地下水污染物迁移数值模拟

DispersiveTransport&Advection-dispersionEquation(ADE)C0C0AdvectiononlyAdvection&DispersiontCCvxii)(v=q/θAssumingparticlestravelatsameaveragelinearvelocityv=q/θInfact,particlestravelatdifferentvelocitiesvq/θorvq/θDerivation（推导）oftheAdvection-DispersionEquation(ADE)Assumptions1.Equivalent（当量）porousmedium(epm)(i.e.,amediumwithconnectedporespaceoradenselyfracturedmediumwithasinglenetworkofconnectedfractures)2.Miscibleflow（混相流动）(i.e.,solutesdissolveinwater;DNAPL’s（重非轻亲水相液体）andLNAPL’s（轻非轻亲水相液体）requireadifferentgoverningequation.Seep.472,note15.5,inZhengandBennett.)3.NodensityeffectsDensity-dependentflowrequiresadifferentgoverningequation.SeeZhengandBennett,Chapter15.FiguresfromFreeze&Cherry(1979)shhKAQ12Darcy’slaw:h1h2q=Q/AadvectivefluxfA=qch1h2f=F/AAdectivefluxh1h2fA=advectiveflux=qcf=fA+fDHowtoquantifythedispersiveflux?21DiffdccFDAxHowaboutFick’slaw（见下一张PPT）ofdiffusion?whereDdistheeffectivediffusioncoefficient.Fick’slawdescribesdiffusionofionsonamolecularscaleasionsdiffusefromareasofhighertolowerconcentrations.(Zheng&Bennett,Fig.3.8.)Transverse：横向Weneedtointroducea“law”todescribedispersion,toaccountfor（解释）thedeviation（偏差）ofvelocitiesfromtheaveragelinearvelocitycalculatedbyDarcy’slaw.AveragelinearvelocityTruevelocitiesWewillassumethatdispersionfollowsFick’slaw,orinotherwords,thatdispersionis“Fickian（费克方程）”.Thisisanimportantassumption;itturnsoutthattheFickianassumptionisnotstrictlyvalid（有效的）nearthesourceofthecontaminant.21DccfDxwhereDisthedispersioncoefficient.porositycvcxhhKcqfxxA][12AdvectivefluxxccDfxD12DispersivefluxAssume1DflowDisthedispersioncoefficient.Itincludestheeffectsofdispersionanddiffusion.DxissometimeswrittenDLandcalledthelongitudinal（纵向的）dispersioncoefficient.porosityCase1*DvDLx0*DDD*istheeffectivemoleculardiffusioncoefficient[L2T-1]isthetortuosity（扭转）factor[-]1Assume1DflowandapointsourceCase2Tracer：示踪剂fA=qxcAdvectivefluxDxrepresentslongitudinaldispersion(&diffusion);Dyrepresentshorizontaltransverse（水平横波）dispersion(&diffusion);Dzrepresentsverticaltransversedispersion(&diffusion).)(12xccDfxDx)(12zccDfzDzDispersivefluxes)(12yccDfyDyFigurefromFreeze&Cherry(1979)ContinuouspointsourceInstantaneous（瞬时的）pointsourceAveragelinearvelocitycenterofmassUniform：均匀的FigurefromWangandAnderson(1982)InstantaneousPointSourcetransversedispersionlongitudinaldispersionGaussianEllipse：椭圆；variance：方差、差异；lateral：横向的Derivation（推导）oftheADEfor1Duniformflowand3Ddispersion(e.g.,apointsourceinauniformflowfield)f=fA+fDMassBalance:Fluxout–Fluxin=changeinmassvx=aconstantvy=vz=0DefinitionoftheDispersionCoefficientina1Duniformflowfieldvx=aconstantvy=vz=0Dx=xvx+DdDy=yvx+DdDz=zvx+Ddwherexyzareknownasdispersivities（弥散度）.Dispersivityisessentially（本质上）a“fudge（蒙混）factor”toaccountforthedeviationsofthetruevelocitiesfromtheaveragelinearvelocitiescalculatedfromDarcy’slaw.Ruleofthumb:y=0.1x;z=0.1ytcxcvzcDycDxcDzyx222222ADEfor1Duniformflowand3DdispersionNosink/sourceterm;nochemicalreactionsQuestion:Ifthereisnosourceterm,howdoesthecontaminantenterthesystem?tcxcvxcD22SimplerformoftheADEUniform1Dflow;longitudinaldispersion;Nosink/sourceterm;nochemicalreactionsThereisafamousanalyticalsolutiontothisformoftheADEwithacontinuouslinesourceboundarycondition.ThesolutioniscalledtheOgata&Bankssolution.Question:Isthisequationvalidforbothpointandlinesourceboundaries?Effectsofdispersionontheconcentrationprofile(Zheng&Bennett,Fig.3.11)nodispersiondispersion(Freeze&Cherry,1979,Fig.9.1)t1t2t3t4EffectsofdispersiononthebreakthroughcurveFigurefromWangandAnderson(1982)InstantaneousPointSourceGaussianBreakthroughCurve（浓度比值和时间的曲线）ConcentrationProfile（浓度比值和距离的曲线）longtailFigurefromFreeze&Cherry(1979)Microscopicorlocal（局部的）scaledispersionMacroscopicDispersion(causedbythepresenceofheterogeneities（异质性）)HomogeneousaquiferHeterogeneousaquifersFigurefromFreeze&Cherry(1979)Plug：栓子；dilution：稀释Dispersivity()isameasureoftheheterogeneitypresentintheaquifer.Averyheterogeneousporousmediumhasahigherdispersivitythanaslightlyheterogeneousporousmedium.Dispersionina3Dflowfieldxzx’z’globallocalKxxKxyKxzKyxKyyKyzKzxKzyKzzK’x000K’y000K’z[K]=[R]-1[K’][R]K=zhKyhKxhKqzhKyhKxhKqzhKyhKxhKqzzzyzxzyzyyyxyxzxyxxxDispersionCoefficient(D)D=D+DdDxxDxyDxzDyxDyyDyzDzxDzyDzzD=Ingeneral:DDdDrepresentsdispersionDdrepresentsmoleculardiffusionzcDycDxcDfzcDycDxcDfzcDycDxcDfzzzyzxDzyzyyyxDyxzxyxxDxIna3Dflowfielditisnotpossibletosimplifythedispersiontensortothreeprincipalcomponents.Ina3Dflowfield,wemustconsiderall9componentsofthedispersiontensor.Thedefinitionofthedispersioncoefficientismorecomplicatedfor2Dor3Dflow.SeeZhengandBennett,eqns.3.37-3.42.Dx=xvx+DdDy=yvx+DdDz=zvx+DdRecall,thatfor1Duniformflow:GeneralformoftheADE:Expandsto9termsExpandsto3terms(Seeeqn.3.48inZ&B)EffectoflongitudinalandtransversedispersivitiesontheplumeconfigurationFigure3.24.fromZheng&Bennett