determination of groundwater ﬂownets, ﬂuxes, velocities

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Engineering Analysis with Boundary Elements 30 (2006) 1030–1044

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Determination of groundwater flownets, fluxes, velocities, and traveltimes using the complex variable boundary element method

Todd C. Rasmussena,�, Guoqing Yub

aWarnell School of Forestry and Natural Resources, University of Georgia, Athens, GA 30602-2152, USAbWater Resources and Hydroelectric Power Institute, Hohai University, Nanjing 210098, China

Received 11 August 2005; accepted 16 January 2006

Available online 7 November 2006

Abstract

The complex variable boundary element method, CVBEM, employs the Cauchy integral with any complex variable (e.g., complex

potential, complex flux, or complex velocity) to solve boundary value problems. The CVBEM formulation is consistent with the primal

and dual solutions of the boundary integral equation, as well as the analytic element method. The resulting problem is overdetermined

because two boundary conditions can be specified at each node. Ordinary least squares provides a unique solution that minimizes

boundary specification errors. Flownets are obtained by noting that the position of fluid–stream potential intersections can be found by

exchanging potential and position in the Cauchy integral, which enhances the determination of travel times along streamlines. Three

regional groundwater flow problems are used to illustrate the CVBEM approach, the original problem as defined by Toth, plus two

related problems as described by Domenico and Paliauskas, and by Nawalany.

r 2006 Elsevier Ltd. All rights reserved.

Keywords: Complex variable boundary integral method; Travel times; Groundwater flow and transport; Toth’s problem

1. Introduction

Mapping groundwater flow patterns is an important partof characterizing subsurface flow and transport behavior.A traditional tool for presenting maps of fluid potentialand streamlines is the flownet. The flownet helps tovisualize the flowfield, to delineate the capture zone orinfluence areas of boundary discharge and recharge, todesign groundwater development and remediation mea-sures, and to evaluate the effects of alternative boundaryconditions during site characterization.

Flownets are also used to compare calculated waterlevels with known observations while at the same timepredicting the path of contaminants along streamlines. Fortransport problems, flowpaths and travel times found byparticle tracking are used to identify outflow locations andthe arrival time of contaminants, as well as to show theadvective advance of a contaminant front within the flowdomain.

e front matter r 2006 Elsevier Ltd. All rights reserved.

ganabound.2006.01.017

ing author. Tel.: +1706 542 4300; fax: +1 706 542 8356.

ess: [email protected] (T.C. Rasmussen).

Extensive effort has been dedicated to the developmentof post-processing algorithms for mapping velocity fields.Among others, Pollock [1] used simple linear interpolationto generate the velocity vector field from a block-centered,finite-difference groundwater flow model. The Pollockapproach finds pathlines within a finite-difference grid cellusing an analytical function of nodal values. While thevelocity component is continuous in the direction of flow,discontinuities arise in the direction transverse to flow dueto the need to discretize the domain into grid cells.A continuous velocity field may also be obtained using

the finite-element method [2], which avoids the transversedirection discontinuity by using a bilinear interpolation forboth head and stream functions. Both finite-difference andfinite-element methods provide pathlines and travel times,and eliminate the need for time-stepping. The shortcomingof both techniques, however, is the need for internaldiscretization of the flow domain.Alternative methods for producing a groundwater

velocity field include mixed finite elements or mixed-hybridfinite elements [3,4]. These methods couple the headformulation with the velocity field by using head and

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ARTICLE IN PRESST.C. Rasmussen, G. Yu / Engineering Analysis with Boundary Elements 30 (2006) 1030–1044 1031

velocity as degrees of freedom, or by applying the sameapproach to both head and velocity fields. Unfortunately,these methods suffer from the need to expend morethan twice as much computational effort as thestandard Galerkin models that use linear or bilinear headinterpolation.

Zijl [5,6] developed finite-element methods based on atransport velocity representation (TVR) of groundwatermotion. This method solves coupled Laplace-type equa-tions (two for two-dimensional flow fields, and three forthree-dimensional flow fields) and by adding auxiliaryboundary conditions to ensure that the problems arewell-posed.

While generally restricted to steady flow, these methodscan be applied to long-term contaminant transportproblems if short-term transients are unimportant andthe steady flow assumption is acceptable [7]. Nonetheless,these methods need to perform either a time-stepping orcontour interpolation when plotting flownets and calculat-ing travel times.

Regardless of the method employed, conventionalgroundwater flow models rely on either potential or streamfunctions as the primary unknowns, and determine fluidvelocities once these unknowns have been calculatednumerically. This dependence on potential or streamfunctions may result in poor velocity field accuracies whencompared to the accuracy of the primary unknowns.

Because the velocity field is usually deduced from theresults of flow modeling, the accuracy of the flow pathsdepends on both the flow model and the interpolationscheme used to construct the velocity field. Apart from theaccuracy of the velocity field, the accuracy of calculatedtravel times is also related to the step size associated withthe time-stepping scheme. Fine discretization in both spaceand time are major constraints to the application of bothflownet and pathline methods to travel time calculations.

A significant improvement in accuracy can be achievedby avoiding the discretization of the flow domain. Theboundary element method (BEM) is an established methodfor describing groundwater flow that reduces the problemdomain dimensionality by transforming a domain probleminto a boundary problem [8,9]. Thus, two-dimensionaldiscretization of the flow domain can be reduced to asimple discretization of the one-dimensional boundary,providing more accurate mapping of fluid potentials andstreamlines.

Another significant improvement was achieved byemploying the complex variable boundary element method(CVBEM), which provides a method for directly obtainingthe flownet [10,11]. CVBEM uses the Cauchy Integral toconvert a differential equation into a boundary integralequation which is further reduced to an algebraic systemafter approximation and discretization.

For many problems, CVBEM yields an overdeterminedsystem of linear algebraic equations that can be readilysolved using ordinary least squares (OLS). OLS incorpo-rates all of the original information, while at the same time

forming a square matrix that is symmetric, positive definiteand diagonally dominant [12,13].Another advance is the use of an inverse mapping

technique for defining the flownet, i.e., streamline andequipotential positions. Rather than solving for complexpotentials, yðzÞ ¼ fðzÞ þ {cðzÞ, as a function of complexposition (z ¼ xþ {y), we invert the map to provide thelocation of specified flownet intersections, zðyÞ ¼ xðyÞþ{yðyÞ.This flownet computation uses a post-processing strategy

that accurately and efficiently provides the flownetgeometry and travel times. Instead of determining traveltimes by summing travel-time increments between selectivelocations along a streamline, Dz, we obtain travel timesusing increments in fluid potential, Df, along eachstreamline.Yet another improvement is obtained by the direct

calculation of fluxes and flow velocities [30]. Two methodsare available; using the complex flux or velocity asboundary conditions, or by employing complex potentialboundary conditions and noting that the flux and velocityinternal to the domain is the derivative of the potentialfunctions. Both methods provide excellent predictions ofthe flux and velocity fields within the flow domain.

2. Mathematical formulation

2.1. Vector representation

The two-dimensional, steady continuity equation forhomogeneous, isotropic porous medium flow withoutinternal sources or sinks, and with negligible compressi-bility of the fluid and medium, can be written in terms ofthe divergence of the flux vector, ~q, using

r �~q ¼qqx

qxþ

qqy

qy¼ 0, (1)

where

~q ¼ ½qx; qy� ¼ rf (2)

is the Darcian flux vector, qx and qy are the compo-nents of flux in x- and y-directions, respectively,and f is the Grinskii, or discharge, potential [14], definedusing

f ¼

�Kh general case;

�Th confined aquifers;

�12

Kh2 unconfined aquifers;

8><>: (3)

where h is the hydraulic head, K is the homogeneous,isotropic hydraulic conductivity, T is the aquifer transmis-sivity, and the Dupuit–Forchheimer assumptions arerequired for the unconfined case.Eq. (1) is written in terms of the discharge potential using

r2f ¼q2fqx2þ

q2fqy2¼ 0. (4)

ARTICLE IN PRESST.C. Rasmussen, G. Yu / Engineering Analysis with Boundary Elements 30 (2006) 1030–10441032

Eq. (1) holds throughout the flow domain for both steadyflow and unsteady, incompressible flow. The curl of the fluxvector for these conditions is

r �~q ¼ det

qqx

qqy

qx qy

24

35 ¼ qqy

qx�

qqx

qy¼ 0 (5)

which is the same as

r � rf ¼qðqf=qyÞ

qx�

qðqf=qxÞ

qy¼ 0. (6)

The fluid velocity, ~v, is related to the flux using ~v ¼ ~q=Z,where Z is the effective porosity of the medium.A velocity potential, j ¼ f=Z, can also be formed ina manner similar to the discharge potential for homo-geneous conditions:

~v ¼ rj ¼ rfZ

� �(7)

for the same conditions as Eq. (3).

2.2. Complex representation

The vector representation can be written in complexform using

q ¼ qx þ {qy. (8)

The derivative of the complex flux is

rq ¼dq

dz¼

qq

qx¼

qqx

qx� {

qqy

qx, (9)

where q ¼ qx � {qy is the complex conjugate of q, and z ¼

xþ {y is the complex position. The derivative can also befound using

rq ¼qq

qð{yÞ¼ �

qqy

qy� {

qqx

qy(10)

because the path of the derivative is immaterial. SettingEqs. (9) and (10) equal yields

rq ¼qqx

qx� {

qqy

qx¼ �

qqy

qy� {

qqx

qy(11)


qqx

qxþ

qqy

qy

� �� {

qqy

qx�

qqx

qy

� �¼ ðr �~qÞ � {ðr �~qÞ ¼ 0

(12)

for steady, irrotational flow.The complex potential for steady, homogeneous flow can

be formed using

y ¼ fþ {c, (13)

where f and c are the fluid potential and streamfunctions, respectively. The derivative of the complexpotential is

dydz¼

qyqx¼

qfqxþ {

qcqx

(14)


dydz¼

qyq ð{yÞ

¼qcqy� {

qfqy

. (15)

Setting Eqs. (14) and (15) equal yields

dydz¼

qfqxþ {

qcqx¼

qcqy� {

qfqy

(16)

which yields the Cauchy–Riemann conditions:

qx ¼qfqx¼

qcqy

,

qy ¼qfqy¼ �

qcqx

ð17Þ

so that q ¼ dy=dz and r2y ¼ 0. While f in Eq. (3) issuitable for general analysis of groundwater flow, thisanalysis shows that one must utilize the complex discharge,q, instead of complex potential, f, for both confined andunconfined aquifers in order for the Cauchy–Riemannconditions to hold.

2.3. Cauchy integral equation

The Cauchy theorem states that if a function, f ðzÞ, isanalytic in a single connected region, O, then the integral off ðzÞ with respect to z along a simple closed contour, G 2 O,must vanish:I

Gf ðzÞdz ¼ 0. (18)

The analytic function can be defined using

f ðzÞ ¼yðzÞ

z� a. (19)

Alternatively, the function can be defined using theconjugate of the complex flux:

f ðzÞ ¼qðzÞ

z� a. (20)

Yet another definition uses the complex velocity,v ¼ ðqx þ {qyÞ=Z, where Z is the effective porosity of themedium:

f ðzÞ ¼vðzÞ

z� a. (21)

The salient feature of the alternative formulations is thedecomposition of the analytic function into a pair oforthogonal components.Point a in Eqs. (19)–(21) can be located either inside or

along the boundary of the domain. For the case wherePoint a is located on the boundary, G:

oðaÞ ¼1

b{

IG

oðzÞz� a

dz; a 2 G (22)

and for Point a located within the domain, O:

oðaÞ ¼1

2p{

IG

oðzÞz� a

dz; a 2 O (23)


where b is the inner angle at Point a made by tangentiallines before and after the point (b ¼ p for a smoothboundary at the point) and o can be taken as the complexpotential, y, complex flux, q ¼ dy=dz, or complex velocity,v ¼ q=Z.

2.4. CVBEM vs. BEM and AEM

A direct relationship exists between the traditional BEMand CVBEM. BEM uses Green’s second identity to find thevalue of potential within the flow domain [15,9]. We startby noting that

dz

z� a¼ d lnðz� aÞ ¼ d ln rþ {dl, (24)

where z� a ¼ re{l and a single-valued branch of ðp;�p� istaken for the argument of the complex logarithmicfunction. This means that Eq. (23), when combined withEq. (19), can be written as

yðaÞ ¼1

2p{

IGðfþ {cÞ � ðd ln rþ {dlÞ

¼1

2p{

IG

fd ln r� cqlqs

ds

� ��

þ{ cd ln rþ fqlqs

ds

� ��, ð25Þ

where n and s are the outwardly directed normal and thecounterclockwise oriented tangent vectors to the boundary,respectively. Noting that q ln r=qn ¼ ql=qs and performingan integration by parts yields

yðaÞ ¼1

2p

IG

fq ln r

qn� ln r

qcqs

� �ds,

þ{

2p

IG

cq ln r

qnþ ln r

qfqs

� �ds ð26Þ

which can be separated to yield

fðaÞ ¼1

2p

IG

fq ln r

qn� ln r

qcqs

� �ds,

cðaÞ ¼1

2p

IG

cq ln r

qnþ ln r

qfqs

� �ds. ð27Þ

These equations indicate that the real and imaginarycomponents of potential at Point a can be calculatedfrom the sum of the double-layer logarithmic potential(with densities f=2p, c=2p) and the single-layerlogarithmic potential (with densities ðqc=qsÞ=2p,�ðqf=qsÞ=2p) over the contour G. This assumesthat the complex potential along the inside of the bou-ndary equals the jump across the boundary, sothat the complex potential vanishes outside theboundary.

Finally, we obtain the conventional real-value boundaryelement equations by using the Cauchy–Riemann

conditions, qf=qn ¼ qc=qs and qf=qs ¼ �qc=qn:

fðaÞ ¼1

2p

IG

fq ln r

qn� ln r

qfqn

� �ds,

cðaÞ ¼1

2p

IG

cq ln r

qn� ln r

qcqn

� �ds ð28Þ

which are the primal solution that solves the divergenceproblem, r �~q ¼ 0, and the dual solution for the curlproblem, r �~q ¼ 0, respectively [7]. Note that thisformulation can also be used to find the complex flux,qðzÞ, and velocity, vðzÞ, using appropriate boundaryconditions.The analytic element method (AEM) can also be

compared to the CVBEM. AEM uses the superpositionprinciple to combine analytic solutions for individualpoints, lines, planes, etc. to construct the response withina domain to the combination of these geometries [16–18].Both the BEM and CVBEM normally place the modeled

domain within the boundary. Yet BEM can also be usedwhen the modeled domain is external to the boundary [9].For CVBEM, we can move a point source withinan infinite domain to the inside of the domain by insertinga cutting line that starts at the point source, extendsto infinity, and surrounds the domain. The distanceto the boundary is infinite except for the point source.This yields:

yðzÞ ¼ �Q

2p

IGdðz0Þd ln z, (29)

where Q is the total flow from the point source and dðz0Þ isthe Dirac delta function. This equation is equivalent to theAEM solution for a point source:

yðzÞ ¼ y0 �Q

2pln z=z0. (30)

3. Numerical implementation

3.1. Boundary value specification

Using the Cauchy integral equation to solve a boundaryvalue problem requires the specification of the boundaryconditions, wðaÞ where a 2 G. AEM is appropriate whenthese boundary conditions can be defined analytically. Forapplications where simple boundary conditions are notavailable, then the flow boundary is discretized in spaceand then constant or interpolated functions are used toapproximate the boundary conditions along each bound-ary segment.In this study, the boundary is approximated using

straight-line segments joined at their ends. Twokinds of boundary-value distributions along the elementare considered—constant and linearly varying boundaryvalues.


For constant boundary values along each element,Eq. (22), becomes

oðziÞ ¼1

p{

Xn

j¼1jai

oj lnzjþ1 � zi

zj � zi

(31)

because b ¼ p for straight-line segments with constantboundary values when zi is defined at the centers ofsegments. Also, i ¼ 1; 2; . . . ; n, o is the complex function atzi, oj is the constant complex function at note zj in elementj, zj and zjþ1 are the complex positions of the endpoints ofelement j, and n is the total number of the boundaryelements.

In a similar manner, the solution within the domain,Eq. (23), becomes

oðziÞ ¼1

2p{

Xn

j¼1

oj lnzjþ1 � zi

zj � zi

. (32)

Separating the real and imaginary parts from Eq. (31), oneobtains 2n equations expressed in terms of nodal values ofo as

RðoðziÞÞ ¼1

p

Xn

j¼1jai

½�bijRðojÞ þ aijIðojÞ�,

IðoðziÞÞ ¼ �1

p

Xn

j¼1jai

½aijRðojÞ þ bijIðojÞ� ð33Þ

in which

aij ¼ lnzjþ1 � zi

zj � zi

��,

bij ¼ � argzjþ1 � zi

zj � zi

� �, ð34Þ

where aii ¼ �P

iaj aij, bii ¼ �P

iaj bij. This can be writtenin matrix form using

A B

�B A

� �ð2n� 2nÞ

RðoÞ

IðoÞ

" #

ð2n� 1Þ

¼

0

0

� �ð2n� 1Þ

. (35)

The complex function for a linear interpolation ofboundary conditions is

oðzÞ ¼ N1ðzÞoj þN2ðzÞojþ1 (36)

for z 2 ½zj ; zjþ1�, and where

N1ðzÞ ¼zjþ1 � z

zjþ1 � zj

,

N2ðzÞ ¼z� zj

zjþ1 � zj

.

After insertion of Eq. (36) into Eq. (22) and integration,we obtain

oðziÞ ¼1

{bi � ln jðziþ1 � ziÞ=ðzi�1 � ziÞj

�Xn

j¼1jai

jai�1

½N1ðziÞoj þN2ðziÞojþ1� lnzjþ1 � zi

zj � zi

� �, ð37Þ

where i ¼ 1; 2; . . . ; n, and the other notation has the samemeaning as before. Eq. (37) can be rewritten as

Xn

j¼1

½aij � { bij�oj ¼ 0, (38)

where i ¼ 1; 2; . . . ; n and

aij ¼ RXGe3zj

ZGe

NjðzÞ

z� zj

dz

24

35,

bij ¼ � IXGe3zj

ZGe

NjðzÞ

z� zj

dz

24

35þ dijbi

in which dij is the Kronecker delta and the integration isperformed for the Cauchy principal value when i ¼ j. Thesingle-valued branch of the argument of complex logarith-mic function is taken in the same manner as in the constantdistribution case. Separating the real and imaginary partsyields a set of equations which are identical to Eq. (35) inshape, but have different matrix entries because of thedifference in interpolation functions.

3.2. Ordinary least squares

It can be observed that Eq. (35) results in a system of 2n

equations, regardless of which kind of boundary valueapproximation method we use. In addition, both methodscontain 2n nodal variables, with n variables for the realpart and n variables for the imaginary part. Theseequations usually require that a single boundary condition(e.g., either fluid potential, f, or stream function, c) beprescribed at every boundary node.By moving the data with known boundary conditions

(and their coefficients) to the right-hand side of theequation, and retaining the unknown nodal variables andcoefficients on the left-hand side, we obtain

Cu

RðoÞ

IðoÞ

" #u

¼ �Ck

RðoÞ

IðoÞ

" #k

, (39)

where the subscripts u and k refer to unknown and knownboundary nodal data, respectively. The unknown matrixhas the dimension ð2n�mÞ, where mo2n.The system of linear equations presented as Eq. (39) is

algebraically overdetermined; i.e., there are more equationsthan unknowns. This results from the specification of either


the real or imaginary components of the flux at each node,thus reducing the number of unknowns.

For a well-posed problem containing known boundaryconditions on m boundary segments, there will generally be2m choices of the equation combinations and thus 2m

alternative solutions, each of which satisfies a subset of theoriginal equations.

The algebraic system of linear equations is commonlysolved using explicit, implicit or hybrid methods [11,19,20].The explicit and implicit methods selectively excluderelevant equations until the matrix is square, resulting indifferent approximate solutions depending upon whichequations were excluded. Although the hybrid methodprovides a minor improvement in solution accuracy, themethod requires a substantial increase in computationalload.

OLS is useful for solving overdetermined systems of[12,21]. The goal is to minimize the global approximationerror; i.e., the L2 norm of the residual error vector isminimized when the solution vector is substituted into thesystem of linear equations. Jankovic and Barnes [22] use asimilar approach to fit higher-order line elements toimprove the accuracy.

Employing the OLS method leads to [12]:

ou ¼ �½CTu Cu�

�1½CTu Ck�ok, (40)

where the superscript, T, refers to the transpose of theoriginal matrix, and the exponent, �1, refers to the inverseof the matrix. Eq. (40) is our working equation system forthe solution vector. The coefficient matrix of Eq. (40) issymmetric, positive definite and diagonally dominant,which simplifies its solution [12,13].

3.3. Flownet mapping

Defining the flownet requires finding the position ofstreamline–equipotential intersections. Our approach is totransform the complex potential problem in the physicalplane, yðzÞ, into a complex position problem in thepotential plane, zðyÞ. Note that the differentiation rule ona compound function f ½fðx; yÞ;cðx; yÞ�, leads to

qfqx

qfqx

24

35 ¼ qf

qxqcqx

qfqy

qcqy

24

35 qf

qf

qfqc

24

35 (41)

or

qfqf

qfqc

24

35 ¼ 1

Jb

qcqy

�qcqx

�qfqy

qfqx

24

35 qf

qx

qfqy

24

35, (42)

where Jb is the Jacobian:

Jb ¼ det

qfqx

qcqx

qfqy

qcqy

24

35 ¼ q2

x þ q2y ¼ kqk

2. (43)

By applying the differentiation rule to x and y, we obtain

qxqf

qxqc

24

35 ¼ 1

kqk2

qfqy

�qcqy

264

375,

qyqf

qyqc

264

375 ¼ 1

kqk2

�qcqx

qfqx

24

35. ð44Þ

If the fluid flux can be determined (i.e., finite and nonzero),we have

qx

qf¼

qy

qc¼

qx

kqk2,

qy

qf¼ �

qx

qc¼

qy

kqk2ð45Þ

and

q2x

qf2þ

q2x

qc2¼ 0,

q2y

qf2þ

q2y

qc2¼ 0 ð46Þ

or, in complex notation:

q2z

qf2þ

q2z

qc2¼ 0. (47)

Eq. (47) implies that the position of the complexpotential can be defined in the potential plane, and isthe starting point for the inverse mapping procedure.This equation can be used to solve for the position offlownet intersections within the flow domain once all ofthe complex potential boundary conditions have beendetermined,As noted above, the complex potential at any point

within the flow domain is found using

oðzÞ ¼1

2p{

Xn

k¼1

ok

zkþ1 � z

zkþ1 � zk

� ��

�okþ1zk � z

zkþ1 � zk

� ��ln

zkþ1 � z

zk � z

� �. ð48Þ

From Eq. (47), it is clear that the position of a target fluidpotential–stream function intersection, zðytÞ, can be foundby term-by-term swapping of the position with thepotential in the Cauchy integral:

zðytÞ ¼1

2p{

IG

zðyÞy� yt

do (49)

which is identical to Eq. (23) except that thecomplex potential and position have been exchanged.Upon discretization with linear interpolation along linear


boundary elements, this becomes

zðoÞ ¼1

2p{

Xn

k¼1

zk

okþ1 � ookþ1 � ok

� ��

�zkþ1ok � o

okþ1 � ok

� ��ln

okþ1 � ook � o

� �. ð50Þ

For cases when the transformed domains in the complexpotential plane include a branch cut (i.e., complex positionsoverlap in the complex potential domain), a prediction–correction procedure can be employed to locate branchcuts (i.e., ground-water divides) that isolate regions of one-to-one correspondence. The procedure can be applied ineither an upstream or downstream direction. This ap-proach is especially useful when trying to determine theextent of capture zones [23].

Note that the inverse mapping procedure can be used forany of the functions specified in Eqs. (19)–(21), i.e.,o ¼ ½y; q; v�.

3.4. Flux and velocity

We present two approaches for solving flux and velocityproblems. The first is to ignore the potential problem andrestrict our solution to flux or velocity, only. To do this wedirectly solve the Cauchy Integral explicitly for flux orvelocity using Eq. (20) or (21), respectively, with theircorresponding boundary conditions.

qðaÞ ¼1

2p{

IG

q

z� adz a 2 O, (51)

vðaÞ ¼1

2p{

IG

v

z� adz a 2 O. (52)

Eqs. (51) and (52) provide direct estimates of the flux andvelocity at any point within the domain, O. Thisapproach is suitable when the fluid potential is notrelevant to the problem, such as when all of theboundary conditions are specified in terms of flux orvelocity. This method takes advantage of the fact that both

Fig. 1. Complex flux problem geometry. The flux vector, q ¼ qx þ {qy, can

be separated into normal, qn, and tangential, qs, components along the

boundary, so that q ¼ qn þ {qs.

flux and velocity can be separated into orthogonalcomponents, allowing them to serve as the primaryvariable in the Cauchy integral.The boundary conditions for this formulation are

specified using the normal and tangential components offlux, q ¼ qn þ {qs, or velocity, v ¼ vn þ {vs, shown in Fig. 1.The boundary normal vector, ~n, is pointed outward fromthe boundary, and the boundary tangential vector, ~s ¼ {~n,is pointed counterclockwise along the boundary, keepingthe domain to the left of the boundary.The use of conventional boundary conditions—i.e., the

specified potential (Dirichlet) and specified flux(Neumann)—implies a specification of the boundaryflux. That is, the first type implies the specificationof the tangential component of the complex flux, whilethe second type implies the specification of the normalcomponent of the complex flux [23]. For example, qs ¼ 0results from the specification of a constant potential alonga boundary, qf=qs ¼ 0, and qn ¼ 0 results from thespecification of an impermeable boundary condition,qf=qn ¼ 0.A second method for determining the flux and velocity

within the flow domain relies on the fact that thecomplex potential is continuously differentiablewithin the flow domain, so that a continuous velocityfield can be defined within the flow domain usingcomplex boundary potentials [24]. Eq. (23) has alreadybeen used to define the complex potential within the flowdomain:

yðaÞ ¼1

2p{

IG

yz� a

dz; a 2 O. (53)

The flux is the derivative of Eq. (53) with respect toposition:

qðaÞ ¼dydz

��a

¼1

2p{

IG

y

ðz� aÞ2dz; a 2 O (54)

which can be approximated using

q ¼1

2p{

Xn

k¼1

gkyk, (55)

where linear interpolation of y along G is used, and:

gk ¼1

zk � zk�1

� �ln

zk � z

zk�1 � z

� �

�1

zkþ1 � zk

� �ln

zkþ1 � z

zk � z

� �. ð56Þ

The fluid velocity at any point can be found by dividing theflux by the effective porosity, v ¼ q=Z. The resulting fluxand velocity fields are continuous because (a) the right-hand side terms of Eq. (56) include elementary functionsonly, (b) the points zj ; zjþ1; etc., are located on theboundary, and (c) the point zi is always located inside thedomain.Eq. (55) can be used once all the complex boundary

conditions have been specified. If some of the boundary


conditions are unknown, then they must first be obtainedbefore Eq. (55) can be used to determine the complex fluxat locations internal to the flow domain. It is clear thatspecification of the flow problem using Eq. (55) results in aunique analytic solution of the complex flux and velocitythroughout the flow domain.

3.5. Travel times

The travel time, t, of a fluid particle moving from Point a

to Point b along a streamline is the path integral of theinverse velocity [13]:

t ¼Z b

a

ds

v, (57)

where s is the distance along the streamline and v ¼

vx þ { vy ¼ q=Z is the fluid velocity, and where q is the fluidflux and Z is the effective porosity. Also note that q ¼ rywhere y ¼ fþ {c, so that

t ¼1

Z

Z fb

fa

dfkvk2

(58)

because qc=qs ¼ 0 along a streamline, and kvk2 ¼ v v.We can also write the travel time as

t ¼ ZZ fb

fa

kuk2 df, (59)

where

u ¼dz

dy¼

qx

qfþ {

qy

qf¼

q

Jb

¼1

q. (60)

We now have an integral equation in the complex potentialplane that provides the travel time of a particle along astreamline between two potentials. The integral can readilybe solved by first transforming the interval from naturalvalues, f 2 ½fa;fb� to x 2 ½�1; 1�:

f ¼1� x2

fa þ1þ x2

fb (61)

so that

t ¼ ZDf2

Z 1

�1

kuk2 dx, (62)

where Df ¼ fb � fa. Quadrature leads to

t ¼ ZDf2

Xj

Gjkuk2j , (63)

where j indicates the jth quadrature point, Gj and kuk2j are

the weighting factor and integrand values at point xj. Thevalues of c and Df are normally prescribed while thevalues of u are calculated using the complex potentialanalog to the complex position derivative:

uðyÞ ¼1

2p{

Xn

k¼1

f kzk, (64)

where

f k ¼ln½ðyk � yÞ=ðyk�1 � yÞ�

yk � yk�1�

ln½ðykþ1 � yÞ=ðyk � yÞ�ykþ1 � yk

.

(65)

4. Applications

Three analytic solutions are used to demonstrate thestrengths and weaknesses of the modeling approachpresented here. These solutions are variations of a solutionpresented by Toth in 1963 [25], which describes regionalgroundwater flow in a rectangular domain. The solution isappropriate for steady, two-dimensional (profile), satu-rated, homogeneous, isotropic flow bounded above by awater table and bounded on the remaining three sides byimpermeable surfaces.The Toth solution is appropriate for a linear trend

plus a sinusoidal variation in water-table elevation.Two additional solutions for this problem are alsoavailable. Domenico and Palciauskas [26] and Nawalany[27] provide additional numerical solutions for slightlydifferent water-table boundary conditions. These solutionsare used to provide the complex potentials and fluxeswithin the flow domain for comparison with the CVBEMsolution.The problem geometry consists of a domain two

horizontal units in distance and a unit thickness. A totalof 60 nodes at equal spacing are distributed alongthe domain boundary. The problem geometry is shown inFig. 2. A representative water-table boundary condition isalso shown. Note that flow between the water-table and therectangular flow domain is not considered.

4.1. Toth’s problem

In 1963, Toth [25] presented a solution for regional, two-dimensional, ðx; yÞ, groundwater flow in an unconfined,homogeneous, and isotropic aquifer. A symmetrical land-scape is assumed, with no-flow boundaries on the verticalsides and horizontal bottom of the domain. x0 and y0 arethe horizontal and vertical extent of the flow domain,respectively. The upper, horizontal surface is specifiedusing

fðx; yÞ ¼ y0 þ cxþ a sin bx, (66)

where c ¼ tan a, a ¼ a0= cos a, and b ¼ b0= cos a, and wherea, is the slope of the regional water table perpendicular tothe valley bottom, and a0 and b0 are the amplitude andfrequency of water-table fluctuations about the regionalslope.Toth’s analytic solution for this problem is [25]

fðx; yÞ ¼ y0 þcx0

2þ

a

bx0ð1� cos bx0Þ

þX1j¼1

b cos ax cosh ay, ð67Þ

ARTICLE IN PRESS

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1 41

11 31

Distance from flood plain

Ele

vatio

n

Fig. 2. Problem domain geometry. Horizontal extent is two units. Vertical extent is one unit. Nodal locations indicated using open circles (�). Sides and

lower boundaries are impermeable. Top boundary condition is specified using a function that changes for each of the three examples (the Toth problem

boundary condition is illustrated).

Potential Contours Analytic

Flux Contours Analytic

(a)

(b)

Fig. 3. Toth analytic solutions for (a) complex potential, y ¼ fþ {c and

(b) complex flux, q ¼ qx þ {qy. Solid and dotted lines show constant values

for the real and imaginary components, respectively.

T.C. Rasmussen, G. Yu / Engineering Analysis with Boundary Elements 30 (2006) 1030–10441038

where a ¼ jp=x0 and

b ¼2

x0 cosh ay0

abð1� cos bx0 cos jpÞ

b2� a2

�

þcðcos jp� 1Þ

a2

�. ð68Þ

The flux vectors are obtained by differentiating withrespect to x and y, respectively:

qx ¼qfqx¼ �

X1j¼1

ab sin ax cosh ay, (69)

qy ¼qfqy¼X1j¼1

ab cos ax sinh ay (70)

and the stream function is readily found to be

cðx; yÞ ¼ c0 �X1j¼1

b sin ax sinh ay (71)

which was obtained using the Cauchy–Riemann condi-tions.

A linear interpolator is used for boundary conditionsbetween the 60 nodes. The first simulation uses prescribedhead boundary conditions, fðxbÞ, on the upper surface andprescribed stream function boundary conditions, c ¼ 0, onthe sides and bottom. The second simulation uses prescribedtangential flux, qsðxbÞ, along the upper boundary, andprescribed normal flux, qn ¼ 0, along the sides and bottom.

ARTICLE IN PRESS

Potential Contours Numeric Potential Error

0.00

02

0.0002

0.0002

Flux Potential Boundary Conditions Flux Error

0.0002

0.0002

0.00020. 0005

0 .00 0 5

0.0005

0 .00050.00 1

0.00

1

0 .001

0 001

0.0

02

0.0

02

0.005

Flux Flux Boundary Conditions Flux Error0.0005

0.001

0.001

0.002

0.002

(a)

(b)

(c) (f )

(e)

(d)

Fig. 4. Toth numeric solutions for (a) complex potential, y ¼ fþ {c, (b) complex flux, q ¼ qx þ {qy, calculated using boundary potentials, yb,

and (c) complex flux calculated using boundary fluxes, qb. Contours of error norms between the analytic and numeric solutions are shown in (d)–(f).

1 1.1 1.2 1.3 1.4 1.50.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

φ

ψ

Position Contours Numeric

1 1 1 3 1 41

42

44

46

48

50

5254

56

58

60

Fig. 5. Contour lines of constant complex position, z ¼ xþ {y, within the complex potential domain, y ¼ fþ {c. Lines of constant horizontal distance, x,

and elevation, y, are shown using solid and dotted lines, respectively. Flownet positions can be found using constant increments in complex potential,

Df and Dc, respectively.

T.C. Rasmussen, G. Yu / Engineering Analysis with Boundary Elements 30 (2006) 1030–1044 1039


Figs. 3 and 4 present the analytic and calculatedcomplex potentials and fluxes for this problem. Fig. 4aillustrates the resulting flownet, which are just lines ofequal complex potential. Fig. 4d shows the magnitude ofthe errors, e ¼ jy� yj, between the analytic and numericsolutions.

To illustrate how the complex position of equi-potential–streamline intersections are obtained, a plot ofthe complex positions as a function of complex po-tential calculated using Eq. (50) is presented as in Fig. 5.Lines of constant horizontal and vertical extent are shownusing the solid and dotted lines, respectively. Travel timescan be obtained using Eq. (63) along each streamline,c ¼ ci.

Horizontal (solid lines) and vertical (dashed lines) fluxcontours are presented for prescribed complex potentialboundary conditions in Fig. 4b. Fig. 4e presents themagnitude of the calculated flux error, e ¼ jq� qj. Notethat the largest errors occur near the boundary of the flowdomain.

Calculated flux contours obtained using complex fluxboundary conditions are shown in Fig. 4c, and thecorresponding errors are shown in Fig. 4f. Note theimproved estimate in flux (i.e., smaller errors) nearthe boundary. This reduction in overall error near the

Potential Cont

11

Flux Contou

11

1 4

1 4

(a)

(b)

Fig. 6. Domenico and Palciauskas analytic solutions for (a) complex potential,

constant values for the real and imaginary components, respectively.

boundary is offset by slightly poorer velocity estimateswithin the interior of the domain.

4.2. Domenico and Palciauskas solution

Domenico and Palciauskas [26] present a simplifiedboundary condition that uses a cosine variation ofhydraulic head prescribed along the top boundary:

f ¼ cospx

2(72)

and no flow on the remainder of the boundaries, and wherex0 ¼ 2. These conditions are equivalent to:

our

rs

y ¼

Top boundary. Nodes 41 to 1: qx ¼ �p=2 sinðpx=2Þ.Left boundary. Nodes 1 to 11: qx ¼ 0.Lower boundary. Nodes 11 to 31: qy ¼ 0.Right boundary. Nodes 31 to 41: qx ¼ 0

where a unit hydraulic conductivity is assumed. Theanalytic solution for this problem is [26]

qx ¼ �p2sin

px

2cosh

py

2� tanh

p2

sinhpy

2

h i,

qy ¼p2cos

px

2sinh

py

2� tanh

p2

coshpy

2

h ið73Þ

s Analytic

31

Analytic

31

1

1

fþ {c, and (b) complex flux, q ¼ qx þ {qy. Solid and dotted lines show

ARTICLE IN PRESS


0.0002

0.0004

0.0006

0.0008

0.0008

0.0010.0010.0012

0.0014


0.0005

0.0005

0.001

0.0010.001

0.0010.001 5

0.0015 0.0015

0 .00150.0 02

0.002 0.002

0.00 20.0 03

0.0030.003

0.003

Flux Flux Boundary Conditions Flux Error

0.001

0.00

1

0 .0 0 1 5 0.0015

0.0020.0020.0025

(a)

(b)

(c)

(d)

(e)

(f)

Fig. 7. Domenico and Palciauskas numeric solutions for (a) complex potential, y ¼ fþ {c, (b) complex flux, q ¼ qx þ {qy, calculated using boundary

potentials, yb, and (c) complex flux calculated using boundary fluxes, qb. Contours of error norms between the analytic and numeric solutions are shown in

(d)–(f).


and the fluid potential and stream functions are

f ¼ cospx

2cosh

py

2� tanh

p2

sinhpy

2

h i,

c ¼ � sinpx

2sinh

py

2� tanh

p2

coshpy

2

h i. ð74Þ

The analytic solution for both the complex potential andflux is shown in Fig. 6. The numerical solution is obtainedby again using linear interpolation between boundarynodes. The results, shown in Fig. 7, are virtually identicalto the analytic solution. Note that the velocity field errorsare practically identical, but the errors near the boundaryare again greater when complex potentials are used as theboundary conditions.

Solutions using the finite-element method based on aTVR, Zijl [5] used 40 and 20 equidistant intervals in the x-and y-directions, respectively, for the discretization of thedomain and obtained similar results. The clear advantageof the boundary integral method lies in the simplicity ofdiscretization, which is limited to a single dimension alongthe boundary.

4.3. Nawalany solution

Another simplification of the Toth problem was devel-oped by Nawalany [27]. In this problem a sinusoidal head

is superimposed on a linear trend along the topboundary:

f ¼ �c x�x0

2psin

2px

x0

� �(75)

and the rest of the boundary conditions are identical. Thisis equivalent to:

Top boundary. Nodes 41 to 1: qx ¼ �cð1� cos 2px=x0Þ.Left boundary. Nodes 1 to 11: qx ¼ 0.Lower boundary. Nodes 11 to 31: qy ¼ 0.Right boundary. Nodes 31 to 41: qx ¼ 0.The analytic solution for this problem is

qx ¼X1m¼1

ab sin ðaxÞ cosh ð�aðy0 � yÞÞ, (76)

qy ¼ �X1m¼1

ab cos ðaxÞ sinh ð�aðy0 � yÞÞ, (77)

where a ¼ mp=x0, and

b ¼8cx0ð1� cos mpÞ

ðmpÞ2ðm2 � 4Þ coshð�ay0Þ(78)

when m is odd and b ¼ 0 when m is even.

ARTICLE IN PRESS

Potential Contours Analytic

11 31

Flux Contours Analytic

1 4 1

1 4 1

11 31

(a)

(b)

Fig. 8. Nawalany analytic solutions for (a) complex potential, y ¼ fþ {c,and (b) complex flux, q ¼ qx þ {qy. Solid and dotted lines show constant

values for the real and imaginary components, respectively.

T.C. Rasmussen, G. Yu / Engineering Analysis with Boundary Elements 30 (2006) 1030–10441042

The equipotentials and streamlines can be obtainedusing integration:

f ¼ f0 �X1m¼1

b cos ðaxÞ cosh ð�aðy0 � yÞÞ, (79)

c ¼ c0 þX1m¼1

b sin ðaxÞ sinh ð�aðy0 � yÞÞ. (80)

For our problem, we set c ¼ 0:1, x0 ¼ 1, and y0 ¼ 1, andagain use 60 elements along the perimeter. Results areshown in Figs. 8 and 9 for the analytic and simulationresults. Note that the results are consistent with both theToth and Nawalany solutions.

Table 1 provides a comparison between the methodpresented here with the analytic solution [26], TVR, andconventional FEM solutions using 88 elements [27]. It isclear that our method is superior to both the FEM andTVR methods, and requires less discretization effort.

5. Discussion and conclusions

Internal discretization of the flow domain can be avoidedfor conditions of steady flow through homogeneous mediausing real-variable BEMs, complex-variable boundaryelement methods (CVBEMs), and analytic element meth-ods (AEMs). The advantage of AEM and CVBEM lies in

their ability to determine both streamlines and velocities atpoints within the flow domain with excellent accuracy andcontinuity. This capability lays the foundation upon whichcontaminant transport problems can be addressed.CVBEM uses the Cauchy integral along with boundary

elements to convert a domain problem into a boundaryproblem, which simplifies the solution of domain-typeproblems by reducing the discretization and computationaleffort. CVBEM is more efficient and accurate than BEMbecause of the reduced number of equations requiringsolution.By reformulating the problem in the complex potential

plane, the same code can be used to compute the complexpositions of specified complex potentials for appropriateboundary conditions. This provides us with the positions ofthe flownet (equipotential–streamline) intersections withgreater accuracy and efficiency.Travel times can be accurately determined using

potential increments in the complex potential plane, thusavoiding the need to perform particle tracking using time-stepping algorithms.CVBEM can also be used with complex flux boundary

conditions to continuously define the complex flux withinthe flow domain. This formulation couples the conserva-tion equation, r �~q ¼ 0, with the irrotational assumption,r �~q ¼ 0, which results from specifying both the normaland tangential fluxes. Most specified head and fluxboundary conditions can be readily transformed intocomplex velocity boundary conditions. The complex fluxboundary conditions allows specification both of normalfluxes, as well as flux that is tangent to the boundary. Thus,boundary fluxes oriented at any angle to the boundary canbe considered.By using the complex discharge or velocity as the basic

unknown, CVBEM provides a more accurate flow fieldthan most domain methods because the fluxes or velocitiesare continuous and differentiable functions throughout theflow domain. This results in strict mass conservation andthe ability to avoid the unintentional loss/gain of mass dueto discretization and numerical approximation.The complex approach results in two possible boundary

conditions for every node—either f or c for complexpotential boundary conditions, or qn or qs for complex fluxboundary conditions. Because more than one boundaryvalue can be applied to each node, the problem becomesoverdetermined. Ordinary least squares (CVBEM-OLS)allows for such conditions, improving the accuracy of thesolution. CVBEM-OLS yields a minimum global approx-imation error, providing a more accurate answer. This is incontrast to other methods, such as the TVR scheme, whichmust employ auxiliary boundary conditions to make theproblem well-posed.Three examples related to Toth’s regional flow geometry

are presented. The first uses the original Toth solution,along with two additional solutions proposed by Domeni-co and Palciauskas, and by Nawalany. In both examples,the complex flux formulation provides a solution that is

ARTICLE IN PRESS


5e005

0.0001 0.0001

0.000150.0002

0.00025


0.0001

0.00

01

0.0001

0.000 1

0 .00 02

0.0002

0.000 2

0 .00020.0003

0.00030.000 3

0.00030. 0 00 5

0 .0005 0.0005

0.0005

Flux Flux Boundary Conditions Flux Error

0.0001

0.00

01

0.0002 0.0002

0.0003

0.0005

(a)

(b)

(c)

(d)

(e)

(f )

Fig. 9. Nawalany numeric solutions for (a) complex potential, y ¼ fþ {c, (b) complex flux, q ¼ qx þ {qy, calculated using boundary potentials, yb,

and (c) complex flux calculated using boundary fluxes, qb. Contours of error norms between the analytic and numeric solutions are shown in (d)–(f).

Table 1

Comparison of four solution methods using Nawanaly boundary conditions: Analytic: analytic solution [26], CVBEM : this study, TVR: transport velocity

representation [27], and FEM : finite-element method [27]

y Method x ¼ 0:25 x ¼ 0:50

qx qy qx qy

0.3 Analytic �0:04587 �0:04757 �0:06881 0.00000

CVBEM �0:04584 �0:04745 �0:06865 0.00000

TVR �0:04529 �0:04862 �0:06763 0.00000

FEM �0:04360 �0:05050 �0:06320 0.00000

0.5 Analytic �0:02577 �0:02405 �0:03705 0.00000

CVBEM �0:02573 �0:02401 �0:03698 0.00000

TVR �0:02539 �0:02447 �0:03646 0.00000

FEM �0:02440 �0:04757 �0:03440 0.00000

0.7 Analytic �0:01527 �0:01130 �0:02169 0.00000

CVBEM �0:01524 �0:01129 �0:02165 0.00000

TVR �0:01503 �0:01148 �0:02135 0.00000

FEM �0:01444 �0:01200 �0:02000 0.00000


accurate, while at the same time being easier to discretize.The use of flux boundary conditions improves the accuracyof velocity distributions, especially near the boundary,which is an important asset in two-dimensional contami-nant transport problems.

This method can also be generalized to include internalsources and sinks, which are first removed from theproblem domain, and then by adding cutting lines betweenthe external boundary and the internal boundary to makethe domains simply connected. Because the velocity


components must be single-valued—unlike stream func-tions, which can be multi-valued—the integral along thecutting lines cancel, providing a unique solution to theproblem.

Heterogeneous media can also be considered by deli-neating zones of homogeneous materials. Each zone islinked to neighboring zones along their interfacial bound-ary using flux boundary conditions, which are oriented inopposite directions for each boundary. A further potentialapplication includes transient flow cases, such as forincompressible fluids in non-deformable media, andperiodic boundary conditions [28,29]. Additional applica-tions to Dupuit flow in regional groundwater flowproblems are possible, as well as two-dimensional flow inopen channels.

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