Directory UMM :Data Elmu:jurnal:A:Advances In Water Resources:Vol23.Issue5.2000:

The method of images for leaky boundaries

Erik I. Anderson

*

SEH, Inc., 421 Frenette Drive, Chippewa Falls, WI 54729, USA Received 15 July 1999; accepted 20 October 1999

Abstract

An analytic solution is presented describing ¯ow to a drain in a semi-in®nite domain bounded by a leaky layer of constant thickness. The solution is developed by applying the method of images to two parallel boundaries: an inhomogeneity boundary and an equipotential boundary. It is then demonstrated that the solution for the problem with the leaky layer approximated by a leaky boundary (a mixed boundary condition) may be obtained by allowing the thickness,h_{, and the hydraulic conductivity,k, of the} leaky layer to vanish while holding the ratioh=k_{constant. A method of images for leaky boundaries is proposed, in which a drain} is imaged with respect to a leaky boundary by an image drain and an image line dipole. The method of images for a leaky boundary is applied to solve the problem of ¯ow to a horizontal drain in a semi-con®ned aquifer. Ó _{2000 Elsevier Science Ltd. All rights}

reserved.

Keywords:Groundwater; Analytic; Method of images; Leaky boundary; Semi-con®ned

1. Introduction

Groundwater ¯ow domains often contain thin layers of low hydraulic conductivity that act as leaky barriers to ¯ow. A typical example, a€ecting the interaction of groundwater and surface water, is a low-permeability layer of sediment or silt lining a streambed. The leaky layer may have a major impact on both heads and dis-charges in an aquifer and cannot be ignored in the mathematical formulation of the problem. Barriers are often included in engineering design. Waste isolation is a common example; engineered systems for waste isola-tion may include geomembranes, bentonite blankets, grout curtains and slurry walls.

The ¯ow within a leaky layer is approximated often as being one-dimensional and the physical layer is re-placed by a mixed boundary condition. We refer to this type of boundary as a leaky boundary. The e€ects of a leaky boundary can be included in numerical studies in an approximate fashion [12], but exact solutions have proven dicult to obtain. Polubarinova-Kochina [13, p. 376] presents a general approach for solving problems with leaky boundaries by conformal mapping; Van der Veer [18] applies the method to solve two problems. The

approach is applicable to problems with horizontal leaky boundaries, and vertical equipotential and im-permeable boundaries. This is a signi®cant restriction on the types of problems that may be investigated analyti-cally; for example, the approach does not apply to problems of ¯ow in semi-con®ned aquifers (i.e. aquifers bounded on top by a leaky boundary and on the bottom by an impermeable base). Van der Veer [19,20] presents an approach for solving problems in semi-con®ned aquifers based on the superposition of basic solutions. However, the resulting solutions do not satisfy the conditions along the leaky boundary; an explanation of the error is provided herein. Bruggeman [4] presents several solutions, expressed in Fourier series, to prob-lems in semi-con®ned aquifers.

Problems of ¯ow in semi-con®ned aquifers are often solved by using the Dupuit approximation. Use of the Dupuit approximation results in a governing di€erential equation which is often simpler to solve than the exact equation. Strack [16] presents several cases. Dupuit so-lutions are widely used and generally accepted in engi-neering practice. Bear and Braester [3], however, demonstrate that application of the Dupuit approxi-mation to cases of semi-con®ned ¯ow may result in signi®cant errors in discharge.

In this paper we develop a general approach, based upon the method of images, for solving problems with leaky boundaries. The approach may be applied to more

*

Fax: +1-715-720-6300.

E-mail address:eanderson@sehinc.com (E.I. Anderson).

general problems than the approach presented by Polubarinova-Kochina and is applicable to problems of semi-con®ned ¯ow. The method of images is a classical technique for solving boundary value problems; the method is frequently used for solving problems gov-erned by Laplace's equation, such as those encountered in the ®elds of groundwater mechanics, hydrodynamics, and electrostatics. Maxwell [9] cites Thomson [17] as the originator of the method of electrical images.

In groundwater mechanics the method of images is applied frequently to solve steady and transient prob-lems in two and three dimensions. Strack [16] presents several steady and transient solutions in two dimensions. Hantush and Jacob [6] solve the problem of transient ¯ow to a well in a leaky strip by applying the method of images; their solution is based on the Dupuit approxi-mation. Muskat [11], Haitjema [5] and Steward [15], solve various three-dimensional ¯ow problems; in each case the author considers ¯ow in a con®ned aquifer to a partially penetrating well. Shan et al. [14] applies the method of images to model air ¯ow to a partially pen-etrating well; in this case the top of the aquifer is modeled as a constant pressure boundary.

The method of images is applied to satisfy conditions along equipotential and impermeable boundaries in most cases. Muskat [11], following Maxwell [9], shows that the method of images may also be applied to in-homogeneity boundaries. Fitts [8] presents an approxi-mate approach, loosely based on the method of images, for solving three-dimensional groundwater ¯ow prob-lems in strati®ed aquifers.

In the following, we extend the method of images to include leaky boundaries. First, we develop the method of images for a leaky layer ± a thin layer of low hy-draulic conductivity bounded on top by an equipoten-tial. Second we demonstrate that the leaky boundary ± a boundary of zero thickness and ®nite resistance ± is a limiting case of the leaky layer. Next, we propose a method of images for a leaky boundary. Finally, we apply the method of images for a leaky boundary to solve the problem of ¯ow to a horizontal drain in a semi-con®ned aquifer.

2. Problem description

We consider steady, two-dimensional groundwater ¯ow to a drain in a semi-in®nite aquifer bounded by a horizontal leaky layer of thickness h, as illustrated in Fig. 1. We adopt a Cartesian coordinate system with the y axis pointing vertically upward; the problem is for-mulated in terms of the complex coordinatezˆx‡iy. The drain, represented by the dot in the ®gure, is located atzˆzd. The boundaryB divides the complexzplane

into two domains, D _and D_{. The domain} D _contains

the aquifer andD _{contains the leaky layer. The upper}

boundary of the leaky layer is an equipotential of head

/₀, and is represented by the dashed line in the ®gure. We de®ne a complex potential in each domain

zin D _{XˆU‡iW; …1a†}

zin D _{XˆU‡iW; …1b†}

where the speci®c discharge potential is

Uˆk/; …2a†

Uˆk/; …2b†

W andW are stream functions andX andX are ana-lytic functions of z; kand /are the hydraulic conduc-tivity and head, respectively, in the aquifer;kand/are the hydraulic conductivity and head in the leaky layer, respectively. The boundary B _{links the two domains}

with the conditions that both the head and the normal component of ¯ow are continuous across B_{. The ®rst}

condition, expressed in terms of the two potentials is

zon B _Uˆ k

kU

_{: …3†}

The second condition, expressed in terms of the two stream functions, is

zon B _{WˆW: …4†}

These are the standard conditions that apply along the boundary of an inhomogeneity (e.g. [16, p. 412]).

It is often assumed that the ¯ow in the leaky layer may be approximated well as one-dimensional ¯ow. The domain D _{is then eliminated from the problem and}

replaced by a boundary condition on D

y ˆ0; qyˆ

/ÿ/₀

c ; …5†

where qy is the speci®c discharge in the y direction,/0

represents the ®xed head above the leaky layer and cis the resistance of the leaky layer

cˆh

k: …6†

Eqs. (5) and (6) de®ne the leaky boundary. The leaky boundary will provide a good approximation of the

e€ects of a leaky layer when the horizontal component of ¯ow in the leaky layer is negligible.

3. The method of images for a leaky layer

We present an analytical solution describing ¯ow to a drain near a leaky layer. The solution is derived by ap-plying the method of images to both an equipotential boundary and an inhomogeneity boundary. Numerous authors apply the method of images to inhomogeneity and impermeable boundaries to model strati®ed for-mations in the ®elds of groundwater mechanics and geophysical prospecting. These papers are summarized by Muskat [10,11] who cites Maxwell [9] as originally solving an analogous problem by the method of images. The method has not been applied previously to prob-lems with leaky layers. Three basic solutions will be used to determine the appropriate forms of the images about each boundary.

The ®rst basic solution contains a drain in the lower-half plane with a horizontal equipotential located a distancehabove the real axis (Fig. 2(a)). The solution is a classical application of the method of images (e.g. [16, p. 44])

Xˆ Q 2p ln…

zÿzd† ÿ Q 2p ln‰

zÿ …zd‡2ih†Š ‡U0; …7†

whereQis the discharge of the drain andU0is the value

of the horizontal equipotential.

Basic solutions 2 and 3 are developed from a single classical solution consisting of a drain in an aquifer of two hydraulic conductivities ([13, p. 372]). In both so-lutions, the lower-half plane is designated as D _{with a}

hydraulic conductivity of kand the upper-half plane is designated as D _{with a hydraulic conductivity of k.} Basic solution 2 has a drain at zˆzd in D (Fig. 2(b))

which gives

Xˆ …1ÿj† Q 2p ln…

zÿzd† ‡C; …8a†

Xˆ Q 2p ln…

zÿzd† ‡j Q 2p ln…

zÿzd† ‡ k

kC: …8b†

The third basic solution contains a drain atzˆzdinD

(Fig. 2(c))

Xˆ Q

2p ln…zÿzd† ÿj Q

2p ln…zÿzd† ‡ k

kD; …9a†

Xˆ …1‡j† Q

2p ln…zÿzd† ‡D: …9b†

In both solutions

jˆkÿk

k‡k …10†

andCandDare real constants.

To create a solution with a leaky layer from the three basic solutions, we image drains about both the inho-mogeneity boundaryB_{and the equipotential boundary}

inD_{(see Fig. 1). First consider a drain of dischargeQ}

atzdand the inhomogeneity boundary. The solutions in

D _andD _{are given by (8a) and (8b). For now, we}

ne-glect the real constants, and write

Xˆ …1ÿj† Q

2p ln…zÿzd†; …11a†

Xˆ Q 2p ln…

zÿzd† ‡j Q 2p ln…

zÿzd†: …11b†

These expressions include the e€ects of the drain at zˆzd and satisfy the conditions (3) and (4) along the

inhomogeneity boundary. The boundary condition,

UˆU₀ for zˆx‡ih is not satis®ed by (11a). We create an equipotential of value Uˆ0 for zˆx‡ih by placing an image drain of opposite strength in D

with respect to the equipotential boundary. We see from (7) that the complex coordinates of the image drain are given by zˆzd‡2ih. We add the drain to (11a) to

obtain

Xˆ …1ÿj† Q 2p ln…

zÿzd†

ÿ…1ÿj† Q 2p ln‰

zÿ …zd‡2ih†Š: …12†

The real constantU₀may be added to the complex po-tential to satisfy the boundary condition; for now, we neglect the constant, leaving the equipotential along zˆx‡ih _{with the valueUˆ0.}

We have satis®ed the boundary conditionUˆ0 for zˆx‡ih_{, but the conditions (3) and (4) along the} in-homogeneity boundary are now violated by addition of the new drain toX. To correct this, bothXandXmust be modi®ed: the new drain inD_{atzˆzd‡2ihmust be} imaged about the inhomogeneity boundary. We obtain the complex coordinates and strength of the image drain by inspection of (9a). From (9a), the image of a drain at zd in D with respect to an inhomogeneity boundary

along the real axis is located at zd. In (12) the drain is

located atzˆzd‡2ihso that the image drain must be

placed at zˆzdÿ2ih. We also see from (9a) that the

image of a drain of strengthQhas a strength ofÿjQ. In (12) the strength of the drain is ÿ…1ÿj†Q so the strength of the image drain must bej…1ÿj†Q. We add the image drain to (12) to obtain

Xˆ …1ÿj† Q 2p ln…

zÿzd†

ÿ …1ÿj† Q 2p ln‰

zÿ …zd‡2ih†Š

‡j…1ÿj† Q

2p ln‰zÿ …zdÿ2ih

_{†Š; …13†}

X now contains the proper imaging across the inho-mogeneity boundary, but inspection of (9b) shows that we must add a new drain toX also. We see from (9b) that the complex potential in D _{feels the e€ect of the}

drain outside of D_{, and so a new drain must also be}

included in the expression forX. We see from (9b) that the new drain must be located atzˆzd‡2ihand have

a strength of ÿ…1ÿj†…1‡j†Q. We add this drain to (11b) to obtain

Xˆ Q

2p ln…zÿzd† ‡j Q

2p ln…zÿzd†

ÿ …1ÿj†…1‡j† Q 2p ln‰

zÿ …zd‡2ih†Š: …14†

We have now taken three steps in the imaging process. We began with the basic solutions (8a) and (8b) which satisfy the conditions along the inhomogeneity bound-ary. In the second step we applied the method of images in D _{to satisfy the equipotential boundary condition.}

This violated the conditions along the inhomogeneity boundary. In the third step,XandXwere modi®ed to satisfy again the conditions along the inhomogeneity boundary. Once again,Xcontains a drain outside ofD

which violates the conditions along the equipotential

boundary; we have returned to conditions similar to those at the conclusion of the ®rst step. However, the location of the last drain added, located atzˆzdÿ2ih,

is farther from the equipotential boundary than the original drain at zˆzd and the strength has changed

from …1ÿj†Q in (11a) to j…1ÿj†Q. We recall that j

equals…kÿk†=…k‡k†and conclude thatjis less than one fork<k; the strength of the drain has decreased. We have established a pattern in the imaging process; repetition of the second and third steps described above will result in drains of decreasing strength being added in

XandX that lie farther from the real axis than in pre-vious steps. The imaging must be continued inde®nitely; the ®nal expressions contain an in®nite number of drains

Xˆ …1ÿj† Q 2p

X1

nˆ0

jn_ln zÿzd‡2ih

_n

zÿzdÿ2ih…n‡1†

‡k/0;

…15a†

Xˆ Q

2p ln…zÿzd† ‡j Q

2p ln…zÿzd† ÿ …1ÿj 2_† Q

2p

X 1

nˆ0

jn_ln‰zÿz

dÿ2ih…n‡1†Š ‡k/0; …15b†

where the constants,k/₀andk/₀, are evaluated from the boundary conditions. Fig. 3(a) and (b) show the distri-bution of drains inXandX, respectively. The complex potentialX, valid inD_{, contains the actual drain atzdand}

an in®nite sum of drains outside of D _{whose locations}

approach in®nity and whose discharges decay exponen-tially with their distance from the real axis. The complex potential X, valid in D_{, contains an in®nite sum of}

drains outside ofD_{and another in®nite sum of drains}

inside ofD_{, but outside the domain of interest. Anderson}

[2] demonstrates that the expressions (15a) and (15b) satisfy the boundary conditions exactly and that the in-®nite series appearing in (15a) and (15b) converge.

4. Limiting case: the leaky boundary

Hantush [7], Van der Veer [18], and Bruggeman [4] solved the problem discussed above with the leaky layer replaced by the boundary condition

y ˆ0; qyˆ

/ÿ/₀

c : …16†

The solution is

Xˆ Q 2p ln

zÿzd zÿzd

ÿQ

p exp

i ck…z

ÿzd†

E1

i ck…z

ÿzd†

‡k/0; …17†

whereE1…z†is the exponential integral [1] de®ned as

E1…z† ˆ

Z 1

z

exp…ÿt†

We will show here that the solution (17) for a leaky boundary is a limiting case of the solution (15b) for a leaky layer obtained by letting the thickness and hy-draulic conductivity of the leaky layer, h _{and k,} ap-proach zero while keeping the ratioh_{=k constant and} equal toc, the resistance of the leaky boundary.

The solution for X (15b), presented graphically in Fig. 3(a), consists of drains of decreasing recharge dis-tributed evenly along the line extending from zˆzd‡2ih to in®nity along the positive imaginary

axis. In the ®gure, the drains are represented by dots; the discharge of each drain is displayed to the right of each dot. The drains in the upper-half plane approach one another as the parameterh is decreased. In the limit as h and k vanish while holding h=k constant, we pass from a discrete sum of drains to a continuous distribu-tion of drains, or a line sink.

We rewrite the series in (15b) in a more convenient form to aid in taking the limit

…1ÿj2_† Q

2p

X1

nˆ0

jn_ln‰zÿz

dÿ2ih…n‡1†Š

ˆX 1

mˆ1

rm

2pln…zÿdm†Dn; …19†

where

dmˆzd‡2ih…n‡1† ˆzd‡2imh: …20†

We express the discharge of thenth, or…mÿ1†th, drain asrmÿ1Dnwhere

rmˆ

Q 2h

1ÿj2

j j

m _21†

and where Dn represents the distance between the mth and …m‡1†th drains and is equal to 2h. We may ex-press the integer min terms of the location of the mth drain,dm. From (20) we obtain

mˆ …dmÿzd†=2ih: …22†

Substituting (22) into (21) we obtain the strength of the mth drain as a function of its location

rmDnˆ

Q 2h

1ÿj2

j j

…dmÿzd†=2ih_Dn

ˆ Q 2h

1ÿj2

j exp

1 2ih lnj

…dm

ÿzd†

Dn: …23†

We obtain the expression for the complex potential upon substitution of (23) and (19) into (15b)

Xˆ Q

2p ln…zÿzd† ‡j Q

2p ln…zÿzd†

ÿ Q 4p

1ÿj2 h_j

X1

mˆ1

exp 1 2ih lnj

…dm

ÿzd†

ln…zÿdm†Dn‡k/0: …24†

The coecients in (24) containingjmay be expanded in Taylor series aboutkˆ0 as follows:

jˆkÿk

k‡kˆ1‡2

X1

nˆ1

ÿk

k

n

; …25a†

1ÿj2

j ˆ4

k k

X1

nˆ0 k

k

2n

; …25b†

lnjˆ ÿ2k

k

X1

nˆ0 k

k

2n

: …25c†

We substitute the above expansions into (24) to obtain an expression valid for k_{=k<1 and h=kˆc. The} complex potential may now be written as

Xˆ Q

In the limit, for h and k approaching zero with h_{=kˆc, the ®rst series appearing in (26) vanishes, the} second series and fourth series are reduced to single terms, and the third series becomes an integral. The expression for the complex potential becomes

lim

where the complex numberddenotes a point of the line extending along the imaginary axis from zd to in®nity

and may be written in terms ofn as

dˆzd‡in; 06n61: …28†

We rewrite the integral in (27), making the following change of variables:

Zˆ i

The integral represents a semi-in®nite line sink of ex-ponentially decaying strength lying along the negative real axis of the Z plane. The integral in (30) may be integrated by parts to give

F…z† ˆ ÿQ

where use is made of

lim

D!ÿ1expDln…ÿD† ˆ0: …31b† The remaining integral represents a line dipole ([16, p. 291]) and may be evaluated as (see (18))

Q

The ®nal form of the complex potential is obtained from (27), (29a), (29b), (31a), (31b), and (32) as

which is identical to (17). The leaky boundary is a lim-iting case of the leaky layer, as asserted. The leaky boundary represents exactly the e€ects of a membrane, where the membrane has a ®nite resistance but no thickness. Fig. 4 shows ¯ow nets for the case of a leaky layer, developed from (15a) and (15b) and for the case of

Fig. 4. Contours of constant head (dashed) and stream function (solid) for ¯ow to a drain from: (a) a leaky layer withk=k_{ˆ10 andh=jz} dj ˆ0:4;

a leaky boundary, developed from (33). In Fig. 4(a), the parameters of the problem are k=k_{ˆ10 and} h_=jz

dj ˆ0:4; in Fig. 4(b), the parameter is ck=jzdj ˆ4.

The ¯ow ®elds in the aquifer are nearly identical; the horizontal component of ¯ow in the leaky layer is in-deed negligible in this case.

5. The method of images for a leaky boundary

We applied the method of images earlier to both an inhomogeneity boundary and an equipotential bound-ary to construct a leaky layer. In passing through the limit forhandkapproaching zero while holdingh=k constant, the two boundaries join to form a single leaky boundary. We introduce in this section the method of images for a leaky boundary; we examine the solution (33) to determine the form of the image, re¯ected across a leaky boundary, of a single drain.

Eq. (33) indicates that a drain beneath a leaky boundary must be imaged across that boundary by a recharge drain, with an extra term added; the extra term has the functional form exp…Z†E1…Z†and we recall from

(32) that the function represents a line dipole of expo-nentially decaying strength distribution lying along the negative real axis of theZplane. The function is ®nite, but not analytic, at in®nity with branch points atZˆ0 and Zˆ ÿ1. The image drain in (33) at zˆzd is the

source for the discharge drain atzˆzd, while the line

dipole corrects the behavior of the complex potential along the leaky boundary without a€ecting the net dis-charge in the aquifer.

This interpretation provides the basis of the method of images for a leaky boundary. The image for a drain of strength Qat zd with respect to a leaky boundary

con-sists of a drain of strengthÿQatzd, and a semi-in®nite

line dipole originating at zd and oriented normal to the

leaky boundary. The line dipole has an exponential strength distribution which vanishes at in®nity.

We present in Fig. 5 the solution from Fig. 4(b) continued in the upper-half plane. The real axis of thez plane, corresponding to the leaky boundary, is shown as the solid horizontal line. The heavy line originating at the drain in the lower-half plane represents the branch cut in the stream function created by the drain. The heavy line originating at the image drain in the upper-half plane represents the branch cut for the image drain and the branch cut for the line dipole.

6. The image of a line dipole

The images, with respect to a leaky boundary, for other features such as line sinks, dipoles, line dipoles and line doublets, may be generated from the basic solution (33) for a single drain beneath a leaky boundary. We present without derivation the image, with respect to a leaky boundary, for a line dipole of the form

Xldˆ Q

p

…ÿ1†n …nÿ1†!

Z ÿ1

0

Dnÿ1expD

ZÿD dD; …nˆ1;2;3. . .†:

…34†

We will use the results to solve the problem of ¯ow to a horizontal drain in a semi-con®ned aquifer. We note, that fornˆ1, (34) has the same form as the line dipole appearing in (32). Eq. (34) may be integrated to express the line dipole in terms of an exponential integral

Xldˆ Q

p exp…Z†En…Z†; …35†

where the function En…z†is related to E1…z†by the

fol-lowing relationships [1]:

En‡1…z† ˆ

1

n‰exp…ÿz† ÿzEn…z†Š …nˆ1;2;3;. . .†; …36a†

dEn…z†

dz ˆ ÿEnÿ1…z† …nˆ1;2;3;. . .†; …36b†

E0…z† ˆ

exp…ÿz†

z : …36c†

The equality of (34) and (35) may be veri®ed, fornP2, by repeated substitution of (36a) into (35) to express En…z† in terms of E1…z†. The function obtained in this

manner, when expanded, may be shown to equal the integral given in (34).

We place the line dipole along the negative imaginary The complex potential for the line dipole is

Xldˆ

The image of this line dipole, with respect to a leaky boundary lying along the real axis, is given by

X…_ldim†ˆQ

We demonstrate that the function

XˆXld‡X… im†

ld ‡k/

0 …40†

satis®es condition (16) along the leaky boundary. We may express (16) in terms of the functionL…z†, de®ned in the following manner ([13, p. 376])

L…z† ˆX‡ickdX

dz; …41†

whereÿdX=dzis the complex speci®c discharge function (e.g. [13, p. 36]) and is equal toqxÿiqy. The condition

(16) along the leaky boundary,yˆ0, may be expressed as ReL…z† ˆk/₀. We di€erentiate (40) and substitute into (41) to obtain

Lˆ ÿQ Using the symmetry relationships [1]

exp…z† ˆexp…z†; …44a†

En…z† ˆEn…z†; …44b†

we see that

Refexp…z†En…z†g ˆRe fexp…z†En…z†g …45†

and therefore, from (42), foryˆ0, the real part of L…z† is equal tok/₀. The condition along the leaky boundary is satis®ed exactly.

As a demonstration, we apply (33) and (39) to iden-tify the image, with respect to a leaky boundary, of a drain and a semi-in®nite line dipole. The resulting so-lution will be used both to derive the soso-lution for ¯ow to a drain in a semi-con®ned aquifer and to identify the error in the work by Van der Veer [19,20]. The leaky boundary lies along the real axis of the complex plane; the drain lies in the lower-half plane at zˆzd and the

line dipole originates at zˆzd and extends to zˆzdÿi1. The portion of the complex potential

as-sociated with the drain is

Xdrˆ Q

2p ln…zÿzd† …46†

and the portion associated with the line dipole, obtained from (35) with nˆ1, is given by

The image of the drain in (46), obtained from (33), is

X…_drim†ˆ ÿ Q

and the image of the line dipole in (47), obtained from (38) and (39), is given by

The complex potential containing Xdr andXld and

sat-isfying the condition (16) is given by

XˆXdr‡Xld‡X…

We combine (46)±(49) to obtain

Xˆ Q

The image of the drain and the line dipole in the lower-half plane consists of a drain and a line dipole in the upper-half plane. The image drain at zˆzd is opposite

in strength of the drain atzˆzd and the image line

di-pole originating atzˆzd and extending tozˆzd‡i1,

is of the form exp…Z†E2…Z†. Van der Veer [19,20]

exp…Z†E2…Z†in (51) is neglected and therefore his

solu-tion does not satisfy the proper condisolu-tion along the leaky boundary.

Van der Veer's approach [19,20] is to superimpose two basic solutions, each satisfying conditions along a leaky boundary lying on the real axis, to satisfy condi-tions along an impermeable base. The ®rst basic solution (17) describes ¯ow to a drain in the lower-half plane from the leaky boundary; the second solution describes ¯ow to a drain in the upper-half plane from the leaky boundary. The second solution is obtained from the ®rst by a simple rotation of the domain. However, this conformal transformation alters the form of the leakage function,L…z†(41), by introducing a minus sign in front of the term dX=dz. The two basic solutions may be su-perimposed, but the appropriate condition along the leaky boundary will not be satis®ed, as demonstrated above.

7. Example: ¯ow to a horizontal drain in a semi-con®ned aquifer

We develop the solution to the problem illustrated in Fig. 6 as an application of the method of images for a leaky boundary. The ®gure shows an aquifer of ®nite depth H, bounded on top by a leaky boundary of re-sistancecand on the bottom by an impermeable base. A horizontal drain of discharge Q exists at zˆzd. We

apply the method of images about the two parallel boundaries to obtain a solution. Bruggeman [4, p. 311] solves the same problem using a di€erent approach.

We begin with the solution (33) for a drain beneath a leaky boundary. The condition along the impermeable base may be satis®ed by imaging the two drains and the line dipole in (33) about zˆxÿiH. The images of a drain and of a line dipole, with respect to an imperme-able boundary, are known (e.g. [16]). The image of a drain is a drain of the same discharge re¯ected across the boundary; the image of a line dipole is a line dipole of the same strength distribution re¯ected across the boundary. We image the terms in (33) about the aquifer base and obtain

Xˆ Q

The third and fourth terms in (52) are the images of the ®rst and second terms with respect to the aquifer base. The condition along the leaky boundary, yˆ0, is now violated; we may satisfy that condition by imaging the two drains and the line dipole, the third and fourth terms in (52), about yˆ0 using (33) and (51) to obtain their images. We obtain

Xˆ Q

Once again, we may satisfy the condition along the aquifer base by imaging the new terms in (53) about zˆxÿiH. The process must be continued inde®nitely resulting in in®nite sums of drains, and line dipoles of the form (38) of increasing order n. The process is te-dious but straightforward. The resulting solution may be expressed as

Xˆ Q

X

m

nˆ1 am;nEn

i ck…z

(

ÿzdÿ2iHm†

)

‡ Q 2p

X Mÿ1

mˆ1

exp

ÿ i

ck…zÿzd‡2iH…m‡1††

X

m

nˆ1 am;nEn

(

ÿ i

ck…zÿzd‡2iH…m‡1††

)

‡k/₀; …54†

where

am;1ˆ

ÿ2 formodd; mP_1;

0 formeven; mP1;

a1;nˆ0 fornP2;

am;nˆ ÿ2amÿ1;nÿ1‡amÿ1;n fornP2; mP2: …55†

The values of the coecientam;nform67 andn67 are

given in Table 1. The expression (54) is written such that for any ®niteMthe condition along the leaky boundary is satis®ed exactly and the condition along the imper-meable base is satis®ed approximately. We demonstrate in Appendix A that expression (54) satis®es exactly the boundary conditions along the aquifer top for anyM.M must become in®nite to satisfy the condition along the impermeable base; we o€er no proof that, forM ! 1, the in®nite series of line dipoles converge. The proof is dicult as each term of the series of line dipoles repre-sents a semi-in®nite line integral which is ®nite but not analytic at in®nity. As Table 1 shows, the coecients am;n grow rapidly and alternate in sign. However, each

individual line dipole has only a local e€ect on the ¯ow ®eld and each successive line dipole in a series is located farther from the domain of interest.

The ®rst two terms in (54), consisting of the drains, converge to a known function forM ! 1

Q 2p

lnzÿzd zÿzd

‡ Q 2p

X1

mˆ1

…ÿ1†m

ln …zÿzdÿ2iHm† …zÿzdÿ2iHm†

…zÿzd‡2iHm†

…zÿzd‡2iHm†

ˆ Q 2p ln

tanh p

4H…zÿzd†

tanh p

4H…zÿzd†

( )

: …56†

This may be seen by writing the left-hand side of (56) as the logarithm of an in®nite product and identifying the in®nite product as the ratio of hyperbolic functions ([1, p. 85]) shown on the right-hand side of (56). By it-self, (56) represents the solution for ¯ow from an equi-potential to a drain in an aquifer of ®nite depth; the upper boundary, lying along the real axis, is an equi-potential and the lower boundary, at zˆxÿiH, is im-permeable. The remaining terms in (54), consisting of the in®nite sums of line dipoles, may be viewed as two semi-in®nite line dipoles with discontinuous strength distributions. One line dipole originates at zd and

ex-tends along the positive imaginary axis and the other originates at zdÿ2iH and extends along the negative

imaginary axis. We interpret the solution obtained by the method of images as consisting of a basic solution (56), which re¯ects an equipotential boundary along the real axis and an impermeable boundary forzˆxÿiH, and line dipoles which correct the behavior along the leaky boundary without changing the net discharge into the aquifer.

Fig. 7 shows a ¯ow ®eld for the caseck=Hˆ2 and jzdj=Hˆ0:4. The ¯ow ®eld is obtained from (54) by

setting M equal to 4. The aquifer top and base are shown as heavier lines than the streamlines. We have extended the contours of head and streamfunction across the impermeable base to demonstrate graphically the approximated lower boundary condition; the con-dition is approximated well. The solution is quite ac-curate, although it was obtained with only a few terms; the e€ect of the line dipoles on the area of interest vanishes rapidly in this example.

Table 1

Values of the coecientam;nform67 andn67

am;n am;n

1 2 3 4 5 6 7

1 ÿ2 0 0 0 0 0 0

2 0 4 0 0 0 0 0

3 ÿ2 4 ÿ8 0 0 0 0

4 0 8 ÿ16 16 0 0 0

5 ÿ2 8 ÿ32 48 ÿ32 0 0

6 0 12 ÿ48 112 ÿ128 64 0

7 ÿ2 12 ÿ72 208 ÿ352 320 ÿ128

8. Conclusions

A new approach for solving problems of groundwater ¯ow with leaky boundaries was presented. The ap-proach is an extension of the classical method of images commonly used to solve groundwater ¯ow problems. The new approach may be applied to more general problems than the approach presented by Polubarinova-Kochina [13, p. 376]. The image for a drain beneath a leaky boundary was presented as well as the images for a speci®c class of line dipoles. The results were applied to solve the problem of ¯ow in a semi-con®ned aquifer to a horizontal drain. The results may be applied to obtain the solutions to other groundwater problems and the approach may be generalized further by considering other boundary types.

The method of images for both leaky layers and leaky boundaries was presented for steady, two-dimensional groundwater ¯ow. The approach is general, however, and may be applied to three-dimensional ¯ows. For example, basic three-dimensional solutions exist for a point sink beneath a planar discontinuity in hydraulic conductivity and for a point sink beneath a planar equipotential [9]. These solutions have forms similar to the basic solutions (7), (8a), (8b), (9a), and (9b) used in the present analysis. It is clear that the approach pre-sented here could be used to generate the solution for a point sink beneath a planar leaky boundary. In a similar fashion, the approach is applicable to some transient problems.

Acknowledgements

I wish to thank an anonymous referee for a particu-larly thorough review of the manuscript and for identi-fying Bruggeman [4] as a necessary reference.

Appendix A

We verify that the solution (54) satis®es condition (41) along the leaky boundary. We partition the po-tential (54) in the following manner:

XˆXdr‡Xld‡k/0; …A:1† Similarly, we partition the leakage function L…z†, given by (41), into a portion due to drains and a portion due to line dipoles

LˆLdr‡Lld‡k/0; …A:8†

Recall that the condition along the leaky boundary is given by ReLˆk/₀foryˆ0.

We di€erentiate (A.2) and substitute into (A.9) to obtain

Lldˆ

We modify the ®rst term in (A.12) into a more conve-nient form; we operate on the indices to represent Z1…z;mÿ1†asZ1…z;m† From (55) we obtain

am‡1;nˆam;nÿ2am;nÿ1 fornP2 …A:14†

and we may write

F ˆa1;1

We use (55) to evaluate the following coecients:

a1;1ˆ ÿ2; …A:17†

Substituting (A.18) and (A.19) into (A.16), we obtain

F ˆ ÿQ

Finally, we use (36c) to obtain

F ˆickQ

We substitute (A.21) and (A.22) into (A.12) to obtainLld

We evaluate the real part of Lld along the real axis

(yˆ0). Note that for yˆ0, from (A.4) to (A.7), we have

Z1ˆZ2; Z3ˆZ4: …A:24†

Using the symmetry relationship (45), we see that the real parts of the terms in (A.23) containing the expo-nential integrals cancel and we have

foryˆ0; ReLldˆRe ick

(A.24) we see that the real part of the logarithmic terms in (A.11) vanishes along the real axis and we have

foryˆ0; ReLdrˆRe ick

We substitute (A.25) and (A.26) into (A.8) to verify the boundary condition along the real axis

for yˆ0; Re LˆRe ick Q

We use the following relationship to evaluate (A.27):

Re 1

The remaining terms in (A.27) vanish except for the constant. We obtain ReL…x† ˆk/₀; the boundary con-dition is satis®ed exactly for any value ofM.

References

[1] Abramowitz M, Stegun, IA. Handbook of mathematical func-tions, New York: Dover, 1965.

[2] Anderson EI. Groundwater ¯ow with leaky boundaries, Ph.D. Thesis, Department of Civil Engineering, University of Minneso-ta-Minneapolis, 1999.

[3] Bear J, Braester C. Flow from in®ltration basins to drains and wells. In: Proceedings of the American Society of Civil Engineers, J Hydra Div 1966;95(5):115±34.

[4] Bruggeman GA. Analytical solutions of geohydrological prob-lems, developments in water science, vol. 46. Amsterdam: Elsevier, 1999.

[5] Haitjema HM. Modeling three-dimensional ¯ow in con®ned aquifers by superposition of both two and three-dimensional analytic functions. Water Resour Res 1985;21(10):1557±66. [6] Hantush MS, Jacob CE. Non-steady Green's functions for an

in®nite strip of leaky aquifer. Trans Am Geophys Union 1955;36(1):101±12.

[7] Hantush MS. Wells near streams with semipervious beds. J Geophys Res 1965;70(12):2829±31.

[8] Fitts CR. Simple analytic functions for modeling three-dimen-sional ¯ow in layered aquifers. Water Resour Res 1989;25(5):943± 8.

[9] Maxwell JC. A treatise on electricity and magnetism, vol. 1. Oxford: Clarendon Press, 1873.

[10] Muskat M. Potential distribution about an electrode on the surface of the earth. Physics 1933;4(4):129±47.

[11] Muskat M. The ¯ow of homogeneous ¯uids through porous media. New York: McGraw-Hill, 1937.

[12] Nield SP, Townley LR, Barr AD. A framework for quantitive analysis of surface water±groundwater interaction: ¯ow geometry in a vertical section. Water Resour Res 1994;30(8):2461±75. [13] Polubarinova-Kochina PY. Theory of ground water movement.

Princeton, NJ: Princeton University Press, 1962 (translated by De Wiest, JMR).

[15] Steward DR. Three-dimensional analysis of the capture of contaminated leachate by fully penetrating, partially penetrating, and horizontal wells. Water Resour Res 1999;35(2):461±8. [16] Strack ODL. Groundwater mechanics. Englewood Cli€s, NJ:

Prentice-Hall, 1989.

[17] Thomson W. On the mathematical theory of electricity in equilibrium. The Cambridge and Dublin Math J 1848;III.

[18] Van der veer P. Exact solutions for two-dimensional groundwater ¯ow problems involving a semi-pervious boundary. J Hydro 1978;37:159±68.

[19] Van der veer P. Exact solutions for two-dimensional groundwater ¯ow in a semicon®ned aquifer. J Hydro 1994; 156:91±9.