temporal moment-generating equations: modeling transport and mass transfer in heterogeneous aquifers

7/29/2019 Temporal moment-generating equations: Modeling transport and mass transfer in heterogeneous aquifers

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WATER RESOURCES RESEARCH, VOL. 31, NO. 8, PAGES 1895-1911, AUGUST 1995

Temporal moment-generating equations: Modeling transportand mass transfer in heterogeneousaquifersCharlesF. Harvey and StevenM. GorelickDepartmentof Geologicaland EnvironmentalSciences, tanfordUniversity,Stanford,CaliforniaAbstract. We presentan efficientmethod or determining emporalmomentsofconcentration or a solute subject o first-order and diffusivemass ransfer n steadyvelocity ields. The differentialequations or the momentsof all ordershave the sameform as the steadystate nonreactive ransportequation.Thus temporal momentscan becalculatedby a solute ransportcode that was written to simulatenonreactivesteadystatetransport,even though he actual ransportsystems reactiveand transient.Higher-ordermomentsare found recursively rom lower-ordermoments.For many casesa smallnumberof momentssufficiently escribe he movementof a soluteplume. The first fourmomentsdescribe he accumulatedmass,mean, spread,and skewness f the concentrationhistoriesat all locations.Actual concentrationhistoriesat any location can beapproximatedrom the momentsby applying he principleof maximumentropy,aconstraintconsistentwith the physicalprocess f dispersion. he forms of the moment-generatingequations or different mass ransfer modelsprovide nsight nto reactivetransport hroughheterogeneous quifers.For the mass ransfermodelswe considered,the zeroth moment n a heterogeneous quifer is independentof the mass ransfercoefficients. hus, if the velocity ield is known, the mass ransportedpast any point, orout any boundary,can be calculatedwithout knowledgeof the spatialpattern of masstransfer coefficients nd, in fact, without knowledgeof whether mass ransfer is occurring.Also, for both first-order and diffusive mass transfer models, the mean arrival timedependson the distributioncoefficient ut is independentof the valuesof the ratecoefficients,egardless f the spatialvariabilityof groundwater elocityand mass ransfercoefficients.

IntroductionAt a location downgradient rom a contaminantspill or atracer njection,soluteconcentrationsise and then fall as thesoluteplume passes y. This concentration istory,or break-throughcurve, s often adequately ummarized y a few low-

order moments. The first four moments describe the accumu-lated mass and the mean, variance, and skewness of thebreakthroughcurve.We present an efficient method for calculating emporalmoments of concentration subject to complex initial andboundaryconditions n heterogeneous quifers.This methodextends he analytic work of Kucera [1965], Schneider ndSmith [1968], Villermaux 1981], Valocchi 1985, 1986],Parkerand Valocchi 1986],GoltzandRoberts1987],and Sardinet al.[1991].These authorsdeterminedexplicitexpressionsor tem-poral moments n infinite and semi-infinitehomogeneous o-mainswith an initial point sourceof mobile concentration. yderivingmoment-generating ifferentialequations,we providea numericalmethod or determining emporalmomentsof anyorder in heterogeneous quifers with complex initial andboundary onditions. erhapsof morevalue, he form of thesedifferential equationsprovides nsight nto solute transportsubject o rate-limited mass ransfer n complexvelocity ields.In the next section, emporalmomentequations re derivedfor a varietyof mass ransfermodels: ocal equilibrium, inearCopyright1995 by the American Geophysical nion.Paper number 95WR01231.0043-1397/95/95WR-01231505.00

rate-limited mass transfer, and diffusive mass transfer withimmobiledomainsof differentgeometries. he local equilib-rium moment equationsare similar to the equations or tem-poral moments of a conservative olute in a turbulent shearflow derived by Tsai and Holley [1978]. For all of the masstransfermodels, he moment-generatingifferentialequationshave the same orm as the steadystate nonreactive ransportequation.Temporal momentscanbe calculated, fter a simplechange of variables, with a solute transport code that waswritten to simulatenonreactive teadystate transport.The moment equations demonstrate some fundamentalpropertiesof solutes hat are subject o rate-limitedmass rans-fer and are transported hroughspatiallyheterogeneous qui-fers. For example, the zeroth moment at all locations n aheterogeneous quifer is independentof mass ransfercoeffi-cients.Thus the mass ransportedpast any point, or out anyboundary,can be calculatedwithout knowledgeof the spatialpattern of mass transfer coefficientsand, in fact, withoutknowledge of whether mass transfer is occurring.Also, themean arrival time of a solute dependson the distributioncoefficient ut not on the rate coefficientsthe first-orderki-neticand mmobiledomaindiffusion oefficients). ecausehemoment-generating quationsare derivedwithout specifyingthe velocity ield, these resultscan be generalized o differentspatialpatternsof velocity,and to a wide range of boundaryconditions.These propertiesare also independentof initialconditions nd spatialheterogeneityn mass ransfercoefficients.

In the third section we apply the moment-generatingmethod o severalhypothetical xamples.We showhow maps1895


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1896 HARVEY AND GORELICK: TEMPORAL MOMENT GENERATING EQUATIONS

of temporal momentscan be calculated or different velocityfields and how solute behavior differs for the different masstransfer models.

In the fourth section we discuss how the moments can beused o estimateconcentrationst specificimesand ocations.We give an example n which the zeroth through fourth mo-mentsprovidea good approximation f a breakthrough urvedespite complex spatial heterogeneities n both velocity andthe mass transfer coefficients.

Differential Equations for Temporal MomentsThe n th temporalmoment of concentrationn at a partic-ular location is defined as

m n = tnc dt (1)

where C is concentration and t is time. From the definition ofthe Laplace transform,

C = e-ptc dt (2)

it follows that,

ran=1)n(anC/- Op p-->o (3)where is heLaplaceransformf concentrationith espectto time andp is the Laplaceparameter [Aris, 1958].By applying 3) directly to the Laplace transformof thesolute ransportequation,we obtain the differentialequationsfor temporalmoments. hesedifferential quations analsobederived through repeated integration by parts, without using(3); however, he algebra s more difficult. n the next threesubsections e will derive differentialequations or the tem-poral momentsof concentration ubject o three differentmasstransfer models: ocal equilibrium, irst-orderand diffusive.Local Equilibrium

When the local equilibriumassumptionLEA) is applied,the three-dimensional dvective-dispersivequation sOCL(C)= ROt (4)

[James nd Rubin, 1979], where R is the retardation actor,which can vary over space.L is the advective-dispersiveper-ator,

m(*)-x gij -- JOXi, (5)vi is thevelocityn the direction, o is the matrixof disper-sioncoefficients, nd summationover repeatedspatial ndicesi and is implied.Finite momentsof arrival time existonly for boundarycon-ditions that do not allow a continuous nput of solute. Withnonhomogeneousoundary onditions, olutemassmay neverdecline to zero. Consequently, emporal moments are onlymeaningful or the three boundaryconditions:

OC D OCC= 0 --= 0 C "" = 0, (6)Or/ v Or/zero concentration, ero gradient, and zero flux, where r/isperpendicular o the boundary.These conditionsmay be dis-tributed over an irregularlyshapedboundary. n somecircum-stances, he principle of linear superposition an be used toconsider nhomogeneous oundaryconditions.The Laplace ransformof the governing quation 4) is

L(') =pRO-RCo (7)Both R and Co are functionsof space.Throughout he paperboth aquifer parametersand initial conditionsmay be anyarbitrary unctionof location.For brevity, his dependencesnot explicitlynotate& The Laplace transformof the boundaryconditions6) is

O=0 O =0 (8)Or/ v Or/Local equilibrium: erothmoment. Applying 3) with n =

0 to both sidesof (7) provideshe differential quation or thezeroth moment:L(mo) = -RCo (9)

Applying 3) to the boundaryconditions 8) yieldsboundaryconditions or the zeroth moment differentialequation:Omo D Omom0 = 0 --= 0 m0 = 0 (t0)Or/ v Or/

Equations 9) and (t0) do not contain he Laplaceparameteror the Laplace ransformof any quantities, o nverseLaplacetransforms re not required.Although he advective-dispersivequation 4) is a partialdifferential equation containingderivatives n both time andspace,he equation or the zerothmoment 9) and subsequentmomentscontainsonly spatial derivatives.Conveniently, heboundaryconditions t0) for the moment equation alwayshave the same orm of the boundary onditions8) for tran-sient concentration. Both the retardation factor and the initialconcentrations ay have any spatialpattern.The zeroth momentat a particular ocation s the area underthe breakthrough urve ecordedat that location.The productof the zeroth moment and the groundwater lux providesameasureof the mass o pass he location. t is the mass luxintegratedover time, and we will refer to it as he accumulatedmass. If the concentrations are resident concentrations, thenthe accumulatedmass s the mass o advectpast hat location.If the concentrations re flux averagedconcentrations,henthe accumulatedmass s the mass o advectand disperse astthe location n the directionof flow [Parker nd van Genuchten,1984;Sposito t al., 1986].Calculatinga map of the zeroth moment by numericallysolving 9) is very efficient.By specifying onstantsourcesequal to R Co, steadystate numericalsolute transportcodescan be used to determine the zeroth moments hroughout hedomain with a computationalexpenseequivalent o a singlesteady state simulation or the first time step of a transientsimulation. n contrast,many time stepsmay be required tocalculate he zeroth momentby transientsimulation.Numerical simulationof the temporal moment-generatingequations s more robust han simulationof transientconcen-trations;however, imulations f temporalmoments an suffer


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HARVEY AND GORELICK: TEMPORAL MOMENT GENERATING EQUATIONS 1897

from the same oscillations s numerical calculationof steadystate ransport.When the grid P6cletnumber s large, oscilla-tions will occur at the sharp front upstream rom a "source."However, since steadystate fronts are not as sharp as early-time fronts, oscillations hat appear in transient simulationsmay not appear in temporal moment simulations.Also, sincewe are only interested n the first five or so moments,propa-gatingerrors may not be as severeas when hundredsof timesteps re calculated. urthermore,since ime is not discretizedin a steadystate simulation, arge Courant numberswill notcausenumerical naccuraciese.g., Sudicky, 1989]. AppendixA shows hat numerical solution of the temporal momentequations greescloselywith an analyticsolution,even for asharp ront.Local equilibrium: irst moment. The differentialequationfor the first moment s found by applying 3) with n = i toboth sidesof the governing quation 7). First (7) is differen-tiated with regard to p'

L =pR+RO (11)From 3) weknow hatC(p - 0) = moandaC/ap(p --) O)= -mi. Applying hese ules to (11) yields

L(ml) = -Rmo (12)Using (3) to determine he boundaryconditions or the firstmomentputs he boundary onditions10) in termsof m i butleaves heir form unchanged.Equation (12) describeshe first moment as a functionoflocationand shows hat the first moment dependsonly on thezeroth moment and the retardation actor. Again, any numer-ical code used to model steadystate solute transportcan beused to find the first moment. By specifying olute sourceseverywhere qual to the Rmo, the solutionof the steadystatetransport roblemgives he spatially istributedirstmoments.The mean arrival time of solute at any location can becalculated rom the zeroth and first momentsas/x 1 = ml/mo.Thus the mean arrival time /& can be found with a totalcomputational xpense quivalent o two time stepsof a stan-dard transport code: one solution to determine the zerothmoments, and a second solution to determine the first mo-ments.With resident concentrations,/xl s the mean time forsoluteparticles o advectpasta location.With flux concentra-tions, /& is the mean time for particles o advector dispersepast a location n the directionof flow.Goode [1994] has recentlyshown hat mean groundwaterage can be calculatedby solvingan equationsimilar to a mo-ment-generating quation.The mean groundwater ge can becalculatedby placinga hypothetical olutemassproportionalto the fluid flux at all inflow boundariesand calculating hemean arrival time of the solute. Because the zeroth moment isuniformover space, he first-moment quationcan be normal-ized by the value of the zeroth moment o give a singleequa-tion for the mean groundwaterage. We will consideran ex-ample of this approach n a subsequent ectionof the paper.Local equilibrium: nth moments. All higher-order mo-mentsare foundrecursively. epeatedapplication f (3) to theLaplace transform of the advective-dispersivequation (7)shows hat the nth momentdependsonly on the (n - 1)thmoment:

L(mn) = -nRmn_l (13)

The boundaryconditions etain the same orm as the originalboundaryconditions:Omn D Omnmn--O --=0 mn =0 (14)

Since he nth momentsare a linear functionof the (n - 1)thmoments,higher-ordermomentsare alwaysa linear functionof the initial conditionsCo. Because 14) wasderived or anyadvective-dispersiveperatorand for generalboundarycondi-tions, 14) is true for anypatternof heterogeneity r boundaryconditions.t alsoholds rue regardless f the pattern of initialconditions.

Combined with the zeroth and first moments, the second,third,and ourthmoments rovide aluabledescriptionsf theconcentrationistory t any ocation. he standard eviationof the concentrationhistory s calculatedwith the secondmo-ment:

mlo- = - (15)m0This provides a measure of the spread of the breakthroughcurve.The asymmetry f the breakthroughcurve s described,through he third moment,by the skewness oefficient:

m3 m lm2= ---3 +2 o 3/ m0 m (16)A positiveskewness oetficientndicates ailing in the break-throughcurve.The kurtosis, which describes he "peakedness"of thebreakthroughcurve, can be determinedwith the fourth mo-ment. Further refinements f the shapeof the breakthroughcurve can be describedby even higher-ordermoments. n asubsequentection f thispaper,we will describe method orestimatingbreakthroughcurves rom moments of successiveorder.First-Order Rate-Limited Mass Transfer

Many researchershave found that mobile and immobilephases f soluteare not at equilibriumduring ransport Bahr,1989; Jamesand Rubin, 1979; Mackay et al., 1985; Valocchi,1986]. n this case, he transport quation 4) becomesOC aSL(C)= - + 13 t (17)

where S is the immobile concentrationand/3 is the distributioncoefficient. he advective-dispersiveperator L (,) remainsthe sameas for equilibrium ransport 5). The rate of transferbetween the mobile and immobile regions s often modeledwith the linear equation:

aSat = a/(C- S) (18)wherea is the rate coefficient.ogether,17) and (18) de-scribesolute transport hrough an aquifer in which transferbetween mobile and immobile regions is first-order rate-limited. For simplicity,we have not includedequilibriumsorp-tion combined with rate-limited mass transfer. However, theequations n this paper can be used to model the effectsofequilibriumsorptioncombinedwith rate-limited mass ransfer(as in the paperby Nkedi-Kizza t al. [1984]) hroughseveralsimplechanges f variables.


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Table 1. SourceTerms for Moment EquationsZerothL(mo) First L(m ) SecondL(m2) nthL(mn)

Local -RCo -Rmo - 2Rm equilibriumLinear /3S0 _transferCo/3S0(1 /3)m0 -2(1/3)ml2/3Diffusive /3S02(1 /3)ml2A1-transfer Co/3S0 (1+/3)m0A1 t at

-nRmn_ SO n i-nmi- l Otf Som0-2/3 --nmn-l--rt!/3if-1)! a=1

SO n i-ni- l Otd Som-2/3"t2r --nmn-'-nil3A(n-i)i 1)! n!Anr=1

Combining the Laplace transform of the governingequa-tions (17) and (18) yields

L(C) O p+ -Co-So +as (19)+as Pwhere Co and So are the initial mobile and immobileconcen-trations.Settingp equal to zero provides he equation or thezeroth moment:

L(mo) -- -Co- 13So (20)This result shows hat the zeroth moment dependsonly on theinitial mass n the systemand that the zeroth moment can bepredictedwithoutknowledge f the mass ransferparameters,if initial sorbedmass s absent.Because 20) is derivedwithoutspecifyinghe velocity ield, this result s generalizableo dif-ferent spatialpatternsof velocity nd o a widerangeof bound-ary conditions. he properties f (20) are also ndependent finitial conditionsand spatial heterogeneity n mass transfercoefficients.

Differentiating 19) with regard to p and setting to zeroprovides he equation or the first moment

L(mO-m0(1/3) So (21)This result demonstratesn importantpropertyof the firstmomentof arrival time in heterogeneous quiferswhere trans-port is subject o first-order ate-limitedmass ransfer. f thereis no initial immobilesolute So = 0), as n the caseof a spillor tracer test, the first moment of arrival time dependson thedistributioncoefficient/3but not on the actualrate coefficients,regardless f nonequilibrium onditionsor spatialheterogene-ities in either the velocity or the rate coefficients. hus, inspatiallyheterogeneousquifers, he first momentcan be de-termined from the distributioncoefficient/3,even though therate parameters re unknown.This resultextends arlierwork[GoltzandRoberts, 987;Valocchi, 985], or point njectionsninfinite homogeneousquifers, o heterogeneousquiferswithcomplexspatialvelocity ieldsand holds rue for one, two, orthree dimensions.

Higher-ordermomentsdependon the valuesof rate coeffi-cients as well as the distribution coefficient. Table 1 containsthe equation or the secondmomentand the generalequationfor the n th moment. Unlike higher-ordermoments or localequilibrium,which dependdirectlyon only the previousmo-ment, the higher-ordermoments or rate-limitedmass ransferdependdirectlyon all lower-ordermoments.Despite this in-crease n complexity, he relation betweenhigher-ordermo-

ments and lower-order moments remains linear. Because theform is the same as that for a steadystate solute transportequation, he equation or the n th moment can still be solvedusing he right-handsideas a constant ource n a steadystatesolute transport simulation.Thus the temporal momentsofconcentration during transport subject to first-order masstransfercanbe determinedwith a code hat wasnot specificallywritten to considermass ransferbut that can solve he steadystate advective-dispersivequation.Diffusion Into and Out of Grains, Aggregates,and Immobile Regions

Models of diffusive mass transfer between mobile and im-mobile domainshave proven useful or describing oth labo-ratory results [Ball and Roberts,1991; van Genuchten t al.,1984;Pellett,1966;Rao et al., 1982] and field results Harmonet al., 1992]. In the mobile domain, solute s transportedbyadvection and dispersion. n the immobile domain, solutemoves only by diffusion.The concentrationwithin differentgeometries, uchas ayers Sudicky t al., 1985],cylinders Pel-lett, 1966],and spheres Millerand Weber, 986;Nkedi-Kizza tal., 1982], s determinedby applyingFick's aw, with the con-centrationvalue at the boundaryof the immobile domain setequal to the mobile concentration.For diffusion-controlled ass ransfer, he transportequa-tion (17) remains he sameas or the first-ordermodel,exceptnow t is coupledwith a diffusivemass ransfermodel and S isdefined as the averageconcentrationwithin the immobile re-gion:

S= -; r(V-i)Sar (22)The actual mmobileconcentration a varieswith the distancer from the center of the immobile region. The shapeof theimmobile region is given by v. If v = 1, then the system slayeredand (22) is the integralof the concentration a acrossan immobile layer. If v = 2 or 3, the immobile region iscomposed f cylinders r spheres,espectively, nd (22) rep-resents ntegrationover a cylinderor sphere.For layeredge-ometry,b is the half width of a layer. For cylindricalor spher-ical geometry,b is the radius.The concentrationSa within the immobilezones s governedby the diffusion quation:

.... r (v-i) -b < r < b (23)Ot r -)Or --r


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where d is the apparentdiffusioncoefficientwithin the immo-bile domain.This differentialequation (23) is subject o two boundaryconditions. At the interface of mobile water and an immobile

zoneor grain r = b), the concentration f the immobile luidis set equal to the concentration f the mobile luid:Sa = C r = b (24a)

The second oundarycondition s a symmetry ondition. n thecenter of the domain,

0 = 0 r = 0 (24b)Laplace domain solution for transport with diffusive masstransfer. Taking the Laplace ransformof (17) and (22) andcombiningyields

L(O) pO Co lip ; r(V-)ar- flSo (25)The Laplace ransformof the equation or immobileconcen-tration (23) is

_ d O r(V_)Xa-S0 1) ' --r/ (26)(V-The solutions f (26) for the layered,cylindrical, nd sphericalgeometries, ubjecto the boundary onditions24a) and (24b)are

(?)a= -- Coshb(p/d)/2] -- v= 1 (27a)( _)o[r(p/d)/2] o= C- io[b(p/d)/2q-- v= 2 (27b)=(_)bSinh[r(p/d)/2] oSinhb(p/d)/2] v= 3 (27c)

where , is the modifiedBessel unctionof order k. Substitut-ing expressions27a)-(27c) into (25) and integratingyieldsLaplace domain equations or transport subject o diffusionthroughvarious mmobile domain geometries.By employingthe relationships between modified Bessel functions offractional order and hyperbolic tangents and cotangents[Abramowitz nd Stegun,1972, p. 443], a singlegeneralizedequation or all geometries an be determined:

I ' /c- /2[P a)L(C) pC Co vl3(pCSo)(adp)/2 .,---,rv2,,/2-[P a) /(28)

where he diffusiveate coefficienta= d/b2. In theory, yusing ractionalvaluesof v, (28) can be used to determineconcentrationsor geometries hat are between ayered,cylin-drical, and spherical.For example, v = 2.3 could describecigar-shaped eometries.Moment equations or transport with diffusive ransfer. Inthis section,differentialequations or the momentsof concen-tration subject o diffusive ransferare derivedwith the sameprocedure used for local equilibrium and first-order masstransfer.Applying (3) to (28) providesdifferentialequationsfor the temporalmomentsof concentration. he results or the

Table 2. Coefficients for Diffusive TransferA1 A2 An1 2 (24n+4 22n+2)Layer _ m (- 1)n B2n23 15 (2n + 2)!1 1 1 1 11 19Cylinder .......8 48 8 ' 2!(24) 3!(512) 4!(1280)1 2 22n 2Sphere -- (--1)n32n2)! 2n+25 315

B, is the kth Bernoulli number.A o - 1.

zeroth, first, second,and n th momentsare given n Table 1.The moment equationshave the same form for all three ge-ometries,but for each geometry he numericalcoefficients redifferent. These coefficientsare listed in Table 2. Below, theequation and the coefficientsor the n th temporal moment isderived or sphericalmmobilegeometry v - 3). The deriva-tion for the layeredcase v = 1) is similarandwill not be givenhere. A generalequationproviding he n th coefficient or thecylindricalcase v = 2) was not derived. nstead, he coeffi-cients necessary o calculate he first four momentswere de-termined and are shown in Table 2.

After replacing he Bessel unctionwith its hyperbolicco-tangent representation, he Laplace domain transport equa-tion (28) for sphericalmmobilezones v = 3) can be rewrit-ten:

L(&)p&Co313(p&S0)() I/2(29)

To find the equation or the nth moment, 29) is differentiatedn times,and the limit is taken asp goes o 0 (i.e., (3) is appliedto both sides f (29)). First, replacing he hyperbolic otangentby its power series,1 z z 3Cothz) = -+ -- 45

and rearrangingyields

2z 5945

2 2nB n(2n)! 2n-1-''(30)1 1 2(0)=p0 Co 3/3(p0So) - -aJ2p- aJ4p2...

22nB2n[_2npn_3-..)(2n)! (3)where B, is the kth Bernoulli number. Because 31) is nowwritten as a polynomial n p, we can easilydetermine he n thderivativewith respect op, then setp equal to zero to find then th moment. The results for the zeroth, first, and second mo-ments and the generalresult for the n th moment are given nTable 1. Although he expressionsn Table 1 appearcomplex,they require very little computationalexpense o evaluatebe-cause hey nvolvemostly ntegerarithmeticand can be calcu-lated once and stored.

As is the case for first-order mass transfer, the zeroth mo-ment dependsonly on the initial concentrations. lso, in theabsence f initial sorbedmass So = 0), the first moment sindependentof the rate parameter


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Comparison of Different Mass Transfer ModelsWe can determine under what conditions the different mass

transfermodelswill produce he samemoments y comparingthe moment-generating quations n Tables 1 and 2. Theseequationsshow hat the n th moment equations or the first-order and diffusivemodelshave he same orm but differ by thecoefficientsn each erm.By settingOff Cd,henth momentequationsare the same,except or the A coefficients, hichdescribehe effects f different mmobiledomaingeometriesnthe diffusivemomentequations.Belowwe examinehow theseA coefficients ffect the temporal moments esulting rom thedifferent mass ransfer models.Because he advectivedisper-sive operator (5) and the boundaryconditions 6) are notspecified, he resultsare generalizable o differentspatialpat-terns of velocity and dispersivity nd different boundarycon-ditions.

Zeroth moment. All of the mass transfer models consid-ered here will produce the same zeroth momentswhen thedistribution of initial mass is the same. Thus, if(R C0)LEA--- C0 q- /3S0)first-order:C0 q- /3S0)diffusion (32)the zeroth moments at all locations will be the same for LEA,first-order, or diffusive mass transfer models. Since the zerothmoment at any location dependsonly on the initial massandadvectionand dispersion through the operatorL (,)), themass ransportedpast any point, or out any boundary,can befound without knowledgeof the mass ransferproperties. fone s not interested n timing,strategieso control he amountof mass o pass hroughparticular ocations an be devisedbymodeling he zerothmomentwithoutconsideringhe effects fmass-limited mass transfer.

First moment. In the absenceof initial immobile mass, hefirstmoments t all locations ill be the samewhen 32) is trueand the distributioncoefficients re equal:RLEA- ] : /3 irst-order- /3diffusion (33)

Second moment. The second moment for the LEA modelmust be less than the second moment for the two rate-limitedmodels if the zeroth and first moments are the same. Thesecondmoment equations or rate-limited mass ransfer con-tain an addition "source" erm (see Table 1) involving heparametersf andaa. However,f this erm s the same ortwo rate-limited models, they will produce the same secondmoments. Thus in the absence of initial immobile concentra-tion, the zeroth, first, and secondmoments will be the same forthe first-order nd he diffusivemodelsf (32) and (33) are trueand

1 1 1 1..... (34)af 3aa, 8aa,2 15ad,3where the subscripts , 2, and 3 indicate ayered, cylindrical,and sphericalgeometries, espectively.Higher-order moments. If the zeroth, first, and secondmo-mentsagree, he third moments rom the different ate-limitedmodels cannot be the same. In order for the third moments toagree,

1 2 1 2..... (3s)off 15aa, 48aa,2 315aa,3wouldhave o be true. But (35) is inconsistent ith (34). Thusthe different mass ransfer models cannot produce the same

zeroth, first, second, and third moments. When the zeroth,first,and secondmomentsagree,higher-ordermomentswill belarger or first-order, ayered,cylindrical, nd sphericalmodels,respectively.Temporal moments when initial sorbed mass exists. Theconditionsunder which different mass-transfermodels pro-duce the same moments are more restrictive when initial im-mobile massexists So > 0). If the zeroth momentsagreebetween he LEA model and the rate-limitedmodels (32) istrue), then the first momentscannotagree because he rate-limited equationscontain an additional "source" erm (seeTable 1). In order for the zeroth and first moments o agreebetween he different ate-limitedmodels, oth (33) and (34)mustbe true. The secondmoments annotbe equalamonganyof the differentmodels ecause oth (34) and (35) wouldneedto be satisfied, nd these equationsare inconsistent.Comparisonwith previous esults. The coefficientsn (34)have been widely used in chemicalengineering e.g., Viller-maux, 1978], and more recently in hydrology, o determinefirst-order mass transfer rate coefficients such that the first-ordermodelwill produce imilar esultso the diffusion od-els, n homogeneous edia.Valocchi 1985]usedan analyticsolution to show that with the 1/15 coefficient the first-orderand spherical iffusionmodels re equivalent i.e., produce hesamezeroth, first, and secondmoments) or a homogeneousinfinite media with a uniform velocity ield. Burr et al. [1994],Rabideauand Miller [1994], Parker and Valocchi 1986], andHarmon et al. [1989],amongothers,haveused his coefficient.Goltz and Roberts 1987] calculated he coefficientsor layeredand cylindrical iffusionmodels n homogeneousnfinite me-dia with a uniform velocity ield. Due to differentdefinitionsofconcentration, he immobile porosityor bulk densityappearsin the equivalence xpressions sedby someof theseauthors.The resultshere indicate two new facts about these equiva-lences.1. In the absence f initial immobilemass, he equivalenceremainscorrect or heterogeneousields of both velocityandmass ransfer coefficients, nd complex nitial and boundaryconditions. his extendsValocchi's1985] and Goltzand Rob-erts' [1987] derivation or a point injection n an infinite uni-form porousmedia to more complexand realisticsituations.2. The equivalences re not true if there is any initial im-mobilemass So > 0) (the second emporalmomentswill notbe the same). However, if the distributioncoefficients reequaland (34) is satisfied,he zerothand irstmomentswill bethe sameamong he different mass ransfer models.Immobile Concentration Moments

The immobile concentrationmomentsat a particular oca-tion are simplealgebraic unctions f the mobileconcentrationmoments and the mass transfer parameters at that location.Thus, after the mobile moments have been calculated, theimmobile moments at a particular location depend only onquantitiesassociatedwith that location. The expressionsorimmobile concentrationmomentsare given in Table 3. Fordiffusive transfer, these expressions equire the coefficientslisted in Table 2.

The immobile concentration moments do not have usefulphysical nterpretations, uchas mean arrival time, that can beattributed to mobile domain moments.They simply describehow immobileconcentration arieswith time. At a point withno initial sorbedmass, he meanof the immobileconcentrationhistory or first-ordermass ransfer s


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Table 3. Immobile Concentration MomentsZeroth tho First th

Linear transfer

Diffusive transfer

So mo So n mi 1 Somom ml - + n! o +n!-=n mi 1 SoSo mo So n'ZA(n-i).ol-n!Ano A m+ A1 + A2 ' otOld old old i = 0

Tildes ndicate mmobilephase.

1 =/x + (36)where the tild ndicates mmobile phase.Thus the mean ofthe immobile concentrationhistory at a particular point in-creases from the mean arrival time of mobile solute as thevalue of the rate coefficientc at that point decreases.UsingTable 3 and 15), a similar esultcanbe found or the mmobileconcentration variance for first-order mass transfer:

12= 0.2a (37)Thus the varianceof the immobile concentrationhistory alsoincreases from the variance of the mobile concentration as themass ransfercoefficient ecreases. ike previous esults, heseresultsare also rue regardless f heterogeneousmass ransferparameters nd nonuniformvelocity.Two Examples

For two hypothetical olute ransportproblems n heteroge-neousvelocity fields, we demonstratehow momentscan becalculated rom the equations n Table 1. In the first caseweconsidera point sourceof solute transportedby a regionalgradient through a simple structuredpattern of conductivity.In the secondcasewe considersolute njected at a well in amore realisticand complexpattern of heterogeneity eneratedfrom a randomprocess.n both examples, oncentrations erecalculated y the Laplace domain inite elementmethod de-scribed n Appendix B.Example 1: A Rectangular Low-ConductivityZone

Flow and transportwere simulatedusing the conductivityfield shown n Figure la and the parameter values isted inTable 4. All lengthdimensions re normalizedby the lengthofthe domain, and all time dimensions re normalized by therepresentativeime to equilibrium /[a(1 +/3)]. Constant eadboundarieswere applied o the left and right sides.The veloc-ity vectors n Figure lb show hat flow is from left to right,avoiding he rectangleof low conductivity. slugof solutewasintroduced t the point markedwith a cross Figure la); thentransportwas simulatedusing three different mass transfermodels: ocal equilibrium (LEA), first order, and sphericaldiffusion.Both the flow and transportequationswere solvedbythe finite element method using square elementsand linearbasis functions.

The same initial mass was used for all three models. Forfirst-orderand sphericaldiffusivemass ransfer, he initial sol-ute masswas all in the mobile phaseor domain (So -- 0).

Mass ransferparameters ere chosen sing 32)-(34), so hatzeroth and first moments are the same for all mass transfermodels, and second moments are the same for the two rate-limited models. (The secondmoments or the LEA modelcannotbe the sameas for the rate-limitedmodels.)By consideringnormalized velocity, we can compare resi-dence ime with the time to reach equilibrium. n the absenceof advection and dispersion,concentrations ubject to first-order mass ransfer approach equilibrium as e -*, where z istime normalized by the representative ime to equilibriuml/a(1 + /3); z - 1 representshe time to 63% of equilibrium[Harvey t al., 1994].The normalizedvelocity domain engthsover representative quilibrium imes) describeshe extent owhich the plume is affectedby rate-limited mass ransfer. Anormalized velocity much greater than 1 implies that theplume will move through he domainwithout significantmasstransferring o the immobile phaseor domain. The solutewillappear nonreactive.A normalized velocity much less than 1indicates hat the local equilibriumassumptions valid and thesystem an be modeledwith a retardationcoefficient.Becausethe velocities n Figure lb are of the order of 1, we expect heplumes subject o rate-limited mass ransfer to fall betweenthese wo cases. hey shouldexhibit significant preadingdueto rate-limited mass transfer.

Figure 2 showssnapshots f the plume at four times, foreach of the three differentmass ransfermodels.The plumesbehavevery differentlydespitehaving denticalzeroth and firstmoments.The LEA plume moves at a velocity equal to theproductof the groundwater elocityand the retardation actor(1 +/3). The spherical iffusion nd first-orderplumeshave acomponent hat movesat nearly the groundwater elocityanda component hat movesmore slowly at approximately heretardedvelocity)due to rate-limited ransfer rom the immo-bile phase.The first-orderplume s separatednto two partsat time z --2.9. The upstreampart represents olute hat is desorbingromthe area of the spill or tracer test. The downstream art movesat roughly he groundwater elocity.The plume separatesntotwo peaksbecauseof rate-limited mass ransfer; he phenom-enon is not a result of heterogeneity. he two peaks developeven n perfectlyhomogeneousmedia. This behaviorhasbeenstudied n one dimensionby Bellin et al. [1991] and has beendiscussedn the work of Quinodozand Valocchi 1993]. Ap-pendixC demonstrateshis effectwith an analyticsolution ortwo-dimensionalransport.Figure 3 showsbreakthroughcurvesat the point markedwith a cross n Figure la. The first-ordermodel produces hehighestpeak and earliest peak arrival. Notice that this peakarrivesat about 1/11 the arrival time of the LEA peak, consis-


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A) Problem Set-upMeasurementLocation.u.eo

B)Velocity 0'32

E) First Moment F) Mean Arrival Time......... -:.::-4:-.;..x.,-;-..-:::::::.:,:,::::...::ii::;...-.....::::?*:'::'............ -:.'.:.:..:Max = 92 ''"::*Uin = 0 ?Mi"':*8.......;......................'

G) Second Moment LEA I) Standard Deviation LEA

::.... ***.,.,.,.q:,-'-::...::-.......-,...,..-:5,.....?q57:::7................... .Max = 2900 .. .... , ,-' Max = 10 ' Min = 1.4Min = 0H) Second Mom Rate-Limited

Max = 4600Min = 0

J) Standard Dev Rate Limited

Max = 30Min = 4.2

C) Zeroth Moment

Max -- 33Min = 0

D) Total Accumulated Mass

Max = 0.11Min = 0.0

K) Third Moment LEA

Max8000in = 0.0L) Third Moment First-Order Max.5in = 3.0'**'.....': .,. - '

, ::::::..t,... '"'"',.,jj;::::,.....Max 470000 ::::.:.:;:::.i = ,M) Third Mom Spherical Diff.

**::;-- ....Max= 300000Min = 0,0

N) Skewness Coef LEA

Max = 4.8Min = 0.69

O) Skewness Coef First-Order

P) Skew Coef Spherical Diff.

Max = 6.8Min = 0.93

Figure 1. Problem setup and maps of moments or example 1. In all contour maps, dark shades epresenthigh values,and the sixcontour ntervalsare equallyspaced etween he minimumand maximumvalues. a)Geometry of problem domain and locationsof sourceand measurement. b) Groundwatervelocityvectorsthroughproblemdomain. c) Map of the zeroth moment or all three mass ransfermodels. d) Map of theaccumulatedotal mass,which s the productof the zeroth moment Figure lc), the velocity Figure lb), andthe porosity. e) Map of the first moment or all three mass ransfermodels. f) Map of the mean arrival ime,first moment (Figure le) divided by zeroth moment (Figure lc). (g) Map of the secondmoment forequilibriummass ransfer. h) Map of the secondmoment for the first-orderand sphericaldiffusion ate-limitedmodels. i) Map of the standard eviation 15) for equilibriummass ransfercalculatedrom the zeroth(Figure lc), first (Figure l e), and second Figure lg) moments. j) Map of the standarddeviation or therate-limitedmodels alculatedrom the zeroth Figure c), first (Figure e), and second Figure h) moments.(k) Third moment or equilibriummass ransfer. 1) Third moment or first-ordermass ransfer. m) Thirdmoment or spherical iffusionmass ransfer. n) Skewnessoefficient16) for equilibriumcalculated rom thezeroth (Figure lc), first (Figure le), second Figure lg), and third (Figure lk) moments. o) Map of theskewnessoefficientor firstordercalculatedrom the zeroth Figure c), first (Figure e), second Figure h),and third (Figure 11) moments. p) Skewness oefficient or sphericaldiffusioncalculated rom the zeroth(Figure lc), first (Figure le), second Figure lh), and third (Figure lm) moments.

tent with the retardation factor of 11. The sphericaldiffusionmodel produces he lowestpeak, which arrives at a time be-tween the first-order and LEA peaks. n this figure the con-centrationcurve rom the sphericaldiffusionmodel crosseshefirst-ordercurve twice: irst at the baseof the first-orderpeak,- = 3, then again at - = 23. These results show that thedifferent mass transfer models can produce very differentbreakthroughcurveseven though he low-order temporal mo-ments are the same.

Figures lc-lp show maps of the zeroth, first, second,andthird temporal moments or the three mass ransfer models.From these maps,we calculateother maps: he accumulated

mass, the mean arrival time, the standard deviation of thebreakthroughcurve,and the skewness oefficientof the break-throughcurve.These mapsare groupedwith the moment mapsto emphasizehow each of these quantities s calculated rommoments.

The map of the zeroth momentof the mobile phase Figurelc) is the same or all three mass ransfermodelsbecause heinitial distribution of solute concentration is the same for allthree models.The map is calculatedby solving he steadystatetransport equation with a constantsourceequal to the initialtotal soluteconcentration 9) and (20), and the zeroth valuesof Table 1. From the zeroth moments, we calculate the accu-


9/17


Table 4. Parameters or Examples1 and 2Example1' Example21-

Length of domain 1.0 26Width of domain 0.6 26Thickness 0.02 1Dimension of element 0.02 x 0.02 1 x 1Left boundary H = 0.04 H - 0Right boundary H = 0.0 H = 0Top boundary OH/Oy = 0 H = 0Bottom boundary OH/Oy = 0 H = 0Pumping ate n.a. 5Porosity 0.2 0.3Conductivity 9.1 and 1.2 ? = - 1.6, 0 2 = 2.0, X = 9.0DispersivityLongitudinal 0.03 1.5Transverse 0.01 0.5Distributionoefficient/3 10.0 Ln(Ko) = 0, 0-2= 0.5, h = 9First-orderatecoefficientf 1/11 0.021/30'7Sphericaliffusionatecoefficienta 1/165 =af/15) n.a.

Here, n.a., not applicable.*All lengthdimensions avebeen normalizedby the lengthof the domain,and all time dimensions avebeen normalized y the characteristic ass ransfer ime 1/[a(1 + /3)].1.All ength units are meters, and all time units are hours.

mulatedmass total mass o pass ach ocation)by determiningthe product of the zeroth moment and the groundwater lux(the magnitude of the velocity vectors shown n Figure lbmultiplied by the porosity).Again, the map of accumulatedmass in Figure l d is the same for all three mass transfermodels. The contours show that the accumulated mass in-creases round he end of the low-conductivity one,where thefluid flux is greatest.The map of the first moment,shown n Figure le, is also hesame for all three mass transfer models because we chose R =

1 + /3 (33). The accompanying ap of the mean arrival timem (Figure if) is calculatedby dividing he first momentsbythe zeroth moments.Figures lg and lh show maps of the secondmoment. Thefirst-order and sphericaldiffusionmass ransfer models havethe same secondmomentseverywhere.Since the LEA modeland the rate-limited modelsproduced he same irst moments,the second moment must be smaller for the LEA model than

for the rate-limited models. This fact can be understoodbyconsideringhow the moments are calculated or the steady

1.3 = 2.9 = 8.8 = 53.0

1

Cma = 4.9..

Cma =1.8

Cma = 1.2

Cma = 2.0

Cma = 0.71

Cma = 0.62 Cma = 0.008

..., : ........%--::....i...:......Cma = 0.017........._--.:--:"- ..................................... *::'::' ' -------.:::,:

........... ,:':":7:.,/,.;:'d..... . :*'.... , . .:-?:'!:

Cma = 0.17 Cma = 0.12 Cma = 0.024Figure 2. Contour maps of mobile domain concentrations t particular dimensionlessimes for the firstexample.Darker shades epresenthigher values.There are six contour ntervalsequally spacedbetween themaximumvalues and zero. Each row represents oncentrations ubject o a different mass ransfer model.Each column representsa particular instant in time. All three plumes produce the same zeroth and firsttemporal momentsat all locations see Figures lc and le). The two rate-limited models, irst-order andspherical,produce he samesecondmomentsat all locations see Figure lh). Note that, for the first-ordermodel, the plume has two peaks at = 2.9.


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0.4

0.3

C 0.2o.1

o 70

LEA0 10 20 30 40 50 60

Figure 3. Breakthrough curves simulated at the measure-ment location marked by a cross n Figure la. Although thecurves rom the different mass ransfer models appear quitedifferent, hey share he samezeroth and first moments Fig-ures lc and le). The first-orderand spherical iffusion urveshave the same secondmoment (Figure lh). The two spatialpeaks n the first-orderplume (Figure 2) do not appear n thebreakthroughcurve.

state transportequation.For LEA the secondmoment s de-termined by using he first moment as a constantsource.Forspherical iffusion nd first-ordermodels he secondmomentis determinedby usingboth the zeroth and the first momentsas sources. Thus the zeroth moment at the initial source loca-tion adds a "source" to the calculation of the second momentsfor the rate-limited models, which does not exist for the LEAmodel.This source reates argervaluesand a longershapeofthe secondmoment "plume." Figures li and lj demonstratehow the spreadof the breakthroughcurves s larger for therate-limited models than for the LEA model. These figuresshowmapsof the standarddeviation 15) of the breakthroughcurvesand thusprovidea spatiallydistributedmeasureof thespreadof the breakthrough urve.Figures lk-lm are maps of the third moment. The thirdmoment and all higher-order moments are different for thedifferent mass-transfermodels. Figures ln-lp show maps ofthe skewness coefficients for the three different models. Thesefigures show that the breakthrough curves are much moreskewed or the rate-limited models.They alsoshow hat for allthree models the skewnesss positive everywhere.Thus allbreakthroughcurvesexhibit asymmetric ailing.Example 2: Complex Spatial Patterns of Conductivityand Mass Transfer Coelficients

In this examplewe model solute ransportduring an injec-tion tracer test. At time zero a slug of solute s introducedbya well injectingwater at a constant ate at the center of thedomain. Transport subject o both LEA and first-order masstransfer is considered. n contrast to the first example, thehypothetical quifer s described y spatiallycomplexmapsofboth the conductivityand the first-order mass transfer ratecoefficientscgf nd/3).This exampledemonstrates ow temporal momentscan becalculatedn complex ealisticsituations. lthough he point ofthis example s not statistical,we employeda statisticalsimu-lation procedure n order to generate a heterogeneous on-ductivity field negativelycorrelated with a /3 field. Both thelog-conductivity and log-distribution oefficient = In (/3)fields were generated rom a multi-Gaussiandistributionandan exponential ovarianceunctionwith a correlation engthof9 m. The Y mean and variance are -1.6 and 2.0, and the Z

mean and variance are 0.0 and 0.5. The cross covariancesbetween Y and Z were calculatedusing a correlation coeffi-cient of -0.707. Thus areas of high conductivity end to cor-respond o areaswith a smalldistribution oefficient. he cor-relation coefficientof -0.707 falls between perfect negativecorrelationas considered y Burr et al. [1994] and zero corre-lation. The cross-correlatedieldswere simulatedby the colo-cation method. Sincewe have specifiedboth Y and Z to havethe same covariance functions, we can model the crosscovari-ance as

CovYi, j)= pCovZi,Zj) Var Z)' (38)[Deutsch nd Journel,1992],where and denote spatial oca-tions and p is the correlation coefficient.We generated a single realization by first constructingacovariance matrix for Y in all elements and Z at all nodes fromthe exponentialcovariance unction and the cross-covariancebetween all Y and Z values from (38). Then the Choleskydecompositionmethod [Johnson, 987] was used to draw asingle ealization of the two correlated ields. The rate coeffi-cientaf wascalculatedrom he distributionoefficients nthe work by Burr et al. [1994]:

a = 0.0211(/3) 0'7 (39)Lee et al. [1988] determined this function empirically romcolumn experiments.The velocity vectors n Figure 4d indicate that the high-conductivity one extending nto the lower right corner servesas a conduit or the injectedwater. Figures4e-4h showmapsof the concentrationat instants n time. As the solute spreadsout from the well, the heterogeneitydistorts he ring of con-centration. The plume subject o rate-limited mass transferspreadsmore than the LEA plume. Figure 5 showsbreak-throughcurves or both the LEA and first-ordercasesat themeasurement ocationsshown n Figure 4c.For an injectingwell without a regional gradient, he zerothmoment s constant n space. t is equal to the initial massofthe slug njectiondividedby the pumping ate and is indepen-dent of heterogeneitiesn aquifer or mass ransferparameters.Thus the area under each of the breakthroughcurvesshown nFigure 5 is the same. The zeroth moment of breakthroughcurves s unaffected y the measurementocationor by spatialvariability n either velocityor mass ransfer coefficients. hiscanbe understood y consideringhe differentialequation orthe zeroth moment. The zeroth moment is calculatedby solv-ing an equationanalogouso the steadystatesolute ransportequation or a constant njectionof both fluid and solutemassat the samepoint, which s equivalent o a constantnjectionoffluid at a particular concentration.At steadystate the "con-centration" (which is really the zeroth moment) will havespreaduniformly throughout he domain.Because the zeroth moment is uniform, we can calculate theaccumulatedmassat each ocation without solving he trans-port equation,even or transportsubject o rate-limited rans-fer:

accumulatedmass qmo= q(M/Q) (40)where q is the groundwater lux, M is the mass of soluteinjected, and Q is the pumpingrate. To determine the accu-mulated mass, he flow equationmust be solved o determineq, but the transportequation need not be solved.Figure 4i


11/17


A) Log-Conductivity- .:-- ',, . ,,,.,-.-.,-,--..--=-.-" ....... :-,.W.,'-

....'.,./,,.,. .' .. ;=;o,.....,.,

....-;::-- - ' 't

-. ...................'-"'"'-'"'-'""ii.'g;'' n=-5.9

E) time = 7.6LEA

B) Log-K C) Well Locations Velocity =1.0

":?"""-'""'""-'"'::'"":":'":"':":"'"B Well:/> ,,:..:.....: .... .... ::/.,"'""':':."._..':'.._..-.A'-:..

":':":':x=1.7 Locations....... :::::'.-.&lt/li=- l .7

Concent rat ionF) time = 100

'=:"" ? .. .....:....,.j? '.' .

First -OrderG) time = 7.6.?'.'

.

Max= 3.3ax= 3.1

I) Accumulated Mass J) Mean Arrival Time K) M2 Ratio (LEA/FO)

H) time = 100'F

. >.,:,:....'?'.z'.....o;.:;.:,....' ' ":i. '.'.":-:'.'...'..':..':::! '

"-'::::" ':"''"-':"'::::Mx=O 15:,,:/,. .L) Skew Ratio (LEA/FO)

Max=145Max= 1 6 0 0Min = 120

':i'..' //d.'. ': .'"'.-'.': :'.:-:..: ,:.....:- :,,.::,,., - : ; ':.'"'-,.":::...'.':::,.." '".-.8::' ......':::....'::'"">-.:::.:.'::_:}......,,,.:-.:.,-::::"---'--..'::-::::,:-.,::,.... . ..........'. ........ ..>.. ..............

Max=O :..:.'":..'-..:...."'::;:::ax= .Mn= 36 v:'"/::4...-'::..... ::.::... ?'::':---:-::':":"Mi n= 0 7 ::::::::::::::::::::::::::::::::::::Figure 4. Problem setup and results or example 2. In all contour maps the dark shades epresent highvalues,and the sixcontour ntervalsare equallyspaced etween he minimumand maximumvalues. a) Mapof the log-conductivityalues. b) Map of the log-/3values.The negativecorrelationbetween og-conductivityand log-/3 s apparent from a comparisonof Figures 4a and 4b. (c) Location of the injecting well andmeasurementocations. d) Groundwater elocities. e, f) Concentrationsor the equilibriummodel at twoinstants n time. (g, h) Concentrationsor the first-ordermodel at two instants n time. (i) The accumulatedtotal mass s the same or both mass ransfermodels.The zeroth moment s not shownbecause,despite hespatialvariabilityshown n Figures4a, 4b, and 4d, the zeroth moment s uniform. (j) The mean arrival timeis the same or both mass ransfermodels. k) Map of the ratio of the equilibriumsecondmoment to thefirst-ordersecondmoment.Far from the well the secondmoments rom the two modelsbecomesimilar. (1)Map of the ratio of the skewness oefficients rom the two models. This map shows hat in the high-conductivity one (near the lower right corner) he skewnessor the equilibriummodel s higher han for thefirst-order model.

shows map of the accumulatedmass 40). Most massmovesthrougha high-conductivity hannelout the lower right corner,althoughsomemass s funneled hrougha channelnear the topof the domain.The map of the mean arrival ime (Figure4j) is the same or

both LEA and irst-order ases. he map s dominated y the atearrival imes n the lower eft comer.Low conductivity nd high/3prevent solute rom reaching his comer until late times. It isinteresting o note that the mean arrival ime in this situation sequivalento the meanageof watersince njection Goode, 994].


12/17


Example 2: Breakthough CurvesLocationA LocationB LocationC

,e FO ,.i vo.g / LEA S0.0 00 450 900 0 10 20 0 10 20

TimeFigure 5. Breakthroughcurvesat the three locationsmarked in Figure 4c.

Since the zeroth moment and first moments are the same forthe LEA and first-order models, the second moments mustdiffer. In order to examine his differencewe plotted a map ofthe ratio of LEA second moments to first-order second mo-ments (Figure 4k). Since he ratio is always ess han 1, thesecondmoment s alwaysgreater or the first-ordermodel. Thespread of the breakthroughcurves s always greater for thefirst-ordermodel. However, he contours n Figure 4k indicatethat this difference becomes smaller with distance from thewell. Thus at the injecting well the first-order plume has amuch larger spread han the LEA plume, but as the plumemoves way rom the well, the spreadapproacheshe spreadofthe LEA plume. The breakthroughcurves n Figure 5 areconsistentwith this trend. The two breakthroughcurves atlocation A are closer together than the two breakthroughcurves at location B, which is closer to the well.

We plotted the ratio of the skewness oefficient or LEA tothe skewness oefficient or first order in Figure 41.The ratio isgreater than i in the high-velocity reas,where the accumu-lated mass s large, and less han i in the low-velocity reas.Thus equilibriummass ransfer eads o a greater skewness rmore asymmetryn the breakthrough urves han rate-limitedmass ransfer n the area where most of the masspasses. hisdoesnot imply that the third moment s larger for LEA. Thethird moment s larger everywhere or the first-ordermodel.Calculating Concentrations From the TemporalMoments

At a particular ocation, he temporal momentsof concen-tration provide nformationabout he concentration istory. nthis sectionwe explore how concentrations an be estimatedfrom known valuesof temporal momentsat a particular oca-tion. To estimate a concentrationhistory from the low-ordertemporal moments,we must first decideupon a function orthe concentrationhistory. Researchers rom different fieldshave considereda variety of functions, ncluding step func-tions,splines,Pade approximates, amma unctions, nd Pear-son distributions.We will focus on two functions: he Edge-worth form of the Gram-Charlier series, and the maximumentropy function. We will show that the maximum entropyfunction for log-time can provide reasonableestimatesof thebreakthroughcurve.Edgeworth Form of the Gram-Charlier Series

The Edgeworth orm of the Gram-Charlier Serieshas beenpresentedas a meansof approximatingunctions rom known

moments n a variety of hydrologicstudies [Chatwin, 1970;Guven et al., 1984; Longuet-Higgins, 963]. The series useshigher-ordermoments o form an expansion round he Gaus-siancurve.With only the zeroth throughsecondmoments, heseries educes o the Gaussian urve.Unfortunately, the seriesfails to produce easonable reakthrough urveswhen the mo-ments of the breakthroughcurve deviate strongly rom themomentsof a Gaussian urve.For example, f only the zeroththrough hird momentsare considered nd the skewness oef-ficient 16) is greater than 3, the seriespredictsnegativecon-centrations n the tail of the curve. Figures ln-lp show thateven for LEA mass transfer, the skewness coefficient can belarger han 3.0 at many ocations. azaviet al. [1978]argue hatthisbehavior s tolerable or applications here he rising imbof the breakthroughcurve s the feature of interest. Becauseaccurateestimationof the tail of the breakthroughcurve isimportant for many contaminantproblems,we have pursuedan alternativeapproach.Maximum Entropy Function

The maximumentropymethod for estimatinga curve romits momentshas been applied in fields ranging rom particlephysicso decisionanalysis Agmonet al., 1979;Levine, 1980;Mead and Papanicolaou, 984;Smith, 1993;Zellner and High-field, 1988;Borwein nd Huang, 1995].The idea s to determinethe function with maximum entropy that agrees with theknown moments,where entropy s defined as

i0= - C(t) In [C(t)] dt (41)Both Mead and Papanicolaou1984]and Smith [1993] provideproofs hat, if a finite entropy unctionexists hat agreeswithn moments, hen the maximumentropy unction s uniqueandis

C(t) = exp .i i (42).=

Thus there are n coefficients i for n moments.The maximumentropy function (42) is attractive because he principle ofmaximum entropy s consistentwith the underlyingphysicalprocessof dispersionand because he function reduces oGaussianwhen only three termsare used.Also, the parametervalues or which (42) agreeswith the knownmoments an befound relativelyeasily.The optimizationproblem required o


13/17


A) Concentration1.5 i Fit itherothhroughI ,",.' fourth omentsI', N Simulatedoncentrationli! ,, "True,,reahroughuwe

0.50

B) Log Concentration10o

10-2

lO-S

10-8

Fit with zeroth through

......ititheroth,irst""c3wr..,.l.andecondomentsrnuatdnceetrations0 0.5 I 1.5 2 0 25 50 75 100

NormalizedTime NormalizedTimeFigure 6. Estimatedbreakthrough urve rom low-ordermomentsusing he log maximumentropy unction.(a) The fit to the simulated true" concentrationss improvedwhen using he zcroth to fourth momentsinsteadof only the zcroth o secondmoments. b) Log concentrationsrc plotted against greaterdurationof time than in Figure 6a. The graphsshow hat the log maximumentropy unctions it the tail very well. Atvery late times the numericalsimulationbreaksdown,but the log maximumentropy unction still providesreasonable estimates of concentration.

determine he parametersX is unbounded nd convex Smith,1993]. Furthermore, n theory, f enoughmomentsarc consid-ered, then very complexbreakthrough urves,suchas multi-modal curves, can be estimated.Unfortunately, 42) cannotbe used or many of the exam-pls ivenhere. Values of the parametersX do not appear oexistsuch hat (42) has he samemoments s he breakthroughcurves n the examples.We believe his is because he break-throughcurveshave fat tails. n order for (42) to have finitemoments he last coefficientX, mustbe negative.At late timesthis coefficientwill cause he curve to fall off at a rate greaterthan the exponential ate. In the next sectionwe give an ex-ample in which the curve appears o always all 1ssapidly

than exponentially. ne approach o this problem s to simplytruncate he curveafter some ate time. However,we pursuedan approach hat skso accuratelymodel the tail.Log Maximum Entropy Function

We found that the log transformof the maximum entropyfunctionprovidesgood estimatesof the breakthroughcurvesfrom known moments.By transforming he maximum entropyfunction to log-time, we obtain

C(t) -exp -i n t)i=0 (43)This function is better able to fit curves with fat tails, like ourbreakthrough urves, ecauset can be very positively kewed.Also, like the breakthrough urves, t is zero at the origin.Thefunction 43) reduces o the lognormal unctionwhen only hezeroth through secondmomentsare used.In order to demonstratehe methodwe calculated he pa-rametersX such hat (43) has the samemomentsas we sim-ulated n the second xample t locationB (Figure5) for LEAmass transfer. The breakthrough curve at this location isstronglyskewed due to heterogeneitiesn the velocity anddistribution oefficient ields. n Figures6a and 6b we comparethe true breakthrough urveat this ocation sameas n Figure5) with the curvesestimatedusing 43). Both the curve esti-mated from the zeroth through secondmoments,and curveestimated rom the zeroth through ourth momentsare shown.Figure 6a demonstrates hat, by fitting the zeroth through

fourth moments,a reasonablygood fit to the breakthrough sobtained. n Figure 6b, log concentrations plotted or a muchlonger duration of time. This figure shows hat the estimatedconcentrationrom (43) continueso fit the concentration ellat very ate timesand, n fact, provides easonable stimates fconcentration ven after the concentration asdroppedso owthat numerical errors have corrupted the simulation. Thegraph also shows hat the breakthroughcurve continues odeclineat a rate less han exponential. hus, unless he slopeof the curve changesat times later than we were able tocalculate,he standardmaximum ntropy unction 42) will failto fit this curve after some late time.Although the breakthroughcurvewe estimated s the result

of a complex ransportsystem, he curve tself is fairly mun-dane. Theoretically, he maximumentropy or log-timemaxi-mum entropy)methodcan be used o estimate ery complexcurves,such as multimodal curves.Complex breakthroughcurves, such as multimodal curves, could occur in the field ifthe solute njection s pulsedor if strongsources xistat severallocations.Also, as Bellin et al. [1991] show, irst-order masstransfercan causebimodal curves o developdownstream f apoint source. n practice, however,estimatingsuch complexbreakthroughcurvesmay be difficult.Higher-order moments,and their corresponding oefficientsn the maximum entropyfunction, must be considered.Determining these coefficientsbecomes n increasingly ifficultnonlinearoptimizationprob-lem.Conclusions

The main contributionof this paper is the development fthe moment-generating quationsshown n Tables 1 and 2.Analysisof these equationsextendsearlier work in chemicalengineering Kucera,1965; Schneider nd Smith, 1968; Viller-maux, 1981] and hydrology Valocchi,1985, 1986; Parker andValocchi,1986;Goltzand Roberts, 987;Sardinet al., 1991] oridealized models of infinite homogeneousmedia to spatiallyvariableporousmedia with complex nitial and boundarycon-ditions. For all of the mass transfer models considered here,the moment-generating quationscan be solvednumericallywith codeswritten to simulatesteadystate nonreactive rans-port. Below we list someconclusions rawn from theseequa-tions, and observations bout applicationof the method.


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1. For nonreactivesolutesor equilibrium mass ransfer,aswell as rate-limitedmass ransfer, he zeroth momentsdependon the initial conditions hrough he advective-dispersivep-erator (5). They do not dependon mass ransfercoefficients.2. Because he zeroth moment at any location dependsonly on the initial conditions nd the advective-dispersivep-erator, the mass ransportedpast any point or out any bound-ary can be found without knowledge of mass ransfer coeffi-cients, or knowledge of whether mass transfer is evenoccurring.3. For nonreactive olutesor equilibriummass ransfer,temporal moments of successive rder can be determinedthrougha recursive elationship 13). For momentsof order 1and greater he nth moment ield dependsonly on the (n -1)th moment field through the advective-dispersiveperator(5}.4. For rate-limited mass transfer, higher-ordermomentsdepend on all lower-order moments and the mass transfercoefficients.

5. In the absenceof initial immobile mass, he first momentfor both rate-limitedmodelsdependson the distribution oef-ficient/3but not on the rate coefficientsT or aa.6. For all mass ransfermodels he relationshipof higher-order moments to lower-order moments is linear.

7. The moment-generating equations for all the rate-limited models have the same form. However, the coefficientsin each term differ. For the diffusion models these coefficientsdepend on the Bernoulli numbers.8. When the mass transfer coefficients are such that therate-limited models have the same zeroth and first moments asthe LEA model, all higher-ordermomentswill be larger or therate-limited models.

9. When no initial immobile mass exists, mass transfercoefficients can be determined such that the first-order modeland the diffusionmodels or layered,cylindrical, nd sphericalgeometriesall produce he samezeroth, first, and secondmo-ments.

10. When the rate-limited models produce he same ze-roth, first, and secondmoments, he higher-ordermomentswillbe larger or first-order ayered,cylindrical, nd sphericalmod-els, respectively.11. If initial immobile mass exists So > 0), only thezeroth and first momentscan agree.Secondmoments rom thedifferent models cannot be the same.12. The temporal momentsof immobile concentrationatany location are a simple algebraic function of the mobile

concentration moments and rate coefficient at the same loca-tion.

13. The two examples demonstrated some interestingpropertiesof solute ransport. n the first example,we showedthat a singlepoint sourceof solutecan develop nto a plumewith two peaks when mass ransfer is modeled as first-orderrate-limited. In the secondexample,we consideredan injec-tion well in a highly heterogeneous quifer. We found that,when an instantaneoussolute source s added at the well, themap of the zeroth moment s uniform n space.Thus a map ofthe accumulatedmassat each point can be calculatedwithoutconsidering dvectionand dispersion.14. The Edgeworth form of the Gram-Charlier seriescanproducevery poor estimatesof breakthroughcurves.Parame-ters cannot alwaysbe found such hat the standard he maxi-mum entropy function agreeswith the known moments.15. The log-transformedmaximumentropy unction 43)

that agreeswith the zeroth through ourth momentsclosely itsa breakthrough urve rom the second xample.Thus the firstfive momentsadequatelysummarize he breakthrough urve,even hough he breakthrough urve s the resultof a complexflow and transportsystem.The primary value of these results s their generality.Themoment-generatingquationswere derivedwithout specifyingthe velocities and dispersioncoefficients n the advective-dispersive perator (5) and without specifyinghe boundaryconditions6). Thus he results re independent f the spatialpatterns of groundwatervelocity, dispersivity, nd boundaryconditions.Similarly, he resultshold true for spatiallyheter-ogeneous mass transfer parameters and initial conditions.There are also somesignificantimitations o the method.Allof our resultswere derived for steadystate groundwaterve-locities,and we only considered inear mass ransfer relation-ships.The two hypotheticalexamplesdemonstratehow maps oftemporal momentscan be used to studysolute movement nheterogeneous quifers.From mapsof the zeroth moment andgroundwater elocities, he accumulatedmass o passeverylocationcan be mapped,without knowledgeabout mass rans-fer. If one s not interested n timing, hismap couldbe used odeviseschemes or controlling he amount of mass to passthrough particular locations.With the zeroth and the firstmoments, he mean arrival time of solute can be mappedthroughoutan aquifer. With the secondand third moments,the standard eviationand skewnessf the breakthrough urvecan be mapped at all locations.Maps of these quantitiespro-vide a concise escription f the historyof a soluteplume andan alternative o usingmaps of concentration o studysolutetransport.The moment-generating quationsprovide a powerful toolfor solvinghe inverseproblem.Calculating he zeroth and irstmomentsof a breakthrough urvehelps o separate he effectsof different aquifer parameters on measured concentrationhistories.Because he zeroth moment s independentof masstransfer coefficients,measurements of the zeroth moment canbe used to estimate the hydraulic conductivity ield withoutconsideringmass transfer. Because he first moment is inde-pendent of the rate coefficients,measurementsof the firstmomentcanbe used o estimated he conductivityield and thedistribution-coefficient ield without considering he rate-coefficient ield. Thus, the moment-generating quations m-prove our ability to use measuredbreakthroughcurves o si-multaneouslyestimate fields of mass ransfer coefficients ndhydraulicconductivity. urthermore,becausemuch of the in-formationcontained n a breakthrough urve s summarizednits low-ordermoments, singmeasurements f thesemomentsis an efficientapproach o the inverseproblem.

Appendix AHere we compare numerical solutions of the moment-generatingequationswith analyticsolutions.We consideredpulse nput of mass n an infinite one-dimensional olumnwithconstant luid velocity.The pulse s placedat the location =0. This problem s similar o the problemof a pulse nput ofmassat the upstream end of a semi-infinitecolumn that hasbeen studiedby Kucera 1965], Goltz and Roberts 1987], andmany others.However, in the infinite case he zeroth moment

is not uniform; upstreamof the point of injection, he zerothmoment falls to zero.


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HARVEY AND GORELICK: TEMPORAL MOMENT GENERATING EQUATIONS 19090.006

0.07

1.2

0 -5 0 5 10x

o-lO -5 o 5 lOx

-10 -5 0 5 10FigureA1. Numericallyndanalyticallyalculatedemporalmoments longa columnwith constant elocity or a pulseinput f mass tx = 0. Dots epresenthenodal alues f thefinite elementcalculations.ines epresenthe analytic olu-tions.

The linearordinary ifferentialquationsor the temporalmomentscan be integrated o giveM{eVX/On0=- 1 -H(x)] + H(x)} (A1)

Mm= x{eVX/O[- H(x)][2D vx] H(x)[2D vx]}(A2)Mrn2 {e*X/O[1H(x)](12D- 6Dvx v2x)+ H(x) 12D2 + 6Dvx + v2x2)} (A3)

whereH(x) is the unit step unction r Heavisideunction.Downstream f the source,H(x) = 1, and theseequationsreduceo theequationsor momentsn a semi-infiniteolumntabulated y GoltzandRoberts1987].FigureA1 showshat henumericalolution f rno, n , andm2 closely atcheshe analytic olution.n these imulations,M = 1, v = 1, andD = 1.5. The nodes re all separated ya distance f one unit, so the grid P6cletnumber s Axv/D =0.67. The finite elementcalculations sedsingle-precision4byte) loating umbers.hesharpestrontexistsor hezerothmoment.The first moment,which is calculatedby using hezerothmomentas a constant ource n the advective-dispersiveequation,s moredispersedn space. hus,f oscillationsueto a largegrid P6cletnumber ccur, heywill occur n thecalculation of the zeroth moment.

Appendix BThe Laplacedomain orm of the coupled ransport-masstransfer quations7), (19), and (28) provides convenientmeansof calculating ctualconcentrations.hey have thesame orm as he equationor steady tatenonreactiverans-port.Thus, imulatingaplace omain oncentrationsubjectto first-order mass ransfer, or diffusivemass ransfer, takeslittle more effort than simulating onreactive ransport.Su-dicky 1989, 990] asused hisapproach,ombined ithnu-

mericalnverse aplaceransformation,o simulate othcon-servativesolutesand solutessubject o rate-limited masstransfer.

The methodconsistsf two steps. irst,a steady tatesolutetransportodeornonreactiveolutessusedo determineheLaplace omain oncentrations;hen heLaplace omain on-centrations re numericallynverted o real-time concentra-tions.For anyvalueof the Laplace arameter , the Laplacedomainconcentrations are calculated y consideringheparametersn the Laplace omain quationssdecay oeffi-

Table B1. TermsUsed o Calculate aplaceConcentrationsith Steady tateSimulationSourceTerm Decay CoefficientFirst lSapoOrder Co p + atVariable

geometrydiffusion

Spherical

Layered

CovBSo' [iv/2_(p/aa)/2/Co3/3So- 1/2oth - ]

p-p - p +''-v/2(p]Otd)l/2_-p-3l(paa)'/2[Coth(/2]-p O(p)'Tanh


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cientsand source erms.The parameter group that multipliesC is entered as a decay coefficient,and the other terms areentered as a constantsource.This simple transform of vari-ables given n Table B1) enablesus to simulateLaplace do-main concentrations. he Laplace domain concentrations rethen inverted to real-time concentrations umerically.We usethe Stehfest1970] algorithmbecauset is easy o implement.Sudicky 1989] has shown hat algorithms, uchas the Crumpand the de Hooge, which considercomplexvalues of Laplacedomain concentrations, ork better for large P6clet numbers.

This solution is the same as the solution for a three-dimensional omain Goltzand Roberts, 986] except B3) hasbeen changed or a two-dimensional omain (M. N. Goltz,personalcommunication, 994). Figure C1 showsconcentra-tionscalculated rom (B3) using he parameters rom the sec-ond exampleat r - 3.0. Because he velocitieswere calculatedusing only the high-conductivityvalue (without the low-conductivity one), the plume migrated further than in theexample Figure 2). In order to reduce he possibility f nu-merical error, the calculationwas performedusing30 digitsofprecision.

Appendix CAnalytic expressionsor a plume subject o first-ordermasstransfer show that, under certain conditions,a single plumecan develop wo peaks n space.Bellin et al. [1991] show hat,in one dimension, wo peakscan developafter the injectionofmobile solute at a single point. After one peak has moveddownstream t roughly he fluid velocity,a secondpeak devel-ops near the point of injection.The secondpeak, which rep-

resentsmass hat mobilizes rom the regionof injection,movesdownstream t roughly he speedpredictedby the retardationfactor (1 + /3).Here we use an analyticsolution o show hat, for the pa-rametersused n example1, the two-dimensional lume devel-ops two peaks or a singlepoint source.The two-dimensionalsolution or a point injection of mobile solute in an infinitehomogeneous omainwith a uniform velocity ield isC(x, y, t) = exp (-a[3t)G(x, y, t)

+ a H(t, r)G(x, y, ,)dr (B1)

H(t, r) = 13/2r xp{-[a(t - r)I1{2[ a213r( -+ al3r]} [r(t- 'r)] /2 (B2)

exp -[(x - vt)2/4Dxxt] (y2/4Dyyt)}G(x, ,r)= 4z.(DxDyy)l/2 (B3)

1.0

0 1.7Figure C1. Concentrationsat t = 3.0 in an infinite homo-geneousmedia as determined y (B1). Dark shades epresenthigherconcentrations, nd the sixcontour ntervalsare equallyspaced between the maximum value and zero. The plumeshows wo peaks. The domain length is normalized by thelength of the domain in example 1.

Acknowledgments. This material is based upon work supportedunder National ScienceFoundation grant EAR-9316040. We wouldlike to thank Carl Renshaw or his help with finite elementsimulation.We would also ike to thank Roy Haggerty,Charles M. Harvey, PaulSwitzer, and Mark Goltz for their assistance.We are grateful to theHewlett-PackardCompany or providing he computerequipment hatmade this work possible.ReferencesAbramowitz,M., and I. A. Stegun Eds.), Handbookof MathematicalFunctionsWith Formulas,Graphs,and MathematicalTables,Dover,Mineola, N.Y., 1972.Agmon, N., Y. Alhassid,and R. D. Levine, An algorithm or findingthe distributionof maximalentropy,J. Cornput.Phys.,30, 250-258,1979.Aris, R., On the dispersionof linear kinematic waves,Proc. R. Soc.London A, 245, 268-277, 1958.Bahr, J. M., Analysisof nonequilibriumdesorptionof volatile organicsduringa field test of aquifer decontamination, . Contam.Hydrol.,4,205-222, 1989.Ball, W. P., and P. V. Roberts, Long-term sorption of halogenatedorganic hemicals, , Environ.Sci. Technol., 5(7), 1237-1249,1991.Bellin, A., A. J. Valocchi,and A. Rinaldo, Double peak formation nreactivesolute ransport n one-dimensional eterogeneous orouscolumns, n Quademi del Dipartimento DR, Dipartimento di Ing.Civ. ed Ambientale, Univ. degli Studi di Trento, 1991.Borwein, J. M., and W. Z. Huang, A fast heuristicmethod for poly-nomial moment problemswith Boltzmann-Shannon ntropy,SIAMJ. ControlOptim.,5(1), 68-99, 1995.Burr, D. T., E. A. Sudicky,and R. L. Naff, Nonreactive and reactivesolute ransport n three-dimensional eterogeneous orousmedia:Mean displacement,plume spreading,and uncertainty, Water Re-sour.Res.,30(3), 791-815, 1994.Chatwin, P. C., The approach o normality of the concentrationdis-tributionof solute n a solvent lowingalonga straightpipe,J. FluidMech.,51(2), 321-352, 1970.Chatwin, P. C., The cumulants of the distribution of concentration ofa solutedispersingn solvent lowing hrougha tube,J. Fluid Mech.,51, 63-67, 1972.Deutsch, C. V., and A. G. Journel, Geostatistical oftwareLibrary andUser Guide, Oxford Univ. Press, New York, 1992.Goltz, M. N., and P. V. Roberts, Three-dimensional solutions forsolute transport n an infinite medium with mobile and immobilezones,WaterResour.Res.,22(7), 1139-1148, 1986.Goltz, M. N., and P. V. Roberts, Using the method of moments oanalyze hree-dimensional iffusion-limitedransport rom temporaland spatialperspectives, aterResour. es.,23(8), 1575-1585,1987.Goode, D. J., Direct simulationof groundwaterage, (abstract),EosTrans.AGU, 75(44), Fall Meet. suppl.,231, 1994.Guven, O., F. J. Molz, and J. G. Melville, An analysis f dispersion na stratifiedaquifer, WaterResour.Res.,20(10), 1337-1354, 1984.Harmon, T. C., L. Semprini,and P. V. Roberts,Nonequilibrium rans-port of organic contaminants n groundwater, n ReactionsandMovementof OrganicChemicals n Soils, edited by B. L. Sawhneyand K. Brown, pp. 405-437, Soil Sci. Soc. of Am., Madison, Wisc.,1989.Harmon, T. C., L. Semprini, and P. V. Roberts, Simulatingground-

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(ReceivedSeptember , 1994; evisedApril 10, 1995;acceptedApril 11, 1995.)

temporal moment-generating equations: modeling transport and mass transfer in heterogeneous aquifers

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