Nonstationary stochastic analysis in well capture zone design using first-order Taylor's series...

Nonstationary stochastic analysis in well capture

zone design using first-order Taylor’s series

approximation

Mahmoud I. Bakr1 and Adrian P. Butler

Department of Civil and Environmental Engineering, Imperial College London, London, UK

Received 15 September 2004; revised 25 October 2004; accepted 3 November 2004; published 12 January 2005.

[1] Nonstationarity of flow fields due to pumping wells and its impact on advectivetransport is of particular interest in well capture zone design and wellhead protection.However, techniques based on Monte Carlo methods to characterize the associated capturezone uncertainty are time consuming and cumbersome. This paper introduces analternative approach. The mean and covariance of system state variables (i.e., head, porewater velocity, and particle trajectory) are approximated using a first-order Taylor’sseries with sensitivity coefficients estimated from the adjoint operator for a system ofdiscrete equations. The approach allows nonstationarity due to several sources (e.g.,transmissivity, pumping, boundary conditions) to be treated. By employing numericalsolution methods, it is able to handle irregular geometry, varying boundary conditions,complicated sink/source terms, and different covariance functions, all of which areimportant factors for real-world applications. A comparison of results for the Taylor’sseries approximation with those from Monte Carlo analysis showed, in general, goodagreement for most of the tested particles. Particle trajectory variance calculated usingTaylor’s series approximation is then used to predict well capture zone probabilities underthe assumption of normality of the mass transport’s state variables. Verification of thisassumption showed that not all particle trajectories (depending on their starting location)are normally or log-normally distributed. However, the risk of using the first-ordermethod to delineate the confidence interval of a well capture zone is minimal since itmarginally overestimates the 2.5% probability contour. Furthermore, this should bebalanced against its greater computation efficiency over the Monte Carlo approach.

Citation: Bakr, M. I., and A. P. Butler (2005), Nonstationary stochastic analysis in well capture zone design using first-order Taylor’s

series approximation, Water Resour. Res., 41, W01004, doi:10.1029/2004WR003648.

1. Introduction

[2] It has been widely recognized that heterogeneitysignificantly affects fluid flow and solute transport in thesubsurface. Of the aquifer properties that control the move-ment of fluid and contaminants in the subsurface, hydraulicconductivity (or transmissivity) is generally the most impor-tant, as its variability in space is considerably higher thanthose of the other properties (e.g., porosity). Hydraulicconductivity values can vary by orders of magnitude overa few meters [e.g., Freeze, 1975; Hoeksema and Kitanidis,1984]. During the last two decades many stochastic theorieshave been developed to relate the statistical moments ofhydraulic head, pore water velocity, and other flow and masstransport quantities to those of transmissivity and mean flowcharacteristics [e.g., Bakr et al., 1978; Gelhar and Axness,1983; Dagan, 1984; Winter et al., 1984; Graham andMcLaughlin, 1989a, 1989b; Rubin, 1990; Shvidler, 1993;

Osnes, 1995; Zhang and Neuman, 1996; Guadagnini andNeuman, 1999; Zhang, 2002].[3] In flow and mass transport, nonstationarity (with

location-dependent statistics) may occur due to nonhomo-geneous statistics of medium properties, complex boundaryconfigurations or source/sink terms. Previous studies ofnonstationary stochastic analysis in saturated/unsaturatedflow fields for a variety of conditions and assumptions,especially for cases of nonstationary hydraulic conduc-tivity and bounded domain, include Naff and Vecchia[1986], Rubin and Dagan [1988, 1989], Loaiciga et al.[1993], Rubin and Seong [1994], Li and McLaughlin [1995],Indelman and Rubin [1995], Osnes [1995], Zhang andWinter [1998], Zhang [1998], Guadagnini and Neuman[1999], Riva et al. [2001], and Tartakovsky et al. [2003].Most of these investigations use various moment equationapproaches. In many cases, the partial differential equationsgoverning the first two moments of flow and mass transportquantities are derived using perturbation techniques, andthen solved numerically (e.g., finite element or finitedifference).[4] In well capture zone design, nonstationary stochastic

analysis provides a method for calculating the probability

1Now at Water Resources Research Institute, National Water ResearchCenter, Cairo, Egypt.

Copyright 2005 by the American Geophysical Union.0043-1397/05/2004WR003648

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WATER RESOURCES RESEARCH, VOL. 41, W01004, doi:10.1029/2004WR003648, 2005

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that water at a particular location from a pumping well iscaptured within a specified time. In this case, pumping is animportant source of nonstationarity. Several authors havestudied the stochastic determination of time-related capturezones for unconditional, spatially stochastic transmissivityfields using the Monte Carlo approach. Varljen and Shafer[1991] have illustrated the effect of transmissivity condi-tioning on well capture zone design and its confidenceinterval. Franzetti and Guadagnini [1996] examined thevariance of a random transmissivity field on the determi-nation of stochastic well catchments. Guadagnini andFranzetti [1999] analyzed the effect of heterogeneity onthe delineation of time-related capture zone and developedan empirical expression for probabilistic capture zoneestimation in a heterogeneous (infinite) 2D aquifer.[5] van Leeuwen et al. [1998] investigated the influence

of both transmissivity variance and correlation length inboth fully confined and leaky aquifers through statisticalevaluation of unconditional Monte Carlo simulations. Theythen extended this approach [van Leeuwen et al., 2000] toexamine the effect on uncertainty of conditioning onregular grids of transmissivity measurements. Feyen etal. [2001] incorporated head observations in stochasticwell capture zone design by weighting unconditionaltransmissivity fields on these observations using General-ized Likelihood Uncertainty Estimation (GLUE). Feyen etal. [2002] further examined the effect of conditioning onregular grids of transmissivity measurements for stochasticwell capture zone design, accounting for uncertainty in thestatistical parameters (i.e., mean, variance and correlationlength) using a Bayesian approach. Feyen et al. [2003a]examined the effect of conditioning on transmissivity andweighting on head observation on well capture zoneuncertainty. Finally, they extended their approach to jointweighting on tracer arrival times, conductivity measure-ments, and hydraulic head observations [Feyen et al.,2003b]. More recently, Bakr and Butler [2004] examinedworth of head data in well capture zone design usinginverse modeling. All of these use Monte Carlo analysis topropagate uncertainty in hydraulic parameters, mainly trans-missivity, to well capture location. However, the computa-tional effort required in this approach can be high andbecomes even more cumbersome when conditioning trans-missivity realizations on state variable measurements (e.g.,head, pore water velocity, travel time), as these requiresolutions of the inverse groundwater flow problem [e.g.,McLaughlin and Townley, 1996]. These limitations (i.e.,high computational requirements and lack of convergence)of the Monte Carlo method have been discussed in theliterature [e.g., Ballio and Guadagnini, 2004; Guadagniniand Neuman, 1999].[6] As an alternative approach,KunstmannandKinzelbach

[2000] used a first-order second moment method to computewell capture zones and their associated uncertainty on thebasis of an Eulerian framework. In their analysis the secondmoment of solute concentration is combined with theKolmogorov backward equation to derive moment equa-tions of flow and transport mass fluxes. They consideredtransmissivity and recharge as the main sources of uncer-tainty and used a zonation method to describe the spatialvariability of these parameters. Unconditional and condi-tional simulation results were presented. However, the

mean and variance of concentration isolines did not showgood agreement with Monte Carlo results for the uncondi-tional case, in spite of the restriction of low variances forlog-transmissivity (0.16) and log-recharge (0.4).[7] Recently, Stauffer et al. [2002] reported an investi-

gation of the uncertainty in two-dimensional, steady statewell catchment boundaries due to spatially varying trans-missivity fields using a first-order approximation. In theirstudy, they used longitudinal and transverse particle dis-placement covariances along and normal to the meanparticle trajectory to approximate well catchment uncer-tainty. However, one of limitations in their work is that thenonstationary velocity covariances are approximated bylocally scaling the stationary velocity covariance derivedby Rubin [1990].[8] More recently, Lu and Zhang [2003] presented a

moment equation approach to derive time-dependent meancapture zones and their associated uncertainties. Flowstatistics were obtained by solving the first two momentsof flow, and mean capture zones determined by backtracking conservative particles released from a small circlearound the pumping well(s). They used a Lagrangianapproach developed for a general stochastic model fortransport of conservative solutes in variably saturated non-stationary flow in randomly heterogeneous porous media[Zhang and Lu, 2002] to derive capture zone uncertainty.They demonstrated their approach for the case of multiplewells and concluded that it showed good agreement withresults obtained from Monte Carlo analysis. To ensure thatthe Monte Carlo and moment equation transport approacheshave the same underlying flow statistics, they have used ahybrid Monte Carlo and moment equation approach where aMonte Carlo approximation for the nonstationary velocitycovariance is used in order to calculate the nonstationaryparticles trajectory covariance. In their recent work [Zhangand Lu, 2004] they compared both the full moment equationand the hybrid approach with a full Monte Carlo simula-tions approach. They showed that the three approaches are inexcellent agreement when the variability of hydraulic con-ductivity is small (s2 = 0.16). However, at a moderatevariability (s2 = 0.5), the hybrid and the full Monte Carloresults are in excellent agreement while the results from thefull moment equation approach deviate slightly from the fullMonte Carlo results. They concluded that the first-ordermoment equation is not sufficiently accurate at moderatevariability in the case of divergent/convergent flow. It shouldbe noted that in both works, they used analytically derivedspatial sensitivity coefficients of the pore water velocitywhich are crucial in the accuracy of the calculated capturezones. The analytical solution poses restrictions on theapplicability of their approach as it is derived for homoge-neous media with infinite extent and a mean uniformregional flow.[9] In this paper, we use an alternative numerical

approach for nonstationary stochastic analysis in wellcapture zone. Although nonstationarity due to boundaryconditions is implicitly included, our focus here is on thenonstationarity due to a pumping well.[10] Here, the original partial differential equation is first

discretized on a specified grid using finite elements and theresulting system of equations, the so-called space-timeequation, is used to derive the statistical moments of the

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W01004 BAKR AND BUTLER: NONSTATIONARY STOCHASTIC ANALYSIS IN ZONE DESIGN W01004

flow and mass transport quantities. The approach is basedon a Taylor’s series approximation of the discrete system ofequations and is often termed the vector space-state/adjointstate approach [e.g., Sykes et al., 1985; Zhang, 2002; Bakret al., 2003]. As will be shown later, the main task of thisapproach is to determine flow and transport sensitivitycoefficients.

2. Mathematical Formulation

2.1. Forward Modeling of Flow andAdvective Mass Transport

[11] A forward operator, which takes the form of a partialdifferential equation, is used to predict the response ofgroundwater systems to imposed stresses, initial and bound-ary conditions, and hydraulic parameters. Here, the flowpartial differential equation is solved numerically using theGalerkin (central) finite element method. To obtain thedistribution of pore water velocities, we examined twoapproaches, as will be shown later. In the first, velocitiesare obtained by direct differentiation of Darcy’s law, whilein the second velocities are obtained by applying theGalerkin finite element method to Darcy’s law. Applyingthe latter to the flow and continuity equations leads to asystem of linear equations that can be stated as

g s; Rð Þ ¼ 0; ð1Þ

where s is a vector of nodal values of state variables (i.e.,head and pore water velocity), and R is a vector of nodallog(natural)-transformed transmissivity values; details of(1) are given in Appendix A. Note that throughout thispaper we use bold small letters to indicate vectors andbold capital letters to indicate matrices, and regular fontsto indicate scalers. Here, we consider transmissivity as theonly uncertain parameter.[12] Solving (1) gives an estimation of nodal values of

state variables, in particular pore water velocity. These canthen be used in a solution of the advective mass transportproblem using particle tracking. This approach is based onthe Lagrangian point of view [e.g., Bear, 1972], in whichconcentration is associated not with fixed points or volumeelements in space, but rather with fluid elements or particleswhich move with the prevailing flow velocity. The pathlinesof contaminants under advection alone are governed by

dp

dt¼ u p; tð Þ; ð2Þ

where p is the position vector, and u is pore water (seepage)velocity vector which is function of particle position andtime. The solution of equation (2) for particle location at anytime t can be expressed by

p tð Þ ¼ p t0ð Þ þZ t

t0u p; tð Þdt; ð3Þ

where p(t0) is the position at time t0.[13] Solution to equation (1) can be used to numerically

integrate (2) using the fourth-order Runge-Kutta method[e.g., Zheng and Bennett, 1995]. This method evaluates thevelocity four times for each particle tracking step: once atthe initial point (at previous time step t), twice at two trial

midpoints (1 and 2), and once at a trial end point (3). Anaverage velocity is then calculated at any time step t + 1 as

utþ1xa

¼ utx þ 2utþ1x1

þ 2utþ1x2

þ utþ1x3

� �=6:0

utþ1ya

¼ uty þ 2utþ1y1

þ 2utþ1y2

þ utþ1y3

� �=6:0:

ð4Þ

The position of the particle at the beginning of the next step(xt+1, yt+1) is determined, such that

xtþ1 ¼ xt þ utþ1xa

Dt

ytþ1 ¼ yt þ utþ1ya

Dt:ð5Þ

2.2. First-Order Taylor’s Series Approximation

[14] The first-order Taylor’s series approximation[Dettinger and Wilson, 1981] is an approach used topropagate uncertainties in model’s parameters (independentvariables) to its state (dependent) variables. Usingequation (1), s can be stated as a function of R, s(R), suchthat its Taylor’s series expansion is given by

s Rð Þ ¼ s �Rð Þ þ ds �Rð ÞdR

R� �Rð Þ þ 1

2

d2s �Rð ÞdR2

R� �Rð Þ2þ . . . ; ð6Þ

where �R is a vector of the mean values of R. Taking theexpected value of (6), ignoring the second and higher-orderterms, we obtain

E s Rð Þ½ � ¼ �s Rð Þ ¼ s �Rð Þ; ð7Þ

where E[R � �R] = 0. This means that, for the level ofapproximation considered here by ignoring the second andthe higher-order terms in (6), vector of mean state variables,�s, can be obtained by substituting �R into s(R).[15] Covariance of s can similarly be obtained by taking

the expectation of (s � �s)(s � �s)T, which can be obtainedusing (6) ignoring the second and the higher-order terms,this leads to

E s Rð Þ � �s Rð Þð Þ s Rð Þ � �s Rð Þð ÞTh i

¼ ds �Rð ÞdR

� E R� �Rð Þ R� �Rð ÞTh i ds �Rð Þ

dR

� �T

;

ð8Þ

in which

E R� �Rð Þ R� �Rð ÞTh i

¼ Cr; ð9Þ

where CR is the covariance of R. Note that, so far we haven’tmade any assumptions considering the statistical propertiesof R. As it can be seen from (8) the main computational taskin calculating the state variable covariance is the calculationof the sensitivity coefficients of s with respect to R.

2.3. Stochastic Capture Zone Design Using First-OrderTaylor’s Series Approximation

[16] In this studywe use back particle tracking to determinethe effects of uncertainty on well capture zones. ApplyingTaylor’s series approximation, outlined in section 2.2, tothe forward model that describes particle movement ingroundwater flow, section 2.1, the mean and variance of a

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particle trajectory can be estimated. The mean position ofa particle, p, at any time, t + 1, can be estimated using

�xtþ1p ¼ �xtp þ �utþ1

xaDt

�ytþ1p ¼ �ytp þ �utþ1

yaDt;

ð10Þ

and the variance is given by

s2xtþ1p

¼dxtþ1

p �Rð ÞdR

Crdxtþ1

p �Rð ÞdR

!T

s2ytþ1p

¼dytþ1

p �Rð ÞdR

Crdytþ1

p �Rð ÞdR

!T

:

ð11Þ

In equation (10), �uxat+1 and �uya

t+1 are the mean pore watervelocity in x and y direction, respectively, that are evaluatedat �R using (4). Note that for sake of notation simplicity, thedependence parentheses are removed hereafter, unless theyare explicitly needed.[17] Equations (10) and (11) can be used to evaluate well

capture zone and its uncertainty. In equation (11) sensitivityof particle trajectory with respect to R is obtained bydifferentiating (5), which gives

dxtþ1

dR¼ dxt

dRþdutþ1

xa

dRDt

dytþ1

dR¼ dyt

dRþdutþ1

ya

dRDt:

ð12Þ

Total derivatives of uxat+1 and uya

t+1 are obtained by differen-tiating (4) with respect to R, such that

dutþ1xa

dR¼ 1

6

@utx@R

þ @utxdxt

dxt

@Rþ @utx

@ytdyt

dR

� �

þ 2

6

@utþ1x1

@Rþ@utþ1

x1

@xtþ11

dxtþ11

dRþ@utþ1

x1

@ytþ11

dytþ11

dR

!

þ 2

6

@utþ1x2

@Rþ@utþ1

x2

@xtþ12

dxtþ12

dRþ@utþ1

x2

@ytþ12

dytþ12

dR

!

þ 1

6

@utþ1x3

@Rþ@utþ1

x3

@x3

dxtþ13

dRþ@utþ1

x3

@ytþ13

dytþ13

dR

!

dutþ1ya

dR¼ 1

6

@uty@R

þ@uty@xt

dxt

dRþ@uty@yt

dyt

dR

� �ð13Þ

þ 2

6

@utþ1y1

@Rþ@utþ1

y1

@xtþ11

dxtþ11

dRþ@utþ1

y1

@ytþ11

dytþ11

dR

!

þ 2

6

@utþ1y2

@Rþ@utþ1

y2

@xtþ12

dxtþ12

dRþ@utþ1

y2

@ytþ12

dytþ12

dR

!

þ 1

6

@utþ1y3

@Rþ@utþ1

y3

@xtþ13

dxtþ13

dRþ@utþ1

y3

@ytþ13

dytþ13

dR

!:

Partial derivatives of pore water velocity with respect to itsx and y coordinate depend on how we calculate andinterpolate velocity within a finite element. In this study,as mentioned before, we present a comparison betweentwo methods of calculating pore water velocity. Both usebilinear interpolation to calculate velocity within a finiteelement, however the first uses pore water velocities

obtained from direct differentiation of Darcy’s law whereasthe second uses nodal velocities obtained from the centralfinite element solution.[18] According to the first approach, we obtain velocity

components by

uex ¼ � Te

deJe beTx he

uey ¼ � Te

deJe beTy he;

ð14Þ

where uxe and uy

e are velocity components in x and y direction(L/T), respectively, for element e; he is a vector of hydraulicheads for the nodes of element e; T e is the transmissivityvalue (L2/T) for element e; Je is the effective porosity (L0)for element e; de is the saturated thickness (L) for element e;bxe and by

e are the spatial derivatives of the shape function ne

in x and y direction, respectively, for element e. This methodleads to a discontinuous velocity field at the nodes and theboundaries of the elements.[19] In this case, sensitivity of pore water velocity with

respect to R is obtained by differentiating (14), such that

@uex@R

¼ � 1

deJe

@Te

@RbeTx he � Te

deJe beTx

@he

@R@uey@R

¼ � 1

deJe

@Te

@RbeTy he � Te

deJe beTy

@he

@R:

ð15Þ

Similarly, spatial derivatives can be obtained; for example,derivative of (14) with respect to x is given by

@uex@x

¼ � @

@x

Te

deJe

� �beTx he

@uey@x

¼ � @

@x

Te

deJe

� �beTy he � Te

deJe

@beTy@x

he:

ð16Þ

Here, for uniform parameters (T, d and J) and bilinear shapefunction, derivative of ux

e with respect to x is zero, whilederivative of uy

e with respect to x is nonzero.[20] In the second method we solve Darcy’s law by the

finite element method to obtain pore water velocity compo-nents at the nodes [e.g., Yeh, 1981]. This can then be used toobtain an estimation of the pore water velocity within afinite element using the element shape function ne, such that

uex ¼ neTuex

uey ¼ neTuey:ð17Þ

Here, it should be noted that uxe and ux

e, and uye and uy

e whereuxe and uy

e are scalers and give the pore water velocitycomponents at a point within element e. ux

e and uye are

vectors of sizes 4 1 and give the pore water velocitycomponents at the nodes of element e.[21] Equation (17) can be used to obtain derivatives ux

e

and uye with respect to R, such that

@uex@R

¼ neT@uex@R

@uey@R

¼ neT@uey@R

:

ð18Þ

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Similarly, spatial derivatives of uxe and uy

e can be obtained.For example, the derivatives with respect to x are given by

@uex@x

¼ @neT

@xuex

@uey@x

¼ @neT

@xuey:

ð19Þ

Unlike equation (16), the derivatives of ux with respect to xand y are both nonzero.[22] In equation (18), the sensitivity coefficients of pore

water velocity with respect to R are estimated using theadjoint method. For example, to calculate sensitivity of porewater velocity in the x direction at point x1, ux1, with respectto R, we assume a performance criteria equal to ux1.Sensitivity coefficients are then calculated using

@ux1@R

¼ � @ux1@s

@g

@s

��1 @g

@R: ð20Þ

Proof of this equation and derivation of sensitivity equationsfor flow and mass transport can be found in the work ofBakr et al. [2003] and in Appendix A.[23] Because it is not possible to back track a particle

located exactly at a pumping well, small radial distance(r > 0) is selected to place particles and back track them totheir original location. Reverse particle tracking has suc-cessfully been used before in time related well capture zonedelineation [e.g., Varljen and Shafer, 1991]. To compensatefor the rapid change in the pore water velocity downstreamof the well, more particles are located in this region thanupstream of it. This is to insure that the boundaries of thewell capture zone lateral to the regional direction areestimated to good accuracy, as these are delineated bythose particles downstream of the well.[24] After placing the particles around the pumping well,

they are back tracked in time to calculate the well capturezone, which is estimated by connecting the particles’ endpoints. Similar to Lu and Zhang [2003], we also expressparticle end points in polar coordinates using the radialdistance to the well (Rp

t+1) and the angle between this lineand the x axis (qp

t+1) such that

Rtþ1p ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffixtþ1p � xw

� �2þ ytþ1

p � yw

� �2rð21Þ

and

qtþ1p ¼ tan�1

ytþ1p � yw

xtþ1p � xw

!; ð22Þ

where xw and yw are coordinate of the pumping well incartesian coordinate system. For any elapsed time (t + 1),the curve that connects the endpoints of the meantrajectories (�Rp

t+1, �qpt+1) of all particles released around the

well is considered the position of the mean capture zone forthis well at the given time.[25] If Rp

t+1 is normally distributed, the capture zoneintervals at the 95% confidence level are constructed usingthe �Rp

t+1 plus and minus 1.96sRpt+1 along the direction

�qpt+1. �Rp

t+1 and �qpt+1 can be obtained, given the order of

approximation considered here, by substituting �xpt+1 and �yp

t+1

for xpt+1 and yp

t+1, respectively, in (21) and (22). To calculatesRp

t+1 we need the sensitivity coefficients of Rpt+1 with respect

to R, which can be obtained using

dRtþ1p

dR¼ 1

Rtþ1p

dxtþ1p

dRxtþ1p � xw

� �þdytþ1

p

dRytþ1p � yw

� � !: ð23Þ

Similarly, if Rpt+1 is log-normal, the capture zone intervals at

95% confidence level are constructed using the ln Rtþ1p

� �plus and minus 1.96 slnRp

t+1. In that case, following Lu andZhang [2003], mean and variance of ln(Rp

t+1) can be calcu-lated using their untransformed mean and variance,

ln Rtþ1p

� �¼ ln �Rtþ1

p

� �� 0:5s2

lnRtþ1p

ð24Þ

and

s2lnRtþ1

p¼ ln 1þ

s2Rtþ1p

�Rtþ1p

� �20B@

1CA; ð25Þ

respectively.

3. Numerical Example

[26] In this work, and to allow comparison with aprevious approach, we will consider the hypothetical prob-lem presented by Lu and Zhang [2003]. A two-dimensionalrectangular flow domain in a saturated heterogeneousporous medium has dimensions of 20 12[L] with spec-ified head boundary conditions east, 10[L], and west,10.5[L], and no-flow boundaries to the north and south.[27] To demonstrate the method outlined in this work we

consider an example that is presented before by Lu andZhang [2003]. This should facilitate, for the reader, com-paring the method presented here and the one developed byLu and Zhang [2003]. We consider a two-dimensionalrectangular domain in a saturated heterogeneous porousmedia. The flow domain has dimensions of 20 12[L]with a prespecified head boundary condition at east 10[L]and west 10.5[L], and no-flow boundary at the north andsouth. The domain is uniformly discretized using finiteelements of size 0.4 0.4[L] which gives 50(columns) 30(rows). Effective porosity is assumed a constant value of0.25[L0]. The statistics of the log(natural)-transmissivity aregiven as mean (�r) = 0.0 (i.e., the geometric mean of thetransmissivity is 1.0[L2/T]), variance (sr

2) = 0.5, correlationlength (lr) = 2.0[L], and an exponential covariance. Apumping well is located at (4.8[L], 6.0[L]) with a pumpingrate of q = 0.16[L3/T]. Figure 1 shows a key map of thestudied domain. Within this example we limit the trackingtime to 20[T] to ensure that during Monte Carlo simulationsall tracked particles remained within the flow domain.[28] For the purpose of validating sensitivity coefficients

and comparing the two (first-order and Monte Carlo)methods, we use 4 particles distributed around the pumpingwell at a distance equal to 0.4[L], and having angles 0�, 90�,


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135� and 170� with the background flow direction,particles 1, 2, 3 and 4, respectively. As can be seen fromequation (8), the covariance of the state variables (h, ux, uy,xp, yp) are a function of the covariance of the uncertainparameters (here log-transmissivity) and their sensitivity tothat uncertain parameter. Since the parameter’s covariance isassumed known a priori, the main computational demandsfor calculating the covariance of the state variable is thecalculation of its sensitivity coefficients with respect to thatparameter. In addition, the accuracy of the calculated statevariable’s covariance is dependent on the accuracy of thesesensitivity coefficients. In this study, we calculate thesecoefficients using the adjoint operator as outlined by Bakret al. [2003]. We have validated the sensitivity coefficientsof the system state variables (i.e., h, ux, uy, xp and yp) usingthe finite difference perturbation method. For a given node, i,ri is perturbed with a small step, Dri, and is used to calculateend point for a particle j, (xj

p, yjp). Sensitivity of end point of

particle j, xj, for example, can then be calculated using

dxj

dri¼

xpj � xj

Dri: ð26Þ

[29] Figure 2 shows sensitivity of end point of particlenumber 2 at time 20[T] with respect to r of the nodesalong a-a, see Figure 1. The figure shows a very goodagreement between the adjoint and the finite differenceperturbation methods. This verifies the sensitivity coeffi-cients for particle end points calculated using the adjointmethod. It also indicates that the sensitivity coefficients ofhead and pore water velocity are correct since they areboth used to calculate sensitivity of particle end pointsusing the adjoint method. Note that the finite differenceperturbation method is used here only to validate thesensitivity coefficients using the adjoint method. The finitedifference perturbation method is in general less accurateand more CPU demanding than the adjoint method.[30] Although accurate calculation of sensitivity coeffi-

cients is crucial to obtaining accurate estimation of cova-riances of state variable(s), it is not enough to guaranteecalculation of correct covariances. This is because of thefact that the first-order method is an approximate methodwhich may fail to estimate these covariances because of thenonlinearity of the problem. Nonlinearity occurs, for exam-ple, because of high variance of the uncertain parameter(s)and/or the presence source/sink term.

[31] In this section we try to validate the first-orderapproximation method (FO) using the Monte Carlo method(MC). The latter is a general method for uncertaintypropagation. It doesn’t require a specific correlation struc-ture or variance but it lacks sometimes convergence of itsresults. To guarantee convergence of the Monte Carloresults, a large number of realizations of the uncertainparameters are generally required. Here, we generated5000 realizations of log-transmissivity fields using thestatistics given earlier. Unconditional sequential Gaussiansimulation method is used to generate these realizationsusing GSTAT [Pebesma, 1999]. For each realizationwe run the forward flow and particle back tracking for the4 particles placed around the well for 20[T]. The particles’trajectories, xp and yp, are recorded for each realization attime interval 0.2[T] and used to calculate their mean andvariance at each time step.[32] Figure 3 shows �xp, ln(sxp

2 ), �yp and ln(syp2 ) versus time

for the four particles. The figure shows good agreementbetween the mean calculated by the FO and MC methods.The figure also shows that sx1

2 is moderately approximatedusing the first-order method, while the method fails toreproduce to the same degree sx2

2 and sx32 and is even less

successful in estimating sx42 . The results for sy

2 are similar,although in this case the method performs best for sy2

2 . Onceagain the poorest comparison occurs with particle 4.[33] In this analysis we obtained pore water velocity by

direct differentiation of Darcy’s law. The draw back of thismethod is that it doesn’t approximate well the spatialderivatives of ux and uy, which are needed to calculate thesensitivity of xp and yp with respect to R (equations (12) and(13)). This explains one of the factors that led to the poorapproximation of the variances of the four particles, espe-cially for particle 4 whose trajectory is through a highlyvariable velocity field. As an example of the problemsassociated with this method, @ux/@x and @uy/@y are calcu-lated to be zero close to the pumping well. The reason forthis is that the method uses bilinear basis functions. Ahigher-order basis function may give a better approximationof these derivatives at the expense of a more complicatedforward model for the flow problem. However, we can alsoobtain a better approximation of these derivatives by solv-

Figure 1. Diagram of model domain.

Figure 2. Sensitivity of particle 2 end point, x2 and y2, attime = 20[T].

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ing Darcy’s law using the central finite element method asdescribed in Appendix A.[34] Figure 4 shows the log-variance of xp and yp,

respectively, for the four examined particles. These tendto be an improvement over those in Figure 3, although theresults for particle 4 still show marked discrepancies. This isbecause its trajectory lies very close to the well stagnationpoint where the variability of the pore water velocity is at amaximum.[35] In order to investigate the effect of resolution of the

grid on the numerical solution, we solved the problem usinga grid spacing of 0.1[L], instead of 0.4[L]. It is expected thatthe finer mesh will provide a better approximation of thehead gradient and spatial derivatives of the pore watervelocity close to the well. Figure 5 shows the effect ofusing a finer mesh on the calculated log-variance of xp andyp for the four particles by solving Darcy’s law using thecentral finite element method. It is clear that the variance ofparticle 4 calculated using FO and MC methods areapproaching. It is important to note that the MC solutionis also dependent on the resolution of the finite elementmesh. This is especially true for the calculation of particlevariances. The conclusion here is that a finer mesh down-stream of the well, where the spatial variability of ux and uyis maximum, is essential for a successful application of theFO method in well capture zone.

[36] It should be noted that the solved example here is abounded aquifer. Previous work [Osnes, 1998] has shownthat the presence of boundaries results in velocity covari-ance, between any two points, being dependent on theirlocations rather than solely on their spatial separation,consequently both fluid velocity and particle trajectory arenonstationarity. Although the combined effect of pumpingand boundaries on pore water velocity and particle trajec-tory statistics needs further investigation, we have focusedon analysis of nonstationarity due to pumping where itcomplicates the problem by introducing high nonlinearity inthe velocity field.[37] The proposed methodology is now used to determine

the uncertainty in time related well capture zones. For therest of this analysis we use both the fine grid mesh, i.e.,0.1[L], and the solution of Darcy’s law using the centralfinite element method. A total of 36 particles was distrib-uted at a radial distance of 0.2[L] from the well, with mostof these (32) located downstream of the well. However,before proceeding we examine normality of ln(R) for four ofthe 36 particles. Figure 6 shows probability distributions ofln(R) after 10[T ] (left) and 20[T ] (right) for 4 particles usedin this analysis. These are particles 1, 10, 15 and 35 whichstart at angles 90�, 141�, 171�, and 18�, respectively. Thefigure has the sample data obtained from the MC analysisdisplayed with symbols. Superimposed on the plot is a line

Figure 3. A comparison between the first-order approximation (FO) and Monte Carlo (MC) methods tocalculate (top left) �xp, (bottom left) ln(sxp

2 ), (top right) �yp, and (bottom right) ln(syp2 ) using a coarse grid

and direct differentiation of Darcy’s law.


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presenting a robust linear fit of the sample order statisticsjoining the first and third quartiles of each particle end pointrealizations. This line is extrapolated out to the ends of thesample to help evaluate the linearity of the data. If the datadoes come from a normal distribution, the plot will appearlinear. Other probability density functions will introducecurvature in the plotted sample data. The figure shows thatnone of the 4 distributions conform to be log-normal, andthat particle 15 shows the largest deviation from log-normality. This is valid for times 10[T] and 20[T], althoughmore conformity to log-normality can be observed forparticles 1 and 35 at time 10[T]. To check this conclusion,we perform a more formal statistical hypothesis test, e.g.,Kolmogrov-Smirnov test, to check log-normality of R. Thenull hypothesis for the Kolmogorov-Smirnov test, in thiscase, is that ln(R) has a standard normal distribution whileits alternative hypothesis is that ln(R) does not have thatdistribution. The test failed at a significant level of 5% fortimes 10[T] and 20[T] except for particle 35 at time 10[T],so we reject the hypothesis of normality of ln(R) for the fourparticles at time 20[T] and for particles 1, 10 and 15 at time10[T], while accepting it only for particle 35 at time 10[T].The same test was carried out to test the hypothesis that R is

normally distributed and the hypothesis was rejected for allcases. Here, a further point that needs more investigation isthe impact of boundaries, especially their relative position,on the particle’s end point probability distribution (i.e.,whether this is normal or not).[38] We finally demonstrate the application of the FO

method to delineate well capture zone uncertainty. Althoughit has been demonstrated that the distribution of R is notalways log-normal, for the examined times, if we adopt thisassumption then we can construct the 95% confidenceinterval of the time related well capture zone, and similarlyif we assume R is normally distributed. Figure 7 shows the95% confidence interval for 10[T] and 20[T] capture zones,left and right, respectively, as estimated using the FOmethod for the two cases of normal and log-normal R.The figure shows locations of the 97.5%, the 50% and the2.5% capture probabilities derived from the 36 particlesused in this exercise. The figure also shows a scatterplot ofthe Monte Carlo results for these 36 particles. The figureindicates that the 95% confidence interval can capture thepossible capture zone to a good accuracy, either for thenormal or log-normal assumption of R for both 10[T] and20[T]. It also shows that the location of the sides of the

Figure 5. A comparison between FO and MC methods to calculate (left) ln(sxp2 ) and (right) ln(syp

2 )using a fine grid and solving Darcy’s law using the central finite element method.

Figure 4. A comparison between FO and MC methods to calculate (left) ln(sxp2 ) and (right) ln(syp

2 ) usinga coarse grid and solving Darcy’s law using the central finite element method.

8 of 12


2.5% probability of capturing is overestimated for time20[T] for both cases of normal and log-normal R; more sofor the log-normal case. Also, the sides of the 97.5%probability for the same time is underestimated, especiallywhen we assume R is normally distributed. This meansthat the FO method produces a conservative confidenceinterval. We relate this to the violation of R to normalityand log-normality; especially for particles downstream thepumping well (see particles 10 and 15 in Figure 6).Although Figure 7 (left) shows a better estimation of thetwo probability levels for time 10[T], a similar trend thatsides of the 2.5% probability contour are overestimatedand sides of the 97.5% probability contour are under-estimated as for time 20[T], is observed. This trend seemsto be related to the variance of ln(R) (or R), whichincreases with time.[39] A final point to mention here is the efficiency of the

FO method as compared to the MC analysis. On one hand,for each of the 5000 realizations used in the MC analysis, aforward model for flow and Darcy’s law have to be solvedbefore back tracking the 36 particles in the resulting

velocity field. This means 5000 calls to the iterative solverfor the head fields, 10000 calls to the iterative solver for thevelocity fields (ux and uy), and 5000 calls to the Runge-Kutta algorithm to back track the 36 particles. On the otherhand, the main computational demands for the FO analysisis the calculation of sensitivity coefficients of the pore watervelocity. The number of nodes at which we need to calculatesensitivities of ux and uy with respect to R depends on theparticle trajectories (and associated elements). In the case ofthe 10[T] capture zone, sensitivity coefficients of porewater velocity were required at 587 nodes. This entailed2 587 calls to the iterative solver. Also, a forward modelrun with 3 calls to the iterative solver using the meanparameter values is needed to calculate the requiredsensitivity coefficients. This means a total of 1177 callsto the iterative solver. The 587 nodes needed are calculatedduring tracking the 36 particles used to delineate the wellcapture zone. For each particle to calculate its end pointsensitivity coefficients we need sensitivity coefficients ofpore water velocities of nodes of elements that the particlemoves through. We accumulate total number of nodes for

Figure 6. Normal probability plot of ln (R) at (left) time = 10[T] and (right) time = 20[T].

Figure 7. Comparison between the 95% confidence interval of (left) 10[T] and (right) 20[T] wellcapture zones calculated assuming R is normal (squares) and assuming R is log-normal (circles), overlaidon a scatterplot of the MC results. Internal symbols are the 97.5% probability contour, middle symbolsare the 50% probability, and external symbols are the 2.5% probability.


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W01004

the 36 particles which gives 587 node for 10[T] capturezone. Similarly, sensitivity of pore water velocities at1043 node for 20[T] capture zone were needed. So, for the20[T] capture zone, a total of 2 1043 + 3 = 2089 calls tothe iterative solver was required. Moreover, only one call tothe Runge-Kutta algorithm for back tracking the 36 particleswas needed. So, in general, the FO method is more compu-tationally efficient than the MC method since, for the studiedcase, it requires around 7–12 times less CPU time than theMC method to call the iterative solver and 5000 times fewcalls to the Runge-Kutta function.[40] It should also be noted that in the case of condition-

ing simulations to state variable measurements [e.g., Bakrand Butler, 2004], the FO method will have an even greaterefficiency compared to the MC method. This is because foreach MC realization, a more computationally demandinginverse algorithm has to be solved to condition on the statevariable measurements, whereas only one such solution isrequired for the FO method.

4. Summary and Conclusions

[41] In this paper we examined the first-order Taylor’sseries approximation to estimate the time related wellcapture zone and its uncertainty. The method requires theestimation of sensitivity coefficients of a particle end pointwith respect to the uncertain parameter (log-transmissivity,R, in this study). These sensitivity coefficients are functionof sensitivity of pore water velocity with respect to R, and xand y coordinate. The latter showed to play an importantrole in estimating the probability of capturing.[42] To estimate the sensitivity coefficients of ux and uy

with respect to R we use the adjoint method. Verification ofthese coefficients with the finite difference perturbationmethod showed accurate calculation of these coefficientsusing the adjoint method.[43] The study has showed the importance of the calcu-

lation of pore water velocity (and its spatial derivative) instochastic well capture design. Obtaining ux and uy by directdifferentiation of Darcy’s law leads to an inaccurate esti-mation of @ux/@x and @uy/@y. However, solving Darcy’s lawby the central finite element method produces a much betterestimation of these derivatives. Since the method is fullynumerical, fine spatial discretization is possible at the areawhere pore water velocity varies rapidly (e.g., close to thewell).[44] The numerical example showed that, in general, the

first-order method can estimate to high accuracy the meantrajectory of a particle as compared to the Monte Carlomethod. However, estimation of the variances showed lessaccuracy. This is because they are function in the spatialderivatives of the pore water velocity which are numericallydegraded close to the well. Such degradation is due to thehigh variability of the pore water velocity close to the well,especially toward its downstream. Solving Darcy’s lawusing the central finite element method produced betterestimation of the pore water velocity and its spatial deriv-ative in this area around the well. Although this led to abetter estimation of the variance, discrepancies between thefirst-order and the Monte Carlo methods, especially forthose particles originating downstream the well, wereobserved. However, solving Darcy’s law using the central

finite element method on a finer grid led to a betterestimation of the variance of a particle trajectory.[45] Applying the first-order method to estimate the

confidence interval of a time related well capture zone,we examined the method and showed that it can produce agood approximation of the mean and variance of particletrajectories, provided that the effects of heterogeneity on theflow are adequately presented. The method can capture thepossible capture zone to good accuracy. As the methodmarginally overestimates the 2.5% probability level, ittherefore tends to be slightly conservative, and hence theassociated risk to the source is minimal. The advantage,however, is that the first-order Taylor’s series approximationis an attractive method in well capture zone design becauseof its ability to deal with complicated geometry, boundaryconditions, and sink/source terms. Also, the method can beused in the context of the groundwater inverse problem.This enables examination of well capture zone uncertaintyand impacts of conditioning on state variable measurements.In such cases the first-order approximation will present agreat saving in CPU time over the Monte Carlo methodwhen combined with inverse solution algorithms [e.g., Bakrand Butler, 2004].

Appendix A: Sensitivity Coefficients of PoreWater Velocity

[46] In this paper we assume a two-dimensional, isotropicaquifer subject to steady state flow condition. We solve theflow problem using the Galerkin (central) finite elementmethod [Pinder and Gray, 1977; Zienkiewicz and Taylor,1991]. We also use the same method to solve Darcy’s law[Yeh, 1981]. The method leads to a system of equations asgiven in (1), which can be written in more details asfollows:

g s; Rð Þ ¼Ah 0 0

0 Ad 0

0 0 Ad

24

35 h

uxuy

8<:

9=;�

rhruxruy

8<:

9=;: ðA1Þ

Here, h, ux and uy are vectors of size nn (number of nodes ina finite element mesh) of nodal values of head (L), and porewater velocity (L/T) in x and y direction, respectively. Otheritems, Ah, Ad, rh, rux, and ruy in this equation are given asfollows:

Ah ¼Xnee¼1

Ze

bexTebeTx þ beyT

ebeTy

� �dA; ðA2Þ

Ad ¼Xnee¼1

Ze

nenTe dA; ðA3Þ

rh ¼Xnee¼1

Ze

qenedAþZGqesn

edS

� �; ðA4Þ

rux ¼Xnee¼1

Ze

�Te

deJe

@he

@xnedA; ðA5Þ

ruy ¼Xnee¼1

Ze

�Te

deJe

@he

@ynedA; ðA6Þ

10 of 12


where Ah is the conductance matrix of size nn nn; Ad is amatrix of size nn nn; rh is a vector of size nn presentingsystem driving force; rux and ruy are vectors of size nn; T

e isthe transmissivity (L2/T); qe is the pump rate (L3/T); qs

e is aconstant volume flux (L3/T) along boundary G; Je is theeffective porosity (L0); de is the aquifer saturated thickness(L); ne is a vector of nodal values of the basis function; bx

e

and bye are derivatives of ne with respect to x and y,

respectively; superscript e is an index for element number;superscript T indicates transpose operation for a vector or amatrix. In these equations, Ah, Ad, rh, rux, and ruy arederived by assembling the system matrices for a number ofelements equals to ne. Integrations have been carried outnumerically using Gauss quadrature. This requires thatproperties that have nodal values are averaged within anelement e, to be estimated at Gauss points, using its shapefunction ne. Effective porosity and aquifer saturatedthickness have been averaged using the arithmetic mean,e.g., in case of effective porosity it is defined as

Je ¼ neT)e: ðA7Þ

Similarly, derivatives of head with respect to x and y in (A5)and (A6), respectively, can be calculated using

@he

@x¼ beTx he

@he

@y¼ beTy he:

ðA8Þ

[47] In the case of two dimensional flow, the geometricmean represents the effective up-scaled transmissivity [deMarsily, 1986]:

ln Te ¼ neT ln Teð Þ: ðA9Þ

[48] Derivatives of g(s, R) with respect to s and R that arerequired to calculate sensitivity coefficients of pore watervelocity at any node with respect to R, as in (20), areobtained by differentiating (A1) with respect to s and R,such that

@g

@s¼

Ah 0 0

� @rux@h

Ad 0

�@ruy@h

0 Ad

26664

37775 ðA10Þ

and

@g

@R¼

@Ah

@Rh

�@Rux@R

�@Ruy@R

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;: ðA11Þ

This is considering that qse is not function in h nor R.

[49] In (A10) derivatives of rux and ruy with respect to h areobtained by differentiating (A5) and (A6), respectively, withrespect to h, and making use of (A8). This gives

@rux@h

¼Xnee¼1

Ze

�Te

deJe

@

@hbeTx he" #

nedA ðA12Þ

and

@ruy@h

¼Xnee¼1

Ze

�Te

deJe

@

@hbey

The� �

nedA: ðA13Þ

[50] Also, derivatives of Ah, rux and ruy with respect to R

are obtained by differentiating (A2) with respect to R, suchthat

@Ah

@R¼Xnee¼1

Ze

bex@Te

@RbeTx þ bey

@Te

@RbeTy

� �dA; ðA14Þ

rux@R

¼Xnee¼1

Ze

�1e

deJe

@Te

@R

@he

@xnedA; ðA15Þ

ruy

@R¼Xnee¼1

Ze

�1e

deJe

@Te

@R

@he

@ynedA; ðA16Þ

in which, using (A9), we have

@Te

@R¼X4j¼1

nej1

Tj

@Tj@T

@T

@R; ðA17Þ

where the derivative of T with respect to R is a diagonalmatrix and its diagonal elements are given by

@T

@r¼ T : ðA18Þ

[51] Acknowledgments. This work was supported by the EuropeanCommission through W-SaHARA (Stochastic Analysis of Wellhead Pro-tection and Risk Assessment) project. We would like to thank AlbertoGuadagnini and Monica Riva for their assistance in the preparation of thispaper. We also thank two anonymous reviewers for providing insightful anduseful suggestions that helped to improve the manuscript.

ReferencesBakr, M. I., and A. P. Butler (2004), Worth of head data in well capture zonedesign using stochastic inverse modelling, J. Hydrol., 290, 202–216.

Bakr, A., L. W. Gelhar, A. L. Gutjahr, and J. R. MacMillan (1978), Sto-chastic analysis of spatial variability in subsurface flow: 1. Comparisonof one- and three-dimensional flows, Water Resour. Res., 14, 263–271.

Bakr, M. I., C. B. M. te Stroet, and A. Meijerink (2003), Stochastic ground-water quality management: Role of spatial variability and conditioning,Water Resour. Res., 39(4), 1078, doi:10.1029/2002WR001745.

Ballio, F., and A. Guadagnini (2004), Convergence assessment of numericalMonte Carlo simulations in groundwater hydrology, Water Resour. Res.,40, W04603, doi:10.1029/2003WR002876.

Bear, J. (1972), Dynamics of Fluids in Porous Media, Elsevier, New York.Dagan, G. (1984), Solute transport in heterogenous formations, J. FluidMech., 145, 151–177.

de Marsily, G. (1986), Quantitative Hydrogeology: Groundwater Hydrol-ogy for Engineers, Elsevier, New York.

Dettinger, M. D., and J. L. Wilson (1981), First order analysis of uncer-tainty in numerical models of groundwater flow: 1. Mathematical devel-opment, Water Resour. Res., 17, 149–161.

Feyen, L., K. J. Beven, F. De Smedt, and J. Freer (2001), Stochastic capturezone delineation within the generalized likelihood uncertainty estimationmethodology: Conditioning on head observations, Water Resour. Res.,37, 625–638.

Feyen, L., P. J. Ribeiro Jr., F. De Smedt, and P. J. Diggle (2002), Bayesianmethodology to stochastic capture zone determination: Conditioning ontransmissivity measurements, Water Resour. Res., 38(9), 1164,doi:10.1029/2001WR000950.

Feyen, L., J. J. Gomez-Hernandez, P. J. Ribeiro Jr., K. J. Beven, andF. De Smedt (2003a), A Bayesian approach to stochastic capture


11 of 12

W01004

zone delineation incorporating tracer arrival times, conductivity mea-surements, and hydraulic head observations, Water Resour. Res.,39(5), 1126, doi:10.1029/2002WR001544.

Feyen, L., P. J. Ribeiro Jr., J. J. Gomez-Hernandez, K. J. Beven, andF. De Smedt (2003b), Bayesian methodology to stochastic capturezone determination incorporating transmissivity and hydraulic headobservations, J. Hydrol., 271, 156–170.

Franzetti, S., and A. Guadagnini (1996), Probabilistic estimation of wellcatchments in heterogeneous aquifers, J. Hydrol., 174, 149–171.

Freeze, R. A. (1975), A stochastic-conceptual analysis of one-dimensionalgroundwater flow in nonuniform homogeneous media, Water Resour.Res., 11, 725–741.

Gelhar, L. W., and L. Axness (1983), Three-dimensional stochastic analysisof macrodispersion in aquifers, Water Resour. Res., 19, 161–180.

Graham, W., and D. McLaughlin (1989a), Stochastic analysis of nonsta-tionary subsurface solute transport: 1. Unconditional moments, WaterResour. Res., 25, 215–232. (Correction, Water Resour. Res., 25, 2398,1989.)

Graham, W., and D. McLaughlin (1989b), Stochastic analysis of nonsta-tionary subsurface solute transport: 2. Conditional moments, WaterResour. Res., 25, 2331–2355.

Guadagnini, A., and S. Franzetti (1999), Time-related capture zones forcontaminants in randomly heterogeneous formations, Ground Water,37, 253–260.

Guadagnini, A., and S. P. Neuman (1999), Nonlocal and localized analysesof conditional mean steady state flow in bounded, randomly nonuniformdomains: 1. Theory and computation approach, Water Resour. Res., 35,2999–3018.

Hoeksema, R. J., and P. K. Kitanidis (1984), An application of the geosta-tistical approach to the inverse problem in two-dimensional groundwatermodeling, Water Resour. Res., 20, 1003–1020.

Indelman, P., and Y. Rubin (1995), Flow in heterogeneous media displayinga linear trend in log conductivity, Water Resour. Res., 31, 1257–1265.

Kunstmann, H., and W. Kinzelbach (2000), Computation of stochastic well-head protection zones by combining the first-order second-momentmethod and Kolmogorov backward equation analysis, J. Hydrol., 237,127–146.

Li, S.-G., and D. McLaughlin (1995), Using the nonstationary spectralmethod to analyze flow through heterogeneous trending media, WaterResour. Res., 31, 541–551.

Loaiciga, H. A., M. A. Leipnik, M. A. Marino, and P. F. Hudak (1993),Stochastic groundwater flow analysis in the presence of trends in hetero-geneous hydraulic fields, Math. Geol., 25, 161–176.

Lu, Z., and D. Zhang (2003), On stochastic study of well capture zones inbounded, randomly heterogeneous media, Water Resour. Res., 39(4),1100, doi:10.1029/2002WR001633.

McLaughlin, D., and L. R. Townley (1996), A reassessment of the ground-water inverse problem, Water Resour. Res., 32, 1131–1161.

Naff, R. L., and A. V. Vecchia (1986), Stochastic analysis of three-dimensional flow in a bounded domain, Water Resour. Res., 22,695–704.

Osnes, H. (1995), Stochastic analysis of head spatial variability inbounded rectangular heterogeneous aquifers, Water Resour. Res., 31,2981–2990.

Osnes, H. (1998), Stochastic analysis of velocity spatial variability inbounded rectangular aquifers, Adv. Water Resour., 21, 203–215.

Pebesma, E. (1999), GSTAT user’s manual, technical report, Dept. of Phys.Geogr. Utrecht Univ., Netherlands.

Pinder, G. F., and W. G. Gray (1977), Finite Element Simulation in Surfaceand Subsurface Hydrology, Elsevier, New York.

Riva, M., A. Guadagnini, S. P. Neuman, and S. Franzetti (2001), Radialflow in a bounded randomly heterogeneous aquifer, Transp. PorousMedia, 45, 139–193.

Rubin, Y. (1990), Stochastic modeling of macrodispersion in heterogeneousporous media, Water Resour. Res., 26, 133–141.

Rubin, Y., and G. Dagan (1988), Stochastic analysis of boundaries effectson head spatial variability in heterogeneous aquifers: 1. Constant headboundary, Water Resour. Res., 24, 1689–1697.

Rubin, Y., and G. Dagan (1989), Stochastic analysis of boundaries effectson head spatial variability in heterogeneous aquifers: 2. Imperviousboundary, Water Resour. Res., 25, 707–712.

Rubin, Y., and K. Seong (1994), Investigation of flow and transport incertain cases of nonstationary conductivity fields, Water Resour. Res.,30, 2901–2911.

Shvidler, M. I. (1993), Correlation model of transport in random fields,Water Resour. Res., 29, 3189–3199.

Stauffer, F., S. Attinger, S. Zimmermann, and W. Kinzelbach (2002),Uncertainty estimation of well catchments in heterogeneous aquifers,Water Resour. Res., 38(11), 1238, doi:10.1029/2001WR000819.

Sykes, J. F., J. L. Wilson, and R. W. Andrews (1985), Sensitivity analysisfor steady state groundwater flow using adjoint operators, Water Resour.Res., 21, 359–371.

Tartakovsky, D. M., A. Guadagnini, and M. Riva (2003), Stochastic aver-aging of nonlinear flows in heterogeneous porous media, J. Fluid Mech.,492, 47–62.

van Leeuwen, M., C. B. M. te Stroet, A. P. Butler, and J. A. Tompkins(1998), Stochastic determination of well capture zones, Water Resour.Res., 34, 2215–2223.

van Leeuwen,M., A. P. Butler, C. B.M. te Stroet, and J. A. Tompkins (2000),Stochastic determination of well capture zones conditioned on regulargrids of transmissivity measurements, Water Resour. Res., 36, 949–957.

Varljen, M. D., and J. M. Shafer (1991), Assessment of uncertainty in time-related capture zones using conditional simulation of hydraulic conduc-tivity, Ground Water, 29, 737–748.

Winter, C. L., M. Newman, and S. P. Neuman (1984), A perturbationexpansion for diffusion in a random velocity field, SIAM J. Appl. Math.,44, 411–422.

Yeh, G.-T. (1981), On the computation of Darcian velocity and massbalance in the finite element modeling of groundwater flow, WaterResour. Res., 17, 1529–1534.

Zhang, D. (1998), Numerical solutions to statistical moment equations ofgroundwater flow in nonstationary, bounded, heterogeneous media,Water Resour. Res., 34, 529–538.

Zhang, D. (2002), Stochastic Methods for Flow in Porous Media: CopingWith Uncertainties, 350 pp., Elsevier, New York.

Zhang, D., and Z. Lu (2002), Stochastic analysis of flow in a heterogeneousunsaturated-saturated system, Water Resour. Res., 38(2), 1018,doi:10.1029/2001WR000515.

Zhang, D., and Z. Lu (2004), Stochastic delineation of well capturezones, Stochastic Environ. Res. Risk Assess., 18, doi:10.1007/s00477-003-0159-z.

Zhang, D., and S. P. Neuman (1996), Head and velocity covariances underquasi-steady state flow and their effects on advective transport, WaterResour. Res., 32, 77–83.

Zhang, D., and C. L. Winter (1998), Nonstationary stochastic analysis ofsteady state flow through variably saturated, heterogeneous media, WaterResour. Res., 34, 1091–1100.

Zheng, C., and G. D. Bennett (1995), Applied Contaminant TransportModeling, Van Nostrand Reinhold, Hoboken, N. J.

Zienkiewicz, O. C., and R. L. Taylor (1991), The Finite Element Method:Solid and Fluid Mechanics Dynamics and Non-linearity, 4th ed., vol. 2,McGraw-Hill, New York.

��M. I. Bakr, Water Resources Research Institute, National Water Research

Center, El-Qanater El-Khairiya, 13621, Cairo, Egypt. ([email protected])

A. P. Butler, Department of Civil and Environmental Engineering,Imperial College London, Exhibition Road, London, SW7 2AZ, UK.([email protected])

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Nonstationary stochastic analysis in well capture zone design using first-order Taylor's series...

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