Numerical simulations of preasymptotic transport in heterogeneous porous...

Numerical simulations of preasymptotic transport in

heterogeneous porous media: Departures from the Gaussian limit

M. G. Trefry

CSIRO Land and Water, Floreat, Western Australia, Australia

F. P. Ruan

Schlumberger GeoQuest, Asker, Norway

D. McLaughlin

Ralph M. Parsons Laboratory, Department of Civil Engineering, Massachusetts Institute of Technology, Cambridge,Massachusetts, USA

Received 30 November 2001; revised 7 March 2002; accepted 7 March 2002; published 19 March 2003.

[1] The objective of this work is to determine whether conventional solutemacrodispersion theories adequately predict the behavior of individual subsurface plumesover time and length scales relevant to practical risk assessment and remediation activities.This issue is studied with a set of high-resolution numerical simulations of conservativetracers moving through saturated two-dimensional heterogeneous conductivity fields.The simulation experiments are designed to mimic long-term field studies. Spatiallycorrelated statistically stationary lognormal conductivity realizations are generated with aFourier transform procedure. Steady state velocity solutions are calculated for these fieldsusing an accurate Darcian solver. Velocity contour plots reveal the presence ofdisconnected networks of preferential pathways over a range of correlation lengths.Reverse flow cells are rare. The velocity probability density functions have exponentialtails and strong longitudinal asymmetries. Solute concentrations are derived from thesimulated velocity fields with an accurate taut-spline transport code that minimizesnumerical dispersion. The resulting plumes are tracked for travel distances of over onehundred spatial correlation lengths, corresponding to scales of practical interest. Firstmoments and macrodispersivities, which measure the location and extent of the plume, arereasonably well approximated by conventional Fickian theories but continue to vary afterlong travel distances. This temporal variability highlights the slow convergence of theplume moments to Gaussian limits. Calculations of the relative dilution index, which is ameasure of the plume mixing state, indicate strongly non-Gaussian behavior. Othermeasures of plume structure suggested by anomalous dispersion theories also reveal thepersistence of non-Gaussian behavior after long travel distances. These measures appear tobe more sensitive to non-Gaussian behavior than the spatial moments. Taken together, thesimulation results suggest that conservative solute plumes moving through statisticallystationary random media may not converge to Gaussian limits even after travelinghundreds of log conductivity correlation scales. INDEX TERMS: 1829 Hydrology: Groundwater

hydrology; 1832 Hydrology: Groundwater transport; 1869 Hydrology: Stochastic processes; KEYWORDS:

groundwater, solute, stochastic, anomalous transport, simulation

Citation: Trefry, M. G., F. P. Ruan, and D. McLaughlin, Numerical simulations of preasymptotic transport in heterogeneous porous

media: Departures from the Gaussian limit, Water Resour. Res., 39(3), 1063, doi:10.1029/2001WR001101, 2003.

1. Introduction

[2] One of the major topics of subsurface solute transportresearch has been the effect of geological heterogeneity onthe movement of conservative tracers. When viewed atsufficiently large scales heterogeneity tends to disperse orspread solutes, a process commonly called macrodispersion.This process changes significantly over time. At first, when

the dimensions of a plume are smaller than local geologicalanomalies its shape is controlled by individual features suchas conductive channels or low permeability barriers. Thesefeatures induce variations in the plume’s concentrationcontours as some parts of the plume advance more rapidlythan others. If geological anomalies are of limited extentand are scattered throughout space in a random fashion theireffects will tend to ‘‘average out’’ and the plume willeventually become more regular.[3] This conceptual picture of macrodispersion is sup-

ported by the Central Limit Theorem, which implies that theCopyright 2003 by the American Geophysical Union.0043-1397/03/2001WR001101$09.00

SBH 10 - 1

WATER RESOURCES RESEARCH, VOL. 39, NO. 3, 1063, doi:10.1029/2001WR001101, 2003

location of a solute particle subject to many small displace-ments has a Gaussian probability distribution in the limit asthe travel time approaches infinity, provided that the dis-placements are drawn at random from a population withfinite moments and a finite correlation scale [Gnedenko andKolmogorov, 1968]. In the infinite time limit the ensemblemean solute concentration has a Gaussian spatial distribu-tion, with moments that depend on the statistics of therandom velocity field [Dagan, 1989]. The differencebetween the concentration of an individual plume and theensemble mean concentration is a random quantity with avariance that approaches zero at infinite time if the CentralLimit Theorem applies [Dagan, 1990]. It follows thatindividual plumes should approach the spatially Gaussianensemble mean in the infinite time limit. This explains theemphasis on the ensemble mean that characterizes stochas-tic transport theory. Under the assumptions of this theorythe ensemble mean should eventually provide an accuratedescription of any particular real world plume.[4] A number of investigators have studied how the

ensemble mean of a conservative tracer approaches theGaussian limit [Gelhar and Axness, 1983; Dagan, 1989,1990; Rubin, 1990]. Eulerian approaches to this problem arebased on the classical advection-dispersion equation, whichdescribes transport at small (e.g., column) scales:

r: DvrCð Þ � v � rC ¼ @C

@tð1Þ

where v is a random velocity field with specified statisticalproperties, Dv is a local dispersion tensor that describes theeffect of velocity fluctuations at scales smaller than thoseresolved by v (emphasized by including the subscript v) andC(x, t) is the random concentration at the location vector xand time t. Here, confining attention to two-dimensionalprocesses, we suppose that the principal axes of Dv arealigned with the x and y coordinates so Dv is diagonal:

Dv ¼vxALv 0

0 vxATv

0@

1A ð2Þ

The coefficients ALvand ATv

are the longitudinal andtransverse local dispersivities, respectively. This formula-tion assumes that the small-scale dispersion process isFickian (i.e. the dispersive flux is proportional to theconcentration gradient).[5] Under certain assumptions the mean of the solute

concentration, C, is described by a larger-scale version of(1):

r: DvrC� �

� v � rC ¼ @C

@tð3Þ

where v is the mean solute velocity (e.g., a known regionaltrend) and Dv is a diagonal macrodispersion tensor thatdescribes the effect of velocity fluctuations at scales smallerthan those resolved by v(emphasized by including thesubscript v):

Dv ¼vxALv

0

0 vxATv

0@

1A ð4Þ

[6] The coefficients ALvand ATv

are the longitudinal andtransverse macrodispersivities, respectively. In (3) macro-scopic dispersion is Fickian, in the sense that the macro-dispersive flux is proportional to the mean concentrationgradient. The macrodispersivities are generally larger thanthe corresponding local dispersivities since v is smootherthan v and the Dv tensor must account for a greater portionof observed solute spreading than Dv . Expressions for themacrodispersivity can be derived from the local dispersiv-ities and the statistics of the small-scale velocity field. Theasymptotic solution to (3) has a Gaussian spatial distributionwhen the velocity and macrodispersion tensors are constant.[7] Spatial moments indicative of Gaussian behavior

have been observed in several controlled field studies inreasonably homogeneous media. Examples include thewidely cited experiments conducted at Borden, Ontario[Sudicky, 1986] and Cape Cod, Massachusetts [Garabedianet al., 1991; Hess et al., 1992]. Fickian behavior has alsobeen simulated in numerical experiments carried out byBoggs et al. [1992], Tompson and Gelhar [1990], Bellin etal. [1992], Salandin and Fiorotto [1998], and McLaughlinand Ruan [2001]. These field and numerical results lendcredibility to the Gaussian model of field-scale transport.On the other hand, evidence at more geologically complexfield sites [Boggs et al., 1992] and from two-dimensionalmultiscale numerical simulations [McLaughlin and Ruan,2001] suggests that individual plumes may not lookGaussian even after reasonably long travel times. Theseplumes may be highly skewed with long leading or trailingedges, may have multiple peaks, or may be divided intodifferent sections following separate preferential pathways.Plume geometry and the spatial distribution of concentra-tion influence chemical and biological reactions and canhave important implications for risk assessment andremedial design. For these reasons it is important to knowhow far particular plumes might deviate from the Gaussianideal.[8] In this paper we investigate plume convergence with

numerical simulations carried out in much the same way asa set of highly controlled field experiments. Numericalsimulations are attractive because they enable us to take aclose look at individual plumes as they evolve over time. Inparticular, they allow us to determine when we canadequately approximate individual plumes by an asymptoticGaussian model based on (3). Since the log hydraulicconductivity and velocity fields used in our experimentshave finite moments and correlation scales the assumptionsof the Central Limit Theorem and traditional Fickian theoryare met. Consequently, our simulated plumes must reach aGaussian state eventually. The primary question in ourinvestigation is how long it takes for this to occur. Asecondary question is whether simple quantitative measuresof plume structure can provide insight about non-Fickianbehavior. Answers to these questions can help us decidewhat methods are most appropriate for describing real-world plumes over the finite timescales of most interestfor practical applications.[9] In the investigations that follow we perform a set of

single replicate simulations rather than Monte Carlo experi-ments because we are interested in the behavior of specificplumes rather than ensemble means. Of course, individualplumes generated from random velocity fields will differ

SBH 10 - 2 TREFRY ET AL.: PREASYMPTOTIC TRANSPORT IN POROUS MEDIA

and care must be taken in generalizing from a small sample,just as in field experimentation. But if ensemble theory is tobe useful in practice it should apply to any given plume, notjust to a large set of plumes and not just at infinite time.[10] Numerical solute transport experiments are much

easier to perform than field experiments of comparablecomplexity. Even so, we do not propose that numericalsimulations should replace either field experiments ortheoretical analyses. Our numerical experiments rely onassumptions that may affect the validity or scope of partic-ular conclusions. Wherever possible, we have tried toanticipate these possibilities in our discussion. However,we believe that our results, when taken as a whole, raiseimportant questions about the applicability of Fickian/Gaussian models. These questions will need to be inves-tigated in more detail in subsequent theoretical, field, andnumerical studies.

2. Experimental Approach

[11] Our experimental approach is based on a simplifiedrepresentation of a plume of conservative solute movingthrough a random time-invariant velocity field. We haveconfined our attention to two-dimensional transport in a rect-angular region lying in the horizontal plane (see Figure 1).We focus on two-dimensional transport primarily for com-putational reasons. As discussed by McLaughlin and Ruan[2001], there is a tradeoff (for a fixed computationalexpenditure) between simulating long-term transport in atwo-dimensional region versus short-term transport in athree-dimensional region. The two-dimensional approachprovides more insight on convergence to Gaussian condi-tions while the three-dimensional approach accounts forvertical velocity variations and transport paths that areignored in two dimensions. We were effectively limited inthis study to problems having O(106) degrees of freedom(feasible to solve on a high-end workstation). This translatesto a two-dimensional region extending in the longitudinaldirection O(103) log transmissivity correlation scales or athree dimensional region extending O(102) log hydraulicconductivity correlation scales. Since our goal is to identifylong-term trends we have adopted the two-dimensionalalternative. Hopefully, improved computational resourcesand numerical algorithms will soon make it feasible tocheck our two-dimensional results with three dimensionalflow and transport models that can handle O(107) or moredegrees of freedom.[12] The velocity fields required in our transport experi-

ments are obtained by solving a steady state flow problem.Fixed head boundary conditions at the upstream and down-stream ends of the domain yield a spatially uniform velocitytrend (or mean) which is parallel to the longest side of a

2048 by 512 grid of square computational cells with length�x = �y = 1.0 (in our experiments length units are metersand time units are days). The other two sides of the domainare no-flow boundaries. The longitudinal hydraulic gradientis J = �0.02 and the uniform porosity n is 0.3.[13] The spatially variable conductivity fields (denoted K,

with F = ln K ) used in our flow simulations are randomrealizations drawn from lognormal probability distributionswith two-dimensional isotropic Gaussian spatial covariancefunctions [McLaughlin and Ruan, 2001]. The Gaussiancovariance yields differentiable conductivity fields that havemore persistence and are somewhat smoother than fieldsobtained from an exponential covariance. The conductivityfields are generated with the Fourier transform proceduredescribed by Ruan and McLaughlin [1998]. Our use ofhydraulic conductivities rather than transmissivities isformally equivalent to expanding the two-dimensionalproblem domain to a three-dimensional domain with unitdepth and material properties that are constant in the verticaldirection. In the remainder of this paper, we maintain theuse of the term conductivity while considering the results ofstrictly two-dimensional simulations.[14] In order to examine the connection between hetero-

geneity and convergence to a Gaussian limit we performed aset of eight simulation experiments which differ only in thevalues used for the nominal variance sF

2 (0.25, 1.0, 2.5, and4.0, see Table 1) and the log conductivity correlation scalelF (2.0 and 8.0 m). The variance and correlation scalevalues used in our simulations are broadly representative offield situations. Reported values of sF

2 in field studies rangefrom below 0.3 [Boggs et al., 1992] to above 4 [Rehfeldt etal., 1992; Christiansen et al., 1998], while lF spans valuesranging from 1 m (our grid cell size) to tens of meters[Jensen et al., 1993]. The number of correlation scalesincluded in our simulation domain is nearly 1024 for lF =2.0 m and nearly 256 for lF = 8.0 m.[15] The eight conductivity fields identified in Table 1

were constructed from two realizations, one for each of thechosen lF values, with nominal means of hF i = ln 6.25 =1.8326. Log conductivity fields associated with the differentvariances were derived from either of these realizationsusing a simple scaling procedure. The sample means,variances, and correlation scales obtained for each of thetwo realizations were within one percent of the nominalvalues.[16] The flow model used in our simulation experiments

is based on a straightforward two-dimensional steady statefinite difference approximation and a preconditioned con-jugate gradient solver. The convergence rate of the iterativesolver was sensitive to the input field variance: approxi-mately 3000 iterations were sufficient for sF

2 = 0.25, whilealmost 9000 iterations were required to achieve conver-

Figure 1. Configuration of the simulation domain used in the numerical experiments.

TREFRY ET AL.: PREASYMPTOTIC TRANSPORT IN POROUS MEDIA SBH 10 - 3

gence at the highest variance level of sF2 = 4. We checked

our flow solver by comparing its simulated head distribu-tions to those calculated by the three-dimensional BIG-FLOW code [Ababou et al., 1988], used in two-dimensionalmode with the conjugate gradient solver option. For well-converged solutions (Euclidean error norm <10�22) thehead distributions were identical to at least 7 digitsprecision.[17] Velocities at the node identified with indices (i,j)

were computed with the following weighted interpolationscheme

v i; jx ¼� 1

2�x

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiKi; jKi�1; j

phi; j�hi�1; j

� �þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiKiþ1; jKi; j

phiþ1; j�hi; j� �

v i; jy ¼� 1

2�y

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiKi; jKi; j�1

phi; j�hi; j�1

� �þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiKi; jþ1Ki; j

phi; jþ1�hi; j� �

ð5Þ

where hi,j is the head, Ki,j is the conductivity, and the(vx

i, j,vyi, j) are the two components of the velocity solution.

[18] The solute concentration distribution associated witheach of the simulations defined in Table 1 was obtainedfrom an accurate taut-spline Eulerian-Lagrangian transportmodel [Ruan and McLaughlin, 1999] that solves theadvection-dispersion equation given in (1) at the 1.0 mscale of our computational grid. That is, the model is able toresolve O(1.0 m) concentration fluctuations caused byvelocity fluctuations with spatial scales of O(1.0 m) andgreater. The Fickian local dispersion term only accounts fordispersion due to unresolved velocity fluctuations withspatial scales smaller than 1.0 m.[19] For consistency with the earlier approximate simu-

lations of McLaughlin and Ruan [2001], the localdispersivities appearing in (2) are set to the prior valuesof ALv

= 0.15 m and ATv= 0.015 m, giving a mean grid

Peclet number vx�x/vxALv 6.7 in the mean flow

(longitudinal) direction. Numerical simulations show thatreducing these values further to limit the influence of thelocal dispersion matrix tends to reinforce the findings of thispaper. The mean velocity vx appearing in (2) is set equal tothe sample mean of the appropriate velocity realization. Allof the sample velocity means were close to 0.42 m/day.Numerical dispersion induced by our taut-spline solutionalgorithm is significantly smaller than the subgrid scaledispersion, especially for local grid Peclet numbers of theorder of the mean value above (see Ruan and McLaughlin[1999] for a detailed discussion).

[20] In our experiments, the solute concentration is speci-fied to be zero on the left boundary of the computationaldomain. Outflow conditions are specified on the remainingboundaries. The outflow conditions are implemented in thedispersive step of the Eulerian-Lagrangian solution algo-rithm by fixing the outflow boundary concentration at thevalue obtained from the preceding advective step [Ruan andMcLaughlin, 1999]. The initial concentration is zeroeverywhere in the computational domain, except over therectangular source region, where it is equal to a specifiedunit source concentration. The rectangular source hasdimensions 10 m by 75 m and is located near the upstream(left) boundary of the simulation domain (see Figure 1).[21] In our computational experiments, we produced

eight different log conductivity fields with our Fouriertransform generator, derived eight corresponding two-dimensional steady state velocity fields from our flowmodel, and then simulated the evolution of the eightcorresponding solute plumes with our transport model.Plume concentration distributions were plotted at selectedtimes and various numerical measures of plume structurewere computed. In this respect, our results are similar inform to those reported for most field experiments. Theseresults are summarized in the following section.

3. Results

3.1. Velocity Fields

[22] The statistical properties of our eight log conductiv-ity and velocity fields are summarized in Table 1. Thelongitudinal velocity means are only weakly dependent onsF2 while the longitudinal and transverse velocity variances

scale approximately with sF2. Note that the velocity means

and variances are consistently higher (albeit slightly) for thelarger correlation scale. A simple interval estimation method[Greene, 2000, p.146] for the velocity variances shows thatthe 95% confidence intervals for the respective lF = 2 and8 m variances overlap substantially when the effectivenumber of correlation scales (Table 1) is used as a measureof sampling degrees of freedom. In other words, finite-domain effects may account for the observed inflations invelocity statistics.[23] Figure 2 shows the contours of log conductivity

and velocity magnitude (vx2 + vy

2)1/2 in a 300 m by 300 mportion of the simulation domain for lF = 2 and 8 m and forsF2 = 0.25, 1 and 4.0. The velocity solutions reveal the

presence of preferential flow channels (shown as red-colored paths), even for moderate sF

2. For sF2 = 0.25, these

Table 1. Means and Variances of the Log Conductivity and Velocity Fields Used for the Numerical Experimentsa

Experiment

Log Conductivity Statistics Velocity Statistics

lF Correlation Lengths in Domain sF2 �vx svx

2 �vy svy2

1 2 1024 0.25 0.419 1.57(�2) 2.12(�4) 5.04(�3)2 2 1024 1.0 0.422 6.37(�2) 3.53(�4) 2.23(�2)3 2 1024 2.5 0.426 1.62(�1) 5.08(�4) 6.43(�2)4 2 1024 4 0.430 2.65(�1) 6.18(�4) 1.13(�1)5 8 256 0.25 0.423 1.74(�2) �1.64(�3) 5.56(�3)6 8 256 1 0.426 7.15(�2) �3.75(�3) 2.54(�2)7 8 256 2.5 0.431 1.85(�1) �6.98(�3) 7.64(�2)8 8 256 4 0.435 3.02(�1) �9.64(�3) 1.39(�1)

aPowers of 10 are indicated in parentheses.


channels are mostly disconnected and of short length. AssF2 increases the number of such channels stays approxi-

mately constant but connectivity improves. At highvariances, the dominant channels that emerge conveysignificant fractions of the total flow, although no singlechannel persists through the length of the grid. In terms ofabsolute accuracy, Salandin and Fiorotto [1998] give arelevant discussion on the optimal choice of grid spatialincrement and correlation scale in discretized models. Inparticular they show that discretization effects on velocity

solutions are worst for small correlation scale and highvariance F fields, but are likely to be small and will tendto suppress extremes in the flow fields. The consistency ofdevelopment of preferential flow paths with increasingvariance between the two columns of Figure 2 givesconfidence that grid discretization effects are not undulyinfluencing the lF = 2 m, sF

2 = 4 velocity solution.[24] The probability distributions of the velocity solutions

are of particular quantitative interest. Bellin et al. [1992]identify and discuss salient features of velocity probability

Figure 2. Comparison of velocity magnitudes calculated for two different F realizations with correlationscales of 2 m (left top) and 8 m (right top). The region plotted is a 300 m � 300 m portion of thecomputational domain. Colocated velocity solutions for F variances of 0.25, 1, and 4 are shown below eachF realization. Mean flow is from left to right. See color version of this figure at back of this issue.


distributions for sF2 � 1.6, showing nonzero third moments

for the longitudinal components and Gaussian behavior forthe transverse components. That study was performed withlog conductivity fields generated with an exponentialcorrelation function. Features of our velocity fields areexamined in Figure 3 by plotting sample density (orfrequency) functions on a decadic logarithm scale. With thisscale a velocity probability density that can be approxi-mated by a decaying exponential function plots as a straightline. A density that can be approximated by a Gaussianfunction plots as a quadratic with a negative slope ofincreasing magnitude and a density that can be approxi-mated by a power law at large velocities (e.g., a Paretodensity) plots as a logarithmic function with a negativeslope of decreasing magnitude.[25] The lF = 2 and 8 m probability densities match

almost exactly for both velocity components shown inFigure 3. The longitudinal densities are strongly asymmetricaround the mean and clearly non-Gaussian. At high positivevelocities the density functions decay exponentially. Only asmall fraction of the longitudinal velocities are negative,indicating that reverse flow cells are rare. The transversevelocity probability densities are symmetric with exponen-tial tails, and cover a greater range of values as sF

2 increases.For comparison purposes, Figures 3b and 3d show idealGaussian densities (dotted lines) with variances set equal tothe lF = 2 m numerical values (Table 1). It is clear that theexponential transverse densities differ from the Gaussiandensities reported by Bellin et al. [1992], and are in generalqualitative agreement with the densities calculated bySalandin and Fiorotto [1998]. Furthermore, it is importantto note that the asymptotically exponential forms obtainedfor our velocity probability densities imply that the velocitymoments of all orders are finite, so the Central LimitTheorem holds. Consequently, the concentration spatial

distributions obtained from our velocities should eventuallybecome Gaussian. The key question is how long it will takefor this to occur.

3.2. Conventional Methods For Characterizing theExperimental Solute Plumes

3.2.1. Plume Maps[26] Contour plots of solute concentration are one of the

best ways to interpret the results of field or numericaltransport experiments. Figure 4 shows the plumes obtainedfor all eight of our simulations at time t = 2000 days. Theplots in each column show the effects of increasing the logconductivity variance (from top to bottom) for a given logconductivity correlation scale (lF = 2 m in the left columnand lF = 8 m in the right column). It is apparent that thesolute plumes become much more elongated (longitudinalmacrodispersion is much greater) when the log conductivityvariance increases. Transverse macrodispersion also in-creases modestly with variance. These effects are consistentwith classical macrodispersion theory. Increasing the logconductivity correlation length results in significant distor-tion of the plume as it moves away from its rectangularsource. When the variance and correlation scale are bothlarge the local anomalies dominate. The plume becomesvery elongated and dispersed, extending tendrils throughpreferential flow channels in advance of the plume.[27] Similar plots could be produced for other random log

conductivity realizations. Each of these would reflect theeffects of its particular set of geological anomalies. How-ever, our primary purpose here is not to describe thebehavior of an ensemble of random replicates but to studythe long-term evolution of a single plume. If macrodisper-sion theories are to be useful for predicting solute transportin practical situations, they must be able to provide reason-able descriptions of individual plumes over the timescales

Figure 3. Comparison of probability density functions (PDFs) for the velocity solutions for sF2 = 2 m

and 8 m and field variance sF2 = 1, 2.5. In Figures 3b and 3d, ideal Gaussian PDFs are also drawn (dotted

curves) for comparison.


of interest for risk assessment and remediation. In order toexamine this topic further we consider in the followingsections some useful bulk measures of plume structure.3.2.2. Spatial Moments[28] The first and second spatial moments of solute

concentration are widely used to measure the convergenceof field plumes to a Gaussian limit. The (i,j)th spatialmoment of concentration C(x, t) is defined as follows:

Mij tð Þ ¼Z Z

xiy jC x; y; tð Þdx dy ð6Þ

The plume mass at any time t is M00(t) and the coordinatesof its center of mass are mx(t) = M10(t)/M00(t) and my(t) =M01(t)/M00(t). In all the simulations performed here, theinitial plume mass M00(0) is conserved to within 5%.[29] Figure 5 plots versus travel time the deviations

(normalized by lF) of the x and y components of the centerof mass from the values mx = vxt and my = 0 that would beexpected if the plume followed the mean velocity vector.Note that at any given time the plume travels morecorrelation lengths (but the same absolute distance) whenlF = 2 m than when lF = 8 m. It takes about 476 days forthe center of mass to travel 100 correlation lengths whenlF = 2 m and about 1904 days for it to travel 100 correlationlengths when lF = 8 m. Figures 5a and 5c show that thelongitudinal center of mass coordinates depart from thecorresponding mean velocity values by approximately fivecorrelation lengths, although this deviation is variable alongthe trajectories. In some regions the plumes are ahead of themean trajectory while elsewhere they lag behind. Thesediscrepancies are affected by sF

2, but are relatively small.

The transverse center of mass deviations shown in Figures5b and 5d are no greater than a few correlation lengths,although their magnitude tends to increase with sF

2. There isno evidence of convergence of the actual center of masstrajectories to the mean values, even after the plumes havetraveled over distances of 100 log conductivity correlationscales or more.[30] The distribution of mass about the plume centroid is

described by the central spatial moments:

Mcij tð Þ ¼

Z Zx� mx tð Þ½ �i y� mx tð Þ½ �jC x; y; tð Þ dx dy ð7Þ

The second central moments M20c (t) and M02

c (t) are ofparticular interest here since their square roots provideconvenient measures of the distances the plume has spreadin the longitudinal and transverse directions, respectively. Ifthe principal axes of the macrodispersion tensor are alignedwith the x and y coordinates then time-averaged effective(plume-specific) values for the longitudinal and transversemacrodispersivities ALv

(t) and ATv(t) can be computed from

the asymptotic relationships [Tompson and Gelhar, 1990]:

ALvtð Þ ¼ 1

t

Z t

0

1

2vxM00

dMc20

dtdt ¼ Mc

20 tð Þ �Mc20 0ð Þ

2vxM00t

ATvtð Þ ¼ 1

t

Z t

0

1

2vxM00

dMc02

dtdt ¼ Mc

02 tð Þ �Mc02 0ð Þ

2vxM00t

ð8Þ

Time histories for both ALv(t) and ATv

(t) (normalized by sF2)

are plotted in Figure 6.

Figure 4. Comparison of solute plumes for the simulated range of lF and sF2, at time t = 2000 days. A

black rectangle is drawn to indicate the position of the initial distribution advecting at mean flow velocityvx. Full-grid (2047 m � 511 m) solutions are plotted using a decadic logarithm concentration scale. Seecolor version of this figure at back of this issue.


[31] Fickian transport theories predict that the longitudi-nal and transverse macrodispersivities should both be con-stant when the plume shape is nearly Gaussian. Theestimated macrodispersivity values plotted in Figure 6 arecharacterized by irregular variations, reflecting the effects offluctuations in the velocity field. These macrodispersivityvariations tend to be magnified as sF

2 increases. Forcomparison, Figures 6a and 6c also show the theoreticalFickian asymptotic values for the longitudinal macrodis-persivities. These theoretical values apply for Gaussian-correlated log conductivity fields and are derived by Ruan[1997, p. 209]:

ALv

s2F¼ 1:25

lF

g2and

ATv

s2F¼ ALv þ 3ATvð Þ

16g2ð9Þ

where g = O(sF2) is a so-called flow factor, intended to

correct for the effects of the linearization process [Gelharand Axness, 1983; Gelhar, 1993, p. 113]. Dagan [1989, p.320] questions the mathematical consistency of the flow

factor derivation. Here, for simplicity, g is assumed to beunity.[32] The normalized longitudinal macrodispersivities

plotted in Figure 6a and 6c track each other, suggestingthat they scale approximately with sF

2. This is in accord withtheoretical predictions. However, the curves do not appearto be converging to the theoretical values even after theplume has traveled approximately 100 correlation lengths.The transverse macrodispersivities in Figures 6b and 6dvary even more than the longitudinal values, although thisresult is of less importance since the theoretical variance-scaled transverse macrodispersivity is very small (0.012m) in our case. In fact, the major benefit of the transversemacrodispersivity plots is to provide an indirect upperbound on numerical dispersion. Inspection of Figure 6shows that the variance-scaled transverse macrodispersiv-ities are overestimated by approximately 0.2 m, adiscrepancy that arises from both numerical dispersionand the detailed dispersive influences of fluctuations in theDarcian flow field. The contribution of numerical dispersionto this discrepancy is difficult to quantify, since the local

Figure 5. Plume centroid locations plotted as functions of travel time for different variances sF2. Figures

5a and 5c show departures from the mean longitudinal travel velocity vx of the plumes while Figures 5band 5d show lateral deflections of the plumes. A travel distance of 100 correlation lengths corresponds toa travel time of 476 days when lF = 2 m and 1904 days when lF = 8 m. See color version of this figure atback of this issue.


grid Peclet numbers vary spatially with v (a mean velocityscale is used in the local dispersion tensor (2)). However thetotal discrepancy is close to the assumed local longitudinaldispersivity ALv

= 0.15 m, suggesting that numerical artifactsaccount for only a small fraction of the longitudinaldispersion observed in our simulations.3.2.3. Dilution Index[33] The lower-order spatial moments discussed above

provide helpful information about the location and overalldimensions of a solute plume but are not sensitive to itsinternal structure. For example, plumes with the samemoments could have either single or multiple peaks andcould have significantly different shapes. Additional insightabout the degree of mixing within a plume can be obtainedby examining the dilution (or entropy) index proposed byKitanidis [1994]. This index is defined as:

I tð Þ ¼ E tð ÞEmax tð Þ ð10Þ

where

E tð Þ ¼ exp �XNk¼1

pk log pk

" #ð11Þ

and Emax(t) is the maximum possible value of E(t), which,for effectively unbounded plumes, is a simple functional ofthe plume second spatial moments [Kitanidis, 1994]. Theratio pk(t) = Ck(t)�x�y/M00(t) is the normalized solute massat time t in the cell centered on node k. The dilution index islargest when the plume is well mixed and smallest when theplume mass is highly concentrated. It can be shown that E(t)is maximized, for a specified set of second central moments,when the spatial distribution of the solute concentration isGaussian [Kitanidis, 1994]. In this sense I(t) measures thedegree to which the plume has approached the idealGaussian limit. The lower the dilution index, the furtherthe plume is from this limit.[34] Figure 7 shows the temporal evolution of the dilution

indices for the eight plumes simulated in our numericalexperiment as well as the deterministic sF

2 = 0 plume, whichprovides a convenient reference. The deterministic plume isinitially non-Gaussian since it starts from a rectangularsource, but its dilution index comes reasonably close to themaximum entropy value of unity after a short time. Thedilution indices for nonzero log conductivity variancesapproach the maximum entropy limit very slowly. The lF =8 m dilution indices are especially variable, suggesting thatthey are still being influenced by local geological anomaliesafter thousands of days (hundreds of log conductivitycorrelation lengths).

Figure 6. Effective plume macrodispersivities plotted as functions of travel time (longitudinal, Figures6a and 6c; transverse, Figures 6b and 6d) calculated using (8). Horizontal black lines in Figures 6a and 6crepresent the theoretical asymptotic limits based on linearized flow + transport theories. The theoreticalasymptotic limit for Figures 6b and 6d is approximately 0.012 m. A travel distance of 100 correlationlengths corresponds to a travel time of 476 days when lF = 2 m and 1904 days when lF = 8 m. See colorversion of this figure at back of this issue.


3.2.4. Summary[35] The concentration contour plots, effective macro-

dispersivities, and dilution indices presented here providea consistent picture of the long-term behavior of solutesmoving through heterogeneous porous media. Generallyspeaking, our experimental plumes approach the Gaussianlimit rather slowly, taking much more than 100 log con-ductivity correlation scales to attain moment or dilutionindex values compatible with the classical Gaussian modelof macrodispersion. When geological anomalies are morepronounced (larger log conductivity variance) or when theypersist for distances comparable to the plume dimensions(larger log conductivity correlation scale) the plume is moreirregular and the moments and dilution indices are more

variable. The spatial second moments seem to be relativelyinsensitive to deviations from Gaussian behavior and pro-vide only an aggregate picture of plume dimensions. This isconfirmed by examining plume contour plots at times whenthe longitudinal dispersivities (or second spatial moments)are relatively near theoretical predictions. Such plots oftenreveal multiple concentration peaks and internal structurethat differs significantly from the regular contours of aGaussian plume. The dilution index appears to be a some-what more sensitive measure of non-Gaussian behavior, asit increases only gradually toward the ideal value of unity asthe plume becomes more well-mixed and Gaussian inappearance.

3.3. Characterization Methods Suggested byAnomalous Dispersion Theory

[36] The results presented above suggest that the traveldistance required for convergence to the Gaussian limit ina stationary heterogeneous medium may be longer (hun-dreds of log conductivity correlation scales or thousands ofmeters) than generally believed. The practical implicationsof this slow convergence are problem-dependent. But it isapparent from our simulations that the peak concentrationsof the plume must drop by orders of magnitude below thesource concentration before the plumes can be reasonablyapproximated by a Gaussian model. The distinctly non-Gaussian character of our preasymptotic plumes suggeststhat it may be useful to investigate the behavior of trans-port measures suggested by non-Fickian (anomalous)transport theories. Although our plumes should eventuallybecome Gaussian they could resemble anomalous plumesin certain respects over the finite timescales simulated inour numerical experiments. We investigate this possibilityby evaluating several relatively simple anomalous transportmeasures.[37] In order to understand these measures it is helpful to

review some basic concepts of anomalous transport. Manytransport equations have common origins in probabilisticrandom walk theories. In these theories, the evolution of asolute plume is modeled by the distribution of a largenumber of particles that undergo random displacements(jumps) in space over random intervals in time (pausingtimes). If each random particle displacement dx and time dtis drawn randomly from the same transition probabilitydensity pxt(dx, dt), the spatial distribution of particlepositions will evolve with time in a well-defined statisticalmanner. This transition probability density is the basis forthe generalized master equation approach used by Berkowitzand Scher [2001] (the reader is referred to Jazwinski [1970]for a useful discussion of random walk theory). Thisapproach gives an anomalous solute transport equation withfractional derivatives:

DraC � v � rC ¼ @bC

@tbð12Þ

where the constant D describes the spreading of the plumein terms of fractional units. The fractional derivatives maybe defined in terms of the spectra of the derivative terms.Note that the anomalous mean transport equation reduces tothe traditional mean equation of (3) when a = 2.0 and b =1.0.

Figure 7. Dilution index (10) versus time for (a) lF = 2 mand (b) lF = 8 m, each for the four different logconductivity variances indicated. An index of 1 correspondsto the maximum entropy (perfect Gaussian) case. A traveldistance of 100 correlation lengths corresponds to a traveltime of 476 days when lF = 2 m and 1904 days when lF =8 m. See color version of this figure at back of this issue.


[38] If all moments of pxt are finite the Central LimitTheorem applies and the asymptotic probability density of atypical particle or, equivalently, the spatial distribution ofthe mean solute concentration, will be Gaussian. If pxt hasfinite spatial moments but infinite temporal moments, thenthe transport is temporally anomalous and b will generallydiffer from 1.0 [Berkowitz and Scher, 2001]. Such fractional(in time) processes, arising from long-tailed pausing timedensities, have been shown to give rise to long-tailedbreakthrough curves and unusual time dependences ofplume lower spatial moments.[39] Similarly, if pxt has finite temporal moments but

infinite spatial moments, then the transport is spatiallyanomalous and a will generally differ from 2.0 [Benson etal., 2001]. In this case the fractional spatial derivativecoefficient can be related to the velocity probability density.Consider a continuous particle velocity process v with afirst-order probability density p(v) that approaches a powerlaw of the form p(v) � |v|�1�a for large v. If the associatedindex of stability, a, is greater than or equal to 2 the velocityprocess has finite lower-order moments. If a is less than 2the velocity is called an a-stable Levy process [Meerschaertet al., 1999]. In this case, it has infinite lower-ordermoments, the Central Limit Theorem does not apply, andthe particle displacement does not approach an asymptoticGaussian limit [Benson et al., 2001]. We discuss belowsome methods for estimating the coefficients a and b fromobserved (or simulated) concentrations. Estimates of thesecoefficients provide additional quantitative measures of thedegree to which a preasymptotic plumes deviates from theGaussian limit.3.3.1. Spatially Anomalous Dispersion[40] First, we examine the values obtained for the stabil-

ity index a associated with the spatially anomalous trans-port model [Benson et al., 2001]. In this case, the meansolute concentration is described by (12), with thedispersion derivative order a < 2 and the temporalderivative order b = 1. The peak mean concentration Cmax

obtained at any given time is given by [Benson et al., 2001]:

Cmax tð Þ ¼ C0

Dtð Þ1=að13Þ

where C0 is the initial concentration of the instantaneoussource, D is a fractional spreading constant (unspecifiedhere) and t is time. This expression suggests that it ispossible to estimate the a value associated with a particularexperimental plume from a log plot of the maximumsimulated concentration versus time. The third column ofTable 2 presents the results of fitting (13) to theconcentration solutions obtained from our eight numericalexperiments, as well as a nominal deterministic experimentwith sF

2 = 0. Figure 8 shows the resulting log plots of Cmax

versus time. Since the zero-variance nominal case includesonly local (Fickian) dispersion it should yield an asymptotica value of 2. The value fit over a finite length of the curve is2.04, a value that is reassuringly close to the Fickian limit.For nonzero log conductivity variances, the fitted a values

Figure 8. Peak plume concentrations, showing thedeparture from Gaussian conditions as the log conductivityvariance sF

2 increases. The fitted asymptotic slopes of thecurves are related to the listed spatial anomaly (a) indices.See color version of this figure at back of this issue.

Table 2. Fitted Fractional Indices a and b of the Spatial and

Temporal Anomaly Modelsa

Experiment Fitted Parameters

lF sF2 a (b = 1) (b,�s) (a = 2)

L = 743, L = 11432 0 2.04 (�0.996, 0.44), (�0.996, 0.44)2 0.25 1.62 (0.992, 0.46), (0.993, 0.46)2 1.0 1.23 (0.973, 0.50), (0.977, 0.48)2 2.5 1.10 (0.964, 0.55), (0.969, 0.52)2 4.0 1.15 (0.943, 0.61), (0.949, 0.60)8 0 2.04 (�0.996, 0.44), (�0.996, 0.44)8 0.25 1.62 (0.984, 0.51), (0.984, 0.48)8 1.0 1.16 (0.958, 0.57), (0.965, 0.54)8 2.5 0.85 (0.918, 0.74), (0.923, 0.71)8 4.0 0.92 (0.897, 0.89), (0.898, 0.90)

aExperiments of Table 1 are referenced by the ensemble correlationlengths and variances of F. Uncertainties lie in the second decimal place fora and in the third and second digits for b and �s, respectively.


are all less than 2, tending to decrease in magnitude withincreasing sF

2. The exception to this trend is for the sF2 = 4

curve, which has a slightly larger a value than the sF2 = 2.5

curve. This reversal is also seen in the lF = 8 m results(Table 2).[41] For nonzero variances the peak concentration curves

in Figure 8 are not smooth, varying irregularly under theinfluence of the spatial heterogeneity of the underlying logconductivity fields. The lF = 8 m curves exhibit greatervariations, making a fits somewhat more subjective.Nevertheless, it is clear from these simple tests that oursimulated plumes do not follow the classical (a = 2) Fickianvariation of peak concentration with time. Thus it may beconcluded that the preasymptotic plumes display character-istics of anomalous Levy motions, even though the inputvelocity fields are not Levy stable.3.3.2. Temporally Anomalous Dispersion[42] The previous subsection focuses on plumes obtained

from a transport equation with a fractional spatial derivativeand a conventional first-order time derivative (b = 1). In thissection, we consider the case where the time derivative is offractional order and the spatial derivative has the conven-tional coefficient a = 2. Berkowitz and Scher [2001] use aone-dimensional analysis to derive the breakthrough curveC(t) = f (L, s, b, t) of a Dirac delta of solute at x = 0 and t = 0for this case, where L is the breakthrough location and s isan unknown transition length [Montroll and Scher, 1973].The normalized cumulative breakthrough curve Manom(t) is:

Manom tð Þ ¼Z t

0

f L; s; b; t0ð Þdt0 ð14Þ

with s and b treated as fitting parameters. This anomalousbreakthrough curve can be compared with the standardFickian result for lognormal correlated fields [Dagan andNguyen, 1989]:

MFick L;ALv; v; t

� �¼ 1

21� erf

L� vtffiffiffiffiffiffiffiffiffiffiffiffiffi4ALv

vtp

!" #ð15Þ

where ALvis the asymptotic longitudinal macrodispersivity

and v is the mean flow velocity (in the x direction). In theFickian case, v and L are known, leaving the macrodis-persivity as the sole free parameter for fitting.[43] Figure 9 shows results from fitting Manom and MFick

to numerical breakthrough curves calculated at two controlsections (L = 743 m and L = 1143 m downgradient of thesource) perpendicular to mean flow in the two-dimensionalsimulation grids. The numerical breakthrough curves wereestimated by integrating the plume mass present in theregion downgradient of the control section for each outputtime, dividing by the total plume mass, and assemblingthese values into discrete time series (represented by squaresin Figure 9). The curves represent best fits to the numericaldata points. For Manom, greatest weight was given tomatching the breakthrough front, rather than the tail. Thisweighting process also yields the closest fits to the data in aleast squares sense.[44] Overall, the Fickian curves match the shape of the

data points well, with fitted dispersion coefficients (ALv/sF

2 =3.42 and 2.62 m for the 743 m and 1143 m control planes,

respectively, for sF2 = 1, and ALv

/sF2 = 3.33 and 3.25 m,

respectively, for sF2 = 4) in good agreement with the

effective longitudinal macrodispersivities plotted in Figure6 and reasonably close to theoretical values derived from(9). The anomalous breakthrough curves do not provide asgood a fit. While these curves can reproduce sharp fronts,they are unable to represent the short tails of thebreakthrough data. Calculations for all fields show that thisdisagreement is greatest for small lF and sF

2 (i.e. near-Fickian conditions), and least for large lF and sF

2 (non-Fickian conditions).

Figure 9. Breakthrough curves calculated at two controlsections, L = 743 m (solid rectangles) and 1143 m (openrectangles), downgradient of the source, for two correlatedfields (Figures 9a and 9b). Solid curves represent best fitFickian breakthrough functions MFick(15, fitted parametersin text), while dashed curves represent best fit temporalanomaly functions Manom (14, fitted parameters in Table 2).


[45] The final column of Table 2 lists fitted values of band s for the range of numerical experiments. Even thoughthese temporal anomaly parameters do not lead to preciseagreement with all the breakthrough data, it is useful to tryto understand what they signify when fit to breakthroughfronts. The zero-variance Fickian limit experiment results ina lower bound for b of 0.996, close to the ideal value ofunity. Transition lengths are no more than unity, in qual-itative agreement with Berkowitz et al. [2000] who, in acolumn study, found s to be less than the field correlationscale and the corresponding transition velocity to be some-what higher than the mean velocity v. Several trends arenoticeable. First, b exponents decline and transition lengthsincrease as sF

2 increases. This may be interpreted as show-ing the increasingly energetic nature of the velocity as thelog conductivity field becomes more heterogeneous. Asecond trend is that, for a given field, the exponentincreases and transition length decreases as attention movesfrom an earlier control section to a later one. These slightchanges reflect the more diffuse nature of the plume as itpasses the second control section. A third trend is thatexponents for lF = 2 m are somewhat higher (less anom-alous) than for lF = 8 m, and the transition lengths areshorter.[46] The suitability of the temporally anomalous model

may also be examined in terms of other measures. Forexample, it is known that the ratio of the plume meanlongitudinal displacement to the standard deviation of theplume varies as t1/2 under Fickian conditions while underanomalous conditions the ratio is constant [Berkowitz andScher, 2001]:

Fickian caseM10 tð Þ �M10 0ð Þffiffiffiffiffiffiffiffiffiffiffiffiffi

Mc20 tð Þ

p � t1=2

Anomalous caseM10 tð Þ �M10 0ð Þffiffiffiffiffiffiffiffiffiffiffiffiffi

Mc20 tð Þ

p � constant:

ð16Þ

[47] Figure 10 shows plots of this moment ratio for oureight experimental plumes. In Figure 10, all of the large-time moment ratio exponents (slopes on this log-log plot)are reasonably close to the theoretical Fickian value of 1/2,although the slopes vary somewhat for higher variances andcorrelation scales. Overall, there is no compelling evidencethat our preasymptotic experimental plumes resemble tem-porally anomalous dispersion.

4. Discussion and Conclusions

[48] The numerical experiments described in this paperindicate that simulated plumes may not reach the Gaussianlimit predicted by a Fickian macrodispersion model evenafter travel distances of hundreds of log conductivitycorrelation scales. This is not to say that the plumes willnot or cannot converge to that limit. In fact, we know fromthe Central Limit Theorem that plumes generated fromvelocity fluctuations with finite moments and finite corre-lation scales must approach a Gaussian limit. Since thevelocities used in our numerical experiments are of thistype, the resulting plumes should eventually become Gaus-sian. What is at issue is how far a plume must travel beforeit can be reasonably approximated by a Gaussian transport

model. Our experiments suggest that this could be fartherthan the distances of interest in some risk assessment andremediation problems.[49] The slow rate of convergence to the Gaussian limit is

clearly related to the nature of the heterogeneities encoun-tered by the plume as it travels downgradient. Log con-ductivity anomalies that deviate significantly from theregional mean and which are comparable in size to theplume dimensions can be expected to have a greater effectthan smaller more localized features. During the preasymp-totic non-Gaussian regime, significant portions of the plumeare diverted into preferential pathways or around lowconductivity zones, leading to irregularities that are dis-tinctly non-Gaussian. These irregularities can be seen in

Figure 10. Moment ratios (16) of plumes for varying logconductivity variance sF

2 and correlation length lF. Thedashed line indicates the ideal Gaussian-asymptotic slope of1/2 for the log-log plot. See color version of this figure atback of this issue.


concentration contour plots but are less evident in temporalplots of spatially aggregated quantities such as spatialmoments and breakthrough curves. The dilution index andpeak concentrations are more sensitive measures of non-Gaussian conditions. In our simulations, both of thesemeasures clearly reveal significant deviations from theGaussian limit at distances of hundreds of log conductivitycorrelation scales.[50] Since our preasymptotic plumes are not Gaussian it

is worth considering to what extent their behavior resemblesthe anomalous dispersion predicted by non-Fickian trans-port theories. This can be determined by estimating thecoefficients a and b that appear in the anomalous transportequation given in (12). The a values estimated from oursimulated plumes indicate our preasymptotic plumes dis-play behavior that resembles spatially anomalous dispersion(a < 2) for a range of log conductivity variance (sF

2) andcorrelation scale (lF) values. The a values decreasestrongly with increasing sF

2 and weakly with increasinglF, although curves for high lF are more variable. Ourestimated b values do not provide clear evidence ofbehavior resembling temporally anomalous dispersion.[51] The results summarized above and in Figure 8

suggest that our simulated log conductivity and velocityfields include features that are sufficiently persistent ordistinctive to produce behavior similar to anomalous dis-persion. Although this behavior will eventually be over-whelmed by the leveling effect of the Central LimitTheorem it is still clearly discernible in plumes that havemoved hundreds of correlation lengths. If individualplumes exhibit anomalous non-Fickian characteristics overtimes and distances of interest for risk assessment andremediation, there is a real need for an alternative totraditional Fickian transport models. In particular, it maybe useful to consider applying the concepts of anomalousdispersion theory to preasymptotic problems with finite(rather than infinite) velocity moments. If models based onthis theory can reproduce observed dilution indices andfractional derivative coefficients, as well as observedspatial moments, they may provide better predictionsduring the critical early stages of plume movement, whenconcentrations are highest and the plume is still relativelywell contained.[52] Non-Fickian transport theories may be even more

important when the log conductivity fields include morethan one distinct scale [McLaughlin and Ruan, 2001]. Insuch cases, conductivity variations at different scales arelikely to interact in complex ways, possibly extending thetime it takes for the plume to approach a nearly Gaussianstate.[53] It is evident from our simulations that we need better

ways to describe the aggregate behavior of individualpreasymptotic plumes, particularly in situations where fac-tors such as peak concentrations and contact areas betweencontaminated and uncontaminated zones are important. Thisneed for improved characterization of real-world transportphenomena should motivate additional theoretical andnumerical investigations of near-source dispersion in heter-ogeneous media.

[54] Acknowledgments. The authors wish to thank P. Van Bemmel(Schlumberger GeoQuest) for support and P. L. Bjerg (Technical Universityof Denmark) for valuable discussions. Comments and criticisms from

anonymous reviewers have led to improvements in this work. Part of thework presented in this paper was performed using facilities of the JointBureau of Meteorology/CSIRO High Performance Computing and Com-munications Centre (Melbourne).

ReferencesAbabou, R., D. McLaughlin, and L. W. Gelhar, Three-dimensional flow inrandom porous media, Tech. Rep. 318, 833 pp., Ralph M. Parsons Lab.,Mass. Inst. of Technol., Cambridge, 1988.

Bellin, A., P. Salandin, and A. Rinaldo, Simulation of dispersion inheterogeneous porous formations: Statistics, first-order theories, con-vergence of computations, Water Resour. Res., 28(9), 2211–2227,1992.

Benson, D. A., R. Schumer, M. M. Meerschaert, and S. W. Wheatcraft,Fractional dispersion, Levy motion, and the MADE tracer tests, Transp.Porous Media, 42, 211–240, 2001.

Berkowitz, B., and H. Scher, The role of probabilistic approaches to trans-port theory in heterogeneous media, Transp. Porous Media, 42, 241–263, 2001.

Berkowitz, B., H. Scher, and S. E. Silliman, Anomalous transport in la-boratory-scale, heterogeneous porous media, Water Resour. Res., 36(1),149–158, 2000.

Boggs, J. M., S. C. Young, L. M. Beard, L. W. Gelhar, K. R. Rehfeldt, andE. E. Adams, Field study of dispersion in a heterogeneous aquifer, 1,Overview and site description, Water Resour. Res., 28(12), 3281–3291,1992.

Christiansen, J. S., P. Engesgaard, and P. L. Bjerg, A physically and che-mically heterogeneous aquifer: Field study and reactive transport model-ing, in Groundwater Quality: Remediation and Protection, IAHS Publ.,250, 329–336, 1998.

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

Dagan, G., Transport in heterogeneous porous formations: Spatial moments,ergodicity, and effective dispersion, Water Resour., 26(6), 1281–1290,1990.

Dagan, G., and V. Nguyen, A comparison of travel time concentrationapproaches to modeling transport by groundwater, J. Contam. Hydrol.,4, 79–91, 1989.

Garabedian, S. P., D. R. LeBlanc, L. W. Gelhar, and M. A. Celia, Large-scale natural gradient tracer test in sand and gravel, Cape Cod, Massa-chusetts, 2, Analysis of spatial moments for a nonreactive tracer, WaterResour. Res., 27(5), 911–924, 1991.

Gelhar, L. W., Stochastic Subsurface Hydrology, Prentice-Hall, Old Tappan,N. J., 1993.

Gelhar, L. W., and G. L. Axness, Three-dimensional stochastic analysis ofmacrodispersion in aquifers, Water Resour. Res., 19(1), 161–180,1983.

Gnedenko, B. V., and A. N. Kolmogorov, Limit Distributions for Sums ofIndependent Random Variables, Addison-Wesley-Longman, Reading,Mass., 1968.

Greene, W. H., Econometric Analysis, 4th ed., Prentice-Hall, Old Tappan,N. J., 2000.

Hess, K. M., S. H. Wolf, and M. A. Celia, Large-scale natural gradienttracer test in sand and gravel, Cape Cod, Massachusetts, Water Resour.Res., 28(8), 2011–2027, 1992.

Jazwinski, A. H., Stochastic Processes and Filtering Theory, Academic,San Diego, Calif., 1970.

Jensen, K. H., K. Bitsch, and P. L. Bjerg, Large-scale dispersion experi-ments in a sandy aquifer in Denmark: Observed tracer movements andnumerical analyses, Water Resour. Res., 29(3), 673–696, 1993.

Kitanidis, P. K., The concept of the dilution index, Water Resour. Res.,30(7), 2011–2026, 1994.

McLaughlin, D., and F. Ruan, Macrodispersivity and large-scale hydrogeo-logic variability, Transp. Porous Media, 42, 133–154, 2001.

Meerschaert, M. M., D. A. Benson, and B. Baumer, Multidimensionaladvection and fractional dispersion, Phys. Rev. E, 59(5), 5026–5028,1999.

Montroll, E. W., and H. Scher, Random walks on lattices, IV, Continuous-time walks and influence of absorbing boundaries, J. Stat. Phys., 9(2),101–135, 1973.

Rehfeldt, K. R., J. M. Boggs, and L. W. Gelhar, Field study of dispersion ina heterogeneous aquifer, 3, Geostatistical analysis of hydraulic conduc-tivity, Water Resour. Res., 28(12), 3309–3324, 1992.

Ruan, F., A numerical investigation of solute plumes in nested multi-scaleporous media, Ph.D. thesis, Mass. Inst. of Technol., Cambridge, Mass.,1997.


Ruan, F., and D. McLaughlin, An efficient multivariate random field gen-erator using the Fourier transform, Adv. Water Resour., 21, 385–399,1998.

Ruan, F., and D. McLaughlin, An investigation of Eulerian-Lagrangianmethods for solving heterogeneous advection-dominated transport pro-blems, Water Resour. Res., 35(8), 2359–2373, 1999.

Rubin, Y., Stochastic modeling of macrodispersion in heterogeneous porousmedia, Water Resour. Res., 26(1), 133–141, 1990. (Correction, WaterResour. Res., 26(10), 2631, 1990.)

Salandin, P., and V. Fiorotto, Solute transport in highly heterogeneousaquifers, Water Resour. Res., 34(5), 949–961, 1998.

Sudicky, E. A., A natural gradient experiment on solute transport in a sandaquifer: Spatial variability of hydraulic conductivity and its role in thedispersion process, Water Resour. Res., 22(13), 2069–2082, 1986.

Tompson, A. F. B., and L. W. Gelhar, Numerical-simulation of solutetransport in three-dimensional, randomly heterogeneous porous-media,Water Resour. Res., 26(10), 2541–2562, 1990.

��M. G. Trefry, CSIRO Land and Water, Underwood Avenue, Floreat,

Western Australia 6014, Australia. ([email protected])

F. P. Ruan, Schlumberger GeoQuest, Solbraaveien 23, Asker, 1372,Norway.

D. McLaughlin, Ralph M. Parsons Laboratory, Department of CivilEngineering, Massachusetts Institute of Technology, Cambridge, MA02139, USA.


Figure 2. Comparison of velocity magnitudes calculated for two different F realizations withcorrelation scales of 2 m (left top) and 8 m (right top). The region plotted is a 300 m � 300 m portion ofthe computational domain. Colocated velocity solutions for F variances of 0.25, 1, and 4 are shownbelow each F realization. Mean flow is from left to right.

TREFRY ET AL.: PREASYMPTOTIC TRANSPORT IN POROUS MEDIA

SBH 10 - 5

Figure 4. Comparison of solute plumes for the simulated range of lF and sF2, at time t = 2000 days. A

black rectangle is drawn to indicate the position of the initial distribution advecting at mean flow velocityvx. Full-grid (2047 m � 511 m) solutions are plotted using a decadic logarithm concentration scale.


SBH 10 - 7

Figure 5. Plume centroid locations plotted as functions of travel time for different variances sF2 . Figures

5a and 5c show departures from the mean longitudinal travel velocity vx of the plumes while Figures 5band 5d show lateral deflections of the plumes. A travel distance of 100 correlation lengths corresponds toa travel time of 476 days when lF = 2 m and 1904 days when lF = 8 m.


SBH 10 - 8

Figure 6. Effective plume macrodispersivities plotted as functions of travel time (longitudinal, Figures6a and 6c; transverse, Figures 6b and 6d) calculated using (8). Horizontal black lines in Figures 6a and 6crepresent the theoretical asymptotic limits based on linearized flow + transport theories. The theoreticalasymptotic limit for Figures 6b and 6d is approximately 0.012 m. A travel distance of 100 correlationlengths corresponds to a travel time of 476 days when lF = 2 m and 1904 days when lF = 8 m.


SBH 10 - 9

Figure 7. Dilution index (10) versus time for (a) lF = 2 m and (b) lF = 8 m, each for the four differentlog conductivity variances indicated. An index of 1 corresponds to the maximum entropy (perfectGaussian) case. A travel distance of 100 correlation lengths corresponds to a travel time of 476 dayswhen lF = 2 m and 1904 days when lF = 8 m.


SBH 10 - 10

Figure 8. Peak plume concentrations, showing the departure from Gaussian conditions as the logconductivity variance sF

2 increases. The fitted asymptotic slopes of the curves are related to the listedspatial anomaly (a) indices.


SBH 10 - 11

Figure 10. Moment ratios (16) of plumes for varying log conductivity variance sF2 and correlation

length lF. The dashed line indicates the ideal Gaussian-asymptotic slope of 1/2 for the log-log plot.


SBH 10 - 13

Numerical simulations of preasymptotic transport in heterogeneous porous...

Documents

Transcript of Numerical simulations of preasymptotic transport in heterogeneous porous...