paper hpsc2006

INTERNATIONAL CONFERENCE ON HIGH PERFORMANCE SCIENTIFIC COMPUTING Modeling, Simulation and Optimization of Complex Processes

March 6-10, 2006 Hanoi, Vietnam

A general mathematical model of two-dimensional horizontal flow of seawater instrusion

Tran Van Minh, Nguyen The Hung Danang University

Abstract - In this paper the authors present a general mathematical model of two dimensional horizontal flow of seawater intrusion into coastal confined and unconfined aquifers. Algorithms and programing of this model are formulated by weak Galrkin finite element method for prediction of the transient effect of pumping wells on seawater intrusion into coastal confined and unconfined aquifers.

The validity of the model is tested by using the analytical solutions. An example was calculated to determine the location of the interface at Hoa Khanh coastal aquifers when one extracts, by pumping wells, a certain discharge of freshwater, during a long time, near the interface.

I. Introduction

At coastal aquifers, a zone of transition from fresh aquifer water to seawater, the water with different concentrations of salt as a result of hydrodynamic dispersion of the dissolved matter. Under certain conditions, it may be narrow, relative the aquifer thickness, and may be approximated as a sharp interface. The intensive extraction of groundwater has upset the long established balance between freshwater and seawater potentials, causing encroachment of seawater into freshwater aquifers, this phenomenon is said to be seawater intrusion.

General, the interface is described by three-dimensional mathematical model; but under certain conditions for simplifying reason, one can described this interface by regional flow model; with integrating the three-dimensional model over the vertical, between the upper and the lower boundaries of a considered domain.

In this paper, the authors derive a general mathematical model of two-dimensional horizontal flow just validitied for coastal confined aquifer and phreatic aquifer.

The algorithms and programing of this model are formulated by standard Garlerkin finite element technique.

II. Governing equations The governing system of coupled differential equation of phreatic aquifer [4]

-.(aT f) + .(aTah) = If + Is+ q'f+ q's (1)

Sh/t -.(aTa )h + .(aTa f) = -Is- q's (2)

where:

f = hf/a ; T = K(H1+hf) ; Ta = T(H1 - h)/(H1+ hf)


March 6-10, 2006 Hanoi, Vietnam confined aquifer [4]

-.(aT f) + .(aTah) = If + Is+ q'f+ q's (3)

Sh/t -.(aTa )h + .(aTa f) = -Is- q's (4)

where: f = ff/a ; T = K(H1 - H2); Ta = T(H1 - h)/(H1- H2)

T: transmissivity of the aquifer.

With different equation system governing for phreatic aquifer (1), (2) and confined

aquifer (3), (4) we rewrite them a general form as follows

's

'fsfayaxyx qqII)y

hT(

y)

x

hT(

x)

y

fT(

y)

x

fT(

x+++=

a

+

a

+

a

+

a

- (5)

,ssayaxayax0 qI)y

fT(

y)

x

fT(

x)

y

hT(

y)

x

hT(

xt

hS --=

a

+

a

+

a

+

a

-

(6)

where:

gpzh

hf

ff

f

fsff

rr

rra

a

bfb+=

-=

+-= ;;

)1(

[ ] [ ]2f1yy2f1xx Hh)1(HKT;Hh)1(HKT b-b-+=b-b-+= ;

]Hh)1(H[

)hH(TT;

]Hh)1(H[

)hH(TT

2f1

1yay

2f1

1xax b-b-+

-=

b-b-+-

=

and:

+ for the case of phreatic aquifer b = 0

+ for the case of confined aquifer b = 1 Where:

rf : density of freshwater; rs: density of seawater; g: gravity acceleration S0: specific storativity; t: time hf: piezometric head for freshwater

ff ,fs: piezometric head (as Hubberts potential) of freshwater and saltwater respectively

If, If:- supply functions, representing a distributed surface supply of fresh and salt water into the aquifer q'f, q's: sources or sinks of freshwater and salt water aquifer respectively Kx, Ky: x, y directional component of hydraulic conductivity



(a)- Phreatic coastal aquifer (b)- Confined coastal aquifer Fig. 1- Coastal aquifer with fresh and salt grounwater

III. Numerical procedure

The system equations (5), (6) are formulated by weak Galerkin finite element method, for any element, with using linear triangular element

Where:

Ni(e) , i =1,2,3 are the linear shape functions L(P)m , m =1,2 are the partial differential equations (5) or (6) A: area of element The interface depths are linearly interpolated in terms of the shape function as

follow:

=

=n

1ii

)e(i

)e( hNh (8)

Where: h (e): the z coordinate of interface or interface depth at any point of an element n: total nodes of an element; hi: interface depth at node i of an element The time derivative is discretized using the finite difference scheme as

t

)t(h)tt(h

t

h

D-D+

@

t

)t(h)tt(h)t()t(h

t

h)t()t(h)(h

D-D+

-e+=

-e+@e

Where: t

)t(

D-e

=w

)()()1()( tththh D++-= wwe

)7()( dAPLN mA

i

Saltwater Saltwater Freshwater

hfh

H h

Freshwater

x

h

H1 x

Interface Interface H

z 'sq

I

'fq

z

'sq I

'fq

H2



After applying transformation of mathematical techniques, we receive the algebraic equation systems

[ ] [ ]

b-b-++b-b-

+

+

+

+

+

+

=

-

n

1

2f1

2f

snfn

1s1f

n)e(sn

)e(fn

1)e(1s

)e(1f

n

1

b

n

1

a

R

.

.

R

]Hh)1(H[

]hHh)1[(

qq

.

.

qq

3

A)II(

.

.3

A)II(

h

.

.

h

K

f

.

.

f

K

(9)

[ ] [ ] [ ] [ ]

D+

-

-=

D

++

-

D+ n

1

sn

1s

n)e(

sn

1)e(

1s

ttn

1

b

n

1

b

h

.

.

h

t

C

q

.

.

q

3

AI

.

.3

AI

h

.

.

h

t

CK

f

.

.

f

K

(10) where:

[ ] [ ] [ ] [ ] [ ] [ ]

[ ] [ ] [ ] [ ] [ ] [ ] )n,1i(RR;qq;qqqq

CC;KK;KK

m

1e

)e(ii

m

1e

)e(sisi

m

1e

)e(si

)e(fisifi

m

1e

m

1e

)e()e(bb

m

1e

)e(aa

===+=+

===

===

= ==

[ ]

a

+

a

=2kjkik

kj2jij

kiji2i

ay)e(

2kjkik

kj2jij

kiji2i

ax)e(

)e(b

ccccc

ccccc

ccccc

A4

T

bbbbb

bbbbb

bbbbb

A4

TK

in which

So(e): specific storativity of a triangular element ai, bi, ci: coefficients of a shape function Ni

The domain pattern for studying illustration is a rectangular, symmetric by pass CD edge; AC is the coast line along the y axis, AB, CD perpendicular to the coast; Q1, Q2 are the charge of fresh water of pumping well (Fig. 2)

[ ] [ ]

=

a

+

a

=100

010

001

3

ASC;

ccccc

ccccc

ccccc

A4

T

bbbbb

bbbbb

bbbbb

A4

TK )e(0

)e(

2kjkik

kj2jij

kiji2i)e(

y

2kjkik

kj2jij

kiji2i

)e(x)e(

a



Fig.2- Discretization of domain using linear triangular elements Initial and boundary conditions:

- Initial conditions: At initial time t = 0 interface depth (h) is given by Ghyben-Herzberg formulas:

ffs

f hhr-r

r-= (11)

- Boundary conditions: on AB, CD: qni = fi(x,yi,t), i =1, 2 any known total flux of the

liquid normal to the boundary respectively on AC: h = f(y,t0) any known interface depth at initial time

on BD: 0=yf

The system of equations (9) and (10) are solved by iterative conjugate gradient method; First, equation (9) is solved for the head fi , and then equation (10) for the depth hi at all the nodes of the domain.

In each cycle, the coefficients , which depend upon hi , are updated. This order of solving the two systems of coupling equations is chosen because the head fi is influenced by all inflow functions.

In each time step the values of the variables hi are updated IV- Model verification Before proceeding to the real case studies, the numerical model is tested with the analytical solutions. 4.1. Computational domain and applied data: The computational discretization domain is shown in Fig. 2 with three pumping wells; only one half of the phreatic aquifer is considered in the program, because of the symmetry axe 0x of the problem.

Coastline

y

x

Q1

Q2

qn1

qn2

A B

C D



Hydraulic conductivity K= 8.1 m/d; supply functions I = 0 m/d; specific storativity S0 = 0.6; impermeable depth Ht = 25 m; injection wells by pumping: Q1(x1,y1)=Q1(1000,0)=120m3/d, Q2(x2,y2)=Q2(1000,100)= 60m3/d, Q3(x3,y3) = Q3(1000,-100) = 60 m3/d; on coast line (AC = 400 m, x = 0): h = 0 m; on AB = CD = 1100 m perpendicular to the coast line: qn = 0; the domain is divided in to 44 elements and 60 nodes. 4.2. Results and comments

The numerical results of time dependent sharp interface is shown in Table 1. Table 1: Location and the interface depth (h) at times (t)

X (m) 0 100 200 300 400 500 600 700 800 900 1000 1100

h (t=0 day) 0 -16.00 -22.80 -25 -25 -25 -25 -25 -25 -25 -25 -25

h(t=8000 d) 0 -8.91 -14.78 -20.17 -23.95 -24.95 -24.99 -25 -25 -25 -25 -25

h(t=14000 d) 0 -7.54 -12.36 -16.78 -20.96 -24.1 -24.95 -24.99 -25 -25 -25 -25

h(t=20000d) 0 -6.76 -10.96 -14.83 -18.56 -22.05 -24.34 -24.95 -24.99 -25 -25 -25 Based on the numerical results of Table 1, the location of time dependent sharp

interface is shown in Fig. 3. One can see that the distance of the interfaces toe from the coast line, at t = 0 day, is 250 m, and from the extraction well is 750 m. After t = 20000 days, this toe will deeply move into the aquifer and these distances respectively are 701.5 m and 298.5m

-30

-25

-20

-15

-10

-5

0

0 200 400 600 800 1000 1200

x

h

Q1

t = 20000 day

t = 0

Fig. 3: Location of time dependent sharp interface

A comparison of the Theis, Strack formulas and the numerical results using finite

element method for calculation distance from the coastal line to the toe of the sharp interface, is shown in Table 2.

Table 2: Comparison of Theis, Strack formulas and numerical results

X (m) FEM Theis Strack Distance from the coastal line to the toe of the

sharp interface at time t = 20000 day. 701,50 701,46 701,60



From the results summarized in Table 2, it follows that the results of these

methods differ very little, but the numerical solutions by finite element method can calculation the shape, location of the sharp interface with different geologies.

V. Application for a real problems Applying for quartier North of Danang City, with an area of 2.88 km2 , one its side along the coastal line of Danang Bay (Fig. 4).

Fig 4: Sketch of pumping wells of two geological layers of

phreatic and confined aquifer This region includes two aquifer layers:

Upper layers (layer 1 & 2) are phreatic aquifer, Lower layers (layer 3 & 4) are confined aquifer And between two aquifer layers is an impermeable thin layer (Fig. 5, 6)

Table 3: Data of hydrogeology of aquifer layers

Hydraulic conductivity Aquifer Average satatic water

level (m)

Average layer

thickness (m)

Layer

Kx (m/day)

Ky (m/day)

Kz (m/day)

1 16 16 1,6 Phreatic aquifer

0,74

20

2 20 20 1,2

3 12 12 1,2 Confined aquifer

0,85

25

4 8 8 0,8



Fig. 5: Geological section of different layers in a Easterly-Westerly

Fig. 6: Geological section of different layers in Northerly-Southerly direction

Sketch of extraction pumping wells Extraction pumping wells are located in four row paralleling the coastal line; the extraction wells of phreatic aquifer are noted HKi. The extraction wells of confined aquifer are noted DKi (Fig. 4). The discharge of pumping wells and well coordinates as shown in Table 4.

Table 4: Location of fresh water extractrion wells


March 6-10, 2006 Hanoi, Vietnam Sketch for study Region for study is a rectangular with one side being the coastal line 1600 m long and another perpendicular to the coastal line (1300 m long for phreatic aquifer and 1700 m for confined aquifer). Phreatic aquifer includes 192 rectangular elements, 221 nodes and confined aquifer include 272 rectangular elements, 306 nodes.

Fig 7: Sketch for studying phreatic aquifer

Fig 8: Sketch for studying confined aquifer


March 6-10, 2006 Hanoi, Vietnam Initial condition At initial time t0, parameters hf, h are determined from steady flow. For phreatic aquifer and confined aquifer they are showed correspondent in Table 5 and Table 6.

Table 5: Parameters hf, h at perpendicular cross section to the coastal line and crossing the pumping well HK6 (at initial time t0 = 0).

Node 9 26 43 60 77 94 111 128 145 162 179 196 213 x 110 200 300 400 500 600 700 800 900 1000 1100 1200 1300 hf 0 0.45 0.55 0.64 0.72 0.78 0.85 0.91 0.96 1.01 1.06 1.11 1.15

h initial 0 -

18.1 -

20.5 -

20.5 -

18.4 -

20.1 -

20.9 -

22.3 -

19.1 -

20.1 -20 -

18.5 -

20.6

Table 6: Parameters ff , h at perpendicular cross section to the coastal line and crossing the pumping well DK3 (at initial time t0 = 0).

Node 10 27 44 61 78 95 112 129 146 163 180 197 214 231 248 265 282 299

X 70 100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500 1600 1700

ff 0 0.36 0.51 0.62 0.72 0.80 0.88 0.95 1.02 1.08 1.14 1.19 1.25 1.30 1.35 1.39 1.44 1.48

h initial -22.3 -34.4 -40.6 -44.9 -44.8 -44.6 -45.8 -44.9 -45.9 -44.2 -44.6 -45.8 -45.8 -44.4 -44.9 -47.3 -46.2 -44.6

Boundary condition For phreatic aquifer

- Boundary AB: qn1 = 0, - Boundary CD: qn2 = 0,

- Boundary AC: h = 0, - Boundary BD: 0=yf

For confined aquifer - Boundary AB: qn1 = 0, - Boundary CD: qn2 = 0.

- Boundary AC: h = -20m, - Bin BD: 0=yf

Results:

For phreatic aquifer, we study two cases: Case 1: no supply function I = 0.00 m / day Case 2: with supply function I = 0.001 m / day The movement of interface at pumping wells of phreatic aquifer without supply functions as in Table 7.

Table 7: Distance from the toe to pumping well of phreatic aquifer without supply functions.

No Pumping well Distance from the toe to pumping well after a time

period (m)



1000 day 8000 day 10000 day 1 HK1 200 98 45 2 HK2 195 96 40 3 HK3 195 96 40 4 HK4 200 98 45 5 HK5 385 200 45 6 HK6 389 198 40 7 HK7 385 200 45 8 DK1 489 240 160 9 DK2 480 230 155 10 DK3 485 230 155 11 DK4 485 240 155 12 DK5 685 390 280 13 DK6 685 390 280

The movement of interface at cross section passing pumping wells of phreatic aquifer with supply function as in Table 8.

Table 8: Distance from the toe to pumping well of phreatic aquifer

with supply function Distance from the toe to pumping well after a time

period (m) No Pumping well

1000 day 8000 day 10000 day 1 HK1 290 275 180 2 HK2 300 285 175 3 HK3 290 275 175 4 HK4 300 285 180 5 HK5 425 390 320 6 HK6 420 395 325 7 HK7 435 390 320

Location of interface at perpendicular plane section to the coastal line across pumping wells HK6, HK3 of phreatic aquifer without supply functions (Table 9, Fig. 9 and Fig.10).

Table 9: Location and depth of interface corresponding the time period (day)

at plane section across pumping well HK6, HK3.

At plane section across pumping well HK6 (Q6=300m3/day)

Node 9 26 43 60 77 94 111 128 145 162 179 196 213 X 110 200 300 400 500 600 700 800 900 1000 1100 1200 1300 h initial 0 -18.1 -20.5 -20.5 -18.4 -20.1 -20.9 -22.3 -19.1 -20.1 -20 -18.5 -20.6 h(t=1000 day) 0 -14 -19.9 -20.4 -17.4 -20.1 -20.9 -22.3 -19.1 -20.1 -20 -18.5 -20.6 h(t=8000 d) 0 -7.5 -12 -15.4 -17.5 -20.2 -20.9 -22.3 -19.1 -20.1 -20 -18.5 -20.6


March 6-10, 2006 Hanoi, Vietnam h(t=10000 d) 0 -4.8 -9.1 -12.5 -15.9 -18.5 -20.8 -22.3 -19.1 -20.1 -20 -18.5 -20.6

At plane section across pumping well HK3 (Q6=300m3/day)

Node 10 27 44 61 78 95 112 129 146 163 180 197 214 X 110 200 300 400 500 600 700 800 900 1000 1100 1200 1300 h initial 0 -18.1 -20. -20.4 -17.9 -20.1 -20.9 -22.3 -19.1 -20.1 -19.5 -19.2 -20.7 h(t=1000 day) 0 -14.3 -18.4 -20.4 -17.9 -20.1 -20.9 -22.3 -19.1 -20.1 -19.5 -19.2 -20.7 h(t=8000 d) 0 -7 -11 -15 -17 -20.1 -20.9 -22.3 -19.1 -20.1 -19.5 -19.2 -20.7 h(t=10000 d) 0 -5 -9 -12 -18.5 -20.1 -20.9 -22.3 -19.1 -20.1 -19.5 -19.2 -20.7

.

Location of interface at perpendicular plane section to the coastal line across the

pumping wells HK6, HK3 of phreatic aquifer with supply function (Table 10, Fig. 11).

Table 10: Location of time dependent sharp interface at HK6 with supply function

At the cross section HK6 with supply function Q6=300m

3/day

-25-20-15-10

-50

0 2 00 4 00 6 00 8 00 10 00 12 00

Q6=300m3/day

Impermeable layer of phreatic aquifer

t=0 1000 8000 10000 day

Fig. 9: Location of time dependent sharp interface at HK6 when phreatic aquifer has not supply functions

-25-20-15-10

-50

0 2 00 4 00 6 00 8 00 10 00 12 00

Q3=300m3/day

t=0 1000 8000 10000 day


Fig. 10: Location of time dependent sharp interface at HK3 when phreatic aquifer has not supply functions



Node 9 26 43 60 77 94 111 128 145 162 179 196 213 X 110 200 300 400 500 600 700 800 900 1000 1100 1200 1300

h initial 0 -

18.1 -

20.5 -

20.5 -

18.7 -

20.1 -

20.9 -

22.3 -

19.1 -

20.1 -20 -

18.5 -

20.6 h(t=1000

) 0 -

14.7 -

18.7 -

20.5 -

18.7 -

20.1 -

20.9 -

22.3 -

19.1 -

20.1 -20 -

18.5 -

20.6 h(t=8000

) 0 -

11.5 -

16.3 -

18.1 -

18.7 -

20.1 -

20.9 -

22.3 -

19.1 -

20.1 -20 -

18.5 -

20.6 h(t=1000

0) 0 -12 -

13.6 -

15.3 -

17.2 -19 -

20.9 -

22.3 -

19.1 -

20.1 -20 -

18.5 -

20.6 At the cross section HK3 with supply function Q6=300m

3/day Node 10 27 44 61 78 95 112 129 146 163 180 197 214 X 110 200 300 400 500 600 700 800 900 1000 1100 1200 1300

h initial 0 -

18.1 -

20.5 -

20.5 -

18.7 -

20.1 -

20.9 -

22.3 -

19.1 -

20.1 -

19.5 -

19.2 -

20.7 h(t=1000

) 0 -

14.7 -

19.9 -

20.5 -

18.7 -

20.1 -

20.9 -

22.3 -

19.1 -

20.1 -

19.5 -

19.2 -

20.7 h(t=8000

) 0 -

12.2 -

17.3 -

19.2 -

18.5 -

20.1 -

20.9 -

22.3 -

19.1 -

20.1 -

19.5 -

19.2 -

20.7 h(t=1000

0) 0 -10 -

14.3 -

16.3 -

17.5 -

20.1 -

20.9 -

22.3 -

19.1 -

20.1 -

19.5 -

19.2 -

20.7

Location of interface at perpendicular plane section to the coastal line which cross pumping wells DK3, DK6 of confined aquifer (Table 11, Fig. 12).

Q6=300m3/ngy

-25-20-15-10

-50

0 200 4 00 60 0 800 1 000 1 20 0

-25-20-15-10

-50

0 200 400 60 0 800 1000 1 200

Q6=300m3/day

Q3=300m3/day

t=0 1000 8000 10000 day

t=0 1000 8000 10000 day



Fig. 11: Location of time dependent sharp interface at HK6, HK3 when phreatic aquifer has supply functions

Cross section at pumping well HK6

Cross section at pumping well HK3



Table 11: Location of time dependent sharp interface at DK3, DK6 of confined aquifer

Node 10 27 44 61 78 95 112 129 146

X 70 100 200 300 400 500 600 700 800 h initial -22.3 -34.4 -40.6 -44.9 -44.8 -44.6 -45.8 -44.9 -45.9 h(t=1000 d) -22.3 -28.9 -37.7 -41.5 -43.1 -44.6 -45.8 -44.9 -45.9 h(t=8000 d) -22.3 -25.1 -32.8 -37.4 -41.2 -42.9 -44.6 -44.7 -45.9 h(t=10000d) -22.3 -23.6 -30.1 -34.2 -37.7 -40.0 -42.5 -44.5 -45.8 Node 163 180 197 214 231 248 265 282 299 X 900 1000 1100 1200 1300 1400 1500 1600 1700 h initial -44.2 -44.6 -45.8 -45.8 -44.4 -44.9 -47.3 -46.2 -44.6 h(t=1000) -44.2 -44.6 -45.8 -45.8 -44.4 -44.9 -47.3 -46.2 -44.6 h(t=8000) -44.2 -44.6 -45.8 -45.8 -44.4 -44.9 -47.3 -46.2 -44.6 ht=10000) -44.2 -44.6 -45.8 -45.8 -44.4 -44.9 -47.3 -46.2 -44.6 Node 11 28 45 62 79 96 113 130 147 X 50 100 200 300 400 500 600 700 800 h initial -19.6 -34.4 -40.4 -44.9 -46.7 -47.2 -47.0 -45.0 -44.0 h(t=1000) -19.6 -28.8 -36.3 -40.0 -43.4 -47.2 -44.6 -45.0 -44.0 h(t=8000) -19.6 -24.7 -32.8 -37.4 -40.6 -44.3 -44.0 -45.0 -44.0 ht=10000) -19.6 -24.1 -30.0 -34.4 -37.3 -40.0 -42.5 -45.0 -44.0 Node 164 181 198 215 232 249 266 283 300 X 900 1000 1100 1200 1300 1400 1500 1600 1700 h initial -46.7 -44.6 -43.4 -45.3 -46.7 -44.9 -43.3 -46.2 -44.6 h(t=1000) -46.7 -44.6 -43.4 -45.3 -46.7 -44.9 -43.3 -46.2 -44.6 h(t=8000) -46.7 -44.6 -43.4 -45.3 -46.7 -44.9 -43.3 -46.2 -44.6 H(t=10000) -46.7 -44.6 -43.4 -45.3 -46.7 -44.9 -43.3 -46.2 -44.6 ht=10000) -46.7 -44.6 -43.4 -45.3 -46.7 -44.9 -43.3 -46.2 -44.6



On the horizontal plane, the toe of interface corresponds the time period of phreatic aquifer when it has supply function (Fig. 13a) and has not supply function (Fig. 13b).

Fig.13a: The moving of toe when the phreatic aquifer has supply functions

Fig. 12: Location of time dependent sharp interface at DK3, DK6 of confined aquifer



Fig. 13b: The movement of toe when the phreatic aquifer havent supply functions

On the horizontal plane, the toe of interface corresponds the time period of confined aquifer (Fig. 14).

Fig.14: The movement of toe of confined aquifer



Based on the results above, we can see that, during the pumping time period 8000 days (@ 22 years), the distance of the nearest toe the pumping well systems (HK1, HK2, HK3, HK4) of phreatic aquifer is 96m (if this aquifer has not supply functions) and is 275m (if this aquifer has supply function), and the confined aquifer (DK1, DK2, DK3, DK4) is 130m.

These results are correct with physical phenomena, because when the phreatic aquifer has a supply function, the goundwater level will arise and the pressure of this aquifer will arise, too; so the toe of the interface will be pushed toward the sea.

VI- Conclusion

This mathematical model of two-dimensional horizontal flow of seawater intrusion is

generally valid for phreatic aquifer and confined aquifer. Algorithm and programing can be applied for any coastal aquifer with many pumping wells for predicting the sharp interface. It is important for the geologists, hydrologists, hydraulic engineers, water resources planners, managers, and governmental policy makers, who are engaged in the sustainable development of coastal fresh groundwater resources. References 1. Phan Ngoc Cu, Ton Si Kinh, Groundwater Mechanics, ed. Dai Hoc & Trung Hoc Chuyen Nghiep, Ha

Noi 1981. 2. NGUYEN The Hung, Finite element method in flow problems, Monograph NXB Xay Dung, Ha Noi

2004. 3. NGUYEN The Hung, Mathematical model of sediment transport two dimensional horizontal flow,

Proceedings of International Conference on Engineering Mechanics Today, Vol1, p.541-548, Hanoi 1995.

4. Jacob Bear and Arnold Verruijt , Modeling Groundwater Flow and Pollution, D. Reidel Publishing Company 1979.

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