WATER RESOURCES RESEARCH, VOL. 32, NO. 4, PAGES 891-899, APRIL 1996

Lagrangian analysis of transport in heterogeneousformations under transient flow conditions

Gedeon Dagan Faculty of Engineering,,Tel Aviv,

Alberto Bellin Dipartamento di IngegneriaCivile ed Ambientale, Universita di Trento, Trent, Italy

Yoram Rubin Department of Civil Engineering,University of California, Berkeley

Abstract. The coupledeffects of porousmedia heterogeneityand flow unsteadinesson transportare investigated.We addressthe problemusing a stochastic-Lagrangian approach.The log condictivityis modeled as a spacerandom function;however, the flow unsteadinessis consideredas deterministic,and it affectsthe magnitudeand directionof the mean head gradient.Using a first-ordersolution in terms of the log conductivity variance,we developeda three-dimensionalsolution that includesthe velocitycovariances, the macrodispersioncoefficients, and the displacementtensor, all as a function of the media heterogeneityand the unsteadymean flow parameters.The model is appliedto the caseof a periodicvariation in the flow direction.We showthat the effectsof unsteadiness on the longitudinalspread are insignificant,while it affectsstrongly the transversespread, in the form of an addedconstant dispersivity. Results obtained in previousstudies are discussedand are shownto be particular casesof the presentresults.

1. Introduction and Brief Discussion of and Zhang [1990] and of Zhang and Neuman [1990], who Previous Work obtained a quasi-linearapproximation of the transport equa- tion basedon Corrsin'sconjecture. While the first-ordertheory Transport of solutesin flow is a subject of great predicts an asymptoticallyvanishing macrodispersion coeffi- interest in variousapplications. Field measurementshave in- cient at great travel times, the Corrsin conjectureleads to a dicated that the dispersionprocess is enhancedby the large- finite one. However,it wasrecently proved [Dagan, 1994] that scaleheterogeneity, i.e., by the spatialvariability of the hydrau- a rigorous derivation of the nonlinear, quadratic in the log lic conductivity K, which is generally present in natural conductivityvariance, term of the macrodispersivitytends to formations.It is customaryto modelK(x) as a randomspace zero as well. This result strengthensconfidence in the linear function,to accountfor its irregularvariation in spaceand for theory and precludesexplaining the aforementioneddiscrep- the uncertaintyaffecting its distribution.Various models,ana- ancy for Borden site, a formation of weak heterogeneity,by lytical or numerical,have been developedin the past in order nonlinear effects. to relate the evolutionof soluteplumes to the heterogeneous In the searchfor a differentexplanation of this discrepancy, structure. Most of the studies were carried out for conditions it was suggested[e.g., Naff et al., 1988] that the presenceof of steadyand uniformaverage flow, the resultof applicationof small transientsof flow around the mean may causethe en- a constanthead drop on the boundariesof the formation. In hanced transversedispersion. Furthermore, a recent experi- addition to simplicity,the justificationwas that under natural ment in the sameaquifer [Farellet al., 1994] hasrevealed that gradient conditions,flows are generallyclose to steady.Since the time variation of flow, as manifested in head measurements the timescaleof transientsis generallyof a seasonalnature, the in piezometers,was quite significant.This finding has stimu- assumptionof steadinessis particularlyappropriate for trans- lated a further examinationof the impact of flow unsteadiness port field testsof relativelyshort duration. upon transport [Farell et al., 1994] in this particular experi- Analysisof field resultsin one of the most elaborateexper- ment. Furthermore,the subjectis of interestfor predictionof iments carried out so far, at the Borden site [Sudicky,1986], plume evolutionover extendedperiods of, say, tens or hun- showedgood agreementbetween linearized models, based on dredsof years,for which transientsare alwayspresent. a first-order approximationin the log conductivityvariance, Although transportunder unsteadyflow conditionshas re- and measurementsfor the longitudinalfield-scale dispersion. ceivedless attention, a few studieswere carriedout in the past, and we shall discussthem briefly here. In contrast,the rate of increaseof the plume transversemo- The problem was addressedby Kinzelbach and Ackerer mentswas underpredictedby the linearizedtheory. A possible [1986] and by Goodeand Konikow[1990] in the conventional explanationfor this effectwas offeredin the works of Neuman frameworkof solvingthe transportequation for a deterministic Copyright1996 by the American GeophysicalUnion. velocityfield, while dispersionis representedby a tensor of Paper number 95WR02497. constantcoefficients. Goode and Konikow [1990] considereda 0043-1397/96/95WR-02497 $05.00 flow which is uniform in spaceand variesperiodically around a

891 892 DAGAN ET AL.: LAGRANGIAN ANALYSIS OF TRANSPORT steadyone. Thus the firstexample in whichthe meantemporal velocityis constantand unidirectional,while a constantfluc- tuation normal to it is appliedperiodically, is particularlyin- structive.In thisexample, the meantrajectory is a zigzagline of a smallangle + 0 and -0 aroundthe horizontalaxis. The main findingis that an observerinspecting the plumeat the timesits centroid crossesthe mean flow path detectsan apparentin- creaseof the transversedispersion coefficient. This increaseis easyto grasp:When the flow suddenlychanges direction, part of the longitudinalspread of the plumein the previousperiods manifestsitself as a transversespread for axesparallel to the Figure 1. Schematicrepresentation of the advectivetrans- instantaneousvelocity or to the mean temporalvelocity. We port of a solutebody in a realizationof a velocityfield uniform shall not dwell here on the more involved cases of transients in space,but randomin time. relatedto aquiferstorativity analyzed by Goodeand Konikow [1990].A point of principleis that the actualdispersion coef- ficients,defined as half the rate of changeof the plume spatial to zero for large t or is finite and very small if pore-scale moments,are thoseof steadyflow for an observermoving with dispersionis accountedfor. Naff et al.'s [1988] analysiswas the instantaneousflow velocity.In other words,the apparent similar,but u is regardedas deterministic,based on pastmea- increaseof the transversemoments does not showup in the surementsof the head gradient. instantaneousdispersion coefficients. Our analysisdiffers in a few respectsfrom that of Rehfeldt In contrastwith this approach,we considerhere the caseof and Gelhat [1992]or of Naff et al. [1988].Thus we definethe a randomvelocity field whose impact upon transport cannot be dispersioncoefficient as D22 = (1/2)d{S22)/dt, whereS22 = encapsulatedin a diffusiveeffect. As we shallshow later, the (1/.Zlo)f.q0(X2 - R2)2 da isthe plume spatial moment with latter is a particularcase of a velocityfield associatedwith a respectto the centroid.Here X2 = a2 + ftoU2(t') dt' is the "Brownianmotion," which may be a good approximationin transversecoordinate of the trajectoryof a particleoriginating casesin which the timescalesof the mean flow are much larger at x = a, R2 = (1/.Zlo).fAoX2 da is thecentroid transverse than those associatedwith heterogeneity. trajectory,and A o is the inputzone of a conservativesolute of Another line of attackwas adopted by Naff et al. [1988]and initial constantconcentration. For ergodic plumes in steady Rehfeldtand Gelhat [1992].They consideredheterogeneous flow throughheterogeneous media, d(S22)/dt is indeedequal formationsof random stationarystructure and a mean head to dX22/dt [see,e.g., Dagan, 1990]. However,this is not the gradientwhich is uniform in space,but fluctuatingin time casefor a spatiallyuniform flow, as impliedby (2). Indeed,in around a constantmean. Rehfeldt and Gelhat [1992] assumed this case,in absenceof pore-scaledispersion, the plumetrans- that thesefluctuations are random and small and neglecttheir lates as a whole in each realization, and it is easyto see that interactionwith velocityfluctuations associated with heteroge- S22(t) = S22(0) = constand D22 = 0. This resultis ex- neity.In otherwords, the three additivecomponents of trans- plainedin Figure 1, in whichA o is a circle.The solutebody port are translationby uniformand steadyflow, macrodisper- wandersalong a randompath, but does not disperse, and Xij in sion associatedwith heterogeneityunder the samecondition, (2) is a measureof uncertainty,i.e., of the envelopeof solute and a temporalfluctuating flow in a homogeneousformation. bodytrajectories. It canbe regardedas a measureof dispersion Sinceonly the last componentis of concernhere, we shall of the mean concentration if measurements of concentration discussit in somedetail. Furthermore,we prefer to castit in a andspatial moments are definedas time averagesover periods Lagrangianrather than an Eulerianframework, since we find muchlarger than the timescaleof fluctuationsof the velocity. the first quite illuminating.Thus, with neglectof pore-scale This was not the casein the aforementionedfield experiments, dispersion,the fluctuationof the trajectoryof a fluid or solute sincemeasurements were carried out over relatively short ti- particle is givenby mescalesand "instantaneous"pictures of the plume were ob- tained.We feel,therefore, that interpreting X•j in (2) asspatial momentsis not justified. X'(t) = u(t') dt' (1) A seconddifference of principleis that we regardthe tem- •0t poralvariation of the headgradient as deterministic. Indeed, in the Borden site field experiment,heads were monitoredat where u is the randomvelocity fluctuation, which is indepen- short time intervals and their temporal changesduring the dent of x and resultssolely from the temporalhead gradient transport experiment could be captured quite accurately variation.The trajectoriescovariance is therefore [Farellet al., 1994].In the caseof prediction,uncertainty may affectthe temporalfluctuations, though basically they are of a Xij(t) = uo(t', t") dt' dt" (i, j = 1, 2, 3) (2) seasonalnature. However, we believe that this uncertainty otfot shouldnot be regardedas resulting in dispersionand the right contextis that of time seriesand centroidmotion (Figure 1). RehfeMtand Gelhar [1992] assumed that the covariancesuij These topicsare not discussedhere. are stationaryand of finite integralscales. Furthermore, their Finally,the randomvelocity temporal variations were incor- solutionis equivalentto assumingthat Xij representsthe spa- poratedin the derivationof the mean concentrationfield by tial momentsof an ergodicplume. As a result,for a uniform Kabala and Sposito[1991]. They usedthe Van Kampencumu- mean flow in the x• direction,D22 = (1/2)dX22/dt tendsto lant expansion,and the velocityfield wasregarded as station- a constant,"Fickian" limit, for large t. This is in contrastwith aryin spaceand time. However, only a generalformulation was D 22associated with steadyflow and heterogeneity, which tends presented,and no attemptwas made to relate the temporal DAGAN ET AL.: LAGRANGIAN ANALYSIS OF TRANSPORT 893 variationsto thoseof the head and to the heterogeneousstruc- ture. ui(x,t) = Ko•- (JiY' -J. axax.02X) = u.(t)vi,.(x) In the presentstudy we employthe Lagrangianapproach we (9) haveused in the pastto investigatetransport in steadyflow. In our analysisthe effectsof unsteadinessand heterogeneityin- tli,a = •ii.y' OXiOX. teract, in the spirit of the studyof Goodeand Konikow[1990].

where 8i. is the Kronecker unit tensor and here and in the 2. Statement of the Flow Problem sequelwe adopt the index summationconvention. and First-Order Solution Next,we multiplyui(x, t) by ui(y, t') andensemble aver- age. In the ensuingexpression we encounterthe covariances Flow takes place in a heterogeneousformation. The log Cxr(r)= (X'(x)Y' (Y)) = -P(r) andC x = (X'(x)x' (Y)) = O(r).The conductivityY(x) = In K is modeledas a stationaryrandom functionsP and Q, satisfying•72p = -Cy and •72Q = P, spacefunction of meanmr = (Y) and covarianceC•,(r) = were usedextensively in the past [Dagan,1989; Rubin, 1990]. (Y' (x)Y' (y)), whereY' = Y - mr, r = x - y, X(Xl, x2, x3) Their expressionsfor an exponential,isotropic Cr are recalled is a Cartesiancoordinate, and anglebrackets stand for ensem- in the appendix.After thesepreparatory steps, the final gen- ble averaging.The head H(x, t) = (H) + h is assumedto eral expressionsof the velocitycovariances are have a giventime dependentmean gradient V(H) = -J(t) (3) uij(x, y, t, t') = (ui(x, t)uj(y, t')) = U.(t)Ut3(t')v½.t3(r) r = x- y in line with previousworks. Equation (3) implies that the (lO) instantaneoushead gradientis constantover the zone covered 02p by the plume,which may be an accurateassumption if the scale characterizingthe spatialvariation of X7(H) is sufficientlylarge %,.t3(r)= (vi,.(x)v•,o(y))= Si.•ijt3C•r) + Sjt30riOr. [Farell et al., 1994]. We can set the problem in a rigorous mathematicalframework by assumingthe flow to be confined 02p o4Q so that H satisfies + 8i,OrjOro OriOr,Or•Oro

V2H + VY. VH = 0 x G P. In words,the tensor vo,.o(x, y)(i, j, a,/3 = 1, 2, 3) yields (4) the velocitycovariance tensor for an arbitrary mean velocity H= -J(t).x x•OP. vector U. The unsteadiness of the motion is evident in the where D. is the flow domainand 0D.is its boundary.Equations dependenceof the latter on time. (3) and (4) yield for the head fluctuationh(x, t) In the pastwe haveconsidered steady flows parallel to thex 1 axis,i.e., U(U, 0, 0) for which V2h + VY. Vh = J. VY x • P. (5) o2p h =0 xE 0D. Uij:vij,ll: 8il•jlCy(r) + •jl OriOrl It is seenthat the solutionof (4) and (5) is identicalwith the 02p o4Q solutionfor steadyuniform flow providedthat at eachinstant J assumesits time dependentvalue. + $i•Or•Or• OriOr•Or2• (11) In line with previouswork on steadyflow, we adopt a first- order approximationfor h by replacing(5) by Theexplicit expressions of Uo (11)are given by Rubin [1990] for two-dimensionalflow and isotropicexponential C•., by V2h(x,t) = J(t)' VY' (x) (6) Zhang and Neuman [1992] for three dimensions,and by Rubin and Dagan [1992] for anisotropicC•.. As a matter of fact, and let the flow domainexpand to infinity. The velocityfield is determinedwith the aid of Darcy'slaw becauseof manysymmetries, the components of vq..o canbe at first order as follows[see, e.g., Dagan, 1984]: expressedsolely with the aidof Uq (seethe appendix). V(x, t) = U(t) + u(x, t) (7) 3. Statement of the Transport Problem and U= {V) =•- J(t) u(xt) K•[J(t)Y'(x) - Vh] First-Order Solution A plume of a conservativesolute is injected at t = 0 in a where Kc is the conductivitygeometric mean and n is the volume (area)Ao, i.e., C(x, 0) = Co for x = a G Ao and porosity,assumed to be constant. C - 0 otherwise.We neglectthe effectof pore-scaledisper- To obtainthe expressionsof the velocitycovariances, a pre- sion,so that the trajectoryx = X(t; a) of a particleoriginating requisiteto solvingthe transportproblem, we recall the pro- at x = a, t = 0 satisfies cedureused in the pastfor steadyflow [e.g.,Dagan, 1984].We observethat by the linearityof (6) we may write dX(t; a) --= dt V(X, t) X(0; a) = a (12) h(x, t) = J(t). VX(X) •72X = Y' (x) (8) and the auxiliaryrandom function X dependsonly on the het- By substituting(7) in (12) and integrating,we get for the erogeneousstructure. By substituting(8) in (7) we obtain mean(X) and fluctuationX' the followingexpressions 894 DAGAN ET AL.' LAGRANGIAN ANALYSIS OF TRANSPORT

U(t) •,xlI • (X(t; a))= a + U(t') dt' •0t (•3) X'(t; a) = øotu[X(t'), t'] dt' Figure 2. Definition sketchfor transportby a periodicmean The spatialmoments of the plume, i.e., the centroid coor- flow (equation(18)). dinateR - (1/Ao) f.40X da andthe secondmoments So = (1/Ao) f.4o(Xi- R•)(Xj - Rj) da withrespect to thecen- troid, are givenin terms of trajectories[see, e.g., Dagan, 1989] depend,on the other. Equations(17) will be employedin the by followingsections for analyzingperiodic flows. We wish to showthat previousresults obtained in the liter- ature are particularcases of (17). Indeed, the steadyflow R(t)= (X(t;0)) +•00 X'(t;a) da solutionswe developedin the past correspondto U constant 0 (14) and (X(t')) - (X(t")) = U(t' - t"). The solutionof Goode andKonikow [1990] results from (17) for vo.,,•t• = do.,,•t•8(t'- t"), So(t) = So(O) + Xo(t) - Ro(t) whered•j,,•t • is a tensorof constantcomponents. Finally, the whereX o = (X•(t; a)Xj(t; a)) is the trajectorycovariance approximation of RehfeMtand Gelhat [1992] for the unsteady andR•j is the R covariance.We assumethat the plumeis term is equivalentto takingvij,,•t • = 8•,•Sjt•p,•t•(t'- t"), ergodic,i.e., its transverseextent is sufficientlylarge compared whereP,•t• is the velocitytemporal autocorrelation. to the Y correlation scale [Dagan, 1990] to warrant taking We proceedwith illustratingour approachby applying(17) Rij • 0 in (14). Hence to a simplecase of unsteadymean flow. R • (R) = (X(t; 0)) 4. Application to Simple Periodic Flows So(t)• (Sij(t)) = Sij(O)+ Xij(t) (•5) Similarlyto previousstudies, we considera flow which fluc- tuates periodicallyaround a steady,uniform one. The most 1 d(So) 1 dX•j generalrepresentation of U(t) is by a Fourier series[Farell et Dq(t)= • •=dt 2 dt al., 1994] whosebasic period is 1 year. To graspthe effectof unsteadinessin a simple manner we consider here a single whereD o is the effectivedispersion tensor. harmonicand defer a more generalrepresentation to a future To deriveX o in a simpleand consistentmanner with the study.Thus the flow is of constantmean U1 in the x i direction first-order approximationadopted for the velocity field, we and fluctuatesin the x2 direction(Figure 2). We adopt here follow the procedureadopted in the past [Dagan, 1989] and and subsequentlyvariables that are made dimensionlesswith replaceX in (13) by its mean, i.e., respectto Iv, the log conductivityintegral scale,and Iy/U 1 as length and timescales,respectively. Hence X'(t) = u[(X(t')), t'] dt' •0t (16) U1 = 1 U2= • cos(t/X) U3= 0 (18) (X•) = t (X2) =/3 sin (t/h) (X3) = 0 Xij(t) = uij[(X(t')), (X(t")), t', t"] dt' dt" The two parameterscharacterizing unsteadiness are/3, the dimensionlessamplitude of the mean trajectory, and X = Finally,we substituteuo (10) in (16), to obtainX o andDo T/2 ,r, whereT is the dimensionlessperiod (Figure 2). For/3 = explicitlyin terms of the velocitycovariances as follows: 0 we recover the case of steadyflow of uniform velocity U• analyzedextensively in previousstudies. We shall denote the covariancespertinent to/3 = 0 byx•O")(t ) and correspondingly Xo(t) = U.(t')U•(t")vq,.•[(X(t')) - (X(t"))] dt' dt" D•)")(t)= (1/2)dX,(ft)/dt. The trajectorycovariance in (17) becomesnow, by substitu- (17) tion of (18), 1 dXq Do(t)= 2 dt Xq(t) = {vo.•(r) + [U2(t') + U2(t")]%•2(r)

5 [U.(t)U•(t')+ U.(t')U•(t)]vo,.•[(X(t))-(X(t'))]dt' + U2(t')U2(t")vo,22(r)}dt' dt" 1'1 = t' -- t" (19) Equations(17) encapsulatethe main result of the last two sections.They expressthe plume secondspatial moment and r2 = fi [sin (t'/X) - sin (t"/X)] dispersioncoefficients as functionsof the unsteadymean ve- = 2fi sin [(t' - t")/2X] cos [(t' + t")/2X] locity U(t) on one hand, and the heterogeneousstructure throughthe functionsP and Q on whichvo,,t•(r ) in (11) r3=0 DAGAN ET AL.: LAGRANGIAN ANALYSIS OF TRANSPORT 895

When treating two-dimensionalflows we have to suppressin 50 (19) the dependenceon %.

To carryout the calculationsin (19), it is beneficialto switch 40 to the variablest' = (v + /x)/2 and t" = ( v - /x)/2, leading to the final expression • 3o

Xij(t)= • 0 •• {vij,11(r)+ 2(/3/X) COS(/X/2A) t• 20

1 ßcos (v/2X)vij,•2(r) + •(/3/X) 2 lO ß[COS (/x/X) + COS(vl•)]vij,22(r)) dv a/x (20)

o ' I ' I ' I ' I with r• = /X, r 2 = 2/3 sin (/X/2X) cos (v/2X), and r3 = 0. 0 6 12 18 24 30 The macrodispersioncoefficients result from the differentia- tU1/IY tion of (20), leavingonly the first integraland replacingv - 2t - /X in the integrand. 0.4 • 1•=0.2 Summarizing,Xi• (20) andthe associatedD i• requirecarry- 0.3 .... •=0 ing out a doubleor a simplequadrature, respectively, for given parametersvalues and given Cv. The resultsof suchcompu- 0.2 tations are given in the next section. 0.1

-0.0 5. A Few Numerical Results -0.1 In order to graspthe implicationsof the developmentsof the previoussection, we have carried out detailed computations -0.2

for a few valuesof the parametersof interest.We haveselected -0.3 the exponentialand isotropic Cy = rr2vexp (-r/Iy), andat first order the velocityand trajectoriescovariances are linear in -0.4 ' i ' I ' I ' I ' rr2v.The basicfunctions P andQ for thisCr werederived in 0 6 12 18 24 30 the past and we give here in the appendix their analytical tu 1/Iy

expressionsfor both two- and three-dimensionalflows. 3.0 First, we have consideredtwo-dimensional flows, pertinent

to verticallyaveraged velocity and concentrationor to regional 2.5 flows. With the velocity and mean trajectory given by (18),

while suppressingx3, we have selectedfirst the value X - 1, 2.0 i.e., TU•/Ir - 2 •r. For I• of the order of metersand U• of the

order of centimetersper day, T is of the order of years.As for 1.5 the only remainingparameter, the dimensionlessamplitude/3,

we have selectedthe three values/3 = 0.1, 0.2, and 0.3. 1.0 The computationsare straightforward:(1) by differentiating

P and Q (equation(A2)) we havederived the three coefficients 0.5 Sij (l l) fori, j = 1, 2, andsubsequently the coefficients vij,o•[ 3

(10),which are simplyrelated to Uij (seethe appendix,equa- 0.0

tions(A4)). Subsequently,we havecomputed Xij (20) by two 0 5 10 15 20 25 30 numericalquadratures and D•j (17) by one. tu 1/Iy In Figure3 werepresent the dependence ofX•i(t;/3 = 0.2) aswell as that of X•f t) ontravel time (or travel distance along Figure 3. Two-dimensionalflow: (a) the unsteadylongitudi- the x• axis).The steadystate values are thoseobtained in the nal trajectorycovariance X• • (equation(20)) for X = 1,/3 = 0.2 past [Dagan, 1984]. The resultsare qualitativelythe samefor andX(• t) (/3= 0); (b) sameas Figure 3a for X•2; and(c) same as Figure 3a for X22. other values of/3. It is seen that the longitudinalX•(t) (Figure 3a) is very closeto X[?(t), thoughsome small fluctuations are present in the former.X•2(t ) (Figure 3b) is an oscillatorycomponent of dence of 0 upon the dimensionlesstravel time (or distance period T = 2•rX and of small amplitude.It also has a small alongx•),as well as that of theangle tan -• (U2/U•) = tan-• drift related to the initial conditions.Last, X22(t ) (Figure 3c) [/3/X cos (t/A)] between the tangent to the mean centroid is definitelylarger than X•2ø(t), andwe shalldiscuss its be- trajectoryand x• (Figure 2). The behavioris similarfor other havior subsequently. values of/3, and the discussionis deferred to the next section. With the aid of X• we determinedthe principalvalues, the The longitudinalcomponent X•,•(t) is veryclose to X•(t) longitudinalXi, I andthe transverseXn, n of the tensorXii, as andis not shown.In Figure5 we givethe transverseXn,n asa well as the angle 0 betweenthe longitudinalaxis xi andx •, the functionof timefor/3 = 0.1,0.2, and 0.3 as well as X•)(t). It directionof the mean flow (Figure 2). is emphasizedthat X22 (Figure 3c) for the samevalues of/3 is To illustrate the results,we present in Figure 4 the depen- not distinguishablefrom Xn,n. 896 DAGAN ET AL.: LAGRANGIAN ANALYSIS OF TRANSPORT

20 5 I • [3:0.1 C) •=0.2 •O-- •=0.3

- 0

-5

-10

-20 , i , , [ , , 0 10 20 30 40 50 60 tU1/IY 0 Figure 4. Two-dimensionalflow: the angle 0 between the 0 25 50 75 100 125 150 longitudinalmean direction of Xii andthe x i axis(solid line) andthe anglebetween the meantrajectory (equation (18)) and tUlffy xl (dotted line). Figure 6. Two-dimensionalflow: the differencebetween X22 (equation(20)) andX• ). As is apparentfrom Figures3-5, the main impactof the pres- enceof the unsteadyharmonic component is uponX22(t; •3) • Xii,ii, whichis largerthan X• ). To graspthe nature of the the longitudinalmacrodispersion coefficient D 11 (Figure7a) is veryclose to D(i•t), exceptfor smallperiodic fluctuations. The incremental effect of unsteadiness,we have representedin Figure6 thedifference X22 - X(2S•) as a functionof time.It is mixed componentD12 (Figure 7b), resultingfrom the differ- entiationof X12 (Figure 3b), is purelyoscillatory and of small seenthat this differencehas a linear trend with slightoscilla- amplitude.The mostsignificant component is D 22 (Figure7c), tions around it. A linear regressionhas led to the following valuesfor halfthe slope Aar = (1/2)[Xii,ii(t) - X•)(t)]/t whichis larger than D• ) anddisplays an oscillatory as well as a constant increment, to be discussedin the next section. Aar = 0.0018(t3 = 0.1, h = 1) Our next set of computationswere for three-dimensional transport,again for an exponentialisotropic Cy and for the Aar = 0.0065(16 = 0.2, X = 1) sameunsteady mean flow of (18) in the horizontalplane x 1,x2. (21) Aar = 0.016(16 = 0.3, X = 1) Thistime the tensorsX• andD• havenine components each. They were computed by the same procedure as for two- Aar = 0.0014(t3 = 0.3, X = 5) dimensionalflows, except that P and Q are givenby (A3). We havecarried out computationsfor the sameparameter values, We have supplementedthe resultsfor X = 1 in (21) by an i.e., X = I and/3 equal to 0.1, 0.2, and 0.3. Many of the results additionalone for a muchlarger period of X = 5, T = 10rr. parallelthose of two-dimensionalflow and are recapitulatedin It is seenthat Aa r increaseswith t3, roughlylike 162,and wordsonly. Thus the longitudinalprincipal value Xi,i(!) isvery diminisheswith increasingX. closeto Xi i(t) for the rangeof parameterswe choseand the In Figure 7 we representthe dependenceon time of the sameis truefor Xiii,iii(t) • X•) (t) = X•3t) (t). Thelatter variousDii aswell as of D•?)(t) forthe largest t3 = 0.3(the has been derived previously[Dagan, 1984] and is also repre- behavioris similarfor other t3). Similarlyto Xll (Figure 3a), sentedhere in Figure 8. The behaviorof 0, the anglebetween thelongitudinal principal axis of X• andthe meanflow direc- tion x i (Figure 2) is very similarto that of Figure 4: At t - 0 4.0 it is the sameas that of the meanvelocity, and it tendsquickly X 22=XII,II ;•=0 3.5 -- Xii,i I ;•=0.1 to zero after a few cycles.The main impact of unsteadinessis 3.0 OXii,ii ;•=0'2 felt again in the horizontal transversedirection x2. This is illustratedin Figure8, whichdisplays the dependenceof Xii,iI •:• 2.5 andofX• ) upontravel time (distance alongx1) for allvalues of • 2.0 The incrementX•(t) - X•(t) canbe approximatedac- ;• 1.5 curately by a linear trend of slope 2Aa•, similarly to the two-dimensionalcase (Figure 6). The half slopeshave the 1.0 followingvalues:

0.5 Aar = 0.0025(t3 = 0.1, X = 1) 0.0 Aar = 0.01(t3 = 0.2, X = 1) (22) 0 5 10 15 20 25 30 tUlffy Aar = 0.022(t3 = 0.3, X = 1) Figure 5. Two-dimensionalflow: the transversetrajectow whichare closeto, thoughsomewhat larger than, thoseof (21) varianceXii,ii (seeFigure 2) for • > 0 andX• ) (• = 0). for two-dimensionalflow and transport. DAGAN ET AL.: LAGRANGIAN ANALYSIS OF TRANSPORT 897

1.0 2.00 x 22=xII,II '1•=0 1.75 0.8 • XH,H ' I]=0'1 1.50 O Xli,ii'[•=0.2 0.6 • XII,II ß 1•:0.3 1.25

0.4 1.oo

75 0.2 O. • 15=0.3 .... 15=0 0.50 0.0 ' I ' I ' I ' I ' I a o 25 50 75 100 125 150 0.25 tU 1/Iy 0.00 0.20 0 5 10 15 20 25 30 -- . 0.15 tU1/IY 0.10 Figure 8. Three-dimensionalflow: the transversetrajectory

0.05 varianceXii,i I forX = 1,/3 > 0 andX(2S:? (/3 = 0).

,•0.00 al. [1988], and it yields a small oscillatorycontribution to $22 -0.05 and D22. We shall adhere here to the definition of S22 with -0.10 respectto the instantaneouscentroid, though the differences betwebnthe two definitionsare seento be quite small. -0.15 We proceednow with discussingthe numericalresults.

-0.20

b 0 25 50 75 100 125 150 6. Discussion of Numerical Results tu lfIy We have examined in sections 4 and 5 the interaction be- 0.200 tween a simple unsteadyflow, of constantmean horizontal -- 1•=0.3 velocity and periodic transverseone, and the heterogeneous 0.175 .... i•=0 structure.Our main simplifyingassumptions were neglecting 0.150 pore-scaledispersion and adoptinga first-orderapproximation in •r} for flowand transport. It is emphasizedthat the results 0.125 for velocityand trajectorycovariances and for macrodispersion coefficientsare consistentleading-order terms in expansionsin O.lOO powersof •r•. 0.075 While we havekept the periodconstant (except in (21)), we 0.050 have examineda range of valuesof the amplitudes/3(made

0.025

0.150 0.000 , I ,' I ' • , I , I ' -- •=0.3 C 0 25 50 75 100 125 150 .... I•--O tU 0.125 Figure 7. Two-dimensionalflow: (a) the unsteadylongitudi- nal macrodispersioncoefficient D 11 = (1 / 2)dX1 •/dt (equa- 0.100 tion(20)) for X = 1,/3 = 0.3 andD• t) (/3 = 0); (b) sameas Figure 7a for D 12; and (c) sameas Figure 7a for D22 supple- • .075- mentedby the approximation(26) (thick-dashedline). 0.050

Finally, in Figure 9 we have representedthe dependenceof 0.025 D22upon time for/3 = 0.3as well as the steady D•:•)(/3 = 0). The othercomponents, Dll andD12, are verysimilar in behavior 0.000 • ' i - -'- to thosepertaining to two-dimensionalflow (Figures7a and7b). 0 10 20 30 40 50 As a lastpoint, one can define the transversesecond spatial tU lffy momentof the plumewith respectto the axisx 1, rather than to Figure 9. Three-dimensionalflow: the unsteadytransverse the centroidof coordinate(R2) = /3 sin (t/X). This resultsin macrodispersioncoefficient D22 - (1/2)dX22/dt (equation theextra term (R2(t)) 2 = •2 sin2(t/X) supplementingX22of (20))for X = 1,/3 - 0.3,D•:? (/3 = 0) andthe approximation Figure 5. This term is essentiallythe one consideredby Naff et (26) (thick-dashedline). 898 DAGAN ET AL.: LAGRANGIAN ANALYSIS OF TRANSPORT dimensionlesswith respectto Iy) up to 0.3, which represents In the steadystate case,i.e., for/3 = 0, the only term left in quite a large deviationof the mean trajectory(18) from the the integrandsof (24) is $22(t' - t", 0, 0) alongthe x• axis, mean (temporal) one. leading to the expressionsderived in the past [e.g., Dagan, A first result,consistent with both field findingsand previous 1984].Since $22(r•, 0, 0) is a holecovariance, D(2S:/) tends to works,is that the longitudinal(as well asthe verticalfor three- zero for t >> 1 in both two- and three-dimensionaltransport, dimensionalflow) trajectoryvariances X• (and X33) and the whileX(2s:/) - In t (twodimensions) and X(2sd ) -• const(three associatedmacrodispersion coefficients D • (and D33) are lit- dimensions).Hence the linear trend present in X22 or the tle influencedby unsteadiness(Figures 3 and 4). The cross constant,Fickian, contribution Aa r (equations(21) and (22)) componentX•2 (Figure 3) is oscillatory,and it impactsmainly to D22 originatesfrom two termsin (24): The first one is U22, theangle 0 betweenthe longitudinal main axis of theXii tensor the transversevelocity covarianceintegrated along the actual and x•. At t = 0 this angle is identicalwith that betweenthe trajectoryrather than the x• axis,and the secondone is from instantaneousmean velocity and thex• axis(Figure 4). This is the longitudinalvelocity covariance U• •, howeverfor the cross expected,since the effectof heterogeneityis not felt then and flow of meanvelocity U2 and with r 2 and r• exchangedin the the plume translateswith the meanvelocity. The angle0 tends, argumentof U•. Theseterms are similar,but more complex, however,quickly to zero, and the plume becomespractically than thoseaccounted for by Goodeand Konikow[1990] (see parallel with the x• axis(Figure 4). This is mainly due to the section1). It is recalledthat X22 andD22 are proportionalto quick increaseof X• with t as comparedwith X•2. tr2•and depend here in a complexmanner on/3 andit. The main effect of the unsteadyflow componentis in in- The half slopesA a r of the linear trend of the increment creasingthe transversetrajectory variance Xii,ii(t) • X22(t) •22(t) -- X(2sd) incorporate in a singleparameter the main and the associatedeffective dispersion coefficient D 22(t) (Fig- unsteadyeffect investigated here. First, it leads to the simple ures5-9) withrespect to thesteady ones X(2S•)(t) and D(2?(t). approximation The effectis more dramaticin three-dimensionalflows (Figure 8), for whichX(2?(t) tendsto a constantvalue, whereas the X22(t;/3, it)• X•)(t) + 2Aar(/3, it)t (25) unsteadycomponent has a linear trend. The rest of our dis- whichneglects the oscillatorypart of X22, seento be negligible cussion is devoted to this effect. in Figure 6. First, we shallrecall in an explicitmanner the relationships By the sametoken, D22 can be approximatedby leadingto X22 (see (20) and (A4)) D22(t;/3, it)• D•)(t) + Aar(/3, it) (26) X22(t; •, it) = {U22(r1,r2, r3) and we have plotted (26) in both Figures7c and 9. It is seen that indeedD22 in (26) averagesthe time oscillationsof D22, which are more pronouncedthan thoseof X22. + [U2(t') + U2(t")]U•2(r2, r•, r3) The mostimportant point, however, is that Aar in (25) and (26) representsa Fickian effectwhich becomes the dominant + U2(t')U2(t")U•(r2, r•, r3)} dt' dt" (23) one for sufficientlylarge t, especiallyin three-dimensional wherer• = t' - t", r2 = 2/3sin [(t' - t")/2X]cos [(t' + t")/2X], flow.As a matter of fact, it is made dimensionlesswith respect to U•Iy, and therefore, A at may be viewed as the ratio be- The differencebetween (19) and (23) is that we have re- tweenthe asymptoticmacrodispersivity and the log conductiv- placedv•i,• bythe components of Ui•, thevelocity covariances ity integralscale. Its numericalvalues (equations (21) and (22)) for steadyflow of unit meanvelocity parallel to x • (A4). These suggestan approximatequadratic dependence upon/3 and a componentsare given in (11) and were derived in the past decaywith it (21). [Rubin, 1990; Rubin and Dagan, 1992; Zhang and Neuman, The approximateresults (equations (23) and (24)) will serve 1992]. The evaluationof the contributionof each of the three in the future for analysisof field testsin which Cv is generally terms of (23) showedthat the midterm is responsiblemainly of an anisotropicstructure which does not lend itselfto simple, for the smalloscillatory part OfXli,iI • X22 (Figures5 and8) analytical,expressions for the functionsP and Q (equations andD22 (Figures7c and 9). The linear trend of the increment (A2) and (A3)). X22(t) - X(2S•)(t)(Figure 6, equations(21) and(22)) is asso- ciatedwith the remainingtwo terms,leading to the following approximateexpressions for X22 and D22.' 7. Summary and Conclusions By usingthe Lagrangianapproach and a first-orderapprox- X22(t) • [U22(r•, r2, r3) imationof flow and transportequations, we havebeen able to foliot derivethe soluteparticles trajectory covariances X• for an averagemean flow uniformin spacebut varyingin time. Under + U2(t')U2(t")U•(r2, r•, r3)] dt' dt (24a) ergodicconditions that applyfor sufficientlylarge plumes com- with r• = t' - t", r2 = 2/3 sin [(t' - t")/2X) cos [(t' + t")/2X], paredto heterogeneityscales, the trajectorycovariances are and r 3 = 0; and equalto the plume spatialmoments, while the centtoldmoves alongthe mean trajectory.In a similarvein, the D•/ = (1/2) dX•/dt representthe effective dispersion coefficients of D22(t) • [U22(rl, r2, r3) + U2(t)U2(t')Uii(r2, ri, r3)} dt' the plume. I0t Similarlyto thesteady flows investigated in the past, the X•/ (24b) resultfrom the integration of thevelocity covariances u•/along with r• = t - t', r2 = 2/3 sin [(t - t')/2it) cos [(t + t')/2it], the meantrajectory •X•, whichdeparts from a straightline and and r 3 = 0. dependsnonlinearly on time. However, u• arethe same as for DAGAN ET AL.: LAGRANGIAN ANALYSIS OF TRANSPORT 899 steadystate flow providedthat the mean velocityis the instan- flowin the /3 direction)as givenby (10) and Uo(r•, r2) = taneousvelocity of the actual flow. vo,• (equation(11)) for two-dimensionalflow. Thus We have illustratedthe procedureby applyingthe general Vll,ll--'U11 t/22,22= U•i method to the particularcase of an isotropicexponential log conductivitycovariance and a periodictime-varying flow. Thus V12,11= V21,11= Ull,21• Ull,12= U12 (A4) the mean velocitycomponent U• is constantwhile the trans- v12,22= v21,22 -- v22,12 = v22,21 = $•2 verse U2 variesharmonically in time. U22,11• Ull,22--'U12,21-- V21,21 = U22 Our main result was that for the selectedvalues of param- eters,the longitudinaldispersion of the plume is little affected whereU•(r•, r2) = Uo.(r2,rO. The relationships arethe same for by the unsteadyflow. The plume rotates by the angle of the three-dimensionalflow with additionaldependence on r 3. mean flow at t - 0, and it becomespractically aligned with x • after a period. Acknowledgment.This studywas supportedthrough NSF grant The major impact of the unsteadycomponent is on trans- 9304481. versedispersion. The effectivedispersion coefficient Dn,n • D 22 canbe viewedas made up from three terms:(1) the steady References state component,which results from the integration of the transversevelocity covariance U22 alongthe x • axisand which Dagan, G., Solute transportin heterogeneousporous formations,J. Fluid Mech., 145, 151-177, 1984. tendsto zero for large travel time; (2) a smallperiodic com- Dagan, G., Flow and Transportin PorousFormations, 465 pp., Spring- ponentstemming mainly from U•2, and (3) a constant,Fick- er-Verlag, New York, 1989. ian, additionalterm. The last term resultsfrom the integration Dagan, G., Transportin heterogeneousformations: Spatial moments, of the transversevelocity covariance U22 alongthe sinusoidal ergodicity,and effectivedispersion, Water Resour. Res., 26, 1281- 1290, 1990. trajectoryand from that of U• in the x2 direction.The third Dagan, G., An exactnonlinear correction to transversemacrodisper- categoryof terms encapsulatesthe main contributionof flow sivityfor transportin heterogeneousformations, Water Resour. Res., unsteadinessupon transport. These terms depend in a complex 30, 2699-2705, 1994. mannerupon heterogeneity, being proportional to rr2v,and on Farell, D. A., A.D. Woodbury, E. A. Sudicky,and M. O. Rivett, the amplitudeand period of the time-varyingU2. Thisresult is Stochasticand deterministicanalysis of dispersionin unsteadyflow at the Borden tracer-testsite, Ontario, Canada,J. Contam.Hydrol., differentfrom that of previousstudies, in which heterogeneity 15, 159-185, 1994. and time dependenceof the meanflow effectswere separated. Goode, D. J., and L. F. Konikow,Apparent dispersionin transient groundwaterflow, WaterResour. Res., 26, 2339-2351, 1990. Kabala, Z. J., and G. Sposito,A stochasticmodel of reactive solute Appendix: A Few Analytical Auxiliary Results transportwith time-varyingvelocity in a heterogeneousaquifer, Wa- ter Resour. Res., 27, 341-350, 1991. The fundamentalfunctions P(r) and Q (r) [see,e.g., Dagan, Kinzelbach,W., and P. Ackerer, Modelisationde la propagationd'un 1989;Rubin, 1990] satisfythe equations contaminantdans un champ d'6coulementtransitoire, Hydrogeolo- gie, 2, 197-205, 1986. V2p= -Cv(r) V2Q= P (A1) Nail?,R. L., T.-C. J. Yeh, and M. W. Kemblowski, A note on the recent natural gradientexperiment at the Borden site, WaterResour. Res., For an isotropic C•,, (A1) become ordinary differential 24, 2099-2104, 1988. equations,with V 2 -= (1/r)d/dr(rd/dr) (twodimensions) and Neuman, S. P., and Y.-K. Zhang, A quasi-lineartheory of non-Fickian •72 • (t/r2)d/dr(r2d/dr) (threedimensions). For easeof and Fickian subsurfacedispersion, 1, Theoreticalanalysis with ap- referencewe give here the final results for Cv = (r2vexp (-r), plicationto isotropicmedia, WaterResour. Res., 26, 867-902, 1990. wherer is made dimensionlesswith respectto I•,: Rehfeldt, K. R., and L. W. Gelhar, Stochasticanalysis of dispersionin unsteadyflow in heterogeneousaquifers, Water Resour.Res., 28, Two dimensions 2085-2099, 1992. Rubin, Y., Stochasticmodeling of macrodispersionin heterogeneous P/(I•r•,) - 1 - e-r- E + Ei(-r) - In (r) porousmedia, WaterResour. Res., 26, 133-142, 1990. (A2) Rubin, Y., and G. Dagan,A note on head and velocitycovariances in (r- 5)e -r three-dimensionalflow throughheterogeneous anisotropic porous Q/(I•r•,) = +•- t- + 5+ Ei(-r) media, Water Resour.Res., 28, 1463-1470, 1992. Sudicky,E. A., A natural-gradientexperiment on solutetransport in a r+6 sandaquifer: Spatial variability of hydraulicconductivity and its role in the dispersionprocess, Water Resour. Res., 22, 2069-2082, 1986. - • In(r) Zhang, D., and S. P. Neuman, Comment on "A note on head and velocity covariancesin three-dimensionalflow through heteroge- Three dimensions neousanisotropic porous media" by Y. Rubin and G. Dagan, Water 2 2e -r Resour. Res., 28, 3343-3344, 1992. P/(l•r•,) = -i + e-r- -- Zhang, Y.-K., and S. P. Neuman,A quasi-lineartheory of non-Fickian r r and Fickian subsurfacedispersion, 2, Application to anisotropic (A3) media and the Borden site, Water Resour.Res., 26, 903-914, 1990. 4 4e -r r 2 Q/(I•tr•,) = -3 - e-r + r r 6 A. Bellin, Dipartamentodi IngegneriaCivile ed Ambientale,Uni- versith di Trento, via Mesiano di Povo, 77, 1-38050,Trento, Italy. where E is the Euler constantand Ei is the exponentialinte- (e-mail: [email protected]) G. Dagan, Faculty of Engineering,Tel Aviv University,P.O. Box gral. It is recalledthat the velocitycovariances (equations (t0) 39040,Ramat Aviv 69978,Israel. (e-mail: [email protected]) and(11)) areexpressed with the aid of the derivativesof P andQ. Y. Rubin, Departmentof Civil Engineering,University of Califor- Next, we present the relationships between the tensor nia, Berkeley,CA 94720. vij,•13(rl,r2) (the covariancebetween the ith componentof the velocityfluctuation at x causedby a unit mean flow in the (ReceivedDecember 5, 1994;revised August 14, 1995; a directionand the jth fluctuationat y = x - r for a unit mean acceptedAugust 17, 1995.)