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DirectporetocoreupscalingofdisplacementprocessesDynamicporenetworkm


ArashAghaeiMohammadPiriDepartmentofChemicalandPetroleumEngineeringUniversityofWyomingLaramieWyomingUSAarticleinfoArticlehistoryReceived18December2014Accepted2January2015Availableonline10January2015Thi

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Document on Subject : "DirectporetocoreupscalingofdisplacementprocessesDynamicporenetworkm"ā€” Transcript:

1 Directpore-to-coreup-scalingofdisplaceme
Directpore-to-coreup-scalingofdisplacementprocesses:Dynamicporenetworkmodelingandexperimentation ArashAghaei,MohammadPiri DepartmentofChemicalandPetroleumEngineering,UniversityofWyoming,Laramie,Wyoming,USA articleinfoArticlehistory:Received18December2014Accepted2January2015Availableonline10January2015ThismanuscriptwashandledbyCorradoCorradini,Editor-in-Chief Pore-scale”owsPorenetworkmodelingCore-”oodingexperiments summaryWepresentanewdynamicporenetworkmodelthatiscapableofup-scalingtwo-phase”owprocessesfromporetocore.Thisdynamicmodelprovidesaplatformtostudyvarious”owprocessesinporous 1.Introduction Multiphase”owinporousmediaoccursinmanynaturalandarti“cialprocessessuchassubsurface”owofhydrocarbonsandbrine,geologicstorageofCO ,non-aqueousphaseliquids(NAPL)migrationinsoil,reactivetransport,andwaterremovalingasdif-fusionlayer(GDL)ofprotonexchangemembrane(PEM)fuelcells.Understandingthedisplacementandtransportprocessesrelevantt

2 omultiphase”owsystemsandpredictingtheass
omultiphase”owsystemsandpredictingtheassociatedmacroscopicpropertiesarecrucialfordesignandpredictionofper-formanceoftheseprocesses(Dullien,1992;Sahimi,2010 permeabilities.Thesefunctionsaremanifestationofmanypore- scalephenomena,andtheyinformthenumericalsolversofmassconservationequationsabouttheunderlyingdisplacementphys-ics.Pore-scaleinvestigationsareoftenusedtodevelopimprovedunderstandingoffundamentalphenomenarelevanttoagivenpro-cessandpredictthepertinent”owandtransportproperties.Thesephysically-basedpropertiesarethenusedtoinformthelarger-scalecontinuummodels. Correspondingauthor.E-mailaddress:(M.Piri). JournalofHydrology522(2015)488…509 ContentslistsavailableatScienceDirectJournalofHydrologyjournalhomepage:www.elsevi modelsarecomputationallyexpensiveandmaynotbesuitableforstudyingmultiphase”owatthecorescale.Inthesecondgroupofpore-scalemodels,i.e.,networkmodels,theporespaceisrepre-sentedbyanetworkofidealizedporesandthroa

3 ts.Pore-scaledis-placementsarecarriedout
ts.Pore-scaledis-placementsarecarriedoutintheporenetworktosimulatemultiphase”ow(Ųrenetal.,1998;Patzek,2001;Bluntetal.,2002;PiriandBlunt,2005a,b Porenetworkmodelingwas“rstintroducedinthe1950sbyFatt(1956a,b,c)whousedanetworkofrealresistorstocalculaterelativepermeabilityandcapillarypressureforadrainageprocess.Networkmodeling,sinceitsintroduction,hasevolvedenormously.Today,onecanmaptheporespaceofarocksamplewithhigh-resolutionimagingtechniquesandextractanequivalentporenetwork(DongandBlunt,2009).Ourknowledgeofthepore-scaledisplacementphysicshasalsoimproveddramaticallyduetomicro-”uidicsandothertypesofexperiments.Porenetworkmod-elscanbedividedintoquasi-staticanddynamic.Themajorityofthepreviously-developednetworkmodelsarequasi-staticinwhichthepore-scaledisplacementstakeplacebasedontheirthresholdcapillarypressure.Thesemodelshavehadsigni“cantsuccessinmodelingtwo-andthree-phase”owinporousmediaundercapillary-dominatedconditions(Ųrene

4 tal.,1998;Patzek,2001;ŲrenandBakke,2003;
tal.,1998;Patzek,2001;ŲrenandBakke,2003;ValvatneandBlunt,2004;PiriandBlunt,2005a,b).However,quasi-staticmodelsdonotincludetheeffectsofviscousandgravityforces,andtherefore,cannotbeusedtostudycasesinwhichcapillary-dominatedassumptionsdonotapply.Viscousandgravityforcesbecomesigni“cantduringmanysubsurface”owprocesses,suchasenhancedoilrecovery(EOR)methodsofpolymerandsurfactant”ooding,andhighvelocity”owregimesthatareencounteredinnaturallyandhydraulicallyinducedfracturesaswellasnearwell-boreareas(Lake,1989;Sahimi,2010).Inthesecases,thecombinedeffectsofcapillary,vis-cous,andgravityforcesdeterminethe”owbehaviorinporousmedia.Indynamicnetworkmodels,ontheotherhand,viscousandinsomecasesgravityforcesaretakenintoaccount.Thesemodelscanbeusedtostudythecasesinwhichviscousorgravityforcesarerelevant.However,duetovariousreasons,suchasdif“-cultiesinimplementingthecomplexpore-scalephysicsandthecomputationalcostsassociatedwiththesemodels,thepre

5 vi-ously-developeddynamicmodelslacksomeo
vi-ously-developeddynamicmodelslacksomeofthecriticalcapabil-itiestosuccessfullysimulatedynamictwo-phase”owprocessesatthecorescale. Tables1and2listthepreviously-developeddynamicporenet-workmodelsalongwiththeirlargestnetworksize,phenomenastudied,andvalidationtechniques.Acomprehensivereviewofdynamicporenetworkmodelsoftwo-phase”owinporousmediacanbefoundelsewhere,see,Joekar-NiasarandHassanizadeh(2012)andAghaei(2014).Here,wediscussthecharacteristicsandpredictivecapabilitiesofeachpreviously-developedmodel.Wethenpresentanoverviewofthedynamicmodeldeveloped underthisstudyandexplainthecriticalrelevanceofitscapabili- tieswithinthecontextofdirectup-scaling. KoplikandLasseter(1985)developedthe“rstdynamicnetworkmodeltostudytheeffectsofmicroscopicporestructureonmacro-scopicphenomena.Theystatedthecomputationaldif“cultiesasso-ciatedwithdynamicporenetworkmodelingasthelimitingconstraintinselectingthenetworksize.Lenormandetal.(1988),Bluntand

6 King(1991),andLeeetal.(1995)developedmod
King(1991),andLeeetal.(1995)developedmodelsinwhichtheporescontainedallthe”uidandthepressuredropstookplaceexclusivelyinthethroats.Lenormandetal.(1988)drainagesimulationsatvariouscapillarynumbersandviscosityratiosandidenti“edthreedistinct”owpatterns.BluntandKingrandrainagesimulationsintwo-andthree-dimensionalrandomnetworksandcalculatedrelativepermeabilitiesbasedonlocal”owratesandpressuredrops.Themodeldevelopedbyetal.(1995)wasaparallelmodelthatwasusedtoperformwater-”oodingandmiscible-”oodingsimulationsinnetworksaslargeas524,288pores.ThismodelwaslaterextendedbyKamathetal.(1996)andXuetal.(1999)whowereabletotoreproducerecoveriesinafullycore-”oodingexperimentinadolomitesample. vanderMarcketal.(1997)extendedthemodelintroducedbyLenormandetal.(1988)byallowinguptotwomenisciinporeMogensenandStenby(1998)developedamodelinwhichthecorner”owandsnap-offdisplacementswereincorporated.Akeretal.(1998)developedamodelwithhourglass-shapedthroats

7 tostudytimedependenciesofpressuredistrib
tostudytimedependenciesofpressuredistributionand”uidfrontindrainageprocesses.Later,KnudsenandHansenmodi“edthismodelbyaddingbiperiodicboundarycondi-DahleandCelia(1999)introducedanewinterfacetrackingmethodinwhich”uidscouldformcompartmentsthatweresepa-ratedby”uid…”uidinterfaces.HughesandBlunt(2000)usedthewetting-phaseviscouspressuredroptoperturbtheorderofdis-placements.Thisquasi-dynamicmodelwasusedtostudytheeffectofthecapillarynumberduringimbibition.ConstantinidesandPayatakes(2000)studiedtheeffectsofwettinglayersondis-connectionofthenon-wettingphaseduringimbibition.Thompson(2002)createdrandomporenetworkswithconverg-ing…diverginggeometryforthroatstostudydrainageandimbibi-tionin“brousmaterials.SinghandMohanty(2003)introducedanewmethodforhandlingthecorner”owinwhichthewettingphasewasremovedfromthelayersinproportiontothelocalcap-illarypressuredrop.Nordhaugetal.(2003)extendedthemodeldevelopedbyBluntandKing(1991)tostudyinterfacial

8 velocitiesandareas.Lųvolletal.(2005)stud
velocitiesandareas.Lųvolletal.(2005)studieddrainageofahigh-viscositywettingphaseandstabilizingeffectofgravityusingadynamicporenetworkmodelandglassbeadsexperiments. Al-GharbiandBlunt(2005)incorporatedlayerswellingandsnap-offdisplacementsinadynamicmodelandusedittostudytheeffectsofcapillarynumberandviscosityratioindrainagein Table1Previously-developeddynamicporenetworkmodels.StudyLargestnetworksizePhenomenastudiedValidationtechniquesKoplikandLasseter(1985)100EffectofontrappingN/ALenormandetal.(1988)10,000FlowregimesMicro-modelsBluntandKing(1991)80,000DrainageLeeetal.(1995)524,288ImbibitionKamathetal.(1996)262,144Saturationpro“les,recoveriesUnsuccessfulcore-”oodingXuetal.(1999)131,072Saturationpro“les,recoveriesRecoveriesinmiscible”oodingvanderMarcketal.(1997)2,401Pressure“eldindrainageMicro-models,viscosityratioofoneMogensenandStenby(1998)3,375Trapping,Akeretal.(1998)4,800Pressure“eldindrainageGlassbeads,viscosityratioofone

9 DahleandCelia(1999)indrainageN/AHughesan
DahleandCelia(1999)indrainageN/AHughesandBlunt(2000),”owpatternsMicro-modelsMaximumnumberofpores.A.Aghaei,M.Piri/JournalofHydrology522(2015)488…509 2Dlattices.Nguyenetal.(2006)introducedanewnetworkmodeltostudythecompetitionbetweensnap-offandpiston-likedis-placementsduringimbibitionanditseffectonrelativepermeabil-itiesandresidualoilsaturation.Theonlydynamiceffectincludedinthismodelwastheviscouspressuredropassociatedwiththewetting“lms.DiCarlo(2006)usedaquasi-dynamicnetworkmodeltostudythesaturationovershootbehindin“ltrationfronts.Themodelincludedtheviscouspressuredropofthewettingphaseduringimbibition.PiriandKarpyn(2007)createdaporenetworkrepresentationoffracturevoidspacefromthefractureaperturemapobtainedthroughX-raymicrotomography.Theyusedaquasi-dynamicnetworkmodeltosimulatetwo-phase”owinasin-glefracture(KarpynandPiri,2007Joekar-Niasaretal.(2010)studiedthequalitativebehaviorofnon-equilibriumcapillarypres-suretheory(Stauffer

10 ,1978;HassanizadehandGray,1990)usingadyn
,1978;HassanizadehandGray,1990)usingadynamicnetworkmodel.Tųråetal.(2012)extendedthemodeldevelopedbyAkeretal.(1998)byincorporatingthedynamicsofthewettinglayersusinganapproachsimilartoSinghandMohanty(2003).Theyusedthismodeltostudysaturationpro“lesduringimbibitionandtheresistivityindexatdifferentcapillaryShengandThompson(2013)extendedthedynamicporenetworkmodeldevelopedbyThompson(2002),andcoupleditwithacontinuum-scalereservoirsimulator.Thecouplingwasper-formedbyembeddingporenetworksinsideafewgridblocksofthereservoirsimulator. Basedontheliteraturereviewpresentedabove,itisclearthatwhiletherehavebeennumerousdynamicnetworkmodelingstud-iesinthepast,thereisstillaneedforaphysically-baseddynamicnetworkmodelcapableofmodelingpore-scaledisplacementsinrandomnetworksandup-scalingthemtothecorescale.Thepre-viously-developeddynamicnetworkmodelsthatincludethecom-plexpore-scalephysics(Al-GharbiandBlunt,2005;MogensenandStenby,1998;DahleandCeli

11 a,1999)lackthesuf“cientcomputa-tionalper
a,1999)lackthesuf“cientcomputa-tionalperformanceandcanmodeltwo-phase”owinnetworksoffewhundredtofewthousandpores,andtheonesthatarecapableofsimulating”owinlargerporenetworks(Leeetal.,1995;Kamathetal.,1996;Xuetal.,1999)lackthenecessarypore-scalephysicssuchasthewetting-phasecorner”owandsnap-offdis-placementstoquantitativelypredictthemultiphase”ow Weintroduceanentirelynewdynamicnetworkmodeltoper-formphysically-basedup-scalingofdisplacementprocessesfromporetocorescale.Themodeltakesintoaccountviscous,capillary,andgravityforcesaswellaswettabilityeffects.Itincorporatesallrelevantdisplacementmechanismsandallowsforvariationsintheorderbywhichtheytakeplaceunderdifferent”owregimesinresponsetochangesinthecapillaryandBondnumbers.Thewet-ting-phasecorner”owwithvariablecornerinterfacelocationresponsivetochangesinlocalcapillarypressureisincludedin themodel.Simultaneousinjectionofwettingandnon-wetting”u- idswithconstant”owratesfromtheinletha

12 sbeenincorporatedinthemodel,andithasenab
sbeenincorporatedinthemodel,andithasenabledustostudythesteady-statetwo-phase”owprocesses.Constantpressureboundaryconditionisusedattheoutlet.Thisplatformisdevelopedtorunonmassivelyparallelcomputerclustersandtobridgethegapbetweenpore-scaledisplacementsandcore-scaleprocesses. Inordertovalidatethemodelpresentedhere,weperformedthreetwo-phaseminiaturecore-”oodingexperiments.Theexperi-mentswereperformedinastate-of-the-artcore-”oodingappara-tusintegratedwithahigh-resolutionmicro-CTimagingsystem.Forthe“rsttimeinpore-scalemodelingliterature,theinsitucon-tactanglesmeasuredfromhigh-resolutionmicro-CTimagesoftheporespaceduringexperimentswereusedtodesigncontactangledistributionsinthedigitalporenetwork.Usingthedynamicmodel,simulationswereperformedwithidentical”owratesand”uidpropertiesastheexperiments.Thedynamicmodelwasthenrigor-ouslyvalidatedbycomparingthelocalsaturationpro“les,relativepermeabilities,andfractional”owcurvesobtainedfroms

13 imula-tionsagainsttheirexperimentalcount
imula-tionsagainsttheirexperimentalcounterparts.Thevalidatedmodelwasthenusedtostudydifferentdynamicprocessesthatoccurin,forinstance,manyenhancedoilrecoveryschemes. Inthispaper,“rst,wegiveadetaileddescriptionofthedynamicmodelandtechniquesusedtomodeltwo-phase”owatthecorescale.Then,theminiaturecore-”oodingexperimentsthatareper-formedforthevalidationofthemodelaredescribedindetail.Thisisfollowedbyarigorousvalidationofthesimulationresultsagainsttheirexperimentalcounterparts.Thevalidatedmodelisthenusedtostudylow-IFTandhighviscosity”owprocesses.Thedynamiceffectsobservedinthesehighcapillarynumbersimula-tionsarediscussedindetail.Finally,weincludeasetofconclusionsand“nalremarks. 2.Modeldescription 2.1.Networkrepresentationoftheporousmedium Weusethree-dimensionalrandomnetworksthataregeneratedfromhigh-resolutionmicro-CTimagesoftherocksamplesusedintheexperimentspresentedhere(seeDongandBlunt(2009)moredetailsonthenetworkgenerationmethod

14 ).Twodifferentporenetworksrepresentingth
).TwodifferentporenetworksrepresentingtheporespaceinBentheimerandBerearocksampleswereusedinthiswork.A17mmlongsectionatthemiddleoftheBereacoresamplewasscannedataresolutionof2.49m.Aporenetworkwithalengthof16.3mmandasquarecrosssectionof311mm wasgenerated.Twentyreplicatesofthe16.3mmlongBereanetworkwereconnectedtobuildalar-gernetwork.Averagepropertiesfromtheoriginalnetwork(e.g., Table2Previously-developeddynamicporenetworkmodels(contd).StudyLargestnetworksizePhenomenastudiedValidationtechniquesConstantinidesandPayatakes(2000)6,000Wettinglayers,Thompson(2002)1,728Imbibition”owpatternsN/ASinghandMohanty(2003)1,920Saturationpro“les,indrainageQualitative,duPrey(1973)Nordhaugetal.(2003)5,000InterfacialvelocitiesSandpack,Schaeferetal.(2000)Lųvolletal.(2005)12,800Effectsofviscosity,gravityindrainageGlassbeadstestsAl-GharbiandBlunt(2005)900DisplacementpatternsindrainageN/ANguyenetal.(2006)12,349ImbibitionOak(1990),ChatzisandMorrow(1

15 984)DiCarlo(2006)12,349Saturationoversho
984)DiCarlo(2006)12,349SaturationovershootDiCarlo(2004)PiriandKarpyn(2007)20,890Two-phase”owinfractureFluidoccupancy,KarpynandPiri(2007)Joekar-Niasaretal.(2010)42,875Non-equilibriumcapillaritytheoryTheoretical,HassanizadehandGray(1990)Tųråetal.(2012)767Saturationpro“les,resistivityindexSandpacktestsShengandThompson(2013)1,532CouplingwithreservoirsimulatorN/AMaximumnumberofpores.A.Aghaei,M.Piri/JournalofHydrology522(2015)488…509 coordinationnumberandthroatradii)wereusedattheconnectionsites.Acylindricalporenetworkwithalengthof76.4mmandadiameterof4.58mmwascutfromthelargernetwork.WerefertothisnetworkastheBereanetworkhereinafter.TheBentheimernetworkwasbuiltusingasimilarapproach. ThedimensionsandpetrophysicalpropertiesoftheBentheimerandBereanetworksarelistedinTables3and4,respectively.Theporesandthroatsofournetworkshaverectangular,scalenetrian-gular,andcircularcrosssections.Fig.1(a)showsavolumeren-deredgrayscaleimageoftheBereas

16 andstonecoresample.Thisgrayscaleimageiss
andstonecoresample.Thisgrayscaleimageissegmentedusingthehistogramthresholdingtechniqueproducingapore-grainseparatedlabeledimageseenFig.1(b),whereredandgraycolorsrepresenttheporeandgrainvoxels,respectively.Thegray-scaleandpore-grainseparatedimagesareusedtoconstructaporenetworkrepresentativeoftheBereacoresample,asshowninFig.1(c).Intheporenetworkimage,poresandthroatsareshownwithredspheresandbluecyl-inders,respectively.Fig.1(d)showsamagni“edimageofasmallsectionoftheBereaporenetwork. 2.2.Modelassumptions Inthismodel,”uidsareassumedtobeNewtonian,incompress-ible,andimmiscible.Fluid…”uidinterfacesareassumedtobesharpwithnodiffusiontakingplacebetweentwophases.Fluid”owinsideporesandthroatsisdescribedbyStokesorcreeping”ow.Theclayinsidethepore-spaceisassumedtobefullysaturatedwithimmobilewater. 2.3.Pore-scaledisplacements Aseriesofpore-scale”uiddisplacementsarecarriedoutintheporenetworktosimulatetwo-phase”owdisplacementsinporousmedia.

17 Threefundamentaltypesofpore-scaledisplac
Threefundamentaltypesofpore-scaledisplacementmech-anismsincludedinthisdynamicmodelarepiston-like,pore-body“lling(cooperative“lling),andsnap-off. Thesimulationsstartwithfullywatersaturatedporesandthroatswhereallpossibledisplacementsfromtheinlethavebeenaddedtothesystem.Displacementsoccurintheorderofhighest-displacementpotential(seeSection)thattakesintoaccounttheeffectsofcapillary,viscous,andgravitationalpressuredrops.Aftereachdisplacement,newdisplacementsareaddedtothesystembasedontheaccessof”uidphasestoeachother. 2.4.Displacementpressuredropandsequence Themultiphase”owbehaviorinporousmediaisdeterminedbytheinterlinkedeffectsofcapillary,viscous,andgravityforces.Thereforeitisessentialtoincludetheeffectsofthesethreebasicforcesinthemodel.Inthiswork,thisisachievedbycombiningthecapillary,viscous,andgravitationalpressuredropsofadis-placementintoanewparametercalledthedisplacementpressureasgivenbyEq. DPdisp ¼DPcap žDPvisc žDPgrav š1

18 ŽNc ¼ lmr š2ŽBo¼ DqgL 2r š3Ž Thecontribu
ŽNc ¼ lmr š2ŽBo¼ DqgL 2r š3Ž ThecontributionofeachterminthedisplacementpressuredroptoitstotalvaluedependsonthecapillaryandBondnumbers(2)and(3)).InthecaseswherethecapillaryandBondnum-bersarebothverylow,capillarypressuredominatesthe”owbehaviorinporousmedia.HoweverbyincreasingthecapillaryandBondnumbersthecontributionsofviscousandgravitationalpressuredrops,respectively,becomemoresigni“cant. Forpiston-likedisplacementsweusetheMayer…Stowe…Prin-cen(MSP)methodtocalculatethecapillarypressuredropofdis-placements(MayerandStowe,1965;Princen,1969a,b,1970;Ųrenetal.,1998;PiriandBlunt,2005a).Inthismethod,anenergybalanceequationiswrittenforthe”uidcon“gurationchangesassociatedwithapore-scaledisplacement.Byassumingtheequi-libriumconditionsatconstanttemperatureandcombiningtheresultantequationwiththeYoung…Laplaceequation,thecapillarypressuredropduringa”uidcon“gurationchangeisobtained. DPvisc invdinlet P invdelm defelm P defoutlet Ž Thevisc

19 ouspressuredropoccursduetothefrictionbet
ouspressuredropoccursduetothefrictionbetweenlay-ersof”uidthatmovewithdifferentvelocities.Wecalculatethevis-couspressuredropfortheinvadinganddefending”uidphases.Fortheinvading”uidphasetheviscouspressuredropiscalculatedfromtheinletoftheporenetworktothedisplacementlocation,andforthedefending”uidphase,itiscalculatedfromthedisplacementlocationtotheoutletoftheporenetworkaswritteninEq. DPgrav invdq defŽghelm š5Ž Table3PropertiesoftheporenetworkrepresentativeoftheBentheimersandstonecoresampleusedinExperiment1.ItemThroatsPoresTotalNumber306,825131,346438,171Porosityexcl.clay(%)9.69713.63123.328Porosityincl.clay(%)11.26113.77225.033AbsolutePermeability(mD)2716Length(mm)12.977Diameter(mm)4.972Minimumcoordinationnumber0Maximumcoordinationnumber150Averagecoordinationnumber4.622Minimumradius(m)0.280.300.28Maximumradius(m)101.00148.48148.48Averageradius(m)13.7125.9617.38Averageshapefactor0.0420.0460.044Triangularcrosssections(%)79.790

20 70.66577.055Squarecrosssections(%)19.590
70.66577.055Squarecrosssections(%)19.59028.95922.399Circularcrosssections(%)0.6200.3760.546Connectedtotheinlet336303363Connectedtotheoutlet324803248Isolatedclusters45Isolated143446589 Table4PropertiesoftheporenetworkrepresentativeoftheBereasandstonecoresampleusedinExperiments2and3.ItemThroatsPoresTotalNumber3,913,2221,891,2605,804,482Porosityexcl.clay(%)8.23612.61620.852Porosityincl.clay(%)8.45812.89221.350Absolutepermeability(mD)638Length(mm)76.4Diameter(mm)4.58Minimumcoordinationnumber0Maximumcoordinationnumber45Averagecoordinationnumber4.017Minimumradius(m)0.210.250.21Maximumradius(m)63.50101.0101.0Averageradius(m)6.9513.959.27Averageshapefactor0.0390.0470.042Triangularcrosssections(%)78.52257.66771.592Squarecrosssections(%)20.95441.59427.813Circularcrosssections(%)0.5240.7390.595Connectedtotheinlet151501515Connectedtotheoutlet150701507Isolatedclusters24Isolated7460779115251A.Aghaei,M.Piri/JournalofHydrology522(2015)4

21 88…509 Thegravitationalpressuredropisthe
88…509 Thegravitationalpressuredropistheresultofthehydrostaticpressuredifferenceoftheinvadinganddefending”uidphasesasgivenbyEq.,where invdandq arethedensitiesoftheinvad-inganddefendingphases,respectively,istheaccelerationofgravity,and istheheightoftheporeorthroatinvolvedinthedisplacementmeasuredfromtheinjectionfaceofthemedium. Udisp ¼P invdinlet P defoutlet DPdisp š6Ž Whetherpore-scaledisplacementscantakeplaceandtheirsequencearedeterminedbyaparametercalledthedisplacement.Thedisplacementpotentialisde“nedasthedifferenceofthedrivingforceandthedisplacementpressuredrop.Thedrivingforceisde“nedasthedifferenceoftheinletpressureoftheinvad- ingphaseandtheoutletpressureofthedefendingphase.Thedis- placementpotentialforanygivendisplacementiscalculatedusing.Displacementscantakeplaceonlyiftheyhaveapositivepotential,andtheytakeplaceintheorderofhighest-to-lowestdis-placementpotential. 2.5.Volumebalanceandpressurecalculations Fluidphase

22 pressure“eldcalculationisregardedastheba
pressure“eldcalculationisregardedastheback-boneofthisdynamicmodel.Thepressure“eldisobtainedbywrit-ingvolumebalanceequationsforcenterandcornerphasesateachporeandsolvingtheresultingsystemoflinearequationswiththeappropriateboundaryconditions. 4.58 mm4.58 mm4.58 mm Fig.1.PorenetworkrepresentationoftheBereasandstonecoresample.(a)Volumerenderedgray-scaleimagesoftheBereacoresamplescannedwithamicro-CTscannerataresolutionof2.49m.(b)Volumerenderedpore-grainseparatedimagesoftheBereacoresample.Redandgraycolorsrepresenttheporeandgrainvoxels,respectively.(cPorenetworkgeneratedfromimages(a)and(b).Redandbluecolorsrepresenttheporesandthroatsinthenetwork,respectively.Forillustrativepurposesporesandthroatsareshownwithcircularcross-sectionhere.(d)Amagni“edimageofasmallsectionoftheBereaporenetwork.(Forinterpretationofthereferencestocolorinthis“gurelegend,thereaderisreferredtothewebversionofthisarticle.) A.Aghaei,M.Piri/JournalofHydrology522(