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This article was downloaded by: [Université de Neuchâtel]On: 26 August 2014, At: 01:49Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House37-41 Mortimer Street, London W1T 3JH, UK
Hydrological Sciences JournalPublication details, including instructions for authors and subscription information:
http://www.tandfonline.com/loi/thsj20
An analytical approach to capture zone delineation fo
a well near a stream with a leaky layerMahdi Asadi-Aghbolaghi
a, Gholam Reza Rakhshandehroo
b & Mazda Kompani-Zare
cd
a Faculty of Agriculture, Water Engineering Department, Shahrekord University, Shahrekor
Iranb Civil Engineering Department, School of Engineering, Shiraz University, Shiraz, Iran
c Department of Desert Region Management, School of Agriculture, Shiraz University, Shira
Irand Department of Geography and Environmental Management, University of Waterloo,
Ontario, N2L 3G1, Canada
Published online: 31 Oct 2013.
To cite this article: Mahdi Asadi-Aghbolaghi, Gholam Reza Rakhshandehroo & Mazda Kompani-Zare (2013) An analytical
approach to capture zone delineation for a well near a stream with a leaky layer, Hydrological Sciences Journal, 58:8,
1813-1823, DOI: 10.1080/02626667.2013.840725
To link to this article: http://dx.doi.org/10.1080/02626667.2013.840725
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1813Hydrological Sciences Journal – Journal des Sciences Hydrologiques, 58 (8) 2013
http://dx.doi.org/10.1080/02626667.2013.840725
An analytical approach to capture zone delineation for a well near a
stream with a leaky layer
Mahdi Asadi-Aghbolaghi1 , Gholam Reza Rakhshandehroo2 and Mazda Kompani-Zare3,4
1 Faculty of Agriculture, Water Engineering Department, Shahrekord University, Shahrekord, Iran
2Civil Engineering Department, School of Engineering, Shiraz University, Shiraz, Iran
3 Department of Desert Region Management, School of Agriculture, Shiraz University, Shiraz, Iran
4 Department of Geography and Environmental Management, University of Waterloo, Ontario N2L 3G1, Canada
Received 17 October 2010; accepted 28 January 2013; open for discussion until 1 April 2014
Editor D. Koutsoyiannis;
Associate editor A. Koussis
Citation Asadi-Aghbolaghi, M., Rakhshandehroo, G.R., and Kompani-Zare, M., 2013. An analytical approach to capture zone
delineation for a well near a stream with a leaky layer. Hydrological Sciences Journal , 58 (8), 1813–1823.
Abstract An analytical solution is developed to delineate the capture zone of a pumping well in an aquifer with aregional flow perpendicular to a stream, assuming a leaky layer between the stream and the aquifer. Three differentscenarios are considered for different pumping rates. At low pumping rates, the capture zone boundary will becompletely contained in the aquifer. At medium pumping rates, the tip of the capture zone boundary will intrudeinto the leaky layer. Under these two scenarios, all the pumped water is supplied from the regional groundwater flow in the aquifer. At high pumping rates, however, the capture zone boundary intersects the stream and pumped water is supplied from both the aquifer and the stream. The two critical pumping rates which separate these threescenarios, as well as the proportion of pumped water from the stream and the aquifer, are determined for differenthydraulic settings.
Key words groundwater regional flow; stream boundary; capture zone delineation; complex potential theory; leaky layer
Une approche analytique pour délimiter la zone de captage d’un puits près d’une rivière surcouche perméableRésumé Nous avons développé une solution analytique pour délimiter la zone de captage d’un puits de pompagedans un aquifère avec un écoulement régional perpendiculaire à une rivière, en supposant l’existence une couche
perméable entre la rivière et la nappe phréatique. Trois scénarios différents ont été envisagés pour différents débitsde pompage. Pour de faibles débits de pompage, la limite de la zone de captage sera entièrement contenue dansl’aquifère. Pour des débits de pompage moyens, la pointe de la limite de la zone de captage atteint la couche
perméable. Dans ces deux scénarios, toute l’eau pompée est fournie par l’écoulement des eaux souterraines dansl’aquifère régional. Pour des débits de pompage élevés, cependant, la limite de la zone de captage recoupe larivière et l’eau pompée est fournie à la fois par l’aquifère et par la rivière Les deux débits de pompage critiquesséparant ces trois scénarios et la proportion de l’eau pompée dans la rivière et l’aquifère ont été déterminés pour différents paramètres hydrauliques.
Mots clefs écoulement régional des eaux souterraines; limite de cours d’eau; délimitation de la zone de captage; zone aride;
théorie du potentiel complexe; couche perméable
INTRODUCTION
Capture zone delineation for pumping wells in
aquifers has been studied by many researchers (Grubb
1993, Fienen et al. 2005, Kompani-Zare et al. 2005,
Intaraprasong and Zhan 2007). This topic may be
considered as a critical subject in water resources
management from different perspectives. From an
environmental perspective, for example, the need
for groundwater pollution prevention in areas with
sources of heavy contamination requires a relatively
accurate mathematical model of the capture zone
© 2013 IAHS Press
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1814 Mahdi Asadi-Aghbolaghi et al.
geometry in the affected aquifer. In the case where
the capture zone of a pumping well interferes with
surface water bodies in and around the aquifer, the
source of the pumped water would be a vital subject
from a resource management perspective.
Analytical delineation of capture zones dates
back to 1946, when Muskat performed a thoroughand detailed analysis of the problem using poten-
tial theory in a complex domain. Then, Dacosta and
Bennett (1960) delineated the capture zone analyti-
cally for two recharge and discharge wells with dif-
ferent regional flow angles with respect to the wells.
Since then, many research studies have developed dif-
ferent mathematical schemes to investigate capture
zone properties (e.g. Javandel and Tsang 1986, Shafer
1987, 1996, Grubb 1993, Faybishenko et al. 1995,
Shan 1999, Christ and Goltz 2002, 2004). In partic-
ular, Javandel and Tsang (1986), Faybishenko et al.
(1995), Shan (1999), and Christ and Goltz (2002,
2004) delineated the capture zone for multiple ver-
tical pumping wells placed at different angles to the
groundwater regional flow.
Different types of hydraulic boundaries influence
groundwater flow in the aquifer, and more specifi-
cally, the pumping well capture zone. Theis (1941)
was among early researchers who considered a fully
penetrating vertical well in a confined aquifer per-
fectly connected to a stream on one side. He uti-
lized the concept of image well theory to incorpo-
rate groundwater–stream interaction in his analyticalderivations. Later, Glover and Balmer (1954) gen-
eralized Theis’s approach and obtained solutions to
the problem based on a series of idealistic assump-
tions. Their method of solution was improved by
Jenkins (1968), who gave dimensionless tables and
curves leading to a scheme for management of the
water resources. Since then, the method of images
has been used by researchers to satisfy various types
of boundary conditions, such as no-flow and constant
head boundaries (Zhan 1999, Zhan and Cao 2000,
Kompani-Zare et al. 2005).
Groundwater flow domains often contain bound-aries or inhomogeneities that act as leaky barri-
ers to flow. Typical examples include problems of
groundwater–surface water interaction, in which an
aquifer is separated from a river by a layer of silt
(Anderson 2003).
In most real cases, a thin layer with low
hydraulic conductivity along the streambed sepa-
rates the aquifer from the stream (Anderson 2000,
2003). Mathematically, finding an analytical solu-
tion for this case is more complicated than the
case with perfect hydraulic connection (Anderson
2000). Hantush (1965) developed an analytical solu-
tion for drawdown of a vertical pumping well near
a stream with a semi-pervious layer between the
stream and aquifer. His approach contained some
limiting assumptions in the definition of the leaky
layer. Anderson (2000) obtained an analytical solu-tion to the problem in a complex domain by dropping
Hantush’s assumption. Recent researchers who have
investigated the interaction between groundwater and
streams through assumption of a leaky layer between
them include Hantush (2005), Intaraprasong and
Zhan (2007), Ha et al. (2007), Rushton (2007), and
Intaraprasong and Zhan (2009).
Some researchers have studied groundwater flow
of a pumping well near a partially penetrating stream
(Hunt 1999, Zlotnik and Huang 1999, Butler et al.
2001, Bakker and Anderson 2003). They ignored the
vertical component of velocity and applied the Dupuit
approximation. Hunt (1999) developed an analytical
solution to transient groundwater flow of a pump-
ing well near a partially penetrating stream with a
clogged streambed. He considered both sides of the
stream in his solution. Bakker and Anderson (2003)
presented an explicit analytical solution for steady,
two-dimensional (2D) groundwater flow to a well near
a leaky streambed that penetrates the aquifer partially.
They assumed that leakage from the stream is approx-
imated as occurring along the centreline of the stream.
In their setting, the problem domain is infinite and pumping on one side of the stream may induce flow
to the other side.
Analytical solutions have been presented for dif-
ferent engineering applications, with governing equa-
tions similar to groundwater flow equations (Cole and
Yen 2001, Lin 2010). Lin (2010) presented analyti-
cal solutions of heat conduction for isotropic media
with finite dimensions. He utilized a Fourier trans-
form together with the image method to find solutions
to a composite-layer medium. He assumed two dif-
ferent boundary types: thermal isolation (Neumann)
and isothermal (Dirichlet), and expressed explicit fullfield solutions as simple closed-forms, which may be
easily used in other engineering applications too.
In this study, a steady-state analytical solution
is developed to delineate capture zone for a fully
penetrating well in an aquifer with regional flow per-
pendicular to a stream, on the assumption that there
is a leaky layer between the stream and the aquifer.
Complex potential theory and superposition law are
used to obtain the analytical solutions to the problem.
Three different scenarios are considered for different
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An analytical approach to capture zone delineation for a well near a stream with a leaky layer 1815
pumping rates. At low pumping rates, the capture
zone boundary will be completely contained in the
aquifer. At medium pumping rates, however, the tip of
the capture zone boundary will intrude into the leaky
layer. Under these first two scenarios, all the pumped
water is supplied from the regional groundwater flow
in the aquifer. However, at high pumping rates, thecapture zone boundary intersects the stream and the
pumped water is supplied from both the aquifer and
the stream. The two critical pumping rates separat-
ing these three scenarios—the proportion of pumped
water from the stream and the aquifer, and the inter-
val during which the water is gaining from the stream,
in the third scenario—are determined for different
hydraulic settings.
MODEL DESCRIPTION
Figure 1 illustrates a schematic plan view of a verti-
cal pumping well in an aquifer that is separated from
a stream boundary by a leaky layer. The origin of
the coordinate system is located at the interface of
the leaky layer and the aquifer. The stream and the
leaky layer fully penetrate the aquifer and are paral-
lel to the x-axis. The well is also fully penetrating
and is located at (0,– a). The thickness of the leaky
layer, h, is constant and groundwater regional flow, q,
is perpendicular to the stream axis.
Complex potential theory is adopted to investi-
gate 2D steady-state groundwater flow in the aquifer and the leaky layer. The complex variable, z , is defined
as z = x+ iy, where i =√ −1, for the coordinate sys-
tem shown in Fig. 1. The entire domain is subdivided
into two domains, D and D∗, corresponding to two
Fig. 1. Schematic view of a fully penetrating pumping wellin an aquifer that is separated from a stream by a leakylayer.
media, the aquifer and the leaky layer, with two differ-
ent hydraulic conductivities, K and K ∗. Two complex
potentials for the domains are introduced as:
Ω = Φ + iΨ in D (1a)
Ω∗ = Φ∗ + iΨ ∗ in D∗ (1b)
where Ω refers to the complex potential, Φ to the
potential function, and Ψ to the stream function, and
superscript ∗ denotes the same parameters for the
leaky layer. In a confined aquifer the potential func-
tion is associated with hydraulic head, φ, via Φ = KBφ + C C , in which B is the aquifer thickness and
C C is an arbitrary constant. For an unconfined aquifer,
Φ = 1/2 K φ2 + C u, where C u is another arbitrary
constant (Strack 1989).Along the stream boundary, hydraulic head is
constant and, therefore, the potential function Φ∗,
should be constant there. Along the leaky layer and
aquifer interface, y = 0, named the inhomogeneity
boundary, the hydraulic head and normal compo-
nent of flow should be constant, and this can be
expressed in terms of potential and stream functions
as (Anderson 2000):
Φ = K
K ∗Φ∗ and Ψ = Ψ ∗ at y = 0 (2)
SOLUTION TO THE PROBLEM
An analytical solution is developed based on super-
position of complex potentials for two components of
the system to delineate the capture zone of a pumping
well near a stream with a leaky layer. The first com-
ponent is the well without any regional groundwater
flow in the aquifer and the second is the regional flow.
Both components have a stream with a leaky layer at
their boundaries.
The first component was presented by Anderson(2000), who utilized three basic solutions to deter-
mine the appropriate forms of the images across each
boundary. The first basic solution contains a drain
near a horizontal equipotential, and the basic solu-
tions (2) and (3) have been developed from a single
classical solution consisting of a drain in an aquifer
with two hydraulic conductivities (Polubarinova-
Kochina 1962). Anderson (2000) established a pattern
in the imaging process, and the final solution may be
expressed as follows:
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1816 Mahdi Asadi-Aghbolaghi et al.
Ω∗ = (1− κ)Q
2π B
∞n=0
κn
ln z + ai+ 2ihn
z −
ai−
2ih (n+
1)+Φ∗0
(3a)
Ω = Q
2π Bln ( z + ai)+ κ
Q
2π Bln ( z − ai)
−
1− κ2 Q
2π B
∞n=0
κn ln
[ z − ai− 2ih (n+ 1)]+Φ0
(3b)
Φ∗0 = K ∗ Bφ0 for confined aquifer (4a)
Φ0 = KBφ0 for confined aquifer (4b)
Φ∗0 = 1
2 K ∗φ02 for unconfined aquifer (4c)
Φ0 = 1
2 K φ02 for unconfined aquifer (4d)
where φ0 is the constant hydraulic head evaluated
from the boundary condition, κ = ( K – K ∗)/( K + K ∗) = (1 – β)/(1 + β) and Q is the discharge of
the pumping well. Anderson (1999) demonstrates that
expressions (3a) and (3b) satisfy the boundary condi-tions exactly and the infinite series appearing in (3a)
and (3b) converge. If hydraulic head at the stream
boundary is set equal to zero, then in equations (3a)
and (3b) Φ∗ = Φ0∗ = 0.
For the second component, the regional
groundwater flow, q, passes through both the
aquifer and the leaky layer perpendicular to the
stream boundary. Applying equation (2) across the
inhomogeneity boundary, the complex potentials are
obtained as:
Ω∗ = qiz +Φ0∗ (5a)
Ω = qiz +Φ0 (5b)
Again, if hydraulic head at the stream boundary
( y = h) is set equal to zero, then in equations (5a)
and (5b) one obtains Φ0∗ = qh, and Φ0
∗ = K K ∗qh,
respectively.
Combining equations (3) and (5) yields the final
solution to the problem as:
Ω∗ = (1− κ)Q
2π B
∞n=0
κn
ln z + ai+ 2ihn
z −
ai−
2ih (n+
1)+ qiz +Φ∗0
(6a)
Ω = Q
2π Bln ( z + ai)+ κ
Q
2π Bln ( z − ai)
−
1− κ2 Q
2π B
∞n=0
κn
ln [ z − ai− 2ih (n+ 1)]+ qiz +Φ0
(6b)
where Φ∗0 and Φ0 are evaluated from boundary con-ditions. Following arguments presented for equations
(3) and (5), if hydraulic head at the stream bound-
ary is zero, then constants in equations (6a) and (6b)
would take the forms Φ0∗ = qh and Φ0
∗ = K K ∗ qh,
respectively.
These equations may be expressed in dimension-
less form as:
Ω D∗ = (1− κ)Q D
∞n=
0
κn ln
z D + i+ 2ih Dn
z D − i− 2ih D (n+ 1)+ iz D +Φ∗ D0
(7a)
Ω D =Q D ln ( z D + i)+ κQ D ln ( z D − i)
−
1− κ2
Q D
∞n=0
κn
ln [ z D−
i−
2ih D (n+
1)]+
iz D+Φ D0
(7b)
where the subscript D denotes the dimensionless
terms and dimensionless parameters are defined as
Ω D =Ω/aq, Q D = Q/2π Baq, x D = x/a, y D = y/a,
z D = z /a, h D = h/a. The branch of functions Ω( z )
and Ω∗( z ) (shown in Fig. 3(a) – (c)) is fixed in the z -
plane with the cut along the imaginary axis from the
point−i to the point ih D. The real and imaginary parts
of Ω D are Φ D and Ψ D, respectively. These parts for
domains D∗ and D would be:
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An analytical approach to capture zone delineation for a well near a stream with a leaky layer 1817
Φ D∗ = (1− k )Q D
∞n=0
κn
ln x D
2 + ( y D + 1+ 2h Dn)2
x D
2
+ [ y D − 1− 2h D (n+ 1)]
2
− y D +Φ∗ D0
(8a)
Ψ D∗ = (1−κ)Q D
∞n=0
κn
tan−1 y D + 1+ 2h Dn
x D
−tan−1 y D − 1− 2h D (n+ 1)
x D
+ x D
(8b)
Φ D
=Q D ln x D
2
+( y D
+1)2
+ κQ D ln x D
2 + ( y D − 1)2
−
1− κ2
Q D
∞n=0
κn ln x D
2 + [ y D − 1
−2h D (n+ 1)]2− y D +Φ D
(8c)
Ψ D =Q D tan−1 y D + 1
x D
+ κQ D tan−1 y D − 1
x D
−
1− κ2
Q D
∞n=0
κntan−1
y D − 1− 2h D (n+ 1)
x D
+ x D
(8d)
Stagnation point
The stagnation point is a key point in delineating the
capture zone of a well. The stagnation point exists
on the streamline forming the capture zone bound-
ary in a flow domain (Strack 1989, Kompani-Zareet al. 2005). In the vicinity of this point, the stream-
lines turn from parallel to the capture zone boundary
to perpendicular to it. In the present case of a uni-
form flow and a single well, only one stagnation point
exists on the capture zone boundary at low pump-
ing rates. At the stagnation point, the flow velocity
and hydraulic gradient are zero, i.e. ∂Ψ D/∂ x D =∂Ψ D/∂ y D = 0. Differentiating the stream functions
given by equations (8b) and (8d), with respect to x D
and y D yields:
∂Ψ D∗
∂ x D
= (1− κ)Q D
∞n=0
κn
− y D + 1+ 2h Dn
x D
2
+( y
D +1+
2h D
n)2
+ y D − 1− 2h D (n+ 1)
x D2 + [ y D − 1− 2h D (n+ 1)]2
+ 1
(9a)
∂Ψ D∗
∂ y D
= (1− κ)Q D
∞n=0
κn
x D
x D2 + ( y D + 1+ 2h Dn)2
− x D
x D2 + [ y D − 1− 2h D (n+ 1)]2
(9b)
∂Ψ D
∂ x D
= −Q D
y D + 1
x D2 + ( y D + 1)2
− κQ D
y D − 1
x D2 + ( y D − 1)2
+
1− κ2
Q D
∞
n=0
κn y D − 1− 2h D (n+ 1)
x D2 + [ y D − 1− 2h D (n+ 1)]
2
+1
(9c)
∂Ψ D
∂ y D
= Q D
x D
x D2 + ( y D + 1)2
+ κQ D
x D
x D2 + ( y D − 1)2
−
1− κ2
Q D
∞n=0
κn x D
x D2 + [ y D − 1− 2h D (n+ 1)]2
(9d)
Based on the location of the stagnation point(s),
equations (9a) to (9d) may be set equal to zero to
find the stagnation point(s) coordinates, x Ds and y Ds.
However, the equations contain series which make
explicit determination of their solution impossible.
Hence, a numerical method is adopted to find the
stagnation point coordinates. It is worth noting that
the stagnation point may be located in the aquifer, in
the leaky layer, or on the stream boundary owing to
different pumping rate values.
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1818 Mahdi Asadi-Aghbolaghi et al.
CAPTURE ZONE DELINEATION
The capture zone boundary can be determined based
on the stagnation point location. For the case in which
a regional flow perpendicular to the stream is consid-
ered the capture zone boundary is symmetric relative
to y-axis. Figure 2 depicts the capture zone bound-ary for the hydraulic configuration shown on Fig. 1,
where a low pumping rate of Q D = 0.4 is applied.
In Fig. 2, h D = 0.4, β = K ∗/ K =0.5, Φ0 = Φ0∗ =
0 and the stagnation point is located at x Ds = 0 and
y Ds = –0.547.
Note that basic configurations are considered
in this research where the well pumping rate, Q D,
together with the thickness of leaky layer, h D, and the
ratio of hydraulic conductivity in the aquifer to that in
the leaky layer, β , vary. However, parameters such as
the aquifer hydraulic conductivity, regional flow rate
and distance between the well and stream are held constant.
Three scenarios for capture zone configuration
For different Q D values, the capture zone configura-
tions create three scenarios. At low pumping rates,
water is withdrawn solely from the aquifer and the
capture zone boundary is completely contained in
the aquifer (Fig. 3(a)). For this scenario, the stream
gains water from the aquifer, or the regional flow,
along its entire length, the tip of the capture zone,
or the stagnation point, is located inside the aquifer,
and equations (9c) and (9d) may be set equal to zero
to find its coordinates. From equation (9d) one gets
–3
–2.5
–2
–1.5
–1
–0.5
0
0.5
1
–1.5 –1 –0.5 0 0.5 1 1.5
x D
y D
Pumping Well
Capture Zone Boundary
Branch Cut
Inhomogeneity Boundary
Stream Boundary
Leaky Layer
Fig. 2. Capture zone boundary for Q D = 0.4, h D = 0.4 and β = 0.5.
Fig. 3. Streamlines (solid), iso-potential lines (dashed),capture zone boundary (solid bold) and branch cut (dashed
bold) for h D = 0.4, β = 0.5, Φ∗ D0 = Φ D0 = 0, and for
different dimensionless pumping rates: (a) Q D = 0.4,(b) Q D = 0.8, and (c) Q D = 1.
x Ds = 0, meaning that the stagnation point is located
on the symmetry axis. Substituting x D = 0 in equa-
tion (9c), the ordinate of the stagnation point would
be obtained. This scenario occurs when Q D is between
zero and a critical value, Q DC 1, where Q DC 1 may be
viewed as the pumping rate at which the stagnation
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An analytical approach to capture zone delineation for a well near a stream with a leaky layer 1819
point would be located on the inhomogeneity
boundary.
As Q D increases, however, the capture zone
boundary crosses the leaky layer, the stagnation point
would be located inside the layer (Fig. 3(b)), and
equations (9a) and (9b) may be used to obtain its
coordinates. Setting equation (9b) equal to zero, x D isobtained to be zero, meaning that the stagnation point
is again located on the symmetry axis. Substituting
x D = 0 in equation (9a) and setting it equal to zero,
the ordinate of the stagnation point can be obtained.
In this scenario, the stream water does not enter the
well and pumped water is still supplied solely by
the aquifer or the regional flow. The second scenario
ends when Q D increases and reaches a value Q DC 2, at
which the capture zone touches the stream at only one
point. This touched point with (0,h D) coordinates is
the stagnation of the capture zone.
The third scenario is for the case in which Q D
is greater than Q DC 2. In this case, the capture zone
boundary crosses the stream at two distinct points and
the pumped water from the well is supplied by both
the aquifer and the stream (Fig. 3(c)). In this scenario,
there would be two stagnation points at the stream
boundary, and equations (9a) and (9b) may be used
to obtain their coordinates. By setting equation (9b)
equal to zero, one may obtain y D = h D, and by sub-
stituting y D = h D in equation (9a) and setting it equal
to zero, two symmetrical abscissas are obtained for
x Ds of the stagnation points. For the third scenario, itis important to determine the proportions of pump-
ing water gain from the stream. For good accuracy,
Fig. 3(a) – (c) has been drawn using n = 1000 in the
series.
It is worth noting that, from equations (8a)–(8d)
and Fig. 3(a) – (c), one may conclude that far away
from the well, iso-potential lines would be parallel
to the stream boundary, groundwater flow direction
would be perpendicular to it, and the specific dis-
charge would be q.
First critical pumping rate, Q DC 1 As men-tioned before, at Q D = Q DC 1 the stagnation point is
located on the inhomogeneity boundary. Therefore, to
find Q DC 1 as a function of h D and κ, the stagnation
point should be placed at the origin of coordinates.
By substituting x D = 0 in equation (9d), ∂Ψ D∂ y D
would equate zero for all values of y D , h D and β.
By setting x D = y D = 0 in equation (9c) and mak-
ing it equal to zero, the following equation will be
determined:
∂Ψ D
∂ x D
=− Q DC 1 + κQ DC 1 −
1− κ2
Q DC 1
∞n=0
κn 1
1+ 2h D (n+ 1)+ 1 = 0
(10)
Therefore,
Q DC 1 =1
1− κ +
1− κ2 ∞
n=0
κn
1+2h D(n+1)
(11)
It must be noted that Q DC 1 depends not only on the
aquifer hydraulic conductivity, but also on the thick-
ness and conductivity of the leaky layer, via h D and κ,
respectively. In general, larger h or smaller K ∗ would
result in greater Q DC 1, which make sense physically.In other words, higher pumping rates are required to
place the stagnation point on the boundary when a less
conductive or thicker leaky layer exists.
Figure 4 shows Q DC 1 as a function of β for
different values of h D. As depicted in Fig. 4, Q DC 1
increases significantly as β decreases and the changes
are greater for smaller β values. For a given β value,
Q DC 1 increases for larger h D values. For the very
small values of β , the leaky layer acts as a barrier for
groundwater to reach the stream as a constant head
boundary; a phenomenon that requires higher Q DC 1
values.For the special case in which K ∗→ K, β →
1 then κ → 0, meaning that the leaky layer no longer
exists or, in other words, is considered as a part of the
aquifer. Then, Q DC 1 would only depend on h D:
Q DC 1 =1+ 2h D
2+ 2h D
(12)
Second critical pumping rate, Q DC 2 As men-
tioned previously, at Q D < Q DC 2 the capture zone
boundary does not cross the stream, and at Q D >Q DC 2 the capture zone boundary crosses the stream
boundary. At Q D = Q DC 2, the stagnation point lies
on the stream boundary and, hence, x Ds = 0 and y Ds
= h D. Therefore, ∂Ψ ∗ D∂ x D, and ∂Ψ ∗ D
∂ y D should
be equal to zero at (0,h D) for Q D = Q DC 2. However,
∂Ψ ∗ D∂ y D = 0 at x D = 0 for all values of β and h D
(equation (9b)). Therefore, Q DC 2 should be obtained
by equation (9a) for x D = 0 and y D = h D. Substituting
coordinate (0,h D) in equation (9a):
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1820 Mahdi Asadi-Aghbolaghi et al.
0
1
2
3
4
5
6
7
0.01 0.1 1 10
Q
D C 1
h D = 0.1
h D = 0.2
h D = 0.3
h D = 0.4
β
Fig. 4. Q DC 1 vs β for different values of h D.
− (1− κ)Q DC 2∞
n=0
2κn
1+ h D + 2h Dn+ 1 = 0 (13)
and
Q DC 2 =1
(1− κ)∞
n=0
2κn
1+h D+2h Dn
(14)
Figure 5 depicts Q DC 2 with respect to β for differ-
ent values of h D. As expected, Q DC 2 is always greater
than Q DC 1 for all values of β and h D. However, com- paring Figs 4 and 5 reveals that both Q DC 1 and Q DC 2
have similar increasing trends as β decreases and /or
h D increases.
For the special case in which K ∗→ K, β→ 1 and
κ → 0, then:
0.5
1.5
2.5
3.5
4.5
5.5
6.5
0.01 0.1 1 10
Q D C 2
h D = 0.1
h D = 0.2
h D = 0.3
h D
= 0.4
β
Fig. 5. Q DC 2 vs β for different values of h D.
Q DC 2 =1+ h D
2(15)
Setting h D → 0 in equation (15), one would get
Q DC 2 = 1/2. This pumping rate has been derived
by researchers such as Strack (1989), who did notconsider the leaky layer at all.
Water pumped from stream for Q D > Q DC 2
When the pumping rate is greater than Q DC 2, a certain
portion of pumped water would be supplied by the
stream, Q DR (to be determined). From water resources
management and contaminant transport view points,
it is vital to know what portion of the pumped water
is actually withdrawn from the stream (Muskat 1946).
To calculate Q DR, it is essential to locate the two
stagnation points on the stream boundary (Fig. 3(c)).
For this purpose, equation (9a) should be set equal tozero at y D = h D, the stream boundary:
∂Ψ D∗
∂ x D
= (1− κ)Q D
∞n=0
−2κn
1+ h D (2n+ 1)
x DS 2 + [1+ h D (2n+ 1)]2
+ 1 = 0
(16)
As seen in the equation, x Ds depends upon Q D, β
(via κ) and h D. Unfortunately, it is not possible to
find an explicit analytical solution for equation (16)and numerical methods are implemented to solve it.
As shown in Fig. 3(c), there are two stagnation points
on the stream boundary, symmetrically located on
both sides of the y-axis, with 2 x Ds distance between
them. Water enters the aquifer from the stream along
the interval between two stagnation points. The prob-
lem has a symmetric axis passing through the pump-
ing well and y-axis.
Monotonically increasing curves of x Ds vs Q D for
different values of h D and β are shown in Fig. 6(a) –
(d). In these figures, curves cross the horizontal axis,
x Ds = 0, where Q D = Q DC 2. In these curves, for Q D
values less than Q DC 2, x Ds also equals zero. As can
be seen in Fig. 6(a)–(d), for Q D > Q DC 2 by increasing
Q D , x Ds also increases. However, the rate of increase
in x Ds with Q D is different at different β and h D val-
ues. For smaller values of Q D, the slope of the curve is
increasing and the maximum slope occurs at x Ds = 0.
With increase in Q D, the curve slope decreases until
it approaches the linear condition at high Q D values.
The linear parts of the curves with smaller values
of β that have higher slopes occur sooner. Also, the
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An analytical approach to capture zone delineation for a well near a stream with a leaky layer 1821
0
1
2
3
4
5
6
7
8
0 2 4 6 8 10
Q D
x D s
β = 0.01β = 0.05β = 0.1β = 0.5β = 2β = 10
h D = 0.1
0
1
2
3
4
5
6
7
8
0 2 4 6 8 10
Q D
x D s
β = 0.05β = 0.1β = 0.5β = 2
β = 0.01
β = 10
h D = 0.2
0
1
2
3
4
5
6
7
8
0 2 4 6 8 10
Q D
x D s
β = 0.05β = 0.1β = 0.5β = 2
β = 0.01
β = 10
h D = 0.3(c)
(b)(a)
0
1
2
3
4
5
6
7
8
0 2 4 6 8 10
Q D
x D s
β = 0.05β = 0.1β = 0.5β = 2
β = 0.01
β = 10
h D = 0.4(d)
Fig. 6. x Ds vs Q D for different values of β , and for h D equal to: (a) 0.1, (b) 0.2, (c) 0.3 and (d) 0.4.
curves with smaller β values cross the horizontal axis
at higher Q D values. The higher linear part slope of
the curves means that a longer interval on the stream
boundary is required to supply the increase in Q D.
Also, the higher Q D for x Ds = 0 in the curves with
smaller β shows that, as the hydraulic conductivity of
the leaky layer declines, larger Q D is needed for the
capture zone to touch the stream boundary.
The half portion of water pumped from the
stream, x DR/2, may be calculated based on the differ-
ence in the stream function values for the streamline
passing through the stagnation point, Ψ Ds∗, and the
streamline passing through x D = 0 and y D = h D,
Ψ Dt ∗. To calculate the stream function for the stream-
line passing through (0,h D), the limits of Ψ D∗ are
calculated (equation (8b)) when x D → 0+:
lim x D→0+
Ψ D∗ = (1− κ)Q D
∞n=
0
κnπ = πQ D (17)
Therefore, f may be introduced as the proportion of
pumped water from the stream to the total pumping
water as:
f = πQ D − Ψ Ds∗
πQ D
(18)
Figure 7(a) – (d) shows f vs Q D for different values
of β and h D. As expected, for the given values of β
and h D , f increases with any increase in Q D; however,
the increase is not linear. The general trend (except
in low conductive and thick leaky layers) is that the
rate of increase in f is larger at low Q D and smaller
at high Q D values. For the given Q D and h D val-
ues, f decreases with any decrease in β. It reflects
the fact that the stream would make less contribu-
tion to the pumped water when the conductivity of the
leaky layer decreases. For any Q D
and β , f decreases
as h D increases. This may be interpreted as thicker
leaky layers lessening the stream contribution to the
pumped water, which makes sense hydraulically. For
low conductive (β = 0.01) and thick leaky layers (h D
≤ 0.3), the stream contribution to pumped water ( f )
becomes very small (<0.13%) even for large values
of Q D (Fig. 7(c) and (d)).
CONCLUSIONS
A steady-state analytical solution was developed to
delineate the capture zone of a fully penetrating pumping well in an aquifer with regional flow per-
pendicular to a stream boundary, assuming a leaky
layer between the stream and the aquifer. Complex
potential theory and superposition law were used to
obtain analytical solutions to the problem. Three dif-
ferent scenarios were considered for different pump-
ing rates. At low pumping rates, the capture zone
boundary was completely contained in the aquifer.
At medium pumping rates, however, the tip of the
capture zone boundary intruded into the leaky layer.
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1822 Mahdi Asadi-Aghbolaghi et al.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 2 4 6 8 10
Q D
f
β = 0.01
β = 0.05
β = 0.1
β = 0.5
h D = 0.1
β = 2
β = 10
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 2 4 6 8 10
Q D
f
β = 0.05
β = 0.1
β = 0.5
β = 2
β = 0.01
β = 10
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 2 4 6 8 10
Q D
f
β = 0.05
β = 0.1
β = 0.5
β = 2
β = 0.01
β = 10
h D = 0.3(c)
(b)(a)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 2 4 6 8 10
Q D
f
β = 0.05
β = 0.1
β = 2
β = 0.5
β = 10
β = 0.01(d) h D = 0.4
h D = 0.2
Fig. 7. Variation of f vs Q D for different values of β , and for h D equal to: (a) 0.1, (b) 0.2, (c) 0.3 and (d) 0.4.
Under these two scenarios all of the pumped water
was supplied from the regional groundwater flow
in the aquifer. Finally, at higher pumping rates, the
capture zone boundary intersected the stream and
pumped water was supplied from both the aquifer
and the stream. The two critical pumping rates which
separate these three scenarios were obtained analyt-
ically. The results show that both critical pumping
rates have similar trends and increase significantly asthe conductivity of the leaky layer decreases and /or
its thickness increases. Furthermore, in the third sce-
nario, the contribution of the stream to the pumped
water and the length of the segment through which
water enters the aquifer from the stream were deter-
mined and investigated for different hydraulic set-
tings. It was found that, for a given pumping rate,
the proportion of pumping water supplied from the
stream would decrease as the conductivity of the leaky
layer decreases or its thickness increases.
Acknowledgements The detailed and useful com-ments and suggestions of the reviewers, associate
editor and co-editor of the journal regarding this
manuscript are hereby acknowledged.
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