Vdensity groundwater flow and solutetransport in porous media containing nonuniform discrete
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Variable-densitygroundwaterflowandsolutetransportinporousmediacontainingnonuniformdiscretefractures
ThomasGraf*,Rene´Therrien
De´partementdeGe´ologieetGe´nieGe´ologique,Universite´Laval,Ste-Foy,Que´bec,CanadaG1K7P4Received10September2004;receivedinrevisedform15April2005;accepted15April2005Availableonline20June2005
AbstractVariationsinfluiddensitycangreatlyaffectfluidflowandsolutetransportinthesubsurface.Heterogeneitiessuchasfracturesplayamajorroleforthemigrationofvariable-densityfluids.Earliermodelingstudiesofdensityeffectsinfracturedmediawererestrictedtoorthogonalfracturenetworks,consistingofonlyverticalandhorizontalfractures.Thepresentstudyaddressesthephe-nomenonof3Dvariable-densityflowandtransportinfracturedporousmedia,wherefracturesofanarbitraryinclinecanoccur.Ageneralformulationofthebodyforcevectorisderived,whichaccountsforvariable-densityflowandtransportinfracturesofanyorientation.Simulationresultsarepresentedthatshowtheverificationofthenewmodelformulation,fortheporousmatrixandforinclinedfractures.Simulationsofvariable-densityflowandsolutetransportarethenconductedforasinglefracture,embeddedinaporousmatrix.Thesimulationsshowthatdensity-drivenflowinthefracturecausesconvectiveflowwithintheporousmatrixandthatthehigh-permeabilityfractureactsasabarrierforconvection.Othersimulationswereruntoinvestigatetheinfluenceoffrac-tureinclineonplumemigration.Finally,tabulardataofthetracerbreakthroughcurveintheinclinedfractureisgiventofacilitatetheverificationofothercodes.Ó2005ElsevierLtd.Allrightsreserved.Keywords:Numericalmodeling;Fracturedrock;Inclinedfracture;Contaminanttransport;Density;Instability;Buoyancyterm
1.Introduction
Insubsurfaceenvironments,contaminantsaremainlytransportedbygroundwater.Thetransportpatternisgreatlyaffectedbythephysicalparametersofboththemediumandthecontaminant.Thetransportpropertiesofthefluid,suchasviscosityandfluiddensity,canalsohaveasignificantimpactoncontaminanttransport.Thefluiddensityq(fluidmassperunitvolumeoffluid)cannotalwaysbeassumeduniform.Itdecreaseswithincreasingtemperature,increaseswithincreasingsalinityandincreaseswithincreasingpressureduetofluidcompressibility.Spatialvariationsoffluiddensityplayanimportantroleincontaminantmigrationwithinvariousgeologicalmedia.If,forexample,afluidofhighdensityoverliesalessdensefluid,thesystemispotentiallyunstableanddensity-drivenflowmaytakeplace,whichlevelsoutthedensitystratificationandeventuallystabilizesthesystem.Additionally,ifflowistransient,therearetem-poralchangesindensity.Examplesfordensity-drivenflowandtransportcanbefoundinmanyareasofsub-surfacehydrology,oceanography,meteorology,geo-physicsandhazardouswastedisposal.Thesafedisposalofhazardouschemicalsiscom-monlyregardedasfeasibleinlow-permeabilitygeologi-calmediaatadepthofupto1000m[4].InCanada,oneconceivablehostrockfortherepositoryofradioactivewasteisthecrystallinemetamorphicrockoftheCana-dianShield.Here,thegroundwateratdepthsgreater
0309-1708/$-seefrontmatterÓ2005ElsevierLtd.Allrightsreserved.
doi:10.1016/j.advwatres.2005.04.011*Correspondingauthor.E-mailaddress:thomas.graf.1@ulaval.ca(T.
Graf).AdvancesinWaterResources28(2005)
1351–1367
www.elsevier.com/locate/advwatresthan800misaCa–Na–Clbrinewithdissolvedsolidsexceeding100,000mglÀ1[8].Therefore,thedensityofsuchdeepfluidsvariesdramaticallywithsoluteconcen-tration.AcutesafetyquestionsregardingnuclearwasteNomenclature
•scalarvariablesaredenotedinnormalitalicletters•vectorvariablesaredenotedinboldsmallletters•matrixvariablesaredenotedinboldcapitalletters
Latinletters2bfractureaperture[L]csoluteconcentration,expressedasrelativeconcentration[–]Ddfree-solutiondiffusioncoefficient[L2TÀ1]Dijcoefficientsofhydrodynamicdispersionten-sor[L2TÀ1]Dfrijhydrodynamicdispersioncoefficientofthefracture[L2TÀ1]fthree-dimensionalfunction,definedoverasurfaceS;f=f(x,y,z)[–]fglobalboundaryfluxvector[L2TÀ1]FfunctionthatdefinesthesurfaceSinthreedimensions;F(x,y,z)=constant[–]gaccelerationduetogravity[LTÀ2]gglobalbodyforcevector[L2TÀ1]h0equivalentfreshwaterhead[L]Hhypotenuseofarectangle[L]Hglobalconductancematrix[LTÀ1]iunitvectorinx-direction[–]I+,IÀfracture–matrixinterface[–]junitvectoriny-direction[–]JJacobianmatrix[–]kunitvectorinz-direction[–]K0ijcoefficientsofhydraulicconductivitytensoroffreshwater[LTÀ1]Kfr0hydraulicfreshwaterconductivityofthefrac-ture[LTÀ1]‘vgeometryofthemodeldomain;v=x,y,z[L]Lvgeometryofablockelement;v=x,y,z[L]LTGcharacteristiclengthscale[L]Nfetotalnumberoffractureelementsinthedo-main[–]pvectorthatisnormaltoregionR[–]Pdynamicpressureofthefluid[MLÀ1TÀ2]PeggridPecletnumber[–]qiDarcyflux[LTÀ1]RverticalprojectionofSonacoordinateplane[–]ReReynoldsnumber[–]saxisalongaslopedtube[L]Sfluidmassmatrix[L]Ssurface,definedbythefunctionF(x,y,z)=constant[L2]SSspecificstorage[LÀ1]Sopspecificpressurestorativity[MÀ1LT2]ttime[T]vilinearflowvelocity[LTÀ1]wiapproximationfunction[–]
