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Advances in Water Resources
journalhomepage:www.elsevier.com/locate/advwatres
Impact of time-dependent wettability alteration on the dynamics of capillary pressure
Abay Molla Kassa
a,b,∗, Sarah Eileen Gasda
a, Kundan Kumar
b, Florin Adrian Radu
baNORCE Norwegian Research Center, Bergen, Norway
bDepartment of Mathematics, University of Bergen, Norway
a r t i c le i n f o
Keywords:
Wettability alteration Dynamic capillary pressure Dynamic wettability Bundle-of-tubes simulation Upscaling
a b s t r a ct
Wettabilityisapore-scalepropertythathasanimportantimpactoncapillarity,residualtrapping,andhysteresis inporousmediasystems.Inmanyapplications,thewettabilityoftherocksurfaceisassumedtobeconstant intimeanduniforminspace.However,manyfluidsarecapableofalteringthewettabilityofrocksurfaces permanentlyanddynamicallyintime.Experimentshaveshownwettabilityalteration(WA)cansignificantly decreasecapillarityinCO2storageapplications.Forthesesystems,thestandardcapillary-pressuremodelthat assumesstaticwettabilityisinsufficienttodescribethephysics.Inthispaper,wedevelopanewdynamiccapillary- pressuremodelthattakesintoaccountchangesinwettabilityatthepore-levelbyaddingadynamictermtothe standardcapillarypressurefunction.Weassumeapore-scaleWAmechanismthatfollowsasorption-basedmodel thatisdependentonexposuretimetoaWAagent.Thismodeliscoupledwithabundle-of-tubes(BoT)model tosimulatetime-dependentWAinducedcapillarypressuredata.Theresultingcapillarypressurecurvesarethen usedtoquantifythedynamiccomponentofthecapillarypressurefunction.Thisstudyshowstheimportanceof time-dependentwettabilityfordeterminingcapillarypressureovertimescalesofmonthstoyears.Theimpact ofwettabilityhasimplicationsforexperimentalmethodologyaswellasmacroscalesimulationofwettability- alteringfluids.
1. Introduction
Wettabilityplaysanimportantroleinmanyindustrialapplications, inparticularsubsurfaceporousmediaapplicationssuchasenhancedoil recovery(EOR)andCO2storage(Blunt,2001;Bonnetal.,2009;Iglauer etal.,2014,2016;Yuetal.,2008).Thewettingpropertyofagivenmul- tiphasesysteminporousmediaisdefinedbythedistributionofcontact angles.Thecontactangle(CA)iscontrolledbysurfacechemistryandas- sociatedforcesactingatthemolecularscalealongthefluid-fluid-solid interface(Bonnetal.,2009).Inporousmediaapplications,microscale wettabilitydetermines thestrength ofpore-scale capillaryforcesand themovementoffluidinterfacesbetweenindividualpores.Atthecore- scale,wettabilityimpactsupscaledquantitiesandconstitutivefunctions suchasresidualsaturation,relativepermeability,andcapillarypressure, whichinturnaffectfield-scalemultiphaseflowbehavior.
Thestandardassumptionisthatwettabilityisastaticpropertyof themultiphasesystem.However,thecompositionofmanyfluidsisca- pableofprovokingthesurfaceswithinporestoundergowettabilityal- teration(WA)viaachangeinCA.CAchangecanaltercapillaryforces at theporescale, and thus affect residual saturationsof the system
∗Correspondingauthorat:NORCENorwegianResearchCenter,Bergen,Norway.
E-mailaddress:[email protected](A.M.Kassa).
(AhmedandPatzek,2003;Blunt,1997).Thiseffecthasbeenexploited extensivelyinthepetroleumindustry,whereoptimalwettingconditions inthereservoirareobtainedthroughavarietyofmeansthatincludes chemicaltreatment,foams,surfactants,andlow-salinitywaterflooding (seeforexampleMorrowetal.,1986;Buckleyetal.,1988;Jadhunandan andMorrow,1995;Haaghetal.,2017;SinghandMohanty,2016).
WettabilityisalsorecognizedasacriticalfactoringeologicalCO2 sequestrationwhichexertsanimportantroleoncaprockperformance (Kimetal.,2012;TokunagaandWan,2013).Thesealingpotentialof thecaprockishighlydependentonCO2beingastronglynon-wetting fluid,andWAmayleadtoconditionsthatallowforbuoyantCO2toleak (Chiquetetal.,2007a;Chalbaudetal.,2009).Besides,WAcanaffect residualsaturationandsubsequentlyimpactthetrappingefficiencyof injectedCO2(Iglaueretal.,2014).Therefore,reliablequantificationof wettabilityisneededforsafeandeffectiveCO2storage.
DespitethefactthatWAisknowntoimpactcore-scalecapillarityand relative permeabilitybehavior,fewdetailedmeasurementsareavail- able tocharacterizethealteration of theconstitutivefunction them- selves.PlugandBruining(2007)havereportedbrine-CO2(gas,liquid) drainage-imbibitionexperimentandshowedcapillaryinstabilityfora
https://doi.org/10.1016/j.advwatres.2020.103631
Received10July2019;Receivedinrevisedform6May2020;Accepted15May2020 Availableonline20May2020
0309-1708/© 2020TheAuthors.PublishedbyElsevierLtd.ThisisanopenaccessarticleundertheCCBYlicense.(http://creativecommons.org/licenses/by/4.0/)
supercriticalCO2-brinesystem,meaningthatthecapillarypressuremea- surementschangesteadilyovertime.Theimbibitioncurvealsoexhib- itedasignificantdeviationfrom theexpectedcurvethatwasnotex- plainedbyclassicalscalingarguments.TheauthorsproposedWAasan explanationbutdidnotexplicitlymeasureanychangesinCA.Addition- ally,recentexperimentsmeasuredcapillarypressurecurvesforasilicate sampleusingafluidpairingofsupercriticalCO2andbrine(Wangand Tokunaga,2015).Repeateddrainage-imbibitioncycleswereperformed for6months,andaclearreductionincapillarypressurewasrecorded foreachsubsequentdrainagecycle.Theauthorsalsoattributedthese deviationsfromtheexpectedcapillarycurvetoachangeinwettability oftherocksampleovertimeduetoCO2exposure,similartoaging.This hypothesiswasconfirmedthroughobservationsofawettinganglein- creasefrom0∘to75∘after6-monthsofexposure.Itisalsoreportedsim- ilar𝑃𝑐−𝑆instabilityanddeviationindolomite/carbonate(Wangetal., 2016),andquartz(Tokunagaetal.,2013;Wangetal.,2016)sandsfor scCO2-brinesystem.MoreliteratureonWAand𝑃𝑐−𝑆measurements canbefoundin(TokunagaandWan,2013).
Theaboveexperimentsrevealthatcapillarypressurecurvesarenot staticforrocksthatundergoWA,despitethefacttheywereperformed followingthestandardmulti-stepprocedure,i.e.where“equilibrium” is obtainedaftereachincrementalstepinpressure.Therefore,thestandard capillarypressuremodelscannotbereadilyappliedwithoutadditional dynamicstocapturethelong-termimpactofWA.Capillarypressuredy- namicsduetoWAaredistinctfromnon-equilibriumflowdynamics(e.g.
Hassanizadehetal.,2002;Dahleetal.,2005;Barenblattetal.,2003) orfromCAhysteresisgeneratedbyrecedingandadvancingangles(e.g.
KrumpferandMcCarthy,2010;Eraletal.,2013).WAalteration isa chemistry-inducedpore-scalephenomenonthataltersthecapillarypres- surefunctionseparatelyfromtheflowconditionsorotherinstabilities.
Thatis,contactanglehasthepotentialtochangeevenwhenthesys- temisatrest.Ontheotherhand,NEmodelsareformulatedtoaddress dynamicsinonlyforsystemsthatareflowing,andthereforeanewap- proachisneededtoaccountforpermanentandcontinualalterationof capillarypressurefunctionsforbothflowingandnon-flowingsystems.
WenotethatcapillarydrivenflowmayinitiateiftheCAchangeislarge.
ThestandardapproachtoWAistoassumeachangeinsurfacechem- istrythatoccursinstantaneously,whichresultsinanimmediateshift insaturationfunctions(Delshadetal.,2009;Lashgarietal.,2016;Yu etal.,2008; Andersen etal.,2015;Adibhatiaetal., 2005).Thisim- plementationentailsaheuristicapproachthatinterpolatesbetweenthe twoendwettingstatesasalinearfunctionofchemicalagentconcen- tration.Lashgarietal.(2016)havederivedaninstantaneousWAmodel fromGibbsenergyandadsorptionisotherms.TheproposedWAmodel iscoupledwith𝑃𝑐−𝑆relationthroughresidualsaturation.Thesemod- elsneglecttheimpactofWAoverlongertimescales(monthstoyears).
Theyalsodonotcapturepore-scaleheterogeneityinwettingproperties.
Intheavailableliterature,onlyonestudy(Al-Mutairietal.,2012)has includedtheeffectofexposuretimeonWAandconstitutiverelationsfor corescalesimulation.Butthisnumericalstudydoesnotsufficientlyin- corporateorupscalepore-scaleprocessestocore-scaleconstitutivelaws.
Toourknowledge,arigorousmathematicalupscalingoflong-term dynamicsin𝑃𝑐−𝑆functionsintroducedbyexposuretoaWAagenthas notbeenpreviouslyperformed.Thefocusofthispaperistoproposea newdynamiccapillarypressuremodelbyupscalingtheWAdynamics fromthepore-tothecore-scale.Section2describesourapproach.We startwithdirectsimulationsof𝑃𝑐−𝑆curvesfromapore-scalemodel representedbyacylindricalbundle-of-tubes.WAisintroducedatthe porescaleusingamechanisticmodelforCAchangeasafunctionofex- posuretimetoareactiveagent.Thismodelisdevelopedbasedonthein- sightsfromlaboratoryexperiments,givingtheflexibilitytoincorporate otherdataasappropriate.WeemphasizetheCAmodelisonlymeantas abasisonwhichtodemonstratetheupscalingapproach.InSection3, wepresenttheresultingcurvesgeneratedbythepore-scalemodelus- ingtwodifferentpore-scalemodelsforCAchange.Thesecurvesarethen usedtocorrelatethedynamictermintheupscaled𝑃𝑐−𝑆function.Fi-
nally,weanalyzethelinkbetweenpore-scaleparametersandupscaled correlations.
2. Approach
Theextended𝑃𝑐−𝑆relationshipintroducesadynamiccomponent thatcapturesthechangingwettabilityasmeasuredbythedeviationof thedynamiccapillarypressurefromtheequilibrium(static)capillary pressure.Thisrelationshipcanbedescribedasfollows:
𝑃𝑐(⋅)−𝑃𝑐st,in∶=𝑓dyn(⋅), (1) where 𝑃𝑐st,in represents thecapillary pressure forthesystem givena staticinitialwettingstate,andfdyn representsthedeviationfromthe staticstate.TheinitialstaticcurvecanbedescribedbytheBrooks-Corey model,
𝑃𝑐st,in=𝑐𝑤
(𝑆𝑤−𝑆𝑤𝑐
1−𝑆𝑤𝑐 )−𝑎𝑤
, (2)
wherecwistheentrypressure,1/awisthepore-sizedistributionindex, whereasSwcistheresidualwatersaturation.
Theobjectiveofthisstudyistocharacterizeandquantifythedy- namictermfdyn,thekeytermofinterestinthe𝑃𝑐−𝑆model,forasys- temthatundergoesWA.Weproposeaninterpolationmodeltohandle theWAdynamicsin𝑃𝑐−𝑆relation.Toobtainaninterpolationmodel, thedynamiccomponentinEq.(1)canbescaledbythedifferencebe- tweentwostaticcurves,eachrepresentingtheinitialandfinalwetting- statecapillarypressurecurves,togiveanon-dimensionalquantity𝜔we callthedynamiccoefficient,whichisdefinedasfollows
𝜔(
𝑃𝑐st,f−𝑃𝑐st,in)
=𝑓dyn, (3)
where𝑃𝑐st,fisthefinalwettingstatecapillarypressure.Intheprevious studies(e.g.seeDelshadetal.,2009;Lashgarietal.,2016;Yuetal., 2008;Andersenetal.,2015;Adibhatiaetal.,2005),thecoefficient𝜔is assumedonlychemistrydependent.Here,𝜔inEq.(3)isassumedtobe afunctionofnotonlythechemistrybutalsotheexposuretimetothe WAagent.
TheexpressioninEq.(3)canbesubstitutedintoEq.(1)toobtaina dynamicinterpolationmodel
𝑃𝑐=(1−𝜔)𝑃𝑐st,in+𝜔𝑃𝑐st,f. (4) Wenotethatinthemodelpresentedabove,wehavedefinedthe“to- tal” capillarypressurePc assimplythemeasureddifferenceinphase pressuresatanypointintime.Inareservoirsimulation,thiswouldbe capillarypressureinagivengridcell,whereasinalaboratoryexper- imentdesignedtomeasure𝑃𝑐−𝑆data,itcorrespondstothepressure dropacrossthesampleatequilibrium.Forquasi-staticdisplacementin abundleofcapillarytubes,Pcisthedifferencebetweenboundarycon- ditionpressures,i.e.,𝑃𝑐=𝑃𝑙𝑟𝑒𝑠−𝑃𝑟𝑟𝑒𝑠,(seeFig.1).
Theexactnatureof𝜔anditsfunctionaldependenciescanonlybe determinedfromafullcharacterizationof𝑃𝑐−𝑆curvesunderdifferent conditions.Thesecurvescanbederivedfromlaboratoryexperiments, butthisapproachiscostlyandtime-consuming.Alternatively,onemay takeamoretheoreticalapproachbysimulating𝑃𝑐−𝑆curvesusinga pore-scalemodelthatincludestheimpactofWA.
ForasystemthatundergoesWA,asignificantchangeinCAcould leadtothewettingphasebecomingnon-wettingandviceversa.Forclar- ity,wewillcontinuetousewsubscriptforthephasethatwasoriginally wettingandnwforthephasethatisoriginallynon-wettingregardless oftheactualstateofwettabilityinthesystem.
2.1. Pore-scalemodel
Therearevariouschoicesofpore-scalemodelsavailable.Theeasiest toimplementandanalyzeisthebundle-of-tubes(BoT)modelwhichis acollectionofcapillarytubeswithadistributionofradii.Herein,we
Fig.1. Fluiddisplacementinanon-interactivebundle-of-tubes(BoT).Here,the leftreservoircontainsnon-wettingfluidthatdisplacesthewettingfluidtothe rightandviceversa.
describethemodelandtheapproachfortheimplementationoftime- dependentWA.
TheBoTmodelisapopularapproachduetothesimplicityofim- plementationandtheabilitytostudythebalanceofenergyandforces directlyatthescaleofinterfaces(BartleyandRuth,1999;Dahleetal., 2005;HellandandSkjæveland,2006).TheaveragebehaviorofaBoT modelcanthenbeusedtoconstructbetterconstitutivefunctionsatthe macroscale(Dahleetal.,2005;HellandandSkjæveland,2006;2007).
There areknownlimitationstothispore-scale approach, i.e. lackof residualsaturation andtortuosity.Weemphasize thatthisstudyis a firststepatstudyingtheimpactoflong-termWAfrompore-tocore- scale,andassuchthesesecondaryaspectsarebeyondthescopeofthis work.
Inthispaper,weconsidercylindricalBoThavinglength,L,andis showninFig.1.These tubesaredesignedtoconnectwetting(right) andnon-wetting(left)reservoirswithpressureslabeledas𝑃rres.and𝑃𝑙res. respectively.Letthereservoirpressuresdifferencebedefinedas
Δ𝑃 =𝑃𝑙𝑟𝑒𝑠−𝑃𝑟𝑟𝑒𝑠. (5)
Initially,thetubesin thebundlearefilled withawettingphase.To displacethewettingphasefluidinthemthtube,thepressuredrophas toexceedthelocalentrypressure,Pc,m,definedas(Dahleetal.,2005)
Δ𝑃 >𝑃𝑐,𝑚(𝑅𝑚,𝜃𝑚), (6)
wherePc,m(Rm,𝜃m)isgivenbytheYoung’sequation 𝑃𝑐,𝑚(𝑅𝑚,𝜃𝑚)=2𝜎cos(𝜃𝑚)
𝑅𝑚 ,𝑚=1,2,⋯,𝑁, (7) whereRmand𝜃marethetuberadiusandCA,respectively,forthemth tube,Nstandsfornumberoftubes,𝜎isfluid-fluidinterfacialtension.
Aslongascondition(6)issatisfied,thefluidmovementacrossthe length of tube m can be approximated bythe Lucas-Washburn flow model(Washburn,1921),
𝑞𝑚= 𝑅2𝑚(Δ𝑃−𝑃𝑐,𝑚(𝑅𝑚,𝜃𝑚))
8(𝜇nw𝑥𝑖𝑛𝑡𝑚 +𝜇w(𝐿−𝑥𝑖𝑛𝑡𝑚)), (8) where,𝜇nw and𝜇w arenon-wettingandwettingfluidviscosities,re- spectively,thesuperscriptintstandsforfluid-fluidinterface,and𝑞𝑚= 𝑑𝑥𝑖𝑛𝑡𝑚∕𝑑𝑡istheinterfacevelocity.Theinterfaceisassumedtobetrapped whenitreachedattheoutletofthetube,thus𝑞𝑚=0.Apositiverateof changein𝑥𝑖𝑛𝑡𝑚 isassociatedwithanincreaseinnon-wettingsaturation fortubem.FromEq.(8),onecanthendeterminetherequiredtimeto reachaspecifiedinterfaceposition.
2.2. Pore-scaletime-dependentWAmodel
Inthispaper,weconsideraWAmechanismatthepore-scalethat evolvessmoothlyfromaninitialtofinalwettingstatethroughexposure
time.Theinitialandfinalwettingstatescanbearbitrarilychosen,i.e.
fromwettingtonon-wettingorviceversa.
TheWAagentisdefinedaseitherthenon-wettingfluiditselforsome reactivecomponenttherein.Weconsideranalterationprocesswithin anygivenpore,ortube,thatcontinuesuntiltheultimatewettingstateis reachedlocallyinthepore.Thealterationispermanent,butcanalsobe haltedatsomeintermediatewettabilitystateiftheWAagentisdisplaced fromthepore.Iftheagentisreintroducedtotheporeatsomelaterpoint, alterationcontinuesuntilthefinalstateisreached.
Tothisend,weintroduceageneralfunctionalformofpore-scaleWA mechanismbyCAchange,
𝜃𝑚(⋅)∶=𝜃𝑚,in+𝜑(⋅)ΔΘ, (9) whereΔΘ =𝜃𝑚,f−𝜃𝑚,in,𝜃m,fand𝜃m,inaretheultimateandinitialcontact anglesrespectively.InEq.(9),𝜃mdecreasesandincreasesbasedonthe choiceoftheinitialandfinalwettingconditions.Theterm𝜑(· )∈[0, 1](when𝜑(⋅)=1theCAattainsitsultimatevalueandCAisfixedatthe initialstatewhen𝜑(⋅)=0)inEq.(9)isresponsibleforgoverningthe WAdynamics.
WA involvescomplexphysical andchemical processeswhose de- scriptionisbeyondthescopeofthiswork.However,weprovideabrief summaryoftheroleofadsorption/desorptionprocessesinCAchange (Blut,2017; Duetal., 2019).Sucha hypothesishasbeen supported by experimentmeasurements. For instance,CO2-water core-flooding experimentsshowadsorption-typerelationsbetweenCAandpressure (Dicksonetal.,2006;JungandWan,2012;Iglaueretal.,2012),and alsowithexposure time(JafariandJung,2016; Sarajietal., 2013).
SimilarCAevolutionsasafunctionofsurfactantconcentrationandex- posuretimearereportedinDavisetal.(2003)and(Mortonetal.,2005) foranoildropletonametalsurfaceimmersedinionicsurfactantsolu- tions.InMortonetal.(2004),aLangmuiradsorptionmodelisproposed topredicttheexperimentobservationsin(Davisetal.,2003).Giventhe insightsabove,weconsideraCAmodelthatevolvesaccordingtothe rateofadsorptionoftheWAagentonthesurfaceofthepores.Follow- ingMcKee(1991),vanErpetal.(2014),thedynamicparameter𝜑in Eq.(9)canbestatedas,
𝑑𝜑
𝑑𝑡 =𝐽+−𝐽−, (10)
where𝐽+and𝐽−representratesofadsorptionanddesorptionofaWA agentrespectivelyatthesolidsurface.InMcKee(1991),𝐽+ istaken tobeproportionaltotheWAagentandthesurfaceunoccupiedbythe adsorbedWAagent,i.e.,
𝐽+=𝑘1𝜒𝑚( 1−𝜑∕𝜑)
, (11)
wherek1isarateconstant,𝜑representsthemaximumsurfacesaturated concentration,and𝜒misameasureofthelocalexposuretimeoftube m.Thedesorptionratecanberelatedwiththecurrentsurfaceconcen- trationandisdefinedas,
𝐽−=𝑘2𝜑 (12)
wherek2isarateconstantfordesorptionrate.CombiningEqs.(10)–(12) wouldgiveus,
𝑑𝜑 𝑑𝑡 =𝑘1𝜒𝑚(
1−𝜑∕𝜑)
−𝑘2𝜑. (13)
Assuming𝜑=1andfollowingMcKee(1991),onecanapplyaperturba- tionanalysistoEq.(13)toobtainafirst-orderapproximationfor𝜑in termsof𝜒m,
𝜑≈ 𝜒𝑚
𝐶+𝜒𝑚, (14)
where𝐶= 𝑘𝑘2
1isaparameterthatcontrolsthespeedandextentofalter- ation.𝜒misdefinedasthetime-integrationofexposuretoaWAagent, heretakentobethelocalnon-wettingsaturationoftubem,
𝜒𝑚∶= 1 𝑇 ∫
𝑡 0
𝑥𝑖𝑛𝑡𝑚
𝐿 𝑑𝜏, (15)
whereTisapre-specifiedcharacteristictime.Inthispaper,thecharac- teristictimeissetasthetimeforonecompletedrainagedisplacement understaticinitialwettingconditions,whichcanbepre-computedac- cordingtoEq.(8).
Asanaside,detailedlaboratoryworkwouldbeneededtofurther enrichthepore-scaleCAmodelbyfittingCtoexperimentaldata.This exerciseisbeyondthescopeofthispaper,andweconsidertheunder- lyingCAchangeinEq.(14)areasonablebasisforwhichtoperformthe upscalingaspectofourstudy.
Herein,weconsidertwomodelsforexposuretimeatthelocalscale.
ForfirstwetakethecasewhereCAmodificationbasedonEq.(14)is strictlydependentonlocalexposuretime𝜒mandleadstoCAvariation fromtubetotube,hereafterreferredtoasthenon-uniformWAmecha- nism.Wenotethatwettabilitygradientswithinindividualtubesarenot consideredinthismodel,andthusthereisnovariationinCAalongthe tube.ThisisduetotheflowmodelinEq.(8),wheretheCAonlyaffects theentrypressureofthetube.
ThesecondisreferredtoasuniformWA,whichisbasedontheas- sumptionthattheWAagentdissolvesintothewettingphasefromthe non-wettingfluidandaffectsalltubessimultaneously.Inthiscase,all tubeshavethesamepropertiesthataregovernedbythebulkexposure timeacrosstheentirebundle(i.e.REV).Wecandefine𝜒
𝜒∶= 1 𝑇∫
𝑡
0 𝑆𝑛𝑤𝑑𝜏. (16)
asthebulkoraverage,exposuretimeasafunctionofaveragesaturation.
TheuniformmodelisimplementedintoEq.(14)bytaking𝜒𝑚=𝜒. Insummary,weintroducetwotypesofWAmechanisms,uniform andnon-uniform,thatserve asendmembersof possibleWA mecha- nismsatthepore-scale.Ontheoneend,non-uniformWArestrictsal- terationtoonlydrainedporesandexcludesanyinteractionoftheWA agentbetweenpores.ThisleadstosignificantheterogeneityinCAfrom oneporetoanother.Attheotherend,theuniformcaseassumestheWA agentcanalterallporessimultaneously.Inreality,WAwillliesome- whereinbetween,butwehavechosensimplerendmemberstoaidin furtheranalysisofsimulateddatainthenextsection.
2.3. Simulationapproach
Theuniformandnon-uniformapproachesarecoupledintotheBoT modelfollowingAlgorithm1forasingledrainage-imbibitioncycle.The objectiveistoperformsimulatedexperimentsthatmimiclaboratory- derivedcapillarypressurecurves,i.e.thepressureisadjustedincremen- tallyupordownaftereachstep dependingonifthebundleisunder drainageorimbibition,respectively.Contactanglesareupdatedcontin- uouslythroughouttheflowprocessesinastep-wisemanneroncethe displacementiscompletedforeachpressureincrement.Thatis,contact anglesthathavebeenalteredaccordingtotheuniformornon-uniform mechanism,areupdatedbeforethenextpressureincrement.Thisisa reasonableapproximationgiventhatitisonlyentrypressuresinindi- vidualtubesthatareaffectedbyCAchange.
WecontrolthepressuredropΔPinthewaythattheWAprocess is completedwithina fewnumbersofdrainage-imbibition cycles.In thefirstdrainage-imbibitioncycles,theΔPincrementissuchthattubes drain/imbibeoneatatimewithapressuredropclosetothenexttube entrypressure.Inthelastdrainage-imbibitioncycle,everyΔPincrement isreducedbytwoandthreeordersofmagnitudeforthenon-uniform anduniformcase,respectively.Consequently,theflowslowsdownby thesamemagnitudeirrespectiveofwhetherthetubedrainsorimbibes.
Atthecompletionofthenumericalexperiment,weobtainasetof𝑃𝑐−𝑆
“datapoints” thatcanbeplottedintheusualway.
Oncethecapillarypressurecurvesaregeneratedforboththeuniform andnon-uniformapproaches,theresultingcurvesareusedtoquantify thedynamiccoefficientintheinterpolationfunctioninEq.(4).Thegoal istodevelopacorrelationmodelthatinvolvesonlyasingleparameter,
Algorithm1 Asingledrainage-imbibitioncycle.Fluidandrockprop- ertiesaregivenaccordingtoTable2
1: Drainagedisplacement
2: setthemaximumcapillarypressure𝑃𝑐max 3: whileΔ𝑃<𝑃𝑐max do
4: increasethenon-wettingpressure,𝑃𝑙res 5: calculatethepressuredropΔ𝑃=𝑃𝑙res−𝑃𝑟res 6: ifΔ𝑃>2𝜎cos𝑅(𝜃𝑚)
𝑚 then
7: draintherespectivetubes
8: calculatetheelapseoftimetodraintubesfromEquation(8) 9: calculateandstoreaveragedquantities𝑆𝑛𝑤and𝜒
10: ifnon-uniformWAthen
11: calculate𝜒𝑚forinvadedporesfromEquation(15) 12: calculate𝜃𝑚fromEquation(9)and(15)
13: elseif uniformWAthen
14: updateeach𝜃𝑚inbundleidenticallyfromEquation(9)
15: and(16)
16: endif
17: endif 18: endwhile
19: Imbibitiondisplacement
20: definetheminimumentrypressure𝑃𝑐min 21: whileΔ𝑃>𝑃𝑐min do
22: decreasethenon-wettingpressure,𝑃𝑙res 23: calculatethepressuredropΔ𝑃=𝑃𝑙res−𝑃𝑟res 24: ifΔ𝑃>𝜎cos𝑅(𝜃𝑚
𝑚 then
25: imbibetherespectivetubes
26: calculatetheelapsedoftimetoimbibetubesfrom 27: Equation(8)
28: calculateandstoreaveragedquantities𝑆𝑛𝑤and𝜒 29: ifnon-uniformWAthen
30: calculate𝜒𝑚fromEquation(15) 31: update𝜃𝑚fromEquation(9)and(15) 32: elseifuniformWAthen
33: updateeach𝜃𝑚identicallyfromEquation(9)and(16)
34: endif
35: endif 36: endwhile
andthis parametershould haveaclearrelationwith changesin the pore-scaleWAmodelparameterC.
3. Results
Inthissection,wepresentthesimulatedcapillarypressureandas- sociatedresultsforeachWAcase.Weformulateacorrelationmodel, whichisthenfittothesimulateddata.Finally,weinvestigatethesen- sitivityofthecorrelatedmodeltothepore-scaleWAparameter.
3.1. Bundleoftubesmodelset-up
The pore scale is described by a BoT model (see Section 2.1).
Each tubein theBoTisassignedadifferent radiusR,withtheradii drawnfromatruncatedtwo-parameterWeibull distribution(Helland andSkjæveland,2006)
𝑓(𝑅)= [𝑅−𝑅
𝑅avmin
]𝜂−1 𝜂 𝑅avexp
(
− [𝑅−𝑅
𝑅avmin
]𝜂) 1−exp(
−[𝑅
max−𝑅min 𝑅av
]𝜂) (17)
where𝑅max,𝑅min,andRavaretheporeradiiofthelargest,smallest,and averageporesizes,respectively,and𝜂isadimensionlessparameter.The averageisobtainedbythemeanof𝑅maxand𝑅min.Therockparameters andfluidpropertiesarelistedinTable1.
Fig.2. Simulated𝑃𝑐−𝑆datafortheinitialandfinalwettingstatescompared toaBrooks-CoreymodelwithcalibratedparametersgiveninTable2.
3.2. Staticcapillarypressureforendwettingstates
Astartingpointforthedynamiccapillary-pressuremodelspresented inSection2(Eqs.(1)and(4))ischaracterizingthecapillarypressure curvesfortheendwettingstates.Giventhesametubegeometryand fluidpairingdescribedabove,thecapillarypressure–saturationdataare simulatedunderstaticconditionsforboththeinitialandfinalwetting states.
Onlyasingledrainageexperimentisneededinthestaticcasetofully characterizethecapillarypressurecurve.Thisisduetothelackofresid- ualtrappinginaBoTmodel.Weemphasizethathysteresisisnotpossible foraBoTifthecontactanglesinthetubes(andotherparameters)are heldconstant.
Thesimulated staticcurves arethen correlatedwiththe Brooks- Coreymodel(2).TheresultingcorrelationscanbefoundinFig.2,while fittedparametersfortheBrooks-CoreymodelcanbefoundinTable2. NotethattheBrooks-Coreymodelisundefinedatzeroirreduciblewet- tingphasesaturation.Thus,weleftafewporesundrainedtoallowfor comparisonbetweentheBrooks-Coreymodelandthesimulated𝑃𝑐−𝑆 data.
Fig.2comparestheBrooks-Coreyformula(2)andthecapillarypres- surecurvesassociatedwithstaticcontactangles(initialandfinalwet- tingstates).
TheBrooks-Coreycorrelationgivesanexcellentmatchtothesim- ulated𝑃𝑐−𝑆dataunderstaticconditions.Weobservethatthepore-
Table1
Parametersusedtosimulatequasi-staticfluiddisplacementinBoT.
parameters values unit parameters values unit 𝜎ow 0.0072 N/m no. radii 500 [-]
𝑅 min 6 μm 𝑅 max 40 μm
𝜃f 80 degree 𝜃in 0.0 degree
𝜇w 0.0015 Pa.s 𝜇nw 0.0015 Pa.s
R av 23 μm L 0.001 m
𝜂 1.5 [-]
Table2
Estimatedcorrelationparametervaluesforinitialandfinalwetting statecapillarypressurecurves.
Initial wetting state Final wetting state
param. value unit param. value unit
c w 360 [Pa] c w 56 [Pa]
a w 0.2778 [-] a w 0.2778 [-]
R 2 1 - R 2 1 -
sizedistributionindexawfortheinitialandfinalwettingstatesarethe same,whichisexpectedsincethesamedistributionoftuberadiiisused inbothcases.Ontheotherhand,thecoefficientcwdecreasesbyafac- torof0.85fromtheinitialtothefinalwettingstatecorrespondingto adecreaseinacore-scalecapillaryentrypressure.TheLeverett-Jscal- ingtheory(Xuetal.,2016)predictsthatentrypressurescalesbycos𝜃, whichagreesnicelywiththereductionincos𝜃byafactorof0.83fora CAchangefrom0to80degrees.
WereiteratethatforthestaticcasewherenoWAoccurs,theBrooks- CoreymodeldescribesbothdrainageandimbibitionfortheBoT.
3.3. Simulatedcapillarypressuredata
Wepresentthesimulatedcapillarypressuredata(seeFig.3),com- paringtheresultsoftheuniformandnon-uniformapproachesdescribed previously.Theuniformdataaregeneratedwiththepore-scaleWApa- rameter𝐶=0.005,whileforthenon-uniformdata𝐶=5× 10−4.Atotal oftwoandfourdrainage-imbibitioncyclescarriedoutfortheuniform andnon-uniformcases,respectively.
Forbothcases,weobserveasteadydecreaseincapillarypressure overtime.Intheend,acompletewettabilitychangehasevolvedfrom theinitialtofinalprescribedstates,whosestaticcurvesareplottedin Fig.3forreference.Havingreachedthefinalwettingstate,anyaddi- tionaldrainage-imbibitioncyclewouldfollowalongthestaticcurvefor thefinalwettingstate.Weremarkthatwettability-induceddynamics alsointroduces anapparenthysteresisinthe𝑃𝑐−𝑆data. Thiseffect isuniquetothecylindricalBoTmodel,whichwerecallcannotexhibit hysteresisunderstaticwettabilityconditions.However,arealporous mediummayexhibitcapillarypressurehysteresiswithstaticwettabil- ity.
Therearenotabledifferencesbetweenthetwosetsofcurves.Forthe uniformcase(Fig.3a),therearedistinctcurvesforeachdrainageand imbibitiondisplacement.Thecapillarypressurebeginstodecreaseim- mediatelyandinacontinuousmannerovertime.ThisisbecausetheCA (Fig.4a)ischangingforalltubessimultaneouslybasedontheaverage exposuretimeovertheentirebundle.TheuniformityresultsintheCA insmallertubesbeingalteredsignificantlyatanearlytime(atthesame rateasthelargertubes),andthusthecapillarypressureisdecreased evenatlowaveragewettingsaturationinthefirstdrainagecurve.The fastdynamicsinCAchangeleadtoanon-monotonecapillarycurveat anearlytime.
Incomparison,thenon-uniformcase(Fig.3b)hasadelayinexhibit- ingtheeffectsofWA.Theinitialdrainagecurveisidenticaltotheinitial wettingstaticcurveandallsubsequentdrainagecurvesfollowalongthe previousimbibitioncurve.ThisisaresultoftherestrictiononCAchange toonlytubesthataredrained.Inotherwords,atthetube-level,there isnochangeinentrypressurefromtheinitialstate(orthestateaftera singledrainage-imbibitioncycle)untilthattubeisdrained.Incontrast totheuniformcase,thecapillarypressureatlowSwisdrawntowards theinitialstate.ThiscanbedescribedbyexaminingtheCApertubera- diusintime,showninFig.4b.Largertubesthatdrainfirstandimbibe last,resultinginlongerlocalexposuretime,andthusmoreextensiveCA change,comparedtothesmallertubesthatdrainlastandimbibefirst.
Therefore,theinitialwettingstatepersistsinthesmallertubes.
Werecallthatboththeuniformandnon-uniformcasesareselected asendmembersofpossibleWAmechanismsinrealporousmedia.In realsystems,WAindifferentsizedporesmayoccurinamorecomplex manner.
WehaveobservedabovethatcapillarypressurecurvesinFig.3aand barenotauniquefunctionofsaturation,thatis,theyexhibithysteresis forthissimpleBoTgeometry.Wenotethatthe𝑃𝑐−𝑆datapointsare color-codedaccordingtotimeevolvedateachdatapoint.Thismotivates atransformationofthedataintothetimedomainbyplottingagainst 𝜒,asshowninFig.5forbothcases.Indoingso,weobtainaunique functionwithrespecttoexposuretimeforboththeuniformandnon- uniformcases.WenotethatthecurvesinFig.5showthatthecapillary
pressureincreasesanddecreaseswitheachdrainage-imbibitioncycle.
Inaddition,thetransformationrevealstheseparatedrainagecurvesfor thenon-uniformcasethatwerehiddeninFig.3b.
3.4. Dynamiccapillarypressuremodeldevelopment
FollowingtheapproachdiscussedinSection2,weappliedEq.(3)to calculate the dynamic coefficient 𝜔 for both the uniform and non- uniformcases.TheresultingcoefficientisplottedinFig.6asafunction ofbothSw(toppanels)and𝜒(bottompanels)forbothWAcases.We recallthat𝜔isacoefficientthatinterpolatesbetweenthecapillarypres- sureattheinitialandfinalwettingstatesatanygivensaturation,where 𝜔=0givestheinitialcapillarypressureand𝜔=1givesthefinal𝑃𝑐−𝑆 curve.
Fortheuniformcase,𝜔isanon-uniquefunctionofwetting-phase saturation,seeFig.6a, butwithvaluesthatarecontinuouslyincreas- ingasthedynamiccapillarypressuremovestowardsthefinalwetting state.Forthenon-uniformcase,thedynamiccoefficientinFig.6balso exhibitsnon-uniquenesswithrespecttosaturation.Reflectingthe𝑃𝑐−𝑆 data,thecapillarypressurepersistsattheinitialstateatlowsaturation.
Thismeansthatthevalueof𝜔decreaseswithdecreasingSwalongthe drainagepathandincreasesonlyalongimbibitionpaths.Thecomplex relationof𝜔insaturationspacemakesitchallengingtoproposeafunc- tionalformfor𝜔−𝑆𝑤relationinbothcases.
Figs.6cand6dshowthat𝜔exhibitsdifferentbehaviorasafunction ofaverageexposuretime.Fortheuniformcase,Fig.6c,𝜔issmoothly increasinganduniquelyrelatedto𝜒,mimickingthefunctionalformof thepore-scalemodelinEq.(9).Ontheotherhand,thecoefficient𝜔in thenon-uniformcase,Fig.6d,isnotmonotonicallyincreasingin𝜒but continuestoriseandfallwithtimedespitethetransformationtothe temporaldomain.
ThecurvesinFig.6giveusimportantinsightintotheformof𝜔best suitedtoeachWAcase.Wetakeeachcaseinturn:
3.4.1. Uniformcase
Thesmoothlyvaryingfunctionalityof𝜔and𝜒inFig.6cmotivates anadsorption-typemodel:
𝜔= 𝜒
𝛽1+𝜒, (18)
where𝛽1isafittingparameterobtainedfromthebestfittothesimu- lateddatainFig.6c.Forthisparticularcase,thecalibratedparameter isestimatedtobe𝛽1=0.01.
Theformofthedynamic𝑃𝑐−𝑆modelfortheuniformcaseisob- tainedbysubstitutingEq.(18)intoEq.(4)togive:
𝑃𝑐= 𝜒 𝛽1+𝜒
(𝑃𝑐st,f−𝑃𝑐st,in)
+𝑃𝑐st,in. (19)
3.4.2. Non-uniformcase
Thenon-trivialbehaviorof𝜔inFig.6dmakesitchallengingtopro- poseafunctionalrelationbetweenthedynamiccoefficient𝜔and𝜒di- rectlyaswedidfortheuniformWAcase.Instead,weobservethat𝜔in Fig.6bhasawell-behavedcurvatureforeachdrainage-imbibitioncy- clealongthesaturationhistory.Further,thecurvatureofeachcycleis increasingwithincreasingexposuretime.Giventheseinsights,wepro- posedamodelforthedynamiccoefficientthathasthefollowingform, 𝜔(𝑆𝑤,𝜒)= 𝑆𝑤
𝛼(𝜒)+𝑆𝑤, (20)
where𝛼controlsthecurvatureofthe𝜔−𝑆𝑤curveforeachdrainage- imbibition cycle. Since 𝜔 is increasingfunction of exposure time, 𝛼 shoulddecreasealongtheaveragedvariable𝜒.
Thefunctionformof𝜔inEq.(20)isthenmatchedwiththe𝜔−𝑆𝑤
datatoanalyzethedynamicsof𝛼along𝜒.Theobtained𝛼 −𝜒relation isdecreasingashypothesizedandinparticularhasthefollwingform,
𝛼(𝜒)=𝛽2∕𝜒, (21)
where𝛽2isnon-dimensionalfittingparameter.Forthisparticularsimu- lationtheparameter𝛽2isestimatedtobe0.004forthefourofdrainage- imbibitioncycles.
Theformofthedynamic𝑃𝑐−𝑆modelforthenon-uniformcaseis thenobtainedbysubstitutingEqs.(20)and(21)intoEq.(4):
𝑃𝑐= 𝜒𝑆𝑤
𝛽2+𝜒𝑆𝑤
(𝑃𝑐st,f−𝑃𝑐st,in)
+𝑃𝑐st,in. (22)
ThecalibrateddynamiccapillarypressuremodelsinEqs.(19)and (22)arecomparedwiththesimulatedcapillarypressuredatainFig.3, withtheresultspresentedinFig.7foreachWAcase.Weobservethat theproposeddynamicmodelsagreewellwithsimulateddynamiccap- illarypressurecurves.Thecorrelationcoefficientforthiscomparison is𝑅2=0.9921and𝑅2=0.98,fortheuniformandnon-uniformcase,re- spectively.Thus,wehaveobtainedasingle-parametermodelinboththe uniformandnon-uniformWAcasesthatdescribetheevolutionofdy- namiccapillarityovermultipledrainage-imbibitioncyclesratherthan usingamodelconsistingofmultipleparametersthatchangewitheach cycle(orhysteresismodels).
3.5. Modelsensitivitytopore-scalemodelparameter
Wehypothesizethatparameters𝛽1inEq.(19)and𝛽2inEq.(22)are dependentontheparameterCinEq.(9)thatcontrolsthedynamicsof wettabilityalterationatthepore-scale.Weinvestigatethissensitivityby repeatingthecapillarypressuresimulationsfordifferentvaluesofthe pore-scaleparameterCanddeterminethecorrelatedvalueof𝛽1and𝛽2
ineachcase.
FortheuniformWAcase,Fig.8ashowsthattheinterpolationmodel parameterislinearlyproportionaltothepore-scalemodelparameter, withaproportionalityconstantof2.Thus,therelationship𝛽1=2𝐶can beusedtopredicttheupscaledparameterdirectlyfromknowledgeof thepore-scaleprocess.Incontrast,thenon-uniformcaseinFig.8bshows apowerlawmodel,where𝛽2=𝑏1𝐶𝑏2iscorrelatedwithestimatedpa- rametersof𝑏1=3.3× 106and𝑏2=1.8.
Thegeneralformofdynamiccapillarypressuremodelcannowbe obtainedfortheuniformWAbyincorporatingtherelationshipfor𝛽1in Eq.(19):
𝑃𝑐= 𝜒 2𝐶+𝜒
(𝑃𝑐st,f−𝑃𝑐st,in)
+𝑃𝑐st,in. (23)
Similarly,weobtainageneralnon-uniformmodelbysubstituting𝛽2in Eq.(22)
𝑃𝑐= 𝜒𝑆𝑤 𝑏1𝐶𝑏2+𝜒𝑆𝑤
(𝑃𝑐st,f−𝑃𝑐st,in)
+𝑃𝑐st,in. (24)
In their final form, the dynamic capillary pressure models in Eqs.(23)and (23)aredependentontwovariables,saturationandtime, andasinglewettabilityparameter,C.Thelattermustbedeterminedby fittingEq.(9)withparameterCtolaboratoryexperimentsforagiven sampleexposedtoaWAagent.
3.6. Applicabilitytoarbitrarysaturationhistory
We note that the saturation historyused togenerate the𝑃𝑐−𝑆 curvesforthetwoWA casesin Figs.3aand3ccanbe thoughtofin eachcaseasasinglearbitrarypathwithinaninfinitenumberofpossi- blepaths.Ifadifferentpathhadbeenchosen,suchasaflowreversalat intermediatesaturationoraprolongedexposuretimeatagivensatura- tion,itwouldresultinentirelydifferentcapillarypressuredynamics.
In order totest thedynamic modelsdeveloped in Eqs. (19) and (22) foranyarbitrarysaturationhistory,wegenerate manydifferent 𝑃𝑐−𝑆curvesbytakingnumerousdifferentpathsinthesaturation-time domain.Theresultingsimulateddataformsasurfacewithrespectto saturationandexposuretimeasshowninFig.9aandb
Fig.3. SimulationresultsforWA-induceddy- namicsofcapillarypressureasafunctionof wettingsaturationgivenuniform(a)andnon- uniform(b)WAatthepore-scale.Thecolorof eachdatapointindicatesthetimeelapsedin months,withelapsedtimeequalto7months fortheuniformcaseand20monthsforthenon- uniformcase.Thedatapointsareobtainedfol- lowingAlgorithm1.Thestatic𝑃𝑐−𝑆 curves fortheendwettingstates,𝜃inand𝜃f,areplot- tedasareferencewithdottedanddashedline, respectively.
Fig.4. CA,𝜃,fortheuniformcase(a)andnon- uniformcase(b).UniformCA,whichisidentical acrossthebundle,isshownasafunctionof av- eragesaturation.TheCAdatashowthedrainage- imbibitionfluidhistorypaths.Non-uniformCAis shownasafunctionoftuberadiusandtime.The colorscaleinbothfiguresindicatestimeelapsedin months.
Fig.5.Capillary pressuredataplottedasa functionof 𝜒 fortheuniform(a)andnon- uniform(b)WAcase.Thecolorofeachdata pointindicatesthetimeelapsedinmonths.
Theinsetplotin(b)istheresolutionofthe capillarypressureforthefirstthreecycles.
Wethenapplythecalibrateddynamicmodelstothesamesaturation- timepathsusedtogeneratethe𝑃𝑐−𝑆−𝜒surface.Thedifferencebe- tweenthecalibratedmodelandthesimulateddataisshowninFig.9c anddfortheuniformandnon-uniformcases,respectively.Agoodcom- parison of thedynamic modelsto simulateddata demonstrates that modelcalibrationtoasinglesaturation-timepathisrobustenoughto beappliedtoanypossiblepath.
3.7. Discussion
Weinvestigatedthepotentialof theinterpolation-basedmodelto predicttheWAinduceddynamicsincapillarypressure–saturationre- lations.Intheinterestof completeness,we alsoexploredothertypes ofmodelstocapturecapillarypressuredynamics,includingthemixed-
wetmodelofSkjævelandetal.(2000).Forbrevity,wedonotreportthe resultsofthatseparatestudyherein.Wefoundthatalthoughothermod- elscouldbecalibratedwithreasonableaccuracy,theyallinvolvedmore thanonecalibrationparameter(uptofour)thatneedtobeadjustedin eachdrainage-imbibitioncycle.Therefore,thesingle-parametersingle- valued interpolation model presented in this study is the preferred modelduetoitsreliabilityforreplicatingthesimulatedBoTdata.
Theproposedinterpolationmodelisanupscaledmodelthatallows forachangeincapillarypressureasafunctionofupscaledvariables, saturation,andexposuretime,toaWAagent.Werecallthatexposure timeissimplytheintegrationofsaturationhistoryovertime.Themodel consistsofthreemaincomponents– twocapillarypressurefunctionsat theinitialandfinalwettingstateandadynamicinterpolationcoefficient thatmovesfromonestatetotheother.Theinitialandfinalcapillary
Fig.6.Plotofthedynamiccoefficient𝜔againstwettingphasesaturation(top)and𝜒(bottom)fortheuniformWAcase(leftpanels)andnon-uniformWAcase(right panels).Thedatapointsarecolor-codedwithexposuretimeinmonths.Theinsetplotin(d)istheresolutionofthedynamiccoefficientforthefirstthreecycles.
Fig.7. ComparisonofthedynamicmodelsinEqs.(19)and(22)withthesimulated𝑃𝑐−𝑆datainFig.3foruniform(a)andnon-uniform(b)WAcases.Thedata pointsarecolor-codedwithexposuretimeinmonths.
pressurefunctionscanbedeterminedapriorifromstaticexperiments usinginertfluids.Inthisstudy,theinitialandfinalstatesarerepresented byclassicalBrooks-Coreyfunctions.Thedynamiccoefficientisthusthe onlyvariablecorrelatedtodynamiccapillarypressuresimulations.In thisstudy,wehaveshownthatthecoefficientcanbeeasilycorrelated tosaturationandexposuretimeviaasingleparameter.
Wehaveobservedthattheformofthedynamictermisdependenton theunderlyingmechanismsforWA.Weemployedtwomodels,uniform andnon-uniform,thatrepresenttwoendmembersofrealsystems.One
endmemberisidenticalCAthroughouttheREV,whiletheotherresults inseverelyheterogeneousCAfromsmalltolargepores.Thedifferences inthetwoWAmechanismschangesthecomplexityoftheresultingcap- illarydynamics.Intheuniformcase,thedynamiccoefficientcanbecor- relatedtoexposuretimethroughasorption-typemodel,whichseemsto beanaturalresultgiventheCAchangeattheporescaleisalsobasedon asorptionmodel.Thisisaninterestingobservationthatrequiresmore analysisinfuturework.Inthenon-uniformcase,thedynamiccoeffi- cienthasnosimilarsorptionformwithincreasedexposuretime,but
Fig.8. Therelationbetweenpore-scalewettabilityparameterCandthecorrelationparameters𝛽1and𝛽2inEqs.(19)and(22)fortheuniform(a)andnon-uniform (b)WAcases,respectively.
Fig.9. Top:Simulatedcapillarypressureobtainedbytakingmultiplepathsinthe𝑆𝑤×𝜒domain,fortheuniform(left)andnon-uniform(right)cases.Bottom:The differencebetweenthedynamicmodel(23)withtherespectivedata,scaledby𝑃𝑐max.
nowwiththeproductofsaturationandexposuretimeasthedynamic variable.Theadditionalcomplexityisneededtodrawthecapillarypres- surecurvebacktotheinitialwettingatlowsaturation(aregionofthe 𝑃𝑐−𝑆curvedominatedbysmallerporeswheretheCAtakeslongerto change).
Animportantresultofthisstudyisquantifyingthelinkbetweenthe pore-andcore-scale.Weshowedthatbyvaryingtheparameterthatal- tersthespeedandextentofCAchangeineachindividualpore,wecould predicttheresultingimpactondynamiccapillarypressure.Infact,in boththeuniformandnon-uniformcases,thereisaverysimplescaling fromthepore-scaleandmacroscaleparameters.Intheuniformcase,
thetwoparametersaredirectlyproportional,whileinthenon-uniform case,themacroscaleparameterscaleswiththepore-scaleparametervia apowerlaw.Theimplicationofthisresultisthatbyknowingthemech- anismthatcontrolsCAatthepore-scale,whichcanbeobtainedbyarel- ativelysimplebatchexperiment,wecanquantifyapriorithemacroscale dynamicswithouthavingtoperformpore-scalesimulations.Thisisan importantgeneralizationandvaluableformakinguseofexperimental datatoinformmacroscaleconstitutivefunctions.
Wehave quantifiedtheabilityof theinterpolationmodeltocap- tureunderlyingWAatthepore-scaleforasimpleBoT.Thisresultisa naturaldevelopmentfrompreviousstudiesthatincorporatetheinterpo-
