Instabilities in variable density flows: Stability ... - Wiley Online Library

WATER RESOURCESRESEARCH,VOL. 33,NO. 1, PAGES31-41, JANUARY 1997

Instabilities invariable density flows: Stability andsensitivity analyses for homogeneous andheterogeneous media RobertA. Schincariol Department of EarthSciences, University of WesternOntario,London,Canada

Franklin W. Schwartz Department of Geological Sciences, OhioStateUniversity, Columbus

Carl A. Mendoza Department of Earth andAtmospheric Sciences, Universityof Alberta,Edmonton,Canada

Abstract.Thisstudyimproves ourunderstanding of instability phenomena thatmay

accompany thetransport of dense. plumes of dissolved contaminants. Onemajorobjective

isto testhowwellanalyticstability theorydeveloped byList [1965]appliesto the transport of dense plumes in bothhomogeneous andheterogeneous media.Thedatato testtheprediction comefromnumerical modelexperiments in whichinstability growthis generated by perturbingthe interfacebetweenfluidsof differingdensity.Stabilitycriteria, asdetermined by the transverse Rayleighnumber,the ratioof transverse to longitudinal Rayleigh numbers,and the nondimensional wavenumber,compareverywell with results observed in the numericalexperiments for isotropicmedia.Comparisons involving correlated randomfieldsweremuchlesssuccessful because plumestabilityis determined on a localbasisas a functionof the changingpermeabilityfield. Instabilitiestend to

dissipate in zonesof lowerpermeability andgrowin zonesof higherpermeability. Another

objective of thestudy isto determine thefactors thatcontribute to stability andinstability in homogeneous andheterogeneous systems. Sensitivity analyses usinga transport model withinthe frameworkof List'sstabilitytheoryshowthat stabilityis promotedby low mediumpermeability,smalldensitydifferences, andsignificant dispersion. In heterogeneous media, stabilityis promotedby increasedcorrelationlengthscalesand increasedlog permeabilityvariance.Furthermore,the simulationsillustratethe intimate relationship that existsbetweeninstabilitygrowthand decayand the heterogeneous nature of the permeabilityfield. Thusstabilitycriteriathat do not incorporatecharacteristics of thepermeabilityfield will not be suitablefor naturalor field-scaleporousmedia. Introduction

the flow andtransportparameters[Schincariol et al., 1994].In geologicallyrelevantsituationsit will be the heterogeneityof Schincariol etal.[1994] illustrated mathematically howsmall the porousmedia,from a pore scaleto the reservoirscale,that perturbations, imposedon the interfacebetweenfluidsof dif- will controlthe perturbations. Here the maximumwavelength feringdensities, couldresultin instabilities thatgrowwithtime. should be somewhat similar to the dimensions of the reservoir Theinterfacialor fingeringinstabilities werestudiedin relation [Marle, 1981].Severalquestionsremainedunanswered, howto groundwater contaminationproblemswhere a more dense ever,whichwe aimto address in thispaper.The firstimportant plumeis enclosed by, and is movingalongin, a bodyof less issueis to definewhat parametersdeterminethe stabilityof densefluid.Their analysis explainedqualitatively howfeatures unstably-stratified flowsin variable densitysystems.Stability apoftheinitialperturbation, in particulartheperturbation wave- criteriahavebeendevelopedusinganalytical-mathematical length,andthe densitycontrastbetweenthe two fluidsdeter- proaches[e.g.,List, 1965]and throughthe analysisof laboraminedthe growthrate of the perturbation. In flowthrougha tory experiments[e.g.,Oostromet al., 1992].In general,hownaturalporousmedium perturbationsor interfacialdistur- ever, there has been no rigorousvalidation of the stability bances arecontinuously generated because of heterogeneitiescriteria to date.Clearly,if the efficacyof the analyticalcriteria

of the medium[Moissis and Wheeler,1990].Theserandom canbe c.onfirmed,theywouldprovidea usefulbasisfor analyzingboth experimentaland natural systems. Unresolvedquestionsremain alsowith respectto the manferences in poregeometryto largerheterogeneities on the scaleof the problemunder consideration. Whether or not ifestationof unstableflowsin heterogeneousmedia.There has these perturbations become unstable depends onwhethertheir been virtually no experimentalwork with these systemsbewavelength exceedsa criticalwavelength,whichdependson causeof the difficultyin creatingan artificialmediumin a flow tank which accuratelyreflectsthe heterogeneityof natural Copyright 1997bytheAmerican Geophysical Union. media. Thus relativelylittle is knownconcerningthe character of mixingpatternsthat developand how featuresof the hetPapernumber96WR02587. 0043-1397/97/96WR-02587509.00 erogeneityimpactstabilityor the growthrate of instabilities.

perturbations to flow occurover many scales,from slightdif-

31

32

SCHINCARIOL ET AL.:INSTABILITIES IN VARIABLEDENSITYFLOWS

verseRayleighnumber;r/, the ratio of the transverse to lon

porousmedium:permeabilityk

gitudinal Rayleigh numbers (•.T/•L);

porosity e

and to, th,

nondimensional wave number of the disturbance. The tw,

Rayleigh numbers are definedmathematically as[List,1965] ""

,, mixing zone•

gkl(p2- p•)

x

AT=

OD TIX

(1)

gk!(p2- p•)

XL--

ODLt. t'

(2)

constant; k is the intrinsicpermeFigure 1. Schematic representation ofthegeometry foran- where!7isthegravitational alyticstabilitycalculations (modifiedfromList [1965]).

ability;p• and92arethe densities Ofthe ambientfluidandthe plume,respectively; 0 isporosity;/x isviscosity; andD 7-andDL

are transverseand longitudinaldispersioncoefficients.AlThe goalof thiswork is to improveour understanding of though Listdefined the dispersion coefficients asD = instability phenomena in variable-density systems andto eval- herewe usethe more presentlyacceptedform of D = a v + uate the criteriathat distinguish betweenstableand unstable D [ whereD is thecoefficient of hydrodynamic dispersion, a is flows.We utilize the numericalmodellingapproachof Schin- the mediadispersivity, v is the averagelinear groundwater cariolet aI. [1994]as the basisfor a stabilityand sensitivity velocity, andD* is thebulkdiffusion coeftScient. In addition, a constant meanviscosity, herewe definea analysis of two-dimensional, homogeneous, andheterogeneousWhileListassumed

systems. Ouranalysis incorporates andbuildsontheorydevel- vi.scosity thatis a function of soluteconcentration. Viscosity opedby List [1965]whichis applicable,in a generalsense,to

effects, fortheflowgeometry •tnder investigation where amore

thesystems considered. We focuson List'sworkasto date,it

dense, viscous fluidd0wnivardly displaces a lessd.ense, viscous

is the onlymajortheoreticalanalysisof interfacialinstabilities one,arestabilizing whiledensity effects aredestabilizing. This

formiscible fluids inporous media withdensity straiification in should notresultin a largedeviation fromList'si'e•ults, asat a horizontal flow field.

typicalgroundwater velocities the densityeffectswill be dominant.The readeris referredto Weltyand Gelhar [19.92]for a thoroughdiscussion of the effectsof viscosityand densityin List's [1965] Stability Analysis variousone-dimengional displacement scenarios.A characterList [1965]conducteda stabilityanalysisof the problem isticlength(1) is definedas [List, 1965]: whoseflowgeometryis shownin Figure 1. The analysisexam-

inesthebehavior ofthemixing zonebetween twohorizontally stratifiedmisciblefluidsof density92 and p•, where Both fluids are assumed to be in uniform

horizontal

motion

1= ( rrDrX/U)u2

(3)

whereX is the distancefrom the sourcewhere local stabilityis

to be e•,aluated, and U is the Darcyflux.The characteristic withan average linearvelocity, v, in a homoge.neous poi•ous length isperhaps themos{ nebulous of allthetermsinvarious medium of known intrinsicpermeability,k, and porosity,0.

Bothfitlidsare assumed to havethe sameviscosity, •. Instabilitiiasalongthe interfacebetweenthe two fluids are developedby perturbingmathematically the fundamentalvariables such as density,pressure,and velocity.When the systemis unstable,these perturbationswill grow; when the systemis stable,the perturbationswill die out. List [1965]developedan analyticalsolutionfor the stability of the interfacebetweenthe fluidsfrom the continuityequation, Darcy'sequation,andthe advection-dispersion equation. In approachingthis problemanalytically,List had to assume that (1) fl0w is steady in the solution of the advection-

dispersion equation,(,2)therateof growthof the mixinglayer or dispersion zoneisverysmalloverSomelength,sothe problem is investigated for a zonewhere the sidesof the mixing zone are parallel, and (3) diffusionis negligible,so that the solutionis valid only at large velocities.Thus with theseassumptions,List presenteda local theory that assesses the stability of an interfaceovera lengthwherethe dispersion zone doesnot widen.His analysis couldnot address, for example, thetwo-dimensional character of anyinstabilities thataregenerated,theproblemof evaluating howan arbitrarydisturbance

formsof theRayleigh number,anditsdefinitionoftenpi'esents

difficulties in applying stability theoryto practical problems. Basically,the characteristic length is needed as a nondimensionalizingparameterfor the Rayleighnumber. In most one-

dimensional problems thecharacteristic lengthisequalio the lengthof the mixingzone [Buesand Aachib, 1991], and its definitionis rather intuitive.In most two-dimensional problems it is somelength associated with fluid movement(for example,Oostromet al. [1992] set the characteristiclength

equalto theplumethickness), anditsdefinition becomes less clearandintuitive. Thewavenumber(to)is definedas[List, 19651 to= 2'rrl/L

(4)

where L isthewavelength oftheunstable wavegenerated at the source.For the simulationexperiments uponwhichour analysis is based,all of thenecessary parameters requiredto calculateXz., Xr, •, and toare known.

List's[1965]analysis alsoprovides a theoretical relationship fortherateof instability growthasa functionof time.The most unstable wavewill growwithtimeaccording to the amplificaatx = 0 decays or grows asit is sweptdownstream, or how tion [List,1965]: instabilities behaveasa functionof transport,wherethe zone A = exp[ (toci)T] (5) of dispersionsignificantlyincreases.This more "realistic"

problemis addressed numericallyhere. whereci is a stabilityparameterthat reflectshow much the List [1965]determinedthat stabilityof the interfacede- nondimensional complex wavevelocity differs fromtheperturpendeduponthree dimension!ess parameters: Xr, the trans- bationvelocity,andT is dimensionless time,definedas

SCHINCARIOL

T = tUo/l

ET AL.: INSTABILITIES

IN VARIABLE

(6)

FLOWS

6c

•=o

whereUo is dimensionless fluxgivenby

33

6h

7•-=o

C=Co

gk(p2- Pi)

u0=

DENSITY

(7)

h=ho •c

h:hi

•-=0

c=O

Theamplification istheratioof theamplitude of theinterfacial

6c

perturbation at dimensionless timesgreater thanzeroto the initialamplitude.At stabilitythe amplification is 1, which meansthat the initialperturbationwill neithergrownor decay.

6h

•-•-=o E•-=o < .............

1.0625

m

';

Figure 2. Model domain and boundaryconditionsfor nu-

List[1965]conducted a detailedtheoreticalanalysis to de- mericalsimulationof tank experiments; Co, hi, and ho are scribehowstabilitydependedupon Xr, •1, and •o.His analysis specifiedvaluesof concentrationor equivalentfreshwater demonstratedthat instabilitygrowthwas favored in systems head, and n designates the directionnormalto the domain withminimaldispersion andsmallwave.numbers. Experimen- boundary. tal work, conductedas part of his study,suggested that the

stability analysis wascorrect; however, thisportionof thestudy wasnot very extensive.

Y sinY[•oft(Y 2 + $2)+ •2(0)2 q_f•2)q_y2•2_ ,02•32] _ 1;cosY

Homogeneous Medium

ßtanh•[wfl(Y2 + 82)+ y2(0)2+ •2) _ (y2•2_ 02j•2)]

This sectionexaminessimplehomogeneous-media systems toestablish (1) howwellList's[1965]stabilityandperturbation

+ Y5 sinY tanh&-(•o+ ft)(Y2 + f;2)- cosY

ß + =0 (8) growthrelationships describe morerealisticsystems (i.e., whereList'sassumptions, discussed in the previoussection,are whereY is definedby (9), 8 by (10), and/3by (11) asfollows: notinvoked),and(2) howvariousparameters of thisproblem

control stability•indperturbation growth.Our approach uti-

¾={[( rø2(1 - •) X•wci)2+X__•] i/2 - w2(1 2 +Xrwci} (9) +r/) u2 2 +XTWCi w2(1 +•1) + [(w2 -5-(i- w)- Xrwci) 2 2 + •.___•] 1/2} 1/2 (10)

lizescomputersimulations to investigate theseissues. Details ofthemodelling approach, modelverification, anda discussion oftimeandspacediscretization andaccuracy criteriaaregiven bySchincariol etal. [1994].Briefly,a two-dimensional, coupled variable-density flowandtransportcodeis usedto simulatethe behaviourof a denseplumethat invadesthe model domain shownin Figure2. The domainsizeandboundaryconditions 8= for thesecomputerexperiments coincidewith thoseof Schincarioland Schwartz's [1990]flow tank experiments. TestingList's Theory

Theapproach to testingList's[1965]theoryisto utilizethe

2

/3 = (w2r/+ Xrwci)i/2

(11)

resultsfrom variousmodellingruns as thoughthey were ex-

For the iterative solution,knownvaluesof Xr, •I, and •o are

perimental data.For eachof the simulationexperiments, we

calculated from the base casevariables and constantsshown in

determine whetherthe denseplumeshouldbe stableusing Table 1. For the varioussimulations,valuesof Xr are variedby thedensity of theplumein relationto ambientwater, List'stheoreticalapproachandthe parameters of the simula- adjusting tiontrial.Stability predictions arethentestedbyinspecting the and valuesof •oare variedby changingthe wavelengthof the at the source, asexplained by Schincaresults of the simulation experiment to determinewhetherthe interfaceperturbation to longitudiamplitude of the initialperturbations, as represented in the rioleta!. [1994].Sincetheratioof the transverse to Dr/Dr, •1is heldconconcentration fields,growsin time. Sinusoidal perturbations nal Rayleighnumbers01) simplifies variablesdetermining areappliedto theloweredgeof theplumeby increasing and stant closeto 2.98. The dimensionless on the theoreticalstabilitydiagram decreasing theheightof theinflowzone,for the densefluid, stabilitycanbe presented byList.Eachof thesolidlinescurving throughan adjustment of the specified concentrations of the (seeFigure3) created lowesttwo nodesformingthe 2i node source.The resulting acrossthis figurerepresentsa stabilityboundary,basedon to a differentvalueof •1.The sinusoidal perturbation hasanamplitude of 2.5mm,whichis List'stheory,whichcorresponds 5% of the sourcewidth.The wavelength of the perturbation line for anygivenvalueof r/, thusdividesit, •o spaceinto wasalteredby defining the concentration (Co,in percentof stableandunstableregionsasshownon Figure3 and similar maximumsourceconcentration) of the lowertwo nodesas a. figures. Onecomplication in calculating wavenumbers andRayleigh functionof time as eitherc0 = 100.0 or Co = 0.0. A more is that one padetaileddiscussion of interfaceperturbation mechanisms is numbersfor all of the modellingexperiments rameter,X, whichis the distancefrom the sourcewherestagivenby Schincariol et aI. [1994]. isunknown. Thisisbecause wearenot With all of the key parameters definedin the simulation bilityisto beevaluated, stability in a theoretical system at a givenpoint, experiments, theparameters affecting stability andgrowthrate tryingto assess canbe calculated usingrelationships (1) through(7). We de- asListdid,buttryingto applyList's[1965]theoryto assessing growor rivec•, whichis requiredfor determining the growthrate the stabilityof a systemwheregivenperturbations decay as they progress through the flow domain. Further, it is factors (wci), throughan iterativesolution of oneof List's not clear howsensitivestabilitycalculations are to X. There[1965]analyticstabilityexpressions:

34

SCHINCARIOLET AL.: INSTABILITIESIN VARIABLEDENSITYFLOWS

study haveaninitialamplitude of 2.5mm.Thewavenumbers

Table 1. Flow and TransportParameterValues

andRayleigh numbers fortheeightexperimental runs,witha 2000mg/Lfluid,arecalculated forX values of5, 10,20,40,and

Value

Parameter

Permeability,* m2

5.7 X 10-l•

80 cm.This combinationof parametersgivesrise to transverse

Porosity*

0.38

Rayleigh numbers of 25.4,35.9,50.8,71.9,and101.6,respectively.Asshown in Figure3,theboundary between stableand

2.75 X 10-6 1.0 • 10-3 2.0 x 10-4

Average lineargroundwater velocity,* m s-• Longitudinaldispersivity,? m Transversedispersivity,? m Tortuosity?(•)

0.35

1.61 • 10-9

Aqueousdiffusion coefficient for diluteNaC1

solution? (Da), m2s-• Bulkdiffusion coefficientS' (•'Da = D ,•), m2 s-1 Compressibility of H20 at 25øC,õ m2 N-•

5.635 x 10-tø 1.002 x 10-3 4.8 x 10-•ø

Viscosityof NaC1solution,$Pa s 1000 mg/L 1500 mg/L 2000 mg/L 3000 mg/L 5000 mg/L

1.004 x 1.005 x 1.006 x 1.008 x 1.011 x

Viscosityof H20 at 20øC,$Pa s

unstable modeltrials(i.e.,actuallyobserved in the experiments asmarkedbythetransition fromopensquares to opencircles)

occurs generally as expected betweenr• = 0.9 and r• = 5.0. Thusstability calculations forthemodelling dataaresomewhat sensitiveto the choice of X. However, one of the values,X =

10 cm,coincides with the theoretically predictedtransitionat

approximately r/ = 2.98.Consequently, in all subsequent cal10-3 10-3 10-3 10-3 10-3

Relativedensity at 20øC,:[: g cm-3 0.9982 0.9989 0.9993 0.9997 1.0004 1.0018

Pure H20 1000mg/L NaC1solution 1500mg/L NaC1solution 2000 mg/L NaC1 solution 3000mg/L NaC1solutionat 20øC 5000 mg/L NaC1solutionat 20øC

culations, X is assumed to be 10 cm. The analysiswasrepeated with resultsfrom the 3000-mg/LNaC1simulationtrials.Wave numbers andRayleighnumbers werecalculatedforX valuesof

5, 10, 15,and20 cm.The resultsof thisanalysis(not shown) againindicate X = 10 cmasprovidinga transitionin stability at r• = 2.98.

Figure4 presents theresultsof stabilitycalculations involving the entiresetof simulation resultsfor the five different relativeplumedensitieslistedin Table 1. The boundarybetweenobserved stabilityandinstability(markedby the transition from open squaresto open circles)is found to be at *Schincariol and Schwartz[1990]. approximately • = 2.98, whichis in agreementwith List's ?Robinson and Stokes[1970]. $Weast [1980]; relative density D• ø (density relative topurewaterat [I965] theory,and illustratesthat the choiceof X = 10 cm,

4øC). õDomenico and Schwartz[1990].

basedon 2000- and 3000-mg/LNaC1 simulations,is valid at other concentrations. Note that plume velocitiessignificantly

variedfromambientfieldvalues(2.75 x 10-6 m/s)at the higherdensities.For thesetrials the actual plume velocities fore, beforedeterminingthe theoreticalstabilityfor the model were used for the stabilitycalculations.These differencesin trialsasa functionof density,a sensitivityanalysis is conducted velocityarise becausethe densitydriving forces become so to determine how the observed transition between stable and significantthat simplevelocityestimatesusingDarcy'sequaunstablesystemsdependsuponX. tion acrossthe domainbecomeerroneous.In summary,then, The numericalsimulationsare basedon the base casepaList's theorydoesdescribestabilityrelationsfor simpleexperrameters describedin Table 1 with boundaryheadsfixed to iments in homogeneous isotropicmedia. create an initial averagelinear groundwatervelocityof 2.75 x 10-6 m/s.By usingeightdifferentperturbing wavelengths Anothervery differenttest examinedthe extent to whichthe analytictheoryis able to predictgrowthrates of instabilities. (3.56, 4.16, 4.75, 5.35, 5.94, 7.15, 9.50, and 11.88 cm), the This analysiscomparesgrowthrates (amplification)as preresultingplumesimulationsrepresenta spectrumof both stadicted by (5) with the growthof perturbationsactuallyobble and unstableflow conditions.All perturbationsfor this

4 3

j

2

m Stable

Stable _./•-•.1'

o Unstable /

t m Stable

4

3

o Unstable

Stable•

2

•11=0.5

g 101

•

11=5.0

$

3

3

2

2

• 101

2

•0 3

4

Ohs,able .• 6 ?

102

2

3

4

Transverse RayleighNumar (•)

lO0

10I

2

3

4 5 6 7

102

2

3

4

Transverse RayleighNumber(•)

Figure 3. StabiliVdiagramillustratingthe resultsof calcula- Figure4. Stability diagram illustrating theresultsof calcula-

tionsinvolvingeight differentperturbingwavelengths for a tionsin which boththedensity difference andtheperturbing 2000m•L plume.Thevegicalcolumns of datapointsbegin- wavelength arevaried.Theverticalcolumns of datapoints ningat Xr = 25.4 correspond to X valuesof 5, 10, 20, 40, and 80 cm,respectively. Solidlinesfrom List [1965].

beginning at Xr = 16.8coincide withplumes having concentrations of1000,1500,2000,3000,and5000mg/L,respectively.

SCHINCARIOL

ET AL.: INSTABILITIES

IN VARIABLE

served in simulationtrialsusingthe basecaseparameters(Table 1), a 2000-mg/LNaC1 plume, and three perturbing wavelengths (5.35,5.94,and7.15cm).The resultsof the com-

parison arepresented in Figure5 wherethecalculated amplification(via (5)) isa straightlinewithslopecoci/2.3.Note that although theactualgrowthratesarenot linearandbeginto fall offafterlongtimes,the theoreticalpredictionsare fairly close. The reasonwhy the trend of observedamplificationsdeviates from the linear resultspredictedby List's [1965] theory is relatedto assumptions inherentin his approach.His theoryis a localone,assessing the stabilityof an interfaceovera length wherethe dispersionzone is assumednot to widen and thus

cannotexplainthe spatialgrowthof perturbations. In effect, with transportin our model system,dispersionreducesthe concentration(density) gradient,which acts to stabilizethe interface. Theseresultssuggest thatList'sequationdescribing the

DENSITY

FLOWS

35

• o

3

Stable Unstable

Stable

•' 2

•

• 101•)

6

3

2 100 101

,-"

•

•

•

, 2

Unstable

• 3

• 4

c.• decreasing dispersivity ,•.... ' 5 6

• 7

• ,• • 8 102

Transverse RayleighNumber(X,r)

amplification of instabilities is usefulin establishing relative Figure 6. The changingpattern of stabilityas a functionof magnitudes of growthratesbut not the actualtrendin time. dispersivityvalues(see Table 2 for values).Anisotropicdata Four additional simulation trials were run to see how well

List's[!965] theorydescribes stabilityrelationships for anisotropicmedia(i.e., kx > ks). In thesetrials,kx remainedat

shownascolumnof data pointsdenotedby "A."

Becausechangingdispersivities causedthe grid Pecletnum-

5.7x 10-• m2,whilek_? was2.5timessmaller. Theperturbing bers to vary, the model grid had to be adjustedto accommo-

wavelength for thesecases,9.5, 8.3, 7.7, and 7.1 cm., yielded wavenumbersof 1.21, 1.39, 1.49, and 1.62, respectively.As shownin Figure 6 (column of data points denotedby "A"), List'sanalytictheory again predictedthe stabilityconditions observed in the model trials.

Sensitivity Studies

In thissection,we interpret the stabilitycalculationsfor our numericalexperimentsto examine,in a mannersimilarto List [1965],howvarioustransportparametersinfluencethe stability of variable-densitysystems.First, we examinethe questionof howstabilitydependsuponthe dispersive characteristics of the medium.The five simulationtrials utilized dispersivityvalues shown in Table 2. The source concentration

in these simula-

tionswas2000mg/LNaC1with perturbationsappliedasa sine wave(wavelengthof 7.15 cm, amplitudeof 2.5 mm), as described by Schincariol et at. [1994].The basesetof parameters werethosefrom Table 1 that applyto an NaC1solutionconcentrationof 2000 mg/L.

I01

c

s

.•

Theoretical

7

©

6

•

Obse

4

datethe smallerdispersivity values.The firstthreesimulations employthe domainshownin Figure2 discretized(iL•:, Az) at 2.5 mm, which resultsin grid Pec!etnumbersbelow 2.4. The last two simulations,with dispersivities smallerthan the basecasevalues,wererun with gridspacings (•c,/lz) of 1.25mm. Thisdiscretization keptgridPecletnumbersbelow2.1.However, becauseof memory and CPU time constraints,reducingthe grid discretization forceda cut of the model domainby one half. Figure6 showsthe stabilitydatapointscalculatedfor the five simulations.Note how instabilityin the systemis promotedby decreasingdispersivities. Decreasingdispersivities reducethe wavenumberandincreasethe Rayleighnumber,both of which reducethe stabilityof the system.This tendencyfor instability to be mainly a propertyof minimally dispersivesystemswas alsoobtainedby List [1965]. The next analysisinvolvesvarying both permeabilityand densityfrom the basecasevalues.Equations(1), (2), and (4) showthat bothpermeabilityand densitydifferenceimpactthe calculationof the Rayleighnumber in the samemanner, but the wavenumberdoesnot dependuponeitherpermeabilityor density difference.Thus increasingthe permeabilitywhile holding the densitydifferenceconstantshouldproducethe sameresultasincreasing the densitydifferenceandholdingthe permeabilityconstant.This hypothesis wastestedby usingresultsfrom simulationswith a 2000-mg/L NaC1 plume, and a perturbingwavelength of 7.15cm.Permeability values,bothkx andkz, are increasedand decreasedby factorsof 1.5 and0.75,

respectively, overthebase-case permeability of 5.7 x 10-• m2.Theoretical waveandRayleighnumbers forthesetrialsare plotted,asbefore,in Xr, 00space(Figure7, pointsA, B, and ß -

ß

0

2

4

6

8

1

12

14

16

1

DimensionlessTime (T)

Figure5. Comparison of theoreticalgrowthrates(amplification)with actualobservedgrowthrates for a 2000 mg/L plumeperturbedbywavelengths corresponding to nondimensional wavenumbers of 3.18(curvea), 2.76(curveb), and2.10 (curvec).

Table 2. Tabulation of Longitudinaland Transverse DispersivityValues

Trial

aL, m X 10-3

at, rn x 10-4

1 2 3

2.0 1.5 1.0 0.75 0.50

4.0 3.0 2.0 1.5 1.0

4 5

36

SCHINCARIOL ET AL.: INSTABILITIES IN VARIABLE DENSITY FLOWS

Character of the Random Permeability Fields 3

[]

Stable

o

Unstable

Stable

Fieldsof synthetic heterogeneous mediaaregenerated using

thespectral techniques developed byRobinet al. [1993].We assume a lognormal permeability distribution [Freeze,1975]

characterized bya mean(Y) andvariance (o-•)thatexhibits a weaklystationary, spatially correlated structure definedbyan exponential-decay covariance function. Thex andz correlation scalesare definedby ,.• and %.

The analysis with the heterogeneous permeability fieldsin-

•

.•

•• 10•

•

A 2

3

4

volvesfive differentrealizations,whosepropertiesare defined in Table 3. Realizations2 and4 differfrom realization1 only in terms of an increasedcorrelation-length scalein the x dim • increasing •eability DB incteeing density diffe•ncerection. Realizations3 and 5 differ from realization 1 only in s

6

•

s

I02

Transverse RayleighNumar (•)

Figure 7. The changingpattern of stabili• as a •nction of pemeabili• and densi• difference.PointsA, B, and C corre-

spondto a 2000m•

NaC1plumeandk = 5.7 x 10- ••

8.55x 10-•, and4.28x 10-• m2,respectively. Points • and E correspondto 3000 and 1500 m•

NaC1plumes,and

k = 5.7 X 10 -x• m 2.

termsof a greatervarianceof the logpermeability. Effective parameters (k•, k33,,4 •) for theseheterogeneous mediaare calculated usingthestochastic analyticsolutions of GeIharand Axness[1983].Shownin Table3 are calculatedvaluesof the effectivepermeabilityask• and k33, assuming that macroscopic flowisparallelto thex axis.Themacrodispersivity value (A • •) represents longitudinaldispersion in the x direction. Becausethey are designedto representthe srnall-scale variabilitythat mightexistin naturalmediaat a scaleof about1 m, the magnitudesof the log permeabilityvarianceand the correlation lengthsare small relative to typical field measured values[Hesset at., 1992;Sudicky,1986].This particularchoice of parametersisjustifiedon the basisof data from the Borden aquifer.Sudicky[!986] foundthat an exponentialcovariance modelwith a varianceof 0.29,an isotropichorizontalcorrela-

C). Theseresultsshowthat systems with smallerpermeabilities tend to be stableand thosewith larger permeabilitiestend to be unstable.This behavioris expected,givena constantwave number,becausethe Rayleighnumberis directlyproportional to the permeability.Alsoplottedon Figure7 (pointsA, D, and E) are theoreticalstabilitycalculationsfor anotherset of three tion lengthof about2.8 m, and a verticalcorrelationlengthof modeltrialswherepermeability isfixedat 5.7 x 10-• m2,but 0.12 m, characterizedthe macroscaleheterogeneousstructure the densitydifferenceis variedequivalentto 1500-,2000-,and of a two-dimensional crosssection.The componentof the 3000-mg/LNaC1plumesin distilledwater.Thesepointsnearly overallvariancedue to small-scale variabilityand anymeasureoverlapwith the variablepermeabilitysetsincethe actualvari- ment error, typicallyregardedasthe "nuggeteffect,"was0.09. ationsin densitycontrastare 1.47 and 0.73 comparedto the Becausethe permeabilitydata were obtainedon a regularbase2000-mg/LNaC1 plume.Not unexpectedly,an increasing spacedgrid with 0.05 m verticaland 1.0 m horizontalspatial densitydifferencedecreases plumestability. discretization, thissmall-scale variabilityis taken as represenThe final analysisconsiders the anisotropiccasewhere the tativeof that in a sampleof porousmedia,overrelativelysmall permeabilityin the z directionchangeswhile that in the x horizontaldistances,such as the simulationdomain repredirection

is held constant. The three simulation

trials that

sented here.

providethe data for this analysisinvolvethe basecaseset of Plotsof realizations1, 2, and3 are shownin Figures9a, 9c, parameters(Table 1) with a plumeof 3000 mg/L and a per- and 9d (overlyingconcentrationdistributions will be discussed turbingwavelengthof 7.15 cm. The three differentvertical

permeability values,5.7 x 10-'•, 2.28 x 10-•', and 1.14x !0-• m2 provide anisotropy ratios(k.,./k.,)of !:1, 2.5:1,and 5:1,respectively. As expected,reducingthe verticalpermeabi!ity or increasingthe anisotropism of the systemenhancesthe overall stability(Figure 8). As transversepermeabilitiesdecrease,it is increasingly difficultto developthe verticalflows necessaryto propagatethe instabilities.Thesecalculationsre-

4

r•Stable l

3

2

.•

o

Stable

Unstable

10•

flect,in a slightlydifferentway,the criticalrole that permeabilityplaysin instabilitydevelopment. q=5.0

HeterogeneousMedia The nextset of simulationresultsare designedto examine issuesof stabilityin heterogeneous media.Most commonly,a sensitivity analysisinvolvingrandomfieldsis carriedout sto-

chastically to capturethe ensemble behaviorof the transport process. However,in our case,a stochastic analysisis infeasi-

• 100 10•

2

•-

•

3

•

O •

4

5

6

decrying

7 8

102

Transverse Rayleigh Number (M)

ble, given the long run times associatedwith the variable-

Figure 8. The changingpatternof stabilityas a functionof density flowandtransport simulations. Thusthe analyses that changing anisotropy ratios.Points A, B, andC correspond to followare basedon a singlerealizationfor eachset of statis- 3000mg/LNaCI plumesandk•,/k..ratiosof 1:1,2.5:1,and5:!, tical parameters.

respectively.

SCHINCARIOL ET AL.: INSTABILITIES IN VARIABLE DENSITY FLOWS

Table 3.

37

SummaryStatisticsand Effective Parametersfor Five Realizations Realization

Variable

kv k', m2

1

- 23.58 5.7 x 10-•x

2

3

-23.58 5.7 x 10-•

- 23.58 5.7 x 10-•

4

-23.58 5.7 x 10-•

5

- 23.58 5.7 x 10-•

o-•

0.05

0.05

0.11

0.05

0.21

z•., m z•, m

0.05 0.01

0.15 0.01

0.05 0.01

0.30 0.01

0.05 0.01

k•, m2 k33,m2 k,/k3

A x•

5.8 X 10-• 5.6X 10-ix 1.03

2.4 X 10-3

5.8 X 10-• 5.6X 10-" 1.04

7.2 X 10-3

5.9 X 10-• 5.6X 10-•t 1.07

5.1 X 10-3

5.9X 10-• 5.6X 10-'t 1.05

1.4X 10-2

6.2X 10-• 5.4X 10-tt 1.14

9.0 x 10-3

Here ky, meanof the log-transformed permeability field{Y = In (k)}; k, meanpermeability; o'•, variance of thelog-transformed permeability field{ Y = In (k) }; r.•,longitudinal (horizontal) correlatit3n length;%, transverse (vertical)correlation length;k•, effective permeability in thext direction; k,3, effectivepermeability in thex3 direction;k •/k3, effectivepermeability ratio;andA•, effectivemacrodispersivityin the x• direction.

later).Realizations4 and 5 are not shown.All fiverealizations weregeneratedfrom the samerandomnumberseed;therefore the permeabilityfields are locallysimilar and differ only in correlationlength and variance.Realization 1 (Figure 9a), whichhas the smallestcorrelationlength of the three (Table 3), is usedasthe basecasefor subsequent analyses. Assigning r• = 0.01 m providesfour nodesper correlationin the z or

the lobe-shapedinterfacialinstabilities.However, becauseof the heterogeneityof the medium, the instabilitiesbecome much more raggedthan is the casefor comparablehomogeneousmedia (see Schincariolet al. [1994] for comparison). With the 2000-mg/Lplume,after 72 hours,the instabilitiesare relativelysmallandobviously slowgrowing(Figure11a).However,aswe wouldexpect,the 3000-mg/Lplumeafter the same transverse direction. This is about the smallest number of time period exhibitsincreasinginstabilityas shownby a more nodesthat shouldbe used to keep discretizationerrors to an marked amplificationof the instabilities(Figure lib). When acceptable level [Ababouet al., 1989].With a varianceof 0.05, the plumeconcentrationis increasedto 5000mg/L, the growth 95.4% of all permeabilityvalueswould fall between -24.03 rates increase,and the denseplume contactsthe lower flow and-23.13, or in arithmeticspacebetween3.7 x 10-• and boundary(Figure 11c). This boundaryeffectis more marked 9.0x 10-•• m2,whichisa contrast invalues of approximately 2.4. after 60 hoursas instabilitiescoalesce(Figure lid). A more detailedanalysisof thesesimulationtrials provides Qualitative Stability Analysis insightinto the influenceof local-scaleheterogeneityon the The five realizations formed the basis for a series of simugrowthand decayof instabilities.In Figures9a and 9b the 0.1 lationtrialswith perturbationshavinga wavelengthof 7.15 cm and 0.5 relative concentration contours for the 2000- and 3000andan amplitudeof 2.5 mm. This particularwavelengthwas mg/L plumesare superimposed on the syntheticpermeability chosenas it representsa somewhatmoderatewavelength,as field (note that high-permeabilityzones correspondto the comparedto the large range of possibleperturbationsthat whiter gideof the grey scale).The growthrate curvesfor the couldbe generatedby the simulatedmedia, and exceedsthe two plumes,in terms of normalizedamplitudeand distance, criticalwavelengthsothat at leastin the initial portionsof the are plotted in Figure 12.Also shownin Figure 12, for comparfield,an unstableperturbationis propagated.A user-defined ison, are the growth rate curvesfor the same plumes in a perturbationis required so its growth characteristicscan be homogeneous andisotropicpermeabilityfield.The normalized properlystudiedas the perturbationinteractswith heterogeamplitudeis definedasthe amplitudeof a travellinginstability, neitiesin the permeabilityfield. To be sure that only the growthand decaycharacteristics of the definedperturbation as referencedby the c/co = 0.5 contour,at a specifiedtime

were studied, a seriesof simulationswere run, with each of the

fiverealizations, to ensurethatthe fieldswereincapableof self propagationof unstableperturbations.Simulationswith permeabilityfieldsthat naturallyperturb the plume have shown thatvarianceslarger than thoseaddressedhere are required for unstableselfpropagation. Althoughthis ongoingresearch intoself propagationis beyondthe scopeof this paper,it is usefulto showone simulation(Figure 10) whichnicelyillustrates the range of perturbations,some smaller and some largerthan the correlationlengthof the media,that maybe generatedby the heterogeneitiesthemselves. As a startingpoint for the stabilityanalysis, we describethe behaviorof plumesof varyingdensityas theymovethrough realization1. Figure 11 illustratesrelative concentrations as plumeswith a densitydifferenceequivalentto 2000-,3000-, and5000-mg/LNaC1flow within distilledwater. All three of thesemodelsystemsare unstableand illustrate

dividedby the originalamplitude(2.5 mm). In the caseof the 2000-mg/Lplume,instabilities grow slowlyand irregularlyas they traversethe permeabilityfield. Interspersedwith this modestgrowthare periodsof increasedgrowththat occurin more permeablezones with fewer low permeabilitylenses. After approximately 0.35 m, just after a maximumnormalized amplitudeof approximately 2.0 is reached(Figure12), instabilities decaymarkedlyas they enter a zone containinglowpermeabilitylenses. When the denseplume initial concentrationis increasedto 3000 mg/L, the plume followsa differenttrajectory.Initially, instabilitygrowthis slowas instabilitiestraversea zoneof low permeability. However,afterapproximately 0.2m of travel,the growthrate abruptlyincreases asinstabilities encounter a higher-permeabilityarea. Farther alongthe flow path, at travel distances of approximately 0.4 to 0.6 m, thishigh-permeability zoneandthe overlyinglow-permeability zoneresultin a more

38

SCHINCARIOL ET AL.: INSTABILITIES IN VARIABLE

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