Journal of Hydrology 381 (2010) 297–307

Contents lists available at ScienceDirect

Journal of Hydrology

journal homepage: www.elsevier .com/ locate / jhydrol

Efficient conceptual framework to quantify flow uncertainty in large-scale,highly nonstationary groundwater systems

Chuen-Fa Ni a,*, Shu-Guang Li b, Chien-Jung Liu c, Shaohua Marko Hsu c

a Graduate Institute of Applied Geology, National Central University, Taiwanb Department of Civil and Environmental Engineering, Michigan State University, USAc Department of Water-Resources Engineering and Conservation, Feng Chia University, Taiwan

a r t i c l e i n f o s u m m a r y

Article history:Received 20 February 2009Received in revised form 17 November 2009Accepted 3 December 2009

This manuscript was handled by P. Baveye,Editor-in-Chief, with the assistance of ChrisSoulsby, Associate Editor

Keywords:Spectral methodFlow uncertaintyNonstationary flowPerturbation-based numerical methodMonte Carlo simulation

0022-1694/$ - see front matter � 2009 Elsevier B.V. Adoi:10.1016/j.jhydrol.2009.12.002

* Corresponding author. Address: Graduate InstituteCentral University, Chungli City, Taoyuan 32004227151x65874; fax: +886 3 4263127.

E-mail address: nichuenfa@geo.ncu.edu.tw (C.-F. N

This study presents a hybrid spectral method (HSM) to estimate flow uncertainty in large-scale highlynonstationary groundwater systems. Taking advantages of spectral theories in solving unmodeledsmall-scale variability in hydraulic conductivity, the proposed HSM integrates analytical and numericalspectral solutions in the calculation procedures to estimate flow uncertainty. More specifically, theHSM involves two major computational steps after the mean flow equation is solved. The first step isto apply an analytical-based approximate spectral method (ASM) to predict nonstationary flow variancesfor entire modeling area. The perturbation-based numerical method, nonstationary spectral method(NSM), is then employed in the second step to correct the regional solution in local areas, where the var-iance dynamics is considered to be highly nonstationary (e.g., around inner boundaries or strong sources/sinks). The boundary conditions for the localized numerical solutions are based on the ASM closed formsolutions at boundary nodes. Since the regional closed form solution is instantaneous and the moreexpensive perturbation-based numerical analysis is only applied locally around the strong stresses, theproposed HSM can be very efficient, making it possible to model strongly nonstationary variance dynam-ics with complex flow situations in large-scale groundwater systems. In this study the analytical-basedASM solutions was first assessed to quantify the solution accuracy under transient and inner boundaryflow conditions. This study then illustrated the HSM accuracy and effectiveness with two synthetic exam-ples. The HSM solutions were systematically compared with the corresponding numerical solutions ofNSM and Monte Carlo simulation (MCS), and the analytical-based solutions of ASM. The simulationresults have revealed that the HSM is computationally efficient and can provide accurate variance esti-mations for highly nonstationary large-scale groundwater flow problems.

� 2009 Elsevier B.V. All rights reserved.

Introduction

Fluid flows in aquifers have been recognized strongly influ-enced by spatial variation in hydraulic conductivity or transmissiv-ity. To characterize flow uncertainty caused by such variation,many investigations generally employ stochastic theories to quan-tify impacts of flow uncertainty on aquifer heterogeneity. In thepast three decades, intensive studies in the field of stochastichydrogeology have produced a number of analytical and numericaltheories to predict flow uncertainty for heterogeneous aquifer sys-tems (see review by Gelhar (1993), McLaughlin and Townley(1996), Dagan and Neuman (1997), Zhang (2002), Rubin (2003)).Stochastic analytical theories assume statistically uniform flow,

ll rights reserved.

of Applied Geology, National1, Taiwan. Tel.: +886 3

i).

ignoring important large-scale dynamics caused by sources andsinks, boundary effects, transient conditions, and trends in aquiferproperties. Stochastic analytical methods are efficient because theclosed form solutions for large-scale problems can be quickly ob-served with low computational cost. Although the analytical solu-tions for many simplified situations can provide general insightinto the impacts of small-scale heterogeneity on variance dynam-ics, the restricted assumptions have significantly limited such solu-tions to be applied to practical, site-specific conditions. Criticalreviews have been made by many investigators (e.g., Dagan andNeuman, 1997; Zhang and Zhang, 2004; Dagan, 2004; Li et al.,2004a,b; Neuman, 2004, 2005; de Marsily et al., 2005; Renard,2007; Miller and Gray, 2008).

Motivated by the needs to relax the restricted conditions forstochastic analytical theories, the stochastic numerical theo-ries have been developed to accommodate site-specific andnonstationary features (e.g., Smith and Freeze, 1979; Grahamand McLaughlin, 1989a,b; Li and McLaughlin, 1991; Zhang, 1999;

298 C.-F. Ni et al. / Journal of Hydrology 381 (2010) 297–307

Winter and Tartakovsky, 2002; Guadagnini and Neuman, 1999a,b;Guadagnini et al., 2003; Ye et al., 2004; Zhang and Lu, 2004). How-ever, recent investigations have revealed that most stochastic the-ories are unable to model efficiently the effects of random, small-scale heterogeneity, especially for large-scale groundwater flowand transport problems. Because the available methods, MonteCarlo simulation (MCS) and perturbation-based methods, requireresolving numerically uncertain small-scale processes before thevariances can be computed (Li et al., 2004a,b), such requirementtranslates into the needs to solve large numbers of partial differen-tial equations on very fine grids. This is one of the important rea-sons why most stochastic numerical solutions reported in theliterature are limited to small and simplistic problems (e.g., Liet al., 2004a,b; Ni and Li, 2006; McLaughlin et al., 2006; Renard,2007).

To break the major computational burden, previous investiga-tions of Li and his co-workers have developed nonstationary spec-tral method (NSM) for predicting nonstationary flows and solutetransport (Li and McLaughlin, 1995; Li et al., 2003, 2004a,b). Thismethod is more efficient than traditional perturbation-basedmethods because the perturbation equations in NSM are evaluatedbased on an extension of the classical spectral method for station-ary flow and transport problems (Mizell et al., 1982; Gelhar, 1993).The main advantage of NSM is that the transformed perturbationequations in the spectral domain can yield smooth solutions.Therefore, such solutions require coarser numerical grids thanthe solutions of traditional perturbation-based methods do to re-solve the unmodeled small-scale heterogeneity (Li and McLaugh-lin, 1995; Li et al., 2003, 2004a,b). However, for problems withrealistic scales and complexity, the computational cost of NSM isstill significantly more expensive than that of deterministic meth-ods, especially for large-scale and unsteady problems, primarilybecause the NSM still requires solving relatively large numbersof partial differential equations in the spectral domain. On the ba-sis of classical spectral theories, Ni and Li (2005, 2006) later fo-cused on identifying the accuracy of the perturbation equationwith an approximated form. They dropped the secondary termsin the perturbation equations and derived approximated closedform solutions for estimating flow variances. Their systematicanalysis showed that the developed approximate spectral method(ASM), which involves simple closed form formulas, is more effi-cient than NSM and can capture mildly nonstationary flow dynam-ics such as hydraulic conductivity trends and composite media (Niand Li, 2005, 2006). However, for strongly nonstationary flowscaused by, for example, specified heads and impermeable bound-aries, the ASM leads to inaccurate estimations of flow variancesin the proximity of strong stresses. The regions are 2 to 3 correla-tion lengths (k) of log hydraulic conductivity (ln K) from specifiedhead and impermeable boundaries. To resolve the variance dynam-ics near such strong stresses with strongly nonstationary flow pat-terns, the numerical-based NSM is suggested (Ni and Li, 2005,2006).

Taking the advantages of NSM and ASM, this study presents ahybrid spectral method (HSM) for predicting flow uncertainty inlarge-scale complex groundwater flow systems. The concept ofthe proposed hybrid method here is based, in part, on ideas pre-sented by Székely (1998, 2008) and Mehl and Hill (2002) for localgrid refinements and by the hierarchical downscale modeling forlarge-scale problems (e.g., Li et al., 2006; Li and Liu, 2006a,b; Milleret al., 2006). The HSM, based on spectral methods in solving sto-chastic perturbation equations, involves two major computationalsteps after the mean flow equation is solved. The first step is to ap-ply an analytical-based method, the ASM, to predict nonstationaryvariances (i.e., the velocity variances) for entire modeling area.Then the perturbation-based numerical spectral method, theNSM, is employed to correct the ‘‘regional solution” in local areas,

where the variance dynamics is considered to be highly non-stationary (e.g., around inner boundaries or strong sources/sinks).The boundary conditions for the local numerical solutions arebased on the ASM closed form solutions at boundary nodes. Sincethe ‘‘regional” closed form solution is instantaneous and the moreexpensive perturbation-based numerical analysis is only appliedlocally around the strong stresses, the proposed HSM can be veryefficient, making it possible to model strongly nonstationary vari-ance dynamics with complex flow situations in complex large-scale groundwater systems.

This paper will present the accuracy and effectiveness of theASM and HSM with synthetically generated two-dimensionalexamples. Extending the study of Ni and Li (2005, 2006), in the firstexample this study will assess the ASM accuracy for predictingflow situations under transient flow, inner boundary, and source/sink conditions. The concept and estimation procedures of HSMwill then be illustrated with the second example. In this examplea pumping well is located at the center of modeling area. In thethird example, we then apply the developed HSM to a more com-plex large-scale flow situation. In the test examples the solutions ofASM and HSM will be systematically compared with the corre-sponding numerical solutions of NSM and MCS. Although thisstudy does not directly address the data issue, the methodologyit produces may indirectly benefit geostatistical parameter estima-tions through inverse stochastic groundwater flow and contami-nant transport modeling.

Stochastic perturbation equations

This study considers an unsteady flow in a confined and heter-ogeneous two-dimensional porous medium with multiple source/sinks and systematic trends. The random logarithm transmissivityfluctuation is approximately related to the hydraulic head andvelocity fluctuations by the following first-order, mean-removedflow equations (e.g., Li and McLaughlin, 1991; Gelhar, 1993; Li etal., 2003, 2004a):

Ss

Tg

@h0

@t¼ @2h0

@xj@xjþ lj

@h0

@xj� Jj

@y0

@xjþ Uy0x 2 D ð1Þ

u0jðx; tÞ ¼ �Tg

B@h0

@xj� Jjy

0� �

x 2 D ð2Þ

The boundary conditions are

h0ðx; tÞ ¼ 0 x 2 CD

rh0ðx; tÞ � nðxÞ ¼ 0 x 2 CN

Eqs. (1) and (2) are written in Cartesian coordinates and x is theposition vector for the point (x1, x2) in domain D. Homogeneousconditions are defined both on specified head boundaries CD andspecified flux boundaries CN. The notations y0, h0, and u0j are, respec-tively, the mean-removed log transmissivity, head, and Darcyvelocity; y0 is assumed to be the only source of randomness inaquifers; h is the mean head that can be obtained by solving thefirst-order (deterministic) mean head equation (i.e., withoutconsidering the effect of random, small-scale heterogeneity);Jj = Jj(x, t) is the deterministic mean head gradient;Ss = Ss(x) is thespecific storage; B = B(x) is the mean aquifer thickness; Tg = Tg(x,t) is the geometric mean transmissivity; lj = lj(x) is the trend slopefor the mean log transmissivity; q = q(x, t) is the sources/sink term;U ¼ ðSs

Tg

@h@t �

qTgÞ in Eq. (1) represents the source term caused by

variation of mean flow; and n is the unit normal vector of the fluxboundary CN.

Note that the assumption that products of fluctuations can beneglected can only be justified when the fluctuation variances

C.-F. Ni et al. / Journal of Hydrology 381 (2010) 297–307 299

are small (Dagan, 1989; Gelhar, 1993; Zhang, 2002; Li et al., 2003;Rubin, 2003). Here the perturbation Eqs. (1) and (2) describes thelinear, nonstationary transformation from y0 to h0 to u0j .

Solutions to the stochastic perturbation equations

Nonstationary spectral method (NSM)

Spectral methods offer a particularly convenient way to derivehead and velocity statistics from linearized fluctuation equationssuch as Eqs. (1) and (2). The traditional stationarity-based spectralapproach (Gelhar, 1993) is applicable only when the independentvariable fluctuation (y0) as well as dependent variable fluctuations(h0, u0j) are wide-sense stationary. Papoulis (1984) showed that theoutput (e.g., h0, u0j) of linear transformations such as Eqs. (1) and (2)are stationary only if the input (e.g., y0) is stationary and the trans-formations are spatially invariant. In the problem of interest here,spatial invariance implies that the fluctuation Eqs. (1) and (2)should have constant coefficients with the boundaries sufficientlydistance to have no effect on head and velocity fluctuations inthe region of interest. Such a spatial invariance requirement isclearly not met because of real-world complexities, the heteroge-neous trending media, boundary effects, and sources/sinks intro-duced into most aquifer systems.

The NSM is a perturbation approach and does not requiredependent fluctuations to be stationary. This method differs fromother classical perturbation methods primarily in the form of thespectral representation of the output variable fluctuation. Moreprecisely, the dependent fluctuations are represented as a stochas-tic integral expanded in terms of a set of unknown complex-valued‘‘transfer functions” why(x, k, t). The fluctuations have the followingFourier–Stieltjes representation (e.g., Priestley, 1981; Li andMcLaughlin, 1991; Li and McLaughlin, 1995; Li et al., 2003):

y0ðxÞ ¼Z 1

�1expðikjxjÞdZyðkÞ ð3Þ

h0ðx; tÞ ¼Z þ1

�1whyðx; k; tÞ expðikjxjÞdZyðkÞ ð4Þ

and

u0jðx; tÞ ¼Z þ1

�1wujyðx; k; tÞ expðikjxjÞdZyðkÞ ð5Þ

where i = (�1)1/2, kj is component j of the wave number k, and dZy(k)is the random Fourier increment of y0(x), evaluated at k. The Fourierrepresentation can be viewed as the continuous version of a Fourierseries expansion of y(x). The random Fourier increment at a partic-ular wave number is analogous to the random amplitude of one ofthe terms in the Fourier integral. The symbols why, wujy

are unknownhead and velocity transfer functions introduced to account for pos-sible nonstationary flow transformations. These transfer functionsmust be selected such that h0 and u0j satisfy the governing perturba-tion equations.

Substituting Eqs. (3) and (4) into Eq. (1) we obtain the followingtransfer function equations:

Ss

Tg

@why

@t¼@2why

@xj@xjþ ð2ikj þ ljÞ

@why

@xjþ ðikjlj � k2Þwhy � ikjJj þ U ð6Þ

wujy¼ � Tg

B@why

@xj� ikjwhy � Jj

� �ð7Þ

Eqs. (6) and (7) are deterministic and complex-valued differen-tial equations. Unlike the classical stationary spectral method,which requires transfer functions to be spatially invariant, thetransfer functions introduced here are spatially variants. The trans-

fer functions why, wujyobtained from Eqs. (6) and (7) can then be

used to derive the head and velocity variances in the same wayas the classical stationary spectral method (e.g., Mizell et al.,1982; Gelhar, 1993):

r2hðx; tÞ ¼

Z þ1

�1whyðx; k; tÞw�hyðx; k; tÞSyyðkÞdk ð8Þ

r2ujðx; tÞ ¼

Z þ1

�1wujyðx; k; tÞw�ujy

ðx; k; tÞSyyðkÞdk ð9Þ

where Syy(k) is the spectral density function of the log transmissiv-ity (Priestley, 1981; Gelhar, 1993). In this study we are especiallyinterested in the velocity variances because the head variance isusually very small and only some special spectral density functionsthat exhibit closed form formula for head variance.

Approximate spectral method (ASM)

Eqs. (1) and (2) represent the full version of perturbation equa-tions. In order to introduce further approximations, one needs toclarify the hydrogeologic properties that are corresponding tothe terms shown in Eq. (1). In Eq. (1), lj reflects the trend effectthat contributes to the small-scale variability. The U term reflectsthe contribution of sources/sinks and the transient mean flow tothe small-scale variability. Mathematically, we can treat theseterms as the extra stresses that are added to a very simple flowsituation.

Follow the observations in Ni and Li (2005, 2006), we can ignorethe secondary terms that are not providing significant impact onthe evaluation of flow uncertainty in a moderately nonstationaryflow situations. Giving these requirements, Eq. (1) can be reducedto the following simple formation:

@2h0

@xj@xj¼ Jj

@y0

@xjx 2 D ð10Þ

u0jðxÞ ¼ �Tg

B@h0

@xj� Jjy

0� �

x 2 D ð11Þ

Note that the time varying term has been taken out from Eqs.(10) and (11). Eqs. (10) and (11) can be evaluated based on the con-cept of quasi-steady flow, i.e., the transient state mean hydraulicheads are evaluated first and then calculated head gradientsJj = Jj(x, t) for perturbation Eqs. (10) and (11). Taking advantage ofscale disparity between the mean and fluctuation processes andinvoking the spectral representation, one can solve Eqs. (10) and(11) to obtain the following expression for predicting velocity vari-ances in heterogeneous porous media (Gelhar, 1993; Ni and Li,2005, 2006):

r2ujðxÞ ¼ r2

yðxÞJ2ðx; tÞ

T2gðxÞ

BðxÞ

Z þ1

�1

Z þ1

�11�xjx1

x2

� �2syyðxÞdx1dx2

ð12Þ

Note syy is the dimensionless spectral density function of y0(x),syy ¼ Syy=r2

y ;r2y is the log transmissivity variance, xj is the wave

number, x2 ¼ x21 þx2

2;r2u1

andr2u2

are respectively the longitudinaland transverse velocity variances. The detailed derivation processof Eq. (12) can be found in Dagan (1989), Gelhar (1993), or Zhang(2002) for statistically homogeneous media and is not repeatedhere.

Eq. (12) can be easily integrated in the polar coordinate system.The results are the following explicit expressions:

r2u1ðxÞ ¼ 0:375r2

yðxÞJ2ðx; tÞ

T2gðxÞ

BðxÞ ð13Þ

41

424344454647

x1 (m)

x 2(m

)

0 20 40 60 800

20

40

60

80

Lakestage = 42 + 2.0*sin(π*Δt)

No Flow

Con

stan

tHea

d=

40m

Con

stan

tHea

d=

48m

No Flow

A'A

Fig. 1. The conceptual model and mean flow distributions (at t = 1.0 day) fortransient, two-dimensional flow example.

4142

434445

46

47

41

x 2(m

)

40

60

80

Co

nst

antH

ead

=40

m

Co

nst

antH

ead

=4

8m

A'

No Flow

Pumping wellQ = -1000 m3/day

A

41

42

43

x1 (m)

x 2(m

)

30 35 40 45 5030

35

40

45

50NSM Correction Area

300 C.-F. Ni et al. / Journal of Hydrology 381 (2010) 297–307

r2u2ðxÞ ¼ 0:125r2

yðxÞJ2ðx; tÞ

T2gðxÞ

BðxÞ ð14Þ

These expressions are independent of the specific form of thelog transmissivity spectrum or covariance function for the isotropiccase.

Hybrid spectral method (HSM)

Previous studies in Ni and Li (2005, 2006) revealed that theclosed form formulas Eqs. (13) and (14) can capture the majorsmall-scale dynamics except for the locally strong stress areas suchas hydrological boundaries or wells. Given these observations, wepropose in this study a hybrid spectral method (HSM) for stochas-tic groundwater modeling. The improved ‘‘hybrid” spectral model-ing will be proceeded in two steps: First, the approximate spectralmethod (ASM) is applied to entire modeling area to obtain a ‘‘re-gional” screening level uncertainty analysis. Second, the non-stationary spectral method (NSM) is then applied to locally refineor correct the solutions in small areas, where the flow situationsare considered to be highly nonstationary. The boundary condi-tions for the local NSM solutions (i.e., the transfer function why inEqs. (6) and (7)) are based on the ‘‘regional” closed form solution(i.e., the ASM solutions). The regional solution and the numericalcorrections are expected to be highly efficient since the former isin a closed form and the latter are applied only in very small areas.

To complete the evaluation of numerical solutions, the bound-aries for each localized numerical model must be defined appropri-ately. Based on the concept of HSM, within an appropriate regionthe known transfer functions at the boundaries of local NSM mod-els have the following formation:

whyðx; k; tÞ ¼�ikjJjðx; tÞ

k2 ð15Þ

where the why can represent the specified transfer function bound-ary condition in the spectral domain.

x1 (m)0 20 40 60 80

0

20

No Flow

Boundary for NSMcorrection area

Fig. 2. The conceptual model, mean flow distributions, and HSM local correctionarea for steady and two-dimensional flow example.

Illustrative examples and numerical considerations

The previous study by Ni and Li (2005, 2006) had showed theaccuracy of the ASM to predict flow uncertainty in trending andcomposite porous media. Their results did not present the capabil-ity of the ASM to predict flow uncertainty that caused by strongstresses such as transient flow, inner boundaries, and sources/sinks. This study will first assess the ASM accuracy for predictingflow situations under transient flow, inner boundary, and source/sink conditions. The concept and estimation procedures of theHSM will then be illustrated with two examples to show the effec-tiveness and the accuracy in predicting flow uncertainty. Thisstudy considers two-dimensional flows in bounded and rectangu-lar areas. The general flow direction for each case is driven bytwo constant head boundary conditions at the east and westboundaries. No flow boundary conditions are specified at the southand north boundaries. The domain sizes for examples are either 80by 80k or 300 by 300k, where k is the correlation length of small-scale ln K fluctuation. Figs. 1–3 show the conceptual models andmean flow patterns for three modeling examples. In Fig. 1 the lakein the central region of the modeling area is specified by constanthead values. Here the lake stages are designed to vary with time tocreate a transient state situation for the model (see Fig. 1 for thetemporal variation equation). Fig. 2 presents the mean head distri-bution for the second example. In this example a pumping well isinstalled at the center portion of the modeling area. The concept ofthe HSM is illustrated with this example because the strong stress(i.e., the pumping well) has locally created a highly nonstationary

flow pattern around the well. As will be shown in the results, theanalytical-based ASM solution is inaccurate around the well regionand such region requires local correction by the numerical-basedNSM. A more complex and large-scale conceptual model is pre-sented in Fig. 3. In this steady state example, the hydraulic conduc-tivity trend is generated randomly by spectral algorithm based onGaussian model (Deutsch and Journel, 1997; Li and Liu, 2006a,b).The stresses in the modeling area include (1) a constant global re-charge applied to entire modeling area and a strongly local re-charge with recharge rate of 1.0 m/day in the area x = 40–90 mand y = 40–90 m, (2) a lake with constant lake stage of 98 m inthe area x = 120–150 m and y = 200–250 m, and (3) three wellswith pumping rates of 1500, 3000, and 6000 m3/day installed atcenter portion of the modeling area (see Fig. 3). In all illustrativeexamples, the small-scale fluctuation is modeled stochasticallyby the exponential spectral density function with a ln K varianceof 1.0 and an isotropic correlation length k of 1.0 m. Note that anuniform aquifer thickness (B = 20 m) in each example is considered

x1 (m)

x 2(m

)

0 50 100 150 200 250 3000

50

100

150

200

250

300

2.52.01.51.00.50.0

-0.5-1.0-1.5-2.0

Local RechargeR =0.01 m/day

LakeStage = 98m

WellQ= - 3000m3/d

WellQ= - 6000m3/d

WellQ= - 1500m3/d

Co

nsta

ntH

ead

=9

8m

Co

nsta

ntH

ead

=1

00m

No Flow

No Flow

Boundary for NSMcorrection area

Boundary for NSMcorrection area

ln K

90

92

94

96

9896

9492

86

82

100 96 92

98

x1 (m)

x 2(m

)

0 50 100 150 200 250 3000

50

100

150

200

250

300

a

b

Fig. 3. The conceptual model and the mean flow distribution for steady andcomplex groundwater flow system: (a) conceptual model and the randomlygenerated hydraulic conductivity trend, (b) the mean head distribution simulatedbased on the hydrogeological conditions in (a).

C.-F. Ni et al. / Journal of Hydrology 381 (2010) 297–307 301

to be known deterministically. For each case the MCS solution isbased on 10,000 equally likely realizations. The grid spaces usedfor ASM and NSM methods are 1k in the first and second examplesand 2k in the third example. In order to provide appropriate gridspace in Monte Carlo simulation to resolve small-scale variability,the grid space for MCS is adjusted to 1/3k for each example.

Results and discussion

To systematically present the proposed HSM, the discussions ofthe simulation results are divided into three different sections.

Mildly nonstationary flow with variations of lake stages

The purpose of the first example is to assess the ASM to predictflow uncertainty in mildly nonstationary flow fields. Based on theconceptual model shown in Figs. 1 and 4 shows the snapshot ofpredicted velocity variances (plotted with standard deviationSTD) at time 1.0 day by ASM and the numerical methods NSMand MCS. In Fig. 4 the solution of ASM agrees well with those ofNSM and MCS. Except for the locations along the edges of lakeboundaries and the specified constant head and no flow bound-

aries, ASM can reproduce well the patterns (Fig. 4) and magnitude(the centerline profiles in Fig. 5 for times at 1.25 and 1.5 days) ofvelocity variances. For specified points at locations (30, 40) and(50, 40), the variations of velocity variances with time are exactlythe same (see Fig. 6). The regions where show ASM inaccuratesolutions are 1–5k from specified head conditions (i.e., the lakeedges and the constant head boundaries) and no flow boundaryconditions, depending on the head gradients near those locations(see Figs. 4 and 5).

Although in this example the lake stage varies in the ranges of2 m, we found that the velocity variances at the upstream location(i.e., the location (30, 40)) are generally higher than that at thedownstream location (i.e., the location (50, 40)). The result reflectsthe closed form solutions of ASM in Eqs. (13) and (14), showingthat the values of variances are proportional to head gradients. Inthe upstream side of the lake, the head gradients are greater thanor equal to that in the downstream side of the lake. Based on theresults shown in Figs. 4–6, the ASM solutions can predict accu-rately the flow uncertainty under flow situations with mildly non-stationarity. For strongly nonstationary flows such as those nearlake edges, local numerical corrections are required to obtain bet-ter predictions of flow uncertainty. The simulation results in thisexample imply that the transient flow situation did not contributemuch on the variance estimation for nonstationary flow. Becausethe quasi-steady approximation had been used in ASM, the tran-sient mean flow gradient Jj(x, t) can represent main nonstationaryflow dynamics.

Highly nonstationay flow with a pumping well

The second test example here is employed to illustrate the con-cept of the HSM to predict highly nonstationary flows. Specifically,we consider a simple flow situation involving a constant head dri-ven flow from west boundary to east boundary and a well locatedat the center point of the modeling area (see Fig. 2 for the concep-tual model and flow pattern). Around the well location, we select acorrection area 20k by 20k, where the flow nonstationarity is con-sidered to be very high and cannot be accurately evaluated by theanalytical-based ASM. The grid space for the background modelingarea and the small correction area are the same. Under this condi-tion, all the ASM, NSM, and HSM solutions will be obtained basedon using same resolution of mean head information. For practicalapplications, the local correction areas for HSM can have finer gridspaces to resolve the local solutions in a more detailed manner.

Fig. 7 shows the velocity variances that are plotted along thedomain center line. The results clearly show that the NSM, HSM,and the MCS obtain identical solutions for entire modleing area.In this example the ASM overestimates the longitudinal velocitySTD and underestimates the transverse velocity STD near the welllocation; however, the inaccuracy for transverse velocity STD is rel-atively small compared with the longitudinal velocity STD. As isshown in Fig. 7, the hybrid method does improve the solutionaround the well location and only require the numerical calcula-tion in a small area (i.e., 20k by 20k). In this example, the gridnodes in numerical calculation for HSM reduces from 81 � 81(i.e., the NSM) to 21 � 21 (i.e., the HSM). Based on our Pentium DCPU 3.4 GHz system, the CPU times for estimating such solutionsare in the order of seconds and minutes for the HSM and NSM,respectively. However, the MCS based on 256 � 256 grid nodesand 10,000 equally likely realizations took several hours to obtainthe corresponding solution.

Large-scale and highly nonstationary flow

Based on the concept and the observations from previous exam-ples, we then extend the modeling area to 300k by 300k in the

x1 (m)0 20 40 60 80

Nonstationary Spectral Method

x1 (m)

x 2(m

)

0 20 40 60 800

20

40

60

80Approximate Spectral Method

x1 (m)

x 2(m

)

0 20 40 60 800

20

40

60

80

x1 (m)0 20 40 60 80

x1 (m)0 20 40 60 80

1.00.90.80.70.60.50.40.30.20.10.0

σu2x1 (m)

0 20 40 60 80

1.00.90.80.70.60.50.40.30.20.10.0

σu1Monte Carlo Simulation

Fig. 4. The distributions of velocity STDs (t = 1.0 day) estimated by ASM (left column), NSM, and MCS (right column).

σ u1

0.0

0.5

1.0

1.5

2.0Approximate Spectral MethodNonstationary Spectral MethodMonte Carlo Simulation

x1 (m)

σ u2

0 20 40 60 800.0

0.2

0.4

0.6

0.8

1.0

σ u1

0.0

0.5

1.0

1.5

2.0

1.5 day

1.25 day1.25 day

1.5 day

x1 (m)

σ u2

0 20 40 60 800.0

0.2

0.4

0.6

0.8

1.0

1.25 day

1.5 day

1.5 day

1.25 day

a

b

Fig. 5. The centerline profiles (y = 40 m) of velocity STDs (at t = 1.25 and 1.5 days) estimated by ASM, NSM, and MCS.

302 C.-F. Ni et al. / Journal of Hydrology 381 (2010) 297–307

third example. To increase the flow complexity in this example, asynthetically generated trend and several sources and sinks areadded in the modeling area. Although the overall background headgradient from west to east boundaries is fixed, the introducedsources and sinks here have significantly changed the flow patterns(see Fig. 3a for the conceptual model and Fig. 3b for mean flowpattern).

In this example we select two correction areas where the flowsare considered to be highly nonstationary. The two correctionareas with dimension 100k by 100k are created for three well loca-tions and the lake (see Fig. 3a). Similar to example 1, the lake stagein this example is also specified by a constant head to represent theextreme situation that the water body is well connected withgroundwater. After the correction areas are defined for HSM, we

Time (day)

σ u1

2 4 6 80.0

0.5

1.0

1.5

2.0Monte Carlo SimulationNonstationary Spectral MethodApproximate Spectral Method

Time (day)

σ u2

2 4 6 80.0

0.2

0.4

0.6

0.8

1.0Time (day)

σ u1

2 4 6 80.0

0.5

1.0

1.5

2.0Monte Carlo SimulationNonstationary Spectral MethodApproximate Spectral Method

a

x1 = 30 m

x1 = 50 m

Time (day)

σ u2

2 4 6 80.0

0.2

0.4

0.6

0.8

1.0b

x1 = 30 m

x1 = 50 m

Fig. 6. The temporal fluctuations of velocity STDs estimated by ASM, NSM, and MCS at points (30, 40) and (50, 40) in the modeling area.

x1 (m)

σ u2

0 20 40 60 800.0

2.0

4.0

6.0

8.0

Hybrid Spectral MethodApproximate Spectral MethodNonstationary Spectral MethodMonte Carlo Simulation Nr=10000

b

σ u1

0.0

2.0

4.0

6.0

8.0a

Fig. 7. Centerline profiles of velocity STDs that are estimated by HSM, ASM, NSM,and MCS.

C.-F. Ni et al. / Journal of Hydrology 381 (2010) 297–307 303

run four different models individually on our Intel Pentium D3.4 GHz Workstation. The orders of CPU times are in seconds, min-utes, hours, and days for ASM, HSM, NSM, and MCS, respectively.

Fig. 8 shows the contour of velocity STDs for ASM, NSM, andMCS. The results reveal that the ASM has captured the major non-stationary pattern of the velocity STDs by comparing with solu-tions of the full version of first-order numerical method, theNSM. However, when the modeling areas close to well fields and

the lake, the analytical-based ASM yields slightly inaccurate solu-tions. Figs. 9–12 show the velocity STDs estimated by four differentmethods in the correction areas. Figs. 9 and 10 show the correctionarea around the well fields, while Figs. 11 and 12 present the localcorrection area near the lake region. With the local correction pro-cedures, the HSM significantly improves the solutions around welland lake locations. There is slightly inaccuracy for HSM solutionsnear the boundaries of the correction areas. Such solution inaccu-racy is caused mainly by the inconsistency of boundary valuesfrom ASM solutions. However, the inaccuracy is limited in a verysmall region and can be removed by extending the distances of cor-rection areas to 2 and 3k.

Conclusion

We have proposed a conceptually improved stochastic methodcalled hybrid spectral method (HSM) that integrates analytical andnumerical spectral methods in the solving procedures to predictflow uncertainties in complex, large-scale heterogeneous porousmedia. Following the observations in Ni and Li (2005, 2006), thisstudy first created an example to evaluate the ASM accuracy forflow with transient and inner boundary conditions (lake stage fluc-tuations). This study then used two examples to illustrate the basicconcept and the applications of the HSM in complex large-scaleflow systems. The first example involving the condition of lakestage fluctuations reveals that the ASM can capture accuratelythe variance dynamics for transient flow situations. Although thereare slightly inaccurate regions along the lake edges as comparedwith the corresponding nonstationary numerical solutions ob-tained from NSM and MCS, those regions are limited in 1–5k fromthe lake edges. Based on the concept of HSM, solutions in suchinaccurate regions can be improved by locally applying numeri-cal-based NSM.

The comparison results in the second and third examplesshowed that the proposed HSM can accurately predict the detaildynamics of flow uncertainty for highly nonstationary groundwater

Fig. 8. The comparison of velocity STDs estimated by ASM (left column), NSM, and MCS (right column) for steady, two-dimensional large-scale problems (the values higherthan 2.0 are drawn in white color).

Fig. 9. The comparison of longitudinal velocity STDs for ASM, NSM, MCS, and HSM in the correction area for the well field (the values higher than 2.0 are drawn in whitecolor).

304 C.-F. Ni et al. / Journal of Hydrology 381 (2010) 297–307

flow problems. The computationally costly numerical method (i.e.,the NSM) is only applied to local areas; the HSM can be very effi-cient for complex large–scale flow problems.

Stochastic theories had been developed for several years butstill have difficulty to be widely used for practical applications,mostly because of the restrictive assumptions and the expensive

Fig. 10. The comparison of transverse velocity STD for ASM, NSM, MCS, and HSM in the correction area for the well field (the values higher than 2.0 are drawn in white color).

Fig. 11. The comparison of longitudinal velocity STDs for ASM, NSM, MCS, and HSM in the correction area for the lake.

C.-F. Ni et al. / Journal of Hydrology 381 (2010) 297–307 305

Fig. 12. The comparison of transverse velocity STDs for ASM, NSM, MCS, and HSM in the correction area for the lake.

306 C.-F. Ni et al. / Journal of Hydrology 381 (2010) 297–307

computation for problems with realistic complexities and sizes.The proposed hybrid method conceptually takes the advantagesof analytical and numerical stochastic theories. Therefore, it pro-vides an opportunity to include stochastic theories in practicalgroundwater modeling problems.

Acknowledgments

This research was supported in part by the National ScienceCouncil of the Republic of China under Contract NSC 97-2116-M-008-002 and NSC 98-2116-M-008-002.

References

Dagan, G., 1989. Flow and Transport in Porous Formations. Springer-Verlag, NewYork.

Dagan, G., 2004. On application of stochastic modeling of groundwater flow andtransport. Stochastic Environmental Research and Risk Assessment 18 (4), 266–267.

Dagan, G., Neuman, S.P., 1997. Subsurface flow and transport: A StochasticApproach. International Hydrology Series. Unesco: IAHS/AISH. CambridgeUniversity Press, New York.

de Marsily, G., Delay, F., Goncalves, J., Renard, P., Teles, V., Violette, S., 2005. Dealingwith spatial heterogeneity. Hydrogeology Journal 13 (1), 161–183.

Deutsch, C.V., Journel, A.G., 1997. GSLIB Geostatistical Software Library and User’sGuide. Oxford University Press, New York.

Gelhar, L.W., 1993. Stochastic Subsurface Hydrology. Prentice-Hall, EnglewoodCliffs, NJ.

Graham, W., McLaughlin, D., 1989a. Stochastic analysis of nonstationary subsurfacesolute transport 1. Unconditional moments. Water Resources Research 25 (2),215–232.

Graham, W., McLaughlin, D., 1989b. Stochastic analysis of nonstationary subsurfacesolute transport 2. Conditional moments. Water Resources Research 25 (11),2331–2355.

Guadagnini, A., Neuman, S.P., 1999a. Nonlocal and localized analyses of conditionalmean steady state flow in bounded, randomly nonuniform domains 1. Theoryand computational approach. Water Resources Research 35 (10), 2999–3018.

Guadagnini, A., Neuman, S.P., 1999b. Nonlocal and localized analyses of conditionalmean steady state flow in bounded, randomly nonuniform domains 2.Computational examples. Water Resources Research 35 (10), 3019–3039.

Guadagnini, A., Riva, M., Neuman, S.P., 2003. Three-dimensional steady state flow toa well in a randomly heterogeneous bounded aquifer. Water ResourcesResearch 39 (3), 1048. doi:10.1029/2002WR001443.

Li, S.G., Liu, Q., 2006a. A real-time, interactive steering environment for integratedground water modeling. Ground Water 44 (5), 758–763.

Li, S.G., Liu, Q., 2006b. Interactive Ground Water (IGW). Environmental Modellingand Software 21 (3), 417–418.

Li, S.G., McLaughlin, D., 1991. A nonstationary spectral method for solvingstochastic groundwater problems – unconditional analysis. Water ResourcesResearch 27 (7), 1589–1605.

Li, S.G., McLaughlin, D., 1995. Using the nonstationary spectral method to analyzeflow through heterogeneous trending media. Water Resources Research 31 (3),541–551.

Li, S.G., McLaughlin, D., Liao, H.S., 2003. A computationally practical method forstochastic groundwater modeling. Advances in Water Resources 26 (11), 1137–1148.

Li, S.G., Liao, H.S., Ni, C.F., 2004a. A computationally practical approach for modelingcomplex mean flows in mildly heterogeneous media. Water Resources Research40 (12), W12405. doi:10.1029/2004WR003469.

Li, S.G., Liao, H.S., Ni, C.F., 2004b. Stochastic modeling of complex nonstationarygroundwater systems. Advances in Water Resources 27 (11), 1087–1104.

Li, S.G., Liu, Q., Afshari, S., 2006. An object-oriented hierarchical patch dynamicsparadigm (HPDP) for modeling complex groundwater systems across multiple-scales. Environmental Modelling and Software 21 (5), 744–749.

McLaughlin, D., Townley, L.R., 1996. A reassessment of the groundwater inverseproblem. Water Resources Research 32 (5), 1131–1161.

McLaughlin, D., Zhou, Y.H., Entekhabi, D., Chatdarong, V., 2006. Computationalissues for large-scale land surface data assimilation problems. Journal ofHydrometeology 7 (3), 494–510.

Mehl, S., Hill, M.C., 2002. Development and evaluation of a local grid refinementmethod for block-centered finite-difference groundwater models using sharednodes. Advances in Water Resources 25 (5), 497–511.

Miller, C.T., Gray, W.G., 2008. Hydrogeological research, education, and practice. Apath to future contributions. Journal of Hydrologic Engineering 13 (1), 7–12.

Miller, C.T., Abhishek, C., Farthing, M.W., 2006. A spatially and temporally adaptivesolution of Richards’ equation. Advances in Water Resources 29 (4), 525–545.

Mizell, S.A., Gutjahr, A.L., Gelhar, L.W., 1982. Stochastic analysis of spatial variabilityin two-dimensional steady groundwater flow assuming stationary andnonstationary heads. Water Resources Research 18 (4), 1053–1067.

Neuman, S.P., 2004. Stochastic groundwater models in practice. StochasticEnvironmental Research and Risk Assessment 18 (4), 268–270.

Neuman, S.P., 2005. Trends, prospects and challenges in quantifying flow andtransport through fractured rocks. Hydrogeology Journal 13 (1), 124–147.

Ni, C.F., Li, S.G., 2005. Simple closed form formulas for predicting groundwater flowmodel uncertainty in complex, heterogeneous trending media. Water ResourcesResearch 41 (11), W11503. doi:10.1029/2005WR004143.

Ni, C.F., Li, S.G., 2006. Modeling groundwater velocity uncertainty in nonstationarycomposite porous media. Advances in Water Resources 29 (12), 1866–1875.

C.-F. Ni et al. / Journal of Hydrology 381 (2010) 297–307 307

Papoulis, A., 1984. Probability Random Variables, and Stochastic Processes.McGraw-Hill, New York, NJ.

Priestley, M.B., 1981. Spectral Analysis and Time Series. Academic Press, London.Renard, P., 2007. Stochastic hydrogeology: what professionals really need? Ground

Water 45 (5), 531–541.Rubin, Y., 2003. Applied Stochastic Hydrogeology. Oxford University Press, New

York.Smith, L., Freeze, R.A., 1979. Stochastic analysis of steady state groundwater flow in

a bounded domain, 2, two-dimensional simulations. Water Resources Research15 (6), 1543–1559.

Székely, F., 1998. Windowed spatial zooming in finite-difference ground water flowmodels. Ground water 36 (5), 718–721.

Székely, F., 2008. Three-dimensional mesh resolution control in finite differencegroundwater flow models through boxed spatial zooming. Journal of Hydrology351 (3–4), 261–267.

Winter, C.L., Tartakovsky, D.M., 2002. Groundwater flow in heterogeneous compositeaquifers. Water Resources Research 38 (8), 1148. doi:10.1029/2001WR000450.

Ye, M., Neuman, S.P., Guadagnini, A., Tartakovsky, D.M., 2004. Nonlocal andlocalized analyses of conditional mean transient flow in bounded, randomlyheterogeneous porous media. Water Resources Research 40, W05104.doi:10.1029/2003WR002099.

Zhang, D., 1999. Nonstationary stochastic analysis of transient unsaturated flow inrandomly heterogeneous media. Water Resources Research 35 (4), 1127–1141.

Zhang, D., 2002. Stochastic Methods for Flow in Porous Media: Coping withUncertainties. Academic Press, London.

Zhang, D., Lu, Z., 2004. An efficient, high-order perturbation approach for flow inrandom porous media via Karhunen–Loeve and polynomial expansions. Journalof Computational Physics 194 (2), 773–794.

Zhang, Y.K., Zhang, D., 2004. Forum: the state of stochastic hydrology. StochasticEnvironmental Research and Risk Assessment 18 (4), 265.