Lagrangian Analysis of Transport in Heterogeneous Formations Under

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 University,Tel Aviv, Israel 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 aquifer 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 random space 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 random in time. relatedto aquiferstorativity analyzed by Goodeand Konikow [1990].A point of principleis that the actualdispersion coefficients,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 the centroidtransverse 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 obtained.We feel,therefore, that interpreting X•j in (2) asspatial momentsis not justified. X'(t) = u(t') dt'

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