Advances in Water Resources 31 (2008) 906–914

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Advances in Water Resources

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Derivation and relative performance of strings of line elements for modeling(un)confined and semi-confined flow

Mark Bakker *

Water Resources Section, Faculty of Civil Engineering and Geosciences, Delft University of Technology, 2628CN Delft, The NetherlandsKiwa Water Research, Nieuwegein, The Netherlands

a r t i c l e i n f o

Article history:Received 27 November 2007Received in revised form 20 February 2008Accepted 24 February 2008Available online 7 March 2008

Keywords:Analytic element methodLine elementsLine-sinkLine-doublet

0309-1708/$ - see front matter � 2008 Elsevier Ltd. Adoi:10.1016/j.advwatres.2008.02.005

* Address: Water Resources Section, Faculty of CivilDelft University of Technology, 2628CN Delft, The Ne

E-mail address: mark.bakker@tudelft.nl.

a b s t r a c t

In the analytic element method, strings of line-sinks may be used to model streams and strings of line-doublets may be used to model impermeable walls or boundaries of inhomogeneities. The resulting solu-tions are analytic, but the boundary conditions are met approximately. Equations for line elements maybe derived in two ways: through integration of point elements (the integral solution) and through appli-cation of separation of variables in elliptical coordinates (the elliptical solution). Using both approaches,two sets of line elements are presented for four flow problems: line-sinks and line-doublets in (un)con-fined flow, and line-sinks and line-doublets in semi-confined flow. Elliptical line elements have theadvantage that they do not need a far-field expansion for accurate evaluation far away from the element.The derivation of elliptical line elements is new and applicable to both (un)confined flow and semi-con-fined flow; only the resulting expressions for elliptical line elements for semi-confined flow have notbeen found in the current groundwater literature. Existing solutions for elliptical line elements for(un)confined flow were intended for the modeling of isolated features. Four examples are presented,one for each flow problem, to demonstrate that strings of elliptical line elements may be used to obtainaccurate solutions; elliptical line-doublets for semi-confined flow can only be strung together in combi-nation with two integral line-doublets. For a zigzag canal in (un)confined flow, a string of elliptical line-sinks performed better than a string of integral line-sinks of the same order when the smallest anglebetween two adjacent segments is less than 130�. Elliptical line-doublets performed better than integralline-doublets for a square inhomogeneity in a uniform, confined flow field; the difference was smaller foran octagonal inhomogeneity. For semi-confined flow, the difference between the integral and ellipticalline-sinks was small when modeling a zigzag canal.

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1. Introduction

The analytic element method for modeling groundwater flow isbased on the superposition of analytic solutions to the governingdifferential equation. Each analytic solution is called an analyticelement. Line elements are the main building blocks of the analyticelement method. Three types of line elements may be distin-guished: line-sinks, line-dipoles, and line-doublets [17,18]. Aline-sink creates a jump in the normal component of flow acrossthe element and has a net discharge. A line-dipole creates a jumpin the normal component of flow across the element and has a zeronet discharge. A line-doublet creates a jump in the tangential com-ponent of flow, and thus the discharge potential, across the ele-ment. As a line-dipole is in essence a special case of a line-sink,this paper only considers the use of line-sinks and line-doublets.

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Engineering and Geosciences,therlands.

Strings of line elements may be used to model a number of aquiferfeatures such as streams and canals (line-sinks), cracks or fissures(line-dipoles), leaky walls (line-doublets), and abrupt changes inthe aquifer properties (line-doublets). For an extensive set of aqui-fer features that may be modeled with line elements, see Strack[17]. Introductions to the analytic element method may also befound in [11] and [9]. A recent overview of applications of the ana-lytic element method is given by Hunt [12].

Line-sinks and line-doublets are commonly derived throughanalytic integration of sinks and doublets along a line in the ana-lytic element method [17,20,18,3], where the strength of the lineelements is varied as a polynomial along the line segment. Thestrength of a line-sink represents inflow into the element and thestrength of a line-doublet represents the jump in potential acrossthe element. The strength may be specified a priori (for examplethe extraction rate of a line-sink), but for most applications thestrength is unknown a priori and is computed from specifiedboundary conditions. For example, a line-sink with a polynomialstrength of order N may be used to model a canal segment with

M. Bakker / Advances in Water Resources 31 (2008) 906–914 907

a known head. A polynomial of order N has N + 1 unknown coeffi-cients, so that the head may be specified at N + 1 collocation pointsleading to a set of N + 1 linear equations. If the model contains Msuch line-sinks, the total number of unknowns is M(N + 1), whichmay be obtained by solving a system of M(N + 1) linear equations.Jankovic and Barnes [13] presented examples to show the advan-tage of grouping the polynomial strengths into Chebyshev polyno-mials; they specified boundary conditions at more control pointsthan the number of unknown coefficients and obtained a solutionin the least-squares sense.

An alternative approach for the derivation of line elements ispresented in this paper. The approach is based on application ofseparation of variables in elliptical coordinates and leads to lineelements that have an entirely different strength than the line ele-ments obtained through integration. For line elements fulfilling La-place’s equation, this approach leads to so called double-rootelements, which Strack [17] derived, using conformal mapping,for the simulation of flow to thin isolated features such as ditches,drains, or cracks. It will be shown here that these elements may becombined effectively in strings, especially when the angle betweentwo adjacent segments is small (this will be quantified later). Bak-ker [4] applied separation of variables to obtain solutions for ellip-tical inhomogeneities in multi-aquifer systems, which consistmathematically of superposition of a solution that fulfills Laplace’sequation and multiple solutions that fulfill the modified Helmholtzequation. Kuhlman and Neuman [14] applied separation of vari-ables to obtain expressions for line-sinks that fulfill the modifiedHelmholtz equation with a complex right-hand side. They usedthese elements to solve transient flow problems in the Laplace do-main following the approach of Furman and Neuman [10].

The objective of this paper is to present two sets of line ele-ments and evaluate their performance when applied to four flowproblems. Two sets of line elements are presented for each flowproblem: One set is derived through integration of point elementsalong a line and one set through application of separation of vari-ables in elliptical coordinates. The four flow problems are: (1) line-sinks in confined or unconfined flow, (2) line-sinks in semi-con-fined flow, (3) line-doublets in confined or unconfined flow, and(4) line-doublets in semi-confined flow. For each flow problem,an example application is presented to assess the relative numer-ical performance of the two sets of line elements by comparingthe accuracy of solutions of the same order.

Fig. 1. The physical z = x + iy coordinates and the local Z = X + iY coordinates; lineelement from X = �1 to X = 1.

2. Problem description

We consider steady, two-dimensional flow in horizontal, piece-wise homogeneous aquifers. When flow in the aquifer is a combi-nation of confined and unconfined flow (from hereon written as(un)confined flow), it is governed by Poisson’s equation

r2U ¼ �Nr ð1Þ

where $2 is the two-dimensional Laplacian, Nr is the areal recharge,and U is the discharge potential defined as [17]

U ¼ 12

kh2 for unconfined flow

U ¼ kHh� 12

kH2 for confined flowð2Þ

where k is the hydraulic conductivity, h is the hydraulic head mea-sured with respect to the aquifer base, and H is the aquifer thick-ness. Solutions to Poisson’s equation may be written as the sumof a solution fulfilling Poisson’s equation plus solutions to Laplace’sequation

r2U ¼ 0 ð3Þ

When considering (un)confined flow in this paper, we will considersolutions to Laplace’s equation, as solutions to Poisson’s equationmay always be superimposed. Solutions are derived in terms of acomplex potential X of which the real part is the discharge potentialU and the imaginary part the stream function W.

Equations for line elements are derived in a local, scaled X, Ycoordinate system where the element lies along the X-axis and ex-tends from X = �1 to X = +1 (Fig. 1). The transformation from thephysical domain to the local domain may be written as [17]

Z ¼ 2z� ðz1 þ z2Þz2 � z1

ð4Þ

where z = x + iy is the complex coordinate in the physical coordinatesystem, Z = X + iY is the complex coordinate in the local coordinatesystem, and z1 and z2 are the complex coordinates of the end pointsin the physical domain. As (4) is a conformal mapping, flow that ful-fills Laplace’s equation in the physical domain also fulfills Laplace’sequation in the Z-domain.

For each problem, one set of line elements will be derived interms of elliptical coordinates. Complex elliptical coordinates xmay also be written as a conformal mapping (e.g. [19,4])

x ¼ gþ iw ¼ ln Z þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðZ � 1ÞðZ þ 1Þ

ph ið5Þ

or conversely

Z ¼ cosh x ð6Þ

Elliptical coordinate g is zero along the element and positive else-where. Elliptical coordinate w varies from �p to p (Fig. 2) and is dis-continuous along the element and the line extending to the left ofthe element (the negative X-axis). As (5) is a conformal mapping,flow that fulfills Laplace’s equation in the physical domain also ful-fills Laplace’s equation in the x-domain.

When the flow is semi-confined, the discharge potential isdefined as [17]

U ¼ kHðh� h�Þ ð7Þ

and fulfills the modified Helmholtz equation

r2U ¼ U=k2 ð8Þ

where h* is the fixed head above the semi-confining unit, and k isthe leakage factor defined as

k ¼ffiffiffiffiffiffiffiffiffikHcp

ð9Þ

where c is the resistance to vertical flow through the semi-confininglayer.

Transformation of (8) to the X,Y coordinate system includes ascaling and a rotation; a line element with a length L in the phys-ical domain has a length 2 in the local coordinate system, so thatthe differential equation for U transforms from (8) to

r2U ¼ U=K2 ð10Þ

Fig. 2. Elliptical coordinates g and w; element is heavy line at center.

Fig. 3. Potential variation along X-axis for first four terms of integral line-sink in(un)confined flow.

908 M. Bakker / Advances in Water Resources 31 (2008) 906–914

where

K ¼ 2k=L ð11Þ

In elliptical coordinates, solutions for U that fulfill (10) are sought inseparated form as [15]

U ¼ FðgÞGðwÞ ð12Þ

where F fulfills

o2Fog2 � aþ 1

2K2 coshð2gÞ� �

F ¼ 0 ð13Þ

and G fulfills

o2G

ow2 þ aþ 12K2 cosð2wÞ

� �G ¼ 0 ð14Þ

where a is a separation constant. Eqs. (13) and (14) are modifiedMathieu differential equations of the radial and circumferentialkind, respectively, and a is also referred to as the Mathieu character-istic number [15,1]. Note that when K approaches infinity, (13) and(14) approach the differential equations that result from separationof variables for Laplace’s equation [15].

3. Line-sinks

We will derive two sets of line-sinks. Derivation of the first setfollows the traditional approach whereby a point-sink is integratedalong a line. The strength r of the line-sink (the extraction rate) isvaried as a polynomial of order N along the line-sink

rðXÞ ¼XN

n¼0

anXn ð15Þ

where an are coefficients. The second set of line-sinks is derivedusing separation of variables in elliptical coordinates.

Both sets of line-sinks have a number of coefficients that may bechosen to meet boundary conditions. Here, we will use strings ofline-sinks to model streams or canals with a specified head (andthus potential) distribution. The potential is specified at a numberof collocation points along each line-sink to obtain linear equationsfor the coefficients. The collocation points are distributed along aline-sink using the cosine rule proposed by [13]. The location ofcollocation point n of a total of N + 1 points is

Xn ¼ cospðn� 1

2ÞN þ 1

� �n ¼ 1; . . . ;N þ 1 ð16Þ

It is noted that locating collocation points with (16) gives a majorimprovement of the solution over a uniform distribution of points.Jankovic and Barnes [13] also suggested to use more control pointsthan coefficients in the solution and to meet the specified potentialat control points in a least-squares sense. The additional improve-

ment of the solution using this overspecification approach is smallfor the problems investigated in this paper as long as the collocationpoints are located according to (16).

3.1. Line-sinks for (un)confined flow

The first set of line-sinks for (un)confined flow is obtainedthrough integration of a point-sink along a line. The complex po-tential for a well (a point-sink) with unit discharge and locatedat Z = D is (e.g. [17])

X ¼ 12p

lnðZ � DÞ ð17Þ

The potential for a line-sink with a strength (15) is obtainedthrough integration of (17)

X ¼XN

n¼0

anL4p

Z 1

�1Dn lnðZ � DÞdD ð18Þ

Note that the component anDn in r results in the discharge of an

incremental section dD of the line-sink equal to anDnLdD/2. Strack

[17,18] presented analytic expressions for the integral (18) as wellas asymptotic expansions (also called far-field expansions) for accu-rate implementation in computer programs. This type of line-sinkwill be referred to here as the ‘integral’ solution.

The contribution to the potential of the first four terms in (18) iscomputed along the X-axis from �2 to +2, while the coefficients an

are taken as one (Fig. 3). The variation from �1 to +1 (between thevertical dotted lines) is the variation along the element. Note thatall even terms have a net discharge as they keep increasing awayfrom the element. The odd terms have a zero net discharge and ap-proach zero away from the element.

The second set of line-sinks for (un)confined flow is derived inelliptical coordinates. The general solution to Laplace’s equationin terms of elliptical coordinates is [15]

U ¼ A0gþ B0wþX1n¼1

ðAneng þ Bne�ngÞ½Cn sinðnwÞ þ Dn cosðnwÞ� ð19Þ

where An, Bn, Cn, Dn are coefficients. To obtain the expression for aline-sink, we neglect the terms eng as we know that the gradientof the potential decreases with distance from the line-sink. Thepotential is continuous everywhere, including across the element,so that we remove the term B0w and all odd terms sin(nw) fromthe solution and are left with

U ¼ b0

2pgþ

XN

n¼1

bne�ng cosðnwÞ ð20Þ

M. Bakker / Advances in Water Resources 31 (2008) 906–914 909

where we have taken the first N terms in the expansion and intro-duced new coefficients bn. The corresponding complex potential is

X ¼ b0

2pxþ

XN

n¼1

bne�nx ð21Þ

Substitution of (5) for x gives an expression in terms of Z

X ¼ b0

2pln Z þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðZ � 1ÞðZ þ 1Þ

ph iþXN

n¼1

bn

½Z þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðZ � 1ÞðZ þ 1Þ

p�nð22Þ

This type of line-sink will be referred to here as the ‘elliptical’ solu-tion. Expression (22) is identical to the line-sink double-root ele-ment without poles ([17], eq. 39.22). Note that (22) may beevaluated at any distance without loss of accuracy, because thesummation already consists of negative powers of Z so that nofar-field expansion is needed.

The potential contribution of the first four terms in (22) alongthe X-axis is shown in Fig. 4. We notice that the term for n = 0 resultsin a constant potential along the element (between the vertical dot-ted lines). The other terms represent a potential variation of Xn

along the element, but with a zero net discharge. In fact, the line-sink with n = 0 is the exact solution for a straight canal with a con-stant potential and a discharge b0 in an infinite field. The exact solu-tion for a straight canal in an otherwise uniform flow field may beobtained through combination of the terms with n = 0 and n = 1, to-gether with a uniform flow term. Elliptical line-sinks are attractivein that they result in a polynomial variation of the potential alongthe element. But since they create a strong singular behavior at theirend points, they may not be effective when used in strings, espe-cially when the adjacent line-sink lies along the same line. It isimportant to note here that one integral line-sink of length L maybe written as two integral line-sinks of length L/2. This is not possi-ble for elliptical line-sinks as the singularities of the two shorterline-sinks will not cancel each other. Another way to look at it isas follows: When adding two adjacent line-sinks with constant in-flow, it is equivalent to one long line-sink with constant inflow.When adding two adjacent line-sinks with constant head (the firstterm in (22)) it does not result in one long line-sink with constanthead, as the boundary conditions are not superimposable. It willbe shown in the following, however, that elliptical line-sinks maybe used in strings and perform well when the angle between twoadjacent line-sinks is significantly less than 180�.

3.2. Example of line-sinks for (un)confined flow

The numerical performance of the integral and elliptical solu-tions are compared when modeling confined flow to a canal with

Fig. 4. Potential variation along X-axis for first four terms of elliptical line-sink in(un)confined flow.

a constant potential in an infinite field; the same number of coef-ficients is used in each solution. We investigate a canal that makesa regular zigzag pattern where the smallest angle between twoadjacent sections is b. We are not interested in the end points ofthe canal, as we know that the elliptical solution performs muchbetter than the integral solution when modeling the singularbehavior at the end of a canal. We consider a canal consisting of18 segments, each represented by one line-sink, plus a referencehead at a distance from the canal. We vary the angle b and studythe two line-sinks in the middle of the canal. Contours of the po-tential for the cases b = 140� and b = 90� are shown in Fig. 5. Theleft side of each contour plot (to the left of the dotted line) repre-sents the integral solution, while the right side represents the ellip-tical solution. All line-sinks have order N = 5. For the integralsolution, the overspecification rate was 2, although that did nothave a significant influence; no overspecification was used forthe elliptical solution. Both solutions perform well for b = 140�,while the elliptical solution performs better for b = 90� (the poten-tial needs to be constant along the line-sink, which it clearly is notfor the integral solution of Fig. 5b).

The error in the boundary condition varies along the elementand is zero at the collocation points. The average error is close tozero and the largest errors occur, as expected, near the singularitiesat the end points of the line elements. The root-mean-square-erroris computed by evaluating the potential at 1000 uniformly spacedpoints along the canal. The magnitude of the error is not that inter-esting as it is a function of the difference between the value of thepotential in the canal and at the reference point. What matters isthe difference in error between the two solutions, and the decreasein error when the order of the line-sinks increases. The root-mean-square-error decreases by approximately an order of magnitude

Fig. 5. Contours of potential for (un)confined flow to a zigzag canal where thesmallest angle between two segments is (a) b = 140� and (b) b = 90�. Integral sol-ution is shown on left of dotted line and elliptical solution on right. Order of all line-sinks is N = 5.

910 M. Bakker / Advances in Water Resources 31 (2008) 906–914

when the order of the line-sink is increased by 6. For b = 140�, theintegral solution performs better; the error in the integral solutionfor order N is approximately equal to the error in the elliptical solu-tion for order N + 2. When the angle is decreased to b = 90�, the er-ror in the integral solution increases significantly while the error inthe elliptical solution increases only slightly. Now the ellipticalsolution is more accurate; the error in the elliptical solution for or-der N is approximately equal to the error in the integral solutionfor order N + 3.5. Further evaluation shows that the root-mean-square-error in the two solutions is approximately equal forb = 130�.

Note that the exact solution for the zigzag canal has singulari-ties at the corner points; the singularity is stronger when the angleb is smaller. When b is large, the singularity is weak and bettermodeled with an integral line-sink. When b is small, the singularityis stronger, and better modeled with an elliptical line-sink.

3.3. Line-sinks for semi-confined flow

Two types of line-sinks are derived for the modified Helmholtzequation (10). The first type is obtained through integration of thepotential of a point-sink. A point-sink with a unit discharge may bewritten as (e.g. [17])

U ¼ � 12p

K0ðr=KÞ ð23Þ

where K0 is the modified Bessel function of the second kind andorder zero, and

r ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðX � DÞ2 þ Y2

qð24Þ

where (X,Y) = (D,0) is the location of the sink. The potential for aline-sink with a strength (15) may be written as (compare (18))

U ¼ �XN

n¼0

anL4p

Z 1

�1DnK0ðr=KÞdD ð25Þ

This integral has not been evaluated in closed form. Bakker andStrack [3] proposed to approximate K0 by a polynomial so thatthe integral may be evaluated analytically. They used a polynomialapproximation that is valid up to r = 8K; beyond this distance it isaccurate to set K0 to zero for problems of semi-confined flow. Thisapproach works well for orders up to 8. Beyond 8, combination ofthe inaccuracy of the polynomial approximation and finite com-puter arithmetic leads to inaccurate results.

The potential variation of the first four terms of (25) along theX-axis is shown in Fig. 6 for L/K = 4; the coefficients in (25) aretaken as one. The head variation is similar to the variation forthe integral line-sink of the Laplace equation (Fig. 3) except that

Fig. 6. Potential variation along X-axis for for first four terms of integral line-sink insemi-confined flow

all terms approach zero away from the line-sink, as may be ex-pected for semi-confined flow.

Next, we derive a solution for a line-sink using separation ofvariables in elliptical coordinates, as stated in (12). The solutionsF and G are called modified Mathieu functions [1]. The solutionsF are radial solutions of which the first kind approaches infinitywhen g approaches infinity, and the second kind approaches zerowhen g approaches zero (e.g. [4]). For semi-confined flow, thepotential approaches zero far away from the line-sink, so radialsolutions of the second kind are used. The solutions G are circum-ferential solutions; solutions of the first kind are periodic whilesolutions of the second kind are not. Hence, only circumferentialsolutions of the first kind are used, because solutions of the secondkind jump across the line w = ±p. First kind solutions come in twotypes: even and odd functions. Since the potential is continuousacross a line-sink, the even solutions are used (similar to theuse of the cosine function in (20)). The solution for an ellipticalline-sink may now be written as

U ¼XN

n¼0

bnQenð1=K;wÞKenð1=K; gÞ=Kenð1=K;0Þ ð26Þ

where Qen(1/K,w) is the even circumferential modified Mathieufunction of order n, Ken(1/K,g) is the even radial modified Mathieufunction of order n, and the terms Ken(1/K,0) are used for scaling(use of symbols for modified Mathieu functions conforms to [1]).

The contribution to the potential of the first four terms in (26) isshown in Fig. 7 for L/k = 4, using bn = 1. The pattern is similar to thepattern of the elliptical line-sink for Laplace’s equation (Fig. 4).They have the same strong singularity at the end points, but thevariation along the element is not exactly a polynomial.

3.4. Example of line-sinks for semi-confined flow

The performance of the integral and elliptical solutions arecompared when modeling the same zigzag canal with a constantpotential, as used in Fig. 5. Along the canal the potential is specifiedas 1 while far away (�8k) from the canal, the potential approacheszero. Only the case for which the smallest angle between two adja-cent segments is 90� is considered, but two values of the leakagefactor are used: L/k = 10 (Fig. 8a) and L/k = 2 (Fig. 8b). The left sideof each contour plot (to the left of the dotted line) represents theintegral solution, while the right side represents the elliptical solu-tion. The order is N = 5 in all solutions, and no overspecification isused. Both solutions perform well, but the integral solution per-forms slightly better for the small value of k (Fig. 8a), while theelliptical solution performs better for the larger value of k. Theroot-mean-square-error is computed along the center two line-

Fig. 7. Potential variation along X-axis for for first four terms of elliptical line-sinkin semi-confined flow.

Fig. 8. Contours of potential for semi-confined flow to zigzag canal: (a) L/k = 2(contour interval 0.1) and (b) L/k = 10 (contour interval 0.05). Integral solution isshown on left of dotted line and elliptical solution on right. All line-sinks have orderN = 5.

M. Bakker / Advances in Water Resources 31 (2008) 906–914 911

sinks by evaluating the error in potential at 1000 uniformly spacedpoints. The error analysis is less interesting, as the integral solu-tions cannot be evaluated beyond order 8. For the case ofL/k = 10, the root-mean-square-error in the elliptical solution de-creases by one order of magnitude when the order of the line-sinkis increased from 5 to 10; the same improvement is obtained forthe case of L/k = 2 when the order is increased from 3 to 10. Theerror in the integral solution decreases at approximately the samerate for L/k = 2, but significantly slower for L/k = 10.

4. Line-doublets

We will derive two sets of line-doublets for (un)confined flowand two sets for semi-confined flow. The structure of this sectionfollows the structure of the previous section on line-sinks. As thederivations are similar, they are presented in less detail. The inte-gral solution is derived through integration of a point-doublet. Thestrength s is varied as a polynomial of order N along the line-dou-blet and is equal to the jump in the potential across the element

sðXÞ ¼XN

n¼0

anXn ð27Þ

where an are coefficients. The elliptical solution is derived usingseparation of variables in elliptical coordinates. We emphasize thatline-doublets create a jump in the potential and thus the tangentialcomponent of flow, but that the normal component of flow is con-tinuous across the element.

Both sets of line-doublets have a number of coefficients thatmay be chosen to meet boundary conditions. Here, we will usestrings of line-doublets to model the boundary of an inhomogene-ity and to model an impermeable wall. For an inhomogeneity, lin-

ear equations for the coefficients are obtained by requiringcontinuity of head across the inhomogeneity boundary at a num-ber of collocation points along each line-doublet. For an imperme-able wall, linear equations for the coefficients are obtained bysetting the normal component of flow equal to zero at a numberof collocation points along each line-doublet [17]. Overspecifica-tion is not used in this section.

4.1. Line-doublets for (un)confined flow

The integral line-doublets for Laplace’s equation are obtainedthrough integration of a point-doublet along a line; the axis ofthe point-doublet is perpendicular to the line-doublet. The com-plex potential for a point-doublet with unit strength, located atZ = D, and with its axis in the Y direction is (e.g. [17])

X ¼ i2p

1Z � D

ð28Þ

The potential for a line-doublet with a strength (27) is obtainedthrough integration of (28)

X ¼XN

n¼0

ani2p

Z 1

�1

Dn

Z � DdD ð29Þ

Strack [17] presented analytic expressions for the integral (29) andits asymptotic expansion. As stated, the integral line-doublet cre-ates a jump across the element equal to (27), which is a regularpolynomial and not plotted here.

The second set of line-doublets for Laplace’s equation are de-rived in elliptical coordinates. The general solution to Laplace’sequation in terms of elliptical coordinates was given in (19). To ob-tain the expression for a line-doublet, we neglect the term g as wellas all terms eng as we know that the effect of the line-doublet iszero at infinity. The term w does indeed create a potential jumpacross the line-doublet, but the jump extends to infinity alongthe negative X-axis. To cancel this jump we can add either a termln(Z � 1) or a term ln(Z + 1). Both are needed as we will see in thefollowing. Furthermore, we want the other terms to create a jumpin the potential across the line-doublet, so we select the odd termssin(nw) rather than the even terms cos(nw). This results in the fol-lowing complex potential for an elliptical line-doublet.

X ¼ b0i2p

lnZ þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðZ � 1ÞðZ þ 1Þ

pZ � 1

!� b1i

2pln

Z þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðZ � 1ÞðZ þ 1Þ

pZ þ 1

!

�XN

n¼2

bn

½Z þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðZ � 1ÞðZ þ 1Þ

p�n�1 ð30Þ

The last term in (30) is identical to the line-doublet double-root ele-ment without poles ([17], eq. 39.26). Strack [17] derived this ele-ment for modeling isolated features. Since we want to use stringsof elliptical line-doublets to model, for example, an inhomogeneityboundary, we added the first two terms in (30), which give a non-zero jump at the end points. The potential jump across the line-dou-blet for the first four terms of an elliptical line-doublet is shown inFig. 9. Note that the first two terms give an almost linear variationof the jump, from zero at one end to one at the other. The otherterms create a jump that is zero at the end points, increases quicklyaway from the end points, and is bounded by plus and minus one.

4.2. Example of line-doublets for confined flow

The performance of the two sets of solutions is compared formodeling a square inhomogeneity in a confined, uniform flow fieldfrom left to right. Each side of the inhomogeneity is modeled withone line element of order N = 5. Contours of the head are shown inFig. 10 for two cases. In Fig. 10a, the transmissivity inside the inho-

Fig. 11. Contours of the head for an octagonal inhomogeneity in a uniform flowfield with Tout/Tin = 100 for order N = 5. Integral solution is shown on left of dottedline and elliptical solution on right.

Fig. 10. Contours of the head for a square inhomogeneity in a uniform flow fieldwith (a) Tout/Tin = 100, (b) Tout/Tin = 0.01 for order N = 5. Integral solution is shownon left of dotted line and elliptical solution on right. Two extra contour lines (da-shed) are shown in (b).

Fig. 9. Potential jump across element for first four terms of elliptical line-doublet in(un)confined flow.

912 M. Bakker / Advances in Water Resources 31 (2008) 906–914

mogeneity is 100 times smaller than outside, while in Fig. 10b thetransmissivity inside the inhomogeneity is 100 times larger thanoutside; note that in Fig. 10b two additional contour lines (dashed)

are shown to illustrate the accuracy of contour lines that cross theinhomogeneity boundary. The integral solution is shown in the leftside of each figure (to the left of the dotted line) and the ellipticalsolution in the right side. It may be seen from Fig. 10a that theelliptical solution performs significantly better for this case thanthe integral solution. For the case of Fig 10b, both solutions aregood, although the elliptical solution is significantly more accuratenear the corner of the inhomogeneity, which can be seen whenzooming in on the corner (not shown). When the order of theline-doublets is increased, the error in both the integral and ellip-tical solutions decreases, as we have seen for the examples withthe line-sinks. It is noted that a formulation using Chebyshev poly-nomials [13] and an overspecification factor of 2 gives results verysimilar to the integral solutions shown in Fig. 10 (these solutionswere obtained with Split [7]).

As is the case for line-sinks, the integral solutions perform betterwhen the smallest angle between two adjacent elements is largerthan 90� . Contours for uniform flow through a regular octagonalinhomogeneity (with inside angles of 135�) and a transmissivitythat is 100 times smaller inside the inhomogeneity than outside,is shown in Fig. 11. The order of all elements is 5. Here we see thatthe difference between the integral solution and the elliptical solu-tion is much smaller, although the elliptical solution still performsbetter; both solutions improve when the order is increased.

4.3. Line-doublets for semi-confined flow

Integral line-doublets are obtained through integration of apoint-doublet along a line, again with the axis of the point-doubletperpendicular to the line. An expression for a point-doublet thatfulfills the modified Helmholtz equation is obtained using the stan-dard approach of taking the limit of two sinks with opposite dis-charge that approach each other while their discharges increase.This results in the following expression for a point-doublet withunit strength and located at (D,0)

U ¼ Y2pK

K1ðr=KÞr

ð31Þ

where K1 is the modified Bessel function of the second kind and or-der one, and r is given by (24). The potential for a line-doublet witha strength (27) is obtained through integration (compare (29))

Fig. 13. Contours of potential for well near impermeable wall in semi-confined flowwith L = 4k: (a) straight impermeable wall with integral solution to left of dottedline and elliptical solution to right, (b) integral solution for impermeable wall co-nsisting of three segments. Order of all line-doublets is N = 5.

M. Bakker / Advances in Water Resources 31 (2008) 906–914 913

U ¼XN

n¼0

anY2pK

Z 1

�1

DnK1ðr=KÞr

dD ð32Þ

As for the integral line-sink that fulfills the modified Helmholtzequation (25), this integral may be carried out analytically whenK1 is approximated by a polynomial, following the same procedureas in [3]. Restrictions on the use of the polynomial approximationare the same as for the line-sink (25). The jump across the elementis equal to the polynomial (27) and is not plotted here.

An expression for an elliptical line-doublet that fulfills the mod-ified Helmholtz equation is obtained using the same procedurethat was applied to obtain the elliptical line-sink (26). The solutionconsists of a sum of products of radial and circumferential modi-fied Mathieu functions. Radial functions of the second kind areused again, as they approach zero for g approaching infinity. Cir-cumferential functions of the first kind are used, since they areperiodic, but this time the odd functions are used, since they createa jump in the potential across the element. The resulting expres-sion for an elliptical line-doublet that fulfills the modified Helm-holtz equation is

U ¼XN

n¼1

bnQonð1=K;wÞKonð1=K; gÞ=Konð1=K;0Þ ð33Þ

where Qon(1/K,w) is the odd circumferential modified Mathieu func-tion of order n, Kon(1/K,g) is the odd radial modified Mathieu func-tion of order n, and the terms Kon(1/K,0) are used for scaling [1].The potential jump across the elliptical line-doublet is shown inFig. 12 for the first four terms of (33). Note that the summation in(33) starts at 1 (odd functions of order 0 do not exist), and that thejump is zero at both end points for all orders. These elements cannotbe strung together, as that requires a non-zero jump at the corners.There are no terms in the general solution that create such as jump.In contrast, such terms do occur in the elliptical solution for Laplace’sEq. (19), which includes the terms g and w. Alternatively, it was at-tempted to create such a solution by taking the limit of two ellipticalline-sinks with opposite strength that approach each other whiletheir strength increases. This results in an infinite jump at the endpoints, which is not useful in practical applications. Another alterna-tive is to add the first two terms of the integral solution; such a cross-pollination of solutions is not considered here further.

4.4. Example of line-doublets for semi-confined flow

Inhomogeneities in semi-confined flow cannot be modeled withline-doublets, because not only the potential jumps, but also thevalue of k in the differential equation (8) (e.g. [6]). Line-doubletsare suitable, however, for modeling leaky or impermeable walls.Because the elliptical line-doublets (33) may not be strung to-

Fig. 12. Potential jump along element for first four terms of elliptical line-doubletin semi-confined flow.

gether, the problem of a well near a straight impermeable wall ismodeled. The components of the discharge vector are obtainedby analytic differentiation of (23), (32) and (33). The impermeablewall is modeled with a single line-doublet of order N = 5. Contoursof the potential are shown in Fig. 13a for the case that L/k = 4. Theintegral solution is shown to the left of the dotted line and theelliptical solution to the right. It is difficult to assess the accuracyof the solution visually, as the contours of the potential need tointersect the impermeable wall at a 90� angle. The error in the nor-mal discharge is computed at 1000 uniformly spaced points alongthe line-doublet. The root-mean-square-error for the elliptical line-doublet is several orders smaller than for the integral line-doubletfor this case; the error along the integral line-doublet is relativelylarge along the last 5% of its length for this case. As for the othersolutions, the error decreases when the order of the line-doubletis increased. It is emphasized that the chosen example highly fa-vors the elliptical solution, which is especially good at the simula-tion of flow around isolated features. The integral solution by itselfmay be used to simulate flow around an impermeable wall thatconsists of multiple straight segments, as is shown in Fig. 13b,where a pumping well is located near an impermeable wall con-sisting of three segments.

5. Discussion and conclusion

Eight sets of line elements were discussed for the analytic ele-ment modeling of groundwater flow. Strings of line elements canbe placed accurately along aquifer features such as streams, canals,impermeable or leaky walls, or geological boundaries. An overview

914 M. Bakker / Advances in Water Resources 31 (2008) 906–914

of practical applications of these elements is given in [12] while rulesfor their application are given in [11]. Four flow problems were stud-ied: a line-sink in (un)confined flow, a line-sink in semi-confinedflow, a line-doublet in (un)confined flow, and a line-doublet insemi-confined flow. For each flow problem, two sets of line elementswere presented. One set was derived through integration of pointelements along a line (referred to as the integral solution), and oneset was derived using separation of variables in elliptical coordinates(referred to as the elliptical solution). Each solution has a number ofunknown coefficients; similar to a polynomial, the number of coef-ficients is generally equal to the order of the element plus one. Linearequations for the coefficients are obtained by applying boundaryconditions at collocation points along the element. As the coeffi-cients of one element depend on the value of the coefficients ofthe other elements, the coefficients are obtained by solving one largesystem of linear equations for all elements simultaneously, using astandard explicit method. The root-mean-square error in the bound-ary condition decreases when the order is increased. For severalcases it was shown that 6 or 7 additional coefficients per element re-sulted in a decrease of the root-mean-square-error by an order ofmagnitude.

Combined confined and unconfined flow to line elements is gov-erned by Laplace’s equation. The integral solutions for line-sinksand line-doublets governed by Laplace’s equation are found in[17]. A new derivation for elliptical elements was presented, whichis also applicable to semi-confined flow. The elliptical solutions arealso found in [17], where they are called double-root elements andwere derived for modeling (un)confined flow with straight, isolatedfeatures. Two additional terms were added to the elliptical line-doublet (30) to model a non-zero potential jump at the end points.It was shown that accurate solutions may be obtained with ellipticalline elements when they are strung together. Relative performancewas evaluated by comparing solutions of the same order. For thecase of a zigzag canal, elliptical line-sinks perform better than inte-gral line-sinks when the angle between two adjacent line-sinks issmaller than approximately 130�. Elliptical line-sinks are especiallyuseful for modeling Dupuit flow in systems of man-made canals andditches, which often intersect at near 90� angles. Natural streamswith continuously varying direction need to be discretized instraight segments, which are better modeled with strings of integralline-sinks. An alternative for streams may be the use of curved lineelements [16]. Another alternative are the single-root elementsdeveloped by ([17], Sec. 40). These elements have the same singu-larity as an elliptical line-sink at one end, but not the other. Strack[17] derived these elements, using conformal mapping, to be usedas the last element of a string of integral line elements. Evaluationof the numerical performance of these elements (which are notavailable for semi-confined flow) was beyond the scope of this pa-per. It would of course be best to build the correct corner singularityinto the solution, as was done for Laplace’s equation by Detournay[8], who developed one-sided elements for power-type singularitiesto be used for the boundary of a domain. Elliptical line-doublets per-formed better than integral line-doublets for modeling a squareinhomogeneity in a uniform flow field, especially when the trans-missivity inside the inhomogeneity is less than outside; the differ-ence was less for a regular octagonal inhomogeneity. Sharpcorners may be more common in modeling inhomogeneity bound-aries, for example when the aquifer is tiled with inhomogeneities ina checkerboard fashion or using polygons of different shapes, assuch representing a piecewise homogeneous aquifer. Anotheradvantage of elliptical line elements is that no asymptotic expan-sion is needed for evaluation in the far field, since elliptical line ele-ments for (un)confined flow are already in a form that consists ofone or two logs plus a series of negative powers.

Semi-confined flow is governed by the modified Helmholtzequation. The presented sets of line elements each have their

own advantages and disadvantages. The integral solutions arebased on the analytic integration of a polynomial approximationof a modified Bessel function. This polynomial approximation isnot valid beyond a distance of 8k, which is accurate enough formodeling semi-confined flow or multi-aquifer flow (e.g. [3]), butnot for analytic element modeling of steady unsaturated flow(e.g. [5]). Furthermore, use of the polynomial approximations lim-its the order that can be computed accurately to approximately 8.The elliptical line elements are based on modified Mathieu func-tions. The elliptical line elements for semi-confined flow presentedin this paper have not been found elsewhere in the groundwaterliterature. These can be evaluated at much larger distance (thereis no practical limit), but their computational cost is relatively high,even while using the efficient subroutines of [2]. Numerical perfor-mance of the two sets of line-sinks was similar for the zigzag canal.Elliptical line-doublets can only be strung together when integralsolutions of order 0 and 1 are added. Comparison of flow arounda single straight impermeable wall showed that the elliptical solu-tion performed much better than the integral solution, as expected.

Acknowledgements

Development of the integral line-doublets for semi-confinedflow was funded in part by WHPA, Inc., Bloomington, IN. Develop-ment of elliptical line-sinks was funded in part by the joint re-search program of the Dutch water companies.

References

[1] Alhargan FA. Algorithms for the computation of all Mathieu functions ofinteger orders. ACM Trans Math Software 2000;26(3):390–407.

[2] Alhargan FA. Algorithm 804: subroutines for the computation of Mathieufunctions of integer orders. ACM Trans Math Software 2000;26(3):408–14.

[3] Bakker M, Strack ODL. Analytic elements for multiaquifer flow. J Hydrol2003;271(1-4):119–29.

[4] Bakker M. Modeling groundwater flow to elliptical lakes and through multi-aquifer elliptical inhomogeneities. Adv Water Resour 2004;27(5):497–506.

[5] Bakker M, Nieber JL. Two-dimensional steady unsaturated flow throughembedded elliptical layers. Water Resour Res 2004;40:W12406. doi:10.1029/2004WR003295.

[6] Bakker M. An analytic element approach for modeling polygonalinhomogeneities in multi-aquifer systems. Adv Water Resour 2006;29(10):1546–55.

[7] Bandilla K, Suribhatla R, Jankovic I. SPLIT, Win32 computer program foranalytic-based modeling of single-layer groundwater flow in heterogeneousaquifers with particle tracking, capture-zone delineation, and parameterestimation. Version 3.2 http://www.groundwater.buffalo.edu/software/VBB/VBBMain.htm; 2006.

[8] Detournay C. A Cauchy integral element for power-type singularities. ApplMath Model 1992;16:450–63.

[9] Fitts CR. Groundwater science. New York: Academic Press; 2002.[10] Furman A, Neuman SP. Laplace-transform analytic element solution of

transient flow in porous media. Adv Water Resour 2003;26:1229–37.[11] Haitjema HM. Analytic element modeling of groundwater flow. San Diego,

CA: Academic Press; 1995.[12] Hunt RJ. Ground water modeling applications using the analytic element

method. Ground Water 2006;44(1):5–15.[13] Jankovic I, Barnes R. High-order line elements in modeling two-dimensional

groundwater flow. J Hydrol 1999;226(3–4):211–23.[14] Kuhlman KL, Neuman, SP. Advances in LT-AEM theory and application to

transient groundwater flow. In: Abstract and poster at AGU fall meeting; 2006.H41B-0397.

[15] Moon P, Spencer DE. Field theory handbook: including coordinate systemsdifferential equations and their solutions. 2nd ed. New York, NY: Springer-Verlag; 1971.

[16] Steward DR, Le Grand P, Jankovic I, Strack ODL. Analytic formulation of Cauchyintegrals for boundaries of curvilinear geometry. Proc Roy Soc A2007;464:223–48.

[17] Strack ODL. Groundwater mechanics. Englewood Cliffs, NJ: Prentice Hall;1989.

[18] Strack ODL. Theory and applications of the Analytic Element Method. RevGeophys 2003;41(2):1005. doi:10.1029/ 2002RG000111.

[19] Suribhatla R, Bakker M, Bandilla K, Jankovic I. Steady two-dimensionalgroundwater flow through many elliptical inhomogeneities. Water ResourRes 2004;40:W04202. doi:10.1029/2003WR002718.

[20] Zaadnoordijk WJ, Strack ODL. Area sinks in the analytic element method fortransient groundwater flow. Water Resour Res 1993;29(12):4121–9.