Asadi-Aghbolaghi et al.pdf

7/18/2019 Asadi-Aghbolaghi et al.pdf

http://slidepdf.com/reader/full/asadi-aghbolaghi-et-alpdf 1/12

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

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) containedin the publications on our platform. However, Taylor & Francis, our agents, and our licensors make norepresentations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of tContent. Any opinions and views expressed in this publication are the opinions and views of the authors, andare not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon ashould be independently verified with primary sources of information. Taylor and Francis shall not be liable forany losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoeveor howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content.

This article may be used for research, teaching, and private study purposes. Any substantial or systematicreproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in anyform to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http://

www.tandfonline.com/page/terms-and-conditions

http://www.tandfonline.com/page/terms-and-conditions

http://www.tandfonline.com/action/showCitFormats?doi=10.1080/02626667.2013.840725



http://dx.doi.org/10.1080/02626667.2013.840725

http://www.tandfonline.com/action/showCitFormats?doi=10.1080/02626667.2013.840725

http://www.tandfonline.com/loi/thsj20



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

[email protected]

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

mailto:[email protected]

mailto:[email protected]



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



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:




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:




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.




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




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):




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




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.




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.

REFERENCES

Anderson, E.I., 1999. Groundwater flow with leaky boundaries.

Thesis (PhD). Department of Civil Engineering, University of

Minnesota, Minneapolis.

Anderson, E.I., 2000. The method of images for leaky boundaries.

Advances in Water Resources, 23 (5), 461–474.

Anderson, E.I., 2003. An approximation for leaky boundaries in

groundwater flow. Journal of Hydrology, 274 (1–4), 160–175.

Bakker, M. and Anderson, E.I., 2003. Steady flow to a well near a

stream with a leaky bed. Ground Water , 41 (6), 833–840.

Butler, J.J., Zlotnik, V.A., and Tsou, M., 2001. Drawdown and stream

depletion produced by pumping in the vicinity of a partially

penetrating stream. Ground Water , 39(5), 651–659.

Christ, J.A. and Goltz, M.N., 2002. Hydraulic containment: analytical

and semi-analytical models for capture zone curve delineation.

Journal of Hydrology, 262 (1–4), 224–244.

Christ, J.A. and Goltz, M.N., 2004. Containment of g roundwater con-

tamination plumes: minimizing drawdown by aligning capturewells parallel to regional flow. Journal of Hydrology, 286 (1–4),

52–68.

Cole, K.D. and Yen, D.H.Y., 2001. Green’s functions, temperature,

and heat flux in the rectangle. International Journal of Heat

Mass Transfer , 44, 3883–3894.

Dacosta, J.A. and Bennett, R.R., 1960. The pattern of flow in the

vicinity of a recharging and discharging pair of wells in an

aquifer having areal parallel flow. In: Subterranean waters

(IAHS, IUGG General Assembly of Helsinki). Wallingford:

IAHS Press, IAHS Publ. 52, 524–536.

Faybishenko, B.A., Javandel, I., and Witherspoon, P.A., 1995.

Hydrodynamics of the capture zone of a partially penetrating

well in a confined aquifer. Water Resources Research, 31 (4),

859–866.

Fienen, M.N., Luo, J., and Kitanidis, P.K., 2005. Semi-analyticalhomogeneous anisotropic capture zone delineation. Journal of

Hydrology, 312 (1–4), 39–50.

Glover, R.E. and Balmer, C.G., 1954. River depletion resulting from

pumping a well near a river. EOS, Transactions of the American

Geophysical Union, 35, 468–470.

Grubb, S., 1993. Analytical model for estimation of steady-state

capture zones of pumping wells in confined and unconfined

aquifers. Ground Water , 31 (1), 27–32.

Ha, K., et al., 2007. Estimation of layered aquifer diffusivity and

river resistance using flood wave response model. Journal of

Hydrology, 337 (3–4), 284–293.

Hantush, M.M., 2005. Modeling stream–aquifer interactions with

linear response functions. Journal of Hydrology, 311 (1–4),

59–79.




Hantush, M.S., 1965. Wells near streams with semipervious beds.

Journal of Geophysical Research, 70, 2829–2838.

Hunt, B.J., 1999. Unsteady stream depletion from ground water

pumping. Ground Water , 37 (1), 98–102.

Intaraprasong, T. and Zhan, H., 2007. Capture zone between two

streams. Journal of Hydrology, 338 (3–4), 297–307.

Intaraprasong, T. and Zhan, H., 2009. A general framework of

stream–aquifer interaction caused by variable stream stages.


Javandel, I. and Tsang, C.F., 1986. Capture zone type curves:

a tool for aquifer cleanup. Ground Water , 24 (5),

616–625.

Jenkins, C.T., 1968. Techniques for computing rate and volume of

stream depletion by wells. Ground Water , 6 (2), 37–46.

Kompani-Zare, M., Zhan, H., and Samani, N., 2005. Analytical study

of capture zone of a horizontal well in a confined aquifer.


Lin, R.L., 2010. Explicit full field analytic solutions for two-

dimensional heat conduction problems with finite dimensions.

International Journal of Heat Mass Transfer , 53, 1882–1892.

Muskat, M., 1946. The flow of homogeneous fluids through porous

media. Ann Arbor, MI: J.W. Edwards.

Polubarinova-Kochina, P.Y., 1962. Theory of ground water movement .

Princeton, NJ: Princeton University Press, (translated by J.M.R.DeWiest).

Rushton, K., 2007. Representation in regional models of saturated

river–aquifer interaction for gaining/losing rivers. Journal of

Hydrology, 334 (1–2), 262–281.

Schafer, D.C., 1996. Determining 3D capture zones in homogeneous,

anisotropic aquifers. Ground Water , 34 (4), 628–639.

Shafer, J.M., 1987. Reverse path line calculation of time-related cap-

ture zones in nonuniform flow. Ground Water , 25 (3), 283–289.

Shan, C., 1999. An analytical solution for the capture zone of two

arbitrarily located wells. Journal of Hydrology, 222 (1–4),

123–128.

Strack, O.T.D., 1989. Groundwater mechanics. Englewood Cliffs, NJ:

Prentice-Hall.

Theis, C.V., 1941. The effect of a well on the flow of a nearby

stream. EOS Transactions of the American Geophysical Union,

22, 734–738.

Zhan, H., 1999. Analytical and numerical modeling of a double-well

capture zone. Mathematical Geology, 31 (2), 175–193.

Zhan, H. and Cao, J., 2000. Analytical and semi-analytical solu-

tions of horizontal well capture times under no-flow and

constant-head boundaries. Advances in Water Resources, 23 (8),

835–848.

Zlotnik, V.A. and Huang, H., 1999. Effects of shallow penetra-

tion and streambed sediments on aquifer response to stream

stage fluctuations (analytical model). Ground Water , 37 (4),599–605.

Documents

Asadi-Aghbolaghi et al.pdf