _ n n i R n~
J J Q f(xy) dx dy = 0 i = 1 N (12)
1 1
Fig 3 The global coordinate function f (xy)
The integration can be performed element by element with the
aid of the coordinate functions f (NEUMAN etal 1974)
For non-stationary problems it is not possible to apply the
Galerkin principle on the time derivative But it is possible
to discretize this derivative and to set up the system of equation
(eq 12) for each finite time step
One obtains in this way a series of sets of laquocriations tha s
an approximation of the non-stationary flow problem
Treatment of boundary conditions
a Initial conditions
In non-stationary flow problern gtne must start with given
values of a at time t = t ri c
b Boundary conditions
- Prescribed head boundary
Nodal points which lie on a prescribed head boundary have a
prescribed value for h bull r Hgte r umber of unknown h s is n n
reduced and accordingly no the number of equations
- Prescribed flux through z-- boundary
If one knows the flux (iuumlr jw or outflow per unit area) and
the area of the boundary oi-r element the inflow or outflow
through the boundary per flemlt=nt can be calculated
One may assume this flow to J bull part of the sink term Q in
eq (12)
12
32 A p p l i c a t i o n o f t h e f i n i t e e l e m e n t
m e t h o d o n t h e f l o w i n g r o u n d w a t e r
b a s i n s
From hydrogeological investigations the geometry and the matrix
of the hydraulic properties are known Knowing the geometry the
flow can be split up into a finite number of layers This holds
also for the nature of the boundaries
Now a nodal grid is superimposed on each layer in such a way
that the nodal points of the different layers are situated right
above each other (see fig 4) and in each layer the nodal point
lies in the middle of this layer
-nodal point
Fig 4 Schematic representation of the nodal points in the
xyz-space
Each layer has N nodal points of which P nodal points have a
prescribed head In each layer remain N-P = N nodal points with
unknown head Eq (12) can be applied to each point with unknown
head in each separate layer
As was stated in the last paragraph for each time step one
can write
13
Ah bdquo ~ -r1 + Q = O in = 12N in n m At yi A h + B (13)
where
A = in
ocircf ocircf ocircf of (k d f mdashri- ~ + k df - ~ -jS) dxdy
J x n ox ox y n oy oy
in (S d + S ) ff dx dy O if i icirc n
s y i n
Q f dx dy
The matrices A B and the vector Q have been defined in in l
elsewhere (NEUMAN etal 1974) A is the conductivity matrix in
B the storage matrix and Q the source vector in l
In order to clarify these rather complicate expressions apply
equation (13) poundo point 4 of fig 5
6 7
Fig 5 Top view of part of the nodal grid
lhl + A42h2 + A43h3 + A44h4 + A45h5 + A46h6 + A47h7 +
h - h
+ B _iuumli i ^ + Q = o 44 t - t x4
n n-1
(14)
where h and h are values of h at time t and t 4n 4n-1 4 n n-1
respectively
14
If the h-values belonging to the A-values are taken at time
t = t we get an equation with one unknown h for time t = t n-1 4n n
The equation can then be solved directly
However this is an explicit formulation which puts strong limitations
on the magnitude of the time step t - t (REMSON etal 1971) n n-1
Values larger than a certain value cause instability With an implicit method the values of h are taken at time
t = t This method is unconditionally stable Now one gets an equation n
with 7 unknowns h through h If one has 7 equations for every 1 n n
point 1 through 7 we can solve the values h i = 17
in
Of all the methods possible the iterative method of Gauss-Seidel
has been chosen The Gauss-Seidel method starts by choosing a first estimate of h at time t = t (usually equal to h ) At each in n j -i in-l iteration step a new value of h is calculated using the last
in calculated values of h If one passes the nodal points from 1 to
i n 7 eq (14) becomes
h-j + 1 = A h^ + 1 + A bdquo h | + 1 + A _ h | + 1 + A h j + A r h j + 4 n 4 1 1 n 4 2 2 n 43 3 n 45 5 n 4 6 6 n
B h B + A h J + - 4 4 4 n ~ + 0 J (A + mdash) A 4 7 h 7 n At n U 4 4 At
(15)
where
h-i = in
j = iteration step or number of iteration
i = number of nodal point n = time = t
n
The iteration process converges if the successive differences k+1 k
between h and h become continuously smaller in in
The iteration process is stopped if at each nodal point the k+1 k
difference between h and h is smaller than a certain prescribed in in
value
15
To accelerate the convergence an overrelaxation factor W is used
(for more information about W see FORSYTHE and WASOW 1960)
k+1 k T7 k+1 k h = h + W (h - h )
in in in in (16)
If W = 1 there is no overrelaxation Usually W varies between 1 and
2 and the optimum value very often is found by trial and error
Eq (15) is valid for points in aquifers The connection with
adjacent layers is made by the term Q because in this term the
flow from adjacent aquitards q is included (two aquifers are always
separated by an aquitard)
Aquitards
We dont apply the finite element method on points in aquitards
As stated before there is no horizontal flow in aquitards So it has
no sense to replace the continuous variation of h in the aquitard
by equation (11)
The nodal points in aquitards represent a certain volume of an
aquitard hich transmit water from one aquifer to another and which
has storage capacity The nodal points are included in the Gauss-
Seidel scheme -s follows (see fig 6 )
1 aauifer
a -_vS laquo 2 [~^-t^yM^^^-^r~ aqutarci l-Z
aquifer
Fig 6 Schematic situation for points in aquitards
Point 2 is situated in the middle of the aquitard with thickness
d storage coefficient S and vertical hvdraulic conductivity k c J v
Suppose the Gauss-Seidel iteration starts in the upper layer and
ends in the bottom-layer Then the flux from point 1 in aquifer 1 to
16
point 2 in the aquitard at time t = t and in iteration step j + 1
is
k q = (hf -hf )-iumlmdash f (17)
1 n 2 n OSxd1
and the flux from point 3 in aquifer 2 to point 2 in the aquitard
is
1 c 1
qbdquo = (h - hf) mdashHmdashr- (18) 2 3 n 2 n OSxd
The net flux ltl-q9 must be equal to the change in storage per unit
area per unit time This change in storage can numerically be
approximated by
te - h-
n n-1
Combina t ion of ( 1 7 ) (18) ana (19) g i v e s
l tr1 - h (hJ + 1 + hj - 2 hj+1) - - - - - - S1 - ilaquoS L u l l = o (20)
] i 3 n 2 n 1 c t - t j j x d n n -1
T + 1
From t h i s e q u a t i o n h^ kan be s o l v e d
3 3 C o m p u t e r o g r a m
The problem outl^ugrave so far has been translated into a computer
program written in FORTRAN IV
The flow chart ot Uis program is given in fig 7 whereas in
appendix A the complete listing of the program can be found
In the flow chart the 5 main parts of the program are
indicated
A Reading and printing of input data
B Determination of the matrices A and B and the vector Q per layer
17
M
1 A
FLOH to unsaturated zono
FLOW to tertiary syster
FLOW to secondnrv system
Leakage to adjacent aquitards
SOLVE equation (14)
yes
CALCULATE flux through constant head boundnr
CALCULATE waterbalance n p r 1 OAro r r J j - ~
PRINT of output
Ier node per layer
k = k+1
t = t + At
7 DUMP variables on tape
-gt-
( STOP )
Fig 7 Flow chart of the program FEMSAT with diviation in 5 main
parts A B C D and E
C START PROGRAM FEMSAT
f READ general information
PRINT general information
f READ element information
PRINT element information 7 READ stationary nodal point information
PRINT stationary nodal point information 7
READ non-stationary nodal point information for t = t
FRINT non-stationary nodal point information for t = t
L 2 (
^ yes R E A D non-stat input for t gt t
J PRINT non-stat input for t gt t (- 1 -- M degY
GENERATE A B and AREA-matrix for confined aquifers
_L
GENERATE A B and AREA-matrix for unconfined aouifer
bull
C Solving h with Gauss-Seidel
D Determination of the water balance terms per layer
E Printing of results
The program exists of a main program with several subroutines
The main program calls the subroutines In the subroutines most
calculations are performed
In this way an easy adaptation to different problems is
obtained (see also section 44) Comments in the program will
help the reader to understand the program
In the next chapter the preparation of the input data executing
of the program and the output will be treated
Acknowledgement
Some parts of the computer program were taken from the program
UNSAT2 of NEUMAN etal (1974)
IV INPUT EXECUTION AND OUTPUT
41 I n p u t
The input is divided into endogenous variables (chosen by the
user or calculated by the program during execution) and exogenous
variables (based on physical data)
411 Endogenous variables
Translation of groundwater basin into a nodal point network
Probably the most important but also the most difficult stage
in the process of simulating groundwater flow is the translation
of the natural groundwater basin into a nodal point network Questions
which arise are
a In how many layer the geo-hydrological system mustcan be
schematized
b The locations of the boundaries and the nature of these boundaries
(impervious prescribed head prescribed flux)
c The configuration of the nodal points
19
The answers of these questions depend on the problem under
consideration
- Schematization into layers
Usually our knowledge about the geo-hydrological situation is
such that no more than 5 different layers can be distinguished
(remember contrast rule)
- Boundary conditions
The top boundary condition of a groundwater basin usually is
a prescribed flux (effective precipitation)
The bottom boundary condition usually is a prescribed flux
boundary with flux zero (impervious hydrological base)
For the boundaries at the other sides of the system one sees
either prescribed head boundaries (eg a fully penetrating
river) or prescribed flux boundaries (including zero flux)
- Nodal point configuration
Nodal points are situated
on the boundaries
along internal boundaries which indicate changes in
hydraulic properties within the layer
along the tertiary surface water system
on places where artificial recharge or discharge wiii tgtke
place
elsewhere
Construction of the finite element mesh
Assume we have schemetized the groundwater basin iatu a finite
number of layers with different direction of flow (horizontal and
vertical) Within one layer the hydrological properties may vary
from place to place
Now a nodal grid is superimposed on each layer in a way already
described in section 32 Thus each layer has the same number of
nodal points situated on the same places in the horizontal plane
but with different z-coordinates
We are free to construct quadrilaterals or triangles because
the program automatically divides a quadrilateral into two triangles
having identical material properties The dimensions of the elements
20
should be small at places where large changes in hydrological
properties occur or where large gradients in hydraulic head are
expected to occur
Coding of nodal points
Each node in the finite element network is assigned an integer
code which indicates the type of calculation for this node
The coding is as follows
- Code = 1 nodal points with described head as function of time
- Code 2 internal nodal points
- Code = 3 internal nodal points which represent secondary surface
water system
- Code = 5 nodal points with prescribed artificial recharge or
discharge and points with prescribed boundary flow
(both as functions of time)
Other important endogenous variables are the magnitude of the
time step the value of the variable which stops the iteration
process and the maximum number of iterations allowed per time step
(see also group A and B of section 413)
412 Exogenous variables
After constructing the nodal grid each node represents part of
the groundwater basin Geo-hydrological properties can be assigned
to each mode based on physical data (group C D and E of section
413) In group F of section 413 time-dependent variables can
be assigned to each node
413 Data input to Program FEMSAT
The data input must be punched on cards according to the
following instructions
GROUP COLUMNS FORMAT SYMBOL DESCRIPTION
A 1-80 20A4 HEAD Desired heading to be printed at
the beginning of the execution
1-5 15 NUMNP number of nodal points
21
GROUP COLUMNS FORMAT SYMBOL DESCRIPTION
6-10 15
11-15 15
16-20 15
21-25 15
26-30 15
31-35 15
NUMEL
NUMLAY
MBAND
NTSPI
NTSPO
MAXIT
1-10 F103 ST
11-20 F103 DELT
21-30 F103 TMAX
31-40 F105 TOL
number of elements
number of layers
maximum difference in node number
between two adjacent nodes
number of time steps between a
change in non-stationary input data
number of time steps between two
successive outputs
maximum number of iterations per
time step
time to start with the execution
initial time interval
maximum time to be reached before
execution is stopped
maximum allowed absolute change in
the values of h between any two
successive iterations in a given
time step
Iterations continue until all changes
are less than or equal to TOL or
until MAXIT is exceeded
1- 5 15
6-10 15
11-15 15
16-20 15
KX(N1)
KX(N2)
KX(N3)
KX(N4)
sequential number of i-th corner of
element N
sequential number of j-th corner
sequential number of k-th corner
sequential number of 1-th corner
(see fig 8)
Fig 8 Numbering of element corner nodes
22
GROUP COLUMNS FORMAcircT SYMBOL DESCRIPTION
One card must be provided for each element in sequential order
starting with N = 1 and ending with N = NVJMEL
I-K 1103 SANG(NLY)
1i-iO MO3 C1(NLY)
21-30 F 103 C2(NLY)
21-4- 1-104 THI(NLY)
angle in degrees between CI and
the laquo-coordinate to he used in
nodal print N in layer LY (see
poundig 9)
First principal conductivity of
material in node N and layer LY
idem second principal conductivity
thickness of layer LY at nodal
point N
ig V Definition of SANGCI and C2
One card icircuiPt be provided for each model point ctarting wih M = 1
and ending with N = NUMNP If there is norlt3 than 1 layer for each
layer a set must be provided starting with the upper layer LY - 1
and ending with the bottom layer LY = NIJMLAY The variables SANG
M and 2 are not relevant tor aquitards but for reasons of
conveniency this procedure is maintained (making SANG CI and C2
dummy variables)
1-10 110 KODE(N)
11-20 II0 NTERR(N)
coding of nodal points (see
paragraph 42)
number of tertiary relation in node N
The q-h relationship of the flow to
23
GROUP COLUMNS FORMAT SYMBOL DESCRIPTION
21-30 FlOl X(N)
31-40 FlOl Y(N)
41-50 FlOl HGL(N)
51-60 FlOl HBSC(N)
61-70 FlOl LESC(N)
the tertiary system as discussed
in par 24 are schematized in a small
number of different relations
x-coardinate of node N
y-coordinate of node N
height ground level in node N
height bottom secondary conduit in
node N If node N is not a secondary
conduit this variable is a dummy
variable
length of secondary conduit in area
represented by node N
One card must be provided for each node starting with N = 1 and
ending with N = NUMNP
E 1-10 FlOl P(NLY1) initial value of h at node N in
layer LY
11-20 FlOl VERRES(NLY) vertical resistance at node N and
layer LY If the layer is an aquifer
this value is a dummy variable
21-30 F101 RADRES(NLY) radial resistance for flow to
secondary system at node N in layer
LY
31-40 FlOl ENRES(NLY) entrance resistance for flow to
secondary system at node N in layer
LY
41-50 FlOl WP(NLY) wet perimeter of secondary system at
node N in layer LY
51-60 FlOl P0R(NLY) specific yield of the material at
node N in layer LY
61-70 FlOl SS(NLY) specific storage of the material
at node N in layer LY
24
GROUP COLUMNS FORMAT SYMBOL DESCRIPTION
One card must be provided for each node starting with N = 1 and
ending with N = NUMNP If there is more than 1 layer for each layer
a set of cards must be provided starting with LY = 1 and ending
with LY = NUMLAY
F 1-10 F101 P(NLY1) value of h at different times at
node N in layer LY Node N lies
on a prescribed head boundary
11-20 F101 WTSC(N) water table in secondary system at
node N at different times
21-30 F 101 ARRED(NLY) artificial recharge to or discharge
from node N in layer LY or
prescribed boundary flow at
different times
31-40 F 101 QPHR(N) flux through phreatic surface at
node N at different times
The data in group F are non-stationary input data One card must be
provided for each node starting with N = 1 and ending with N = NUMNP
In case of more than 1 layer for each layer a set must be provided
Not relevant variables are taken as dummy variables
After a certain number of time steps (NTSPI) the data of group E
are read So the set cards E must contain NxLYxC cards where C =
(TMAXNTSPIXDELT) + 1 if DELT is a constant
42 R u n n i n g t h e p r o g r a m o n t h e C y b e r 7200
The program has been executed on a Cyber 7200 computer of
IWIS-TNO in the Hague
- The demanded input is punched on cards according to the instructions
of the last paragraph and is put on a permanent file
- For each specific problem inthe program only the dimension statements
in the main program have to be adjusted according to the instructions
in the lines of comment
- After compilation the program together with the input file is
executed
25
- Intermediate data necessary for continuation of the calculations
are saved on another permanent file This gives the opportunity
to restart the program at a time the calculation was stopped
The only change in the input file is to adjust the starting time
Remark
With the aid of a terminal small changes in the program and in the
input file can be made easily and the output immediately appears
on the terminal In this way it is possible to stop erroneous
calculations This is especially useful when being in the testing
phase of the program
43 O u t p u t
Immediately after reading the input data they are printed so
that possible errors can be detected During execution after each
certain number of time steps (NTSPO) calculated values of h and
water balance terms are printed In appendix B an example is given
The output data can be devided into two groups
A Data per node per layer
- Coding of the nodal points KODE
- Hydraulic head HEAD
- Flow through phreatic surface at the time indicated in head
of table FLOWUN
- Flow to tertiary surface water system FLOWTS
- Flow to secondary surface water system FLOWSS
- Flow to adjacent layers LEAKAGE
- Artificial recharge or discharge and prescribed boundary
flow ARRED
- Change in storage during last time step STORAGE
- Flow through prescribed head boundary LATFLOW
B Data per layer
Summation of the terms already mentioned under A gives the
magnitude of the water balance terms per layer at the time
indicated (instanteneous water balance terms) Summation of
these instanteneous terms multiplied by their respective time
intervals since time is zero gives the cumulative water balance
terms per layer
26
After each print out we are able to check the calculations
because the sum of the water balance terms per layer must be
approximately zero
44 C o m p u t i n g t i m e
The computing time depends on the number of nodal points and
the number of layers But it depends also on the magnitude of the
changes in hydraulic head and on the value which terminates the
Gauss-Seidel iteration (TOL) To give an impression TOL = 001 m
Costs per day simulation (in hlfl) on the Cyber 7200
number of nodes number of layer bull bdquo bull bull _ 1mdash x a x b (21)
where
a = costs in Dutch guilders per system second (ƒ 027)
b = empirical factor with magnitude about 001
V NUMERICAL EXPERIMENTS
In general there are two possibilities to test the validity of
numerical models The first and most commonly used method is to
compare numerical results with analytical solutions The second
possibility is to compare numerical results with measured field
data Since the last alternative requires many data that are
difficult to obtain we restrict ourselves for the time being to
comparison with the analytical solutions
5 1 F l o w t o a w e l l i n a n u n c o n f i n e d
c i r c u l a r - s h a p e d a q u i f e r
Flow to a well can be divided into steady-state flow and
unsteady-state flow (KRUSEMAN and DE RIDDER 1976)
27
511 Steady state flow in an unconfined aquifer
For this case the formula of Thiem-Dupuit is valid (see fig 10)
Q bull k lth2-hicircgt ln(r2r])
(22)
where
hhbdquo = hydraulic head in 1 and 2 respectively
Q = discharge from well
rbdquo = distance from well
CL31
hmdash^ri --r2
Fig 10 Schematic cross-section of a pumped unconfined aquifer
Fig 11 gives the nodal grid superimposed on a well in a
circular shaped aquifer Near the well the nodal point distance
is made smaller Quadrilaterals as well as triangles are used
Numerical calculations with this finite element network and 3 -1
an extraction rate of 200 m day were performed for 10 days
At that moment a steady-state situation was almost achieved
28
2tmdashnodal point number
160 180 X-axesim)
Fig 11 Nodal point configuration for flow to a well in a circular-
shaped aquifer The outer boundary is taken to be constant
The well is situated in nodal point 42
29
In fig 12 the results of the simulation are compared with the
analytical solution The agreement between numerical results and
the analytical solution is excellent
draw- 15irS (m) down
analytical solution imdashx calculated values
distance from well r (m)
Fig 12 Comparison between analyt ical and calculated s teady-state
drawdown for the problem of f ig 11 with 1
-1 Q bull - 200 m 3 d a y _ 1
k = 10 mday
d = 100 m
S = 2 y
S = 0 s
512 Unsteady-state flow in an unconfined aquifer
Analytical solutions for unsteady-state flow in an unconfined
aquifer only exist if the aquifer extends infinitely For the
aquifer given in fig 11 this condition is not fulfilled However
during the first one or two days after starting the extraction the
influence of a constant hydraulic head at the outer boundary on the
30
drawdown near the well can be neglected (see fig 13) After one
day the lateral inflow is only 25 of the extraction the remainder
being the change in storage due to drawdown
2 0 0
m 3 day 1
Q=200m 3 day
100-
vraquomdash Lateral inflow
N amp storage
I I ^ V - W 10
t (days)
Fig 13 Values of Q lateral inflow and change in storage as
functions of time
The nonsteady-state or Theis equation can be written as
i _ Q s 4iTkD
dy = 4irkD
W(u) (23)
where
u = r2S
4irkDt
r = distance from well
S = storage coefficient
Q = constant discharge from well
D = thickness of the waterbearing layer
at the beginning of pumping
t = time after starting the extraction
s = corrected drawdown
= s - 322D
s = calculated drawdown
W(u) = Theiss well function
[J
CIAT1
[T]
]
31
In fig 14 the analytical unsteady-state solution for the case
described in section 511 is given together with the numerical
results at times 05 and 10 days The agreement between numerical
results and the analytical solution is rather good
Drawdown s1(m)
10
10
10
Theiss solution
x caculated values t=Q5
bdquo t= lO
10 10- 10 10 t r2(daym2)
Fig 14 Comparison between analytical and calculated steady-state
drawdown for the problem of fig 11 with
Q
k
d
S y
s s
=
=
=
=
=
200
1
10
3 m
0
0
2
0
-1 day
m day
m
These results indicate that the assumption of a linear variation
of h between two nodal points (see fig 3 ) even with the rather
rough element network of fig 11 is acceptable So there is no
need to improve for example the finite element network by introducing
higher order coordinate functions
32
52 F l o w t o w e l l i n 3-1 a y e r e d s y s t e m
In fig 15 a schematic cross-section of a 3-layered system of
two aquifers separated by an aquitard is given A well of
infinitisemal radius is completed in the lower aquifer and discharges
at a constant rate QTT Each layer is homogeneous isotropic and
extending radially infinite
bdquo bull - bull bull ~ i
mdash Q II
aquifer I
lugrave^i~IcircIuml7i-i~~ aquitard --r-rl~-
-II--I i - i
aquifer
Fig 15 Schematic cross-section of a 3-layered system
For this situation NEUMAN and WITHERSPOON (1969a) give rather
complicated analytical solutions In the special case where the
hydraulic properties of the two aquifers are identical they give
graphical illustrations of their solutions for three different
cases
These graphical solutions give the opportunity to verify
roughly numerical results of a 3-layered system Therefore a nodal
grid is superimposed on a circular shaped groundwater flow system
with the extraction in the centre of the lower aquifer and with
a constant head on the outer boundary Fig 16 gives the nodal
point situation in the x - y plane of 1 layer The nodes in the
other two layers are situated in the middle of the layers at the
same places in the x - y plane only their z-coordinate is different
33
A
-Jy
bull33
V 5
iumlff
Fig 16 Nodal point configuration for the 3-layered problem (top view)
34
In fig 17 numerical results are compared with the analytical
solution The agreement is rather bad especially for the two
aquifers But in terms of the water balance the difference between
aquifers and aquitard is not so big because the aquitard has
about 16 times bigger storage capacity than the aquifers The
small value of the specific storage of the aquifers is the reason
that small errors in the components of the water balance have a
big influence on the calculated values of the hydraulic head
Eig 17 Dimensionless drawdown vs dimensionless time in a
3-layered system with
II = 50 m day
3 m (distance from well)
aquifer I
aquitard
aquifer II
Thi (m)
100
25
100
S s
001
0642
001
k(mday )
1
0277
1
The discrepancies are partly caused by errors in discretization
In the z-direction these errors can be eliminated by taking the
35
vertical resistance of the aquitard to be infinite Then the
calculated drawdown in the pumped aquifer can be compared with
the analytical Theiss solution
This is done in fig 18 for different distances from the well
One sees that the agreement is better when the distance is bigger
Apparently the linearization of the logarithmic drawdown curve
near the well is responsible for the differences The differences
for r = 3 m are of the same magnitude as for the pumped aquifer in
the 3-layered case
joo gt A 50
20
Theiss solution
bull calculated values r=3m (distance f rom well) x bdquo bdquo r=7m
1 L
I l_ 001 003 0 0 5 Ol Q2
- I I 5 10 t - K t
Fig 18 Dimensionless drawdown vs dimensionless time in a
confined aquifer with
Q =500 m day
k = 10 mday
D = 100 m
S = 001 s
The differences between numerical results and analytical solutions
are not caused by taking a finite area because changing the outer
boundary into an impermeable boundary has only minor effects
36
VI SUMMARY AND CONCLUSIONS
In this report a numerical model for non-stationary saturated
groundwater flow in a multi-layered system has been treated
Groundwater flow can be described in different ways
a Physical-mathematical by combining Darcys law and the law of
continuity (section 21)
b Geo-hydrological the flow in a natural groundwater basin takes
place in layers with different hydraulic properties Definitions
of the most important hydraulic properties are given in section
22
c Schematical groundwater flows horizontally in aquifers and
vertically in aquitards (section 23)
d Approximating by dividing the region of flow into a finite
number of elements and applying Darcys law and the law of
continuity on each of these elements In these ways one gets
a set of equations which can be solved given (initial and)
boundary conditions (section 25)
e Numerical the set of equations has been solved using a Galerkin
type finite element method and the implicit Gauss-Seidel method
with overrelaxation factor (section 31 and 32)
f Computer-solvable the numerical solution has been translated
in a FORTRAN IV program called FEMSAT (section 33)
In chapter IV the input execution and output of the program
FEMSAT are treated
Finally in chapter V some numerical applications are given
simulation of flow to a well in unconfined aquifer and simulation
of flow to a well in a 3-layered configuration
Comparison of hydraulic heads obtained by numerical calculations
and analytical true solutions showed an excellent agreement in case
of stationary flow and a good to moderate agreement in case of non-
stationary flow However the diviations caused by the discretization
process must be seen in relation to the errors made in the
establishment of the hydraulic properties
37
As was stated in the introduction numerical models can be
valuable tools in the management of water resources Therefore
numerical models must be suitable to simulate effects of certain
operations upon the spreading of water over saturated groundwater
unsaturated groundwater and surface water The model described in
this report is a model for saturated groundwater flow The relation
with the surface water can be taken into account via radial and
entrance resistances or via a relation between flow to the surface
water system and the hydraulic head The relation with the unsaturated
zone has been restricted to the input factor effective precipitation
In practise however the unsaturated and saturated system are
interconnected Therefore the model should be improved by taking
into account the unsaturated zone This can be done in at least
two ways
a Replacing the unsaturated zone by a parametrical model
b Coupling the unsaturated zone in an iterative way with the
saturated system
In order to find out the usefulness of the numerical model for
water resources management it will be applied to actual regional
groundwater systemslaquoAlso sensitivity analyses will be carried out
The computer model itself can be improved by applying standard
programs eg automatical generation of a nodal grid and graphical
plotting of the results
38
VII REFERENCES
ERNST LF 1962 Grondwaterstromingen in de verzadigde zone en hun
berekening bij aanwezigheid van horizontale evenwijdige open
leidingen Thesis University Utrecht pp 189
1978 Drainage of undulating soils with high groundwater
tables (in press)
FORSYTHE GE and WR WASON 1960 Finite Difference Methods for
Partial Differential Equations John Wiley and Sons New York
pp 444
KRUSEMAN GP and NA DE RIDDER 1976 Analysis and evaluation of
pumping test data ILRI-bulletin No 11 pp 200
LAAT PJM DE and C VAN DEN AKKER 1976 Simulatiemodel voor grond-
waterstroming en verdamping in Modelonderzoek 1971-1974 ten
behoeve van de waterhuishouding in Gelderland deel II
Grondslagen pp 145-197
NEUMAN SP RA FEDDES and E BRESLER 1974 Finite element
simulation of flow in saturated -unsaturated soils considering
water uptake by plants Hydraulic Engineering Laboratory
Technion Israel pp 104
and PA WITHERSPOON 1969a Theory of Flow in a confined
two Aquifer System Water Resources Research Vol 5 No 4
pp 803-816
^_ and PA WITHERSPOON 1969b Applicability of Current Theories
of Flow in leaky aquifers Water Resources Research Vol 5
No 4 pp 817-829
PINDER GF and EO FRIND 1972 Application of Galerkins Procedure
to Aquifer Analysis Water Resources Research Vol 8 no 1
pp 108-120
REMSON I GE HORNBERGER and FJ MOLZ 1971 Numerical Methods
in Subsurface Hydrology John Wiley and Sons New York
pp 389
WESSELING JG 1977 Toepassing van Finite Difference- en Finite
Element Methoden voor het oplossen van een aantal grondwatershy
stromingsproblemen (met computerprogrammas) ICW-nota 944
pp 63
39
ZIENKIEWICZ OC 1971 Finite Element Method in Engineering Science
McGraw-Hill London pp 521
40
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APPENDIX 8 EXAMPLE OF OUTPUT
HEAD AND TERMS OF UATERBALANCE PER NODE PER LAYER
AT TIME 260 IN II AND I13DAYS
T
LAYER NODE KODE HEAD FLOUUN FLOUTS FLOUSS LEAKAGE STORAGE LAT FLOU
13 19 25 31 37 43
125008 125000 12S000 12S000 12S000 12S000 12S000 12S000
0000 0000 0000 0000 0000 0000 0000 0000
0000 0000 0000 0000 0000 0000 0000 0000
0000 0000 0000 0000 0000 0000 0000 0000
- 000 - 000 -000 - 000 - 000 - 000 - 000 0000
0000 0000 0000 0000 0000 0000 0000 0000
0000 000 000 000 000 000 000 000
000 0000 0000 0P100 0000 0000 0000 0000
1 7
13 19 25 31 37
12S000 125000 125000 125000 125000 12S000 125000 125000
0000 0000 0000 0000 0000 0000 0000 0000
- 001 -007 - 023 - 020 - 009 - 003 0000
0000 0000 0000 0000 0000 0000 0000 0000
000
001
007
023
020
009
003
003
0000 0000 0000 0000 0000 0000 0000 0000
1 7
13 19 25 31 37 43
125000 125000 124998 124972 124895 124787 124647 124358
0000 0000 0000 0000 0000 0000 0000 0000
- 000 001 007 023 020 009 003
0000
0000 0000 0000 0000 0000 0000 0000
50000
0000 182
1982 3862 1674 419 078 041
0000 0000 0000 0000 0000 0000 0000 0000
TERMS OF WATERBALANCE PER LAYER DURING LAST TIMESTEP AND
AND SINCE BEGINNING IN M3DAY M3
FLOU TO UNSATURATED ZONE
FLOU TO TERTIARY SYSTEM
0000
0000
0000
0000
LAYER NO 1
FLOU TO SECONDARY SYSTEM
FLOU TO AD3ACENT LAYERCS)
ARTIFICIAL FLOU AND PRESCRIBED BOUNDARY FLOU
ACCUMULATION TERMS
LATERAL FLOU THROUGH PRESCR HEAD BOUNDARY
0000
-000
0000
000
000
0000
-000
0000
000
000
LAYER NO 2
FLOU TO SECONDARY SYSTEM
FLOU TO AD3ACENT LAYERCS1
ARTIFICIAL FLOU AND PRESCRIBED BOUNDARY FLOU
ACCUMULATION TERMS
LATERAL FLOU THROUGH PRESCR HEAD BOUNDARY
0000
-378
0000
381
0000
0000
-055
0000
056
0000
LAYER NO 3
FLOU TO SECONDARY SYSTEM
FLOU TO AD3ACENT LAYERCS5
ARTIFICIAL FLOU AND PRESCRIBED BOUNDARY FLOU
ACCUMULATION TERMS
LATERAL FLOU THROUGH PRESCR HEAD BOUNDARY
0000
370
50000
49376
0000
0000
055
12979
12909
0000
f Artificial recharge or discharge q and prescribed boundary
flux qfo
The magnitudes of these flows are not influenced by the conditions
inside the groundwater basin but remain on a prescribed level
Thus q is an internal boundary condition q an external boundary cl D
condition
For confined aquifers some of the above mentioned terms dont
exist These terms are the flow through the phreatic surface qf
and the flow to the tertiary surface water system q
Aquitards are considered only as transmitters of water from one
aquifer to another with possibilities of storage (eq 6 ) This means
that for aquitards the water balance terms ar through f are set
equal to zero
25 S o l u t i o n o f g r o u n d w a t e r f l o w p r o b l e m s
The basic equations for saturated groundwater flow are eq (5)
for aquifers and eq (6) for aquitards The solution of these
equations require knowledge of the physical system and the auxiliary
conditions describing the system constraints
These auxiliary conditions are (REMSON etal 1971)
a Geometry of the system
Eg the diviation of the system into a restricted number of
layers and the horizontal extension of the system under
consideration
b The matrix of hydraulic properties including in-homogeneity
and an-isotropy
c Initial conditions Because we are dealing with non-stationary
flow the values of pertinent system variables (such as hydraulic
head at the initial time) must be known
d Boundary conditions They describe the conditions at the boundaries
pf the system as a function of time
One can distinguish between
- hydraulic head boundary
- flow (including zero flow) boundary
- both head and flow boundary
After specifying the auxiliary conditions an unique solution of
the variation of the hydraulic head in space and time can be
obtained
For special shapes of flow and assumptions exact analytical
solutions exist In nature however one is usually confronted with
problems like
- spacial variation in the hydraulic properties of the soil
- irregular shape of the groundwater basin
- empirical relations between q and h
- d not being a constant
For such situations by numerical and analogy models only an
approximation of the actual situation can be obtained In the next
chapter a numerical solution called the finite element method is
treated
III NUMERICAL SOLUTIONS
In this chapter the numerical solution of the aquifer equation
(eq 5) and the aquitard equation (eq 6) will be treated The
method of solving is the finite element method In section 31 only
a short mathematical description of this method will be given For
more mathematical background the reader is referred to publications
on this subject (eg ZIENKIEWICZ 1971 NEUMAN etal 1974 and
WESSELING 1976)
In section 32 the application of the finite element method on
the flow in groundwater basins is treated while in section 33 the
translation into a computer program is given
3 1 F i n i t e e l e m e n t m e t h o d
For reasons of simplicity the finite element method will be
explained for the stationary version of eq 5 ie
10
(k d^) + (k d ^ ) - Q = O (10) ocircx x 6x 6y y Sy
The basic idea of each numerical solution of eq (10) is to
apply this equation on a small but finite volume The flow region
of interest R is divided into a finite number of volumes called
elements The elements can have different shapes but for reasons
of conveniency we only use triangles- Quadrilaterals can also be
used because they can be divided into two triangles The sizes of
the triangles may vary They can be adapted according to the
geometry of the region and the accuracy desired (fig 11 gives an
example of the diviation of a region into triangles and quadrilaterals)
The corner points of the elements represent the nodal points
If there are N nodal points then the continuous variation of h
in region R is approximated by the expression
h (x_v) ~ h f (xy) (11) a n n
where
h (AV) = approximation of u (xy) a
h = value of h in nodal point n n
f ltxgty) = global coordinate function
In nodal point n a must be equal to h so f (x y ) = 1 and a n n n n
f (xy) = 0 i n Between f (x v ) and f (xy) of the nodal n j ] n n n n j j
points surrounding nodal point n f (xy) varies linearly from
1 to 0
In fig 3 a picture of f (xy) is given n
If one knows the values h in the nodal points n we know the approximation h of h We can find these values of b bv using for
a n
example the Galerkin method
Galerkins principle states that an approximate solution of h in
eq (10) can be obtained from the following system of equations
N ~~ (k d -A + -^- (k d T-y E h f (xy) f(xy)dxdy +
-ox x ox ocircy y oy - _ n n i R n~
J J Q f(xy) dx dy = 0 i = 1 N (12)
1 1
Fig 3 The global coordinate function f (xy)
The integration can be performed element by element with the
aid of the coordinate functions f (NEUMAN etal 1974)
For non-stationary problems it is not possible to apply the
Galerkin principle on the time derivative But it is possible
to discretize this derivative and to set up the system of equation
(eq 12) for each finite time step
One obtains in this way a series of sets of laquocriations tha s
an approximation of the non-stationary flow problem
Treatment of boundary conditions
a Initial conditions
In non-stationary flow problern gtne must start with given
values of a at time t = t ri c
b Boundary conditions
- Prescribed head boundary
Nodal points which lie on a prescribed head boundary have a
prescribed value for h bull r Hgte r umber of unknown h s is n n
reduced and accordingly no the number of equations
- Prescribed flux through z-- boundary
If one knows the flux (iuumlr jw or outflow per unit area) and
the area of the boundary oi-r element the inflow or outflow
through the boundary per flemlt=nt can be calculated
One may assume this flow to J bull part of the sink term Q in
eq (12)
12
32 A p p l i c a t i o n o f t h e f i n i t e e l e m e n t
m e t h o d o n t h e f l o w i n g r o u n d w a t e r
b a s i n s
From hydrogeological investigations the geometry and the matrix
of the hydraulic properties are known Knowing the geometry the
flow can be split up into a finite number of layers This holds
also for the nature of the boundaries
Now a nodal grid is superimposed on each layer in such a way
that the nodal points of the different layers are situated right
above each other (see fig 4) and in each layer the nodal point
lies in the middle of this layer
-nodal point
Fig 4 Schematic representation of the nodal points in the
xyz-space
Each layer has N nodal points of which P nodal points have a
prescribed head In each layer remain N-P = N nodal points with
unknown head Eq (12) can be applied to each point with unknown
head in each separate layer
As was stated in the last paragraph for each time step one
can write
13
Ah bdquo ~ -r1 + Q = O in = 12N in n m At yi A h + B (13)
where
A = in
ocircf ocircf ocircf of (k d f mdashri- ~ + k df - ~ -jS) dxdy
J x n ox ox y n oy oy
in (S d + S ) ff dx dy O if i icirc n
s y i n
Q f dx dy
The matrices A B and the vector Q have been defined in in l
elsewhere (NEUMAN etal 1974) A is the conductivity matrix in
B the storage matrix and Q the source vector in l
In order to clarify these rather complicate expressions apply
equation (13) poundo point 4 of fig 5
6 7
Fig 5 Top view of part of the nodal grid
lhl + A42h2 + A43h3 + A44h4 + A45h5 + A46h6 + A47h7 +
h - h
+ B _iuumli i ^ + Q = o 44 t - t x4
n n-1
(14)
where h and h are values of h at time t and t 4n 4n-1 4 n n-1
respectively
14
If the h-values belonging to the A-values are taken at time
t = t we get an equation with one unknown h for time t = t n-1 4n n
The equation can then be solved directly
However this is an explicit formulation which puts strong limitations
on the magnitude of the time step t - t (REMSON etal 1971) n n-1
Values larger than a certain value cause instability With an implicit method the values of h are taken at time
t = t This method is unconditionally stable Now one gets an equation n
with 7 unknowns h through h If one has 7 equations for every 1 n n
point 1 through 7 we can solve the values h i = 17
in
Of all the methods possible the iterative method of Gauss-Seidel
has been chosen The Gauss-Seidel method starts by choosing a first estimate of h at time t = t (usually equal to h ) At each in n j -i in-l iteration step a new value of h is calculated using the last
in calculated values of h If one passes the nodal points from 1 to
i n 7 eq (14) becomes
h-j + 1 = A h^ + 1 + A bdquo h | + 1 + A _ h | + 1 + A h j + A r h j + 4 n 4 1 1 n 4 2 2 n 43 3 n 45 5 n 4 6 6 n
B h B + A h J + - 4 4 4 n ~ + 0 J (A + mdash) A 4 7 h 7 n At n U 4 4 At
(15)
where
h-i = in
j = iteration step or number of iteration
i = number of nodal point n = time = t
n
The iteration process converges if the successive differences k+1 k
between h and h become continuously smaller in in
The iteration process is stopped if at each nodal point the k+1 k
difference between h and h is smaller than a certain prescribed in in
value
15
To accelerate the convergence an overrelaxation factor W is used
(for more information about W see FORSYTHE and WASOW 1960)
k+1 k T7 k+1 k h = h + W (h - h )
in in in in (16)
If W = 1 there is no overrelaxation Usually W varies between 1 and
2 and the optimum value very often is found by trial and error
Eq (15) is valid for points in aquifers The connection with
adjacent layers is made by the term Q because in this term the
flow from adjacent aquitards q is included (two aquifers are always
separated by an aquitard)
Aquitards
We dont apply the finite element method on points in aquitards
As stated before there is no horizontal flow in aquitards So it has
no sense to replace the continuous variation of h in the aquitard
by equation (11)
The nodal points in aquitards represent a certain volume of an
aquitard hich transmit water from one aquifer to another and which
has storage capacity The nodal points are included in the Gauss-
Seidel scheme -s follows (see fig 6 )
1 aauifer
a -_vS laquo 2 [~^-t^yM^^^-^r~ aqutarci l-Z
aquifer
Fig 6 Schematic situation for points in aquitards
Point 2 is situated in the middle of the aquitard with thickness
d storage coefficient S and vertical hvdraulic conductivity k c J v
Suppose the Gauss-Seidel iteration starts in the upper layer and
ends in the bottom-layer Then the flux from point 1 in aquifer 1 to
16
point 2 in the aquitard at time t = t and in iteration step j + 1
is
k q = (hf -hf )-iumlmdash f (17)
1 n 2 n OSxd1
and the flux from point 3 in aquifer 2 to point 2 in the aquitard
is
1 c 1
qbdquo = (h - hf) mdashHmdashr- (18) 2 3 n 2 n OSxd
The net flux ltl-q9 must be equal to the change in storage per unit
area per unit time This change in storage can numerically be
approximated by
te - h-
n n-1
Combina t ion of ( 1 7 ) (18) ana (19) g i v e s
l tr1 - h (hJ + 1 + hj - 2 hj+1) - - - - - - S1 - ilaquoS L u l l = o (20)
] i 3 n 2 n 1 c t - t j j x d n n -1
T + 1
From t h i s e q u a t i o n h^ kan be s o l v e d
3 3 C o m p u t e r o g r a m
The problem outl^ugrave so far has been translated into a computer
program written in FORTRAN IV
The flow chart ot Uis program is given in fig 7 whereas in
appendix A the complete listing of the program can be found
In the flow chart the 5 main parts of the program are
indicated
A Reading and printing of input data
B Determination of the matrices A and B and the vector Q per layer
17
M
1 A
FLOH to unsaturated zono
FLOW to tertiary syster
FLOW to secondnrv system
Leakage to adjacent aquitards
SOLVE equation (14)
yes
CALCULATE flux through constant head boundnr
CALCULATE waterbalance n p r 1 OAro r r J j - ~
PRINT of output
Ier node per layer
k = k+1
t = t + At
7 DUMP variables on tape
-gt-
( STOP )
Fig 7 Flow chart of the program FEMSAT with diviation in 5 main
parts A B C D and E
C START PROGRAM FEMSAT
f READ general information
PRINT general information
f READ element information
PRINT element information 7 READ stationary nodal point information
PRINT stationary nodal point information 7
READ non-stationary nodal point information for t = t
FRINT non-stationary nodal point information for t = t
L 2 (
^ yes R E A D non-stat input for t gt t
J PRINT non-stat input for t gt t (- 1 -- M degY
GENERATE A B and AREA-matrix for confined aquifers
_L
GENERATE A B and AREA-matrix for unconfined aouifer
bull
C Solving h with Gauss-Seidel
D Determination of the water balance terms per layer
E Printing of results
The program exists of a main program with several subroutines
The main program calls the subroutines In the subroutines most
calculations are performed
In this way an easy adaptation to different problems is
obtained (see also section 44) Comments in the program will
help the reader to understand the program
In the next chapter the preparation of the input data executing
of the program and the output will be treated
Acknowledgement
Some parts of the computer program were taken from the program
UNSAT2 of NEUMAN etal (1974)
IV INPUT EXECUTION AND OUTPUT
41 I n p u t
The input is divided into endogenous variables (chosen by the
user or calculated by the program during execution) and exogenous
variables (based on physical data)
411 Endogenous variables
Translation of groundwater basin into a nodal point network
Probably the most important but also the most difficult stage
in the process of simulating groundwater flow is the translation
of the natural groundwater basin into a nodal point network Questions
which arise are
a In how many layer the geo-hydrological system mustcan be
schematized
b The locations of the boundaries and the nature of these boundaries
(impervious prescribed head prescribed flux)
c The configuration of the nodal points
19
The answers of these questions depend on the problem under
consideration
- Schematization into layers
Usually our knowledge about the geo-hydrological situation is
such that no more than 5 different layers can be distinguished
(remember contrast rule)
- Boundary conditions
The top boundary condition of a groundwater basin usually is
a prescribed flux (effective precipitation)
The bottom boundary condition usually is a prescribed flux
boundary with flux zero (impervious hydrological base)
For the boundaries at the other sides of the system one sees
either prescribed head boundaries (eg a fully penetrating
river) or prescribed flux boundaries (including zero flux)
- Nodal point configuration
Nodal points are situated
on the boundaries
along internal boundaries which indicate changes in
hydraulic properties within the layer
along the tertiary surface water system
on places where artificial recharge or discharge wiii tgtke
place
elsewhere
Construction of the finite element mesh
Assume we have schemetized the groundwater basin iatu a finite
number of layers with different direction of flow (horizontal and
vertical) Within one layer the hydrological properties may vary
from place to place
Now a nodal grid is superimposed on each layer in a way already
described in section 32 Thus each layer has the same number of
nodal points situated on the same places in the horizontal plane
but with different z-coordinates
We are free to construct quadrilaterals or triangles because
the program automatically divides a quadrilateral into two triangles
having identical material properties The dimensions of the elements
20
should be small at places where large changes in hydrological
properties occur or where large gradients in hydraulic head are
expected to occur
Coding of nodal points
Each node in the finite element network is assigned an integer
code which indicates the type of calculation for this node
The coding is as follows
- Code = 1 nodal points with described head as function of time
- Code 2 internal nodal points
- Code = 3 internal nodal points which represent secondary surface
water system
- Code = 5 nodal points with prescribed artificial recharge or
discharge and points with prescribed boundary flow
(both as functions of time)
Other important endogenous variables are the magnitude of the
time step the value of the variable which stops the iteration
process and the maximum number of iterations allowed per time step
(see also group A and B of section 413)
412 Exogenous variables
After constructing the nodal grid each node represents part of
the groundwater basin Geo-hydrological properties can be assigned
to each mode based on physical data (group C D and E of section
413) In group F of section 413 time-dependent variables can
be assigned to each node
413 Data input to Program FEMSAT
The data input must be punched on cards according to the
following instructions
GROUP COLUMNS FORMAT SYMBOL DESCRIPTION
A 1-80 20A4 HEAD Desired heading to be printed at
the beginning of the execution
1-5 15 NUMNP number of nodal points
21
GROUP COLUMNS FORMAT SYMBOL DESCRIPTION
6-10 15
11-15 15
16-20 15
21-25 15
26-30 15
31-35 15
NUMEL
NUMLAY
MBAND
NTSPI
NTSPO
MAXIT
1-10 F103 ST
11-20 F103 DELT
21-30 F103 TMAX
31-40 F105 TOL
number of elements
number of layers
maximum difference in node number
between two adjacent nodes
number of time steps between a
change in non-stationary input data
number of time steps between two
successive outputs
maximum number of iterations per
time step
time to start with the execution
initial time interval
maximum time to be reached before
execution is stopped
maximum allowed absolute change in
the values of h between any two
successive iterations in a given
time step
Iterations continue until all changes
are less than or equal to TOL or
until MAXIT is exceeded
1- 5 15
6-10 15
11-15 15
16-20 15
KX(N1)
KX(N2)
KX(N3)
KX(N4)
sequential number of i-th corner of
element N
sequential number of j-th corner
sequential number of k-th corner
sequential number of 1-th corner
(see fig 8)
Fig 8 Numbering of element corner nodes
22
GROUP COLUMNS FORMAcircT SYMBOL DESCRIPTION
One card must be provided for each element in sequential order
starting with N = 1 and ending with N = NVJMEL
I-K 1103 SANG(NLY)
1i-iO MO3 C1(NLY)
21-30 F 103 C2(NLY)
21-4- 1-104 THI(NLY)
angle in degrees between CI and
the laquo-coordinate to he used in
nodal print N in layer LY (see
poundig 9)
First principal conductivity of
material in node N and layer LY
idem second principal conductivity
thickness of layer LY at nodal
point N
ig V Definition of SANGCI and C2
One card icircuiPt be provided for each model point ctarting wih M = 1
and ending with N = NUMNP If there is norlt3 than 1 layer for each
layer a set must be provided starting with the upper layer LY - 1
and ending with the bottom layer LY = NIJMLAY The variables SANG
M and 2 are not relevant tor aquitards but for reasons of
conveniency this procedure is maintained (making SANG CI and C2
dummy variables)
1-10 110 KODE(N)
11-20 II0 NTERR(N)
coding of nodal points (see
paragraph 42)
number of tertiary relation in node N
The q-h relationship of the flow to
23
GROUP COLUMNS FORMAT SYMBOL DESCRIPTION
21-30 FlOl X(N)
31-40 FlOl Y(N)
41-50 FlOl HGL(N)
51-60 FlOl HBSC(N)
61-70 FlOl LESC(N)
the tertiary system as discussed
in par 24 are schematized in a small
number of different relations
x-coardinate of node N
y-coordinate of node N
height ground level in node N
height bottom secondary conduit in
node N If node N is not a secondary
conduit this variable is a dummy
variable
length of secondary conduit in area
represented by node N
One card must be provided for each node starting with N = 1 and
ending with N = NUMNP
E 1-10 FlOl P(NLY1) initial value of h at node N in
layer LY
11-20 FlOl VERRES(NLY) vertical resistance at node N and
layer LY If the layer is an aquifer
this value is a dummy variable
21-30 F101 RADRES(NLY) radial resistance for flow to
secondary system at node N in layer
LY
31-40 FlOl ENRES(NLY) entrance resistance for flow to
secondary system at node N in layer
LY
41-50 FlOl WP(NLY) wet perimeter of secondary system at
node N in layer LY
51-60 FlOl P0R(NLY) specific yield of the material at
node N in layer LY
61-70 FlOl SS(NLY) specific storage of the material
at node N in layer LY
24
GROUP COLUMNS FORMAT SYMBOL DESCRIPTION
One card must be provided for each node starting with N = 1 and
ending with N = NUMNP If there is more than 1 layer for each layer
a set of cards must be provided starting with LY = 1 and ending
with LY = NUMLAY
F 1-10 F101 P(NLY1) value of h at different times at
node N in layer LY Node N lies
on a prescribed head boundary
11-20 F101 WTSC(N) water table in secondary system at
node N at different times
21-30 F 101 ARRED(NLY) artificial recharge to or discharge
from node N in layer LY or
prescribed boundary flow at
different times
31-40 F 101 QPHR(N) flux through phreatic surface at
node N at different times
The data in group F are non-stationary input data One card must be
provided for each node starting with N = 1 and ending with N = NUMNP
In case of more than 1 layer for each layer a set must be provided
Not relevant variables are taken as dummy variables
After a certain number of time steps (NTSPI) the data of group E
are read So the set cards E must contain NxLYxC cards where C =
(TMAXNTSPIXDELT) + 1 if DELT is a constant
42 R u n n i n g t h e p r o g r a m o n t h e C y b e r 7200
The program has been executed on a Cyber 7200 computer of
IWIS-TNO in the Hague
- The demanded input is punched on cards according to the instructions
of the last paragraph and is put on a permanent file
- For each specific problem inthe program only the dimension statements
in the main program have to be adjusted according to the instructions
in the lines of comment
- After compilation the program together with the input file is
executed
25
- Intermediate data necessary for continuation of the calculations
are saved on another permanent file This gives the opportunity
to restart the program at a time the calculation was stopped
The only change in the input file is to adjust the starting time
Remark
With the aid of a terminal small changes in the program and in the
input file can be made easily and the output immediately appears
on the terminal In this way it is possible to stop erroneous
calculations This is especially useful when being in the testing
phase of the program
43 O u t p u t
Immediately after reading the input data they are printed so
that possible errors can be detected During execution after each
certain number of time steps (NTSPO) calculated values of h and
water balance terms are printed In appendix B an example is given
The output data can be devided into two groups
A Data per node per layer
- Coding of the nodal points KODE
- Hydraulic head HEAD
- Flow through phreatic surface at the time indicated in head
of table FLOWUN
- Flow to tertiary surface water system FLOWTS
- Flow to secondary surface water system FLOWSS
- Flow to adjacent layers LEAKAGE
- Artificial recharge or discharge and prescribed boundary
flow ARRED
- Change in storage during last time step STORAGE
- Flow through prescribed head boundary LATFLOW
B Data per layer
Summation of the terms already mentioned under A gives the
magnitude of the water balance terms per layer at the time
indicated (instanteneous water balance terms) Summation of
these instanteneous terms multiplied by their respective time
intervals since time is zero gives the cumulative water balance
terms per layer
26
After each print out we are able to check the calculations
because the sum of the water balance terms per layer must be
approximately zero
44 C o m p u t i n g t i m e
The computing time depends on the number of nodal points and
the number of layers But it depends also on the magnitude of the
changes in hydraulic head and on the value which terminates the
Gauss-Seidel iteration (TOL) To give an impression TOL = 001 m
Costs per day simulation (in hlfl) on the Cyber 7200
number of nodes number of layer bull bdquo bull bull _ 1mdash x a x b (21)
where
a = costs in Dutch guilders per system second (ƒ 027)
b = empirical factor with magnitude about 001
V NUMERICAL EXPERIMENTS
In general there are two possibilities to test the validity of
numerical models The first and most commonly used method is to
compare numerical results with analytical solutions The second
possibility is to compare numerical results with measured field
data Since the last alternative requires many data that are
difficult to obtain we restrict ourselves for the time being to
comparison with the analytical solutions
5 1 F l o w t o a w e l l i n a n u n c o n f i n e d
c i r c u l a r - s h a p e d a q u i f e r
Flow to a well can be divided into steady-state flow and
unsteady-state flow (KRUSEMAN and DE RIDDER 1976)
27
511 Steady state flow in an unconfined aquifer
For this case the formula of Thiem-Dupuit is valid (see fig 10)
Q bull k lth2-hicircgt ln(r2r])
(22)
where
hhbdquo = hydraulic head in 1 and 2 respectively
Q = discharge from well
rbdquo = distance from well
CL31
hmdash^ri --r2
Fig 10 Schematic cross-section of a pumped unconfined aquifer
Fig 11 gives the nodal grid superimposed on a well in a
circular shaped aquifer Near the well the nodal point distance
is made smaller Quadrilaterals as well as triangles are used
Numerical calculations with this finite element network and 3 -1
an extraction rate of 200 m day were performed for 10 days
At that moment a steady-state situation was almost achieved
28
2tmdashnodal point number
160 180 X-axesim)
Fig 11 Nodal point configuration for flow to a well in a circular-
shaped aquifer The outer boundary is taken to be constant
The well is situated in nodal point 42
29
In fig 12 the results of the simulation are compared with the
analytical solution The agreement between numerical results and
the analytical solution is excellent
draw- 15irS (m) down
analytical solution imdashx calculated values
distance from well r (m)
Fig 12 Comparison between analyt ical and calculated s teady-state
drawdown for the problem of f ig 11 with 1
-1 Q bull - 200 m 3 d a y _ 1
k = 10 mday
d = 100 m
S = 2 y
S = 0 s
512 Unsteady-state flow in an unconfined aquifer
Analytical solutions for unsteady-state flow in an unconfined
aquifer only exist if the aquifer extends infinitely For the
aquifer given in fig 11 this condition is not fulfilled However
during the first one or two days after starting the extraction the
influence of a constant hydraulic head at the outer boundary on the
30
drawdown near the well can be neglected (see fig 13) After one
day the lateral inflow is only 25 of the extraction the remainder
being the change in storage due to drawdown
2 0 0
m 3 day 1
Q=200m 3 day
100-
vraquomdash Lateral inflow
N amp storage
I I ^ V - W 10
t (days)
Fig 13 Values of Q lateral inflow and change in storage as
functions of time
The nonsteady-state or Theis equation can be written as
i _ Q s 4iTkD
dy = 4irkD
W(u) (23)
where
u = r2S
4irkDt
r = distance from well
S = storage coefficient
Q = constant discharge from well
D = thickness of the waterbearing layer
at the beginning of pumping
t = time after starting the extraction
s = corrected drawdown
= s - 322D
s = calculated drawdown
W(u) = Theiss well function
[J
CIAT1
[T]
]
31
In fig 14 the analytical unsteady-state solution for the case
described in section 511 is given together with the numerical
results at times 05 and 10 days The agreement between numerical
results and the analytical solution is rather good
Drawdown s1(m)
10
10
10
Theiss solution
x caculated values t=Q5
bdquo t= lO
10 10- 10 10 t r2(daym2)
Fig 14 Comparison between analytical and calculated steady-state
drawdown for the problem of fig 11 with
Q
k
d
S y
s s
=
=
=
=
=
200
1
10
3 m
0
0
2
0
-1 day
m day
m
These results indicate that the assumption of a linear variation
of h between two nodal points (see fig 3 ) even with the rather
rough element network of fig 11 is acceptable So there is no
need to improve for example the finite element network by introducing
higher order coordinate functions
32
52 F l o w t o w e l l i n 3-1 a y e r e d s y s t e m
In fig 15 a schematic cross-section of a 3-layered system of
two aquifers separated by an aquitard is given A well of
infinitisemal radius is completed in the lower aquifer and discharges
at a constant rate QTT Each layer is homogeneous isotropic and
extending radially infinite
bdquo bull - bull bull ~ i
mdash Q II
aquifer I
lugrave^i~IcircIuml7i-i~~ aquitard --r-rl~-
-II--I i - i
aquifer
Fig 15 Schematic cross-section of a 3-layered system
For this situation NEUMAN and WITHERSPOON (1969a) give rather
complicated analytical solutions In the special case where the
hydraulic properties of the two aquifers are identical they give
graphical illustrations of their solutions for three different
cases
These graphical solutions give the opportunity to verify
roughly numerical results of a 3-layered system Therefore a nodal
grid is superimposed on a circular shaped groundwater flow system
with the extraction in the centre of the lower aquifer and with
a constant head on the outer boundary Fig 16 gives the nodal
point situation in the x - y plane of 1 layer The nodes in the
other two layers are situated in the middle of the layers at the
same places in the x - y plane only their z-coordinate is different
33
A
-Jy
bull33
V 5
iumlff
Fig 16 Nodal point configuration for the 3-layered problem (top view)
34
In fig 17 numerical results are compared with the analytical
solution The agreement is rather bad especially for the two
aquifers But in terms of the water balance the difference between
aquifers and aquitard is not so big because the aquitard has
about 16 times bigger storage capacity than the aquifers The
small value of the specific storage of the aquifers is the reason
that small errors in the components of the water balance have a
big influence on the calculated values of the hydraulic head
Eig 17 Dimensionless drawdown vs dimensionless time in a
3-layered system with
II = 50 m day
3 m (distance from well)
aquifer I
aquitard
aquifer II
Thi (m)
100
25
100
S s
001
0642
001
k(mday )
1
0277
1
The discrepancies are partly caused by errors in discretization
In the z-direction these errors can be eliminated by taking the
35
vertical resistance of the aquitard to be infinite Then the
calculated drawdown in the pumped aquifer can be compared with
the analytical Theiss solution
This is done in fig 18 for different distances from the well
One sees that the agreement is better when the distance is bigger
Apparently the linearization of the logarithmic drawdown curve
near the well is responsible for the differences The differences
for r = 3 m are of the same magnitude as for the pumped aquifer in
the 3-layered case
joo gt A 50
20
Theiss solution
bull calculated values r=3m (distance f rom well) x bdquo bdquo r=7m
1 L
I l_ 001 003 0 0 5 Ol Q2
- I I 5 10 t - K t
Fig 18 Dimensionless drawdown vs dimensionless time in a
confined aquifer with
Q =500 m day
k = 10 mday
D = 100 m
S = 001 s
The differences between numerical results and analytical solutions
are not caused by taking a finite area because changing the outer
boundary into an impermeable boundary has only minor effects
36
VI SUMMARY AND CONCLUSIONS
In this report a numerical model for non-stationary saturated
groundwater flow in a multi-layered system has been treated
Groundwater flow can be described in different ways
a Physical-mathematical by combining Darcys law and the law of
continuity (section 21)
b Geo-hydrological the flow in a natural groundwater basin takes
place in layers with different hydraulic properties Definitions
of the most important hydraulic properties are given in section
22
c Schematical groundwater flows horizontally in aquifers and
vertically in aquitards (section 23)
d Approximating by dividing the region of flow into a finite
number of elements and applying Darcys law and the law of
continuity on each of these elements In these ways one gets
a set of equations which can be solved given (initial and)
boundary conditions (section 25)
e Numerical the set of equations has been solved using a Galerkin
type finite element method and the implicit Gauss-Seidel method
with overrelaxation factor (section 31 and 32)
f Computer-solvable the numerical solution has been translated
in a FORTRAN IV program called FEMSAT (section 33)
In chapter IV the input execution and output of the program
FEMSAT are treated
Finally in chapter V some numerical applications are given
simulation of flow to a well in unconfined aquifer and simulation
of flow to a well in a 3-layered configuration
Comparison of hydraulic heads obtained by numerical calculations
and analytical true solutions showed an excellent agreement in case
of stationary flow and a good to moderate agreement in case of non-
stationary flow However the diviations caused by the discretization
process must be seen in relation to the errors made in the
establishment of the hydraulic properties
37
As was stated in the introduction numerical models can be
valuable tools in the management of water resources Therefore
numerical models must be suitable to simulate effects of certain
operations upon the spreading of water over saturated groundwater
unsaturated groundwater and surface water The model described in
this report is a model for saturated groundwater flow The relation
with the surface water can be taken into account via radial and
entrance resistances or via a relation between flow to the surface
water system and the hydraulic head The relation with the unsaturated
zone has been restricted to the input factor effective precipitation
In practise however the unsaturated and saturated system are
interconnected Therefore the model should be improved by taking
into account the unsaturated zone This can be done in at least
two ways
a Replacing the unsaturated zone by a parametrical model
b Coupling the unsaturated zone in an iterative way with the
saturated system
In order to find out the usefulness of the numerical model for
water resources management it will be applied to actual regional
groundwater systemslaquoAlso sensitivity analyses will be carried out
The computer model itself can be improved by applying standard
programs eg automatical generation of a nodal grid and graphical
plotting of the results
38
VII REFERENCES
ERNST LF 1962 Grondwaterstromingen in de verzadigde zone en hun
berekening bij aanwezigheid van horizontale evenwijdige open
leidingen Thesis University Utrecht pp 189
1978 Drainage of undulating soils with high groundwater
tables (in press)
FORSYTHE GE and WR WASON 1960 Finite Difference Methods for
Partial Differential Equations John Wiley and Sons New York
pp 444
KRUSEMAN GP and NA DE RIDDER 1976 Analysis and evaluation of
pumping test data ILRI-bulletin No 11 pp 200
LAAT PJM DE and C VAN DEN AKKER 1976 Simulatiemodel voor grond-
waterstroming en verdamping in Modelonderzoek 1971-1974 ten
behoeve van de waterhuishouding in Gelderland deel II
Grondslagen pp 145-197
NEUMAN SP RA FEDDES and E BRESLER 1974 Finite element
simulation of flow in saturated -unsaturated soils considering
water uptake by plants Hydraulic Engineering Laboratory
Technion Israel pp 104
and PA WITHERSPOON 1969a Theory of Flow in a confined
two Aquifer System Water Resources Research Vol 5 No 4
pp 803-816
^_ and PA WITHERSPOON 1969b Applicability of Current Theories
of Flow in leaky aquifers Water Resources Research Vol 5
No 4 pp 817-829
PINDER GF and EO FRIND 1972 Application of Galerkins Procedure
to Aquifer Analysis Water Resources Research Vol 8 no 1
pp 108-120
REMSON I GE HORNBERGER and FJ MOLZ 1971 Numerical Methods
in Subsurface Hydrology John Wiley and Sons New York
pp 389
WESSELING JG 1977 Toepassing van Finite Difference- en Finite
Element Methoden voor het oplossen van een aantal grondwatershy
stromingsproblemen (met computerprogrammas) ICW-nota 944
pp 63
39
ZIENKIEWICZ OC 1971 Finite Element Method in Engineering Science
McGraw-Hill London pp 521
40
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X - e - o
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LUX
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M II II II II II II
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z o r s n r z u icirc N H O - 3 3 0 3 3 m f - O f J l w i c ^ ^ c m z ^ raquo f x mdash 33 r- 0 - 1 p s | -O bull-raquo 3 bulllt 3gt - - - raquo - lt - lt
3 J M raquo H _ - 3 Z 3 Z t n 3gt H Xraquo- -H S - 1 -H
I 03 f - NI r -fc copy bull 3D -lt t n t n -il X ~i X ~i -raquo - n - n - T U mdash n r z r - - t r c c 0 cn a z 3 c - e - - n a i gt -s - n - - n -c 0 r 0 r
mdashf c - i cn c z m 2 33 en H xgt i - H 3gt C 33 33 -lt X
_ 3
-k
33 s c e s t n
j t i T T i - i i n n A Z O S O C O O X d 33
U) 3gt V Z N O bull
t o
Z n m
00 m 0
z z
z C l
z
raquo J J
- -n bullbull-gt r w 33 J Z 2 1gt n bullbull r n
3 Z 3 3 - 1 -H 1 gt - H 1 gt B H H H
V t n Ni
mdash1 m 33 3 t n
- n
c xgt
m x 55
f j
M zgt -n
O)
_ D t 3 - T) C bulln c 0 3= 0 t o 0 x
3= w gt0 3 raquo-i 3 3 r x raquo i gt r - r 0 3 Z u 3 X 3 Z 33 -lt 33
- 1 - i Z Z m bull 33 lt-raquo -lt t n iuml gt
o ^ r - r - c c o z m z m z CD t o t o F I H
t n t n
- n as - T] S) -raquobull bull œ w raquo - t o ro bull-gt CD X
en - n
es
u bull-bull
bull-1 m c -z bull ^ ^ - z r w --lt - 0 - raquo w yO 1mdash1
3gt Z Xgt n - f n 0 n 1-1 [Tl r-i Z 03 Z
-lt -lt icirc-f
- z o 0 - f-1 bull-- z z
J
0 11
w z -0 c 33 3 w Z Z O H -
t gt bull t o -
CS
r
-lt z
7Z O O
H - t S) 3gt - 1
- 1 r o r - r - lt-raquo r o
h ) ^ f m n f i V n i gt t n
J X I - 3gt z c i r c m i gt 3 ~lt r m bulljgt raquo
-lt rgt z raquo 0 3 3 0 m m 0
3 0 0 m
t o H
^ 00 X
iuml gt - 4
-H
3 m
- n
t n
t o
bullbull0 x
z
3
3gt
C -lt Igt - t Ci
APPENDIX 8 EXAMPLE OF OUTPUT
HEAD AND TERMS OF UATERBALANCE PER NODE PER LAYER
AT TIME 260 IN II AND I13DAYS
T
LAYER NODE KODE HEAD FLOUUN FLOUTS FLOUSS LEAKAGE STORAGE LAT FLOU
13 19 25 31 37 43
125008 125000 12S000 12S000 12S000 12S000 12S000 12S000
0000 0000 0000 0000 0000 0000 0000 0000
0000 0000 0000 0000 0000 0000 0000 0000
0000 0000 0000 0000 0000 0000 0000 0000
- 000 - 000 -000 - 000 - 000 - 000 - 000 0000
0000 0000 0000 0000 0000 0000 0000 0000
0000 000 000 000 000 000 000 000
000 0000 0000 0P100 0000 0000 0000 0000
1 7
13 19 25 31 37
12S000 125000 125000 125000 125000 12S000 125000 125000
0000 0000 0000 0000 0000 0000 0000 0000
- 001 -007 - 023 - 020 - 009 - 003 0000
0000 0000 0000 0000 0000 0000 0000 0000
000
001
007
023
020
009
003
003
0000 0000 0000 0000 0000 0000 0000 0000
1 7
13 19 25 31 37 43
125000 125000 124998 124972 124895 124787 124647 124358
0000 0000 0000 0000 0000 0000 0000 0000
- 000 001 007 023 020 009 003
0000
0000 0000 0000 0000 0000 0000 0000
50000
0000 182
1982 3862 1674 419 078 041
0000 0000 0000 0000 0000 0000 0000 0000
TERMS OF WATERBALANCE PER LAYER DURING LAST TIMESTEP AND
AND SINCE BEGINNING IN M3DAY M3
FLOU TO UNSATURATED ZONE
FLOU TO TERTIARY SYSTEM
0000
0000
0000
0000
LAYER NO 1
FLOU TO SECONDARY SYSTEM
FLOU TO AD3ACENT LAYERCS)
ARTIFICIAL FLOU AND PRESCRIBED BOUNDARY FLOU
ACCUMULATION TERMS
LATERAL FLOU THROUGH PRESCR HEAD BOUNDARY
0000
-000
0000
000
000
0000
-000
0000
000
000
LAYER NO 2
FLOU TO SECONDARY SYSTEM
FLOU TO AD3ACENT LAYERCS1
ARTIFICIAL FLOU AND PRESCRIBED BOUNDARY FLOU
ACCUMULATION TERMS
LATERAL FLOU THROUGH PRESCR HEAD BOUNDARY
0000
-378
0000
381
0000
0000
-055
0000
056
0000
LAYER NO 3
FLOU TO SECONDARY SYSTEM
FLOU TO AD3ACENT LAYERCS5
ARTIFICIAL FLOU AND PRESCRIBED BOUNDARY FLOU
ACCUMULATION TERMS
LATERAL FLOU THROUGH PRESCR HEAD BOUNDARY
0000
370
50000
49376
0000
0000
055
12979
12909
0000
- flow (including zero flow) boundary
- both head and flow boundary
After specifying the auxiliary conditions an unique solution of
the variation of the hydraulic head in space and time can be
obtained
For special shapes of flow and assumptions exact analytical
solutions exist In nature however one is usually confronted with
problems like
- spacial variation in the hydraulic properties of the soil
- irregular shape of the groundwater basin
- empirical relations between q and h
- d not being a constant
For such situations by numerical and analogy models only an
approximation of the actual situation can be obtained In the next
chapter a numerical solution called the finite element method is
treated
III NUMERICAL SOLUTIONS
In this chapter the numerical solution of the aquifer equation
(eq 5) and the aquitard equation (eq 6) will be treated The
method of solving is the finite element method In section 31 only
a short mathematical description of this method will be given For
more mathematical background the reader is referred to publications
on this subject (eg ZIENKIEWICZ 1971 NEUMAN etal 1974 and
WESSELING 1976)
In section 32 the application of the finite element method on
the flow in groundwater basins is treated while in section 33 the
translation into a computer program is given
3 1 F i n i t e e l e m e n t m e t h o d
For reasons of simplicity the finite element method will be
explained for the stationary version of eq 5 ie
10
(k d^) + (k d ^ ) - Q = O (10) ocircx x 6x 6y y Sy
The basic idea of each numerical solution of eq (10) is to
apply this equation on a small but finite volume The flow region
of interest R is divided into a finite number of volumes called
elements The elements can have different shapes but for reasons
of conveniency we only use triangles- Quadrilaterals can also be
used because they can be divided into two triangles The sizes of
the triangles may vary They can be adapted according to the
geometry of the region and the accuracy desired (fig 11 gives an
example of the diviation of a region into triangles and quadrilaterals)
The corner points of the elements represent the nodal points
If there are N nodal points then the continuous variation of h
in region R is approximated by the expression
h (x_v) ~ h f (xy) (11) a n n
where
h (AV) = approximation of u (xy) a
h = value of h in nodal point n n
f ltxgty) = global coordinate function
In nodal point n a must be equal to h so f (x y ) = 1 and a n n n n
f (xy) = 0 i n Between f (x v ) and f (xy) of the nodal n j ] n n n n j j
points surrounding nodal point n f (xy) varies linearly from
1 to 0
In fig 3 a picture of f (xy) is given n
If one knows the values h in the nodal points n we know the approximation h of h We can find these values of b bv using for
a n
example the Galerkin method
Galerkins principle states that an approximate solution of h in
eq (10) can be obtained from the following system of equations
N ~~ (k d -A + -^- (k d T-y E h f (xy) f(xy)dxdy +
-ox x ox ocircy y oy - _ n n i R n~
J J Q f(xy) dx dy = 0 i = 1 N (12)
1 1
Fig 3 The global coordinate function f (xy)
The integration can be performed element by element with the
aid of the coordinate functions f (NEUMAN etal 1974)
For non-stationary problems it is not possible to apply the
Galerkin principle on the time derivative But it is possible
to discretize this derivative and to set up the system of equation
(eq 12) for each finite time step
One obtains in this way a series of sets of laquocriations tha s
an approximation of the non-stationary flow problem
Treatment of boundary conditions
a Initial conditions
In non-stationary flow problern gtne must start with given
values of a at time t = t ri c
b Boundary conditions
- Prescribed head boundary
Nodal points which lie on a prescribed head boundary have a
prescribed value for h bull r Hgte r umber of unknown h s is n n
reduced and accordingly no the number of equations
- Prescribed flux through z-- boundary
If one knows the flux (iuumlr jw or outflow per unit area) and
the area of the boundary oi-r element the inflow or outflow
through the boundary per flemlt=nt can be calculated
One may assume this flow to J bull part of the sink term Q in
eq (12)
12
32 A p p l i c a t i o n o f t h e f i n i t e e l e m e n t
m e t h o d o n t h e f l o w i n g r o u n d w a t e r
b a s i n s
From hydrogeological investigations the geometry and the matrix
of the hydraulic properties are known Knowing the geometry the
flow can be split up into a finite number of layers This holds
also for the nature of the boundaries
Now a nodal grid is superimposed on each layer in such a way
that the nodal points of the different layers are situated right
above each other (see fig 4) and in each layer the nodal point
lies in the middle of this layer
-nodal point
Fig 4 Schematic representation of the nodal points in the
xyz-space
Each layer has N nodal points of which P nodal points have a
prescribed head In each layer remain N-P = N nodal points with
unknown head Eq (12) can be applied to each point with unknown
head in each separate layer
As was stated in the last paragraph for each time step one
can write
13
Ah bdquo ~ -r1 + Q = O in = 12N in n m At yi A h + B (13)
where
A = in
ocircf ocircf ocircf of (k d f mdashri- ~ + k df - ~ -jS) dxdy
J x n ox ox y n oy oy
in (S d + S ) ff dx dy O if i icirc n
s y i n
Q f dx dy
The matrices A B and the vector Q have been defined in in l
elsewhere (NEUMAN etal 1974) A is the conductivity matrix in
B the storage matrix and Q the source vector in l
In order to clarify these rather complicate expressions apply
equation (13) poundo point 4 of fig 5
6 7
Fig 5 Top view of part of the nodal grid
lhl + A42h2 + A43h3 + A44h4 + A45h5 + A46h6 + A47h7 +
h - h
+ B _iuumli i ^ + Q = o 44 t - t x4
n n-1
(14)
where h and h are values of h at time t and t 4n 4n-1 4 n n-1
respectively
14
If the h-values belonging to the A-values are taken at time
t = t we get an equation with one unknown h for time t = t n-1 4n n
The equation can then be solved directly
However this is an explicit formulation which puts strong limitations
on the magnitude of the time step t - t (REMSON etal 1971) n n-1
Values larger than a certain value cause instability With an implicit method the values of h are taken at time
t = t This method is unconditionally stable Now one gets an equation n
with 7 unknowns h through h If one has 7 equations for every 1 n n
point 1 through 7 we can solve the values h i = 17
in
Of all the methods possible the iterative method of Gauss-Seidel
has been chosen The Gauss-Seidel method starts by choosing a first estimate of h at time t = t (usually equal to h ) At each in n j -i in-l iteration step a new value of h is calculated using the last
in calculated values of h If one passes the nodal points from 1 to
i n 7 eq (14) becomes
h-j + 1 = A h^ + 1 + A bdquo h | + 1 + A _ h | + 1 + A h j + A r h j + 4 n 4 1 1 n 4 2 2 n 43 3 n 45 5 n 4 6 6 n
B h B + A h J + - 4 4 4 n ~ + 0 J (A + mdash) A 4 7 h 7 n At n U 4 4 At
(15)
where
h-i = in
j = iteration step or number of iteration
i = number of nodal point n = time = t
n
The iteration process converges if the successive differences k+1 k
between h and h become continuously smaller in in
The iteration process is stopped if at each nodal point the k+1 k
difference between h and h is smaller than a certain prescribed in in
value
15
To accelerate the convergence an overrelaxation factor W is used
(for more information about W see FORSYTHE and WASOW 1960)
k+1 k T7 k+1 k h = h + W (h - h )
in in in in (16)
If W = 1 there is no overrelaxation Usually W varies between 1 and
2 and the optimum value very often is found by trial and error
Eq (15) is valid for points in aquifers The connection with
adjacent layers is made by the term Q because in this term the
flow from adjacent aquitards q is included (two aquifers are always
separated by an aquitard)
Aquitards
We dont apply the finite element method on points in aquitards
As stated before there is no horizontal flow in aquitards So it has
no sense to replace the continuous variation of h in the aquitard
by equation (11)
The nodal points in aquitards represent a certain volume of an
aquitard hich transmit water from one aquifer to another and which
has storage capacity The nodal points are included in the Gauss-
Seidel scheme -s follows (see fig 6 )
1 aauifer
a -_vS laquo 2 [~^-t^yM^^^-^r~ aqutarci l-Z
aquifer
Fig 6 Schematic situation for points in aquitards
Point 2 is situated in the middle of the aquitard with thickness
d storage coefficient S and vertical hvdraulic conductivity k c J v
Suppose the Gauss-Seidel iteration starts in the upper layer and
ends in the bottom-layer Then the flux from point 1 in aquifer 1 to
16
point 2 in the aquitard at time t = t and in iteration step j + 1
is
k q = (hf -hf )-iumlmdash f (17)
1 n 2 n OSxd1
and the flux from point 3 in aquifer 2 to point 2 in the aquitard
is
1 c 1
qbdquo = (h - hf) mdashHmdashr- (18) 2 3 n 2 n OSxd
The net flux ltl-q9 must be equal to the change in storage per unit
area per unit time This change in storage can numerically be
approximated by
te - h-
n n-1
Combina t ion of ( 1 7 ) (18) ana (19) g i v e s
l tr1 - h (hJ + 1 + hj - 2 hj+1) - - - - - - S1 - ilaquoS L u l l = o (20)
] i 3 n 2 n 1 c t - t j j x d n n -1
T + 1
From t h i s e q u a t i o n h^ kan be s o l v e d
3 3 C o m p u t e r o g r a m
The problem outl^ugrave so far has been translated into a computer
program written in FORTRAN IV
The flow chart ot Uis program is given in fig 7 whereas in
appendix A the complete listing of the program can be found
In the flow chart the 5 main parts of the program are
indicated
A Reading and printing of input data
B Determination of the matrices A and B and the vector Q per layer
17
M
1 A
FLOH to unsaturated zono
FLOW to tertiary syster
FLOW to secondnrv system
Leakage to adjacent aquitards
SOLVE equation (14)
yes
CALCULATE flux through constant head boundnr
CALCULATE waterbalance n p r 1 OAro r r J j - ~
PRINT of output
Ier node per layer
k = k+1
t = t + At
7 DUMP variables on tape
-gt-
( STOP )
Fig 7 Flow chart of the program FEMSAT with diviation in 5 main
parts A B C D and E
C START PROGRAM FEMSAT
f READ general information
PRINT general information
f READ element information
PRINT element information 7 READ stationary nodal point information
PRINT stationary nodal point information 7
READ non-stationary nodal point information for t = t
FRINT non-stationary nodal point information for t = t
L 2 (
^ yes R E A D non-stat input for t gt t
J PRINT non-stat input for t gt t (- 1 -- M degY
GENERATE A B and AREA-matrix for confined aquifers
_L
GENERATE A B and AREA-matrix for unconfined aouifer
bull
C Solving h with Gauss-Seidel
D Determination of the water balance terms per layer
E Printing of results
The program exists of a main program with several subroutines
The main program calls the subroutines In the subroutines most
calculations are performed
In this way an easy adaptation to different problems is
obtained (see also section 44) Comments in the program will
help the reader to understand the program
In the next chapter the preparation of the input data executing
of the program and the output will be treated
Acknowledgement
Some parts of the computer program were taken from the program
UNSAT2 of NEUMAN etal (1974)
IV INPUT EXECUTION AND OUTPUT
41 I n p u t
The input is divided into endogenous variables (chosen by the
user or calculated by the program during execution) and exogenous
variables (based on physical data)
411 Endogenous variables
Translation of groundwater basin into a nodal point network
Probably the most important but also the most difficult stage
in the process of simulating groundwater flow is the translation
of the natural groundwater basin into a nodal point network Questions
which arise are
a In how many layer the geo-hydrological system mustcan be
schematized
b The locations of the boundaries and the nature of these boundaries
(impervious prescribed head prescribed flux)
c The configuration of the nodal points
19
The answers of these questions depend on the problem under
consideration
- Schematization into layers
Usually our knowledge about the geo-hydrological situation is
such that no more than 5 different layers can be distinguished
(remember contrast rule)
- Boundary conditions
The top boundary condition of a groundwater basin usually is
a prescribed flux (effective precipitation)
The bottom boundary condition usually is a prescribed flux
boundary with flux zero (impervious hydrological base)
For the boundaries at the other sides of the system one sees
either prescribed head boundaries (eg a fully penetrating
river) or prescribed flux boundaries (including zero flux)
- Nodal point configuration
Nodal points are situated
on the boundaries
along internal boundaries which indicate changes in
hydraulic properties within the layer
along the tertiary surface water system
on places where artificial recharge or discharge wiii tgtke
place
elsewhere
Construction of the finite element mesh
Assume we have schemetized the groundwater basin iatu a finite
number of layers with different direction of flow (horizontal and
vertical) Within one layer the hydrological properties may vary
from place to place
Now a nodal grid is superimposed on each layer in a way already
described in section 32 Thus each layer has the same number of
nodal points situated on the same places in the horizontal plane
but with different z-coordinates
We are free to construct quadrilaterals or triangles because
the program automatically divides a quadrilateral into two triangles
having identical material properties The dimensions of the elements
20
should be small at places where large changes in hydrological
properties occur or where large gradients in hydraulic head are
expected to occur
Coding of nodal points
Each node in the finite element network is assigned an integer
code which indicates the type of calculation for this node
The coding is as follows
- Code = 1 nodal points with described head as function of time
- Code 2 internal nodal points
- Code = 3 internal nodal points which represent secondary surface
water system
- Code = 5 nodal points with prescribed artificial recharge or
discharge and points with prescribed boundary flow
(both as functions of time)
Other important endogenous variables are the magnitude of the
time step the value of the variable which stops the iteration
process and the maximum number of iterations allowed per time step
(see also group A and B of section 413)
412 Exogenous variables
After constructing the nodal grid each node represents part of
the groundwater basin Geo-hydrological properties can be assigned
to each mode based on physical data (group C D and E of section
413) In group F of section 413 time-dependent variables can
be assigned to each node
413 Data input to Program FEMSAT
The data input must be punched on cards according to the
following instructions
GROUP COLUMNS FORMAT SYMBOL DESCRIPTION
A 1-80 20A4 HEAD Desired heading to be printed at
the beginning of the execution
1-5 15 NUMNP number of nodal points
21
GROUP COLUMNS FORMAT SYMBOL DESCRIPTION
6-10 15
11-15 15
16-20 15
21-25 15
26-30 15
31-35 15
NUMEL
NUMLAY
MBAND
NTSPI
NTSPO
MAXIT
1-10 F103 ST
11-20 F103 DELT
21-30 F103 TMAX
31-40 F105 TOL
number of elements
number of layers
maximum difference in node number
between two adjacent nodes
number of time steps between a
change in non-stationary input data
number of time steps between two
successive outputs
maximum number of iterations per
time step
time to start with the execution
initial time interval
maximum time to be reached before
execution is stopped
maximum allowed absolute change in
the values of h between any two
successive iterations in a given
time step
Iterations continue until all changes
are less than or equal to TOL or
until MAXIT is exceeded
1- 5 15
6-10 15
11-15 15
16-20 15
KX(N1)
KX(N2)
KX(N3)
KX(N4)
sequential number of i-th corner of
element N
sequential number of j-th corner
sequential number of k-th corner
sequential number of 1-th corner
(see fig 8)
Fig 8 Numbering of element corner nodes
22
GROUP COLUMNS FORMAcircT SYMBOL DESCRIPTION
One card must be provided for each element in sequential order
starting with N = 1 and ending with N = NVJMEL
I-K 1103 SANG(NLY)
1i-iO MO3 C1(NLY)
21-30 F 103 C2(NLY)
21-4- 1-104 THI(NLY)
angle in degrees between CI and
the laquo-coordinate to he used in
nodal print N in layer LY (see
poundig 9)
First principal conductivity of
material in node N and layer LY
idem second principal conductivity
thickness of layer LY at nodal
point N
ig V Definition of SANGCI and C2
One card icircuiPt be provided for each model point ctarting wih M = 1
and ending with N = NUMNP If there is norlt3 than 1 layer for each
layer a set must be provided starting with the upper layer LY - 1
and ending with the bottom layer LY = NIJMLAY The variables SANG
M and 2 are not relevant tor aquitards but for reasons of
conveniency this procedure is maintained (making SANG CI and C2
dummy variables)
1-10 110 KODE(N)
11-20 II0 NTERR(N)
coding of nodal points (see
paragraph 42)
number of tertiary relation in node N
The q-h relationship of the flow to
23
GROUP COLUMNS FORMAT SYMBOL DESCRIPTION
21-30 FlOl X(N)
31-40 FlOl Y(N)
41-50 FlOl HGL(N)
51-60 FlOl HBSC(N)
61-70 FlOl LESC(N)
the tertiary system as discussed
in par 24 are schematized in a small
number of different relations
x-coardinate of node N
y-coordinate of node N
height ground level in node N
height bottom secondary conduit in
node N If node N is not a secondary
conduit this variable is a dummy
variable
length of secondary conduit in area
represented by node N
One card must be provided for each node starting with N = 1 and
ending with N = NUMNP
E 1-10 FlOl P(NLY1) initial value of h at node N in
layer LY
11-20 FlOl VERRES(NLY) vertical resistance at node N and
layer LY If the layer is an aquifer
this value is a dummy variable
21-30 F101 RADRES(NLY) radial resistance for flow to
secondary system at node N in layer
LY
31-40 FlOl ENRES(NLY) entrance resistance for flow to
secondary system at node N in layer
LY
41-50 FlOl WP(NLY) wet perimeter of secondary system at
node N in layer LY
51-60 FlOl P0R(NLY) specific yield of the material at
node N in layer LY
61-70 FlOl SS(NLY) specific storage of the material
at node N in layer LY
24
GROUP COLUMNS FORMAT SYMBOL DESCRIPTION
One card must be provided for each node starting with N = 1 and
ending with N = NUMNP If there is more than 1 layer for each layer
a set of cards must be provided starting with LY = 1 and ending
with LY = NUMLAY
F 1-10 F101 P(NLY1) value of h at different times at
node N in layer LY Node N lies
on a prescribed head boundary
11-20 F101 WTSC(N) water table in secondary system at
node N at different times
21-30 F 101 ARRED(NLY) artificial recharge to or discharge
from node N in layer LY or
prescribed boundary flow at
different times
31-40 F 101 QPHR(N) flux through phreatic surface at
node N at different times
The data in group F are non-stationary input data One card must be
provided for each node starting with N = 1 and ending with N = NUMNP
In case of more than 1 layer for each layer a set must be provided
Not relevant variables are taken as dummy variables
After a certain number of time steps (NTSPI) the data of group E
are read So the set cards E must contain NxLYxC cards where C =
(TMAXNTSPIXDELT) + 1 if DELT is a constant
42 R u n n i n g t h e p r o g r a m o n t h e C y b e r 7200
The program has been executed on a Cyber 7200 computer of
IWIS-TNO in the Hague
- The demanded input is punched on cards according to the instructions
of the last paragraph and is put on a permanent file
- For each specific problem inthe program only the dimension statements
in the main program have to be adjusted according to the instructions
in the lines of comment
- After compilation the program together with the input file is
executed
25
- Intermediate data necessary for continuation of the calculations
are saved on another permanent file This gives the opportunity
to restart the program at a time the calculation was stopped
The only change in the input file is to adjust the starting time
Remark
With the aid of a terminal small changes in the program and in the
input file can be made easily and the output immediately appears
on the terminal In this way it is possible to stop erroneous
calculations This is especially useful when being in the testing
phase of the program
43 O u t p u t
Immediately after reading the input data they are printed so
that possible errors can be detected During execution after each
certain number of time steps (NTSPO) calculated values of h and
water balance terms are printed In appendix B an example is given
The output data can be devided into two groups
A Data per node per layer
- Coding of the nodal points KODE
- Hydraulic head HEAD
- Flow through phreatic surface at the time indicated in head
of table FLOWUN
- Flow to tertiary surface water system FLOWTS
- Flow to secondary surface water system FLOWSS
- Flow to adjacent layers LEAKAGE
- Artificial recharge or discharge and prescribed boundary
flow ARRED
- Change in storage during last time step STORAGE
- Flow through prescribed head boundary LATFLOW
B Data per layer
Summation of the terms already mentioned under A gives the
magnitude of the water balance terms per layer at the time
indicated (instanteneous water balance terms) Summation of
these instanteneous terms multiplied by their respective time
intervals since time is zero gives the cumulative water balance
terms per layer
26
After each print out we are able to check the calculations
because the sum of the water balance terms per layer must be
approximately zero
44 C o m p u t i n g t i m e
The computing time depends on the number of nodal points and
the number of layers But it depends also on the magnitude of the
changes in hydraulic head and on the value which terminates the
Gauss-Seidel iteration (TOL) To give an impression TOL = 001 m
Costs per day simulation (in hlfl) on the Cyber 7200
number of nodes number of layer bull bdquo bull bull _ 1mdash x a x b (21)
where
a = costs in Dutch guilders per system second (ƒ 027)
b = empirical factor with magnitude about 001
V NUMERICAL EXPERIMENTS
In general there are two possibilities to test the validity of
numerical models The first and most commonly used method is to
compare numerical results with analytical solutions The second
possibility is to compare numerical results with measured field
data Since the last alternative requires many data that are
difficult to obtain we restrict ourselves for the time being to
comparison with the analytical solutions
5 1 F l o w t o a w e l l i n a n u n c o n f i n e d
c i r c u l a r - s h a p e d a q u i f e r
Flow to a well can be divided into steady-state flow and
unsteady-state flow (KRUSEMAN and DE RIDDER 1976)
27
511 Steady state flow in an unconfined aquifer
For this case the formula of Thiem-Dupuit is valid (see fig 10)
Q bull k lth2-hicircgt ln(r2r])
(22)
where
hhbdquo = hydraulic head in 1 and 2 respectively
Q = discharge from well
rbdquo = distance from well
CL31
hmdash^ri --r2
Fig 10 Schematic cross-section of a pumped unconfined aquifer
Fig 11 gives the nodal grid superimposed on a well in a
circular shaped aquifer Near the well the nodal point distance
is made smaller Quadrilaterals as well as triangles are used
Numerical calculations with this finite element network and 3 -1
an extraction rate of 200 m day were performed for 10 days
At that moment a steady-state situation was almost achieved
28
2tmdashnodal point number
160 180 X-axesim)
Fig 11 Nodal point configuration for flow to a well in a circular-
shaped aquifer The outer boundary is taken to be constant
The well is situated in nodal point 42
29
In fig 12 the results of the simulation are compared with the
analytical solution The agreement between numerical results and
the analytical solution is excellent
draw- 15irS (m) down
analytical solution imdashx calculated values
distance from well r (m)
Fig 12 Comparison between analyt ical and calculated s teady-state
drawdown for the problem of f ig 11 with 1
-1 Q bull - 200 m 3 d a y _ 1
k = 10 mday
d = 100 m
S = 2 y
S = 0 s
512 Unsteady-state flow in an unconfined aquifer
Analytical solutions for unsteady-state flow in an unconfined
aquifer only exist if the aquifer extends infinitely For the
aquifer given in fig 11 this condition is not fulfilled However
during the first one or two days after starting the extraction the
influence of a constant hydraulic head at the outer boundary on the
30
drawdown near the well can be neglected (see fig 13) After one
day the lateral inflow is only 25 of the extraction the remainder
being the change in storage due to drawdown
2 0 0
m 3 day 1
Q=200m 3 day
100-
vraquomdash Lateral inflow
N amp storage
I I ^ V - W 10
t (days)
Fig 13 Values of Q lateral inflow and change in storage as
functions of time
The nonsteady-state or Theis equation can be written as
i _ Q s 4iTkD
dy = 4irkD
W(u) (23)
where
u = r2S
4irkDt
r = distance from well
S = storage coefficient
Q = constant discharge from well
D = thickness of the waterbearing layer
at the beginning of pumping
t = time after starting the extraction
s = corrected drawdown
= s - 322D
s = calculated drawdown
W(u) = Theiss well function
[J
CIAT1
[T]
]
31
In fig 14 the analytical unsteady-state solution for the case
described in section 511 is given together with the numerical
results at times 05 and 10 days The agreement between numerical
results and the analytical solution is rather good
Drawdown s1(m)
10
10
10
Theiss solution
x caculated values t=Q5
bdquo t= lO
10 10- 10 10 t r2(daym2)
Fig 14 Comparison between analytical and calculated steady-state
drawdown for the problem of fig 11 with
Q
k
d
S y
s s
=
=
=
=
=
200
1
10
3 m
0
0
2
0
-1 day
m day
m
These results indicate that the assumption of a linear variation
of h between two nodal points (see fig 3 ) even with the rather
rough element network of fig 11 is acceptable So there is no
need to improve for example the finite element network by introducing
higher order coordinate functions
32
52 F l o w t o w e l l i n 3-1 a y e r e d s y s t e m
In fig 15 a schematic cross-section of a 3-layered system of
two aquifers separated by an aquitard is given A well of
infinitisemal radius is completed in the lower aquifer and discharges
at a constant rate QTT Each layer is homogeneous isotropic and
extending radially infinite
bdquo bull - bull bull ~ i
mdash Q II
aquifer I
lugrave^i~IcircIuml7i-i~~ aquitard --r-rl~-
-II--I i - i
aquifer
Fig 15 Schematic cross-section of a 3-layered system
For this situation NEUMAN and WITHERSPOON (1969a) give rather
complicated analytical solutions In the special case where the
hydraulic properties of the two aquifers are identical they give
graphical illustrations of their solutions for three different
cases
These graphical solutions give the opportunity to verify
roughly numerical results of a 3-layered system Therefore a nodal
grid is superimposed on a circular shaped groundwater flow system
with the extraction in the centre of the lower aquifer and with
a constant head on the outer boundary Fig 16 gives the nodal
point situation in the x - y plane of 1 layer The nodes in the
other two layers are situated in the middle of the layers at the
same places in the x - y plane only their z-coordinate is different
33
A
-Jy
bull33
V 5
iumlff
Fig 16 Nodal point configuration for the 3-layered problem (top view)
34
In fig 17 numerical results are compared with the analytical
solution The agreement is rather bad especially for the two
aquifers But in terms of the water balance the difference between
aquifers and aquitard is not so big because the aquitard has
about 16 times bigger storage capacity than the aquifers The
small value of the specific storage of the aquifers is the reason
that small errors in the components of the water balance have a
big influence on the calculated values of the hydraulic head
Eig 17 Dimensionless drawdown vs dimensionless time in a
3-layered system with
II = 50 m day
3 m (distance from well)
aquifer I
aquitard
aquifer II
Thi (m)
100
25
100
S s
001
0642
001
k(mday )
1
0277
1
The discrepancies are partly caused by errors in discretization
In the z-direction these errors can be eliminated by taking the
35
vertical resistance of the aquitard to be infinite Then the
calculated drawdown in the pumped aquifer can be compared with
the analytical Theiss solution
This is done in fig 18 for different distances from the well
One sees that the agreement is better when the distance is bigger
Apparently the linearization of the logarithmic drawdown curve
near the well is responsible for the differences The differences
for r = 3 m are of the same magnitude as for the pumped aquifer in
the 3-layered case
joo gt A 50
20
Theiss solution
bull calculated values r=3m (distance f rom well) x bdquo bdquo r=7m
1 L
I l_ 001 003 0 0 5 Ol Q2
- I I 5 10 t - K t
Fig 18 Dimensionless drawdown vs dimensionless time in a
confined aquifer with
Q =500 m day
k = 10 mday
D = 100 m
S = 001 s
The differences between numerical results and analytical solutions
are not caused by taking a finite area because changing the outer
boundary into an impermeable boundary has only minor effects
36
VI SUMMARY AND CONCLUSIONS
In this report a numerical model for non-stationary saturated
groundwater flow in a multi-layered system has been treated
Groundwater flow can be described in different ways
a Physical-mathematical by combining Darcys law and the law of
continuity (section 21)
b Geo-hydrological the flow in a natural groundwater basin takes
place in layers with different hydraulic properties Definitions
of the most important hydraulic properties are given in section
22
c Schematical groundwater flows horizontally in aquifers and
vertically in aquitards (section 23)
d Approximating by dividing the region of flow into a finite
number of elements and applying Darcys law and the law of
continuity on each of these elements In these ways one gets
a set of equations which can be solved given (initial and)
boundary conditions (section 25)
e Numerical the set of equations has been solved using a Galerkin
type finite element method and the implicit Gauss-Seidel method
with overrelaxation factor (section 31 and 32)
f Computer-solvable the numerical solution has been translated
in a FORTRAN IV program called FEMSAT (section 33)
In chapter IV the input execution and output of the program
FEMSAT are treated
Finally in chapter V some numerical applications are given
simulation of flow to a well in unconfined aquifer and simulation
of flow to a well in a 3-layered configuration
Comparison of hydraulic heads obtained by numerical calculations
and analytical true solutions showed an excellent agreement in case
of stationary flow and a good to moderate agreement in case of non-
stationary flow However the diviations caused by the discretization
process must be seen in relation to the errors made in the
establishment of the hydraulic properties
37
As was stated in the introduction numerical models can be
valuable tools in the management of water resources Therefore
numerical models must be suitable to simulate effects of certain
operations upon the spreading of water over saturated groundwater
unsaturated groundwater and surface water The model described in
this report is a model for saturated groundwater flow The relation
with the surface water can be taken into account via radial and
entrance resistances or via a relation between flow to the surface
water system and the hydraulic head The relation with the unsaturated
zone has been restricted to the input factor effective precipitation
In practise however the unsaturated and saturated system are
interconnected Therefore the model should be improved by taking
into account the unsaturated zone This can be done in at least
two ways
a Replacing the unsaturated zone by a parametrical model
b Coupling the unsaturated zone in an iterative way with the
saturated system
In order to find out the usefulness of the numerical model for
water resources management it will be applied to actual regional
groundwater systemslaquoAlso sensitivity analyses will be carried out
The computer model itself can be improved by applying standard
programs eg automatical generation of a nodal grid and graphical
plotting of the results
38
VII REFERENCES
ERNST LF 1962 Grondwaterstromingen in de verzadigde zone en hun
berekening bij aanwezigheid van horizontale evenwijdige open
leidingen Thesis University Utrecht pp 189
1978 Drainage of undulating soils with high groundwater
tables (in press)
FORSYTHE GE and WR WASON 1960 Finite Difference Methods for
Partial Differential Equations John Wiley and Sons New York
pp 444
KRUSEMAN GP and NA DE RIDDER 1976 Analysis and evaluation of
pumping test data ILRI-bulletin No 11 pp 200
LAAT PJM DE and C VAN DEN AKKER 1976 Simulatiemodel voor grond-
waterstroming en verdamping in Modelonderzoek 1971-1974 ten
behoeve van de waterhuishouding in Gelderland deel II
Grondslagen pp 145-197
NEUMAN SP RA FEDDES and E BRESLER 1974 Finite element
simulation of flow in saturated -unsaturated soils considering
water uptake by plants Hydraulic Engineering Laboratory
Technion Israel pp 104
and PA WITHERSPOON 1969a Theory of Flow in a confined
two Aquifer System Water Resources Research Vol 5 No 4
pp 803-816
^_ and PA WITHERSPOON 1969b Applicability of Current Theories
of Flow in leaky aquifers Water Resources Research Vol 5
No 4 pp 817-829
PINDER GF and EO FRIND 1972 Application of Galerkins Procedure
to Aquifer Analysis Water Resources Research Vol 8 no 1
pp 108-120
REMSON I GE HORNBERGER and FJ MOLZ 1971 Numerical Methods
in Subsurface Hydrology John Wiley and Sons New York
pp 389
WESSELING JG 1977 Toepassing van Finite Difference- en Finite
Element Methoden voor het oplossen van een aantal grondwatershy
stromingsproblemen (met computerprogrammas) ICW-nota 944
pp 63
39
ZIENKIEWICZ OC 1971 Finite Element Method in Engineering Science
McGraw-Hill London pp 521
40
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APPENDIX 8 EXAMPLE OF OUTPUT
HEAD AND TERMS OF UATERBALANCE PER NODE PER LAYER
AT TIME 260 IN II AND I13DAYS
T
LAYER NODE KODE HEAD FLOUUN FLOUTS FLOUSS LEAKAGE STORAGE LAT FLOU
13 19 25 31 37 43
125008 125000 12S000 12S000 12S000 12S000 12S000 12S000
0000 0000 0000 0000 0000 0000 0000 0000
0000 0000 0000 0000 0000 0000 0000 0000
0000 0000 0000 0000 0000 0000 0000 0000
- 000 - 000 -000 - 000 - 000 - 000 - 000 0000
0000 0000 0000 0000 0000 0000 0000 0000
0000 000 000 000 000 000 000 000
000 0000 0000 0P100 0000 0000 0000 0000
1 7
13 19 25 31 37
12S000 125000 125000 125000 125000 12S000 125000 125000
0000 0000 0000 0000 0000 0000 0000 0000
- 001 -007 - 023 - 020 - 009 - 003 0000
0000 0000 0000 0000 0000 0000 0000 0000
000
001
007
023
020
009
003
003
0000 0000 0000 0000 0000 0000 0000 0000
1 7
13 19 25 31 37 43
125000 125000 124998 124972 124895 124787 124647 124358
0000 0000 0000 0000 0000 0000 0000 0000
- 000 001 007 023 020 009 003
0000
0000 0000 0000 0000 0000 0000 0000
50000
0000 182
1982 3862 1674 419 078 041
0000 0000 0000 0000 0000 0000 0000 0000
TERMS OF WATERBALANCE PER LAYER DURING LAST TIMESTEP AND
AND SINCE BEGINNING IN M3DAY M3
FLOU TO UNSATURATED ZONE
FLOU TO TERTIARY SYSTEM
0000
0000
0000
0000
LAYER NO 1
FLOU TO SECONDARY SYSTEM
FLOU TO AD3ACENT LAYERCS)
ARTIFICIAL FLOU AND PRESCRIBED BOUNDARY FLOU
ACCUMULATION TERMS
LATERAL FLOU THROUGH PRESCR HEAD BOUNDARY
0000
-000
0000
000
000
0000
-000
0000
000
000
LAYER NO 2
FLOU TO SECONDARY SYSTEM
FLOU TO AD3ACENT LAYERCS1
ARTIFICIAL FLOU AND PRESCRIBED BOUNDARY FLOU
ACCUMULATION TERMS
LATERAL FLOU THROUGH PRESCR HEAD BOUNDARY
0000
-378
0000
381
0000
0000
-055
0000
056
0000
LAYER NO 3
FLOU TO SECONDARY SYSTEM
FLOU TO AD3ACENT LAYERCS5
ARTIFICIAL FLOU AND PRESCRIBED BOUNDARY FLOU
ACCUMULATION TERMS
LATERAL FLOU THROUGH PRESCR HEAD BOUNDARY
0000
370
50000
49376
0000
0000
055
12979
12909
0000
(k d^) + (k d ^ ) - Q = O (10) ocircx x 6x 6y y Sy
The basic idea of each numerical solution of eq (10) is to
apply this equation on a small but finite volume The flow region
of interest R is divided into a finite number of volumes called
elements The elements can have different shapes but for reasons
of conveniency we only use triangles- Quadrilaterals can also be
used because they can be divided into two triangles The sizes of
the triangles may vary They can be adapted according to the
geometry of the region and the accuracy desired (fig 11 gives an
example of the diviation of a region into triangles and quadrilaterals)
The corner points of the elements represent the nodal points
If there are N nodal points then the continuous variation of h
in region R is approximated by the expression
h (x_v) ~ h f (xy) (11) a n n
where
h (AV) = approximation of u (xy) a
h = value of h in nodal point n n
f ltxgty) = global coordinate function
In nodal point n a must be equal to h so f (x y ) = 1 and a n n n n
f (xy) = 0 i n Between f (x v ) and f (xy) of the nodal n j ] n n n n j j
points surrounding nodal point n f (xy) varies linearly from
1 to 0
In fig 3 a picture of f (xy) is given n
If one knows the values h in the nodal points n we know the approximation h of h We can find these values of b bv using for
a n
example the Galerkin method
Galerkins principle states that an approximate solution of h in
eq (10) can be obtained from the following system of equations
N ~~ (k d -A + -^- (k d T-y E h f (xy) f(xy)dxdy +
-ox x ox ocircy y oy - _ n n i R n~
J J Q f(xy) dx dy = 0 i = 1 N (12)
1 1
Fig 3 The global coordinate function f (xy)
The integration can be performed element by element with the
aid of the coordinate functions f (NEUMAN etal 1974)
For non-stationary problems it is not possible to apply the
Galerkin principle on the time derivative But it is possible
to discretize this derivative and to set up the system of equation
(eq 12) for each finite time step
One obtains in this way a series of sets of laquocriations tha s
an approximation of the non-stationary flow problem
Treatment of boundary conditions
a Initial conditions
In non-stationary flow problern gtne must start with given
values of a at time t = t ri c
b Boundary conditions
- Prescribed head boundary
Nodal points which lie on a prescribed head boundary have a
prescribed value for h bull r Hgte r umber of unknown h s is n n
reduced and accordingly no the number of equations
- Prescribed flux through z-- boundary
If one knows the flux (iuumlr jw or outflow per unit area) and
the area of the boundary oi-r element the inflow or outflow
through the boundary per flemlt=nt can be calculated
One may assume this flow to J bull part of the sink term Q in
eq (12)
12
32 A p p l i c a t i o n o f t h e f i n i t e e l e m e n t
m e t h o d o n t h e f l o w i n g r o u n d w a t e r
b a s i n s
From hydrogeological investigations the geometry and the matrix
of the hydraulic properties are known Knowing the geometry the
flow can be split up into a finite number of layers This holds
also for the nature of the boundaries
Now a nodal grid is superimposed on each layer in such a way
that the nodal points of the different layers are situated right
above each other (see fig 4) and in each layer the nodal point
lies in the middle of this layer
-nodal point
Fig 4 Schematic representation of the nodal points in the
xyz-space
Each layer has N nodal points of which P nodal points have a
prescribed head In each layer remain N-P = N nodal points with
unknown head Eq (12) can be applied to each point with unknown
head in each separate layer
As was stated in the last paragraph for each time step one
can write
13
Ah bdquo ~ -r1 + Q = O in = 12N in n m At yi A h + B (13)
where
A = in
ocircf ocircf ocircf of (k d f mdashri- ~ + k df - ~ -jS) dxdy
J x n ox ox y n oy oy
in (S d + S ) ff dx dy O if i icirc n
s y i n
Q f dx dy
The matrices A B and the vector Q have been defined in in l
elsewhere (NEUMAN etal 1974) A is the conductivity matrix in
B the storage matrix and Q the source vector in l
In order to clarify these rather complicate expressions apply
equation (13) poundo point 4 of fig 5
6 7
Fig 5 Top view of part of the nodal grid
lhl + A42h2 + A43h3 + A44h4 + A45h5 + A46h6 + A47h7 +
h - h
+ B _iuumli i ^ + Q = o 44 t - t x4
n n-1
(14)
where h and h are values of h at time t and t 4n 4n-1 4 n n-1
respectively
14
If the h-values belonging to the A-values are taken at time
t = t we get an equation with one unknown h for time t = t n-1 4n n
The equation can then be solved directly
However this is an explicit formulation which puts strong limitations
on the magnitude of the time step t - t (REMSON etal 1971) n n-1
Values larger than a certain value cause instability With an implicit method the values of h are taken at time
t = t This method is unconditionally stable Now one gets an equation n
with 7 unknowns h through h If one has 7 equations for every 1 n n
point 1 through 7 we can solve the values h i = 17
in
Of all the methods possible the iterative method of Gauss-Seidel
has been chosen The Gauss-Seidel method starts by choosing a first estimate of h at time t = t (usually equal to h ) At each in n j -i in-l iteration step a new value of h is calculated using the last
in calculated values of h If one passes the nodal points from 1 to
i n 7 eq (14) becomes
h-j + 1 = A h^ + 1 + A bdquo h | + 1 + A _ h | + 1 + A h j + A r h j + 4 n 4 1 1 n 4 2 2 n 43 3 n 45 5 n 4 6 6 n
B h B + A h J + - 4 4 4 n ~ + 0 J (A + mdash) A 4 7 h 7 n At n U 4 4 At
(15)
where
h-i = in
j = iteration step or number of iteration
i = number of nodal point n = time = t
n
The iteration process converges if the successive differences k+1 k
between h and h become continuously smaller in in
The iteration process is stopped if at each nodal point the k+1 k
difference between h and h is smaller than a certain prescribed in in
value
15
To accelerate the convergence an overrelaxation factor W is used
(for more information about W see FORSYTHE and WASOW 1960)
k+1 k T7 k+1 k h = h + W (h - h )
in in in in (16)
If W = 1 there is no overrelaxation Usually W varies between 1 and
2 and the optimum value very often is found by trial and error
Eq (15) is valid for points in aquifers The connection with
adjacent layers is made by the term Q because in this term the
flow from adjacent aquitards q is included (two aquifers are always
separated by an aquitard)
Aquitards
We dont apply the finite element method on points in aquitards
As stated before there is no horizontal flow in aquitards So it has
no sense to replace the continuous variation of h in the aquitard
by equation (11)
The nodal points in aquitards represent a certain volume of an
aquitard hich transmit water from one aquifer to another and which
has storage capacity The nodal points are included in the Gauss-
Seidel scheme -s follows (see fig 6 )
1 aauifer
a -_vS laquo 2 [~^-t^yM^^^-^r~ aqutarci l-Z
aquifer
Fig 6 Schematic situation for points in aquitards
Point 2 is situated in the middle of the aquitard with thickness
d storage coefficient S and vertical hvdraulic conductivity k c J v
Suppose the Gauss-Seidel iteration starts in the upper layer and
ends in the bottom-layer Then the flux from point 1 in aquifer 1 to
16
point 2 in the aquitard at time t = t and in iteration step j + 1
is
k q = (hf -hf )-iumlmdash f (17)
1 n 2 n OSxd1
and the flux from point 3 in aquifer 2 to point 2 in the aquitard
is
1 c 1
qbdquo = (h - hf) mdashHmdashr- (18) 2 3 n 2 n OSxd
The net flux ltl-q9 must be equal to the change in storage per unit
area per unit time This change in storage can numerically be
approximated by
te - h-
n n-1
Combina t ion of ( 1 7 ) (18) ana (19) g i v e s
l tr1 - h (hJ + 1 + hj - 2 hj+1) - - - - - - S1 - ilaquoS L u l l = o (20)
] i 3 n 2 n 1 c t - t j j x d n n -1
T + 1
From t h i s e q u a t i o n h^ kan be s o l v e d
3 3 C o m p u t e r o g r a m
The problem outl^ugrave so far has been translated into a computer
program written in FORTRAN IV
The flow chart ot Uis program is given in fig 7 whereas in
appendix A the complete listing of the program can be found
In the flow chart the 5 main parts of the program are
indicated
A Reading and printing of input data
B Determination of the matrices A and B and the vector Q per layer
17
M
1 A
FLOH to unsaturated zono
FLOW to tertiary syster
FLOW to secondnrv system
Leakage to adjacent aquitards
SOLVE equation (14)
yes
CALCULATE flux through constant head boundnr
CALCULATE waterbalance n p r 1 OAro r r J j - ~
PRINT of output
Ier node per layer
k = k+1
t = t + At
7 DUMP variables on tape
-gt-
( STOP )
Fig 7 Flow chart of the program FEMSAT with diviation in 5 main
parts A B C D and E
C START PROGRAM FEMSAT
f READ general information
PRINT general information
f READ element information
PRINT element information 7 READ stationary nodal point information
PRINT stationary nodal point information 7
READ non-stationary nodal point information for t = t
FRINT non-stationary nodal point information for t = t
L 2 (
^ yes R E A D non-stat input for t gt t
J PRINT non-stat input for t gt t (- 1 -- M degY
GENERATE A B and AREA-matrix for confined aquifers
_L
GENERATE A B and AREA-matrix for unconfined aouifer
bull
C Solving h with Gauss-Seidel
D Determination of the water balance terms per layer
E Printing of results
The program exists of a main program with several subroutines
The main program calls the subroutines In the subroutines most
calculations are performed
In this way an easy adaptation to different problems is
obtained (see also section 44) Comments in the program will
help the reader to understand the program
In the next chapter the preparation of the input data executing
of the program and the output will be treated
Acknowledgement
Some parts of the computer program were taken from the program
UNSAT2 of NEUMAN etal (1974)
IV INPUT EXECUTION AND OUTPUT
41 I n p u t
The input is divided into endogenous variables (chosen by the
user or calculated by the program during execution) and exogenous
variables (based on physical data)
411 Endogenous variables
Translation of groundwater basin into a nodal point network
Probably the most important but also the most difficult stage
in the process of simulating groundwater flow is the translation
of the natural groundwater basin into a nodal point network Questions
which arise are
a In how many layer the geo-hydrological system mustcan be
schematized
b The locations of the boundaries and the nature of these boundaries
(impervious prescribed head prescribed flux)
c The configuration of the nodal points
19
The answers of these questions depend on the problem under
consideration
- Schematization into layers
Usually our knowledge about the geo-hydrological situation is
such that no more than 5 different layers can be distinguished
(remember contrast rule)
- Boundary conditions
The top boundary condition of a groundwater basin usually is
a prescribed flux (effective precipitation)
The bottom boundary condition usually is a prescribed flux
boundary with flux zero (impervious hydrological base)
For the boundaries at the other sides of the system one sees
either prescribed head boundaries (eg a fully penetrating
river) or prescribed flux boundaries (including zero flux)
- Nodal point configuration
Nodal points are situated
on the boundaries
along internal boundaries which indicate changes in
hydraulic properties within the layer
along the tertiary surface water system
on places where artificial recharge or discharge wiii tgtke
place
elsewhere
Construction of the finite element mesh
Assume we have schemetized the groundwater basin iatu a finite
number of layers with different direction of flow (horizontal and
vertical) Within one layer the hydrological properties may vary
from place to place
Now a nodal grid is superimposed on each layer in a way already
described in section 32 Thus each layer has the same number of
nodal points situated on the same places in the horizontal plane
but with different z-coordinates
We are free to construct quadrilaterals or triangles because
the program automatically divides a quadrilateral into two triangles
having identical material properties The dimensions of the elements
20
should be small at places where large changes in hydrological
properties occur or where large gradients in hydraulic head are
expected to occur
Coding of nodal points
Each node in the finite element network is assigned an integer
code which indicates the type of calculation for this node
The coding is as follows
- Code = 1 nodal points with described head as function of time
- Code 2 internal nodal points
- Code = 3 internal nodal points which represent secondary surface
water system
- Code = 5 nodal points with prescribed artificial recharge or
discharge and points with prescribed boundary flow
(both as functions of time)
Other important endogenous variables are the magnitude of the
time step the value of the variable which stops the iteration
process and the maximum number of iterations allowed per time step
(see also group A and B of section 413)
412 Exogenous variables
After constructing the nodal grid each node represents part of
the groundwater basin Geo-hydrological properties can be assigned
to each mode based on physical data (group C D and E of section
413) In group F of section 413 time-dependent variables can
be assigned to each node
413 Data input to Program FEMSAT
The data input must be punched on cards according to the
following instructions
GROUP COLUMNS FORMAT SYMBOL DESCRIPTION
A 1-80 20A4 HEAD Desired heading to be printed at
the beginning of the execution
1-5 15 NUMNP number of nodal points
21
GROUP COLUMNS FORMAT SYMBOL DESCRIPTION
6-10 15
11-15 15
16-20 15
21-25 15
26-30 15
31-35 15
NUMEL
NUMLAY
MBAND
NTSPI
NTSPO
MAXIT
1-10 F103 ST
11-20 F103 DELT
21-30 F103 TMAX
31-40 F105 TOL
number of elements
number of layers
maximum difference in node number
between two adjacent nodes
number of time steps between a
change in non-stationary input data
number of time steps between two
successive outputs
maximum number of iterations per
time step
time to start with the execution
initial time interval
maximum time to be reached before
execution is stopped
maximum allowed absolute change in
the values of h between any two
successive iterations in a given
time step
Iterations continue until all changes
are less than or equal to TOL or
until MAXIT is exceeded
1- 5 15
6-10 15
11-15 15
16-20 15
KX(N1)
KX(N2)
KX(N3)
KX(N4)
sequential number of i-th corner of
element N
sequential number of j-th corner
sequential number of k-th corner
sequential number of 1-th corner
(see fig 8)
Fig 8 Numbering of element corner nodes
22
GROUP COLUMNS FORMAcircT SYMBOL DESCRIPTION
One card must be provided for each element in sequential order
starting with N = 1 and ending with N = NVJMEL
I-K 1103 SANG(NLY)
1i-iO MO3 C1(NLY)
21-30 F 103 C2(NLY)
21-4- 1-104 THI(NLY)
angle in degrees between CI and
the laquo-coordinate to he used in
nodal print N in layer LY (see
poundig 9)
First principal conductivity of
material in node N and layer LY
idem second principal conductivity
thickness of layer LY at nodal
point N
ig V Definition of SANGCI and C2
One card icircuiPt be provided for each model point ctarting wih M = 1
and ending with N = NUMNP If there is norlt3 than 1 layer for each
layer a set must be provided starting with the upper layer LY - 1
and ending with the bottom layer LY = NIJMLAY The variables SANG
M and 2 are not relevant tor aquitards but for reasons of
conveniency this procedure is maintained (making SANG CI and C2
dummy variables)
1-10 110 KODE(N)
11-20 II0 NTERR(N)
coding of nodal points (see
paragraph 42)
number of tertiary relation in node N
The q-h relationship of the flow to
23
GROUP COLUMNS FORMAT SYMBOL DESCRIPTION
21-30 FlOl X(N)
31-40 FlOl Y(N)
41-50 FlOl HGL(N)
51-60 FlOl HBSC(N)
61-70 FlOl LESC(N)
the tertiary system as discussed
in par 24 are schematized in a small
number of different relations
x-coardinate of node N
y-coordinate of node N
height ground level in node N
height bottom secondary conduit in
node N If node N is not a secondary
conduit this variable is a dummy
variable
length of secondary conduit in area
represented by node N
One card must be provided for each node starting with N = 1 and
ending with N = NUMNP
E 1-10 FlOl P(NLY1) initial value of h at node N in
layer LY
11-20 FlOl VERRES(NLY) vertical resistance at node N and
layer LY If the layer is an aquifer
this value is a dummy variable
21-30 F101 RADRES(NLY) radial resistance for flow to
secondary system at node N in layer
LY
31-40 FlOl ENRES(NLY) entrance resistance for flow to
secondary system at node N in layer
LY
41-50 FlOl WP(NLY) wet perimeter of secondary system at
node N in layer LY
51-60 FlOl P0R(NLY) specific yield of the material at
node N in layer LY
61-70 FlOl SS(NLY) specific storage of the material
at node N in layer LY
24
GROUP COLUMNS FORMAT SYMBOL DESCRIPTION
One card must be provided for each node starting with N = 1 and
ending with N = NUMNP If there is more than 1 layer for each layer
a set of cards must be provided starting with LY = 1 and ending
with LY = NUMLAY
F 1-10 F101 P(NLY1) value of h at different times at
node N in layer LY Node N lies
on a prescribed head boundary
11-20 F101 WTSC(N) water table in secondary system at
node N at different times
21-30 F 101 ARRED(NLY) artificial recharge to or discharge
from node N in layer LY or
prescribed boundary flow at
different times
31-40 F 101 QPHR(N) flux through phreatic surface at
node N at different times
The data in group F are non-stationary input data One card must be
provided for each node starting with N = 1 and ending with N = NUMNP
In case of more than 1 layer for each layer a set must be provided
Not relevant variables are taken as dummy variables
After a certain number of time steps (NTSPI) the data of group E
are read So the set cards E must contain NxLYxC cards where C =
(TMAXNTSPIXDELT) + 1 if DELT is a constant
42 R u n n i n g t h e p r o g r a m o n t h e C y b e r 7200
The program has been executed on a Cyber 7200 computer of
IWIS-TNO in the Hague
- The demanded input is punched on cards according to the instructions
of the last paragraph and is put on a permanent file
- For each specific problem inthe program only the dimension statements
in the main program have to be adjusted according to the instructions
in the lines of comment
- After compilation the program together with the input file is
executed
25
- Intermediate data necessary for continuation of the calculations
are saved on another permanent file This gives the opportunity
to restart the program at a time the calculation was stopped
The only change in the input file is to adjust the starting time
Remark
With the aid of a terminal small changes in the program and in the
input file can be made easily and the output immediately appears
on the terminal In this way it is possible to stop erroneous
calculations This is especially useful when being in the testing
phase of the program
43 O u t p u t
Immediately after reading the input data they are printed so
that possible errors can be detected During execution after each
certain number of time steps (NTSPO) calculated values of h and
water balance terms are printed In appendix B an example is given
The output data can be devided into two groups
A Data per node per layer
- Coding of the nodal points KODE
- Hydraulic head HEAD
- Flow through phreatic surface at the time indicated in head
of table FLOWUN
- Flow to tertiary surface water system FLOWTS
- Flow to secondary surface water system FLOWSS
- Flow to adjacent layers LEAKAGE
- Artificial recharge or discharge and prescribed boundary
flow ARRED
- Change in storage during last time step STORAGE
- Flow through prescribed head boundary LATFLOW
B Data per layer
Summation of the terms already mentioned under A gives the
magnitude of the water balance terms per layer at the time
indicated (instanteneous water balance terms) Summation of
these instanteneous terms multiplied by their respective time
intervals since time is zero gives the cumulative water balance
terms per layer
26
After each print out we are able to check the calculations
because the sum of the water balance terms per layer must be
approximately zero
44 C o m p u t i n g t i m e
The computing time depends on the number of nodal points and
the number of layers But it depends also on the magnitude of the
changes in hydraulic head and on the value which terminates the
Gauss-Seidel iteration (TOL) To give an impression TOL = 001 m
Costs per day simulation (in hlfl) on the Cyber 7200
number of nodes number of layer bull bdquo bull bull _ 1mdash x a x b (21)
where
a = costs in Dutch guilders per system second (ƒ 027)
b = empirical factor with magnitude about 001
V NUMERICAL EXPERIMENTS
In general there are two possibilities to test the validity of
numerical models The first and most commonly used method is to
compare numerical results with analytical solutions The second
possibility is to compare numerical results with measured field
data Since the last alternative requires many data that are
difficult to obtain we restrict ourselves for the time being to
comparison with the analytical solutions
5 1 F l o w t o a w e l l i n a n u n c o n f i n e d
c i r c u l a r - s h a p e d a q u i f e r
Flow to a well can be divided into steady-state flow and
unsteady-state flow (KRUSEMAN and DE RIDDER 1976)
27
511 Steady state flow in an unconfined aquifer
For this case the formula of Thiem-Dupuit is valid (see fig 10)
Q bull k lth2-hicircgt ln(r2r])
(22)
where
hhbdquo = hydraulic head in 1 and 2 respectively
Q = discharge from well
rbdquo = distance from well
CL31
hmdash^ri --r2
Fig 10 Schematic cross-section of a pumped unconfined aquifer
Fig 11 gives the nodal grid superimposed on a well in a
circular shaped aquifer Near the well the nodal point distance
is made smaller Quadrilaterals as well as triangles are used
Numerical calculations with this finite element network and 3 -1
an extraction rate of 200 m day were performed for 10 days
At that moment a steady-state situation was almost achieved
28
2tmdashnodal point number
160 180 X-axesim)
Fig 11 Nodal point configuration for flow to a well in a circular-
shaped aquifer The outer boundary is taken to be constant
The well is situated in nodal point 42
29
In fig 12 the results of the simulation are compared with the
analytical solution The agreement between numerical results and
the analytical solution is excellent
draw- 15irS (m) down
analytical solution imdashx calculated values
distance from well r (m)
Fig 12 Comparison between analyt ical and calculated s teady-state
drawdown for the problem of f ig 11 with 1
-1 Q bull - 200 m 3 d a y _ 1
k = 10 mday
d = 100 m
S = 2 y
S = 0 s
512 Unsteady-state flow in an unconfined aquifer
Analytical solutions for unsteady-state flow in an unconfined
aquifer only exist if the aquifer extends infinitely For the
aquifer given in fig 11 this condition is not fulfilled However
during the first one or two days after starting the extraction the
influence of a constant hydraulic head at the outer boundary on the
30
drawdown near the well can be neglected (see fig 13) After one
day the lateral inflow is only 25 of the extraction the remainder
being the change in storage due to drawdown
2 0 0
m 3 day 1
Q=200m 3 day
100-
vraquomdash Lateral inflow
N amp storage
I I ^ V - W 10
t (days)
Fig 13 Values of Q lateral inflow and change in storage as
functions of time
The nonsteady-state or Theis equation can be written as
i _ Q s 4iTkD
dy = 4irkD
W(u) (23)
where
u = r2S
4irkDt
r = distance from well
S = storage coefficient
Q = constant discharge from well
D = thickness of the waterbearing layer
at the beginning of pumping
t = time after starting the extraction
s = corrected drawdown
= s - 322D
s = calculated drawdown
W(u) = Theiss well function
[J
CIAT1
[T]
]
31
In fig 14 the analytical unsteady-state solution for the case
described in section 511 is given together with the numerical
results at times 05 and 10 days The agreement between numerical
results and the analytical solution is rather good
Drawdown s1(m)
10
10
10
Theiss solution
x caculated values t=Q5
bdquo t= lO
10 10- 10 10 t r2(daym2)
Fig 14 Comparison between analytical and calculated steady-state
drawdown for the problem of fig 11 with
Q
k
d
S y
s s
=
=
=
=
=
200
1
10
3 m
0
0
2
0
-1 day
m day
m
These results indicate that the assumption of a linear variation
of h between two nodal points (see fig 3 ) even with the rather
rough element network of fig 11 is acceptable So there is no
need to improve for example the finite element network by introducing
higher order coordinate functions
32
52 F l o w t o w e l l i n 3-1 a y e r e d s y s t e m
In fig 15 a schematic cross-section of a 3-layered system of
two aquifers separated by an aquitard is given A well of
infinitisemal radius is completed in the lower aquifer and discharges
at a constant rate QTT Each layer is homogeneous isotropic and
extending radially infinite
bdquo bull - bull bull ~ i
mdash Q II
aquifer I
lugrave^i~IcircIuml7i-i~~ aquitard --r-rl~-
-II--I i - i
aquifer
Fig 15 Schematic cross-section of a 3-layered system
For this situation NEUMAN and WITHERSPOON (1969a) give rather
complicated analytical solutions In the special case where the
hydraulic properties of the two aquifers are identical they give
graphical illustrations of their solutions for three different
cases
These graphical solutions give the opportunity to verify
roughly numerical results of a 3-layered system Therefore a nodal
grid is superimposed on a circular shaped groundwater flow system
with the extraction in the centre of the lower aquifer and with
a constant head on the outer boundary Fig 16 gives the nodal
point situation in the x - y plane of 1 layer The nodes in the
other two layers are situated in the middle of the layers at the
same places in the x - y plane only their z-coordinate is different
33
A
-Jy
bull33
V 5
iumlff
Fig 16 Nodal point configuration for the 3-layered problem (top view)
34
In fig 17 numerical results are compared with the analytical
solution The agreement is rather bad especially for the two
aquifers But in terms of the water balance the difference between
aquifers and aquitard is not so big because the aquitard has
about 16 times bigger storage capacity than the aquifers The
small value of the specific storage of the aquifers is the reason
that small errors in the components of the water balance have a
big influence on the calculated values of the hydraulic head
Eig 17 Dimensionless drawdown vs dimensionless time in a
3-layered system with
II = 50 m day
3 m (distance from well)
aquifer I
aquitard
aquifer II
Thi (m)
100
25
100
S s
001
0642
001
k(mday )
1
0277
1
The discrepancies are partly caused by errors in discretization
In the z-direction these errors can be eliminated by taking the
35
vertical resistance of the aquitard to be infinite Then the
calculated drawdown in the pumped aquifer can be compared with
the analytical Theiss solution
This is done in fig 18 for different distances from the well
One sees that the agreement is better when the distance is bigger
Apparently the linearization of the logarithmic drawdown curve
near the well is responsible for the differences The differences
for r = 3 m are of the same magnitude as for the pumped aquifer in
the 3-layered case
joo gt A 50
20
Theiss solution
bull calculated values r=3m (distance f rom well) x bdquo bdquo r=7m
1 L
I l_ 001 003 0 0 5 Ol Q2
- I I 5 10 t - K t
Fig 18 Dimensionless drawdown vs dimensionless time in a
confined aquifer with
Q =500 m day
k = 10 mday
D = 100 m
S = 001 s
The differences between numerical results and analytical solutions
are not caused by taking a finite area because changing the outer
boundary into an impermeable boundary has only minor effects
36
VI SUMMARY AND CONCLUSIONS
In this report a numerical model for non-stationary saturated
groundwater flow in a multi-layered system has been treated
Groundwater flow can be described in different ways
a Physical-mathematical by combining Darcys law and the law of
continuity (section 21)
b Geo-hydrological the flow in a natural groundwater basin takes
place in layers with different hydraulic properties Definitions
of the most important hydraulic properties are given in section
22
c Schematical groundwater flows horizontally in aquifers and
vertically in aquitards (section 23)
d Approximating by dividing the region of flow into a finite
number of elements and applying Darcys law and the law of
continuity on each of these elements In these ways one gets
a set of equations which can be solved given (initial and)
boundary conditions (section 25)
e Numerical the set of equations has been solved using a Galerkin
type finite element method and the implicit Gauss-Seidel method
with overrelaxation factor (section 31 and 32)
f Computer-solvable the numerical solution has been translated
in a FORTRAN IV program called FEMSAT (section 33)
In chapter IV the input execution and output of the program
FEMSAT are treated
Finally in chapter V some numerical applications are given
simulation of flow to a well in unconfined aquifer and simulation
of flow to a well in a 3-layered configuration
Comparison of hydraulic heads obtained by numerical calculations
and analytical true solutions showed an excellent agreement in case
of stationary flow and a good to moderate agreement in case of non-
stationary flow However the diviations caused by the discretization
process must be seen in relation to the errors made in the
establishment of the hydraulic properties
37
As was stated in the introduction numerical models can be
valuable tools in the management of water resources Therefore
numerical models must be suitable to simulate effects of certain
operations upon the spreading of water over saturated groundwater
unsaturated groundwater and surface water The model described in
this report is a model for saturated groundwater flow The relation
with the surface water can be taken into account via radial and
entrance resistances or via a relation between flow to the surface
water system and the hydraulic head The relation with the unsaturated
zone has been restricted to the input factor effective precipitation
In practise however the unsaturated and saturated system are
interconnected Therefore the model should be improved by taking
into account the unsaturated zone This can be done in at least
two ways
a Replacing the unsaturated zone by a parametrical model
b Coupling the unsaturated zone in an iterative way with the
saturated system
In order to find out the usefulness of the numerical model for
water resources management it will be applied to actual regional
groundwater systemslaquoAlso sensitivity analyses will be carried out
The computer model itself can be improved by applying standard
programs eg automatical generation of a nodal grid and graphical
plotting of the results
38
VII REFERENCES
ERNST LF 1962 Grondwaterstromingen in de verzadigde zone en hun
berekening bij aanwezigheid van horizontale evenwijdige open
leidingen Thesis University Utrecht pp 189
1978 Drainage of undulating soils with high groundwater
tables (in press)
FORSYTHE GE and WR WASON 1960 Finite Difference Methods for
Partial Differential Equations John Wiley and Sons New York
pp 444
KRUSEMAN GP and NA DE RIDDER 1976 Analysis and evaluation of
pumping test data ILRI-bulletin No 11 pp 200
LAAT PJM DE and C VAN DEN AKKER 1976 Simulatiemodel voor grond-
waterstroming en verdamping in Modelonderzoek 1971-1974 ten
behoeve van de waterhuishouding in Gelderland deel II
Grondslagen pp 145-197
NEUMAN SP RA FEDDES and E BRESLER 1974 Finite element
simulation of flow in saturated -unsaturated soils considering
water uptake by plants Hydraulic Engineering Laboratory
Technion Israel pp 104
and PA WITHERSPOON 1969a Theory of Flow in a confined
two Aquifer System Water Resources Research Vol 5 No 4
pp 803-816
^_ and PA WITHERSPOON 1969b Applicability of Current Theories
of Flow in leaky aquifers Water Resources Research Vol 5
No 4 pp 817-829
PINDER GF and EO FRIND 1972 Application of Galerkins Procedure
to Aquifer Analysis Water Resources Research Vol 8 no 1
pp 108-120
REMSON I GE HORNBERGER and FJ MOLZ 1971 Numerical Methods
in Subsurface Hydrology John Wiley and Sons New York
pp 389
WESSELING JG 1977 Toepassing van Finite Difference- en Finite
Element Methoden voor het oplossen van een aantal grondwatershy
stromingsproblemen (met computerprogrammas) ICW-nota 944
pp 63
39
ZIENKIEWICZ OC 1971 Finite Element Method in Engineering Science
McGraw-Hill London pp 521
40
z ltc laquo LU U EC LU ltC
c z O E 0C 3 GLZ
X - e - o
gt-o-
LUX
E 2 3 - -Z o
O U K raquo-lt-laquo-
lt-gt z o z
laquoC - L
M II II II II II II
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LJ - 0 - z 0 o c m lt-bull 3 -z z -0 c -0 -3 - Z Z Z C O C 3 - 3 Z z r -0 c xgt -3 -lt Z ~ -J C 3gt - 3 -lt -H ~ s-gt -H 3gt - Z -lt c 0 -
laquo bull
T 1 T 1 3 H H T 1 H z m n - n 0 - n
X O H lt-bull r- 33 r- O C C TJ 3 TJ
33 m gt-raquo i gt - Z 33 Z - 1 Z
33 - -gt -m raquo v j t o - -raquo -laquo-bull W CD ro Z u x laquo_
r - c i -c r --lt H - 1
bull X Z 3 ) Z C l m r - z u bull 1-1 0 -o (S Z O 33 bull u f t l M w w bull z C t H H 3 m m z - w c i ro t o Iuml gt j j -11 - X z
a ro 3 U N m ii t n Cl X xgt c i t n 1 - 3 r o - 3gt bull z I -
s t 33 r- Z 33 i-t
rt 3gt m r- X - t Z - 33 -lt -1
w - z -0 -bull ro - r w f~ f Ti r - xraquo -n -c
O Uuml I H o i-t - n Z C l H 3 P H-l Xgt bull z u r C tO H
C l - 3 w 3gt Cl
r o n c f l H H c ouml lt-laquo - n
z z a - 1 u i - H 3 T
+ Z II C l ^ I C l f l H
11 m w bull ro c i z
3 C 3 3gt 3
T C O II Z
+ 3 l
O
r o
3
r o
3
) - 03
o z
o o gt c ^ c ^ o ^ o ^ t n t n o i i 0 t n t n y i t n t n i K ^ ^ ^ ^ ^ ^ gt ^ t j c o u c o o J W u x t o w c n t c ^ ^ S s 3 X N ^ o T gt 4 ^ w ^ œ N O O N j ( gt c n t J w ^ a D s 3 œ N i o ^ c n gt c ugrave r o ^ c a s 3 œ N j O W I S l B S Iuml S S iuml I B l S f f l C B l S O O O S I S Œ l S l S i Œ J I S S l C D l S S l S l S icirc S l B i S l I B C C l B C D S i S C D l S l B C D i S i i S Il I Il I raquo Il II li II M li II II II II li H II 11 II II II il II II II h 11 l II II II II II II t |t jj raquo Il M il
t n iS
m -x r -Z T- O c - t z
ja- w ro j s es S (S CD C3 IS
- n - 0 - r j T j - n x T i - O T icirc - o T i - D
an CD
rf 0 n -o - n -0 x C X O 3 3 O 3 3 O 3 3 O 3 0 O 3 3 3 O 3 3 O 3 3 V 3 3 ^ - I 3 3 - 3 3 raquo - I 3 3 gt - I 3 3 ) - ( 3 3 I - I
C - I 3 Z 3 Z 3 Z 3 Z 3 Z 3 Z 33 1-1 Z Z
ff
I gt - f X - - l 3 gt - H X raquo - I gt - H 3 gt - i bull-i mdashl mdash1 mdash1 mdashi mdash1 laquo | J gt - j gt - j ^ - f c gt 0 4 N ( r o - gt j gt i S i N C D r n Œ c i reg t n s t n reg t n s ^o -x - x - x - x - x - x r
r - w 2 gt uuml l X gt ] gt T i - n T i - r j r -raquo z n n icirc s j r r r r raquo H o n c - i iuml o r o i - lt m c c s w m c n c c n m raquo bull H j ^ - n o xgt ^ 3 3 Tgt lt-r tz mdash w iuml- i - t m H r r - r - r - - lt r r c r - o - lt z
bulllt x _bull s- -lt -lt _ C3 -n _ - i - _ rgt _ bull t n - -i - - M rlt r - c - m n -gt o n o œ n u n n - n ^ c - z - i ^ 3 gt 3 gt - n o r - c a
z o r s n r z u icirc N H O - 3 3 0 3 3 m f - O f J l w i c ^ ^ c m z ^ raquo f x mdash 33 r- 0 - 1 p s | -O bull-raquo 3 bulllt 3gt - - - raquo - lt - lt
3 J M raquo H _ - 3 Z 3 Z t n 3gt H Xraquo- -H S - 1 -H
I 03 f - NI r -fc copy bull 3D -lt t n t n -il X ~i X ~i -raquo - n - n - T U mdash n r z r - - t r c c 0 cn a z 3 c - e - - n a i gt -s - n - - n -c 0 r 0 r
mdashf c - i cn c z m 2 33 en H xgt i - H 3gt C 33 33 -lt X
_ 3
-k
33 s c e s t n
j t i T T i - i i n n A Z O S O C O O X d 33
U) 3gt V Z N O bull
t o
Z n m
00 m 0
z z
z C l
z
raquo J J
- -n bullbull-gt r w 33 J Z 2 1gt n bullbull r n
3 Z 3 3 - 1 -H 1 gt - H 1 gt B H H H
V t n Ni
mdash1 m 33 3 t n
- n
c xgt
m x 55
f j
M zgt -n
O)
_ D t 3 - T) C bulln c 0 3= 0 t o 0 x
3= w gt0 3 raquo-i 3 3 r x raquo i gt r - r 0 3 Z u 3 X 3 Z 33 -lt 33
- 1 - i Z Z m bull 33 lt-raquo -lt t n iuml gt
o ^ r - r - c c o z m z m z CD t o t o F I H
t n t n
- n as - T] S) -raquobull bull œ w raquo - t o ro bull-gt CD X
en - n
es
u bull-bull
bull-1 m c -z bull ^ ^ - z r w --lt - 0 - raquo w yO 1mdash1
3gt Z Xgt n - f n 0 n 1-1 [Tl r-i Z 03 Z
-lt -lt icirc-f
- z o 0 - f-1 bull-- z z
J
0 11
w z -0 c 33 3 w Z Z O H -
t gt bull t o -
CS
r
-lt z
7Z O O
H - t S) 3gt - 1
- 1 r o r - r - lt-raquo r o
h ) ^ f m n f i V n i gt t n
J X I - 3gt z c i r c m i gt 3 ~lt r m bulljgt raquo
-lt rgt z raquo 0 3 3 0 m m 0
3 0 0 m
t o H
^ 00 X
iuml gt - 4
-H
3 m
- n
t n
t o
bullbull0 x
z
3
3gt
C -lt Igt - t Ci
APPENDIX 8 EXAMPLE OF OUTPUT
HEAD AND TERMS OF UATERBALANCE PER NODE PER LAYER
AT TIME 260 IN II AND I13DAYS
T
LAYER NODE KODE HEAD FLOUUN FLOUTS FLOUSS LEAKAGE STORAGE LAT FLOU
13 19 25 31 37 43
125008 125000 12S000 12S000 12S000 12S000 12S000 12S000
0000 0000 0000 0000 0000 0000 0000 0000
0000 0000 0000 0000 0000 0000 0000 0000
0000 0000 0000 0000 0000 0000 0000 0000
- 000 - 000 -000 - 000 - 000 - 000 - 000 0000
0000 0000 0000 0000 0000 0000 0000 0000
0000 000 000 000 000 000 000 000
000 0000 0000 0P100 0000 0000 0000 0000
1 7
13 19 25 31 37
12S000 125000 125000 125000 125000 12S000 125000 125000
0000 0000 0000 0000 0000 0000 0000 0000
- 001 -007 - 023 - 020 - 009 - 003 0000
0000 0000 0000 0000 0000 0000 0000 0000
000
001
007
023
020
009
003
003
0000 0000 0000 0000 0000 0000 0000 0000
1 7
13 19 25 31 37 43
125000 125000 124998 124972 124895 124787 124647 124358
0000 0000 0000 0000 0000 0000 0000 0000
- 000 001 007 023 020 009 003
0000
0000 0000 0000 0000 0000 0000 0000
50000
0000 182
1982 3862 1674 419 078 041
0000 0000 0000 0000 0000 0000 0000 0000
TERMS OF WATERBALANCE PER LAYER DURING LAST TIMESTEP AND
AND SINCE BEGINNING IN M3DAY M3
FLOU TO UNSATURATED ZONE
FLOU TO TERTIARY SYSTEM
0000
0000
0000
0000
LAYER NO 1
FLOU TO SECONDARY SYSTEM
FLOU TO AD3ACENT LAYERCS)
ARTIFICIAL FLOU AND PRESCRIBED BOUNDARY FLOU
ACCUMULATION TERMS
LATERAL FLOU THROUGH PRESCR HEAD BOUNDARY
0000
-000
0000
000
000
0000
-000
0000
000
000
LAYER NO 2
FLOU TO SECONDARY SYSTEM
FLOU TO AD3ACENT LAYERCS1
ARTIFICIAL FLOU AND PRESCRIBED BOUNDARY FLOU
ACCUMULATION TERMS
LATERAL FLOU THROUGH PRESCR HEAD BOUNDARY
0000
-378
0000
381
0000
0000
-055
0000
056
0000
LAYER NO 3
FLOU TO SECONDARY SYSTEM
FLOU TO AD3ACENT LAYERCS5
ARTIFICIAL FLOU AND PRESCRIBED BOUNDARY FLOU
ACCUMULATION TERMS
LATERAL FLOU THROUGH PRESCR HEAD BOUNDARY
0000
370
50000
49376
0000
0000
055
12979
12909
0000
Fig 3 The global coordinate function f (xy)
The integration can be performed element by element with the
aid of the coordinate functions f (NEUMAN etal 1974)
For non-stationary problems it is not possible to apply the
Galerkin principle on the time derivative But it is possible
to discretize this derivative and to set up the system of equation
(eq 12) for each finite time step
One obtains in this way a series of sets of laquocriations tha s
an approximation of the non-stationary flow problem
Treatment of boundary conditions
a Initial conditions
In non-stationary flow problern gtne must start with given
values of a at time t = t ri c
b Boundary conditions
- Prescribed head boundary
Nodal points which lie on a prescribed head boundary have a
prescribed value for h bull r Hgte r umber of unknown h s is n n
reduced and accordingly no the number of equations
- Prescribed flux through z-- boundary
If one knows the flux (iuumlr jw or outflow per unit area) and
the area of the boundary oi-r element the inflow or outflow
through the boundary per flemlt=nt can be calculated
One may assume this flow to J bull part of the sink term Q in
eq (12)
12
32 A p p l i c a t i o n o f t h e f i n i t e e l e m e n t
m e t h o d o n t h e f l o w i n g r o u n d w a t e r
b a s i n s
From hydrogeological investigations the geometry and the matrix
of the hydraulic properties are known Knowing the geometry the
flow can be split up into a finite number of layers This holds
also for the nature of the boundaries
Now a nodal grid is superimposed on each layer in such a way
that the nodal points of the different layers are situated right
above each other (see fig 4) and in each layer the nodal point
lies in the middle of this layer
-nodal point
Fig 4 Schematic representation of the nodal points in the
xyz-space
Each layer has N nodal points of which P nodal points have a
prescribed head In each layer remain N-P = N nodal points with
unknown head Eq (12) can be applied to each point with unknown
head in each separate layer
As was stated in the last paragraph for each time step one
can write
13
Ah bdquo ~ -r1 + Q = O in = 12N in n m At yi A h + B (13)
where
A = in
ocircf ocircf ocircf of (k d f mdashri- ~ + k df - ~ -jS) dxdy
J x n ox ox y n oy oy
in (S d + S ) ff dx dy O if i icirc n
s y i n
Q f dx dy
The matrices A B and the vector Q have been defined in in l
elsewhere (NEUMAN etal 1974) A is the conductivity matrix in
B the storage matrix and Q the source vector in l
In order to clarify these rather complicate expressions apply
equation (13) poundo point 4 of fig 5
6 7
Fig 5 Top view of part of the nodal grid
lhl + A42h2 + A43h3 + A44h4 + A45h5 + A46h6 + A47h7 +
h - h
+ B _iuumli i ^ + Q = o 44 t - t x4
n n-1
(14)
where h and h are values of h at time t and t 4n 4n-1 4 n n-1
respectively
14
If the h-values belonging to the A-values are taken at time
t = t we get an equation with one unknown h for time t = t n-1 4n n
The equation can then be solved directly
However this is an explicit formulation which puts strong limitations
on the magnitude of the time step t - t (REMSON etal 1971) n n-1
Values larger than a certain value cause instability With an implicit method the values of h are taken at time
t = t This method is unconditionally stable Now one gets an equation n
with 7 unknowns h through h If one has 7 equations for every 1 n n
point 1 through 7 we can solve the values h i = 17
in
Of all the methods possible the iterative method of Gauss-Seidel
has been chosen The Gauss-Seidel method starts by choosing a first estimate of h at time t = t (usually equal to h ) At each in n j -i in-l iteration step a new value of h is calculated using the last
in calculated values of h If one passes the nodal points from 1 to
i n 7 eq (14) becomes
h-j + 1 = A h^ + 1 + A bdquo h | + 1 + A _ h | + 1 + A h j + A r h j + 4 n 4 1 1 n 4 2 2 n 43 3 n 45 5 n 4 6 6 n
B h B + A h J + - 4 4 4 n ~ + 0 J (A + mdash) A 4 7 h 7 n At n U 4 4 At
(15)
where
h-i = in
j = iteration step or number of iteration
i = number of nodal point n = time = t
n
The iteration process converges if the successive differences k+1 k
between h and h become continuously smaller in in
The iteration process is stopped if at each nodal point the k+1 k
difference between h and h is smaller than a certain prescribed in in
value
15
To accelerate the convergence an overrelaxation factor W is used
(for more information about W see FORSYTHE and WASOW 1960)
k+1 k T7 k+1 k h = h + W (h - h )
in in in in (16)
If W = 1 there is no overrelaxation Usually W varies between 1 and
2 and the optimum value very often is found by trial and error
Eq (15) is valid for points in aquifers The connection with
adjacent layers is made by the term Q because in this term the
flow from adjacent aquitards q is included (two aquifers are always
separated by an aquitard)
Aquitards
We dont apply the finite element method on points in aquitards
As stated before there is no horizontal flow in aquitards So it has
no sense to replace the continuous variation of h in the aquitard
by equation (11)
The nodal points in aquitards represent a certain volume of an
aquitard hich transmit water from one aquifer to another and which
has storage capacity The nodal points are included in the Gauss-
Seidel scheme -s follows (see fig 6 )
1 aauifer
a -_vS laquo 2 [~^-t^yM^^^-^r~ aqutarci l-Z
aquifer
Fig 6 Schematic situation for points in aquitards
Point 2 is situated in the middle of the aquitard with thickness
d storage coefficient S and vertical hvdraulic conductivity k c J v
Suppose the Gauss-Seidel iteration starts in the upper layer and
ends in the bottom-layer Then the flux from point 1 in aquifer 1 to
16
point 2 in the aquitard at time t = t and in iteration step j + 1
is
k q = (hf -hf )-iumlmdash f (17)
1 n 2 n OSxd1
and the flux from point 3 in aquifer 2 to point 2 in the aquitard
is
1 c 1
qbdquo = (h - hf) mdashHmdashr- (18) 2 3 n 2 n OSxd
The net flux ltl-q9 must be equal to the change in storage per unit
area per unit time This change in storage can numerically be
approximated by
te - h-
n n-1
Combina t ion of ( 1 7 ) (18) ana (19) g i v e s
l tr1 - h (hJ + 1 + hj - 2 hj+1) - - - - - - S1 - ilaquoS L u l l = o (20)
] i 3 n 2 n 1 c t - t j j x d n n -1
T + 1
From t h i s e q u a t i o n h^ kan be s o l v e d
3 3 C o m p u t e r o g r a m
The problem outl^ugrave so far has been translated into a computer
program written in FORTRAN IV
The flow chart ot Uis program is given in fig 7 whereas in
appendix A the complete listing of the program can be found
In the flow chart the 5 main parts of the program are
indicated
A Reading and printing of input data
B Determination of the matrices A and B and the vector Q per layer
17
M
1 A
FLOH to unsaturated zono
FLOW to tertiary syster
FLOW to secondnrv system
Leakage to adjacent aquitards
SOLVE equation (14)
yes
CALCULATE flux through constant head boundnr
CALCULATE waterbalance n p r 1 OAro r r J j - ~
PRINT of output
Ier node per layer
k = k+1
t = t + At
7 DUMP variables on tape
-gt-
( STOP )
Fig 7 Flow chart of the program FEMSAT with diviation in 5 main
parts A B C D and E
C START PROGRAM FEMSAT
f READ general information
PRINT general information
f READ element information
PRINT element information 7 READ stationary nodal point information
PRINT stationary nodal point information 7
READ non-stationary nodal point information for t = t
FRINT non-stationary nodal point information for t = t
L 2 (
^ yes R E A D non-stat input for t gt t
J PRINT non-stat input for t gt t (- 1 -- M degY
GENERATE A B and AREA-matrix for confined aquifers
_L
GENERATE A B and AREA-matrix for unconfined aouifer
bull
C Solving h with Gauss-Seidel
D Determination of the water balance terms per layer
E Printing of results
The program exists of a main program with several subroutines
The main program calls the subroutines In the subroutines most
calculations are performed
In this way an easy adaptation to different problems is
obtained (see also section 44) Comments in the program will
help the reader to understand the program
In the next chapter the preparation of the input data executing
of the program and the output will be treated
Acknowledgement
Some parts of the computer program were taken from the program
UNSAT2 of NEUMAN etal (1974)
IV INPUT EXECUTION AND OUTPUT
41 I n p u t
The input is divided into endogenous variables (chosen by the
user or calculated by the program during execution) and exogenous
variables (based on physical data)
411 Endogenous variables
Translation of groundwater basin into a nodal point network
Probably the most important but also the most difficult stage
in the process of simulating groundwater flow is the translation
of the natural groundwater basin into a nodal point network Questions
which arise are
a In how many layer the geo-hydrological system mustcan be
schematized
b The locations of the boundaries and the nature of these boundaries
(impervious prescribed head prescribed flux)
c The configuration of the nodal points
19
The answers of these questions depend on the problem under
consideration
- Schematization into layers
Usually our knowledge about the geo-hydrological situation is
such that no more than 5 different layers can be distinguished
(remember contrast rule)
- Boundary conditions
The top boundary condition of a groundwater basin usually is
a prescribed flux (effective precipitation)
The bottom boundary condition usually is a prescribed flux
boundary with flux zero (impervious hydrological base)
For the boundaries at the other sides of the system one sees
either prescribed head boundaries (eg a fully penetrating
river) or prescribed flux boundaries (including zero flux)
- Nodal point configuration
Nodal points are situated
on the boundaries
along internal boundaries which indicate changes in
hydraulic properties within the layer
along the tertiary surface water system
on places where artificial recharge or discharge wiii tgtke
place
elsewhere
Construction of the finite element mesh
Assume we have schemetized the groundwater basin iatu a finite
number of layers with different direction of flow (horizontal and
vertical) Within one layer the hydrological properties may vary
from place to place
Now a nodal grid is superimposed on each layer in a way already
described in section 32 Thus each layer has the same number of
nodal points situated on the same places in the horizontal plane
but with different z-coordinates
We are free to construct quadrilaterals or triangles because
the program automatically divides a quadrilateral into two triangles
having identical material properties The dimensions of the elements
20
should be small at places where large changes in hydrological
properties occur or where large gradients in hydraulic head are
expected to occur
Coding of nodal points
Each node in the finite element network is assigned an integer
code which indicates the type of calculation for this node
The coding is as follows
- Code = 1 nodal points with described head as function of time
- Code 2 internal nodal points
- Code = 3 internal nodal points which represent secondary surface
water system
- Code = 5 nodal points with prescribed artificial recharge or
discharge and points with prescribed boundary flow
(both as functions of time)
Other important endogenous variables are the magnitude of the
time step the value of the variable which stops the iteration
process and the maximum number of iterations allowed per time step
(see also group A and B of section 413)
412 Exogenous variables
After constructing the nodal grid each node represents part of
the groundwater basin Geo-hydrological properties can be assigned
to each mode based on physical data (group C D and E of section
413) In group F of section 413 time-dependent variables can
be assigned to each node
413 Data input to Program FEMSAT
The data input must be punched on cards according to the
following instructions
GROUP COLUMNS FORMAT SYMBOL DESCRIPTION
A 1-80 20A4 HEAD Desired heading to be printed at
the beginning of the execution
1-5 15 NUMNP number of nodal points
21
GROUP COLUMNS FORMAT SYMBOL DESCRIPTION
6-10 15
11-15 15
16-20 15
21-25 15
26-30 15
31-35 15
NUMEL
NUMLAY
MBAND
NTSPI
NTSPO
MAXIT
1-10 F103 ST
11-20 F103 DELT
21-30 F103 TMAX
31-40 F105 TOL
number of elements
number of layers
maximum difference in node number
between two adjacent nodes
number of time steps between a
change in non-stationary input data
number of time steps between two
successive outputs
maximum number of iterations per
time step
time to start with the execution
initial time interval
maximum time to be reached before
execution is stopped
maximum allowed absolute change in
the values of h between any two
successive iterations in a given
time step
Iterations continue until all changes
are less than or equal to TOL or
until MAXIT is exceeded
1- 5 15
6-10 15
11-15 15
16-20 15
KX(N1)
KX(N2)
KX(N3)
KX(N4)
sequential number of i-th corner of
element N
sequential number of j-th corner
sequential number of k-th corner
sequential number of 1-th corner
(see fig 8)
Fig 8 Numbering of element corner nodes
22
GROUP COLUMNS FORMAcircT SYMBOL DESCRIPTION
One card must be provided for each element in sequential order
starting with N = 1 and ending with N = NVJMEL
I-K 1103 SANG(NLY)
1i-iO MO3 C1(NLY)
21-30 F 103 C2(NLY)
21-4- 1-104 THI(NLY)
angle in degrees between CI and
the laquo-coordinate to he used in
nodal print N in layer LY (see
poundig 9)
First principal conductivity of
material in node N and layer LY
idem second principal conductivity
thickness of layer LY at nodal
point N
ig V Definition of SANGCI and C2
One card icircuiPt be provided for each model point ctarting wih M = 1
and ending with N = NUMNP If there is norlt3 than 1 layer for each
layer a set must be provided starting with the upper layer LY - 1
and ending with the bottom layer LY = NIJMLAY The variables SANG
M and 2 are not relevant tor aquitards but for reasons of
conveniency this procedure is maintained (making SANG CI and C2
dummy variables)
1-10 110 KODE(N)
11-20 II0 NTERR(N)
coding of nodal points (see
paragraph 42)
number of tertiary relation in node N
The q-h relationship of the flow to
23
GROUP COLUMNS FORMAT SYMBOL DESCRIPTION
21-30 FlOl X(N)
31-40 FlOl Y(N)
41-50 FlOl HGL(N)
51-60 FlOl HBSC(N)
61-70 FlOl LESC(N)
the tertiary system as discussed
in par 24 are schematized in a small
number of different relations
x-coardinate of node N
y-coordinate of node N
height ground level in node N
height bottom secondary conduit in
node N If node N is not a secondary
conduit this variable is a dummy
variable
length of secondary conduit in area
represented by node N
One card must be provided for each node starting with N = 1 and
ending with N = NUMNP
E 1-10 FlOl P(NLY1) initial value of h at node N in
layer LY
11-20 FlOl VERRES(NLY) vertical resistance at node N and
layer LY If the layer is an aquifer
this value is a dummy variable
21-30 F101 RADRES(NLY) radial resistance for flow to
secondary system at node N in layer
LY
31-40 FlOl ENRES(NLY) entrance resistance for flow to
secondary system at node N in layer
LY
41-50 FlOl WP(NLY) wet perimeter of secondary system at
node N in layer LY
51-60 FlOl P0R(NLY) specific yield of the material at
node N in layer LY
61-70 FlOl SS(NLY) specific storage of the material
at node N in layer LY
24
GROUP COLUMNS FORMAT SYMBOL DESCRIPTION
One card must be provided for each node starting with N = 1 and
ending with N = NUMNP If there is more than 1 layer for each layer
a set of cards must be provided starting with LY = 1 and ending
with LY = NUMLAY
F 1-10 F101 P(NLY1) value of h at different times at
node N in layer LY Node N lies
on a prescribed head boundary
11-20 F101 WTSC(N) water table in secondary system at
node N at different times
21-30 F 101 ARRED(NLY) artificial recharge to or discharge
from node N in layer LY or
prescribed boundary flow at
different times
31-40 F 101 QPHR(N) flux through phreatic surface at
node N at different times
The data in group F are non-stationary input data One card must be
provided for each node starting with N = 1 and ending with N = NUMNP
In case of more than 1 layer for each layer a set must be provided
Not relevant variables are taken as dummy variables
After a certain number of time steps (NTSPI) the data of group E
are read So the set cards E must contain NxLYxC cards where C =
(TMAXNTSPIXDELT) + 1 if DELT is a constant
42 R u n n i n g t h e p r o g r a m o n t h e C y b e r 7200
The program has been executed on a Cyber 7200 computer of
IWIS-TNO in the Hague
- The demanded input is punched on cards according to the instructions
of the last paragraph and is put on a permanent file
- For each specific problem inthe program only the dimension statements
in the main program have to be adjusted according to the instructions
in the lines of comment
- After compilation the program together with the input file is
executed
25
- Intermediate data necessary for continuation of the calculations
are saved on another permanent file This gives the opportunity
to restart the program at a time the calculation was stopped
The only change in the input file is to adjust the starting time
Remark
With the aid of a terminal small changes in the program and in the
input file can be made easily and the output immediately appears
on the terminal In this way it is possible to stop erroneous
calculations This is especially useful when being in the testing
phase of the program
43 O u t p u t
Immediately after reading the input data they are printed so
that possible errors can be detected During execution after each
certain number of time steps (NTSPO) calculated values of h and
water balance terms are printed In appendix B an example is given
The output data can be devided into two groups
A Data per node per layer
- Coding of the nodal points KODE
- Hydraulic head HEAD
- Flow through phreatic surface at the time indicated in head
of table FLOWUN
- Flow to tertiary surface water system FLOWTS
- Flow to secondary surface water system FLOWSS
- Flow to adjacent layers LEAKAGE
- Artificial recharge or discharge and prescribed boundary
flow ARRED
- Change in storage during last time step STORAGE
- Flow through prescribed head boundary LATFLOW
B Data per layer
Summation of the terms already mentioned under A gives the
magnitude of the water balance terms per layer at the time
indicated (instanteneous water balance terms) Summation of
these instanteneous terms multiplied by their respective time
intervals since time is zero gives the cumulative water balance
terms per layer
26
After each print out we are able to check the calculations
because the sum of the water balance terms per layer must be
approximately zero
44 C o m p u t i n g t i m e
The computing time depends on the number of nodal points and
the number of layers But it depends also on the magnitude of the
changes in hydraulic head and on the value which terminates the
Gauss-Seidel iteration (TOL) To give an impression TOL = 001 m
Costs per day simulation (in hlfl) on the Cyber 7200
number of nodes number of layer bull bdquo bull bull _ 1mdash x a x b (21)
where
a = costs in Dutch guilders per system second (ƒ 027)
b = empirical factor with magnitude about 001
V NUMERICAL EXPERIMENTS
In general there are two possibilities to test the validity of
numerical models The first and most commonly used method is to
compare numerical results with analytical solutions The second
possibility is to compare numerical results with measured field
data Since the last alternative requires many data that are
difficult to obtain we restrict ourselves for the time being to
comparison with the analytical solutions
5 1 F l o w t o a w e l l i n a n u n c o n f i n e d
c i r c u l a r - s h a p e d a q u i f e r
Flow to a well can be divided into steady-state flow and
unsteady-state flow (KRUSEMAN and DE RIDDER 1976)
27
511 Steady state flow in an unconfined aquifer
For this case the formula of Thiem-Dupuit is valid (see fig 10)
Q bull k lth2-hicircgt ln(r2r])
(22)
where
hhbdquo = hydraulic head in 1 and 2 respectively
Q = discharge from well
rbdquo = distance from well
CL31
hmdash^ri --r2
Fig 10 Schematic cross-section of a pumped unconfined aquifer
Fig 11 gives the nodal grid superimposed on a well in a
circular shaped aquifer Near the well the nodal point distance
is made smaller Quadrilaterals as well as triangles are used
Numerical calculations with this finite element network and 3 -1
an extraction rate of 200 m day were performed for 10 days
At that moment a steady-state situation was almost achieved
28
2tmdashnodal point number
160 180 X-axesim)
Fig 11 Nodal point configuration for flow to a well in a circular-
shaped aquifer The outer boundary is taken to be constant
The well is situated in nodal point 42
29
In fig 12 the results of the simulation are compared with the
analytical solution The agreement between numerical results and
the analytical solution is excellent
draw- 15irS (m) down
analytical solution imdashx calculated values
distance from well r (m)
Fig 12 Comparison between analyt ical and calculated s teady-state
drawdown for the problem of f ig 11 with 1
-1 Q bull - 200 m 3 d a y _ 1
k = 10 mday
d = 100 m
S = 2 y
S = 0 s
512 Unsteady-state flow in an unconfined aquifer
Analytical solutions for unsteady-state flow in an unconfined
aquifer only exist if the aquifer extends infinitely For the
aquifer given in fig 11 this condition is not fulfilled However
during the first one or two days after starting the extraction the
influence of a constant hydraulic head at the outer boundary on the
30
drawdown near the well can be neglected (see fig 13) After one
day the lateral inflow is only 25 of the extraction the remainder
being the change in storage due to drawdown
2 0 0
m 3 day 1
Q=200m 3 day
100-
vraquomdash Lateral inflow
N amp storage
I I ^ V - W 10
t (days)
Fig 13 Values of Q lateral inflow and change in storage as
functions of time
The nonsteady-state or Theis equation can be written as
i _ Q s 4iTkD
dy = 4irkD
W(u) (23)
where
u = r2S
4irkDt
r = distance from well
S = storage coefficient
Q = constant discharge from well
D = thickness of the waterbearing layer
at the beginning of pumping
t = time after starting the extraction
s = corrected drawdown
= s - 322D
s = calculated drawdown
W(u) = Theiss well function
[J
CIAT1
[T]
]
31
In fig 14 the analytical unsteady-state solution for the case
described in section 511 is given together with the numerical
results at times 05 and 10 days The agreement between numerical
results and the analytical solution is rather good
Drawdown s1(m)
10
10
10
Theiss solution
x caculated values t=Q5
bdquo t= lO
10 10- 10 10 t r2(daym2)
Fig 14 Comparison between analytical and calculated steady-state
drawdown for the problem of fig 11 with
Q
k
d
S y
s s
=
=
=
=
=
200
1
10
3 m
0
0
2
0
-1 day
m day
m
These results indicate that the assumption of a linear variation
of h between two nodal points (see fig 3 ) even with the rather
rough element network of fig 11 is acceptable So there is no
need to improve for example the finite element network by introducing
higher order coordinate functions
32
52 F l o w t o w e l l i n 3-1 a y e r e d s y s t e m
In fig 15 a schematic cross-section of a 3-layered system of
two aquifers separated by an aquitard is given A well of
infinitisemal radius is completed in the lower aquifer and discharges
at a constant rate QTT Each layer is homogeneous isotropic and
extending radially infinite
bdquo bull - bull bull ~ i
mdash Q II
aquifer I
lugrave^i~IcircIuml7i-i~~ aquitard --r-rl~-
-II--I i - i
aquifer
Fig 15 Schematic cross-section of a 3-layered system
For this situation NEUMAN and WITHERSPOON (1969a) give rather
complicated analytical solutions In the special case where the
hydraulic properties of the two aquifers are identical they give
graphical illustrations of their solutions for three different
cases
These graphical solutions give the opportunity to verify
roughly numerical results of a 3-layered system Therefore a nodal
grid is superimposed on a circular shaped groundwater flow system
with the extraction in the centre of the lower aquifer and with
a constant head on the outer boundary Fig 16 gives the nodal
point situation in the x - y plane of 1 layer The nodes in the
other two layers are situated in the middle of the layers at the
same places in the x - y plane only their z-coordinate is different
33
A
-Jy
bull33
V 5
iumlff
Fig 16 Nodal point configuration for the 3-layered problem (top view)
34
In fig 17 numerical results are compared with the analytical
solution The agreement is rather bad especially for the two
aquifers But in terms of the water balance the difference between
aquifers and aquitard is not so big because the aquitard has
about 16 times bigger storage capacity than the aquifers The
small value of the specific storage of the aquifers is the reason
that small errors in the components of the water balance have a
big influence on the calculated values of the hydraulic head
Eig 17 Dimensionless drawdown vs dimensionless time in a
3-layered system with
II = 50 m day
3 m (distance from well)
aquifer I
aquitard
aquifer II
Thi (m)
100
25
100
S s
001
0642
001
k(mday )
1
0277
1
The discrepancies are partly caused by errors in discretization
In the z-direction these errors can be eliminated by taking the
35
vertical resistance of the aquitard to be infinite Then the
calculated drawdown in the pumped aquifer can be compared with
the analytical Theiss solution
This is done in fig 18 for different distances from the well
One sees that the agreement is better when the distance is bigger
Apparently the linearization of the logarithmic drawdown curve
near the well is responsible for the differences The differences
for r = 3 m are of the same magnitude as for the pumped aquifer in
the 3-layered case
joo gt A 50
20
Theiss solution
bull calculated values r=3m (distance f rom well) x bdquo bdquo r=7m
1 L
I l_ 001 003 0 0 5 Ol Q2
- I I 5 10 t - K t
Fig 18 Dimensionless drawdown vs dimensionless time in a
confined aquifer with
Q =500 m day
k = 10 mday
D = 100 m
S = 001 s
The differences between numerical results and analytical solutions
are not caused by taking a finite area because changing the outer
boundary into an impermeable boundary has only minor effects
36
VI SUMMARY AND CONCLUSIONS
In this report a numerical model for non-stationary saturated
groundwater flow in a multi-layered system has been treated
Groundwater flow can be described in different ways
a Physical-mathematical by combining Darcys law and the law of
continuity (section 21)
b Geo-hydrological the flow in a natural groundwater basin takes
place in layers with different hydraulic properties Definitions
of the most important hydraulic properties are given in section
22
c Schematical groundwater flows horizontally in aquifers and
vertically in aquitards (section 23)
d Approximating by dividing the region of flow into a finite
number of elements and applying Darcys law and the law of
continuity on each of these elements In these ways one gets
a set of equations which can be solved given (initial and)
boundary conditions (section 25)
e Numerical the set of equations has been solved using a Galerkin
type finite element method and the implicit Gauss-Seidel method
with overrelaxation factor (section 31 and 32)
f Computer-solvable the numerical solution has been translated
in a FORTRAN IV program called FEMSAT (section 33)
In chapter IV the input execution and output of the program
FEMSAT are treated
Finally in chapter V some numerical applications are given
simulation of flow to a well in unconfined aquifer and simulation
of flow to a well in a 3-layered configuration
Comparison of hydraulic heads obtained by numerical calculations
and analytical true solutions showed an excellent agreement in case
of stationary flow and a good to moderate agreement in case of non-
stationary flow However the diviations caused by the discretization
process must be seen in relation to the errors made in the
establishment of the hydraulic properties
37
As was stated in the introduction numerical models can be
valuable tools in the management of water resources Therefore
numerical models must be suitable to simulate effects of certain
operations upon the spreading of water over saturated groundwater
unsaturated groundwater and surface water The model described in
this report is a model for saturated groundwater flow The relation
with the surface water can be taken into account via radial and
entrance resistances or via a relation between flow to the surface
water system and the hydraulic head The relation with the unsaturated
zone has been restricted to the input factor effective precipitation
In practise however the unsaturated and saturated system are
interconnected Therefore the model should be improved by taking
into account the unsaturated zone This can be done in at least
two ways
a Replacing the unsaturated zone by a parametrical model
b Coupling the unsaturated zone in an iterative way with the
saturated system
In order to find out the usefulness of the numerical model for
water resources management it will be applied to actual regional
groundwater systemslaquoAlso sensitivity analyses will be carried out
The computer model itself can be improved by applying standard
programs eg automatical generation of a nodal grid and graphical
plotting of the results
38
VII REFERENCES
ERNST LF 1962 Grondwaterstromingen in de verzadigde zone en hun
berekening bij aanwezigheid van horizontale evenwijdige open
leidingen Thesis University Utrecht pp 189
1978 Drainage of undulating soils with high groundwater
tables (in press)
FORSYTHE GE and WR WASON 1960 Finite Difference Methods for
Partial Differential Equations John Wiley and Sons New York
pp 444
KRUSEMAN GP and NA DE RIDDER 1976 Analysis and evaluation of
pumping test data ILRI-bulletin No 11 pp 200
LAAT PJM DE and C VAN DEN AKKER 1976 Simulatiemodel voor grond-
waterstroming en verdamping in Modelonderzoek 1971-1974 ten
behoeve van de waterhuishouding in Gelderland deel II
Grondslagen pp 145-197
NEUMAN SP RA FEDDES and E BRESLER 1974 Finite element
simulation of flow in saturated -unsaturated soils considering
water uptake by plants Hydraulic Engineering Laboratory
Technion Israel pp 104
and PA WITHERSPOON 1969a Theory of Flow in a confined
two Aquifer System Water Resources Research Vol 5 No 4
pp 803-816
^_ and PA WITHERSPOON 1969b Applicability of Current Theories
of Flow in leaky aquifers Water Resources Research Vol 5
No 4 pp 817-829
PINDER GF and EO FRIND 1972 Application of Galerkins Procedure
to Aquifer Analysis Water Resources Research Vol 8 no 1
pp 108-120
REMSON I GE HORNBERGER and FJ MOLZ 1971 Numerical Methods
in Subsurface Hydrology John Wiley and Sons New York
pp 389
WESSELING JG 1977 Toepassing van Finite Difference- en Finite
Element Methoden voor het oplossen van een aantal grondwatershy
stromingsproblemen (met computerprogrammas) ICW-nota 944
pp 63
39
ZIENKIEWICZ OC 1971 Finite Element Method in Engineering Science
McGraw-Hill London pp 521
40
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APPENDIX 8 EXAMPLE OF OUTPUT
HEAD AND TERMS OF UATERBALANCE PER NODE PER LAYER
AT TIME 260 IN II AND I13DAYS
T
LAYER NODE KODE HEAD FLOUUN FLOUTS FLOUSS LEAKAGE STORAGE LAT FLOU
13 19 25 31 37 43
125008 125000 12S000 12S000 12S000 12S000 12S000 12S000
0000 0000 0000 0000 0000 0000 0000 0000
0000 0000 0000 0000 0000 0000 0000 0000
0000 0000 0000 0000 0000 0000 0000 0000
- 000 - 000 -000 - 000 - 000 - 000 - 000 0000
0000 0000 0000 0000 0000 0000 0000 0000
0000 000 000 000 000 000 000 000
000 0000 0000 0P100 0000 0000 0000 0000
1 7
13 19 25 31 37
12S000 125000 125000 125000 125000 12S000 125000 125000
0000 0000 0000 0000 0000 0000 0000 0000
- 001 -007 - 023 - 020 - 009 - 003 0000
0000 0000 0000 0000 0000 0000 0000 0000
000
001
007
023
020
009
003
003
0000 0000 0000 0000 0000 0000 0000 0000
1 7
13 19 25 31 37 43
125000 125000 124998 124972 124895 124787 124647 124358
0000 0000 0000 0000 0000 0000 0000 0000
- 000 001 007 023 020 009 003
0000
0000 0000 0000 0000 0000 0000 0000
50000
0000 182
1982 3862 1674 419 078 041
0000 0000 0000 0000 0000 0000 0000 0000
TERMS OF WATERBALANCE PER LAYER DURING LAST TIMESTEP AND
AND SINCE BEGINNING IN M3DAY M3
FLOU TO UNSATURATED ZONE
FLOU TO TERTIARY SYSTEM
0000
0000
0000
0000
LAYER NO 1
FLOU TO SECONDARY SYSTEM
FLOU TO AD3ACENT LAYERCS)
ARTIFICIAL FLOU AND PRESCRIBED BOUNDARY FLOU
ACCUMULATION TERMS
LATERAL FLOU THROUGH PRESCR HEAD BOUNDARY
0000
-000
0000
000
000
0000
-000
0000
000
000
LAYER NO 2
FLOU TO SECONDARY SYSTEM
FLOU TO AD3ACENT LAYERCS1
ARTIFICIAL FLOU AND PRESCRIBED BOUNDARY FLOU
ACCUMULATION TERMS
LATERAL FLOU THROUGH PRESCR HEAD BOUNDARY
0000
-378
0000
381
0000
0000
-055
0000
056
0000
LAYER NO 3
FLOU TO SECONDARY SYSTEM
FLOU TO AD3ACENT LAYERCS5
ARTIFICIAL FLOU AND PRESCRIBED BOUNDARY FLOU
ACCUMULATION TERMS
LATERAL FLOU THROUGH PRESCR HEAD BOUNDARY
0000
370
50000
49376
0000
0000
055
12979
12909
0000
32 A p p l i c a t i o n o f t h e f i n i t e e l e m e n t
m e t h o d o n t h e f l o w i n g r o u n d w a t e r
b a s i n s
From hydrogeological investigations the geometry and the matrix
of the hydraulic properties are known Knowing the geometry the
flow can be split up into a finite number of layers This holds
also for the nature of the boundaries
Now a nodal grid is superimposed on each layer in such a way
that the nodal points of the different layers are situated right
above each other (see fig 4) and in each layer the nodal point
lies in the middle of this layer
-nodal point
Fig 4 Schematic representation of the nodal points in the
xyz-space
Each layer has N nodal points of which P nodal points have a
prescribed head In each layer remain N-P = N nodal points with
unknown head Eq (12) can be applied to each point with unknown
head in each separate layer
As was stated in the last paragraph for each time step one
can write
13
Ah bdquo ~ -r1 + Q = O in = 12N in n m At yi A h + B (13)
where
A = in
ocircf ocircf ocircf of (k d f mdashri- ~ + k df - ~ -jS) dxdy
J x n ox ox y n oy oy
in (S d + S ) ff dx dy O if i icirc n
s y i n
Q f dx dy
The matrices A B and the vector Q have been defined in in l
elsewhere (NEUMAN etal 1974) A is the conductivity matrix in
B the storage matrix and Q the source vector in l
In order to clarify these rather complicate expressions apply
equation (13) poundo point 4 of fig 5
6 7
Fig 5 Top view of part of the nodal grid
lhl + A42h2 + A43h3 + A44h4 + A45h5 + A46h6 + A47h7 +
h - h
+ B _iuumli i ^ + Q = o 44 t - t x4
n n-1
(14)
where h and h are values of h at time t and t 4n 4n-1 4 n n-1
respectively
14
If the h-values belonging to the A-values are taken at time
t = t we get an equation with one unknown h for time t = t n-1 4n n
The equation can then be solved directly
However this is an explicit formulation which puts strong limitations
on the magnitude of the time step t - t (REMSON etal 1971) n n-1
Values larger than a certain value cause instability With an implicit method the values of h are taken at time
t = t This method is unconditionally stable Now one gets an equation n
with 7 unknowns h through h If one has 7 equations for every 1 n n
point 1 through 7 we can solve the values h i = 17
in
Of all the methods possible the iterative method of Gauss-Seidel
has been chosen The Gauss-Seidel method starts by choosing a first estimate of h at time t = t (usually equal to h ) At each in n j -i in-l iteration step a new value of h is calculated using the last
in calculated values of h If one passes the nodal points from 1 to
i n 7 eq (14) becomes
h-j + 1 = A h^ + 1 + A bdquo h | + 1 + A _ h | + 1 + A h j + A r h j + 4 n 4 1 1 n 4 2 2 n 43 3 n 45 5 n 4 6 6 n
B h B + A h J + - 4 4 4 n ~ + 0 J (A + mdash) A 4 7 h 7 n At n U 4 4 At
(15)
where
h-i = in
j = iteration step or number of iteration
i = number of nodal point n = time = t
n
The iteration process converges if the successive differences k+1 k
between h and h become continuously smaller in in
The iteration process is stopped if at each nodal point the k+1 k
difference between h and h is smaller than a certain prescribed in in
value
15
To accelerate the convergence an overrelaxation factor W is used
(for more information about W see FORSYTHE and WASOW 1960)
k+1 k T7 k+1 k h = h + W (h - h )
in in in in (16)
If W = 1 there is no overrelaxation Usually W varies between 1 and
2 and the optimum value very often is found by trial and error
Eq (15) is valid for points in aquifers The connection with
adjacent layers is made by the term Q because in this term the
flow from adjacent aquitards q is included (two aquifers are always
separated by an aquitard)
Aquitards
We dont apply the finite element method on points in aquitards
As stated before there is no horizontal flow in aquitards So it has
no sense to replace the continuous variation of h in the aquitard
by equation (11)
The nodal points in aquitards represent a certain volume of an
aquitard hich transmit water from one aquifer to another and which
has storage capacity The nodal points are included in the Gauss-
Seidel scheme -s follows (see fig 6 )
1 aauifer
a -_vS laquo 2 [~^-t^yM^^^-^r~ aqutarci l-Z
aquifer
Fig 6 Schematic situation for points in aquitards
Point 2 is situated in the middle of the aquitard with thickness
d storage coefficient S and vertical hvdraulic conductivity k c J v
Suppose the Gauss-Seidel iteration starts in the upper layer and
ends in the bottom-layer Then the flux from point 1 in aquifer 1 to
16
point 2 in the aquitard at time t = t and in iteration step j + 1
is
k q = (hf -hf )-iumlmdash f (17)
1 n 2 n OSxd1
and the flux from point 3 in aquifer 2 to point 2 in the aquitard
is
1 c 1
qbdquo = (h - hf) mdashHmdashr- (18) 2 3 n 2 n OSxd
The net flux ltl-q9 must be equal to the change in storage per unit
area per unit time This change in storage can numerically be
approximated by
te - h-
n n-1
Combina t ion of ( 1 7 ) (18) ana (19) g i v e s
l tr1 - h (hJ + 1 + hj - 2 hj+1) - - - - - - S1 - ilaquoS L u l l = o (20)
] i 3 n 2 n 1 c t - t j j x d n n -1
T + 1
From t h i s e q u a t i o n h^ kan be s o l v e d
3 3 C o m p u t e r o g r a m
The problem outl^ugrave so far has been translated into a computer
program written in FORTRAN IV
The flow chart ot Uis program is given in fig 7 whereas in
appendix A the complete listing of the program can be found
In the flow chart the 5 main parts of the program are
indicated
A Reading and printing of input data
B Determination of the matrices A and B and the vector Q per layer
17
M
1 A
FLOH to unsaturated zono
FLOW to tertiary syster
FLOW to secondnrv system
Leakage to adjacent aquitards
SOLVE equation (14)
yes
CALCULATE flux through constant head boundnr
CALCULATE waterbalance n p r 1 OAro r r J j - ~
PRINT of output
Ier node per layer
k = k+1
t = t + At
7 DUMP variables on tape
-gt-
( STOP )
Fig 7 Flow chart of the program FEMSAT with diviation in 5 main
parts A B C D and E
C START PROGRAM FEMSAT
f READ general information
PRINT general information
f READ element information
PRINT element information 7 READ stationary nodal point information
PRINT stationary nodal point information 7
READ non-stationary nodal point information for t = t
FRINT non-stationary nodal point information for t = t
L 2 (
^ yes R E A D non-stat input for t gt t
J PRINT non-stat input for t gt t (- 1 -- M degY
GENERATE A B and AREA-matrix for confined aquifers
_L
GENERATE A B and AREA-matrix for unconfined aouifer
bull
C Solving h with Gauss-Seidel
D Determination of the water balance terms per layer
E Printing of results
The program exists of a main program with several subroutines
The main program calls the subroutines In the subroutines most
calculations are performed
In this way an easy adaptation to different problems is
obtained (see also section 44) Comments in the program will
help the reader to understand the program
In the next chapter the preparation of the input data executing
of the program and the output will be treated
Acknowledgement
Some parts of the computer program were taken from the program
UNSAT2 of NEUMAN etal (1974)
IV INPUT EXECUTION AND OUTPUT
41 I n p u t
The input is divided into endogenous variables (chosen by the
user or calculated by the program during execution) and exogenous
variables (based on physical data)
411 Endogenous variables
Translation of groundwater basin into a nodal point network
Probably the most important but also the most difficult stage
in the process of simulating groundwater flow is the translation
of the natural groundwater basin into a nodal point network Questions
which arise are
a In how many layer the geo-hydrological system mustcan be
schematized
b The locations of the boundaries and the nature of these boundaries
(impervious prescribed head prescribed flux)
c The configuration of the nodal points
19
The answers of these questions depend on the problem under
consideration
- Schematization into layers
Usually our knowledge about the geo-hydrological situation is
such that no more than 5 different layers can be distinguished
(remember contrast rule)
- Boundary conditions
The top boundary condition of a groundwater basin usually is
a prescribed flux (effective precipitation)
The bottom boundary condition usually is a prescribed flux
boundary with flux zero (impervious hydrological base)
For the boundaries at the other sides of the system one sees
either prescribed head boundaries (eg a fully penetrating
river) or prescribed flux boundaries (including zero flux)
- Nodal point configuration
Nodal points are situated
on the boundaries
along internal boundaries which indicate changes in
hydraulic properties within the layer
along the tertiary surface water system
on places where artificial recharge or discharge wiii tgtke
place
elsewhere
Construction of the finite element mesh
Assume we have schemetized the groundwater basin iatu a finite
number of layers with different direction of flow (horizontal and
vertical) Within one layer the hydrological properties may vary
from place to place
Now a nodal grid is superimposed on each layer in a way already
described in section 32 Thus each layer has the same number of
nodal points situated on the same places in the horizontal plane
but with different z-coordinates
We are free to construct quadrilaterals or triangles because
the program automatically divides a quadrilateral into two triangles
having identical material properties The dimensions of the elements
20
should be small at places where large changes in hydrological
properties occur or where large gradients in hydraulic head are
expected to occur
Coding of nodal points
Each node in the finite element network is assigned an integer
code which indicates the type of calculation for this node
The coding is as follows
- Code = 1 nodal points with described head as function of time
- Code 2 internal nodal points
- Code = 3 internal nodal points which represent secondary surface
water system
- Code = 5 nodal points with prescribed artificial recharge or
discharge and points with prescribed boundary flow
(both as functions of time)
Other important endogenous variables are the magnitude of the
time step the value of the variable which stops the iteration
process and the maximum number of iterations allowed per time step
(see also group A and B of section 413)
412 Exogenous variables
After constructing the nodal grid each node represents part of
the groundwater basin Geo-hydrological properties can be assigned
to each mode based on physical data (group C D and E of section
413) In group F of section 413 time-dependent variables can
be assigned to each node
413 Data input to Program FEMSAT
The data input must be punched on cards according to the
following instructions
GROUP COLUMNS FORMAT SYMBOL DESCRIPTION
A 1-80 20A4 HEAD Desired heading to be printed at
the beginning of the execution
1-5 15 NUMNP number of nodal points
21
GROUP COLUMNS FORMAT SYMBOL DESCRIPTION
6-10 15
11-15 15
16-20 15
21-25 15
26-30 15
31-35 15
NUMEL
NUMLAY
MBAND
NTSPI
NTSPO
MAXIT
1-10 F103 ST
11-20 F103 DELT
21-30 F103 TMAX
31-40 F105 TOL
number of elements
number of layers
maximum difference in node number
between two adjacent nodes
number of time steps between a
change in non-stationary input data
number of time steps between two
successive outputs
maximum number of iterations per
time step
time to start with the execution
initial time interval
maximum time to be reached before
execution is stopped
maximum allowed absolute change in
the values of h between any two
successive iterations in a given
time step
Iterations continue until all changes
are less than or equal to TOL or
until MAXIT is exceeded
1- 5 15
6-10 15
11-15 15
16-20 15
KX(N1)
KX(N2)
KX(N3)
KX(N4)
sequential number of i-th corner of
element N
sequential number of j-th corner
sequential number of k-th corner
sequential number of 1-th corner
(see fig 8)
Fig 8 Numbering of element corner nodes
22
GROUP COLUMNS FORMAcircT SYMBOL DESCRIPTION
One card must be provided for each element in sequential order
starting with N = 1 and ending with N = NVJMEL
I-K 1103 SANG(NLY)
1i-iO MO3 C1(NLY)
21-30 F 103 C2(NLY)
21-4- 1-104 THI(NLY)
angle in degrees between CI and
the laquo-coordinate to he used in
nodal print N in layer LY (see
poundig 9)
First principal conductivity of
material in node N and layer LY
idem second principal conductivity
thickness of layer LY at nodal
point N
ig V Definition of SANGCI and C2
One card icircuiPt be provided for each model point ctarting wih M = 1
and ending with N = NUMNP If there is norlt3 than 1 layer for each
layer a set must be provided starting with the upper layer LY - 1
and ending with the bottom layer LY = NIJMLAY The variables SANG
M and 2 are not relevant tor aquitards but for reasons of
conveniency this procedure is maintained (making SANG CI and C2
dummy variables)
1-10 110 KODE(N)
11-20 II0 NTERR(N)
coding of nodal points (see
paragraph 42)
number of tertiary relation in node N
The q-h relationship of the flow to
23
GROUP COLUMNS FORMAT SYMBOL DESCRIPTION
21-30 FlOl X(N)
31-40 FlOl Y(N)
41-50 FlOl HGL(N)
51-60 FlOl HBSC(N)
61-70 FlOl LESC(N)
the tertiary system as discussed
in par 24 are schematized in a small
number of different relations
x-coardinate of node N
y-coordinate of node N
height ground level in node N
height bottom secondary conduit in
node N If node N is not a secondary
conduit this variable is a dummy
variable
length of secondary conduit in area
represented by node N
One card must be provided for each node starting with N = 1 and
ending with N = NUMNP
E 1-10 FlOl P(NLY1) initial value of h at node N in
layer LY
11-20 FlOl VERRES(NLY) vertical resistance at node N and
layer LY If the layer is an aquifer
this value is a dummy variable
21-30 F101 RADRES(NLY) radial resistance for flow to
secondary system at node N in layer
LY
31-40 FlOl ENRES(NLY) entrance resistance for flow to
secondary system at node N in layer
LY
41-50 FlOl WP(NLY) wet perimeter of secondary system at
node N in layer LY
51-60 FlOl P0R(NLY) specific yield of the material at
node N in layer LY
61-70 FlOl SS(NLY) specific storage of the material
at node N in layer LY
24
GROUP COLUMNS FORMAT SYMBOL DESCRIPTION
One card must be provided for each node starting with N = 1 and
ending with N = NUMNP If there is more than 1 layer for each layer
a set of cards must be provided starting with LY = 1 and ending
with LY = NUMLAY
F 1-10 F101 P(NLY1) value of h at different times at
node N in layer LY Node N lies
on a prescribed head boundary
11-20 F101 WTSC(N) water table in secondary system at
node N at different times
21-30 F 101 ARRED(NLY) artificial recharge to or discharge
from node N in layer LY or
prescribed boundary flow at
different times
31-40 F 101 QPHR(N) flux through phreatic surface at
node N at different times
The data in group F are non-stationary input data One card must be
provided for each node starting with N = 1 and ending with N = NUMNP
In case of more than 1 layer for each layer a set must be provided
Not relevant variables are taken as dummy variables
After a certain number of time steps (NTSPI) the data of group E
are read So the set cards E must contain NxLYxC cards where C =
(TMAXNTSPIXDELT) + 1 if DELT is a constant
42 R u n n i n g t h e p r o g r a m o n t h e C y b e r 7200
The program has been executed on a Cyber 7200 computer of
IWIS-TNO in the Hague
- The demanded input is punched on cards according to the instructions
of the last paragraph and is put on a permanent file
- For each specific problem inthe program only the dimension statements
in the main program have to be adjusted according to the instructions
in the lines of comment
- After compilation the program together with the input file is
executed
25
- Intermediate data necessary for continuation of the calculations
are saved on another permanent file This gives the opportunity
to restart the program at a time the calculation was stopped
The only change in the input file is to adjust the starting time
Remark
With the aid of a terminal small changes in the program and in the
input file can be made easily and the output immediately appears
on the terminal In this way it is possible to stop erroneous
calculations This is especially useful when being in the testing
phase of the program
43 O u t p u t
Immediately after reading the input data they are printed so
that possible errors can be detected During execution after each
certain number of time steps (NTSPO) calculated values of h and
water balance terms are printed In appendix B an example is given
The output data can be devided into two groups
A Data per node per layer
- Coding of the nodal points KODE
- Hydraulic head HEAD
- Flow through phreatic surface at the time indicated in head
of table FLOWUN
- Flow to tertiary surface water system FLOWTS
- Flow to secondary surface water system FLOWSS
- Flow to adjacent layers LEAKAGE
- Artificial recharge or discharge and prescribed boundary
flow ARRED
- Change in storage during last time step STORAGE
- Flow through prescribed head boundary LATFLOW
B Data per layer
Summation of the terms already mentioned under A gives the
magnitude of the water balance terms per layer at the time
indicated (instanteneous water balance terms) Summation of
these instanteneous terms multiplied by their respective time
intervals since time is zero gives the cumulative water balance
terms per layer
26
After each print out we are able to check the calculations
because the sum of the water balance terms per layer must be
approximately zero
44 C o m p u t i n g t i m e
The computing time depends on the number of nodal points and
the number of layers But it depends also on the magnitude of the
changes in hydraulic head and on the value which terminates the
Gauss-Seidel iteration (TOL) To give an impression TOL = 001 m
Costs per day simulation (in hlfl) on the Cyber 7200
number of nodes number of layer bull bdquo bull bull _ 1mdash x a x b (21)
where
a = costs in Dutch guilders per system second (ƒ 027)
b = empirical factor with magnitude about 001
V NUMERICAL EXPERIMENTS
In general there are two possibilities to test the validity of
numerical models The first and most commonly used method is to
compare numerical results with analytical solutions The second
possibility is to compare numerical results with measured field
data Since the last alternative requires many data that are
difficult to obtain we restrict ourselves for the time being to
comparison with the analytical solutions
5 1 F l o w t o a w e l l i n a n u n c o n f i n e d
c i r c u l a r - s h a p e d a q u i f e r
Flow to a well can be divided into steady-state flow and
unsteady-state flow (KRUSEMAN and DE RIDDER 1976)
27
511 Steady state flow in an unconfined aquifer
For this case the formula of Thiem-Dupuit is valid (see fig 10)
Q bull k lth2-hicircgt ln(r2r])
(22)
where
hhbdquo = hydraulic head in 1 and 2 respectively
Q = discharge from well
rbdquo = distance from well
CL31
hmdash^ri --r2
Fig 10 Schematic cross-section of a pumped unconfined aquifer
Fig 11 gives the nodal grid superimposed on a well in a
circular shaped aquifer Near the well the nodal point distance
is made smaller Quadrilaterals as well as triangles are used
Numerical calculations with this finite element network and 3 -1
an extraction rate of 200 m day were performed for 10 days
At that moment a steady-state situation was almost achieved
28
2tmdashnodal point number
160 180 X-axesim)
Fig 11 Nodal point configuration for flow to a well in a circular-
shaped aquifer The outer boundary is taken to be constant
The well is situated in nodal point 42
29
In fig 12 the results of the simulation are compared with the
analytical solution The agreement between numerical results and
the analytical solution is excellent
draw- 15irS (m) down
analytical solution imdashx calculated values
distance from well r (m)
Fig 12 Comparison between analyt ical and calculated s teady-state
drawdown for the problem of f ig 11 with 1
-1 Q bull - 200 m 3 d a y _ 1
k = 10 mday
d = 100 m
S = 2 y
S = 0 s
512 Unsteady-state flow in an unconfined aquifer
Analytical solutions for unsteady-state flow in an unconfined
aquifer only exist if the aquifer extends infinitely For the
aquifer given in fig 11 this condition is not fulfilled However
during the first one or two days after starting the extraction the
influence of a constant hydraulic head at the outer boundary on the
30
drawdown near the well can be neglected (see fig 13) After one
day the lateral inflow is only 25 of the extraction the remainder
being the change in storage due to drawdown
2 0 0
m 3 day 1
Q=200m 3 day
100-
vraquomdash Lateral inflow
N amp storage
I I ^ V - W 10
t (days)
Fig 13 Values of Q lateral inflow and change in storage as
functions of time
The nonsteady-state or Theis equation can be written as
i _ Q s 4iTkD
dy = 4irkD
W(u) (23)
where
u = r2S
4irkDt
r = distance from well
S = storage coefficient
Q = constant discharge from well
D = thickness of the waterbearing layer
at the beginning of pumping
t = time after starting the extraction
s = corrected drawdown
= s - 322D
s = calculated drawdown
W(u) = Theiss well function
[J
CIAT1
[T]
]
31
In fig 14 the analytical unsteady-state solution for the case
described in section 511 is given together with the numerical
results at times 05 and 10 days The agreement between numerical
results and the analytical solution is rather good
Drawdown s1(m)
10
10
10
Theiss solution
x caculated values t=Q5
bdquo t= lO
10 10- 10 10 t r2(daym2)
Fig 14 Comparison between analytical and calculated steady-state
drawdown for the problem of fig 11 with
Q
k
d
S y
s s
=
=
=
=
=
200
1
10
3 m
0
0
2
0
-1 day
m day
m
These results indicate that the assumption of a linear variation
of h between two nodal points (see fig 3 ) even with the rather
rough element network of fig 11 is acceptable So there is no
need to improve for example the finite element network by introducing
higher order coordinate functions
32
52 F l o w t o w e l l i n 3-1 a y e r e d s y s t e m
In fig 15 a schematic cross-section of a 3-layered system of
two aquifers separated by an aquitard is given A well of
infinitisemal radius is completed in the lower aquifer and discharges
at a constant rate QTT Each layer is homogeneous isotropic and
extending radially infinite
bdquo bull - bull bull ~ i
mdash Q II
aquifer I
lugrave^i~IcircIuml7i-i~~ aquitard --r-rl~-
-II--I i - i
aquifer
Fig 15 Schematic cross-section of a 3-layered system
For this situation NEUMAN and WITHERSPOON (1969a) give rather
complicated analytical solutions In the special case where the
hydraulic properties of the two aquifers are identical they give
graphical illustrations of their solutions for three different
cases
These graphical solutions give the opportunity to verify
roughly numerical results of a 3-layered system Therefore a nodal
grid is superimposed on a circular shaped groundwater flow system
with the extraction in the centre of the lower aquifer and with
a constant head on the outer boundary Fig 16 gives the nodal
point situation in the x - y plane of 1 layer The nodes in the
other two layers are situated in the middle of the layers at the
same places in the x - y plane only their z-coordinate is different
33
A
-Jy
bull33
V 5
iumlff
Fig 16 Nodal point configuration for the 3-layered problem (top view)
34
In fig 17 numerical results are compared with the analytical
solution The agreement is rather bad especially for the two
aquifers But in terms of the water balance the difference between
aquifers and aquitard is not so big because the aquitard has
about 16 times bigger storage capacity than the aquifers The
small value of the specific storage of the aquifers is the reason
that small errors in the components of the water balance have a
big influence on the calculated values of the hydraulic head
Eig 17 Dimensionless drawdown vs dimensionless time in a
3-layered system with
II = 50 m day
3 m (distance from well)
aquifer I
aquitard
aquifer II
Thi (m)
100
25
100
S s
001
0642
001
k(mday )
1
0277
1
The discrepancies are partly caused by errors in discretization
In the z-direction these errors can be eliminated by taking the
35
vertical resistance of the aquitard to be infinite Then the
calculated drawdown in the pumped aquifer can be compared with
the analytical Theiss solution
This is done in fig 18 for different distances from the well
One sees that the agreement is better when the distance is bigger
Apparently the linearization of the logarithmic drawdown curve
near the well is responsible for the differences The differences
for r = 3 m are of the same magnitude as for the pumped aquifer in
the 3-layered case
joo gt A 50
20
Theiss solution
bull calculated values r=3m (distance f rom well) x bdquo bdquo r=7m
1 L
I l_ 001 003 0 0 5 Ol Q2
- I I 5 10 t - K t
Fig 18 Dimensionless drawdown vs dimensionless time in a
confined aquifer with
Q =500 m day
k = 10 mday
D = 100 m
S = 001 s
The differences between numerical results and analytical solutions
are not caused by taking a finite area because changing the outer
boundary into an impermeable boundary has only minor effects
36
VI SUMMARY AND CONCLUSIONS
In this report a numerical model for non-stationary saturated
groundwater flow in a multi-layered system has been treated
Groundwater flow can be described in different ways
a Physical-mathematical by combining Darcys law and the law of
continuity (section 21)
b Geo-hydrological the flow in a natural groundwater basin takes
place in layers with different hydraulic properties Definitions
of the most important hydraulic properties are given in section
22
c Schematical groundwater flows horizontally in aquifers and
vertically in aquitards (section 23)
d Approximating by dividing the region of flow into a finite
number of elements and applying Darcys law and the law of
continuity on each of these elements In these ways one gets
a set of equations which can be solved given (initial and)
boundary conditions (section 25)
e Numerical the set of equations has been solved using a Galerkin
type finite element method and the implicit Gauss-Seidel method
with overrelaxation factor (section 31 and 32)
f Computer-solvable the numerical solution has been translated
in a FORTRAN IV program called FEMSAT (section 33)
In chapter IV the input execution and output of the program
FEMSAT are treated
Finally in chapter V some numerical applications are given
simulation of flow to a well in unconfined aquifer and simulation
of flow to a well in a 3-layered configuration
Comparison of hydraulic heads obtained by numerical calculations
and analytical true solutions showed an excellent agreement in case
of stationary flow and a good to moderate agreement in case of non-
stationary flow However the diviations caused by the discretization
process must be seen in relation to the errors made in the
establishment of the hydraulic properties
37
As was stated in the introduction numerical models can be
valuable tools in the management of water resources Therefore
numerical models must be suitable to simulate effects of certain
operations upon the spreading of water over saturated groundwater
unsaturated groundwater and surface water The model described in
this report is a model for saturated groundwater flow The relation
with the surface water can be taken into account via radial and
entrance resistances or via a relation between flow to the surface
water system and the hydraulic head The relation with the unsaturated
zone has been restricted to the input factor effective precipitation
In practise however the unsaturated and saturated system are
interconnected Therefore the model should be improved by taking
into account the unsaturated zone This can be done in at least
two ways
a Replacing the unsaturated zone by a parametrical model
b Coupling the unsaturated zone in an iterative way with the
saturated system
In order to find out the usefulness of the numerical model for
water resources management it will be applied to actual regional
groundwater systemslaquoAlso sensitivity analyses will be carried out
The computer model itself can be improved by applying standard
programs eg automatical generation of a nodal grid and graphical
plotting of the results
38
VII REFERENCES
ERNST LF 1962 Grondwaterstromingen in de verzadigde zone en hun
berekening bij aanwezigheid van horizontale evenwijdige open
leidingen Thesis University Utrecht pp 189
1978 Drainage of undulating soils with high groundwater
tables (in press)
FORSYTHE GE and WR WASON 1960 Finite Difference Methods for
Partial Differential Equations John Wiley and Sons New York
pp 444
KRUSEMAN GP and NA DE RIDDER 1976 Analysis and evaluation of
pumping test data ILRI-bulletin No 11 pp 200
LAAT PJM DE and C VAN DEN AKKER 1976 Simulatiemodel voor grond-
waterstroming en verdamping in Modelonderzoek 1971-1974 ten
behoeve van de waterhuishouding in Gelderland deel II
Grondslagen pp 145-197
NEUMAN SP RA FEDDES and E BRESLER 1974 Finite element
simulation of flow in saturated -unsaturated soils considering
water uptake by plants Hydraulic Engineering Laboratory
Technion Israel pp 104
and PA WITHERSPOON 1969a Theory of Flow in a confined
two Aquifer System Water Resources Research Vol 5 No 4
pp 803-816
^_ and PA WITHERSPOON 1969b Applicability of Current Theories
of Flow in leaky aquifers Water Resources Research Vol 5
No 4 pp 817-829
PINDER GF and EO FRIND 1972 Application of Galerkins Procedure
to Aquifer Analysis Water Resources Research Vol 8 no 1
pp 108-120
REMSON I GE HORNBERGER and FJ MOLZ 1971 Numerical Methods
in Subsurface Hydrology John Wiley and Sons New York
pp 389
WESSELING JG 1977 Toepassing van Finite Difference- en Finite
Element Methoden voor het oplossen van een aantal grondwatershy
stromingsproblemen (met computerprogrammas) ICW-nota 944
pp 63
39
ZIENKIEWICZ OC 1971 Finite Element Method in Engineering Science
McGraw-Hill London pp 521
40
z ltc laquo LU U EC LU ltC
c z O E 0C 3 GLZ
X - e - o
gt-o-
LUX
E 2 3 - -Z o
O U K raquo-lt-laquo-
lt-gt z o z
laquoC - L
M II II II II II II
IS - Z w - U U - bull II II bull II - a - laquo - o o bulllaquo- bullbull mdash bullmdashbull Il bullmdash II f M Ol II mdashbullgt -gt -ltmdashgt -
O bulllaquobull gt- Il f - U J U J gt - - ^ U J - gt Œ ) gt gt - raquo J _ l gt - _ J U J Œ l Œ I LU _ l Z bull laquo - 3 3 - 1 laquo - 3 - laquo - bull laquo - _ i _J raquo-J UJ _J w 3 bull
bull O -raquo Z Z lt-gtZgt-gt w ltT ucirc w h Z II II h - Z I S S I raquo-H KH ( raquo gt-i O C D t O L U U I C C 3 ) - i Z t O 0 J M l f l ucirc l - l - K ucirc h ucirc h ( h laquo J 0 C O O h 3 l -
w 3 laquo z z ltrzltt J j o c i - z z J j U l i J O O l i i O O O U I O U i O O U l L 4 I A H O l L l i H K Ucirc Ucirc Iuml U U Ucirc I Iuml U K U Ucirc U U U U U U U U
ltc oc
z
u
o z ltr
UJ O X M K O H E h laquo CC Lu laquo _ l o a i-t z LU - J - y + gt - 0 Li K ^ O + H tn ii CD i -O II Zgt Il _J CO O gt bull- 01 _J II Q Icirc H Ucirc h lt I I -Ugrave Ucirc M H U W
D o s e 1 C K ( 3 laquoO 0 laquo
CD O CD (S CD CD U L O - O t ^ CO CK U O U - -II il H II H H II tf II II II II II II II II M II II II II II II II II II II II
^ reg ^ N f O W ^ O K Œ C ^ C S ^ N n L O O r v C O I gt - O D - laquo - f y P Iuml T L O gt 0 ^ h s K K K K r s K K K K œ œ œ œ œ œ œ œ œ œ c K gt o - C K i gt i gt - o laquo
V lt C -l - U oi-gt o u ( S O X laquo laquo t- CD U Q h-iUJ LU 1 bull lt-gtz
h K Lgt laquo 03 05 Ui CU Il LU OC 1 -
h a c L H a ) w - _ LU U 1 - Lu K H t f l H
CO fy
X raquo H
OC H-ltE E 1
ltX
LU 1 -ltr ce LU z LU O
Cu 3 _J Q_ gt-- Z E Z _l
LU - 3 E -o t n z 3 z O U i - 2 w 5C OC bulllaquobull - O u CD bull c OC II ^ Il 09 4 E Lli gt- H n bullbull CDOC LU Zgt J Z CMUJUumlJ + S LU _ j - D 3 0 3 laquo- h-L U O ) C O C D gt - Z Z I - C D H bull -
l - L O lt - J laquo - t w laquo - t II W II J H T r - H- h- || OC U OC _ - Z Z Z O Iuml LU U Ul laquo i - i O O w O O h - l - M H U X O Ucirc L U U H H Ucirc H
CD CD CD LO CEI H bull LOLO - 0
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I gt - f X - - l 3 gt - H X raquo - I gt - H 3 gt - i bull-i mdashl mdash1 mdash1 mdashi mdash1 laquo | J gt - j gt - j ^ - f c gt 0 4 N ( r o - gt j gt i S i N C D r n Œ c i reg t n s t n reg t n s ^o -x - x - x - x - x - x r
r - w 2 gt uuml l X gt ] gt T i - n T i - r j r -raquo z n n icirc s j r r r r raquo H o n c - i iuml o r o i - lt m c c s w m c n c c n m raquo bull H j ^ - n o xgt ^ 3 3 Tgt lt-r tz mdash w iuml- i - t m H r r - r - r - - lt r r c r - o - lt z
bulllt x _bull s- -lt -lt _ C3 -n _ - i - _ rgt _ bull t n - -i - - M rlt r - c - m n -gt o n o œ n u n n - n ^ c - z - i ^ 3 gt 3 gt - n o r - c a
z o r s n r z u icirc N H O - 3 3 0 3 3 m f - O f J l w i c ^ ^ c m z ^ raquo f x mdash 33 r- 0 - 1 p s | -O bull-raquo 3 bulllt 3gt - - - raquo - lt - lt
3 J M raquo H _ - 3 Z 3 Z t n 3gt H Xraquo- -H S - 1 -H
I 03 f - NI r -fc copy bull 3D -lt t n t n -il X ~i X ~i -raquo - n - n - T U mdash n r z r - - t r c c 0 cn a z 3 c - e - - n a i gt -s - n - - n -c 0 r 0 r
mdashf c - i cn c z m 2 33 en H xgt i - H 3gt C 33 33 -lt X
_ 3
-k
33 s c e s t n
j t i T T i - i i n n A Z O S O C O O X d 33
U) 3gt V Z N O bull
t o
Z n m
00 m 0
z z
z C l
z
raquo J J
- -n bullbull-gt r w 33 J Z 2 1gt n bullbull r n
3 Z 3 3 - 1 -H 1 gt - H 1 gt B H H H
V t n Ni
mdash1 m 33 3 t n
- n
c xgt
m x 55
f j
M zgt -n
O)
_ D t 3 - T) C bulln c 0 3= 0 t o 0 x
3= w gt0 3 raquo-i 3 3 r x raquo i gt r - r 0 3 Z u 3 X 3 Z 33 -lt 33
- 1 - i Z Z m bull 33 lt-raquo -lt t n iuml gt
o ^ r - r - c c o z m z m z CD t o t o F I H
t n t n
- n as - T] S) -raquobull bull œ w raquo - t o ro bull-gt CD X
en - n
es
u bull-bull
bull-1 m c -z bull ^ ^ - z r w --lt - 0 - raquo w yO 1mdash1
3gt Z Xgt n - f n 0 n 1-1 [Tl r-i Z 03 Z
-lt -lt icirc-f
- z o 0 - f-1 bull-- z z
J
0 11
w z -0 c 33 3 w Z Z O H -
t gt bull t o -
CS
r
-lt z
7Z O O
H - t S) 3gt - 1
- 1 r o r - r - lt-raquo r o
h ) ^ f m n f i V n i gt t n
J X I - 3gt z c i r c m i gt 3 ~lt r m bulljgt raquo
-lt rgt z raquo 0 3 3 0 m m 0
3 0 0 m
t o H
^ 00 X
iuml gt - 4
-H
3 m
- n
t n
t o
bullbull0 x
z
3
3gt
C -lt Igt - t Ci
APPENDIX 8 EXAMPLE OF OUTPUT
HEAD AND TERMS OF UATERBALANCE PER NODE PER LAYER
AT TIME 260 IN II AND I13DAYS
T
LAYER NODE KODE HEAD FLOUUN FLOUTS FLOUSS LEAKAGE STORAGE LAT FLOU
13 19 25 31 37 43
125008 125000 12S000 12S000 12S000 12S000 12S000 12S000
0000 0000 0000 0000 0000 0000 0000 0000
0000 0000 0000 0000 0000 0000 0000 0000
0000 0000 0000 0000 0000 0000 0000 0000
- 000 - 000 -000 - 000 - 000 - 000 - 000 0000
0000 0000 0000 0000 0000 0000 0000 0000
0000 000 000 000 000 000 000 000
000 0000 0000 0P100 0000 0000 0000 0000
1 7
13 19 25 31 37
12S000 125000 125000 125000 125000 12S000 125000 125000
0000 0000 0000 0000 0000 0000 0000 0000
- 001 -007 - 023 - 020 - 009 - 003 0000
0000 0000 0000 0000 0000 0000 0000 0000
000
001
007
023
020
009
003
003
0000 0000 0000 0000 0000 0000 0000 0000
1 7
13 19 25 31 37 43
125000 125000 124998 124972 124895 124787 124647 124358
0000 0000 0000 0000 0000 0000 0000 0000
- 000 001 007 023 020 009 003
0000
0000 0000 0000 0000 0000 0000 0000
50000
0000 182
1982 3862 1674 419 078 041
0000 0000 0000 0000 0000 0000 0000 0000
TERMS OF WATERBALANCE PER LAYER DURING LAST TIMESTEP AND
AND SINCE BEGINNING IN M3DAY M3
FLOU TO UNSATURATED ZONE
FLOU TO TERTIARY SYSTEM
0000
0000
0000
0000
LAYER NO 1
FLOU TO SECONDARY SYSTEM
FLOU TO AD3ACENT LAYERCS)
ARTIFICIAL FLOU AND PRESCRIBED BOUNDARY FLOU
ACCUMULATION TERMS
LATERAL FLOU THROUGH PRESCR HEAD BOUNDARY
0000
-000
0000
000
000
0000
-000
0000
000
000
LAYER NO 2
FLOU TO SECONDARY SYSTEM
FLOU TO AD3ACENT LAYERCS1
ARTIFICIAL FLOU AND PRESCRIBED BOUNDARY FLOU
ACCUMULATION TERMS
LATERAL FLOU THROUGH PRESCR HEAD BOUNDARY
0000
-378
0000
381
0000
0000
-055
0000
056
0000
LAYER NO 3
FLOU TO SECONDARY SYSTEM
FLOU TO AD3ACENT LAYERCS5
ARTIFICIAL FLOU AND PRESCRIBED BOUNDARY FLOU
ACCUMULATION TERMS
LATERAL FLOU THROUGH PRESCR HEAD BOUNDARY
0000
370
50000
49376
0000
0000
055
12979
12909
0000
Ah bdquo ~ -r1 + Q = O in = 12N in n m At yi A h + B (13)
where
A = in
ocircf ocircf ocircf of (k d f mdashri- ~ + k df - ~ -jS) dxdy
J x n ox ox y n oy oy
in (S d + S ) ff dx dy O if i icirc n
s y i n
Q f dx dy
The matrices A B and the vector Q have been defined in in l
elsewhere (NEUMAN etal 1974) A is the conductivity matrix in
B the storage matrix and Q the source vector in l
In order to clarify these rather complicate expressions apply
equation (13) poundo point 4 of fig 5
6 7
Fig 5 Top view of part of the nodal grid
lhl + A42h2 + A43h3 + A44h4 + A45h5 + A46h6 + A47h7 +
h - h
+ B _iuumli i ^ + Q = o 44 t - t x4
n n-1
(14)
where h and h are values of h at time t and t 4n 4n-1 4 n n-1
respectively
14
If the h-values belonging to the A-values are taken at time
t = t we get an equation with one unknown h for time t = t n-1 4n n
The equation can then be solved directly
However this is an explicit formulation which puts strong limitations
on the magnitude of the time step t - t (REMSON etal 1971) n n-1
Values larger than a certain value cause instability With an implicit method the values of h are taken at time
t = t This method is unconditionally stable Now one gets an equation n
with 7 unknowns h through h If one has 7 equations for every 1 n n
point 1 through 7 we can solve the values h i = 17
in
Of all the methods possible the iterative method of Gauss-Seidel
has been chosen The Gauss-Seidel method starts by choosing a first estimate of h at time t = t (usually equal to h ) At each in n j -i in-l iteration step a new value of h is calculated using the last
in calculated values of h If one passes the nodal points from 1 to
i n 7 eq (14) becomes
h-j + 1 = A h^ + 1 + A bdquo h | + 1 + A _ h | + 1 + A h j + A r h j + 4 n 4 1 1 n 4 2 2 n 43 3 n 45 5 n 4 6 6 n
B h B + A h J + - 4 4 4 n ~ + 0 J (A + mdash) A 4 7 h 7 n At n U 4 4 At
(15)
where
h-i = in
j = iteration step or number of iteration
i = number of nodal point n = time = t
n
The iteration process converges if the successive differences k+1 k
between h and h become continuously smaller in in
The iteration process is stopped if at each nodal point the k+1 k
difference between h and h is smaller than a certain prescribed in in
value
15
To accelerate the convergence an overrelaxation factor W is used
(for more information about W see FORSYTHE and WASOW 1960)
k+1 k T7 k+1 k h = h + W (h - h )
in in in in (16)
If W = 1 there is no overrelaxation Usually W varies between 1 and
2 and the optimum value very often is found by trial and error
Eq (15) is valid for points in aquifers The connection with
adjacent layers is made by the term Q because in this term the
flow from adjacent aquitards q is included (two aquifers are always
separated by an aquitard)
Aquitards
We dont apply the finite element method on points in aquitards
As stated before there is no horizontal flow in aquitards So it has
no sense to replace the continuous variation of h in the aquitard
by equation (11)
The nodal points in aquitards represent a certain volume of an
aquitard hich transmit water from one aquifer to another and which
has storage capacity The nodal points are included in the Gauss-
Seidel scheme -s follows (see fig 6 )
1 aauifer
a -_vS laquo 2 [~^-t^yM^^^-^r~ aqutarci l-Z
aquifer
Fig 6 Schematic situation for points in aquitards
Point 2 is situated in the middle of the aquitard with thickness
d storage coefficient S and vertical hvdraulic conductivity k c J v
Suppose the Gauss-Seidel iteration starts in the upper layer and
ends in the bottom-layer Then the flux from point 1 in aquifer 1 to
16
point 2 in the aquitard at time t = t and in iteration step j + 1
is
k q = (hf -hf )-iumlmdash f (17)
1 n 2 n OSxd1
and the flux from point 3 in aquifer 2 to point 2 in the aquitard
is
1 c 1
qbdquo = (h - hf) mdashHmdashr- (18) 2 3 n 2 n OSxd
The net flux ltl-q9 must be equal to the change in storage per unit
area per unit time This change in storage can numerically be
approximated by
te - h-
n n-1
Combina t ion of ( 1 7 ) (18) ana (19) g i v e s
l tr1 - h (hJ + 1 + hj - 2 hj+1) - - - - - - S1 - ilaquoS L u l l = o (20)
] i 3 n 2 n 1 c t - t j j x d n n -1
T + 1
From t h i s e q u a t i o n h^ kan be s o l v e d
3 3 C o m p u t e r o g r a m
The problem outl^ugrave so far has been translated into a computer
program written in FORTRAN IV
The flow chart ot Uis program is given in fig 7 whereas in
appendix A the complete listing of the program can be found
In the flow chart the 5 main parts of the program are
indicated
A Reading and printing of input data
B Determination of the matrices A and B and the vector Q per layer
17
M
1 A
FLOH to unsaturated zono
FLOW to tertiary syster
FLOW to secondnrv system
Leakage to adjacent aquitards
SOLVE equation (14)
yes
CALCULATE flux through constant head boundnr
CALCULATE waterbalance n p r 1 OAro r r J j - ~
PRINT of output
Ier node per layer
k = k+1
t = t + At
7 DUMP variables on tape
-gt-
( STOP )
Fig 7 Flow chart of the program FEMSAT with diviation in 5 main
parts A B C D and E
C START PROGRAM FEMSAT
f READ general information
PRINT general information
f READ element information
PRINT element information 7 READ stationary nodal point information
PRINT stationary nodal point information 7
READ non-stationary nodal point information for t = t
FRINT non-stationary nodal point information for t = t
L 2 (
^ yes R E A D non-stat input for t gt t
J PRINT non-stat input for t gt t (- 1 -- M degY
GENERATE A B and AREA-matrix for confined aquifers
_L
GENERATE A B and AREA-matrix for unconfined aouifer
bull
C Solving h with Gauss-Seidel
D Determination of the water balance terms per layer
E Printing of results
The program exists of a main program with several subroutines
The main program calls the subroutines In the subroutines most
calculations are performed
In this way an easy adaptation to different problems is
obtained (see also section 44) Comments in the program will
help the reader to understand the program
In the next chapter the preparation of the input data executing
of the program and the output will be treated
Acknowledgement
Some parts of the computer program were taken from the program
UNSAT2 of NEUMAN etal (1974)
IV INPUT EXECUTION AND OUTPUT
41 I n p u t
The input is divided into endogenous variables (chosen by the
user or calculated by the program during execution) and exogenous
variables (based on physical data)
411 Endogenous variables
Translation of groundwater basin into a nodal point network
Probably the most important but also the most difficult stage
in the process of simulating groundwater flow is the translation
of the natural groundwater basin into a nodal point network Questions
which arise are
a In how many layer the geo-hydrological system mustcan be
schematized
b The locations of the boundaries and the nature of these boundaries
(impervious prescribed head prescribed flux)
c The configuration of the nodal points
19
The answers of these questions depend on the problem under
consideration
- Schematization into layers
Usually our knowledge about the geo-hydrological situation is
such that no more than 5 different layers can be distinguished
(remember contrast rule)
- Boundary conditions
The top boundary condition of a groundwater basin usually is
a prescribed flux (effective precipitation)
The bottom boundary condition usually is a prescribed flux
boundary with flux zero (impervious hydrological base)
For the boundaries at the other sides of the system one sees
either prescribed head boundaries (eg a fully penetrating
river) or prescribed flux boundaries (including zero flux)
- Nodal point configuration
Nodal points are situated
on the boundaries
along internal boundaries which indicate changes in
hydraulic properties within the layer
along the tertiary surface water system
on places where artificial recharge or discharge wiii tgtke
place
elsewhere
Construction of the finite element mesh
Assume we have schemetized the groundwater basin iatu a finite
number of layers with different direction of flow (horizontal and
vertical) Within one layer the hydrological properties may vary
from place to place
Now a nodal grid is superimposed on each layer in a way already
described in section 32 Thus each layer has the same number of
nodal points situated on the same places in the horizontal plane
but with different z-coordinates
We are free to construct quadrilaterals or triangles because
the program automatically divides a quadrilateral into two triangles
having identical material properties The dimensions of the elements
20
should be small at places where large changes in hydrological
properties occur or where large gradients in hydraulic head are
expected to occur
Coding of nodal points
Each node in the finite element network is assigned an integer
code which indicates the type of calculation for this node
The coding is as follows
- Code = 1 nodal points with described head as function of time
- Code 2 internal nodal points
- Code = 3 internal nodal points which represent secondary surface
water system
- Code = 5 nodal points with prescribed artificial recharge or
discharge and points with prescribed boundary flow
(both as functions of time)
Other important endogenous variables are the magnitude of the
time step the value of the variable which stops the iteration
process and the maximum number of iterations allowed per time step
(see also group A and B of section 413)
412 Exogenous variables
After constructing the nodal grid each node represents part of
the groundwater basin Geo-hydrological properties can be assigned
to each mode based on physical data (group C D and E of section
413) In group F of section 413 time-dependent variables can
be assigned to each node
413 Data input to Program FEMSAT
The data input must be punched on cards according to the
following instructions
GROUP COLUMNS FORMAT SYMBOL DESCRIPTION
A 1-80 20A4 HEAD Desired heading to be printed at
the beginning of the execution
1-5 15 NUMNP number of nodal points
21
GROUP COLUMNS FORMAT SYMBOL DESCRIPTION
6-10 15
11-15 15
16-20 15
21-25 15
26-30 15
31-35 15
NUMEL
NUMLAY
MBAND
NTSPI
NTSPO
MAXIT
1-10 F103 ST
11-20 F103 DELT
21-30 F103 TMAX
31-40 F105 TOL
number of elements
number of layers
maximum difference in node number
between two adjacent nodes
number of time steps between a
change in non-stationary input data
number of time steps between two
successive outputs
maximum number of iterations per
time step
time to start with the execution
initial time interval
maximum time to be reached before
execution is stopped
maximum allowed absolute change in
the values of h between any two
successive iterations in a given
time step
Iterations continue until all changes
are less than or equal to TOL or
until MAXIT is exceeded
1- 5 15
6-10 15
11-15 15
16-20 15
KX(N1)
KX(N2)
KX(N3)
KX(N4)
sequential number of i-th corner of
element N
sequential number of j-th corner
sequential number of k-th corner
sequential number of 1-th corner
(see fig 8)
Fig 8 Numbering of element corner nodes
22
GROUP COLUMNS FORMAcircT SYMBOL DESCRIPTION
One card must be provided for each element in sequential order
starting with N = 1 and ending with N = NVJMEL
I-K 1103 SANG(NLY)
1i-iO MO3 C1(NLY)
21-30 F 103 C2(NLY)
21-4- 1-104 THI(NLY)
angle in degrees between CI and
the laquo-coordinate to he used in
nodal print N in layer LY (see
poundig 9)
First principal conductivity of
material in node N and layer LY
idem second principal conductivity
thickness of layer LY at nodal
point N
ig V Definition of SANGCI and C2
One card icircuiPt be provided for each model point ctarting wih M = 1
and ending with N = NUMNP If there is norlt3 than 1 layer for each
layer a set must be provided starting with the upper layer LY - 1
and ending with the bottom layer LY = NIJMLAY The variables SANG
M and 2 are not relevant tor aquitards but for reasons of
conveniency this procedure is maintained (making SANG CI and C2
dummy variables)
1-10 110 KODE(N)
11-20 II0 NTERR(N)
coding of nodal points (see
paragraph 42)
number of tertiary relation in node N
The q-h relationship of the flow to
23
GROUP COLUMNS FORMAT SYMBOL DESCRIPTION
21-30 FlOl X(N)
31-40 FlOl Y(N)
41-50 FlOl HGL(N)
51-60 FlOl HBSC(N)
61-70 FlOl LESC(N)
the tertiary system as discussed
in par 24 are schematized in a small
number of different relations
x-coardinate of node N
y-coordinate of node N
height ground level in node N
height bottom secondary conduit in
node N If node N is not a secondary
conduit this variable is a dummy
variable
length of secondary conduit in area
represented by node N
One card must be provided for each node starting with N = 1 and
ending with N = NUMNP
E 1-10 FlOl P(NLY1) initial value of h at node N in
layer LY
11-20 FlOl VERRES(NLY) vertical resistance at node N and
layer LY If the layer is an aquifer
this value is a dummy variable
21-30 F101 RADRES(NLY) radial resistance for flow to
secondary system at node N in layer
LY
31-40 FlOl ENRES(NLY) entrance resistance for flow to
secondary system at node N in layer
LY
41-50 FlOl WP(NLY) wet perimeter of secondary system at
node N in layer LY
51-60 FlOl P0R(NLY) specific yield of the material at
node N in layer LY
61-70 FlOl SS(NLY) specific storage of the material
at node N in layer LY
24
GROUP COLUMNS FORMAT SYMBOL DESCRIPTION
One card must be provided for each node starting with N = 1 and
ending with N = NUMNP If there is more than 1 layer for each layer
a set of cards must be provided starting with LY = 1 and ending
with LY = NUMLAY
F 1-10 F101 P(NLY1) value of h at different times at
node N in layer LY Node N lies
on a prescribed head boundary
11-20 F101 WTSC(N) water table in secondary system at
node N at different times
21-30 F 101 ARRED(NLY) artificial recharge to or discharge
from node N in layer LY or
prescribed boundary flow at
different times
31-40 F 101 QPHR(N) flux through phreatic surface at
node N at different times
The data in group F are non-stationary input data One card must be
provided for each node starting with N = 1 and ending with N = NUMNP
In case of more than 1 layer for each layer a set must be provided
Not relevant variables are taken as dummy variables
After a certain number of time steps (NTSPI) the data of group E
are read So the set cards E must contain NxLYxC cards where C =
(TMAXNTSPIXDELT) + 1 if DELT is a constant
42 R u n n i n g t h e p r o g r a m o n t h e C y b e r 7200
The program has been executed on a Cyber 7200 computer of
IWIS-TNO in the Hague
- The demanded input is punched on cards according to the instructions
of the last paragraph and is put on a permanent file
- For each specific problem inthe program only the dimension statements
in the main program have to be adjusted according to the instructions
in the lines of comment
- After compilation the program together with the input file is
executed
25
- Intermediate data necessary for continuation of the calculations
are saved on another permanent file This gives the opportunity
to restart the program at a time the calculation was stopped
The only change in the input file is to adjust the starting time
Remark
With the aid of a terminal small changes in the program and in the
input file can be made easily and the output immediately appears
on the terminal In this way it is possible to stop erroneous
calculations This is especially useful when being in the testing
phase of the program
43 O u t p u t
Immediately after reading the input data they are printed so
that possible errors can be detected During execution after each
certain number of time steps (NTSPO) calculated values of h and
water balance terms are printed In appendix B an example is given
The output data can be devided into two groups
A Data per node per layer
- Coding of the nodal points KODE
- Hydraulic head HEAD
- Flow through phreatic surface at the time indicated in head
of table FLOWUN
- Flow to tertiary surface water system FLOWTS
- Flow to secondary surface water system FLOWSS
- Flow to adjacent layers LEAKAGE
- Artificial recharge or discharge and prescribed boundary
flow ARRED
- Change in storage during last time step STORAGE
- Flow through prescribed head boundary LATFLOW
B Data per layer
Summation of the terms already mentioned under A gives the
magnitude of the water balance terms per layer at the time
indicated (instanteneous water balance terms) Summation of
these instanteneous terms multiplied by their respective time
intervals since time is zero gives the cumulative water balance
terms per layer
26
After each print out we are able to check the calculations
because the sum of the water balance terms per layer must be
approximately zero
44 C o m p u t i n g t i m e
The computing time depends on the number of nodal points and
the number of layers But it depends also on the magnitude of the
changes in hydraulic head and on the value which terminates the
Gauss-Seidel iteration (TOL) To give an impression TOL = 001 m
Costs per day simulation (in hlfl) on the Cyber 7200
number of nodes number of layer bull bdquo bull bull _ 1mdash x a x b (21)
where
a = costs in Dutch guilders per system second (ƒ 027)
b = empirical factor with magnitude about 001
V NUMERICAL EXPERIMENTS
In general there are two possibilities to test the validity of
numerical models The first and most commonly used method is to
compare numerical results with analytical solutions The second
possibility is to compare numerical results with measured field
data Since the last alternative requires many data that are
difficult to obtain we restrict ourselves for the time being to
comparison with the analytical solutions
5 1 F l o w t o a w e l l i n a n u n c o n f i n e d
c i r c u l a r - s h a p e d a q u i f e r
Flow to a well can be divided into steady-state flow and
unsteady-state flow (KRUSEMAN and DE RIDDER 1976)
27
511 Steady state flow in an unconfined aquifer
For this case the formula of Thiem-Dupuit is valid (see fig 10)
Q bull k lth2-hicircgt ln(r2r])
(22)
where
hhbdquo = hydraulic head in 1 and 2 respectively
Q = discharge from well
rbdquo = distance from well
CL31
hmdash^ri --r2
Fig 10 Schematic cross-section of a pumped unconfined aquifer
Fig 11 gives the nodal grid superimposed on a well in a
circular shaped aquifer Near the well the nodal point distance
is made smaller Quadrilaterals as well as triangles are used
Numerical calculations with this finite element network and 3 -1
an extraction rate of 200 m day were performed for 10 days
At that moment a steady-state situation was almost achieved
28
2tmdashnodal point number
160 180 X-axesim)
Fig 11 Nodal point configuration for flow to a well in a circular-
shaped aquifer The outer boundary is taken to be constant
The well is situated in nodal point 42
29
In fig 12 the results of the simulation are compared with the
analytical solution The agreement between numerical results and
the analytical solution is excellent
draw- 15irS (m) down
analytical solution imdashx calculated values
distance from well r (m)
Fig 12 Comparison between analyt ical and calculated s teady-state
drawdown for the problem of f ig 11 with 1
-1 Q bull - 200 m 3 d a y _ 1
k = 10 mday
d = 100 m
S = 2 y
S = 0 s
512 Unsteady-state flow in an unconfined aquifer
Analytical solutions for unsteady-state flow in an unconfined
aquifer only exist if the aquifer extends infinitely For the
aquifer given in fig 11 this condition is not fulfilled However
during the first one or two days after starting the extraction the
influence of a constant hydraulic head at the outer boundary on the
30
drawdown near the well can be neglected (see fig 13) After one
day the lateral inflow is only 25 of the extraction the remainder
being the change in storage due to drawdown
2 0 0
m 3 day 1
Q=200m 3 day
100-
vraquomdash Lateral inflow
N amp storage
I I ^ V - W 10
t (days)
Fig 13 Values of Q lateral inflow and change in storage as
functions of time
The nonsteady-state or Theis equation can be written as
i _ Q s 4iTkD
dy = 4irkD
W(u) (23)
where
u = r2S
4irkDt
r = distance from well
S = storage coefficient
Q = constant discharge from well
D = thickness of the waterbearing layer
at the beginning of pumping
t = time after starting the extraction
s = corrected drawdown
= s - 322D
s = calculated drawdown
W(u) = Theiss well function
[J
CIAT1
[T]
]
31
In fig 14 the analytical unsteady-state solution for the case
described in section 511 is given together with the numerical
results at times 05 and 10 days The agreement between numerical
results and the analytical solution is rather good
Drawdown s1(m)
10
10
10
Theiss solution
x caculated values t=Q5
bdquo t= lO
10 10- 10 10 t r2(daym2)
Fig 14 Comparison between analytical and calculated steady-state
drawdown for the problem of fig 11 with
Q
k
d
S y
s s
=
=
=
=
=
200
1
10
3 m
0
0
2
0
-1 day
m day
m
These results indicate that the assumption of a linear variation
of h between two nodal points (see fig 3 ) even with the rather
rough element network of fig 11 is acceptable So there is no
need to improve for example the finite element network by introducing
higher order coordinate functions
32
52 F l o w t o w e l l i n 3-1 a y e r e d s y s t e m
In fig 15 a schematic cross-section of a 3-layered system of
two aquifers separated by an aquitard is given A well of
infinitisemal radius is completed in the lower aquifer and discharges
at a constant rate QTT Each layer is homogeneous isotropic and
extending radially infinite
bdquo bull - bull bull ~ i
mdash Q II
aquifer I
lugrave^i~IcircIuml7i-i~~ aquitard --r-rl~-
-II--I i - i
aquifer
Fig 15 Schematic cross-section of a 3-layered system
For this situation NEUMAN and WITHERSPOON (1969a) give rather
complicated analytical solutions In the special case where the
hydraulic properties of the two aquifers are identical they give
graphical illustrations of their solutions for three different
cases
These graphical solutions give the opportunity to verify
roughly numerical results of a 3-layered system Therefore a nodal
grid is superimposed on a circular shaped groundwater flow system
with the extraction in the centre of the lower aquifer and with
a constant head on the outer boundary Fig 16 gives the nodal
point situation in the x - y plane of 1 layer The nodes in the
other two layers are situated in the middle of the layers at the
same places in the x - y plane only their z-coordinate is different
33
A
-Jy
bull33
V 5
iumlff
Fig 16 Nodal point configuration for the 3-layered problem (top view)
34
In fig 17 numerical results are compared with the analytical
solution The agreement is rather bad especially for the two
aquifers But in terms of the water balance the difference between
aquifers and aquitard is not so big because the aquitard has
about 16 times bigger storage capacity than the aquifers The
small value of the specific storage of the aquifers is the reason
that small errors in the components of the water balance have a
big influence on the calculated values of the hydraulic head
Eig 17 Dimensionless drawdown vs dimensionless time in a
3-layered system with
II = 50 m day
3 m (distance from well)
aquifer I
aquitard
aquifer II
Thi (m)
100
25
100
S s
001
0642
001
k(mday )
1
0277
1
The discrepancies are partly caused by errors in discretization
In the z-direction these errors can be eliminated by taking the
35
vertical resistance of the aquitard to be infinite Then the
calculated drawdown in the pumped aquifer can be compared with
the analytical Theiss solution
This is done in fig 18 for different distances from the well
One sees that the agreement is better when the distance is bigger
Apparently the linearization of the logarithmic drawdown curve
near the well is responsible for the differences The differences
for r = 3 m are of the same magnitude as for the pumped aquifer in
the 3-layered case
joo gt A 50
20
Theiss solution
bull calculated values r=3m (distance f rom well) x bdquo bdquo r=7m
1 L
I l_ 001 003 0 0 5 Ol Q2
- I I 5 10 t - K t
Fig 18 Dimensionless drawdown vs dimensionless time in a
confined aquifer with
Q =500 m day
k = 10 mday
D = 100 m
S = 001 s
The differences between numerical results and analytical solutions
are not caused by taking a finite area because changing the outer
boundary into an impermeable boundary has only minor effects
36
VI SUMMARY AND CONCLUSIONS
In this report a numerical model for non-stationary saturated
groundwater flow in a multi-layered system has been treated
Groundwater flow can be described in different ways
a Physical-mathematical by combining Darcys law and the law of
continuity (section 21)
b Geo-hydrological the flow in a natural groundwater basin takes
place in layers with different hydraulic properties Definitions
of the most important hydraulic properties are given in section
22
c Schematical groundwater flows horizontally in aquifers and
vertically in aquitards (section 23)
d Approximating by dividing the region of flow into a finite
number of elements and applying Darcys law and the law of
continuity on each of these elements In these ways one gets
a set of equations which can be solved given (initial and)
boundary conditions (section 25)
e Numerical the set of equations has been solved using a Galerkin
type finite element method and the implicit Gauss-Seidel method
with overrelaxation factor (section 31 and 32)
f Computer-solvable the numerical solution has been translated
in a FORTRAN IV program called FEMSAT (section 33)
In chapter IV the input execution and output of the program
FEMSAT are treated
Finally in chapter V some numerical applications are given
simulation of flow to a well in unconfined aquifer and simulation
of flow to a well in a 3-layered configuration
Comparison of hydraulic heads obtained by numerical calculations
and analytical true solutions showed an excellent agreement in case
of stationary flow and a good to moderate agreement in case of non-
stationary flow However the diviations caused by the discretization
process must be seen in relation to the errors made in the
establishment of the hydraulic properties
37
As was stated in the introduction numerical models can be
valuable tools in the management of water resources Therefore
numerical models must be suitable to simulate effects of certain
operations upon the spreading of water over saturated groundwater
unsaturated groundwater and surface water The model described in
this report is a model for saturated groundwater flow The relation
with the surface water can be taken into account via radial and
entrance resistances or via a relation between flow to the surface
water system and the hydraulic head The relation with the unsaturated
zone has been restricted to the input factor effective precipitation
In practise however the unsaturated and saturated system are
interconnected Therefore the model should be improved by taking
into account the unsaturated zone This can be done in at least
two ways
a Replacing the unsaturated zone by a parametrical model
b Coupling the unsaturated zone in an iterative way with the
saturated system
In order to find out the usefulness of the numerical model for
water resources management it will be applied to actual regional
groundwater systemslaquoAlso sensitivity analyses will be carried out
The computer model itself can be improved by applying standard
programs eg automatical generation of a nodal grid and graphical
plotting of the results
38
VII REFERENCES
ERNST LF 1962 Grondwaterstromingen in de verzadigde zone en hun
berekening bij aanwezigheid van horizontale evenwijdige open
leidingen Thesis University Utrecht pp 189
1978 Drainage of undulating soils with high groundwater
tables (in press)
FORSYTHE GE and WR WASON 1960 Finite Difference Methods for
Partial Differential Equations John Wiley and Sons New York
pp 444
KRUSEMAN GP and NA DE RIDDER 1976 Analysis and evaluation of
pumping test data ILRI-bulletin No 11 pp 200
LAAT PJM DE and C VAN DEN AKKER 1976 Simulatiemodel voor grond-
waterstroming en verdamping in Modelonderzoek 1971-1974 ten
behoeve van de waterhuishouding in Gelderland deel II
Grondslagen pp 145-197
NEUMAN SP RA FEDDES and E BRESLER 1974 Finite element
simulation of flow in saturated -unsaturated soils considering
water uptake by plants Hydraulic Engineering Laboratory
Technion Israel pp 104
and PA WITHERSPOON 1969a Theory of Flow in a confined
two Aquifer System Water Resources Research Vol 5 No 4
pp 803-816
^_ and PA WITHERSPOON 1969b Applicability of Current Theories
of Flow in leaky aquifers Water Resources Research Vol 5
No 4 pp 817-829
PINDER GF and EO FRIND 1972 Application of Galerkins Procedure
to Aquifer Analysis Water Resources Research Vol 8 no 1
pp 108-120
REMSON I GE HORNBERGER and FJ MOLZ 1971 Numerical Methods
in Subsurface Hydrology John Wiley and Sons New York
pp 389
WESSELING JG 1977 Toepassing van Finite Difference- en Finite
Element Methoden voor het oplossen van een aantal grondwatershy
stromingsproblemen (met computerprogrammas) ICW-nota 944
pp 63
39
ZIENKIEWICZ OC 1971 Finite Element Method in Engineering Science
McGraw-Hill London pp 521
40
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APPENDIX 8 EXAMPLE OF OUTPUT
HEAD AND TERMS OF UATERBALANCE PER NODE PER LAYER
AT TIME 260 IN II AND I13DAYS
T
LAYER NODE KODE HEAD FLOUUN FLOUTS FLOUSS LEAKAGE STORAGE LAT FLOU
13 19 25 31 37 43
125008 125000 12S000 12S000 12S000 12S000 12S000 12S000
0000 0000 0000 0000 0000 0000 0000 0000
0000 0000 0000 0000 0000 0000 0000 0000
0000 0000 0000 0000 0000 0000 0000 0000
- 000 - 000 -000 - 000 - 000 - 000 - 000 0000
0000 0000 0000 0000 0000 0000 0000 0000
0000 000 000 000 000 000 000 000
000 0000 0000 0P100 0000 0000 0000 0000
1 7
13 19 25 31 37
12S000 125000 125000 125000 125000 12S000 125000 125000
0000 0000 0000 0000 0000 0000 0000 0000
- 001 -007 - 023 - 020 - 009 - 003 0000
0000 0000 0000 0000 0000 0000 0000 0000
000
001
007
023
020
009
003
003
0000 0000 0000 0000 0000 0000 0000 0000
1 7
13 19 25 31 37 43
125000 125000 124998 124972 124895 124787 124647 124358
0000 0000 0000 0000 0000 0000 0000 0000
- 000 001 007 023 020 009 003
0000
0000 0000 0000 0000 0000 0000 0000
50000
0000 182
1982 3862 1674 419 078 041
0000 0000 0000 0000 0000 0000 0000 0000
TERMS OF WATERBALANCE PER LAYER DURING LAST TIMESTEP AND
AND SINCE BEGINNING IN M3DAY M3
FLOU TO UNSATURATED ZONE
FLOU TO TERTIARY SYSTEM
0000
0000
0000
0000
LAYER NO 1
FLOU TO SECONDARY SYSTEM
FLOU TO AD3ACENT LAYERCS)
ARTIFICIAL FLOU AND PRESCRIBED BOUNDARY FLOU
ACCUMULATION TERMS
LATERAL FLOU THROUGH PRESCR HEAD BOUNDARY
0000
-000
0000
000
000
0000
-000
0000
000
000
LAYER NO 2
FLOU TO SECONDARY SYSTEM
FLOU TO AD3ACENT LAYERCS1
ARTIFICIAL FLOU AND PRESCRIBED BOUNDARY FLOU
ACCUMULATION TERMS
LATERAL FLOU THROUGH PRESCR HEAD BOUNDARY
0000
-378
0000
381
0000
0000
-055
0000
056
0000
LAYER NO 3
FLOU TO SECONDARY SYSTEM
FLOU TO AD3ACENT LAYERCS5
ARTIFICIAL FLOU AND PRESCRIBED BOUNDARY FLOU
ACCUMULATION TERMS
LATERAL FLOU THROUGH PRESCR HEAD BOUNDARY
0000
370
50000
49376
0000
0000
055
12979
12909
0000
If the h-values belonging to the A-values are taken at time
t = t we get an equation with one unknown h for time t = t n-1 4n n
The equation can then be solved directly
However this is an explicit formulation which puts strong limitations
on the magnitude of the time step t - t (REMSON etal 1971) n n-1
Values larger than a certain value cause instability With an implicit method the values of h are taken at time
t = t This method is unconditionally stable Now one gets an equation n
with 7 unknowns h through h If one has 7 equations for every 1 n n
point 1 through 7 we can solve the values h i = 17
in
Of all the methods possible the iterative method of Gauss-Seidel
has been chosen The Gauss-Seidel method starts by choosing a first estimate of h at time t = t (usually equal to h ) At each in n j -i in-l iteration step a new value of h is calculated using the last
in calculated values of h If one passes the nodal points from 1 to
i n 7 eq (14) becomes
h-j + 1 = A h^ + 1 + A bdquo h | + 1 + A _ h | + 1 + A h j + A r h j + 4 n 4 1 1 n 4 2 2 n 43 3 n 45 5 n 4 6 6 n
B h B + A h J + - 4 4 4 n ~ + 0 J (A + mdash) A 4 7 h 7 n At n U 4 4 At
(15)
where
h-i = in
j = iteration step or number of iteration
i = number of nodal point n = time = t
n
The iteration process converges if the successive differences k+1 k
between h and h become continuously smaller in in
The iteration process is stopped if at each nodal point the k+1 k
difference between h and h is smaller than a certain prescribed in in
value
15
To accelerate the convergence an overrelaxation factor W is used
(for more information about W see FORSYTHE and WASOW 1960)
k+1 k T7 k+1 k h = h + W (h - h )
in in in in (16)
If W = 1 there is no overrelaxation Usually W varies between 1 and
2 and the optimum value very often is found by trial and error
Eq (15) is valid for points in aquifers The connection with
adjacent layers is made by the term Q because in this term the
flow from adjacent aquitards q is included (two aquifers are always
separated by an aquitard)
Aquitards
We dont apply the finite element method on points in aquitards
As stated before there is no horizontal flow in aquitards So it has
no sense to replace the continuous variation of h in the aquitard
by equation (11)
The nodal points in aquitards represent a certain volume of an
aquitard hich transmit water from one aquifer to another and which
has storage capacity The nodal points are included in the Gauss-
Seidel scheme -s follows (see fig 6 )
1 aauifer
a -_vS laquo 2 [~^-t^yM^^^-^r~ aqutarci l-Z
aquifer
Fig 6 Schematic situation for points in aquitards
Point 2 is situated in the middle of the aquitard with thickness
d storage coefficient S and vertical hvdraulic conductivity k c J v
Suppose the Gauss-Seidel iteration starts in the upper layer and
ends in the bottom-layer Then the flux from point 1 in aquifer 1 to
16
point 2 in the aquitard at time t = t and in iteration step j + 1
is
k q = (hf -hf )-iumlmdash f (17)
1 n 2 n OSxd1
and the flux from point 3 in aquifer 2 to point 2 in the aquitard
is
1 c 1
qbdquo = (h - hf) mdashHmdashr- (18) 2 3 n 2 n OSxd
The net flux ltl-q9 must be equal to the change in storage per unit
area per unit time This change in storage can numerically be
approximated by
te - h-
n n-1
Combina t ion of ( 1 7 ) (18) ana (19) g i v e s
l tr1 - h (hJ + 1 + hj - 2 hj+1) - - - - - - S1 - ilaquoS L u l l = o (20)
] i 3 n 2 n 1 c t - t j j x d n n -1
T + 1
From t h i s e q u a t i o n h^ kan be s o l v e d
3 3 C o m p u t e r o g r a m
The problem outl^ugrave so far has been translated into a computer
program written in FORTRAN IV
The flow chart ot Uis program is given in fig 7 whereas in
appendix A the complete listing of the program can be found
In the flow chart the 5 main parts of the program are
indicated
A Reading and printing of input data
B Determination of the matrices A and B and the vector Q per layer
17
M
1 A
FLOH to unsaturated zono
FLOW to tertiary syster
FLOW to secondnrv system
Leakage to adjacent aquitards
SOLVE equation (14)
yes
CALCULATE flux through constant head boundnr
CALCULATE waterbalance n p r 1 OAro r r J j - ~
PRINT of output
Ier node per layer
k = k+1
t = t + At
7 DUMP variables on tape
-gt-
( STOP )
Fig 7 Flow chart of the program FEMSAT with diviation in 5 main
parts A B C D and E
C START PROGRAM FEMSAT
f READ general information
PRINT general information
f READ element information
PRINT element information 7 READ stationary nodal point information
PRINT stationary nodal point information 7
READ non-stationary nodal point information for t = t
FRINT non-stationary nodal point information for t = t
L 2 (
^ yes R E A D non-stat input for t gt t
J PRINT non-stat input for t gt t (- 1 -- M degY
GENERATE A B and AREA-matrix for confined aquifers
_L
GENERATE A B and AREA-matrix for unconfined aouifer
bull
C Solving h with Gauss-Seidel
D Determination of the water balance terms per layer
E Printing of results
The program exists of a main program with several subroutines
The main program calls the subroutines In the subroutines most
calculations are performed
In this way an easy adaptation to different problems is
obtained (see also section 44) Comments in the program will
help the reader to understand the program
In the next chapter the preparation of the input data executing
of the program and the output will be treated
Acknowledgement
Some parts of the computer program were taken from the program
UNSAT2 of NEUMAN etal (1974)
IV INPUT EXECUTION AND OUTPUT
41 I n p u t
The input is divided into endogenous variables (chosen by the
user or calculated by the program during execution) and exogenous
variables (based on physical data)
411 Endogenous variables
Translation of groundwater basin into a nodal point network
Probably the most important but also the most difficult stage
in the process of simulating groundwater flow is the translation
of the natural groundwater basin into a nodal point network Questions
which arise are
a In how many layer the geo-hydrological system mustcan be
schematized
b The locations of the boundaries and the nature of these boundaries
(impervious prescribed head prescribed flux)
c The configuration of the nodal points
19
The answers of these questions depend on the problem under
consideration
- Schematization into layers
Usually our knowledge about the geo-hydrological situation is
such that no more than 5 different layers can be distinguished
(remember contrast rule)
- Boundary conditions
The top boundary condition of a groundwater basin usually is
a prescribed flux (effective precipitation)
The bottom boundary condition usually is a prescribed flux
boundary with flux zero (impervious hydrological base)
For the boundaries at the other sides of the system one sees
either prescribed head boundaries (eg a fully penetrating
river) or prescribed flux boundaries (including zero flux)
- Nodal point configuration
Nodal points are situated
on the boundaries
along internal boundaries which indicate changes in
hydraulic properties within the layer
along the tertiary surface water system
on places where artificial recharge or discharge wiii tgtke
place
elsewhere
Construction of the finite element mesh
Assume we have schemetized the groundwater basin iatu a finite
number of layers with different direction of flow (horizontal and
vertical) Within one layer the hydrological properties may vary
from place to place
Now a nodal grid is superimposed on each layer in a way already
described in section 32 Thus each layer has the same number of
nodal points situated on the same places in the horizontal plane
but with different z-coordinates
We are free to construct quadrilaterals or triangles because
the program automatically divides a quadrilateral into two triangles
having identical material properties The dimensions of the elements
20
should be small at places where large changes in hydrological
properties occur or where large gradients in hydraulic head are
expected to occur
Coding of nodal points
Each node in the finite element network is assigned an integer
code which indicates the type of calculation for this node
The coding is as follows
- Code = 1 nodal points with described head as function of time
- Code 2 internal nodal points
- Code = 3 internal nodal points which represent secondary surface
water system
- Code = 5 nodal points with prescribed artificial recharge or
discharge and points with prescribed boundary flow
(both as functions of time)
Other important endogenous variables are the magnitude of the
time step the value of the variable which stops the iteration
process and the maximum number of iterations allowed per time step
(see also group A and B of section 413)
412 Exogenous variables
After constructing the nodal grid each node represents part of
the groundwater basin Geo-hydrological properties can be assigned
to each mode based on physical data (group C D and E of section
413) In group F of section 413 time-dependent variables can
be assigned to each node
413 Data input to Program FEMSAT
The data input must be punched on cards according to the
following instructions
GROUP COLUMNS FORMAT SYMBOL DESCRIPTION
A 1-80 20A4 HEAD Desired heading to be printed at
the beginning of the execution
1-5 15 NUMNP number of nodal points
21
GROUP COLUMNS FORMAT SYMBOL DESCRIPTION
6-10 15
11-15 15
16-20 15
21-25 15
26-30 15
31-35 15
NUMEL
NUMLAY
MBAND
NTSPI
NTSPO
MAXIT
1-10 F103 ST
11-20 F103 DELT
21-30 F103 TMAX
31-40 F105 TOL
number of elements
number of layers
maximum difference in node number
between two adjacent nodes
number of time steps between a
change in non-stationary input data
number of time steps between two
successive outputs
maximum number of iterations per
time step
time to start with the execution
initial time interval
maximum time to be reached before
execution is stopped
maximum allowed absolute change in
the values of h between any two
successive iterations in a given
time step
Iterations continue until all changes
are less than or equal to TOL or
until MAXIT is exceeded
1- 5 15
6-10 15
11-15 15
16-20 15
KX(N1)
KX(N2)
KX(N3)
KX(N4)
sequential number of i-th corner of
element N
sequential number of j-th corner
sequential number of k-th corner
sequential number of 1-th corner
(see fig 8)
Fig 8 Numbering of element corner nodes
22
GROUP COLUMNS FORMAcircT SYMBOL DESCRIPTION
One card must be provided for each element in sequential order
starting with N = 1 and ending with N = NVJMEL
I-K 1103 SANG(NLY)
1i-iO MO3 C1(NLY)
21-30 F 103 C2(NLY)
21-4- 1-104 THI(NLY)
angle in degrees between CI and
the laquo-coordinate to he used in
nodal print N in layer LY (see
poundig 9)
First principal conductivity of
material in node N and layer LY
idem second principal conductivity
thickness of layer LY at nodal
point N
ig V Definition of SANGCI and C2
One card icircuiPt be provided for each model point ctarting wih M = 1
and ending with N = NUMNP If there is norlt3 than 1 layer for each
layer a set must be provided starting with the upper layer LY - 1
and ending with the bottom layer LY = NIJMLAY The variables SANG
M and 2 are not relevant tor aquitards but for reasons of
conveniency this procedure is maintained (making SANG CI and C2
dummy variables)
1-10 110 KODE(N)
11-20 II0 NTERR(N)
coding of nodal points (see
paragraph 42)
number of tertiary relation in node N
The q-h relationship of the flow to
23
GROUP COLUMNS FORMAT SYMBOL DESCRIPTION
21-30 FlOl X(N)
31-40 FlOl Y(N)
41-50 FlOl HGL(N)
51-60 FlOl HBSC(N)
61-70 FlOl LESC(N)
the tertiary system as discussed
in par 24 are schematized in a small
number of different relations
x-coardinate of node N
y-coordinate of node N
height ground level in node N
height bottom secondary conduit in
node N If node N is not a secondary
conduit this variable is a dummy
variable
length of secondary conduit in area
represented by node N
One card must be provided for each node starting with N = 1 and
ending with N = NUMNP
E 1-10 FlOl P(NLY1) initial value of h at node N in
layer LY
11-20 FlOl VERRES(NLY) vertical resistance at node N and
layer LY If the layer is an aquifer
this value is a dummy variable
21-30 F101 RADRES(NLY) radial resistance for flow to
secondary system at node N in layer
LY
31-40 FlOl ENRES(NLY) entrance resistance for flow to
secondary system at node N in layer
LY
41-50 FlOl WP(NLY) wet perimeter of secondary system at
node N in layer LY
51-60 FlOl P0R(NLY) specific yield of the material at
node N in layer LY
61-70 FlOl SS(NLY) specific storage of the material
at node N in layer LY
24
GROUP COLUMNS FORMAT SYMBOL DESCRIPTION
One card must be provided for each node starting with N = 1 and
ending with N = NUMNP If there is more than 1 layer for each layer
a set of cards must be provided starting with LY = 1 and ending
with LY = NUMLAY
F 1-10 F101 P(NLY1) value of h at different times at
node N in layer LY Node N lies
on a prescribed head boundary
11-20 F101 WTSC(N) water table in secondary system at
node N at different times
21-30 F 101 ARRED(NLY) artificial recharge to or discharge
from node N in layer LY or
prescribed boundary flow at
different times
31-40 F 101 QPHR(N) flux through phreatic surface at
node N at different times
The data in group F are non-stationary input data One card must be
provided for each node starting with N = 1 and ending with N = NUMNP
In case of more than 1 layer for each layer a set must be provided
Not relevant variables are taken as dummy variables
After a certain number of time steps (NTSPI) the data of group E
are read So the set cards E must contain NxLYxC cards where C =
(TMAXNTSPIXDELT) + 1 if DELT is a constant
42 R u n n i n g t h e p r o g r a m o n t h e C y b e r 7200
The program has been executed on a Cyber 7200 computer of
IWIS-TNO in the Hague
- The demanded input is punched on cards according to the instructions
of the last paragraph and is put on a permanent file
- For each specific problem inthe program only the dimension statements
in the main program have to be adjusted according to the instructions
in the lines of comment
- After compilation the program together with the input file is
executed
25
- Intermediate data necessary for continuation of the calculations
are saved on another permanent file This gives the opportunity
to restart the program at a time the calculation was stopped
The only change in the input file is to adjust the starting time
Remark
With the aid of a terminal small changes in the program and in the
input file can be made easily and the output immediately appears
on the terminal In this way it is possible to stop erroneous
calculations This is especially useful when being in the testing
phase of the program
43 O u t p u t
Immediately after reading the input data they are printed so
that possible errors can be detected During execution after each
certain number of time steps (NTSPO) calculated values of h and
water balance terms are printed In appendix B an example is given
The output data can be devided into two groups
A Data per node per layer
- Coding of the nodal points KODE
- Hydraulic head HEAD
- Flow through phreatic surface at the time indicated in head
of table FLOWUN
- Flow to tertiary surface water system FLOWTS
- Flow to secondary surface water system FLOWSS
- Flow to adjacent layers LEAKAGE
- Artificial recharge or discharge and prescribed boundary
flow ARRED
- Change in storage during last time step STORAGE
- Flow through prescribed head boundary LATFLOW
B Data per layer
Summation of the terms already mentioned under A gives the
magnitude of the water balance terms per layer at the time
indicated (instanteneous water balance terms) Summation of
these instanteneous terms multiplied by their respective time
intervals since time is zero gives the cumulative water balance
terms per layer
26
After each print out we are able to check the calculations
because the sum of the water balance terms per layer must be
approximately zero
44 C o m p u t i n g t i m e
The computing time depends on the number of nodal points and
the number of layers But it depends also on the magnitude of the
changes in hydraulic head and on the value which terminates the
Gauss-Seidel iteration (TOL) To give an impression TOL = 001 m
Costs per day simulation (in hlfl) on the Cyber 7200
number of nodes number of layer bull bdquo bull bull _ 1mdash x a x b (21)
where
a = costs in Dutch guilders per system second (ƒ 027)
b = empirical factor with magnitude about 001
V NUMERICAL EXPERIMENTS
In general there are two possibilities to test the validity of
numerical models The first and most commonly used method is to
compare numerical results with analytical solutions The second
possibility is to compare numerical results with measured field
data Since the last alternative requires many data that are
difficult to obtain we restrict ourselves for the time being to
comparison with the analytical solutions
5 1 F l o w t o a w e l l i n a n u n c o n f i n e d
c i r c u l a r - s h a p e d a q u i f e r
Flow to a well can be divided into steady-state flow and
unsteady-state flow (KRUSEMAN and DE RIDDER 1976)
27
511 Steady state flow in an unconfined aquifer
For this case the formula of Thiem-Dupuit is valid (see fig 10)
Q bull k lth2-hicircgt ln(r2r])
(22)
where
hhbdquo = hydraulic head in 1 and 2 respectively
Q = discharge from well
rbdquo = distance from well
CL31
hmdash^ri --r2
Fig 10 Schematic cross-section of a pumped unconfined aquifer
Fig 11 gives the nodal grid superimposed on a well in a
circular shaped aquifer Near the well the nodal point distance
is made smaller Quadrilaterals as well as triangles are used
Numerical calculations with this finite element network and 3 -1
an extraction rate of 200 m day were performed for 10 days
At that moment a steady-state situation was almost achieved
28
2tmdashnodal point number
160 180 X-axesim)
Fig 11 Nodal point configuration for flow to a well in a circular-
shaped aquifer The outer boundary is taken to be constant
The well is situated in nodal point 42
29
In fig 12 the results of the simulation are compared with the
analytical solution The agreement between numerical results and
the analytical solution is excellent
draw- 15irS (m) down
analytical solution imdashx calculated values
distance from well r (m)
Fig 12 Comparison between analyt ical and calculated s teady-state
drawdown for the problem of f ig 11 with 1
-1 Q bull - 200 m 3 d a y _ 1
k = 10 mday
d = 100 m
S = 2 y
S = 0 s
512 Unsteady-state flow in an unconfined aquifer
Analytical solutions for unsteady-state flow in an unconfined
aquifer only exist if the aquifer extends infinitely For the
aquifer given in fig 11 this condition is not fulfilled However
during the first one or two days after starting the extraction the
influence of a constant hydraulic head at the outer boundary on the
30
drawdown near the well can be neglected (see fig 13) After one
day the lateral inflow is only 25 of the extraction the remainder
being the change in storage due to drawdown
2 0 0
m 3 day 1
Q=200m 3 day
100-
vraquomdash Lateral inflow
N amp storage
I I ^ V - W 10
t (days)
Fig 13 Values of Q lateral inflow and change in storage as
functions of time
The nonsteady-state or Theis equation can be written as
i _ Q s 4iTkD
dy = 4irkD
W(u) (23)
where
u = r2S
4irkDt
r = distance from well
S = storage coefficient
Q = constant discharge from well
D = thickness of the waterbearing layer
at the beginning of pumping
t = time after starting the extraction
s = corrected drawdown
= s - 322D
s = calculated drawdown
W(u) = Theiss well function
[J
CIAT1
[T]
]
31
In fig 14 the analytical unsteady-state solution for the case
described in section 511 is given together with the numerical
results at times 05 and 10 days The agreement between numerical
results and the analytical solution is rather good
Drawdown s1(m)
10
10
10
Theiss solution
x caculated values t=Q5
bdquo t= lO
10 10- 10 10 t r2(daym2)
Fig 14 Comparison between analytical and calculated steady-state
drawdown for the problem of fig 11 with
Q
k
d
S y
s s
=
=
=
=
=
200
1
10
3 m
0
0
2
0
-1 day
m day
m
These results indicate that the assumption of a linear variation
of h between two nodal points (see fig 3 ) even with the rather
rough element network of fig 11 is acceptable So there is no
need to improve for example the finite element network by introducing
higher order coordinate functions
32
52 F l o w t o w e l l i n 3-1 a y e r e d s y s t e m
In fig 15 a schematic cross-section of a 3-layered system of
two aquifers separated by an aquitard is given A well of
infinitisemal radius is completed in the lower aquifer and discharges
at a constant rate QTT Each layer is homogeneous isotropic and
extending radially infinite
bdquo bull - bull bull ~ i
mdash Q II
aquifer I
lugrave^i~IcircIuml7i-i~~ aquitard --r-rl~-
-II--I i - i
aquifer
Fig 15 Schematic cross-section of a 3-layered system
For this situation NEUMAN and WITHERSPOON (1969a) give rather
complicated analytical solutions In the special case where the
hydraulic properties of the two aquifers are identical they give
graphical illustrations of their solutions for three different
cases
These graphical solutions give the opportunity to verify
roughly numerical results of a 3-layered system Therefore a nodal
grid is superimposed on a circular shaped groundwater flow system
with the extraction in the centre of the lower aquifer and with
a constant head on the outer boundary Fig 16 gives the nodal
point situation in the x - y plane of 1 layer The nodes in the
other two layers are situated in the middle of the layers at the
same places in the x - y plane only their z-coordinate is different
33
A
-Jy
bull33
V 5
iumlff
Fig 16 Nodal point configuration for the 3-layered problem (top view)
34
In fig 17 numerical results are compared with the analytical
solution The agreement is rather bad especially for the two
aquifers But in terms of the water balance the difference between
aquifers and aquitard is not so big because the aquitard has
about 16 times bigger storage capacity than the aquifers The
small value of the specific storage of the aquifers is the reason
that small errors in the components of the water balance have a
big influence on the calculated values of the hydraulic head
Eig 17 Dimensionless drawdown vs dimensionless time in a
3-layered system with
II = 50 m day
3 m (distance from well)
aquifer I
aquitard
aquifer II
Thi (m)
100
25
100
S s
001
0642
001
k(mday )
1
0277
1
The discrepancies are partly caused by errors in discretization
In the z-direction these errors can be eliminated by taking the
35
vertical resistance of the aquitard to be infinite Then the
calculated drawdown in the pumped aquifer can be compared with
the analytical Theiss solution
This is done in fig 18 for different distances from the well
One sees that the agreement is better when the distance is bigger
Apparently the linearization of the logarithmic drawdown curve
near the well is responsible for the differences The differences
for r = 3 m are of the same magnitude as for the pumped aquifer in
the 3-layered case
joo gt A 50
20
Theiss solution
bull calculated values r=3m (distance f rom well) x bdquo bdquo r=7m
1 L
I l_ 001 003 0 0 5 Ol Q2
- I I 5 10 t - K t
Fig 18 Dimensionless drawdown vs dimensionless time in a
confined aquifer with
Q =500 m day
k = 10 mday
D = 100 m
S = 001 s
The differences between numerical results and analytical solutions
are not caused by taking a finite area because changing the outer
boundary into an impermeable boundary has only minor effects
36
VI SUMMARY AND CONCLUSIONS
In this report a numerical model for non-stationary saturated
groundwater flow in a multi-layered system has been treated
Groundwater flow can be described in different ways
a Physical-mathematical by combining Darcys law and the law of
continuity (section 21)
b Geo-hydrological the flow in a natural groundwater basin takes
place in layers with different hydraulic properties Definitions
of the most important hydraulic properties are given in section
22
c Schematical groundwater flows horizontally in aquifers and
vertically in aquitards (section 23)
d Approximating by dividing the region of flow into a finite
number of elements and applying Darcys law and the law of
continuity on each of these elements In these ways one gets
a set of equations which can be solved given (initial and)
boundary conditions (section 25)
e Numerical the set of equations has been solved using a Galerkin
type finite element method and the implicit Gauss-Seidel method
with overrelaxation factor (section 31 and 32)
f Computer-solvable the numerical solution has been translated
in a FORTRAN IV program called FEMSAT (section 33)
In chapter IV the input execution and output of the program
FEMSAT are treated
Finally in chapter V some numerical applications are given
simulation of flow to a well in unconfined aquifer and simulation
of flow to a well in a 3-layered configuration
Comparison of hydraulic heads obtained by numerical calculations
and analytical true solutions showed an excellent agreement in case
of stationary flow and a good to moderate agreement in case of non-
stationary flow However the diviations caused by the discretization
process must be seen in relation to the errors made in the
establishment of the hydraulic properties
37
As was stated in the introduction numerical models can be
valuable tools in the management of water resources Therefore
numerical models must be suitable to simulate effects of certain
operations upon the spreading of water over saturated groundwater
unsaturated groundwater and surface water The model described in
this report is a model for saturated groundwater flow The relation
with the surface water can be taken into account via radial and
entrance resistances or via a relation between flow to the surface
water system and the hydraulic head The relation with the unsaturated
zone has been restricted to the input factor effective precipitation
In practise however the unsaturated and saturated system are
interconnected Therefore the model should be improved by taking
into account the unsaturated zone This can be done in at least
two ways
a Replacing the unsaturated zone by a parametrical model
b Coupling the unsaturated zone in an iterative way with the
saturated system
In order to find out the usefulness of the numerical model for
water resources management it will be applied to actual regional
groundwater systemslaquoAlso sensitivity analyses will be carried out
The computer model itself can be improved by applying standard
programs eg automatical generation of a nodal grid and graphical
plotting of the results
38
VII REFERENCES
ERNST LF 1962 Grondwaterstromingen in de verzadigde zone en hun
berekening bij aanwezigheid van horizontale evenwijdige open
leidingen Thesis University Utrecht pp 189
1978 Drainage of undulating soils with high groundwater
tables (in press)
FORSYTHE GE and WR WASON 1960 Finite Difference Methods for
Partial Differential Equations John Wiley and Sons New York
pp 444
KRUSEMAN GP and NA DE RIDDER 1976 Analysis and evaluation of
pumping test data ILRI-bulletin No 11 pp 200
LAAT PJM DE and C VAN DEN AKKER 1976 Simulatiemodel voor grond-
waterstroming en verdamping in Modelonderzoek 1971-1974 ten
behoeve van de waterhuishouding in Gelderland deel II
Grondslagen pp 145-197
NEUMAN SP RA FEDDES and E BRESLER 1974 Finite element
simulation of flow in saturated -unsaturated soils considering
water uptake by plants Hydraulic Engineering Laboratory
Technion Israel pp 104
and PA WITHERSPOON 1969a Theory of Flow in a confined
two Aquifer System Water Resources Research Vol 5 No 4
pp 803-816
^_ and PA WITHERSPOON 1969b Applicability of Current Theories
of Flow in leaky aquifers Water Resources Research Vol 5
No 4 pp 817-829
PINDER GF and EO FRIND 1972 Application of Galerkins Procedure
to Aquifer Analysis Water Resources Research Vol 8 no 1
pp 108-120
REMSON I GE HORNBERGER and FJ MOLZ 1971 Numerical Methods
in Subsurface Hydrology John Wiley and Sons New York
pp 389
WESSELING JG 1977 Toepassing van Finite Difference- en Finite
Element Methoden voor het oplossen van een aantal grondwatershy
stromingsproblemen (met computerprogrammas) ICW-nota 944
pp 63
39
ZIENKIEWICZ OC 1971 Finite Element Method in Engineering Science
McGraw-Hill London pp 521
40
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APPENDIX 8 EXAMPLE OF OUTPUT
HEAD AND TERMS OF UATERBALANCE PER NODE PER LAYER
AT TIME 260 IN II AND I13DAYS
T
LAYER NODE KODE HEAD FLOUUN FLOUTS FLOUSS LEAKAGE STORAGE LAT FLOU
13 19 25 31 37 43
125008 125000 12S000 12S000 12S000 12S000 12S000 12S000
0000 0000 0000 0000 0000 0000 0000 0000
0000 0000 0000 0000 0000 0000 0000 0000
0000 0000 0000 0000 0000 0000 0000 0000
- 000 - 000 -000 - 000 - 000 - 000 - 000 0000
0000 0000 0000 0000 0000 0000 0000 0000
0000 000 000 000 000 000 000 000
000 0000 0000 0P100 0000 0000 0000 0000
1 7
13 19 25 31 37
12S000 125000 125000 125000 125000 12S000 125000 125000
0000 0000 0000 0000 0000 0000 0000 0000
- 001 -007 - 023 - 020 - 009 - 003 0000
0000 0000 0000 0000 0000 0000 0000 0000
000
001
007
023
020
009
003
003
0000 0000 0000 0000 0000 0000 0000 0000
1 7
13 19 25 31 37 43
125000 125000 124998 124972 124895 124787 124647 124358
0000 0000 0000 0000 0000 0000 0000 0000
- 000 001 007 023 020 009 003
0000
0000 0000 0000 0000 0000 0000 0000
50000
0000 182
1982 3862 1674 419 078 041
0000 0000 0000 0000 0000 0000 0000 0000
TERMS OF WATERBALANCE PER LAYER DURING LAST TIMESTEP AND
AND SINCE BEGINNING IN M3DAY M3
FLOU TO UNSATURATED ZONE
FLOU TO TERTIARY SYSTEM
0000
0000
0000
0000
LAYER NO 1
FLOU TO SECONDARY SYSTEM
FLOU TO AD3ACENT LAYERCS)
ARTIFICIAL FLOU AND PRESCRIBED BOUNDARY FLOU
ACCUMULATION TERMS
LATERAL FLOU THROUGH PRESCR HEAD BOUNDARY
0000
-000
0000
000
000
0000
-000
0000
000
000
LAYER NO 2
FLOU TO SECONDARY SYSTEM
FLOU TO AD3ACENT LAYERCS1
ARTIFICIAL FLOU AND PRESCRIBED BOUNDARY FLOU
ACCUMULATION TERMS
LATERAL FLOU THROUGH PRESCR HEAD BOUNDARY
0000
-378
0000
381
0000
0000
-055
0000
056
0000
LAYER NO 3
FLOU TO SECONDARY SYSTEM
FLOU TO AD3ACENT LAYERCS5
ARTIFICIAL FLOU AND PRESCRIBED BOUNDARY FLOU
ACCUMULATION TERMS
LATERAL FLOU THROUGH PRESCR HEAD BOUNDARY
0000
370
50000
49376
0000
0000
055
12979
12909
0000
To accelerate the convergence an overrelaxation factor W is used
(for more information about W see FORSYTHE and WASOW 1960)
k+1 k T7 k+1 k h = h + W (h - h )
in in in in (16)
If W = 1 there is no overrelaxation Usually W varies between 1 and
2 and the optimum value very often is found by trial and error
Eq (15) is valid for points in aquifers The connection with
adjacent layers is made by the term Q because in this term the
flow from adjacent aquitards q is included (two aquifers are always
separated by an aquitard)
Aquitards
We dont apply the finite element method on points in aquitards
As stated before there is no horizontal flow in aquitards So it has
no sense to replace the continuous variation of h in the aquitard
by equation (11)
The nodal points in aquitards represent a certain volume of an
aquitard hich transmit water from one aquifer to another and which
has storage capacity The nodal points are included in the Gauss-
Seidel scheme -s follows (see fig 6 )
1 aauifer
a -_vS laquo 2 [~^-t^yM^^^-^r~ aqutarci l-Z
aquifer
Fig 6 Schematic situation for points in aquitards
Point 2 is situated in the middle of the aquitard with thickness
d storage coefficient S and vertical hvdraulic conductivity k c J v
Suppose the Gauss-Seidel iteration starts in the upper layer and
ends in the bottom-layer Then the flux from point 1 in aquifer 1 to
16
point 2 in the aquitard at time t = t and in iteration step j + 1
is
k q = (hf -hf )-iumlmdash f (17)
1 n 2 n OSxd1
and the flux from point 3 in aquifer 2 to point 2 in the aquitard
is
1 c 1
qbdquo = (h - hf) mdashHmdashr- (18) 2 3 n 2 n OSxd
The net flux ltl-q9 must be equal to the change in storage per unit
area per unit time This change in storage can numerically be
approximated by
te - h-
n n-1
Combina t ion of ( 1 7 ) (18) ana (19) g i v e s
l tr1 - h (hJ + 1 + hj - 2 hj+1) - - - - - - S1 - ilaquoS L u l l = o (20)
] i 3 n 2 n 1 c t - t j j x d n n -1
T + 1
From t h i s e q u a t i o n h^ kan be s o l v e d
3 3 C o m p u t e r o g r a m
The problem outl^ugrave so far has been translated into a computer
program written in FORTRAN IV
The flow chart ot Uis program is given in fig 7 whereas in
appendix A the complete listing of the program can be found
In the flow chart the 5 main parts of the program are
indicated
A Reading and printing of input data
B Determination of the matrices A and B and the vector Q per layer
17
M
1 A
FLOH to unsaturated zono
FLOW to tertiary syster
FLOW to secondnrv system
Leakage to adjacent aquitards
SOLVE equation (14)
yes
CALCULATE flux through constant head boundnr
CALCULATE waterbalance n p r 1 OAro r r J j - ~
PRINT of output
Ier node per layer
k = k+1
t = t + At
7 DUMP variables on tape
-gt-
( STOP )
Fig 7 Flow chart of the program FEMSAT with diviation in 5 main
parts A B C D and E
C START PROGRAM FEMSAT
f READ general information
PRINT general information
f READ element information
PRINT element information 7 READ stationary nodal point information
PRINT stationary nodal point information 7
READ non-stationary nodal point information for t = t
FRINT non-stationary nodal point information for t = t
L 2 (
^ yes R E A D non-stat input for t gt t
J PRINT non-stat input for t gt t (- 1 -- M degY
GENERATE A B and AREA-matrix for confined aquifers
_L
GENERATE A B and AREA-matrix for unconfined aouifer
bull
C Solving h with Gauss-Seidel
D Determination of the water balance terms per layer
E Printing of results
The program exists of a main program with several subroutines
The main program calls the subroutines In the subroutines most
calculations are performed
In this way an easy adaptation to different problems is
obtained (see also section 44) Comments in the program will
help the reader to understand the program
In the next chapter the preparation of the input data executing
of the program and the output will be treated
Acknowledgement
Some parts of the computer program were taken from the program
UNSAT2 of NEUMAN etal (1974)
IV INPUT EXECUTION AND OUTPUT
41 I n p u t
The input is divided into endogenous variables (chosen by the
user or calculated by the program during execution) and exogenous
variables (based on physical data)
411 Endogenous variables
Translation of groundwater basin into a nodal point network
Probably the most important but also the most difficult stage
in the process of simulating groundwater flow is the translation
of the natural groundwater basin into a nodal point network Questions
which arise are
a In how many layer the geo-hydrological system mustcan be
schematized
b The locations of the boundaries and the nature of these boundaries
(impervious prescribed head prescribed flux)
c The configuration of the nodal points
19
The answers of these questions depend on the problem under
consideration
- Schematization into layers
Usually our knowledge about the geo-hydrological situation is
such that no more than 5 different layers can be distinguished
(remember contrast rule)
- Boundary conditions
The top boundary condition of a groundwater basin usually is
a prescribed flux (effective precipitation)
The bottom boundary condition usually is a prescribed flux
boundary with flux zero (impervious hydrological base)
For the boundaries at the other sides of the system one sees
either prescribed head boundaries (eg a fully penetrating
river) or prescribed flux boundaries (including zero flux)
- Nodal point configuration
Nodal points are situated
on the boundaries
along internal boundaries which indicate changes in
hydraulic properties within the layer
along the tertiary surface water system
on places where artificial recharge or discharge wiii tgtke
place
elsewhere
Construction of the finite element mesh
Assume we have schemetized the groundwater basin iatu a finite
number of layers with different direction of flow (horizontal and
vertical) Within one layer the hydrological properties may vary
from place to place
Now a nodal grid is superimposed on each layer in a way already
described in section 32 Thus each layer has the same number of
nodal points situated on the same places in the horizontal plane
but with different z-coordinates
We are free to construct quadrilaterals or triangles because
the program automatically divides a quadrilateral into two triangles
having identical material properties The dimensions of the elements
20
should be small at places where large changes in hydrological
properties occur or where large gradients in hydraulic head are
expected to occur
Coding of nodal points
Each node in the finite element network is assigned an integer
code which indicates the type of calculation for this node
The coding is as follows
- Code = 1 nodal points with described head as function of time
- Code 2 internal nodal points
- Code = 3 internal nodal points which represent secondary surface
water system
- Code = 5 nodal points with prescribed artificial recharge or
discharge and points with prescribed boundary flow
(both as functions of time)
Other important endogenous variables are the magnitude of the
time step the value of the variable which stops the iteration
process and the maximum number of iterations allowed per time step
(see also group A and B of section 413)
412 Exogenous variables
After constructing the nodal grid each node represents part of
the groundwater basin Geo-hydrological properties can be assigned
to each mode based on physical data (group C D and E of section
413) In group F of section 413 time-dependent variables can
be assigned to each node
413 Data input to Program FEMSAT
The data input must be punched on cards according to the
following instructions
GROUP COLUMNS FORMAT SYMBOL DESCRIPTION
A 1-80 20A4 HEAD Desired heading to be printed at
the beginning of the execution
1-5 15 NUMNP number of nodal points
21
GROUP COLUMNS FORMAT SYMBOL DESCRIPTION
6-10 15
11-15 15
16-20 15
21-25 15
26-30 15
31-35 15
NUMEL
NUMLAY
MBAND
NTSPI
NTSPO
MAXIT
1-10 F103 ST
11-20 F103 DELT
21-30 F103 TMAX
31-40 F105 TOL
number of elements
number of layers
maximum difference in node number
between two adjacent nodes
number of time steps between a
change in non-stationary input data
number of time steps between two
successive outputs
maximum number of iterations per
time step
time to start with the execution
initial time interval
maximum time to be reached before
execution is stopped
maximum allowed absolute change in
the values of h between any two
successive iterations in a given
time step
Iterations continue until all changes
are less than or equal to TOL or
until MAXIT is exceeded
1- 5 15
6-10 15
11-15 15
16-20 15
KX(N1)
KX(N2)
KX(N3)
KX(N4)
sequential number of i-th corner of
element N
sequential number of j-th corner
sequential number of k-th corner
sequential number of 1-th corner
(see fig 8)
Fig 8 Numbering of element corner nodes
22
GROUP COLUMNS FORMAcircT SYMBOL DESCRIPTION
One card must be provided for each element in sequential order
starting with N = 1 and ending with N = NVJMEL
I-K 1103 SANG(NLY)
1i-iO MO3 C1(NLY)
21-30 F 103 C2(NLY)
21-4- 1-104 THI(NLY)
angle in degrees between CI and
the laquo-coordinate to he used in
nodal print N in layer LY (see
poundig 9)
First principal conductivity of
material in node N and layer LY
idem second principal conductivity
thickness of layer LY at nodal
point N
ig V Definition of SANGCI and C2
One card icircuiPt be provided for each model point ctarting wih M = 1
and ending with N = NUMNP If there is norlt3 than 1 layer for each
layer a set must be provided starting with the upper layer LY - 1
and ending with the bottom layer LY = NIJMLAY The variables SANG
M and 2 are not relevant tor aquitards but for reasons of
conveniency this procedure is maintained (making SANG CI and C2
dummy variables)
1-10 110 KODE(N)
11-20 II0 NTERR(N)
coding of nodal points (see
paragraph 42)
number of tertiary relation in node N
The q-h relationship of the flow to
23
GROUP COLUMNS FORMAT SYMBOL DESCRIPTION
21-30 FlOl X(N)
31-40 FlOl Y(N)
41-50 FlOl HGL(N)
51-60 FlOl HBSC(N)
61-70 FlOl LESC(N)
the tertiary system as discussed
in par 24 are schematized in a small
number of different relations
x-coardinate of node N
y-coordinate of node N
height ground level in node N
height bottom secondary conduit in
node N If node N is not a secondary
conduit this variable is a dummy
variable
length of secondary conduit in area
represented by node N
One card must be provided for each node starting with N = 1 and
ending with N = NUMNP
E 1-10 FlOl P(NLY1) initial value of h at node N in
layer LY
11-20 FlOl VERRES(NLY) vertical resistance at node N and
layer LY If the layer is an aquifer
this value is a dummy variable
21-30 F101 RADRES(NLY) radial resistance for flow to
secondary system at node N in layer
LY
31-40 FlOl ENRES(NLY) entrance resistance for flow to
secondary system at node N in layer
LY
41-50 FlOl WP(NLY) wet perimeter of secondary system at
node N in layer LY
51-60 FlOl P0R(NLY) specific yield of the material at
node N in layer LY
61-70 FlOl SS(NLY) specific storage of the material
at node N in layer LY
24
GROUP COLUMNS FORMAT SYMBOL DESCRIPTION
One card must be provided for each node starting with N = 1 and
ending with N = NUMNP If there is more than 1 layer for each layer
a set of cards must be provided starting with LY = 1 and ending
with LY = NUMLAY
F 1-10 F101 P(NLY1) value of h at different times at
node N in layer LY Node N lies
on a prescribed head boundary
11-20 F101 WTSC(N) water table in secondary system at
node N at different times
21-30 F 101 ARRED(NLY) artificial recharge to or discharge
from node N in layer LY or
prescribed boundary flow at
different times
31-40 F 101 QPHR(N) flux through phreatic surface at
node N at different times
The data in group F are non-stationary input data One card must be
provided for each node starting with N = 1 and ending with N = NUMNP
In case of more than 1 layer for each layer a set must be provided
Not relevant variables are taken as dummy variables
After a certain number of time steps (NTSPI) the data of group E
are read So the set cards E must contain NxLYxC cards where C =
(TMAXNTSPIXDELT) + 1 if DELT is a constant
42 R u n n i n g t h e p r o g r a m o n t h e C y b e r 7200
The program has been executed on a Cyber 7200 computer of
IWIS-TNO in the Hague
- The demanded input is punched on cards according to the instructions
of the last paragraph and is put on a permanent file
- For each specific problem inthe program only the dimension statements
in the main program have to be adjusted according to the instructions
in the lines of comment
- After compilation the program together with the input file is
executed
25
- Intermediate data necessary for continuation of the calculations
are saved on another permanent file This gives the opportunity
to restart the program at a time the calculation was stopped
The only change in the input file is to adjust the starting time
Remark
With the aid of a terminal small changes in the program and in the
input file can be made easily and the output immediately appears
on the terminal In this way it is possible to stop erroneous
calculations This is especially useful when being in the testing
phase of the program
43 O u t p u t
Immediately after reading the input data they are printed so
that possible errors can be detected During execution after each
certain number of time steps (NTSPO) calculated values of h and
water balance terms are printed In appendix B an example is given
The output data can be devided into two groups
A Data per node per layer
- Coding of the nodal points KODE
- Hydraulic head HEAD
- Flow through phreatic surface at the time indicated in head
of table FLOWUN
- Flow to tertiary surface water system FLOWTS
- Flow to secondary surface water system FLOWSS
- Flow to adjacent layers LEAKAGE
- Artificial recharge or discharge and prescribed boundary
flow ARRED
- Change in storage during last time step STORAGE
- Flow through prescribed head boundary LATFLOW
B Data per layer
Summation of the terms already mentioned under A gives the
magnitude of the water balance terms per layer at the time
indicated (instanteneous water balance terms) Summation of
these instanteneous terms multiplied by their respective time
intervals since time is zero gives the cumulative water balance
terms per layer
26
After each print out we are able to check the calculations
because the sum of the water balance terms per layer must be
approximately zero
44 C o m p u t i n g t i m e
The computing time depends on the number of nodal points and
the number of layers But it depends also on the magnitude of the
changes in hydraulic head and on the value which terminates the
Gauss-Seidel iteration (TOL) To give an impression TOL = 001 m
Costs per day simulation (in hlfl) on the Cyber 7200
number of nodes number of layer bull bdquo bull bull _ 1mdash x a x b (21)
where
a = costs in Dutch guilders per system second (ƒ 027)
b = empirical factor with magnitude about 001
V NUMERICAL EXPERIMENTS
In general there are two possibilities to test the validity of
numerical models The first and most commonly used method is to
compare numerical results with analytical solutions The second
possibility is to compare numerical results with measured field
data Since the last alternative requires many data that are
difficult to obtain we restrict ourselves for the time being to
comparison with the analytical solutions
5 1 F l o w t o a w e l l i n a n u n c o n f i n e d
c i r c u l a r - s h a p e d a q u i f e r
Flow to a well can be divided into steady-state flow and
unsteady-state flow (KRUSEMAN and DE RIDDER 1976)
27
511 Steady state flow in an unconfined aquifer
For this case the formula of Thiem-Dupuit is valid (see fig 10)
Q bull k lth2-hicircgt ln(r2r])
(22)
where
hhbdquo = hydraulic head in 1 and 2 respectively
Q = discharge from well
rbdquo = distance from well
CL31
hmdash^ri --r2
Fig 10 Schematic cross-section of a pumped unconfined aquifer
Fig 11 gives the nodal grid superimposed on a well in a
circular shaped aquifer Near the well the nodal point distance
is made smaller Quadrilaterals as well as triangles are used
Numerical calculations with this finite element network and 3 -1
an extraction rate of 200 m day were performed for 10 days
At that moment a steady-state situation was almost achieved
28
2tmdashnodal point number
160 180 X-axesim)
Fig 11 Nodal point configuration for flow to a well in a circular-
shaped aquifer The outer boundary is taken to be constant
The well is situated in nodal point 42
29
In fig 12 the results of the simulation are compared with the
analytical solution The agreement between numerical results and
the analytical solution is excellent
draw- 15irS (m) down
analytical solution imdashx calculated values
distance from well r (m)
Fig 12 Comparison between analyt ical and calculated s teady-state
drawdown for the problem of f ig 11 with 1
-1 Q bull - 200 m 3 d a y _ 1
k = 10 mday
d = 100 m
S = 2 y
S = 0 s
512 Unsteady-state flow in an unconfined aquifer
Analytical solutions for unsteady-state flow in an unconfined
aquifer only exist if the aquifer extends infinitely For the
aquifer given in fig 11 this condition is not fulfilled However
during the first one or two days after starting the extraction the
influence of a constant hydraulic head at the outer boundary on the
30
drawdown near the well can be neglected (see fig 13) After one
day the lateral inflow is only 25 of the extraction the remainder
being the change in storage due to drawdown
2 0 0
m 3 day 1
Q=200m 3 day
100-
vraquomdash Lateral inflow
N amp storage
I I ^ V - W 10
t (days)
Fig 13 Values of Q lateral inflow and change in storage as
functions of time
The nonsteady-state or Theis equation can be written as
i _ Q s 4iTkD
dy = 4irkD
W(u) (23)
where
u = r2S
4irkDt
r = distance from well
S = storage coefficient
Q = constant discharge from well
D = thickness of the waterbearing layer
at the beginning of pumping
t = time after starting the extraction
s = corrected drawdown
= s - 322D
s = calculated drawdown
W(u) = Theiss well function
[J
CIAT1
[T]
]
31
In fig 14 the analytical unsteady-state solution for the case
described in section 511 is given together with the numerical
results at times 05 and 10 days The agreement between numerical
results and the analytical solution is rather good
Drawdown s1(m)
10
10
10
Theiss solution
x caculated values t=Q5
bdquo t= lO
10 10- 10 10 t r2(daym2)
Fig 14 Comparison between analytical and calculated steady-state
drawdown for the problem of fig 11 with
Q
k
d
S y
s s
=
=
=
=
=
200
1
10
3 m
0
0
2
0
-1 day
m day
m
These results indicate that the assumption of a linear variation
of h between two nodal points (see fig 3 ) even with the rather
rough element network of fig 11 is acceptable So there is no
need to improve for example the finite element network by introducing
higher order coordinate functions
32
52 F l o w t o w e l l i n 3-1 a y e r e d s y s t e m
In fig 15 a schematic cross-section of a 3-layered system of
two aquifers separated by an aquitard is given A well of
infinitisemal radius is completed in the lower aquifer and discharges
at a constant rate QTT Each layer is homogeneous isotropic and
extending radially infinite
bdquo bull - bull bull ~ i
mdash Q II
aquifer I
lugrave^i~IcircIuml7i-i~~ aquitard --r-rl~-
-II--I i - i
aquifer
Fig 15 Schematic cross-section of a 3-layered system
For this situation NEUMAN and WITHERSPOON (1969a) give rather
complicated analytical solutions In the special case where the
hydraulic properties of the two aquifers are identical they give
graphical illustrations of their solutions for three different
cases
These graphical solutions give the opportunity to verify
roughly numerical results of a 3-layered system Therefore a nodal
grid is superimposed on a circular shaped groundwater flow system
with the extraction in the centre of the lower aquifer and with
a constant head on the outer boundary Fig 16 gives the nodal
point situation in the x - y plane of 1 layer The nodes in the
other two layers are situated in the middle of the layers at the
same places in the x - y plane only their z-coordinate is different
33
A
-Jy
bull33
V 5
iumlff
Fig 16 Nodal point configuration for the 3-layered problem (top view)
34
In fig 17 numerical results are compared with the analytical
solution The agreement is rather bad especially for the two
aquifers But in terms of the water balance the difference between
aquifers and aquitard is not so big because the aquitard has
about 16 times bigger storage capacity than the aquifers The
small value of the specific storage of the aquifers is the reason
that small errors in the components of the water balance have a
big influence on the calculated values of the hydraulic head
Eig 17 Dimensionless drawdown vs dimensionless time in a
3-layered system with
II = 50 m day
3 m (distance from well)
aquifer I
aquitard
aquifer II
Thi (m)
100
25
100
S s
001
0642
001
k(mday )
1
0277
1
The discrepancies are partly caused by errors in discretization
In the z-direction these errors can be eliminated by taking the
35
vertical resistance of the aquitard to be infinite Then the
calculated drawdown in the pumped aquifer can be compared with
the analytical Theiss solution
This is done in fig 18 for different distances from the well
One sees that the agreement is better when the distance is bigger
Apparently the linearization of the logarithmic drawdown curve
near the well is responsible for the differences The differences
for r = 3 m are of the same magnitude as for the pumped aquifer in
the 3-layered case
joo gt A 50
20
Theiss solution
bull calculated values r=3m (distance f rom well) x bdquo bdquo r=7m
1 L
I l_ 001 003 0 0 5 Ol Q2
- I I 5 10 t - K t
Fig 18 Dimensionless drawdown vs dimensionless time in a
confined aquifer with
Q =500 m day
k = 10 mday
D = 100 m
S = 001 s
The differences between numerical results and analytical solutions
are not caused by taking a finite area because changing the outer
boundary into an impermeable boundary has only minor effects
36
VI SUMMARY AND CONCLUSIONS
In this report a numerical model for non-stationary saturated
groundwater flow in a multi-layered system has been treated
Groundwater flow can be described in different ways
a Physical-mathematical by combining Darcys law and the law of
continuity (section 21)
b Geo-hydrological the flow in a natural groundwater basin takes
place in layers with different hydraulic properties Definitions
of the most important hydraulic properties are given in section
22
c Schematical groundwater flows horizontally in aquifers and
vertically in aquitards (section 23)
d Approximating by dividing the region of flow into a finite
number of elements and applying Darcys law and the law of
continuity on each of these elements In these ways one gets
a set of equations which can be solved given (initial and)
boundary conditions (section 25)
e Numerical the set of equations has been solved using a Galerkin
type finite element method and the implicit Gauss-Seidel method
with overrelaxation factor (section 31 and 32)
f Computer-solvable the numerical solution has been translated
in a FORTRAN IV program called FEMSAT (section 33)
In chapter IV the input execution and output of the program
FEMSAT are treated
Finally in chapter V some numerical applications are given
simulation of flow to a well in unconfined aquifer and simulation
of flow to a well in a 3-layered configuration
Comparison of hydraulic heads obtained by numerical calculations
and analytical true solutions showed an excellent agreement in case
of stationary flow and a good to moderate agreement in case of non-
stationary flow However the diviations caused by the discretization
process must be seen in relation to the errors made in the
establishment of the hydraulic properties
37
As was stated in the introduction numerical models can be
valuable tools in the management of water resources Therefore
numerical models must be suitable to simulate effects of certain
operations upon the spreading of water over saturated groundwater
unsaturated groundwater and surface water The model described in
this report is a model for saturated groundwater flow The relation
with the surface water can be taken into account via radial and
entrance resistances or via a relation between flow to the surface
water system and the hydraulic head The relation with the unsaturated
zone has been restricted to the input factor effective precipitation
In practise however the unsaturated and saturated system are
interconnected Therefore the model should be improved by taking
into account the unsaturated zone This can be done in at least
two ways
a Replacing the unsaturated zone by a parametrical model
b Coupling the unsaturated zone in an iterative way with the
saturated system
In order to find out the usefulness of the numerical model for
water resources management it will be applied to actual regional
groundwater systemslaquoAlso sensitivity analyses will be carried out
The computer model itself can be improved by applying standard
programs eg automatical generation of a nodal grid and graphical
plotting of the results
38
VII REFERENCES
ERNST LF 1962 Grondwaterstromingen in de verzadigde zone en hun
berekening bij aanwezigheid van horizontale evenwijdige open
leidingen Thesis University Utrecht pp 189
1978 Drainage of undulating soils with high groundwater
tables (in press)
FORSYTHE GE and WR WASON 1960 Finite Difference Methods for
Partial Differential Equations John Wiley and Sons New York
pp 444
KRUSEMAN GP and NA DE RIDDER 1976 Analysis and evaluation of
pumping test data ILRI-bulletin No 11 pp 200
LAAT PJM DE and C VAN DEN AKKER 1976 Simulatiemodel voor grond-
waterstroming en verdamping in Modelonderzoek 1971-1974 ten
behoeve van de waterhuishouding in Gelderland deel II
Grondslagen pp 145-197
NEUMAN SP RA FEDDES and E BRESLER 1974 Finite element
simulation of flow in saturated -unsaturated soils considering
water uptake by plants Hydraulic Engineering Laboratory
Technion Israel pp 104
and PA WITHERSPOON 1969a Theory of Flow in a confined
two Aquifer System Water Resources Research Vol 5 No 4
pp 803-816
^_ and PA WITHERSPOON 1969b Applicability of Current Theories
of Flow in leaky aquifers Water Resources Research Vol 5
No 4 pp 817-829
PINDER GF and EO FRIND 1972 Application of Galerkins Procedure
to Aquifer Analysis Water Resources Research Vol 8 no 1
pp 108-120
REMSON I GE HORNBERGER and FJ MOLZ 1971 Numerical Methods
in Subsurface Hydrology John Wiley and Sons New York
pp 389
WESSELING JG 1977 Toepassing van Finite Difference- en Finite
Element Methoden voor het oplossen van een aantal grondwatershy
stromingsproblemen (met computerprogrammas) ICW-nota 944
pp 63
39
ZIENKIEWICZ OC 1971 Finite Element Method in Engineering Science
McGraw-Hill London pp 521
40
z ltc laquo LU U EC LU ltC
c z O E 0C 3 GLZ
X - e - o
gt-o-
LUX
E 2 3 - -Z o
O U K raquo-lt-laquo-
lt-gt z o z
laquoC - L
M II II II II II II
IS - Z w - U U - bull II II bull II - a - laquo - o o bulllaquo- bullbull mdash bullmdashbull Il bullmdash II f M Ol II mdashbullgt -gt -ltmdashgt -
O bulllaquobull gt- Il f - U J U J gt - - ^ U J - gt Œ ) gt gt - raquo J _ l gt - _ J U J Œ l Œ I LU _ l Z bull laquo - 3 3 - 1 laquo - 3 - laquo - bull laquo - _ i _J raquo-J UJ _J w 3 bull
bull O -raquo Z Z lt-gtZgt-gt w ltT ucirc w h Z II II h - Z I S S I raquo-H KH ( raquo gt-i O C D t O L U U I C C 3 ) - i Z t O 0 J M l f l ucirc l - l - K ucirc h ucirc h ( h laquo J 0 C O O h 3 l -
w 3 laquo z z ltrzltt J j o c i - z z J j U l i J O O l i i O O O U I O U i O O U l L 4 I A H O l L l i H K Ucirc Ucirc Iuml U U Ucirc I Iuml U K U Ucirc U U U U U U U U
ltc oc
z
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O Uuml I H o i-t - n Z C l H 3 P H-l Xgt bull z u r C tO H
C l - 3 w 3gt Cl
r o n c f l H H c ouml lt-laquo - n
z z a - 1 u i - H 3 T
+ Z II C l ^ I C l f l H
11 m w bull ro c i z
3 C 3 3gt 3
T C O II Z
+ 3 l
O
r o
3
r o
3
) - 03
o z
o o gt c ^ c ^ o ^ o ^ t n t n o i i 0 t n t n y i t n t n i K ^ ^ ^ ^ ^ ^ gt ^ t j c o u c o o J W u x t o w c n t c ^ ^ S s 3 X N ^ o T gt 4 ^ w ^ œ N O O N j ( gt c n t J w ^ a D s 3 œ N i o ^ c n gt c ugrave r o ^ c a s 3 œ N j O W I S l B S Iuml S S iuml I B l S f f l C B l S O O O S I S Œ l S l S i Œ J I S S l C D l S S l S l S icirc S l B i S l I B C C l B C D S i S C D l S l B C D i S i i S Il I Il I raquo Il II li II M li II II II II li H II 11 II II II il II II II h 11 l II II II II II II t |t jj raquo Il M il
t n iS
m -x r -Z T- O c - t z
ja- w ro j s es S (S CD C3 IS
- n - 0 - r j T j - n x T i - O T icirc - o T i - D
an CD
rf 0 n -o - n -0 x C X O 3 3 O 3 3 O 3 3 O 3 0 O 3 3 3 O 3 3 O 3 3 V 3 3 ^ - I 3 3 - 3 3 raquo - I 3 3 gt - I 3 3 ) - ( 3 3 I - I
C - I 3 Z 3 Z 3 Z 3 Z 3 Z 3 Z 33 1-1 Z Z
ff
I gt - f X - - l 3 gt - H X raquo - I gt - H 3 gt - i bull-i mdashl mdash1 mdash1 mdashi mdash1 laquo | J gt - j gt - j ^ - f c gt 0 4 N ( r o - gt j gt i S i N C D r n Œ c i reg t n s t n reg t n s ^o -x - x - x - x - x - x r
r - w 2 gt uuml l X gt ] gt T i - n T i - r j r -raquo z n n icirc s j r r r r raquo H o n c - i iuml o r o i - lt m c c s w m c n c c n m raquo bull H j ^ - n o xgt ^ 3 3 Tgt lt-r tz mdash w iuml- i - t m H r r - r - r - - lt r r c r - o - lt z
bulllt x _bull s- -lt -lt _ C3 -n _ - i - _ rgt _ bull t n - -i - - M rlt r - c - m n -gt o n o œ n u n n - n ^ c - z - i ^ 3 gt 3 gt - n o r - c a
z o r s n r z u icirc N H O - 3 3 0 3 3 m f - O f J l w i c ^ ^ c m z ^ raquo f x mdash 33 r- 0 - 1 p s | -O bull-raquo 3 bulllt 3gt - - - raquo - lt - lt
3 J M raquo H _ - 3 Z 3 Z t n 3gt H Xraquo- -H S - 1 -H
I 03 f - NI r -fc copy bull 3D -lt t n t n -il X ~i X ~i -raquo - n - n - T U mdash n r z r - - t r c c 0 cn a z 3 c - e - - n a i gt -s - n - - n -c 0 r 0 r
mdashf c - i cn c z m 2 33 en H xgt i - H 3gt C 33 33 -lt X
_ 3
-k
33 s c e s t n
j t i T T i - i i n n A Z O S O C O O X d 33
U) 3gt V Z N O bull
t o
Z n m
00 m 0
z z
z C l
z
raquo J J
- -n bullbull-gt r w 33 J Z 2 1gt n bullbull r n
3 Z 3 3 - 1 -H 1 gt - H 1 gt B H H H
V t n Ni
mdash1 m 33 3 t n
- n
c xgt
m x 55
f j
M zgt -n
O)
_ D t 3 - T) C bulln c 0 3= 0 t o 0 x
3= w gt0 3 raquo-i 3 3 r x raquo i gt r - r 0 3 Z u 3 X 3 Z 33 -lt 33
- 1 - i Z Z m bull 33 lt-raquo -lt t n iuml gt
o ^ r - r - c c o z m z m z CD t o t o F I H
t n t n
- n as - T] S) -raquobull bull œ w raquo - t o ro bull-gt CD X
en - n
es
u bull-bull
bull-1 m c -z bull ^ ^ - z r w --lt - 0 - raquo w yO 1mdash1
3gt Z Xgt n - f n 0 n 1-1 [Tl r-i Z 03 Z
-lt -lt icirc-f
- z o 0 - f-1 bull-- z z
J
0 11
w z -0 c 33 3 w Z Z O H -
t gt bull t o -
CS
r
-lt z
7Z O O
H - t S) 3gt - 1
- 1 r o r - r - lt-raquo r o
h ) ^ f m n f i V n i gt t n
J X I - 3gt z c i r c m i gt 3 ~lt r m bulljgt raquo
-lt rgt z raquo 0 3 3 0 m m 0
3 0 0 m
t o H
^ 00 X
iuml gt - 4
-H
3 m
- n
t n
t o
bullbull0 x
z
3
3gt
C -lt Igt - t Ci
APPENDIX 8 EXAMPLE OF OUTPUT
HEAD AND TERMS OF UATERBALANCE PER NODE PER LAYER
AT TIME 260 IN II AND I13DAYS
T
LAYER NODE KODE HEAD FLOUUN FLOUTS FLOUSS LEAKAGE STORAGE LAT FLOU
13 19 25 31 37 43
125008 125000 12S000 12S000 12S000 12S000 12S000 12S000
0000 0000 0000 0000 0000 0000 0000 0000
0000 0000 0000 0000 0000 0000 0000 0000
0000 0000 0000 0000 0000 0000 0000 0000
- 000 - 000 -000 - 000 - 000 - 000 - 000 0000
0000 0000 0000 0000 0000 0000 0000 0000
0000 000 000 000 000 000 000 000
000 0000 0000 0P100 0000 0000 0000 0000
1 7
13 19 25 31 37
12S000 125000 125000 125000 125000 12S000 125000 125000
0000 0000 0000 0000 0000 0000 0000 0000
- 001 -007 - 023 - 020 - 009 - 003 0000
0000 0000 0000 0000 0000 0000 0000 0000
000
001
007
023
020
009
003
003
0000 0000 0000 0000 0000 0000 0000 0000
1 7
13 19 25 31 37 43
125000 125000 124998 124972 124895 124787 124647 124358
0000 0000 0000 0000 0000 0000 0000 0000
- 000 001 007 023 020 009 003
0000
0000 0000 0000 0000 0000 0000 0000
50000
0000 182
1982 3862 1674 419 078 041
0000 0000 0000 0000 0000 0000 0000 0000
TERMS OF WATERBALANCE PER LAYER DURING LAST TIMESTEP AND
AND SINCE BEGINNING IN M3DAY M3
FLOU TO UNSATURATED ZONE
FLOU TO TERTIARY SYSTEM
0000
0000
0000
0000
LAYER NO 1
FLOU TO SECONDARY SYSTEM
FLOU TO AD3ACENT LAYERCS)
ARTIFICIAL FLOU AND PRESCRIBED BOUNDARY FLOU
ACCUMULATION TERMS
LATERAL FLOU THROUGH PRESCR HEAD BOUNDARY
0000
-000
0000
000
000
0000
-000
0000
000
000
LAYER NO 2
FLOU TO SECONDARY SYSTEM
FLOU TO AD3ACENT LAYERCS1
ARTIFICIAL FLOU AND PRESCRIBED BOUNDARY FLOU
ACCUMULATION TERMS
LATERAL FLOU THROUGH PRESCR HEAD BOUNDARY
0000
-378
0000
381
0000
0000
-055
0000
056
0000
LAYER NO 3
FLOU TO SECONDARY SYSTEM
FLOU TO AD3ACENT LAYERCS5
ARTIFICIAL FLOU AND PRESCRIBED BOUNDARY FLOU
ACCUMULATION TERMS
LATERAL FLOU THROUGH PRESCR HEAD BOUNDARY
0000
370
50000
49376
0000
0000
055
12979
12909
0000
point 2 in the aquitard at time t = t and in iteration step j + 1
is
k q = (hf -hf )-iumlmdash f (17)
1 n 2 n OSxd1
and the flux from point 3 in aquifer 2 to point 2 in the aquitard
is
1 c 1
qbdquo = (h - hf) mdashHmdashr- (18) 2 3 n 2 n OSxd
The net flux ltl-q9 must be equal to the change in storage per unit
area per unit time This change in storage can numerically be
approximated by
te - h-
n n-1
Combina t ion of ( 1 7 ) (18) ana (19) g i v e s
l tr1 - h (hJ + 1 + hj - 2 hj+1) - - - - - - S1 - ilaquoS L u l l = o (20)
] i 3 n 2 n 1 c t - t j j x d n n -1
T + 1
From t h i s e q u a t i o n h^ kan be s o l v e d
3 3 C o m p u t e r o g r a m
The problem outl^ugrave so far has been translated into a computer
program written in FORTRAN IV
The flow chart ot Uis program is given in fig 7 whereas in
appendix A the complete listing of the program can be found
In the flow chart the 5 main parts of the program are
indicated
A Reading and printing of input data
B Determination of the matrices A and B and the vector Q per layer
17
M
1 A
FLOH to unsaturated zono
FLOW to tertiary syster
FLOW to secondnrv system
Leakage to adjacent aquitards
SOLVE equation (14)
yes
CALCULATE flux through constant head boundnr
CALCULATE waterbalance n p r 1 OAro r r J j - ~
PRINT of output
Ier node per layer
k = k+1
t = t + At
7 DUMP variables on tape
-gt-
( STOP )
Fig 7 Flow chart of the program FEMSAT with diviation in 5 main
parts A B C D and E
C START PROGRAM FEMSAT
f READ general information
PRINT general information
f READ element information
PRINT element information 7 READ stationary nodal point information
PRINT stationary nodal point information 7
READ non-stationary nodal point information for t = t
FRINT non-stationary nodal point information for t = t
L 2 (
^ yes R E A D non-stat input for t gt t
J PRINT non-stat input for t gt t (- 1 -- M degY
GENERATE A B and AREA-matrix for confined aquifers
_L
GENERATE A B and AREA-matrix for unconfined aouifer
bull
C Solving h with Gauss-Seidel
D Determination of the water balance terms per layer
E Printing of results
The program exists of a main program with several subroutines
The main program calls the subroutines In the subroutines most
calculations are performed
In this way an easy adaptation to different problems is
obtained (see also section 44) Comments in the program will
help the reader to understand the program
In the next chapter the preparation of the input data executing
of the program and the output will be treated
Acknowledgement
Some parts of the computer program were taken from the program
UNSAT2 of NEUMAN etal (1974)
IV INPUT EXECUTION AND OUTPUT
41 I n p u t
The input is divided into endogenous variables (chosen by the
user or calculated by the program during execution) and exogenous
variables (based on physical data)
411 Endogenous variables
Translation of groundwater basin into a nodal point network
Probably the most important but also the most difficult stage
in the process of simulating groundwater flow is the translation
of the natural groundwater basin into a nodal point network Questions
which arise are
a In how many layer the geo-hydrological system mustcan be
schematized
b The locations of the boundaries and the nature of these boundaries
(impervious prescribed head prescribed flux)
c The configuration of the nodal points
19
The answers of these questions depend on the problem under
consideration
- Schematization into layers
Usually our knowledge about the geo-hydrological situation is
such that no more than 5 different layers can be distinguished
(remember contrast rule)
- Boundary conditions
The top boundary condition of a groundwater basin usually is
a prescribed flux (effective precipitation)
The bottom boundary condition usually is a prescribed flux
boundary with flux zero (impervious hydrological base)
For the boundaries at the other sides of the system one sees
either prescribed head boundaries (eg a fully penetrating
river) or prescribed flux boundaries (including zero flux)
- Nodal point configuration
Nodal points are situated
on the boundaries
along internal boundaries which indicate changes in
hydraulic properties within the layer
along the tertiary surface water system
on places where artificial recharge or discharge wiii tgtke
place
elsewhere
Construction of the finite element mesh
Assume we have schemetized the groundwater basin iatu a finite
number of layers with different direction of flow (horizontal and
vertical) Within one layer the hydrological properties may vary
from place to place
Now a nodal grid is superimposed on each layer in a way already
described in section 32 Thus each layer has the same number of
nodal points situated on the same places in the horizontal plane
but with different z-coordinates
We are free to construct quadrilaterals or triangles because
the program automatically divides a quadrilateral into two triangles
having identical material properties The dimensions of the elements
20
should be small at places where large changes in hydrological
properties occur or where large gradients in hydraulic head are
expected to occur
Coding of nodal points
Each node in the finite element network is assigned an integer
code which indicates the type of calculation for this node
The coding is as follows
- Code = 1 nodal points with described head as function of time
- Code 2 internal nodal points
- Code = 3 internal nodal points which represent secondary surface
water system
- Code = 5 nodal points with prescribed artificial recharge or
discharge and points with prescribed boundary flow
(both as functions of time)
Other important endogenous variables are the magnitude of the
time step the value of the variable which stops the iteration
process and the maximum number of iterations allowed per time step
(see also group A and B of section 413)
412 Exogenous variables
After constructing the nodal grid each node represents part of
the groundwater basin Geo-hydrological properties can be assigned
to each mode based on physical data (group C D and E of section
413) In group F of section 413 time-dependent variables can
be assigned to each node
413 Data input to Program FEMSAT
The data input must be punched on cards according to the
following instructions
GROUP COLUMNS FORMAT SYMBOL DESCRIPTION
A 1-80 20A4 HEAD Desired heading to be printed at
the beginning of the execution
1-5 15 NUMNP number of nodal points
21
GROUP COLUMNS FORMAT SYMBOL DESCRIPTION
6-10 15
11-15 15
16-20 15
21-25 15
26-30 15
31-35 15
NUMEL
NUMLAY
MBAND
NTSPI
NTSPO
MAXIT
1-10 F103 ST
11-20 F103 DELT
21-30 F103 TMAX
31-40 F105 TOL
number of elements
number of layers
maximum difference in node number
between two adjacent nodes
number of time steps between a
change in non-stationary input data
number of time steps between two
successive outputs
maximum number of iterations per
time step
time to start with the execution
initial time interval
maximum time to be reached before
execution is stopped
maximum allowed absolute change in
the values of h between any two
successive iterations in a given
time step
Iterations continue until all changes
are less than or equal to TOL or
until MAXIT is exceeded
1- 5 15
6-10 15
11-15 15
16-20 15
KX(N1)
KX(N2)
KX(N3)
KX(N4)
sequential number of i-th corner of
element N
sequential number of j-th corner
sequential number of k-th corner
sequential number of 1-th corner
(see fig 8)
Fig 8 Numbering of element corner nodes
22
GROUP COLUMNS FORMAcircT SYMBOL DESCRIPTION
One card must be provided for each element in sequential order
starting with N = 1 and ending with N = NVJMEL
I-K 1103 SANG(NLY)
1i-iO MO3 C1(NLY)
21-30 F 103 C2(NLY)
21-4- 1-104 THI(NLY)
angle in degrees between CI and
the laquo-coordinate to he used in
nodal print N in layer LY (see
poundig 9)
First principal conductivity of
material in node N and layer LY
idem second principal conductivity
thickness of layer LY at nodal
point N
ig V Definition of SANGCI and C2
One card icircuiPt be provided for each model point ctarting wih M = 1
and ending with N = NUMNP If there is norlt3 than 1 layer for each
layer a set must be provided starting with the upper layer LY - 1
and ending with the bottom layer LY = NIJMLAY The variables SANG
M and 2 are not relevant tor aquitards but for reasons of
conveniency this procedure is maintained (making SANG CI and C2
dummy variables)
1-10 110 KODE(N)
11-20 II0 NTERR(N)
coding of nodal points (see
paragraph 42)
number of tertiary relation in node N
The q-h relationship of the flow to
23
GROUP COLUMNS FORMAT SYMBOL DESCRIPTION
21-30 FlOl X(N)
31-40 FlOl Y(N)
41-50 FlOl HGL(N)
51-60 FlOl HBSC(N)
61-70 FlOl LESC(N)
the tertiary system as discussed
in par 24 are schematized in a small
number of different relations
x-coardinate of node N
y-coordinate of node N
height ground level in node N
height bottom secondary conduit in
node N If node N is not a secondary
conduit this variable is a dummy
variable
length of secondary conduit in area
represented by node N
One card must be provided for each node starting with N = 1 and
ending with N = NUMNP
E 1-10 FlOl P(NLY1) initial value of h at node N in
layer LY
11-20 FlOl VERRES(NLY) vertical resistance at node N and
layer LY If the layer is an aquifer
this value is a dummy variable
21-30 F101 RADRES(NLY) radial resistance for flow to
secondary system at node N in layer
LY
31-40 FlOl ENRES(NLY) entrance resistance for flow to
secondary system at node N in layer
LY
41-50 FlOl WP(NLY) wet perimeter of secondary system at
node N in layer LY
51-60 FlOl P0R(NLY) specific yield of the material at
node N in layer LY
61-70 FlOl SS(NLY) specific storage of the material
at node N in layer LY
24
GROUP COLUMNS FORMAT SYMBOL DESCRIPTION
One card must be provided for each node starting with N = 1 and
ending with N = NUMNP If there is more than 1 layer for each layer
a set of cards must be provided starting with LY = 1 and ending
with LY = NUMLAY
F 1-10 F101 P(NLY1) value of h at different times at
node N in layer LY Node N lies
on a prescribed head boundary
11-20 F101 WTSC(N) water table in secondary system at
node N at different times
21-30 F 101 ARRED(NLY) artificial recharge to or discharge
from node N in layer LY or
prescribed boundary flow at
different times
31-40 F 101 QPHR(N) flux through phreatic surface at
node N at different times
The data in group F are non-stationary input data One card must be
provided for each node starting with N = 1 and ending with N = NUMNP
In case of more than 1 layer for each layer a set must be provided
Not relevant variables are taken as dummy variables
After a certain number of time steps (NTSPI) the data of group E
are read So the set cards E must contain NxLYxC cards where C =
(TMAXNTSPIXDELT) + 1 if DELT is a constant
42 R u n n i n g t h e p r o g r a m o n t h e C y b e r 7200
The program has been executed on a Cyber 7200 computer of
IWIS-TNO in the Hague
- The demanded input is punched on cards according to the instructions
of the last paragraph and is put on a permanent file
- For each specific problem inthe program only the dimension statements
in the main program have to be adjusted according to the instructions
in the lines of comment
- After compilation the program together with the input file is
executed
25
- Intermediate data necessary for continuation of the calculations
are saved on another permanent file This gives the opportunity
to restart the program at a time the calculation was stopped
The only change in the input file is to adjust the starting time
Remark
With the aid of a terminal small changes in the program and in the
input file can be made easily and the output immediately appears
on the terminal In this way it is possible to stop erroneous
calculations This is especially useful when being in the testing
phase of the program
43 O u t p u t
Immediately after reading the input data they are printed so
that possible errors can be detected During execution after each
certain number of time steps (NTSPO) calculated values of h and
water balance terms are printed In appendix B an example is given
The output data can be devided into two groups
A Data per node per layer
- Coding of the nodal points KODE
- Hydraulic head HEAD
- Flow through phreatic surface at the time indicated in head
of table FLOWUN
- Flow to tertiary surface water system FLOWTS
- Flow to secondary surface water system FLOWSS
- Flow to adjacent layers LEAKAGE
- Artificial recharge or discharge and prescribed boundary
flow ARRED
- Change in storage during last time step STORAGE
- Flow through prescribed head boundary LATFLOW
B Data per layer
Summation of the terms already mentioned under A gives the
magnitude of the water balance terms per layer at the time
indicated (instanteneous water balance terms) Summation of
these instanteneous terms multiplied by their respective time
intervals since time is zero gives the cumulative water balance
terms per layer
26
After each print out we are able to check the calculations
because the sum of the water balance terms per layer must be
approximately zero
44 C o m p u t i n g t i m e
The computing time depends on the number of nodal points and
the number of layers But it depends also on the magnitude of the
changes in hydraulic head and on the value which terminates the
Gauss-Seidel iteration (TOL) To give an impression TOL = 001 m
Costs per day simulation (in hlfl) on the Cyber 7200
number of nodes number of layer bull bdquo bull bull _ 1mdash x a x b (21)
where
a = costs in Dutch guilders per system second (ƒ 027)
b = empirical factor with magnitude about 001
V NUMERICAL EXPERIMENTS
In general there are two possibilities to test the validity of
numerical models The first and most commonly used method is to
compare numerical results with analytical solutions The second
possibility is to compare numerical results with measured field
data Since the last alternative requires many data that are
difficult to obtain we restrict ourselves for the time being to
comparison with the analytical solutions
5 1 F l o w t o a w e l l i n a n u n c o n f i n e d
c i r c u l a r - s h a p e d a q u i f e r
Flow to a well can be divided into steady-state flow and
unsteady-state flow (KRUSEMAN and DE RIDDER 1976)
27
511 Steady state flow in an unconfined aquifer
For this case the formula of Thiem-Dupuit is valid (see fig 10)
Q bull k lth2-hicircgt ln(r2r])
(22)
where
hhbdquo = hydraulic head in 1 and 2 respectively
Q = discharge from well
rbdquo = distance from well
CL31
hmdash^ri --r2
Fig 10 Schematic cross-section of a pumped unconfined aquifer
Fig 11 gives the nodal grid superimposed on a well in a
circular shaped aquifer Near the well the nodal point distance
is made smaller Quadrilaterals as well as triangles are used
Numerical calculations with this finite element network and 3 -1
an extraction rate of 200 m day were performed for 10 days
At that moment a steady-state situation was almost achieved
28
2tmdashnodal point number
160 180 X-axesim)
Fig 11 Nodal point configuration for flow to a well in a circular-
shaped aquifer The outer boundary is taken to be constant
The well is situated in nodal point 42
29
In fig 12 the results of the simulation are compared with the
analytical solution The agreement between numerical results and
the analytical solution is excellent
draw- 15irS (m) down
analytical solution imdashx calculated values
distance from well r (m)
Fig 12 Comparison between analyt ical and calculated s teady-state
drawdown for the problem of f ig 11 with 1
-1 Q bull - 200 m 3 d a y _ 1
k = 10 mday
d = 100 m
S = 2 y
S = 0 s
512 Unsteady-state flow in an unconfined aquifer
Analytical solutions for unsteady-state flow in an unconfined
aquifer only exist if the aquifer extends infinitely For the
aquifer given in fig 11 this condition is not fulfilled However
during the first one or two days after starting the extraction the
influence of a constant hydraulic head at the outer boundary on the
30
drawdown near the well can be neglected (see fig 13) After one
day the lateral inflow is only 25 of the extraction the remainder
being the change in storage due to drawdown
2 0 0
m 3 day 1
Q=200m 3 day
100-
vraquomdash Lateral inflow
N amp storage
I I ^ V - W 10
t (days)
Fig 13 Values of Q lateral inflow and change in storage as
functions of time
The nonsteady-state or Theis equation can be written as
i _ Q s 4iTkD
dy = 4irkD
W(u) (23)
where
u = r2S
4irkDt
r = distance from well
S = storage coefficient
Q = constant discharge from well
D = thickness of the waterbearing layer
at the beginning of pumping
t = time after starting the extraction
s = corrected drawdown
= s - 322D
s = calculated drawdown
W(u) = Theiss well function
[J
CIAT1
[T]
]
31
In fig 14 the analytical unsteady-state solution for the case
described in section 511 is given together with the numerical
results at times 05 and 10 days The agreement between numerical
results and the analytical solution is rather good
Drawdown s1(m)
10
10
10
Theiss solution
x caculated values t=Q5
bdquo t= lO
10 10- 10 10 t r2(daym2)
Fig 14 Comparison between analytical and calculated steady-state
drawdown for the problem of fig 11 with
Q
k
d
S y
s s
=
=
=
=
=
200
1
10
3 m
0
0
2
0
-1 day
m day
m
These results indicate that the assumption of a linear variation
of h between two nodal points (see fig 3 ) even with the rather
rough element network of fig 11 is acceptable So there is no
need to improve for example the finite element network by introducing
higher order coordinate functions
32
52 F l o w t o w e l l i n 3-1 a y e r e d s y s t e m
In fig 15 a schematic cross-section of a 3-layered system of
two aquifers separated by an aquitard is given A well of
infinitisemal radius is completed in the lower aquifer and discharges
at a constant rate QTT Each layer is homogeneous isotropic and
extending radially infinite
bdquo bull - bull bull ~ i
mdash Q II
aquifer I
lugrave^i~IcircIuml7i-i~~ aquitard --r-rl~-
-II--I i - i
aquifer
Fig 15 Schematic cross-section of a 3-layered system
For this situation NEUMAN and WITHERSPOON (1969a) give rather
complicated analytical solutions In the special case where the
hydraulic properties of the two aquifers are identical they give
graphical illustrations of their solutions for three different
cases
These graphical solutions give the opportunity to verify
roughly numerical results of a 3-layered system Therefore a nodal
grid is superimposed on a circular shaped groundwater flow system
with the extraction in the centre of the lower aquifer and with
a constant head on the outer boundary Fig 16 gives the nodal
point situation in the x - y plane of 1 layer The nodes in the
other two layers are situated in the middle of the layers at the
same places in the x - y plane only their z-coordinate is different
33
A
-Jy
bull33
V 5
iumlff
Fig 16 Nodal point configuration for the 3-layered problem (top view)
34
In fig 17 numerical results are compared with the analytical
solution The agreement is rather bad especially for the two
aquifers But in terms of the water balance the difference between
aquifers and aquitard is not so big because the aquitard has
about 16 times bigger storage capacity than the aquifers The
small value of the specific storage of the aquifers is the reason
that small errors in the components of the water balance have a
big influence on the calculated values of the hydraulic head
Eig 17 Dimensionless drawdown vs dimensionless time in a
3-layered system with
II = 50 m day
3 m (distance from well)
aquifer I
aquitard
aquifer II
Thi (m)
100
25
100
S s
001
0642
001
k(mday )
1
0277
1
The discrepancies are partly caused by errors in discretization
In the z-direction these errors can be eliminated by taking the
35
vertical resistance of the aquitard to be infinite Then the
calculated drawdown in the pumped aquifer can be compared with
the analytical Theiss solution
This is done in fig 18 for different distances from the well
One sees that the agreement is better when the distance is bigger
Apparently the linearization of the logarithmic drawdown curve
near the well is responsible for the differences The differences
for r = 3 m are of the same magnitude as for the pumped aquifer in
the 3-layered case
joo gt A 50
20
Theiss solution
bull calculated values r=3m (distance f rom well) x bdquo bdquo r=7m
1 L
I l_ 001 003 0 0 5 Ol Q2
- I I 5 10 t - K t
Fig 18 Dimensionless drawdown vs dimensionless time in a
confined aquifer with
Q =500 m day
k = 10 mday
D = 100 m
S = 001 s
The differences between numerical results and analytical solutions
are not caused by taking a finite area because changing the outer
boundary into an impermeable boundary has only minor effects
36
VI SUMMARY AND CONCLUSIONS
In this report a numerical model for non-stationary saturated
groundwater flow in a multi-layered system has been treated
Groundwater flow can be described in different ways
a Physical-mathematical by combining Darcys law and the law of
continuity (section 21)
b Geo-hydrological the flow in a natural groundwater basin takes
place in layers with different hydraulic properties Definitions
of the most important hydraulic properties are given in section
22
c Schematical groundwater flows horizontally in aquifers and
vertically in aquitards (section 23)
d Approximating by dividing the region of flow into a finite
number of elements and applying Darcys law and the law of
continuity on each of these elements In these ways one gets
a set of equations which can be solved given (initial and)
boundary conditions (section 25)
e Numerical the set of equations has been solved using a Galerkin
type finite element method and the implicit Gauss-Seidel method
with overrelaxation factor (section 31 and 32)
f Computer-solvable the numerical solution has been translated
in a FORTRAN IV program called FEMSAT (section 33)
In chapter IV the input execution and output of the program
FEMSAT are treated
Finally in chapter V some numerical applications are given
simulation of flow to a well in unconfined aquifer and simulation
of flow to a well in a 3-layered configuration
Comparison of hydraulic heads obtained by numerical calculations
and analytical true solutions showed an excellent agreement in case
of stationary flow and a good to moderate agreement in case of non-
stationary flow However the diviations caused by the discretization
process must be seen in relation to the errors made in the
establishment of the hydraulic properties
37
As was stated in the introduction numerical models can be
valuable tools in the management of water resources Therefore
numerical models must be suitable to simulate effects of certain
operations upon the spreading of water over saturated groundwater
unsaturated groundwater and surface water The model described in
this report is a model for saturated groundwater flow The relation
with the surface water can be taken into account via radial and
entrance resistances or via a relation between flow to the surface
water system and the hydraulic head The relation with the unsaturated
zone has been restricted to the input factor effective precipitation
In practise however the unsaturated and saturated system are
interconnected Therefore the model should be improved by taking
into account the unsaturated zone This can be done in at least
two ways
a Replacing the unsaturated zone by a parametrical model
b Coupling the unsaturated zone in an iterative way with the
saturated system
In order to find out the usefulness of the numerical model for
water resources management it will be applied to actual regional
groundwater systemslaquoAlso sensitivity analyses will be carried out
The computer model itself can be improved by applying standard
programs eg automatical generation of a nodal grid and graphical
plotting of the results
38
VII REFERENCES
ERNST LF 1962 Grondwaterstromingen in de verzadigde zone en hun
berekening bij aanwezigheid van horizontale evenwijdige open
leidingen Thesis University Utrecht pp 189
1978 Drainage of undulating soils with high groundwater
tables (in press)
FORSYTHE GE and WR WASON 1960 Finite Difference Methods for
Partial Differential Equations John Wiley and Sons New York
pp 444
KRUSEMAN GP and NA DE RIDDER 1976 Analysis and evaluation of
pumping test data ILRI-bulletin No 11 pp 200
LAAT PJM DE and C VAN DEN AKKER 1976 Simulatiemodel voor grond-
waterstroming en verdamping in Modelonderzoek 1971-1974 ten
behoeve van de waterhuishouding in Gelderland deel II
Grondslagen pp 145-197
NEUMAN SP RA FEDDES and E BRESLER 1974 Finite element
simulation of flow in saturated -unsaturated soils considering
water uptake by plants Hydraulic Engineering Laboratory
Technion Israel pp 104
and PA WITHERSPOON 1969a Theory of Flow in a confined
two Aquifer System Water Resources Research Vol 5 No 4
pp 803-816
^_ and PA WITHERSPOON 1969b Applicability of Current Theories
of Flow in leaky aquifers Water Resources Research Vol 5
No 4 pp 817-829
PINDER GF and EO FRIND 1972 Application of Galerkins Procedure
to Aquifer Analysis Water Resources Research Vol 8 no 1
pp 108-120
REMSON I GE HORNBERGER and FJ MOLZ 1971 Numerical Methods
in Subsurface Hydrology John Wiley and Sons New York
pp 389
WESSELING JG 1977 Toepassing van Finite Difference- en Finite
Element Methoden voor het oplossen van een aantal grondwatershy
stromingsproblemen (met computerprogrammas) ICW-nota 944
pp 63
39
ZIENKIEWICZ OC 1971 Finite Element Method in Engineering Science
McGraw-Hill London pp 521
40
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APPENDIX 8 EXAMPLE OF OUTPUT
HEAD AND TERMS OF UATERBALANCE PER NODE PER LAYER
AT TIME 260 IN II AND I13DAYS
T
LAYER NODE KODE HEAD FLOUUN FLOUTS FLOUSS LEAKAGE STORAGE LAT FLOU
13 19 25 31 37 43
125008 125000 12S000 12S000 12S000 12S000 12S000 12S000
0000 0000 0000 0000 0000 0000 0000 0000
0000 0000 0000 0000 0000 0000 0000 0000
0000 0000 0000 0000 0000 0000 0000 0000
- 000 - 000 -000 - 000 - 000 - 000 - 000 0000
0000 0000 0000 0000 0000 0000 0000 0000
0000 000 000 000 000 000 000 000
000 0000 0000 0P100 0000 0000 0000 0000
1 7
13 19 25 31 37
12S000 125000 125000 125000 125000 12S000 125000 125000
0000 0000 0000 0000 0000 0000 0000 0000
- 001 -007 - 023 - 020 - 009 - 003 0000
0000 0000 0000 0000 0000 0000 0000 0000
000
001
007
023
020
009
003
003
0000 0000 0000 0000 0000 0000 0000 0000
1 7
13 19 25 31 37 43
125000 125000 124998 124972 124895 124787 124647 124358
0000 0000 0000 0000 0000 0000 0000 0000
- 000 001 007 023 020 009 003
0000
0000 0000 0000 0000 0000 0000 0000
50000
0000 182
1982 3862 1674 419 078 041
0000 0000 0000 0000 0000 0000 0000 0000
TERMS OF WATERBALANCE PER LAYER DURING LAST TIMESTEP AND
AND SINCE BEGINNING IN M3DAY M3
FLOU TO UNSATURATED ZONE
FLOU TO TERTIARY SYSTEM
0000
0000
0000
0000
LAYER NO 1
FLOU TO SECONDARY SYSTEM
FLOU TO AD3ACENT LAYERCS)
ARTIFICIAL FLOU AND PRESCRIBED BOUNDARY FLOU
ACCUMULATION TERMS
LATERAL FLOU THROUGH PRESCR HEAD BOUNDARY
0000
-000
0000
000
000
0000
-000
0000
000
000
LAYER NO 2
FLOU TO SECONDARY SYSTEM
FLOU TO AD3ACENT LAYERCS1
ARTIFICIAL FLOU AND PRESCRIBED BOUNDARY FLOU
ACCUMULATION TERMS
LATERAL FLOU THROUGH PRESCR HEAD BOUNDARY
0000
-378
0000
381
0000
0000
-055
0000
056
0000
LAYER NO 3
FLOU TO SECONDARY SYSTEM
FLOU TO AD3ACENT LAYERCS5
ARTIFICIAL FLOU AND PRESCRIBED BOUNDARY FLOU
ACCUMULATION TERMS
LATERAL FLOU THROUGH PRESCR HEAD BOUNDARY
0000
370
50000
49376
0000
0000
055
12979
12909
0000
M
1 A
FLOH to unsaturated zono
FLOW to tertiary syster
FLOW to secondnrv system
Leakage to adjacent aquitards
SOLVE equation (14)
yes
CALCULATE flux through constant head boundnr
CALCULATE waterbalance n p r 1 OAro r r J j - ~
PRINT of output
Ier node per layer
k = k+1
t = t + At
7 DUMP variables on tape
-gt-
( STOP )
Fig 7 Flow chart of the program FEMSAT with diviation in 5 main
parts A B C D and E
C START PROGRAM FEMSAT
f READ general information
PRINT general information
f READ element information
PRINT element information 7 READ stationary nodal point information
PRINT stationary nodal point information 7
READ non-stationary nodal point information for t = t
FRINT non-stationary nodal point information for t = t
L 2 (
^ yes R E A D non-stat input for t gt t
J PRINT non-stat input for t gt t (- 1 -- M degY
GENERATE A B and AREA-matrix for confined aquifers
_L
GENERATE A B and AREA-matrix for unconfined aouifer
bull
C Solving h with Gauss-Seidel
D Determination of the water balance terms per layer
E Printing of results
The program exists of a main program with several subroutines
The main program calls the subroutines In the subroutines most
calculations are performed
In this way an easy adaptation to different problems is
obtained (see also section 44) Comments in the program will
help the reader to understand the program
In the next chapter the preparation of the input data executing
of the program and the output will be treated
Acknowledgement
Some parts of the computer program were taken from the program
UNSAT2 of NEUMAN etal (1974)
IV INPUT EXECUTION AND OUTPUT
41 I n p u t
The input is divided into endogenous variables (chosen by the
user or calculated by the program during execution) and exogenous
variables (based on physical data)
411 Endogenous variables
Translation of groundwater basin into a nodal point network
Probably the most important but also the most difficult stage
in the process of simulating groundwater flow is the translation
of the natural groundwater basin into a nodal point network Questions
which arise are
a In how many layer the geo-hydrological system mustcan be
schematized
b The locations of the boundaries and the nature of these boundaries
(impervious prescribed head prescribed flux)
c The configuration of the nodal points
19
The answers of these questions depend on the problem under
consideration
- Schematization into layers
Usually our knowledge about the geo-hydrological situation is
such that no more than 5 different layers can be distinguished
(remember contrast rule)
- Boundary conditions
The top boundary condition of a groundwater basin usually is
a prescribed flux (effective precipitation)
The bottom boundary condition usually is a prescribed flux
boundary with flux zero (impervious hydrological base)
For the boundaries at the other sides of the system one sees
either prescribed head boundaries (eg a fully penetrating
river) or prescribed flux boundaries (including zero flux)
- Nodal point configuration
Nodal points are situated
on the boundaries
along internal boundaries which indicate changes in
hydraulic properties within the layer
along the tertiary surface water system
on places where artificial recharge or discharge wiii tgtke
place
elsewhere
Construction of the finite element mesh
Assume we have schemetized the groundwater basin iatu a finite
number of layers with different direction of flow (horizontal and
vertical) Within one layer the hydrological properties may vary
from place to place
Now a nodal grid is superimposed on each layer in a way already
described in section 32 Thus each layer has the same number of
nodal points situated on the same places in the horizontal plane
but with different z-coordinates
We are free to construct quadrilaterals or triangles because
the program automatically divides a quadrilateral into two triangles
having identical material properties The dimensions of the elements
20
should be small at places where large changes in hydrological
properties occur or where large gradients in hydraulic head are
expected to occur
Coding of nodal points
Each node in the finite element network is assigned an integer
code which indicates the type of calculation for this node
The coding is as follows
- Code = 1 nodal points with described head as function of time
- Code 2 internal nodal points
- Code = 3 internal nodal points which represent secondary surface
water system
- Code = 5 nodal points with prescribed artificial recharge or
discharge and points with prescribed boundary flow
(both as functions of time)
Other important endogenous variables are the magnitude of the
time step the value of the variable which stops the iteration
process and the maximum number of iterations allowed per time step
(see also group A and B of section 413)
412 Exogenous variables
After constructing the nodal grid each node represents part of
the groundwater basin Geo-hydrological properties can be assigned
to each mode based on physical data (group C D and E of section
413) In group F of section 413 time-dependent variables can
be assigned to each node
413 Data input to Program FEMSAT
The data input must be punched on cards according to the
following instructions
GROUP COLUMNS FORMAT SYMBOL DESCRIPTION
A 1-80 20A4 HEAD Desired heading to be printed at
the beginning of the execution
1-5 15 NUMNP number of nodal points
21
GROUP COLUMNS FORMAT SYMBOL DESCRIPTION
6-10 15
11-15 15
16-20 15
21-25 15
26-30 15
31-35 15
NUMEL
NUMLAY
MBAND
NTSPI
NTSPO
MAXIT
1-10 F103 ST
11-20 F103 DELT
21-30 F103 TMAX
31-40 F105 TOL
number of elements
number of layers
maximum difference in node number
between two adjacent nodes
number of time steps between a
change in non-stationary input data
number of time steps between two
successive outputs
maximum number of iterations per
time step
time to start with the execution
initial time interval
maximum time to be reached before
execution is stopped
maximum allowed absolute change in
the values of h between any two
successive iterations in a given
time step
Iterations continue until all changes
are less than or equal to TOL or
until MAXIT is exceeded
1- 5 15
6-10 15
11-15 15
16-20 15
KX(N1)
KX(N2)
KX(N3)
KX(N4)
sequential number of i-th corner of
element N
sequential number of j-th corner
sequential number of k-th corner
sequential number of 1-th corner
(see fig 8)
Fig 8 Numbering of element corner nodes
22
GROUP COLUMNS FORMAcircT SYMBOL DESCRIPTION
One card must be provided for each element in sequential order
starting with N = 1 and ending with N = NVJMEL
I-K 1103 SANG(NLY)
1i-iO MO3 C1(NLY)
21-30 F 103 C2(NLY)
21-4- 1-104 THI(NLY)
angle in degrees between CI and
the laquo-coordinate to he used in
nodal print N in layer LY (see
poundig 9)
First principal conductivity of
material in node N and layer LY
idem second principal conductivity
thickness of layer LY at nodal
point N
ig V Definition of SANGCI and C2
One card icircuiPt be provided for each model point ctarting wih M = 1
and ending with N = NUMNP If there is norlt3 than 1 layer for each
layer a set must be provided starting with the upper layer LY - 1
and ending with the bottom layer LY = NIJMLAY The variables SANG
M and 2 are not relevant tor aquitards but for reasons of
conveniency this procedure is maintained (making SANG CI and C2
dummy variables)
1-10 110 KODE(N)
11-20 II0 NTERR(N)
coding of nodal points (see
paragraph 42)
number of tertiary relation in node N
The q-h relationship of the flow to
23
GROUP COLUMNS FORMAT SYMBOL DESCRIPTION
21-30 FlOl X(N)
31-40 FlOl Y(N)
41-50 FlOl HGL(N)
51-60 FlOl HBSC(N)
61-70 FlOl LESC(N)
the tertiary system as discussed
in par 24 are schematized in a small
number of different relations
x-coardinate of node N
y-coordinate of node N
height ground level in node N
height bottom secondary conduit in
node N If node N is not a secondary
conduit this variable is a dummy
variable
length of secondary conduit in area
represented by node N
One card must be provided for each node starting with N = 1 and
ending with N = NUMNP
E 1-10 FlOl P(NLY1) initial value of h at node N in
layer LY
11-20 FlOl VERRES(NLY) vertical resistance at node N and
layer LY If the layer is an aquifer
this value is a dummy variable
21-30 F101 RADRES(NLY) radial resistance for flow to
secondary system at node N in layer
LY
31-40 FlOl ENRES(NLY) entrance resistance for flow to
secondary system at node N in layer
LY
41-50 FlOl WP(NLY) wet perimeter of secondary system at
node N in layer LY
51-60 FlOl P0R(NLY) specific yield of the material at
node N in layer LY
61-70 FlOl SS(NLY) specific storage of the material
at node N in layer LY
24
GROUP COLUMNS FORMAT SYMBOL DESCRIPTION
One card must be provided for each node starting with N = 1 and
ending with N = NUMNP If there is more than 1 layer for each layer
a set of cards must be provided starting with LY = 1 and ending
with LY = NUMLAY
F 1-10 F101 P(NLY1) value of h at different times at
node N in layer LY Node N lies
on a prescribed head boundary
11-20 F101 WTSC(N) water table in secondary system at
node N at different times
21-30 F 101 ARRED(NLY) artificial recharge to or discharge
from node N in layer LY or
prescribed boundary flow at
different times
31-40 F 101 QPHR(N) flux through phreatic surface at
node N at different times
The data in group F are non-stationary input data One card must be
provided for each node starting with N = 1 and ending with N = NUMNP
In case of more than 1 layer for each layer a set must be provided
Not relevant variables are taken as dummy variables
After a certain number of time steps (NTSPI) the data of group E
are read So the set cards E must contain NxLYxC cards where C =
(TMAXNTSPIXDELT) + 1 if DELT is a constant
42 R u n n i n g t h e p r o g r a m o n t h e C y b e r 7200
The program has been executed on a Cyber 7200 computer of
IWIS-TNO in the Hague
- The demanded input is punched on cards according to the instructions
of the last paragraph and is put on a permanent file
- For each specific problem inthe program only the dimension statements
in the main program have to be adjusted according to the instructions
in the lines of comment
- After compilation the program together with the input file is
executed
25
- Intermediate data necessary for continuation of the calculations
are saved on another permanent file This gives the opportunity
to restart the program at a time the calculation was stopped
The only change in the input file is to adjust the starting time
Remark
With the aid of a terminal small changes in the program and in the
input file can be made easily and the output immediately appears
on the terminal In this way it is possible to stop erroneous
calculations This is especially useful when being in the testing
phase of the program
43 O u t p u t
Immediately after reading the input data they are printed so
that possible errors can be detected During execution after each
certain number of time steps (NTSPO) calculated values of h and
water balance terms are printed In appendix B an example is given
The output data can be devided into two groups
A Data per node per layer
- Coding of the nodal points KODE
- Hydraulic head HEAD
- Flow through phreatic surface at the time indicated in head
of table FLOWUN
- Flow to tertiary surface water system FLOWTS
- Flow to secondary surface water system FLOWSS
- Flow to adjacent layers LEAKAGE
- Artificial recharge or discharge and prescribed boundary
flow ARRED
- Change in storage during last time step STORAGE
- Flow through prescribed head boundary LATFLOW
B Data per layer
Summation of the terms already mentioned under A gives the
magnitude of the water balance terms per layer at the time
indicated (instanteneous water balance terms) Summation of
these instanteneous terms multiplied by their respective time
intervals since time is zero gives the cumulative water balance
terms per layer
26
After each print out we are able to check the calculations
because the sum of the water balance terms per layer must be
approximately zero
44 C o m p u t i n g t i m e
The computing time depends on the number of nodal points and
the number of layers But it depends also on the magnitude of the
changes in hydraulic head and on the value which terminates the
Gauss-Seidel iteration (TOL) To give an impression TOL = 001 m
Costs per day simulation (in hlfl) on the Cyber 7200
number of nodes number of layer bull bdquo bull bull _ 1mdash x a x b (21)
where
a = costs in Dutch guilders per system second (ƒ 027)
b = empirical factor with magnitude about 001
V NUMERICAL EXPERIMENTS
In general there are two possibilities to test the validity of
numerical models The first and most commonly used method is to
compare numerical results with analytical solutions The second
possibility is to compare numerical results with measured field
data Since the last alternative requires many data that are
difficult to obtain we restrict ourselves for the time being to
comparison with the analytical solutions
5 1 F l o w t o a w e l l i n a n u n c o n f i n e d
c i r c u l a r - s h a p e d a q u i f e r
Flow to a well can be divided into steady-state flow and
unsteady-state flow (KRUSEMAN and DE RIDDER 1976)
27
511 Steady state flow in an unconfined aquifer
For this case the formula of Thiem-Dupuit is valid (see fig 10)
Q bull k lth2-hicircgt ln(r2r])
(22)
where
hhbdquo = hydraulic head in 1 and 2 respectively
Q = discharge from well
rbdquo = distance from well
CL31
hmdash^ri --r2
Fig 10 Schematic cross-section of a pumped unconfined aquifer
Fig 11 gives the nodal grid superimposed on a well in a
circular shaped aquifer Near the well the nodal point distance
is made smaller Quadrilaterals as well as triangles are used
Numerical calculations with this finite element network and 3 -1
an extraction rate of 200 m day were performed for 10 days
At that moment a steady-state situation was almost achieved
28
2tmdashnodal point number
160 180 X-axesim)
Fig 11 Nodal point configuration for flow to a well in a circular-
shaped aquifer The outer boundary is taken to be constant
The well is situated in nodal point 42
29
In fig 12 the results of the simulation are compared with the
analytical solution The agreement between numerical results and
the analytical solution is excellent
draw- 15irS (m) down
analytical solution imdashx calculated values
distance from well r (m)
Fig 12 Comparison between analyt ical and calculated s teady-state
drawdown for the problem of f ig 11 with 1
-1 Q bull - 200 m 3 d a y _ 1
k = 10 mday
d = 100 m
S = 2 y
S = 0 s
512 Unsteady-state flow in an unconfined aquifer
Analytical solutions for unsteady-state flow in an unconfined
aquifer only exist if the aquifer extends infinitely For the
aquifer given in fig 11 this condition is not fulfilled However
during the first one or two days after starting the extraction the
influence of a constant hydraulic head at the outer boundary on the
30
drawdown near the well can be neglected (see fig 13) After one
day the lateral inflow is only 25 of the extraction the remainder
being the change in storage due to drawdown
2 0 0
m 3 day 1
Q=200m 3 day
100-
vraquomdash Lateral inflow
N amp storage
I I ^ V - W 10
t (days)
Fig 13 Values of Q lateral inflow and change in storage as
functions of time
The nonsteady-state or Theis equation can be written as
i _ Q s 4iTkD
dy = 4irkD
W(u) (23)
where
u = r2S
4irkDt
r = distance from well
S = storage coefficient
Q = constant discharge from well
D = thickness of the waterbearing layer
at the beginning of pumping
t = time after starting the extraction
s = corrected drawdown
= s - 322D
s = calculated drawdown
W(u) = Theiss well function
[J
CIAT1
[T]
]
31
In fig 14 the analytical unsteady-state solution for the case
described in section 511 is given together with the numerical
results at times 05 and 10 days The agreement between numerical
results and the analytical solution is rather good
Drawdown s1(m)
10
10
10
Theiss solution
x caculated values t=Q5
bdquo t= lO
10 10- 10 10 t r2(daym2)
Fig 14 Comparison between analytical and calculated steady-state
drawdown for the problem of fig 11 with
Q
k
d
S y
s s
=
=
=
=
=
200
1
10
3 m
0
0
2
0
-1 day
m day
m
These results indicate that the assumption of a linear variation
of h between two nodal points (see fig 3 ) even with the rather
rough element network of fig 11 is acceptable So there is no
need to improve for example the finite element network by introducing
higher order coordinate functions
32
52 F l o w t o w e l l i n 3-1 a y e r e d s y s t e m
In fig 15 a schematic cross-section of a 3-layered system of
two aquifers separated by an aquitard is given A well of
infinitisemal radius is completed in the lower aquifer and discharges
at a constant rate QTT Each layer is homogeneous isotropic and
extending radially infinite
bdquo bull - bull bull ~ i
mdash Q II
aquifer I
lugrave^i~IcircIuml7i-i~~ aquitard --r-rl~-
-II--I i - i
aquifer
Fig 15 Schematic cross-section of a 3-layered system
For this situation NEUMAN and WITHERSPOON (1969a) give rather
complicated analytical solutions In the special case where the
hydraulic properties of the two aquifers are identical they give
graphical illustrations of their solutions for three different
cases
These graphical solutions give the opportunity to verify
roughly numerical results of a 3-layered system Therefore a nodal
grid is superimposed on a circular shaped groundwater flow system
with the extraction in the centre of the lower aquifer and with
a constant head on the outer boundary Fig 16 gives the nodal
point situation in the x - y plane of 1 layer The nodes in the
other two layers are situated in the middle of the layers at the
same places in the x - y plane only their z-coordinate is different
33
A
-Jy
bull33
V 5
iumlff
Fig 16 Nodal point configuration for the 3-layered problem (top view)
34
In fig 17 numerical results are compared with the analytical
solution The agreement is rather bad especially for the two
aquifers But in terms of the water balance the difference between
aquifers and aquitard is not so big because the aquitard has
about 16 times bigger storage capacity than the aquifers The
small value of the specific storage of the aquifers is the reason
that small errors in the components of the water balance have a
big influence on the calculated values of the hydraulic head
Eig 17 Dimensionless drawdown vs dimensionless time in a
3-layered system with
II = 50 m day
3 m (distance from well)
aquifer I
aquitard
aquifer II
Thi (m)
100
25
100
S s
001
0642
001
k(mday )
1
0277
1
The discrepancies are partly caused by errors in discretization
In the z-direction these errors can be eliminated by taking the
35
vertical resistance of the aquitard to be infinite Then the
calculated drawdown in the pumped aquifer can be compared with
the analytical Theiss solution
This is done in fig 18 for different distances from the well
One sees that the agreement is better when the distance is bigger
Apparently the linearization of the logarithmic drawdown curve
near the well is responsible for the differences The differences
for r = 3 m are of the same magnitude as for the pumped aquifer in
the 3-layered case
joo gt A 50
20
Theiss solution
bull calculated values r=3m (distance f rom well) x bdquo bdquo r=7m
1 L
I l_ 001 003 0 0 5 Ol Q2
- I I 5 10 t - K t
Fig 18 Dimensionless drawdown vs dimensionless time in a
confined aquifer with
Q =500 m day
k = 10 mday
D = 100 m
S = 001 s
The differences between numerical results and analytical solutions
are not caused by taking a finite area because changing the outer
boundary into an impermeable boundary has only minor effects
36
VI SUMMARY AND CONCLUSIONS
In this report a numerical model for non-stationary saturated
groundwater flow in a multi-layered system has been treated
Groundwater flow can be described in different ways
a Physical-mathematical by combining Darcys law and the law of
continuity (section 21)
b Geo-hydrological the flow in a natural groundwater basin takes
place in layers with different hydraulic properties Definitions
of the most important hydraulic properties are given in section
22
c Schematical groundwater flows horizontally in aquifers and
vertically in aquitards (section 23)
d Approximating by dividing the region of flow into a finite
number of elements and applying Darcys law and the law of
continuity on each of these elements In these ways one gets
a set of equations which can be solved given (initial and)
boundary conditions (section 25)
e Numerical the set of equations has been solved using a Galerkin
type finite element method and the implicit Gauss-Seidel method
with overrelaxation factor (section 31 and 32)
f Computer-solvable the numerical solution has been translated
in a FORTRAN IV program called FEMSAT (section 33)
In chapter IV the input execution and output of the program
FEMSAT are treated
Finally in chapter V some numerical applications are given
simulation of flow to a well in unconfined aquifer and simulation
of flow to a well in a 3-layered configuration
Comparison of hydraulic heads obtained by numerical calculations
and analytical true solutions showed an excellent agreement in case
of stationary flow and a good to moderate agreement in case of non-
stationary flow However the diviations caused by the discretization
process must be seen in relation to the errors made in the
establishment of the hydraulic properties
37
As was stated in the introduction numerical models can be
valuable tools in the management of water resources Therefore
numerical models must be suitable to simulate effects of certain
operations upon the spreading of water over saturated groundwater
unsaturated groundwater and surface water The model described in
this report is a model for saturated groundwater flow The relation
with the surface water can be taken into account via radial and
entrance resistances or via a relation between flow to the surface
water system and the hydraulic head The relation with the unsaturated
zone has been restricted to the input factor effective precipitation
In practise however the unsaturated and saturated system are
interconnected Therefore the model should be improved by taking
into account the unsaturated zone This can be done in at least
two ways
a Replacing the unsaturated zone by a parametrical model
b Coupling the unsaturated zone in an iterative way with the
saturated system
In order to find out the usefulness of the numerical model for
water resources management it will be applied to actual regional
groundwater systemslaquoAlso sensitivity analyses will be carried out
The computer model itself can be improved by applying standard
programs eg automatical generation of a nodal grid and graphical
plotting of the results
38
VII REFERENCES
ERNST LF 1962 Grondwaterstromingen in de verzadigde zone en hun
berekening bij aanwezigheid van horizontale evenwijdige open
leidingen Thesis University Utrecht pp 189
1978 Drainage of undulating soils with high groundwater
tables (in press)
FORSYTHE GE and WR WASON 1960 Finite Difference Methods for
Partial Differential Equations John Wiley and Sons New York
pp 444
KRUSEMAN GP and NA DE RIDDER 1976 Analysis and evaluation of
pumping test data ILRI-bulletin No 11 pp 200
LAAT PJM DE and C VAN DEN AKKER 1976 Simulatiemodel voor grond-
waterstroming en verdamping in Modelonderzoek 1971-1974 ten
behoeve van de waterhuishouding in Gelderland deel II
Grondslagen pp 145-197
NEUMAN SP RA FEDDES and E BRESLER 1974 Finite element
simulation of flow in saturated -unsaturated soils considering
water uptake by plants Hydraulic Engineering Laboratory
Technion Israel pp 104
and PA WITHERSPOON 1969a Theory of Flow in a confined
two Aquifer System Water Resources Research Vol 5 No 4
pp 803-816
^_ and PA WITHERSPOON 1969b Applicability of Current Theories
of Flow in leaky aquifers Water Resources Research Vol 5
No 4 pp 817-829
PINDER GF and EO FRIND 1972 Application of Galerkins Procedure
to Aquifer Analysis Water Resources Research Vol 8 no 1
pp 108-120
REMSON I GE HORNBERGER and FJ MOLZ 1971 Numerical Methods
in Subsurface Hydrology John Wiley and Sons New York
pp 389
WESSELING JG 1977 Toepassing van Finite Difference- en Finite
Element Methoden voor het oplossen van een aantal grondwatershy
stromingsproblemen (met computerprogrammas) ICW-nota 944
pp 63
39
ZIENKIEWICZ OC 1971 Finite Element Method in Engineering Science
McGraw-Hill London pp 521
40
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APPENDIX 8 EXAMPLE OF OUTPUT
HEAD AND TERMS OF UATERBALANCE PER NODE PER LAYER
AT TIME 260 IN II AND I13DAYS
T
LAYER NODE KODE HEAD FLOUUN FLOUTS FLOUSS LEAKAGE STORAGE LAT FLOU
13 19 25 31 37 43
125008 125000 12S000 12S000 12S000 12S000 12S000 12S000
0000 0000 0000 0000 0000 0000 0000 0000
0000 0000 0000 0000 0000 0000 0000 0000
0000 0000 0000 0000 0000 0000 0000 0000
- 000 - 000 -000 - 000 - 000 - 000 - 000 0000
0000 0000 0000 0000 0000 0000 0000 0000
0000 000 000 000 000 000 000 000
000 0000 0000 0P100 0000 0000 0000 0000
1 7
13 19 25 31 37
12S000 125000 125000 125000 125000 12S000 125000 125000
0000 0000 0000 0000 0000 0000 0000 0000
- 001 -007 - 023 - 020 - 009 - 003 0000
0000 0000 0000 0000 0000 0000 0000 0000
000
001
007
023
020
009
003
003
0000 0000 0000 0000 0000 0000 0000 0000
1 7
13 19 25 31 37 43
125000 125000 124998 124972 124895 124787 124647 124358
0000 0000 0000 0000 0000 0000 0000 0000
- 000 001 007 023 020 009 003
0000
0000 0000 0000 0000 0000 0000 0000
50000
0000 182
1982 3862 1674 419 078 041
0000 0000 0000 0000 0000 0000 0000 0000
TERMS OF WATERBALANCE PER LAYER DURING LAST TIMESTEP AND
AND SINCE BEGINNING IN M3DAY M3
FLOU TO UNSATURATED ZONE
FLOU TO TERTIARY SYSTEM
0000
0000
0000
0000
LAYER NO 1
FLOU TO SECONDARY SYSTEM
FLOU TO AD3ACENT LAYERCS)
ARTIFICIAL FLOU AND PRESCRIBED BOUNDARY FLOU
ACCUMULATION TERMS
LATERAL FLOU THROUGH PRESCR HEAD BOUNDARY
0000
-000
0000
000
000
0000
-000
0000
000
000
LAYER NO 2
FLOU TO SECONDARY SYSTEM
FLOU TO AD3ACENT LAYERCS1
ARTIFICIAL FLOU AND PRESCRIBED BOUNDARY FLOU
ACCUMULATION TERMS
LATERAL FLOU THROUGH PRESCR HEAD BOUNDARY
0000
-378
0000
381
0000
0000
-055
0000
056
0000
LAYER NO 3
FLOU TO SECONDARY SYSTEM
FLOU TO AD3ACENT LAYERCS5
ARTIFICIAL FLOU AND PRESCRIBED BOUNDARY FLOU
ACCUMULATION TERMS
LATERAL FLOU THROUGH PRESCR HEAD BOUNDARY
0000
370
50000
49376
0000
0000
055
12979
12909
0000
C START PROGRAM FEMSAT
f READ general information
PRINT general information
f READ element information
PRINT element information 7 READ stationary nodal point information
PRINT stationary nodal point information 7
READ non-stationary nodal point information for t = t
FRINT non-stationary nodal point information for t = t
L 2 (
^ yes R E A D non-stat input for t gt t
J PRINT non-stat input for t gt t (- 1 -- M degY
GENERATE A B and AREA-matrix for confined aquifers
_L
GENERATE A B and AREA-matrix for unconfined aouifer
bull
C Solving h with Gauss-Seidel
D Determination of the water balance terms per layer
E Printing of results
The program exists of a main program with several subroutines
The main program calls the subroutines In the subroutines most
calculations are performed
In this way an easy adaptation to different problems is
obtained (see also section 44) Comments in the program will
help the reader to understand the program
In the next chapter the preparation of the input data executing
of the program and the output will be treated
Acknowledgement
Some parts of the computer program were taken from the program
UNSAT2 of NEUMAN etal (1974)
IV INPUT EXECUTION AND OUTPUT
41 I n p u t
The input is divided into endogenous variables (chosen by the
user or calculated by the program during execution) and exogenous
variables (based on physical data)
411 Endogenous variables
Translation of groundwater basin into a nodal point network
Probably the most important but also the most difficult stage
in the process of simulating groundwater flow is the translation
of the natural groundwater basin into a nodal point network Questions
which arise are
a In how many layer the geo-hydrological system mustcan be
schematized
b The locations of the boundaries and the nature of these boundaries
(impervious prescribed head prescribed flux)
c The configuration of the nodal points
19
The answers of these questions depend on the problem under
consideration
- Schematization into layers
Usually our knowledge about the geo-hydrological situation is
such that no more than 5 different layers can be distinguished
(remember contrast rule)
- Boundary conditions
The top boundary condition of a groundwater basin usually is
a prescribed flux (effective precipitation)
The bottom boundary condition usually is a prescribed flux
boundary with flux zero (impervious hydrological base)
For the boundaries at the other sides of the system one sees
either prescribed head boundaries (eg a fully penetrating
river) or prescribed flux boundaries (including zero flux)
- Nodal point configuration
Nodal points are situated
on the boundaries
along internal boundaries which indicate changes in
hydraulic properties within the layer
along the tertiary surface water system
on places where artificial recharge or discharge wiii tgtke
place
elsewhere
Construction of the finite element mesh
Assume we have schemetized the groundwater basin iatu a finite
number of layers with different direction of flow (horizontal and
vertical) Within one layer the hydrological properties may vary
from place to place
Now a nodal grid is superimposed on each layer in a way already
described in section 32 Thus each layer has the same number of
nodal points situated on the same places in the horizontal plane
but with different z-coordinates
We are free to construct quadrilaterals or triangles because
the program automatically divides a quadrilateral into two triangles
having identical material properties The dimensions of the elements
20
should be small at places where large changes in hydrological
properties occur or where large gradients in hydraulic head are
expected to occur
Coding of nodal points
Each node in the finite element network is assigned an integer
code which indicates the type of calculation for this node
The coding is as follows
- Code = 1 nodal points with described head as function of time
- Code 2 internal nodal points
- Code = 3 internal nodal points which represent secondary surface
water system
- Code = 5 nodal points with prescribed artificial recharge or
discharge and points with prescribed boundary flow
(both as functions of time)
Other important endogenous variables are the magnitude of the
time step the value of the variable which stops the iteration
process and the maximum number of iterations allowed per time step
(see also group A and B of section 413)
412 Exogenous variables
After constructing the nodal grid each node represents part of
the groundwater basin Geo-hydrological properties can be assigned
to each mode based on physical data (group C D and E of section
413) In group F of section 413 time-dependent variables can
be assigned to each node
413 Data input to Program FEMSAT
The data input must be punched on cards according to the
following instructions
GROUP COLUMNS FORMAT SYMBOL DESCRIPTION
A 1-80 20A4 HEAD Desired heading to be printed at
the beginning of the execution
1-5 15 NUMNP number of nodal points
21
GROUP COLUMNS FORMAT SYMBOL DESCRIPTION
6-10 15
11-15 15
16-20 15
21-25 15
26-30 15
31-35 15
NUMEL
NUMLAY
MBAND
NTSPI
NTSPO
MAXIT
1-10 F103 ST
11-20 F103 DELT
21-30 F103 TMAX
31-40 F105 TOL
number of elements
number of layers
maximum difference in node number
between two adjacent nodes
number of time steps between a
change in non-stationary input data
number of time steps between two
successive outputs
maximum number of iterations per
time step
time to start with the execution
initial time interval
maximum time to be reached before
execution is stopped
maximum allowed absolute change in
the values of h between any two
successive iterations in a given
time step
Iterations continue until all changes
are less than or equal to TOL or
until MAXIT is exceeded
1- 5 15
6-10 15
11-15 15
16-20 15
KX(N1)
KX(N2)
KX(N3)
KX(N4)
sequential number of i-th corner of
element N
sequential number of j-th corner
sequential number of k-th corner
sequential number of 1-th corner
(see fig 8)
Fig 8 Numbering of element corner nodes
22
GROUP COLUMNS FORMAcircT SYMBOL DESCRIPTION
One card must be provided for each element in sequential order
starting with N = 1 and ending with N = NVJMEL
I-K 1103 SANG(NLY)
1i-iO MO3 C1(NLY)
21-30 F 103 C2(NLY)
21-4- 1-104 THI(NLY)
angle in degrees between CI and
the laquo-coordinate to he used in
nodal print N in layer LY (see
poundig 9)
First principal conductivity of
material in node N and layer LY
idem second principal conductivity
thickness of layer LY at nodal
point N
ig V Definition of SANGCI and C2
One card icircuiPt be provided for each model point ctarting wih M = 1
and ending with N = NUMNP If there is norlt3 than 1 layer for each
layer a set must be provided starting with the upper layer LY - 1
and ending with the bottom layer LY = NIJMLAY The variables SANG
M and 2 are not relevant tor aquitards but for reasons of
conveniency this procedure is maintained (making SANG CI and C2
dummy variables)
1-10 110 KODE(N)
11-20 II0 NTERR(N)
coding of nodal points (see
paragraph 42)
number of tertiary relation in node N
The q-h relationship of the flow to
23
GROUP COLUMNS FORMAT SYMBOL DESCRIPTION
21-30 FlOl X(N)
31-40 FlOl Y(N)
41-50 FlOl HGL(N)
51-60 FlOl HBSC(N)
61-70 FlOl LESC(N)
the tertiary system as discussed
in par 24 are schematized in a small
number of different relations
x-coardinate of node N
y-coordinate of node N
height ground level in node N
height bottom secondary conduit in
node N If node N is not a secondary
conduit this variable is a dummy
variable
length of secondary conduit in area
represented by node N
One card must be provided for each node starting with N = 1 and
ending with N = NUMNP
E 1-10 FlOl P(NLY1) initial value of h at node N in
layer LY
11-20 FlOl VERRES(NLY) vertical resistance at node N and
layer LY If the layer is an aquifer
this value is a dummy variable
21-30 F101 RADRES(NLY) radial resistance for flow to
secondary system at node N in layer
LY
31-40 FlOl ENRES(NLY) entrance resistance for flow to
secondary system at node N in layer
LY
41-50 FlOl WP(NLY) wet perimeter of secondary system at
node N in layer LY
51-60 FlOl P0R(NLY) specific yield of the material at
node N in layer LY
61-70 FlOl SS(NLY) specific storage of the material
at node N in layer LY
24
GROUP COLUMNS FORMAT SYMBOL DESCRIPTION
One card must be provided for each node starting with N = 1 and
ending with N = NUMNP If there is more than 1 layer for each layer
a set of cards must be provided starting with LY = 1 and ending
with LY = NUMLAY
F 1-10 F101 P(NLY1) value of h at different times at
node N in layer LY Node N lies
on a prescribed head boundary
11-20 F101 WTSC(N) water table in secondary system at
node N at different times
21-30 F 101 ARRED(NLY) artificial recharge to or discharge
from node N in layer LY or
prescribed boundary flow at
different times
31-40 F 101 QPHR(N) flux through phreatic surface at
node N at different times
The data in group F are non-stationary input data One card must be
provided for each node starting with N = 1 and ending with N = NUMNP
In case of more than 1 layer for each layer a set must be provided
Not relevant variables are taken as dummy variables
After a certain number of time steps (NTSPI) the data of group E
are read So the set cards E must contain NxLYxC cards where C =
(TMAXNTSPIXDELT) + 1 if DELT is a constant
42 R u n n i n g t h e p r o g r a m o n t h e C y b e r 7200
The program has been executed on a Cyber 7200 computer of
IWIS-TNO in the Hague
- The demanded input is punched on cards according to the instructions
of the last paragraph and is put on a permanent file
- For each specific problem inthe program only the dimension statements
in the main program have to be adjusted according to the instructions
in the lines of comment
- After compilation the program together with the input file is
executed
25
- Intermediate data necessary for continuation of the calculations
are saved on another permanent file This gives the opportunity
to restart the program at a time the calculation was stopped
The only change in the input file is to adjust the starting time
Remark
With the aid of a terminal small changes in the program and in the
input file can be made easily and the output immediately appears
on the terminal In this way it is possible to stop erroneous
calculations This is especially useful when being in the testing
phase of the program
43 O u t p u t
Immediately after reading the input data they are printed so
that possible errors can be detected During execution after each
certain number of time steps (NTSPO) calculated values of h and
water balance terms are printed In appendix B an example is given
The output data can be devided into two groups
A Data per node per layer
- Coding of the nodal points KODE
- Hydraulic head HEAD
- Flow through phreatic surface at the time indicated in head
of table FLOWUN
- Flow to tertiary surface water system FLOWTS
- Flow to secondary surface water system FLOWSS
- Flow to adjacent layers LEAKAGE
- Artificial recharge or discharge and prescribed boundary
flow ARRED
- Change in storage during last time step STORAGE
- Flow through prescribed head boundary LATFLOW
B Data per layer
Summation of the terms already mentioned under A gives the
magnitude of the water balance terms per layer at the time
indicated (instanteneous water balance terms) Summation of
these instanteneous terms multiplied by their respective time
intervals since time is zero gives the cumulative water balance
terms per layer
26
After each print out we are able to check the calculations
because the sum of the water balance terms per layer must be
approximately zero
44 C o m p u t i n g t i m e
The computing time depends on the number of nodal points and
the number of layers But it depends also on the magnitude of the
changes in hydraulic head and on the value which terminates the
Gauss-Seidel iteration (TOL) To give an impression TOL = 001 m
Costs per day simulation (in hlfl) on the Cyber 7200
number of nodes number of layer bull bdquo bull bull _ 1mdash x a x b (21)
where
a = costs in Dutch guilders per system second (ƒ 027)
b = empirical factor with magnitude about 001
V NUMERICAL EXPERIMENTS
In general there are two possibilities to test the validity of
numerical models The first and most commonly used method is to
compare numerical results with analytical solutions The second
possibility is to compare numerical results with measured field
data Since the last alternative requires many data that are
difficult to obtain we restrict ourselves for the time being to
comparison with the analytical solutions
5 1 F l o w t o a w e l l i n a n u n c o n f i n e d
c i r c u l a r - s h a p e d a q u i f e r
Flow to a well can be divided into steady-state flow and
unsteady-state flow (KRUSEMAN and DE RIDDER 1976)
27
511 Steady state flow in an unconfined aquifer
For this case the formula of Thiem-Dupuit is valid (see fig 10)
Q bull k lth2-hicircgt ln(r2r])
(22)
where
hhbdquo = hydraulic head in 1 and 2 respectively
Q = discharge from well
rbdquo = distance from well
CL31
hmdash^ri --r2
Fig 10 Schematic cross-section of a pumped unconfined aquifer
Fig 11 gives the nodal grid superimposed on a well in a
circular shaped aquifer Near the well the nodal point distance
is made smaller Quadrilaterals as well as triangles are used
Numerical calculations with this finite element network and 3 -1
an extraction rate of 200 m day were performed for 10 days
At that moment a steady-state situation was almost achieved
28
2tmdashnodal point number
160 180 X-axesim)
Fig 11 Nodal point configuration for flow to a well in a circular-
shaped aquifer The outer boundary is taken to be constant
The well is situated in nodal point 42
29
In fig 12 the results of the simulation are compared with the
analytical solution The agreement between numerical results and
the analytical solution is excellent
draw- 15irS (m) down
analytical solution imdashx calculated values
distance from well r (m)
Fig 12 Comparison between analyt ical and calculated s teady-state
drawdown for the problem of f ig 11 with 1
-1 Q bull - 200 m 3 d a y _ 1
k = 10 mday
d = 100 m
S = 2 y
S = 0 s
512 Unsteady-state flow in an unconfined aquifer
Analytical solutions for unsteady-state flow in an unconfined
aquifer only exist if the aquifer extends infinitely For the
aquifer given in fig 11 this condition is not fulfilled However
during the first one or two days after starting the extraction the
influence of a constant hydraulic head at the outer boundary on the
30
drawdown near the well can be neglected (see fig 13) After one
day the lateral inflow is only 25 of the extraction the remainder
being the change in storage due to drawdown
2 0 0
m 3 day 1
Q=200m 3 day
100-
vraquomdash Lateral inflow
N amp storage
I I ^ V - W 10
t (days)
Fig 13 Values of Q lateral inflow and change in storage as
functions of time
The nonsteady-state or Theis equation can be written as
i _ Q s 4iTkD
dy = 4irkD
W(u) (23)
where
u = r2S
4irkDt
r = distance from well
S = storage coefficient
Q = constant discharge from well
D = thickness of the waterbearing layer
at the beginning of pumping
t = time after starting the extraction
s = corrected drawdown
= s - 322D
s = calculated drawdown
W(u) = Theiss well function
[J
CIAT1
[T]
]
31
In fig 14 the analytical unsteady-state solution for the case
described in section 511 is given together with the numerical
results at times 05 and 10 days The agreement between numerical
results and the analytical solution is rather good
Drawdown s1(m)
10
10
10
Theiss solution
x caculated values t=Q5
bdquo t= lO
10 10- 10 10 t r2(daym2)
Fig 14 Comparison between analytical and calculated steady-state
drawdown for the problem of fig 11 with
Q
k
d
S y
s s
=
=
=
=
=
200
1
10
3 m
0
0
2
0
-1 day
m day
m
These results indicate that the assumption of a linear variation
of h between two nodal points (see fig 3 ) even with the rather
rough element network of fig 11 is acceptable So there is no
need to improve for example the finite element network by introducing
higher order coordinate functions
32
52 F l o w t o w e l l i n 3-1 a y e r e d s y s t e m
In fig 15 a schematic cross-section of a 3-layered system of
two aquifers separated by an aquitard is given A well of
infinitisemal radius is completed in the lower aquifer and discharges
at a constant rate QTT Each layer is homogeneous isotropic and
extending radially infinite
bdquo bull - bull bull ~ i
mdash Q II
aquifer I
lugrave^i~IcircIuml7i-i~~ aquitard --r-rl~-
-II--I i - i
aquifer
Fig 15 Schematic cross-section of a 3-layered system
For this situation NEUMAN and WITHERSPOON (1969a) give rather
complicated analytical solutions In the special case where the
hydraulic properties of the two aquifers are identical they give
graphical illustrations of their solutions for three different
cases
These graphical solutions give the opportunity to verify
roughly numerical results of a 3-layered system Therefore a nodal
grid is superimposed on a circular shaped groundwater flow system
with the extraction in the centre of the lower aquifer and with
a constant head on the outer boundary Fig 16 gives the nodal
point situation in the x - y plane of 1 layer The nodes in the
other two layers are situated in the middle of the layers at the
same places in the x - y plane only their z-coordinate is different
33
A
-Jy
bull33
V 5
iumlff
Fig 16 Nodal point configuration for the 3-layered problem (top view)
34
In fig 17 numerical results are compared with the analytical
solution The agreement is rather bad especially for the two
aquifers But in terms of the water balance the difference between
aquifers and aquitard is not so big because the aquitard has
about 16 times bigger storage capacity than the aquifers The
small value of the specific storage of the aquifers is the reason
that small errors in the components of the water balance have a
big influence on the calculated values of the hydraulic head
Eig 17 Dimensionless drawdown vs dimensionless time in a
3-layered system with
II = 50 m day
3 m (distance from well)
aquifer I
aquitard
aquifer II
Thi (m)
100
25
100
S s
001
0642
001
k(mday )
1
0277
1
The discrepancies are partly caused by errors in discretization
In the z-direction these errors can be eliminated by taking the
35
vertical resistance of the aquitard to be infinite Then the
calculated drawdown in the pumped aquifer can be compared with
the analytical Theiss solution
This is done in fig 18 for different distances from the well
One sees that the agreement is better when the distance is bigger
Apparently the linearization of the logarithmic drawdown curve
near the well is responsible for the differences The differences
for r = 3 m are of the same magnitude as for the pumped aquifer in
the 3-layered case
joo gt A 50
20
Theiss solution
bull calculated values r=3m (distance f rom well) x bdquo bdquo r=7m
1 L
I l_ 001 003 0 0 5 Ol Q2
- I I 5 10 t - K t
Fig 18 Dimensionless drawdown vs dimensionless time in a
confined aquifer with
Q =500 m day
k = 10 mday
D = 100 m
S = 001 s
The differences between numerical results and analytical solutions
are not caused by taking a finite area because changing the outer
boundary into an impermeable boundary has only minor effects
36
VI SUMMARY AND CONCLUSIONS
In this report a numerical model for non-stationary saturated
groundwater flow in a multi-layered system has been treated
Groundwater flow can be described in different ways
a Physical-mathematical by combining Darcys law and the law of
continuity (section 21)
b Geo-hydrological the flow in a natural groundwater basin takes
place in layers with different hydraulic properties Definitions
of the most important hydraulic properties are given in section
22
c Schematical groundwater flows horizontally in aquifers and
vertically in aquitards (section 23)
d Approximating by dividing the region of flow into a finite
number of elements and applying Darcys law and the law of
continuity on each of these elements In these ways one gets
a set of equations which can be solved given (initial and)
boundary conditions (section 25)
e Numerical the set of equations has been solved using a Galerkin
type finite element method and the implicit Gauss-Seidel method
with overrelaxation factor (section 31 and 32)
f Computer-solvable the numerical solution has been translated
in a FORTRAN IV program called FEMSAT (section 33)
In chapter IV the input execution and output of the program
FEMSAT are treated
Finally in chapter V some numerical applications are given
simulation of flow to a well in unconfined aquifer and simulation
of flow to a well in a 3-layered configuration
Comparison of hydraulic heads obtained by numerical calculations
and analytical true solutions showed an excellent agreement in case
of stationary flow and a good to moderate agreement in case of non-
stationary flow However the diviations caused by the discretization
process must be seen in relation to the errors made in the
establishment of the hydraulic properties
37
As was stated in the introduction numerical models can be
valuable tools in the management of water resources Therefore
numerical models must be suitable to simulate effects of certain
operations upon the spreading of water over saturated groundwater
unsaturated groundwater and surface water The model described in
this report is a model for saturated groundwater flow The relation
with the surface water can be taken into account via radial and
entrance resistances or via a relation between flow to the surface
water system and the hydraulic head The relation with the unsaturated
zone has been restricted to the input factor effective precipitation
In practise however the unsaturated and saturated system are
interconnected Therefore the model should be improved by taking
into account the unsaturated zone This can be done in at least
two ways
a Replacing the unsaturated zone by a parametrical model
b Coupling the unsaturated zone in an iterative way with the
saturated system
In order to find out the usefulness of the numerical model for
water resources management it will be applied to actual regional
groundwater systemslaquoAlso sensitivity analyses will be carried out
The computer model itself can be improved by applying standard
programs eg automatical generation of a nodal grid and graphical
plotting of the results
38
VII REFERENCES
ERNST LF 1962 Grondwaterstromingen in de verzadigde zone en hun
berekening bij aanwezigheid van horizontale evenwijdige open
leidingen Thesis University Utrecht pp 189
1978 Drainage of undulating soils with high groundwater
tables (in press)
FORSYTHE GE and WR WASON 1960 Finite Difference Methods for
Partial Differential Equations John Wiley and Sons New York
pp 444
KRUSEMAN GP and NA DE RIDDER 1976 Analysis and evaluation of
pumping test data ILRI-bulletin No 11 pp 200
LAAT PJM DE and C VAN DEN AKKER 1976 Simulatiemodel voor grond-
waterstroming en verdamping in Modelonderzoek 1971-1974 ten
behoeve van de waterhuishouding in Gelderland deel II
Grondslagen pp 145-197
NEUMAN SP RA FEDDES and E BRESLER 1974 Finite element
simulation of flow in saturated -unsaturated soils considering
water uptake by plants Hydraulic Engineering Laboratory
Technion Israel pp 104
and PA WITHERSPOON 1969a Theory of Flow in a confined
two Aquifer System Water Resources Research Vol 5 No 4
pp 803-816
^_ and PA WITHERSPOON 1969b Applicability of Current Theories
of Flow in leaky aquifers Water Resources Research Vol 5
No 4 pp 817-829
PINDER GF and EO FRIND 1972 Application of Galerkins Procedure
to Aquifer Analysis Water Resources Research Vol 8 no 1
pp 108-120
REMSON I GE HORNBERGER and FJ MOLZ 1971 Numerical Methods
in Subsurface Hydrology John Wiley and Sons New York
pp 389
WESSELING JG 1977 Toepassing van Finite Difference- en Finite
Element Methoden voor het oplossen van een aantal grondwatershy
stromingsproblemen (met computerprogrammas) ICW-nota 944
pp 63
39
ZIENKIEWICZ OC 1971 Finite Element Method in Engineering Science
McGraw-Hill London pp 521
40
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APPENDIX 8 EXAMPLE OF OUTPUT
HEAD AND TERMS OF UATERBALANCE PER NODE PER LAYER
AT TIME 260 IN II AND I13DAYS
T
LAYER NODE KODE HEAD FLOUUN FLOUTS FLOUSS LEAKAGE STORAGE LAT FLOU
13 19 25 31 37 43
125008 125000 12S000 12S000 12S000 12S000 12S000 12S000
0000 0000 0000 0000 0000 0000 0000 0000
0000 0000 0000 0000 0000 0000 0000 0000
0000 0000 0000 0000 0000 0000 0000 0000
- 000 - 000 -000 - 000 - 000 - 000 - 000 0000
0000 0000 0000 0000 0000 0000 0000 0000
0000 000 000 000 000 000 000 000
000 0000 0000 0P100 0000 0000 0000 0000
1 7
13 19 25 31 37
12S000 125000 125000 125000 125000 12S000 125000 125000
0000 0000 0000 0000 0000 0000 0000 0000
- 001 -007 - 023 - 020 - 009 - 003 0000
0000 0000 0000 0000 0000 0000 0000 0000
000
001
007
023
020
009
003
003
0000 0000 0000 0000 0000 0000 0000 0000
1 7
13 19 25 31 37 43
125000 125000 124998 124972 124895 124787 124647 124358
0000 0000 0000 0000 0000 0000 0000 0000
- 000 001 007 023 020 009 003
0000
0000 0000 0000 0000 0000 0000 0000
50000
0000 182
1982 3862 1674 419 078 041
0000 0000 0000 0000 0000 0000 0000 0000
TERMS OF WATERBALANCE PER LAYER DURING LAST TIMESTEP AND
AND SINCE BEGINNING IN M3DAY M3
FLOU TO UNSATURATED ZONE
FLOU TO TERTIARY SYSTEM
0000
0000
0000
0000
LAYER NO 1
FLOU TO SECONDARY SYSTEM
FLOU TO AD3ACENT LAYERCS)
ARTIFICIAL FLOU AND PRESCRIBED BOUNDARY FLOU
ACCUMULATION TERMS
LATERAL FLOU THROUGH PRESCR HEAD BOUNDARY
0000
-000
0000
000
000
0000
-000
0000
000
000
LAYER NO 2
FLOU TO SECONDARY SYSTEM
FLOU TO AD3ACENT LAYERCS1
ARTIFICIAL FLOU AND PRESCRIBED BOUNDARY FLOU
ACCUMULATION TERMS
LATERAL FLOU THROUGH PRESCR HEAD BOUNDARY
0000
-378
0000
381
0000
0000
-055
0000
056
0000
LAYER NO 3
FLOU TO SECONDARY SYSTEM
FLOU TO AD3ACENT LAYERCS5
ARTIFICIAL FLOU AND PRESCRIBED BOUNDARY FLOU
ACCUMULATION TERMS
LATERAL FLOU THROUGH PRESCR HEAD BOUNDARY
0000
370
50000
49376
0000
0000
055
12979
12909
0000
C Solving h with Gauss-Seidel
D Determination of the water balance terms per layer
E Printing of results
The program exists of a main program with several subroutines
The main program calls the subroutines In the subroutines most
calculations are performed
In this way an easy adaptation to different problems is
obtained (see also section 44) Comments in the program will
help the reader to understand the program
In the next chapter the preparation of the input data executing
of the program and the output will be treated
Acknowledgement
Some parts of the computer program were taken from the program
UNSAT2 of NEUMAN etal (1974)
IV INPUT EXECUTION AND OUTPUT
41 I n p u t
The input is divided into endogenous variables (chosen by the
user or calculated by the program during execution) and exogenous
variables (based on physical data)
411 Endogenous variables
Translation of groundwater basin into a nodal point network
Probably the most important but also the most difficult stage
in the process of simulating groundwater flow is the translation
of the natural groundwater basin into a nodal point network Questions
which arise are
a In how many layer the geo-hydrological system mustcan be
schematized
b The locations of the boundaries and the nature of these boundaries
(impervious prescribed head prescribed flux)
c The configuration of the nodal points
19
The answers of these questions depend on the problem under
consideration
- Schematization into layers
Usually our knowledge about the geo-hydrological situation is
such that no more than 5 different layers can be distinguished
(remember contrast rule)
- Boundary conditions
The top boundary condition of a groundwater basin usually is
a prescribed flux (effective precipitation)
The bottom boundary condition usually is a prescribed flux
boundary with flux zero (impervious hydrological base)
For the boundaries at the other sides of the system one sees
either prescribed head boundaries (eg a fully penetrating
river) or prescribed flux boundaries (including zero flux)
- Nodal point configuration
Nodal points are situated
on the boundaries
along internal boundaries which indicate changes in
hydraulic properties within the layer
along the tertiary surface water system
on places where artificial recharge or discharge wiii tgtke
place
elsewhere
Construction of the finite element mesh
Assume we have schemetized the groundwater basin iatu a finite
number of layers with different direction of flow (horizontal and
vertical) Within one layer the hydrological properties may vary
from place to place
Now a nodal grid is superimposed on each layer in a way already
described in section 32 Thus each layer has the same number of
nodal points situated on the same places in the horizontal plane
but with different z-coordinates
We are free to construct quadrilaterals or triangles because
the program automatically divides a quadrilateral into two triangles
having identical material properties The dimensions of the elements
20
should be small at places where large changes in hydrological
properties occur or where large gradients in hydraulic head are
expected to occur
Coding of nodal points
Each node in the finite element network is assigned an integer
code which indicates the type of calculation for this node
The coding is as follows
- Code = 1 nodal points with described head as function of time
- Code 2 internal nodal points
- Code = 3 internal nodal points which represent secondary surface
water system
- Code = 5 nodal points with prescribed artificial recharge or
discharge and points with prescribed boundary flow
(both as functions of time)
Other important endogenous variables are the magnitude of the
time step the value of the variable which stops the iteration
process and the maximum number of iterations allowed per time step
(see also group A and B of section 413)
412 Exogenous variables
After constructing the nodal grid each node represents part of
the groundwater basin Geo-hydrological properties can be assigned
to each mode based on physical data (group C D and E of section
413) In group F of section 413 time-dependent variables can
be assigned to each node
413 Data input to Program FEMSAT
The data input must be punched on cards according to the
following instructions
GROUP COLUMNS FORMAT SYMBOL DESCRIPTION
A 1-80 20A4 HEAD Desired heading to be printed at
the beginning of the execution
1-5 15 NUMNP number of nodal points
21
GROUP COLUMNS FORMAT SYMBOL DESCRIPTION
6-10 15
11-15 15
16-20 15
21-25 15
26-30 15
31-35 15
NUMEL
NUMLAY
MBAND
NTSPI
NTSPO
MAXIT
1-10 F103 ST
11-20 F103 DELT
21-30 F103 TMAX
31-40 F105 TOL
number of elements
number of layers
maximum difference in node number
between two adjacent nodes
number of time steps between a
change in non-stationary input data
number of time steps between two
successive outputs
maximum number of iterations per
time step
time to start with the execution
initial time interval
maximum time to be reached before
execution is stopped
maximum allowed absolute change in
the values of h between any two
successive iterations in a given
time step
Iterations continue until all changes
are less than or equal to TOL or
until MAXIT is exceeded
1- 5 15
6-10 15
11-15 15
16-20 15
KX(N1)
KX(N2)
KX(N3)
KX(N4)
sequential number of i-th corner of
element N
sequential number of j-th corner
sequential number of k-th corner
sequential number of 1-th corner
(see fig 8)
Fig 8 Numbering of element corner nodes
22
GROUP COLUMNS FORMAcircT SYMBOL DESCRIPTION
One card must be provided for each element in sequential order
starting with N = 1 and ending with N = NVJMEL
I-K 1103 SANG(NLY)
1i-iO MO3 C1(NLY)
21-30 F 103 C2(NLY)
21-4- 1-104 THI(NLY)
angle in degrees between CI and
the laquo-coordinate to he used in
nodal print N in layer LY (see
poundig 9)
First principal conductivity of
material in node N and layer LY
idem second principal conductivity
thickness of layer LY at nodal
point N
ig V Definition of SANGCI and C2
One card icircuiPt be provided for each model point ctarting wih M = 1
and ending with N = NUMNP If there is norlt3 than 1 layer for each
layer a set must be provided starting with the upper layer LY - 1
and ending with the bottom layer LY = NIJMLAY The variables SANG
M and 2 are not relevant tor aquitards but for reasons of
conveniency this procedure is maintained (making SANG CI and C2
dummy variables)
1-10 110 KODE(N)
11-20 II0 NTERR(N)
coding of nodal points (see
paragraph 42)
number of tertiary relation in node N
The q-h relationship of the flow to
23
GROUP COLUMNS FORMAT SYMBOL DESCRIPTION
21-30 FlOl X(N)
31-40 FlOl Y(N)
41-50 FlOl HGL(N)
51-60 FlOl HBSC(N)
61-70 FlOl LESC(N)
the tertiary system as discussed
in par 24 are schematized in a small
number of different relations
x-coardinate of node N
y-coordinate of node N
height ground level in node N
height bottom secondary conduit in
node N If node N is not a secondary
conduit this variable is a dummy
variable
length of secondary conduit in area
represented by node N
One card must be provided for each node starting with N = 1 and
ending with N = NUMNP
E 1-10 FlOl P(NLY1) initial value of h at node N in
layer LY
11-20 FlOl VERRES(NLY) vertical resistance at node N and
layer LY If the layer is an aquifer
this value is a dummy variable
21-30 F101 RADRES(NLY) radial resistance for flow to
secondary system at node N in layer
LY
31-40 FlOl ENRES(NLY) entrance resistance for flow to
secondary system at node N in layer
LY
41-50 FlOl WP(NLY) wet perimeter of secondary system at
node N in layer LY
51-60 FlOl P0R(NLY) specific yield of the material at
node N in layer LY
61-70 FlOl SS(NLY) specific storage of the material
at node N in layer LY
24
GROUP COLUMNS FORMAT SYMBOL DESCRIPTION
One card must be provided for each node starting with N = 1 and
ending with N = NUMNP If there is more than 1 layer for each layer
a set of cards must be provided starting with LY = 1 and ending
with LY = NUMLAY
F 1-10 F101 P(NLY1) value of h at different times at
node N in layer LY Node N lies
on a prescribed head boundary
11-20 F101 WTSC(N) water table in secondary system at
node N at different times
21-30 F 101 ARRED(NLY) artificial recharge to or discharge
from node N in layer LY or
prescribed boundary flow at
different times
31-40 F 101 QPHR(N) flux through phreatic surface at
node N at different times
The data in group F are non-stationary input data One card must be
provided for each node starting with N = 1 and ending with N = NUMNP
In case of more than 1 layer for each layer a set must be provided
Not relevant variables are taken as dummy variables
After a certain number of time steps (NTSPI) the data of group E
are read So the set cards E must contain NxLYxC cards where C =
(TMAXNTSPIXDELT) + 1 if DELT is a constant
42 R u n n i n g t h e p r o g r a m o n t h e C y b e r 7200
The program has been executed on a Cyber 7200 computer of
IWIS-TNO in the Hague
- The demanded input is punched on cards according to the instructions
of the last paragraph and is put on a permanent file
- For each specific problem inthe program only the dimension statements
in the main program have to be adjusted according to the instructions
in the lines of comment
- After compilation the program together with the input file is
executed
25
- Intermediate data necessary for continuation of the calculations
are saved on another permanent file This gives the opportunity
to restart the program at a time the calculation was stopped
The only change in the input file is to adjust the starting time
Remark
With the aid of a terminal small changes in the program and in the
input file can be made easily and the output immediately appears
on the terminal In this way it is possible to stop erroneous
calculations This is especially useful when being in the testing
phase of the program
43 O u t p u t
Immediately after reading the input data they are printed so
that possible errors can be detected During execution after each
certain number of time steps (NTSPO) calculated values of h and
water balance terms are printed In appendix B an example is given
The output data can be devided into two groups
A Data per node per layer
- Coding of the nodal points KODE
- Hydraulic head HEAD
- Flow through phreatic surface at the time indicated in head
of table FLOWUN
- Flow to tertiary surface water system FLOWTS
- Flow to secondary surface water system FLOWSS
- Flow to adjacent layers LEAKAGE
- Artificial recharge or discharge and prescribed boundary
flow ARRED
- Change in storage during last time step STORAGE
- Flow through prescribed head boundary LATFLOW
B Data per layer
Summation of the terms already mentioned under A gives the
magnitude of the water balance terms per layer at the time
indicated (instanteneous water balance terms) Summation of
these instanteneous terms multiplied by their respective time
intervals since time is zero gives the cumulative water balance
terms per layer
26
After each print out we are able to check the calculations
because the sum of the water balance terms per layer must be
approximately zero
44 C o m p u t i n g t i m e
The computing time depends on the number of nodal points and
the number of layers But it depends also on the magnitude of the
changes in hydraulic head and on the value which terminates the
Gauss-Seidel iteration (TOL) To give an impression TOL = 001 m
Costs per day simulation (in hlfl) on the Cyber 7200
number of nodes number of layer bull bdquo bull bull _ 1mdash x a x b (21)
where
a = costs in Dutch guilders per system second (ƒ 027)
b = empirical factor with magnitude about 001
V NUMERICAL EXPERIMENTS
In general there are two possibilities to test the validity of
numerical models The first and most commonly used method is to
compare numerical results with analytical solutions The second
possibility is to compare numerical results with measured field
data Since the last alternative requires many data that are
difficult to obtain we restrict ourselves for the time being to
comparison with the analytical solutions
5 1 F l o w t o a w e l l i n a n u n c o n f i n e d
c i r c u l a r - s h a p e d a q u i f e r
Flow to a well can be divided into steady-state flow and
unsteady-state flow (KRUSEMAN and DE RIDDER 1976)
27
511 Steady state flow in an unconfined aquifer
For this case the formula of Thiem-Dupuit is valid (see fig 10)
Q bull k lth2-hicircgt ln(r2r])
(22)
where
hhbdquo = hydraulic head in 1 and 2 respectively
Q = discharge from well
rbdquo = distance from well
CL31
hmdash^ri --r2
Fig 10 Schematic cross-section of a pumped unconfined aquifer
Fig 11 gives the nodal grid superimposed on a well in a
circular shaped aquifer Near the well the nodal point distance
is made smaller Quadrilaterals as well as triangles are used
Numerical calculations with this finite element network and 3 -1
an extraction rate of 200 m day were performed for 10 days
At that moment a steady-state situation was almost achieved
28
2tmdashnodal point number
160 180 X-axesim)
Fig 11 Nodal point configuration for flow to a well in a circular-
shaped aquifer The outer boundary is taken to be constant
The well is situated in nodal point 42
29
In fig 12 the results of the simulation are compared with the
analytical solution The agreement between numerical results and
the analytical solution is excellent
draw- 15irS (m) down
analytical solution imdashx calculated values
distance from well r (m)
Fig 12 Comparison between analyt ical and calculated s teady-state
drawdown for the problem of f ig 11 with 1
-1 Q bull - 200 m 3 d a y _ 1
k = 10 mday
d = 100 m
S = 2 y
S = 0 s
512 Unsteady-state flow in an unconfined aquifer
Analytical solutions for unsteady-state flow in an unconfined
aquifer only exist if the aquifer extends infinitely For the
aquifer given in fig 11 this condition is not fulfilled However
during the first one or two days after starting the extraction the
influence of a constant hydraulic head at the outer boundary on the
30
drawdown near the well can be neglected (see fig 13) After one
day the lateral inflow is only 25 of the extraction the remainder
being the change in storage due to drawdown
2 0 0
m 3 day 1
Q=200m 3 day
100-
vraquomdash Lateral inflow
N amp storage
I I ^ V - W 10
t (days)
Fig 13 Values of Q lateral inflow and change in storage as
functions of time
The nonsteady-state or Theis equation can be written as
i _ Q s 4iTkD
dy = 4irkD
W(u) (23)
where
u = r2S
4irkDt
r = distance from well
S = storage coefficient
Q = constant discharge from well
D = thickness of the waterbearing layer
at the beginning of pumping
t = time after starting the extraction
s = corrected drawdown
= s - 322D
s = calculated drawdown
W(u) = Theiss well function
[J
CIAT1
[T]
]
31
In fig 14 the analytical unsteady-state solution for the case
described in section 511 is given together with the numerical
results at times 05 and 10 days The agreement between numerical
results and the analytical solution is rather good
Drawdown s1(m)
10
10
10
Theiss solution
x caculated values t=Q5
bdquo t= lO
10 10- 10 10 t r2(daym2)
Fig 14 Comparison between analytical and calculated steady-state
drawdown for the problem of fig 11 with
Q
k
d
S y
s s
=
=
=
=
=
200
1
10
3 m
0
0
2
0
-1 day
m day
m
These results indicate that the assumption of a linear variation
of h between two nodal points (see fig 3 ) even with the rather
rough element network of fig 11 is acceptable So there is no
need to improve for example the finite element network by introducing
higher order coordinate functions
32
52 F l o w t o w e l l i n 3-1 a y e r e d s y s t e m
In fig 15 a schematic cross-section of a 3-layered system of
two aquifers separated by an aquitard is given A well of
infinitisemal radius is completed in the lower aquifer and discharges
at a constant rate QTT Each layer is homogeneous isotropic and
extending radially infinite
bdquo bull - bull bull ~ i
mdash Q II
aquifer I
lugrave^i~IcircIuml7i-i~~ aquitard --r-rl~-
-II--I i - i
aquifer
Fig 15 Schematic cross-section of a 3-layered system
For this situation NEUMAN and WITHERSPOON (1969a) give rather
complicated analytical solutions In the special case where the
hydraulic properties of the two aquifers are identical they give
graphical illustrations of their solutions for three different
cases
These graphical solutions give the opportunity to verify
roughly numerical results of a 3-layered system Therefore a nodal
grid is superimposed on a circular shaped groundwater flow system
with the extraction in the centre of the lower aquifer and with
a constant head on the outer boundary Fig 16 gives the nodal
point situation in the x - y plane of 1 layer The nodes in the
other two layers are situated in the middle of the layers at the
same places in the x - y plane only their z-coordinate is different
33
A
-Jy
bull33
V 5
iumlff
Fig 16 Nodal point configuration for the 3-layered problem (top view)
34
In fig 17 numerical results are compared with the analytical
solution The agreement is rather bad especially for the two
aquifers But in terms of the water balance the difference between
aquifers and aquitard is not so big because the aquitard has
about 16 times bigger storage capacity than the aquifers The
small value of the specific storage of the aquifers is the reason
that small errors in the components of the water balance have a
big influence on the calculated values of the hydraulic head
Eig 17 Dimensionless drawdown vs dimensionless time in a
3-layered system with
II = 50 m day
3 m (distance from well)
aquifer I
aquitard
aquifer II
Thi (m)
100
25
100
S s
001
0642
001
k(mday )
1
0277
1
The discrepancies are partly caused by errors in discretization
In the z-direction these errors can be eliminated by taking the
35
vertical resistance of the aquitard to be infinite Then the
calculated drawdown in the pumped aquifer can be compared with
the analytical Theiss solution
This is done in fig 18 for different distances from the well
One sees that the agreement is better when the distance is bigger
Apparently the linearization of the logarithmic drawdown curve
near the well is responsible for the differences The differences
for r = 3 m are of the same magnitude as for the pumped aquifer in
the 3-layered case
joo gt A 50
20
Theiss solution
bull calculated values r=3m (distance f rom well) x bdquo bdquo r=7m
1 L
I l_ 001 003 0 0 5 Ol Q2
- I I 5 10 t - K t
Fig 18 Dimensionless drawdown vs dimensionless time in a
confined aquifer with
Q =500 m day
k = 10 mday
D = 100 m
S = 001 s
The differences between numerical results and analytical solutions
are not caused by taking a finite area because changing the outer
boundary into an impermeable boundary has only minor effects
36
VI SUMMARY AND CONCLUSIONS
In this report a numerical model for non-stationary saturated
groundwater flow in a multi-layered system has been treated
Groundwater flow can be described in different ways
a Physical-mathematical by combining Darcys law and the law of
continuity (section 21)
b Geo-hydrological the flow in a natural groundwater basin takes
place in layers with different hydraulic properties Definitions
of the most important hydraulic properties are given in section
22
c Schematical groundwater flows horizontally in aquifers and
vertically in aquitards (section 23)
d Approximating by dividing the region of flow into a finite
number of elements and applying Darcys law and the law of
continuity on each of these elements In these ways one gets
a set of equations which can be solved given (initial and)
boundary conditions (section 25)
e Numerical the set of equations has been solved using a Galerkin
type finite element method and the implicit Gauss-Seidel method
with overrelaxation factor (section 31 and 32)
f Computer-solvable the numerical solution has been translated
in a FORTRAN IV program called FEMSAT (section 33)
In chapter IV the input execution and output of the program
FEMSAT are treated
Finally in chapter V some numerical applications are given
simulation of flow to a well in unconfined aquifer and simulation
of flow to a well in a 3-layered configuration
Comparison of hydraulic heads obtained by numerical calculations
and analytical true solutions showed an excellent agreement in case
of stationary flow and a good to moderate agreement in case of non-
stationary flow However the diviations caused by the discretization
process must be seen in relation to the errors made in the
establishment of the hydraulic properties
37
As was stated in the introduction numerical models can be
valuable tools in the management of water resources Therefore
numerical models must be suitable to simulate effects of certain
operations upon the spreading of water over saturated groundwater
unsaturated groundwater and surface water The model described in
this report is a model for saturated groundwater flow The relation
with the surface water can be taken into account via radial and
entrance resistances or via a relation between flow to the surface
water system and the hydraulic head The relation with the unsaturated
zone has been restricted to the input factor effective precipitation
In practise however the unsaturated and saturated system are
interconnected Therefore the model should be improved by taking
into account the unsaturated zone This can be done in at least
two ways
a Replacing the unsaturated zone by a parametrical model
b Coupling the unsaturated zone in an iterative way with the
saturated system
In order to find out the usefulness of the numerical model for
water resources management it will be applied to actual regional
groundwater systemslaquoAlso sensitivity analyses will be carried out
The computer model itself can be improved by applying standard
programs eg automatical generation of a nodal grid and graphical
plotting of the results
38
VII REFERENCES
ERNST LF 1962 Grondwaterstromingen in de verzadigde zone en hun
berekening bij aanwezigheid van horizontale evenwijdige open
leidingen Thesis University Utrecht pp 189
1978 Drainage of undulating soils with high groundwater
tables (in press)
FORSYTHE GE and WR WASON 1960 Finite Difference Methods for
Partial Differential Equations John Wiley and Sons New York
pp 444
KRUSEMAN GP and NA DE RIDDER 1976 Analysis and evaluation of
pumping test data ILRI-bulletin No 11 pp 200
LAAT PJM DE and C VAN DEN AKKER 1976 Simulatiemodel voor grond-
waterstroming en verdamping in Modelonderzoek 1971-1974 ten
behoeve van de waterhuishouding in Gelderland deel II
Grondslagen pp 145-197
NEUMAN SP RA FEDDES and E BRESLER 1974 Finite element
simulation of flow in saturated -unsaturated soils considering
water uptake by plants Hydraulic Engineering Laboratory
Technion Israel pp 104
and PA WITHERSPOON 1969a Theory of Flow in a confined
two Aquifer System Water Resources Research Vol 5 No 4
pp 803-816
^_ and PA WITHERSPOON 1969b Applicability of Current Theories
of Flow in leaky aquifers Water Resources Research Vol 5
No 4 pp 817-829
PINDER GF and EO FRIND 1972 Application of Galerkins Procedure
to Aquifer Analysis Water Resources Research Vol 8 no 1
pp 108-120
REMSON I GE HORNBERGER and FJ MOLZ 1971 Numerical Methods
in Subsurface Hydrology John Wiley and Sons New York
pp 389
WESSELING JG 1977 Toepassing van Finite Difference- en Finite
Element Methoden voor het oplossen van een aantal grondwatershy
stromingsproblemen (met computerprogrammas) ICW-nota 944
pp 63
39
ZIENKIEWICZ OC 1971 Finite Element Method in Engineering Science
McGraw-Hill London pp 521
40
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X - e - o
gt-o-
LUX
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J X I - 3gt z c i r c m i gt 3 ~lt r m bulljgt raquo
-lt rgt z raquo 0 3 3 0 m m 0
3 0 0 m
t o H
^ 00 X
iuml gt - 4
-H
3 m
- n
t n
t o
bullbull0 x
z
3
3gt
C -lt Igt - t Ci
APPENDIX 8 EXAMPLE OF OUTPUT
HEAD AND TERMS OF UATERBALANCE PER NODE PER LAYER
AT TIME 260 IN II AND I13DAYS
T
LAYER NODE KODE HEAD FLOUUN FLOUTS FLOUSS LEAKAGE STORAGE LAT FLOU
13 19 25 31 37 43
125008 125000 12S000 12S000 12S000 12S000 12S000 12S000
0000 0000 0000 0000 0000 0000 0000 0000
0000 0000 0000 0000 0000 0000 0000 0000
0000 0000 0000 0000 0000 0000 0000 0000
- 000 - 000 -000 - 000 - 000 - 000 - 000 0000
0000 0000 0000 0000 0000 0000 0000 0000
0000 000 000 000 000 000 000 000
000 0000 0000 0P100 0000 0000 0000 0000
1 7
13 19 25 31 37
12S000 125000 125000 125000 125000 12S000 125000 125000
0000 0000 0000 0000 0000 0000 0000 0000
- 001 -007 - 023 - 020 - 009 - 003 0000
0000 0000 0000 0000 0000 0000 0000 0000
000
001
007
023
020
009
003
003
0000 0000 0000 0000 0000 0000 0000 0000
1 7
13 19 25 31 37 43
125000 125000 124998 124972 124895 124787 124647 124358
0000 0000 0000 0000 0000 0000 0000 0000
- 000 001 007 023 020 009 003
0000
0000 0000 0000 0000 0000 0000 0000
50000
0000 182
1982 3862 1674 419 078 041
0000 0000 0000 0000 0000 0000 0000 0000
TERMS OF WATERBALANCE PER LAYER DURING LAST TIMESTEP AND
AND SINCE BEGINNING IN M3DAY M3
FLOU TO UNSATURATED ZONE
FLOU TO TERTIARY SYSTEM
0000
0000
0000
0000
LAYER NO 1
FLOU TO SECONDARY SYSTEM
FLOU TO AD3ACENT LAYERCS)
ARTIFICIAL FLOU AND PRESCRIBED BOUNDARY FLOU
ACCUMULATION TERMS
LATERAL FLOU THROUGH PRESCR HEAD BOUNDARY
0000
-000
0000
000
000
0000
-000
0000
000
000
LAYER NO 2
FLOU TO SECONDARY SYSTEM
FLOU TO AD3ACENT LAYERCS1
ARTIFICIAL FLOU AND PRESCRIBED BOUNDARY FLOU
ACCUMULATION TERMS
LATERAL FLOU THROUGH PRESCR HEAD BOUNDARY
0000
-378
0000
381
0000
0000
-055
0000
056
0000
LAYER NO 3
FLOU TO SECONDARY SYSTEM
FLOU TO AD3ACENT LAYERCS5
ARTIFICIAL FLOU AND PRESCRIBED BOUNDARY FLOU
ACCUMULATION TERMS
LATERAL FLOU THROUGH PRESCR HEAD BOUNDARY
0000
370
50000
49376
0000
0000
055
12979
12909
0000