多孔介质对流、扩散与导电性

Diffusion,ElectricalConduction,andFlowinPorousMedia,ContinuumPercolationTheoryforTruncatedRandomFractalMedia,TwoBasicApplicationsofPercolationTheorytoFlow,NearPerc.ThresholdGivesscalingofhydraulic,electricalconductivity,airpermeabilitywithdistancefromthresholdmoisturecontent.Connectivity/tortuosity,FarfromPerc.ThresholdGivesdependenceofsamepropertiesonmoisturecontentthroughdependenceof“bottleneck”poresize.Pore-sizedependence,NOTE:Thesetwoapplicationsarenotindependent(samecriticalpercolationprobabilityforboth)!,AndOnetoAccessibility(inHysteresis),Theprobabilitythatagivensite(orvolume)isconnectedtotheinfiniteclusterabovethepercolationthresholdisgivenbyauniversalfunctionofpercolationtheory.,ApplytoRandom(Truncated)FractalModelofPorousMedia,Whatisoptimalformofpercolationtheory?Continuum.Whatarerelevantvariables?Criticalvolumefraction(moisturecontent,air-filledporosity).Whataboutpotentialnon-universalbehavior?Much,thoughnotallbehaviorisuniversal.,BasicResultsforSaturationDependenceofProperties,K(S)pore-sizedistributiondominatesathighsat.,connectivity/tortuosityatlowsat.ElectricalConductivitysimilarAirpermeabilityconnectivity/tortuositydominatesthroughoutSoluteandgasdiffusion,connectivity/tortuositydominatethroughout,ConsiderFirstUnsaturatedKonLatticeofTubes,AppropriateAnalogyinContinuumPercolationRepresentation,ResultforK(),Resultfrombottleneckpore.,Resultfromaverageresistancetoflowoveralllargerpores.,Noteargumentisnot-t(unless=1)!,EffectsofTortuosity,Connectivity,HydraulicConductivity,ElectricalConductivity,t=1.88(3D)=1.28(2D),RequirebothKanddK/dtobecontinuousatunknownx;thisgeneratesprefactorandx.,Invicinityoftconductivitymustscaleas,Hanfordsitedata(Rockholdetal.,1988),Connectivity/tortuosity,Pore-sizes,ComparisonswithOtherResults,Balberg,Phil.Mag.1987ifW(g)continuestog=0,Butifdistributioncontinuestog=0,thismeansthatsmallestporehaszeroradius.Thatmeansthat=1inRieuandSpositomodeland,yieldsBalbergresultsince,at=1(alsofor),TofindlimitofvalidityofArchieslawset,Archieslaw,Evaluateatsaturation,Useproportionalityofcriticalvolumefractiontoporosity(Hunt,AdWR),ArchiesLaw,Experimentfinds,m=1.86Thompsonetal.,1987,Kuentzetal.,2000,find(in2-Dsimulations)m=1.28,WhathappensifArchieslawisnotquitevalid(cross-overoccursbeforesaturation)?,Electricalconductivityatsaturationisenhancedby,R2increasestoover0.3ifsinglepointiseliminated,DatafromKatzandThompson,AdvancesinPhysics,1987,ApplicationtoLomaPrietaEarthquakePrecursorSignal,Ultra-lowfrequencymagneticfieldeffectswereinterpreted(MerzerandKlemperer)asduetoincreaseinfaultzoneconductivitybyfactor15.AuthorssuggestedArchieslawexponentchangedfrom2to1.Check:Achangefrom1.88to1.28isexpectedforachangeindimensionalityfrom3to2.Thisisconsistentwithdevelopmentof2-Dnetworkofinterconnectedmicro-fracturesinhoursbeforeearthquake.,Theoryhasconsequencesforsaturationdependenceofelectricalconductivity,DatafromTushengRenforsiltloams(twoadjustableparametersforentirefamilyofcurves).,DatafromTushengRenforsands.Powerca.2.5,not1.88;Balbergnon-universality?,AdditionaldatafromAndrewBinley,Bothcasesoneadjustableparameter,otherparametersfromCassiani,DatafromJeffreyRoberts,WhatAboutAirPermeability?,ResultobservedbyMoldrupetal.,(2003)withpowerequalto1.840.54.Notethattheexpectedpowerin2-dimensionsis1.28(DerridaandVannimenus,1982,J.Phys.)(caseswithpowernear1observedinclay-richmedia,otherwisenear2).,Two-dimensionalconfigurationSteriotisetal.(1999)comparedwithpowerof1.28;oneadjustableparameter.,SoluteandGasDiffusion,Typicaldefinition,ResultsofnumericalsimulationsofEwingandHortonx=systemlength,Finite-sizescaling(Fisher,1970),Valueofuniversalexponentofpercolation,=0.88,Finalformsubstitutingmoisturecontentforporosity,HowtoModifyforGasDiffusion?,Trivialmodification,Butnotallairallowableporeshaveair,onlythoseaccessibletotheinfinitepercolationcluster.,Extrafactorisfrompercolationtheoryandrepresentsthefractionalvolumeattachedtotheinfinitecluster(noadjustableparameters).,DatacompiledbyWerneretal.,2004(VZJ),“TheMoldruprelationship,Dpm/Da=2.5/,originallyproposedforsievedandrepackedsoils,gavethebestpredictionsofseveralporosity-basedrelationships”(Werneretal.,2004),Repackedsoils(presumablywithoutstructure),RieuandSpositoWaterRetentionCurves,Continuoustruncatedrandomfractal;analogytoRieuandSpositomodel,Waterretentioncurve,Porosity,HydraulicConductivityLimitedEquilibration,Cross-overtoregimeofrapidlydiminishingKwillhaveconsequencesforequilibrationonceramicplates,Hypothesis:AtMoistureContentsApproachingThresholdActualWaterRemovedisreducedbyratioof,to,Thisiseffectivelyazero-dimensionmodelwithcorrectconstitutiverelationsratherthanacolumnmodelwithinaccurateconstitutiverelations.,Thedeviationsinthefractalscalingofthewaterretentioncurvesarepredictedbythehydraulicconductivityderivedfromthefractalmodelusingpercolationtheoryforthehydraulicconductivity.Thismeansthatthefractalmodelactuallypredictsthedeviationsfromthefractalmodel.Notethatthedeviationsoccurredatamoisturecontentrelatedtothemoisturecontentatwhichsolutediffusionvanishedinsoilsfromanothercontintent(Moldrupetal.,2001).,HysteresishasTwoComponents,Waterremovedfromaporemustpassthroughaporethroat;waterimbibedmustfilltheentireporebody.Pressureratioisratiooftworadii(typicallyabout2).Allporesthatallowwaterwillcontainthatwaterduringdrainage,butonlythoseconnectedtotheinfinitepercolationclusterwillcontainwaterduringimbibition.,PercolationComponent,RieuandSpositorandomfractalwaterretentioncurve.,Sameaccessibilityfactorasforgasdiffusion.,Productoftwogivesimbibitioncurve.,DatafromBautersetal.(1998),Airentrypressure=11cm,Characteristicpressure=5.5cm,Ratioofpressuresconsistentwithotherexperiments.,Summary,KSCompetitionK(S)MainlyCPA(S)MainlyScalingka(=0)Competitionka()EntirelyScalingDEntirelyScaling,Conclusions,ApplicationofcontinuumpercolationtheorytofractalmodelsyieldsresultsforsaturationscalingofHydraulicconductivityAirpermeabilityElectricalconductivityPressure-saturationcurvesincludinghysteresisandlackofequilibrationSolutediffusionGasdiffusion(worstcomparisonwithexpt.,stillbesttheory)inagreementoraccordwithexperiment.Theoryyieldsconsistentinterpretations,parameters,analysisofrelevanceofporesizedistributionscomparedwith,e.g.,connectivity/tortuosityissues.Theoryreinterpretsfundamentalsoilphysics.,RelevantReferences,Hunt,A.G.,andGee,G.W.,2002,ApplicationofCriticalPathAnalysistoFractalPorousMedia:ComparisonwithExamplesfromtheHanfordSite,AdvancesinWaterResources,25,129-146.Hunt,A.G.,andGee,G.W.,2002,WaterRetentionofFractalSoilModelsUsingContinuumPercolationTheory:TestsofHanfordSiteSoils,VadoseZoneJournal,1,252-260.Hunt,A.G.,2003,PercolativeTransportandFractalPorousMedia,Chaos,Solitons,andFractals,19,309-325.Hunt,A.G.,andEwing,R.P.,2003,OnTheVanishingofSoluteDiffusioninPorousMediaataThresholdMoistureContent,SoilScienceSocietyofAmericaJournal,67,1701-1702,2003.Hunt,A.G.,andGee,G.W.,2003,Wet-EndDeviationsfromScalingoftheWaterRetentionCharacteristicsofFractalPorousMedia,VadoseZoneJournal,2,759-765.Hunt,A.G.,2004,ContinuumPercolationTheoryforWaterRetentionandHydraulicConductivityofFractalSoils:1.EstimationoftheCriticalVolumeFractionforPercolation,AdvancesinWaterResources,27,175-183.Hunt,A.G.,2004,ContinuumPercolationTheor