Download - hydro chapter_7_groundwater_by louy Al hami
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12/26/2012 1
Chapter 7
Civil Engineering Department
Prof. Majed Abu-Zreig
Hydraulics and Hydrology – CE 352
Groundwater Hydraulics
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Hydrologic cycle
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Occurrence of Ground Water
• Ground water occurs when water recharges a porous subsurface geological formation “called aquifers” through cracks and pores in soil and rock
• it is the water below the water table where all of the pore spaces are filled with water.
• The area above the water table where the pore spaces are only partially filled with water is called the capillary fringe or unsaturated zone.
• Shallow water level is called the water table
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Groundwater Basics -
Definitions
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Recharge
Natural
• Precipitation
• Melting snow
• Infiltration by streams and lakes
Artificial • Recharge wells
• Water spread over land in pits, furrows, ditches
• Small dams in stream channels to detain and deflect water
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Aquifers
Definition: A geological unit which can store and supply significant quantities of water.
Principal aquifers by rock type:
Unconsolidated
Sandstone
Sandstone and Carbonate
Semiconsolidated
Carbonate-rock
Volcanic
Other rocks
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Example Layered Aquifer System
Bedient et al., 1999.
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Other Aquifer Features
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Groundwater occurrence in confined and
unconfined aquifer
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Potentiometric Surfaces
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Eastern Aquifer
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Growndwater
basins
in Jordan
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Unconfined Aquifers
• GW occurring in aquifers: water fills partly an
aquifer: upper surface free to rise and decline:
UNCONFINED or water-table aquifer: unsaturated
or vadose zone
• Near surface material not saturated
• Water table: at zero gage pressure: separates saturated
and unsaturated zones: free surface rise of water in a
well
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Confined Aquifer
• Artesian condition
• Permeable material overlain by relatively
impermeable material
• Piezometric or potentiometric surface
• Water level in the piezometer is a measure of
water pressure in the aquifer
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Groundwater Basics -
Definitions • Aquifer Confining Layer or Aquitard
– A layer of relatively impermeable material which restricts vertical
water movement from an aquifer located above or below.
– Typically clay or unfractured bedrock.
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Aquifer Characteristics
• Porosity
– The ratio of pore/void volume
to total volume, i.e. space
available for occupation by air
or water.
– Measured by taking a known
volume of material and adding
water.
– Usually expressed in units of
percent.
– Typical values for gravel are
25% to 45%.
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Bedient et al., 1999.,
Typical Values of Porosity
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Aquifer Properties • Porosity: maximum amount of water that a rock
can contain when saturated.
• Permeability: Ease with which water will flow through a porous material
• Specific Yield: Portion of the GW: draining under influence of gravity:
• Specific Retention: Portion of the GW: retained as a film on rock surfaces and in very small openings:
• Storativity: Portion of the GW: draining when the piezometric head dropped a unit depth
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Figures from Hornberger et al. (1998)
Unconfined aquifer
Specific yield = Sy
Confined aquifer
Storativity = S
b
h h
Storage Terms
S = V / A h
S = Ss b
Ss = specific storage
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Aquifer Characteristics
• Hydraulic Conductivity – Measure of the ease with which water can flow through an
aquifer.
– Higher conductivity means more water flows through an
aquifer at the same hydraulic gradient.
– Measured by well draw down or lab test.
– Expressed in units of mm/day, ft/day or gpd/ft2.
– Typical values for sand/gravel are 2.5 cm/day to 33 m/day
m1 (1 to 100 ft/day).
– Typical values for clay are 0.3 mm/day (0.001 ft/day). That
is why is is an aquifer confining layer.
• Transmissivity (T = Kb) is the rate of flow through a
vertical strip of aquifer (thickness b) of unit width
under a unit hydraulic gradient
•
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Aquifer Characteristics
• Hydraulic gradient – Steepness of the slope of the water table.
– Groundwater flows from higher elevations to lower elevations
(i.e. downgradient).
– Measured by taking the difference in elevation between two
wells and dividing by the distance separating them.
– Expressed in units of ft/ft or ft/mi.
– Typical values for groundwater are .0001 to .01 m/m.
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Aquifer Characteristics
• Groundwater Velocity – How fast groundwater is moving.
– Calculated by conductivity multiplied by gradient divided by
porosity.
– Expressed in units of ft/day.
– Typical values for gravel or sand are 0.15 to 16 m/day (1 to 50
ft/day).
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• Water table: the
surface separating
the vadose zone
from the saturated
zone.
• Measured using
water level in well
The Water Table
Fig. 11.1
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• Precipitation
• Infiltration
• Ground-water recharge
• Ground-water flow
• Ground-water discharge to
– Springs
– Streams and
– Wells
Ground-Water Flow
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• Velocity is
proportional to
– Permeability
– Slope of the water
table
• Inversely
Proportional to
– porosity
Ground-Water Flow
Fast (e.g., cm per day)
Slow (e.g., mm per day)
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• Infiltration
– Recharges ground
water
– Raises water table
– Provides water to
springs, streams
and wells
• Reduction of
infiltration causes
water table to drop
Natural Water
Table Fluctuations
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• Reduction of infiltration causes water table to drop
– Wells go dry
– Springs go dry
– Discharge of rivers drops
• Artificial causes
– Pavement
– Drainage
Natural Water
Table Fluctuations
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• Pumping wells
– Accelerates flow
near well
– May reverse
ground-water flow
– Causes water table
drawdown
– Forms a cone of
depression
Effects of
Pumping Wells
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• Pumping wells
– Accelerate flow
– Reverse flow
– Cause water
table drawdown
– Form cones of
depression Low river
Gaining
Stream
Gaining
Stream
Pumping well
Low well
Low well
Cone of
Depression
Water Table
Drawdown
Dry Spring
Effects of
Pumping Wells
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Dry river
Dry well
Effects of
Pumping Wells
Dry well
Dry well
Losing
Stream • Continued water-
table drawdown
– May dry up
springs and wells
– May reverse flow
of rivers (and
may contaminate
aquifer)
– May dry up rivers
and wetlands
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Ground-Water/
Surface-Water
Interactions
• Gaining streams
– Humid regions
– Wet season
• Loosing streams
– Humid regions, smaller
streams, dry season
– Arid regions
• Dry stream bed
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Figure taken from Hornberger et al. (1998)
Darcy column
h/L = grad h
q = Q/A
Q is proportional
to grad h
x
hAKQ
x
hAQ
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Darcy’s Law Henry Darcy’s Experiment (Dijon, France 1856)
AQxQhQ ,1,
x
hAKQ
x
hAQ
Q
Q: Volumetric flow rate [L3/T]
Darcy investigated ground water flow under controlled conditions
h
h1 h2
h
x
h1
Slope = h/x
~ dh/dx h
x h2
x1 x2
K: The proportionality constant is added to form the following equation:
K units [L/T]
A
: Hydraulic Gradient xhh
A: Cross Sectional Area (Perp. to flow)
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Calculating Velocity with Darcy’s
Law • Q= Vw/t
– Q: volumetric flow rate in m3/sec
– Vw: Is the volume of water passing through area “a” during
– t: the period of measurement (or unit time).
• Q= Vw/t = H∙W∙D/t = a∙v
– a: the area available to flow
– D: the distance traveled during t
– v : Average linear velocity
• In a porous medium: a = A∙n
– A: cross sectional area (perpendicular to flow)
– n: porous For media of porosity
• Q = A∙n∙v
• v = Q/(n∙A)=q/n
Vw
v
x
h
n
Kv
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Darcy’s Law (cont.)
• Other useful forms of Darcy’s Law
dx
dhKq
Q
A =
Q
A.n = q
n = dx
dh
n
Kv
Volumetric Flux (a.k.a. Darcy Flux or
Specific discharge)
Ave. Linear
Velocity
Used for calculating
Q given A
Used for calculating
average velocity of
groundwater transport
(e.g., contaminant
transport Assumptions: Laminar, saturated flow
dx
dhAKQ Volumetric Flow Rate
Used for calculating
Volumes of groundwater
flowing during period of
time
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Figure from Hornberger et al. (1998)
Linear flow
paths assumed
in Darcy’s law
True flow paths
Average linear velocity
v = Q/An= q/n
n = effective porosity
Specific discharge
q = Q/A
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Steady Flow to Wells in Confined Aquifers
• Radial flow towered wells
• Aquifers are homogeneous (properties are uniform)
• Aquifers are isotropic (permeability is independent of flow direction)
• Drawdown is the vertical distance measured from the original to the lowered water table due to pumping
• Cone of depression the axismmetric drawdown curve forms a conic geometry
• Area of influence is the outer limit of the cone of depression
• Radius of Influence (ro) for a well is the maximum horizontal extent of the cone of depression when the well is in equilibrium with inflows
• Steady state is when the cone of depression does not change with time
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Horizontal and Vertical Head Gradients
Freeze and Cherry, 1979.
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Flow to Wells
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Steady Radial Flow to a Well-
Confined
Q
Cone of Depression
s = drawdown
h r
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Steady Radial Flow to a Well-
Confined
• In a confined aquifer, the drawdown
curve or cone of depression varies with
distance from a pumping well.
• For horizontal flow, Q at any radius r
equals, from Darcy’s law,
Q = -2πrbK dh/dr
for steady radial flow to
a well where Q,b,K are
const
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Steady Radial Flow to a Well-
Confined • Integrating after separation of variables, with
h = hw at r = rw at the well, yields Thiem Eqn
Q = 2πKb[(h-hw)/(ln(r/rw ))]
Note, h increases
indefinitely with
increasing r, yet
the maximum head
is h0.
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Steady Radial Flow to a Well-
Confined
• Near the well, transmissivity, T, may be
estimated by observing heads h1 and h2
at two adjacent observation wells
located at r1 and r2, respectively, from
the pumping well
T = Kb = Q ln(r2 / r1)
2π(h2 - h1)
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Steady Radial Flow to a Well-
Unconfined
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Steady Radial Flow to a Well-
Unconfined
• Using Dupuit’s assumptions and applying Darcy’s law
for radial flow in an unconfined, homogeneous,
isotropic, and horizontal aquifer yields:
Q = -2πKh dh/dr
integrating,
Q = πK[(h22 - h1
2)/ln(r2/ r1)
solving for K,
K = [Q/π(h22 - h1
2)]ln (r2/ r1)
where heads h1 and h2 are observed at adjacent
wells located distances r1 and r2 from the pumping
well respectively.
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Steady Flow to a Well in a Confined
Aquifer
2rw
Ground surface
Bedrock
Confined
aquifer
Q
h0
Pre-pumping
head
Confining Layer
b
r1
r2
h2
h1
hw
Observation
wells
Drawdown curve
Q
Pumping
well
Q = Aq = (2prb)Kdh
dr
rdh
dr=
Q
2pT
h2 = h1 +Q
2pTln(
r2r1
)
Theim Equation
In terms of head (we can write it in terms of drawdown also)
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Example - Theim Equation
• Q = 400 m3/hr
• b = 40 m.
• Two observation wells,
1. r1 = 25 m; h1 = 85.3 m
2. r2 = 75 m; h2 = 89.6 m
• Find: Transmissivity (T)
T =Q
2p h2 - h1( )ln
r2r1
æ
è ç
ö
ø ÷ =
400 m3/hr
2p 89.6 m - 85.3m( )ln
75 m
25 m
æ
è ç
ö
ø ÷ =16.3 m2 /hr
h2 = h1 +Q
2pTln(
r2r1
)2rw
Ground surface
Bedrock
Confine
d
aquifer
Q
h0
Confining Layer
b
r1
r2
h2 h1
hw
Q
Pumping
well
Steady Flow to a Well in a Confined Aquifer
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Steady Radial Flow in a Confined
Aquifer
• Head
• Drawdown
h r( ) = h0 +Q
2pTln
r
R
æ
è ç
ö
ø ÷
s r( ) =Q
2pTln
R
r
æ
è ç
ö
ø ÷
s(r) = h0 - h r( )
Steady Flow to a Well in a Confined Aquifer
Theim Equation
In terms of drawdown (we can write it in terms of head also)
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Example - Theim Equation
• 1-m diameter well
• Q = 113 m3/hr
• b = 30 m
• h0= 40 m
• Two observation wells, 1. r1 = 15 m; h1 = 38.2 m
2. r2 = 50 m; h2 = 39.5 m
• Find: Head and drawdown in the well
2rw
Ground surface
Bedrock
Confine
d
aquifer
Q
h0
Confining Layer
b
r1
r2
h2 h1
hw
Q
Pumping
well Drawdown
Adapted from Todd and Mays, Groundwater Hydrology
T =Q
2p s1 - s2( )ln
r2r1
æ
è ç
ö
ø ÷ =
113m3/hr
2p 1.8 m - 0.5 m( )ln
50 m
15 m
æ
è ç
ö
ø ÷ =16.66 m2 /hr
s r( ) =Q
2pTln
R
r
æ
è ç
ö
ø ÷
Steady Flow to a Well in a Confined Aquifer
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Example - Theim Equation
2rw
Ground surface
Bedrock
Confine
d
aquifer
Q
h0
Confining Layer
b
r1
r2
h2 h1
hw
Q
Drawdown
@ well
Adapted from Todd and Mays, Groundwater Hydrology
hw = h2 +Q
2pTln(
rwr2
) = 39.5 m +113m3 /hr
2p *16.66 m2 /hrln(
0.5 m
50 m) = 34.5 m
sw = h0 - hw = 40 m- 34.5 m = 5.5 m
h2 = h1 +Q
2pTln(
r2r1
)
Steady Flow to a Well in a Confined Aquifer
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Steady Flow to Wells in
Unconfined Aquifers
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Steady Flow to a Well in an Unconfined
Aquifer
Q = Aq = (2prh)Kdh
dr
= prKdh2
dr
2rw
Ground surface
Bedrock
Unconfined
aquifer
Q
h0
Pre-pumping
Water level
r1
r2
h2 h1
hw
Observation
wells
Water Table
Q
Pumping
well
rd h2( )
dr=
Q
pK
h02 - h2 =
Q
pKln
R
r
æ
è ç
ö
ø ÷
h2(r) = h02 +
Q
pKln
r
R
æ
è ç
ö
ø ÷
h2 = h1 +Q
2pTln(
r2r1
)
Confined aquifer Unconfined aquifer
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Steady Flow to a Well in an Unconfined
Aquifer
2rw
Ground surface
Bedrock
Unconfined
aquifer
Q
h0
Prepumping
Water level
r1
r2
h2 h1
hw
Observation
wells
Water Table
Q
Pumping
well
2 observation wells: h1 m @ r1 m h2 m @ r2 m
K =Q
p h22 - h1
2( )ln
r2r1
æ
è ç
ö
ø ÷
h2(r) = h02 +
Q
pKln
r
R
æ
è ç
ö
ø ÷
h22 = h1
2 +Q
pKln
r2r1
æ
è ç
ö
ø ÷
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• Given:
– Q = 300 m3/hr
– Unconfined aquifer
– 2 observation wells,
• r1 = 50 m, h = 40 m
• r2 = 100 m, h = 43 m
• Find: K
K =Q
p h22 - h1
2( )ln
r2r1
æ
è ç
ö
ø ÷ =
300 m3 /hr / 3600 s /hr
p (43m)2 - (40 m)2[ ]ln
100 m
50 m
æ
è ç
ö
ø ÷ = 7.3x10-5 m /sec
Example – Two Observation Wells in an
Unconfined Aquifer
2rw
Ground surface
Bedrock
Unconfined
aquifer
Q
h0
Prepumping
Water level
r1
r2
h2 h1
hw
Observation
wells
Water Table
Q
Pumping
well
Steady Flow to a Well in an Unconfined Aquifer
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Pump Test in Confined
Aquifers
Jacob Method
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Cooper-Jacob Method of Solution
Cooper and Jacob noted that for small values of r
and large values of t, the parameter u = r2S/4Tt
becomes very small so that the infinite series can be
approx. by: W(u) = – 0.5772 – ln(u) (neglect higher terms)
Thus s' = (Q/4πT) [– 0.5772 – ln(r2S/4Tt)]
Further rearrangement and conversion to decimal logs yields:
s' = (2.3Q/4πT) log[(2.25Tt)/ (r2S)]
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Cooper-Jacob Method of Solution
A plot of drawdown s' vs.
log of t forms a straight line
as seen in adjacent figure.
A projection of the line back
to s' = 0, where t = t0 yields
the following relation:
0 = (2.3Q/4πT) log[(2.25Tt0)/ (r2S)]
Semi-log plot
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Cooper-Jacob Method of Solution
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Cooper-Jacob Method of Solution
So, since log(1) = 0, rearrangement yields
S = 2.25Tt0 /r2
Replacing s' by s', where s' is the drawdown
difference per unit log cycle of t:
T = 2.3Q/4πs'
The Cooper-Jacob method first solves for T and
then for S and is only applicable for small
values of u < 0.01
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Cooper-Jacob Example
For the data given in the Fig.
t0 = 1.6 min and s’ = 0.65 m
Q = 0.2 m3/sec and r = 100 m
Thus:
T = 2.3Q/4πs’ = 5.63 x 10-2 m2/sec
T = 4864 m2/sec
Finally, S = 2.25Tt0 /r2
and S = 1.22 x 10-3
Indicating a confined aquifer
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Jacob Approximation
• Drawdown, s
• Well Function, W(u)
• Series
approximation of
W(u)
• Approximation of s
s u( ) =Q
4pTW u( )
W u( ) =e-h
hu
¥
ò dh » -0.5772 - ln(u)+ u -u2
2!+
u =r2S
4Tt
W u( ) » -0.5772 - ln(u) for small u < 0.01
s(r,t) »Q
4pT-0.5772 - ln
r2S
4Tt
æ
è ç
ö
ø ÷
é
ë ê ê
ù
û ú ú
s(r,t) =2.3Q
4pTlog10(
2.25Tt
r2S)
Pump Test Analysis – Jacob Method
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Jacob Approximation
s =2.3Q
4pTlog(
2.25Tt
r2S)
0 =2.3Q
4pTlog(
2.25Tt0
r2S)
t0
1=2.25Tt0
r2S
S =2.25Tt0
r2
Pump Test Analysis – Jacob Method
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Jacob Approximation
t0
S =2.25Tt0
r2
t1 t2
s1
s2
s
logt2
t1
æ
è ç
ö
ø ÷ = log
10* t1t1
æ
è ç
ö
ø ÷ =1
1 LOG CYCLE
1 LOG CYCLE
Pump Test Analysis – Jacob Method
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Jacob Approximation
S =2.25Tt0
r2=
2.25(76.26 m2/hr)(8 min*1 hr /60 min)
(1000 m)2
= 2.29x10-5
t0
t1 t2
s1
s2
s
t0 = 8 min
s2 = 5 m s1 = 2.6 m s = 2.4 m
Pump Test Analysis – Jacob Method
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Multiple-Well Systems
• For multiple wells with drawdowns that overlap, the principle of superposition may be used for governing flows:
• drawdowns at any point in the area of influence of several pumping wells is equal to the sum of drawdowns from each well in a confined aquifer
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Multiple-Well Systems
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Injection-Pumping Pair of Wells
Pump Inject
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Multiple-Well Systems
• The previously mentioned principles also
apply for well flow near a boundary
• Image wells placed on the other side of the
boundary at a distance xw can be used to
represent the equivalent hydraulic condition
– The use of image wells allows an aquifer of
finite extent to be transformed into an
infinite aquifer so that closed-form solution
methods can be applied
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Multiple-Well Systems
•A flow net for a pumping
well and a recharging
image well
-indicates a line of
constant head
between the two wells
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Three-Wells Pumping
A
Total Drawdown at A is sum of drawdowns from each well
Q1
Q3
Q2
r
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Multiple-Well Systems
The steady-state drawdown
s' at any point (x,y) is given
by:
s’ = (Q/4πT)ln
where (±xw,yw) are the
locations of the recharge and
discharge wells. For this
case, yw= 0.
(x + xw)2 + (y - yw)2
(x - xw)2 + (y - yw)2
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Multiple-Well Systems
The steady-state drawdown s' at any point (x,y) is given by
s’ = (Q/4πT)[ ln {(x + xw)2 + y2} – ln {(x – xw)2 + y2} ]
where the positive term is for the pumping well and the
negative term is for the injection well. In terms of head,
h = (Q/4πT)[ ln {(x – xw)2 + y2} – ln {(x + xw)2 + y2 }] + H
Where H is the background head value before pumping.
Note how the signs reverse since s’ = H – h
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7.5 Aquifer Boundaries
The same principle
applies for well
flow near a
boundary
– Example:
pumping near a
fixed head stream
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well near an impermeable boundary