Dynamic foam behavior in the entrance region of a porous medium

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ColloidsandSurfacesA:Physicochemicaland

EngineeringAspects

journalhomepage:www.elsevier.com/locate/colsurfa

Dynamicfoambehaviorintheentranceregionofaporousmedium

E.Ashooria,∗,D.Marchesinb,W.R.Rossena

ab

DelftUniversityofTechnology,Netherlands

InstitutoNacionaldeMatemáticaPuraeAplicada,RiodeJaneiro,Brazil

articleinfoabstract

InfoamEOR,complexdynamicsofbubblecreationanddestructioncontrolsfoamproperties.Herewereconsiderwhetherandwhennon-equilibriumeffectsareimportant,focusingspecificallyontheentranceregion,whereinjectedgasandliquidaretransformedintofoam.

Wesolveforwatersaturationandfoamtextureintheentranceregionusingthepopulation-balancefoammodelofKam(2008),whichfeaturesthreesteadystates(nofoam,strongfoam,andanunstableintermediatestate)atsomeinjectionrates,asseeninexperiments.Wederiveandsolveequationsforwatersaturationandfoampropertiesalongtheentranceregionatsteadystate.Mathematicalconditionsontheentranceregionitselfcancontrolwhichoftheseveralpossiblesteadystatesisultimatelytakendownstreambyfoam.Forinstance,iffoamisnotpre-generated,andcapillary-pressuregradientsareneglected,asinmanypublishedsimulationstudies,thefinalsteadystatedownstreamistheonewithhighestwatersaturation–theweakestfoam.Simulationsneglectingcapillarypressurethereforemayleadtoinferenceofthewrongfoamstateintheformationorcore.Insomecases,inthepresenceofcapillarypressure,analysisoftheasymptoticdynamicbehaviorinthevicinityofpossibledownstreamsteadystatesmayruleoutsomepossiblesteadystates.

Weshowthattheapparentlengthofentranceregioncanbequitedifferentifonemeasureswatersaturationorpressuregradient.Finally,wefitfoamkineticparameterstothelengthoftheentranceregionseeninsomeexperiments;acompanionpaper[18]investigatestheeffectoftheseparametersonthetravelingwaveattheshockfrontdownstream.

© 2011 Elsevier B.V. All rights reserved.

Articlehistory:

Received30September2010Accepted30December2010

Available online 13 January 2011Keywords:

EnhancedoilrecoveryFoaminporousmediaEntranceregionCapillarypressure

Populationbalancefoammodel

1.Introduction

Foamisagglomerationofgasbubblesseparatedfromeachotherbythinliquidfilmscalledlamellae[1].Foamcanimprovethesweepefficiencyofinjectedgasesforenhancedoilrecov-erybymitigatingorreducingtheeffectsoflowgasviscosityandreservoirlayering.Thepredictionoffoamperformancereliesonmodeling.Population-balancemodels[2–7]andlocal-equilibriummodels[8–13]arethemainfoam-modelingapproaches.Withsomeadditionalassumptions,aone-dimensionaldisplacementwithlocal-equilibriummodels(LE)canlenditselftoanalysisbyfractional-flowmethods,anapplicationofthemethodofcharac-teristics[14–17].

Inthepopulation-balanceapproach,foamtexture(inverselyrelatedtobubblesize)ismodeledexplicitly,usingabalanceequationforthelamellaethatseparatebubbles.Thisequationis

∗Correspondingauthorat:DepartmentofGeotechnology,Stevinweg1,2628CN,Delft,Room:2.120Netherlands.Tel.:+31152789248;fax:+31152781189.

E-mailaddresses:e.ashoori@tudelft.nl(E.Ashoori),marchesi@impa.br(D.Marchesin),w.r.rossen@tudelft.nl(W.R.Rossen).

0927-7757/$–seefrontmatter© 2011 Elsevier B.V. All rights reserved.doi:10.1016/j.colsurfa.2010.12.043

similartothemassbalanceforsurfactantorwater.Themech-anismsforlamellaecreationanddestructionarerepresentedexplicitlyinthebalanceequationonbubbletexture.Analterna-tivetopopulation-balancemodelingistoassumelocalequilibrium(LE)(equalgenerationanddestructionrates)atalllocationsintheformation.Inthisapproach,foamtexturecanberepresentedexplicitlyorimplicitly,forinstanceinagas-mobility-reductionfactor.

Thefractional-flowmethodbasedonLEmodelsisanapproachthatprovidesusefulinsightsandeaseofuse.Thisapproachincludessomesimplifyingassumptions:incompressiblephases;Newtonianmobilities;one-dimensionalflow;absenceofdispersion,gradientofcapillarypressure,andviscousfingering;andimmediateattain-mentofLE.

Rossenetal.[15]showthat,insomecases,population-balancesimulationscanbemodelednearlyequivalentlywithfractional-flowmethodsthatassumeLE.Chenetal.[7]showexampleswheresimulationassumingLEcloselymatchesasimulationwithfullpop-ulationbalance.Inallthesestudiesthemainmismatchbetweenthetwoapproacheshappensattheveryentranceofthecore.

WeassumethatonasufficientlylargescalefoamiseverywhereatLEwithtwoexceptions:thefirstexceptionisshockfronts,where

218E.Ashoorietal./ColloidsandSurfacesA:Physicochem.Eng.Aspects377 (2011) 217–227

Nomenclature

Forequationsinthistext,itisonlyimportantthataconsistentsetofunitsbeemployed.WegiveSIunitsbelow.ForappendicesAandB,themodelequationsassumecertainunitsasspecifiedthere.Afoamparameterinfirst-order-kineticfoammodelcparameterinEq.(A.7)CcfoamparameterinKammodel,(m−3s−1)CffoamparameterinKammodel

Cg

foamparameterinKammodel,(m−3s−1)Einjectionpointfwwaterfractionalflowfunction(excluding

capillarity-drivenflow)

IinitialstateJLEstatedownstreamoftheentranceregionKcfoamparameterinfirst-order-kineticfoammodel,

(s−1)

k0rggasrelativepermeabilityintheabsenceoffoam

kfrggasrelativepermeabilityinpresenceoffoamkrwwaterrelativepermeability

MRF

foamparameterinfirst-order-kineticfoammodel(mobilityreductionfactor)nfoamparameterinKammodel

nDdimensionlessfoamtexture(=nf/nmax)

nLEdimensionlesslocal-equilibriumfoamtexture

nDffoamtexture(numberoflamellaeperunitvolume),(m−3)

nLEflocal-equilibriumfoamtexture,(m−3)nmaximumfoamtexture,(m−3)

∇max∇ppressuregradientinKammodel,(psift−1)p0foamparameterinKammodel,(psift−1)Pcgas–watercapillarypressure,(Pa)rcfoamcoalescencefunction,(m−3s−1)rgfoamgenerationfunction,(m−3s−1)Sggassaturation

Sgrresidualgassaturationinawater-gassystemSwwatersaturation

Swcconnatewatersaturationinawater-gassystemS∗wlimitingwatersaturationttimes

utotalsuperficialvelocity,(m−3s−1)uggassuperficialvelocity,(m−3s−1)uw

watersuperficialvelocity,(m−3s−1)

vdimensionlessinterstitialvelocity,(m−3s−1)

vs

slopeoftheshocklineonthefractional-flowdia-gram

x

standingcoordinatefordisplacementin1-D,(m)

SuperscriptsandsubscriptsEinjectionpointIinitialstateJlocal-equilibriumstatedownstreamoftheentrance

region

−upstreamoftheshock+downstreamoftheshockGreekSymbolsεsmallvalue(seeAppendixC)Ámovingcoordinate(withshockvelocity)fordis-placementin1-D,(m)󰀆porosity󰀁1,󰀁2eigenvectors󰀃1,󰀃2eigenvalues

󰀃gmobilityofgas,(m2(Pas)−1)󰀃wmobilityofwater,(m2(Pas)−1)

󰀄fggasviscosityinpresenceoffoam,(Pas)󰀄0ggasviscosityinabsenceoffoam,(Pas)󰀄wviscositiesofwater,(Pas)

󰀅

gas–waterinterfacialtension,(Nm−1)

foamtexturechangesabruptly;see[18].Inthispaperwefocus

onthesecond,theentranceregionnearthewellorinjectionfaceofacoreflood,wheregas,waterandsurfactantcreatefoam,orwherefinelytexturedpre-generatedfoammaycometoacoarsertexture.

Therehavebeenanumberofexperimentalstudiesdesignedtoinvestigatewhetherandhowfoamgenerationbeforeinjectionaffectsfoamformationandpropagationinacore.Withgreatcare,Fallsetal.[19]wereabletopreventfoamtexturechangingoveradistanceof60cminabeadpack,using3mmbeads.Somestudiesfindthattheporousmediumcreatesorreshapesthefoamoverarelativelyshortdistance,regardlessofthestateoftheinjectedfluids[10,12,20,21].

Ifthereisnofoamgenerationupstreamofthecore,theentranceregionmaybeobservedasaregionofgreaterwatersaturationSw,neartheentrance,measuredforinstancebyCTimaging[6,7,22,23],oraregionofsmallerpressuregradient󰀁pthanthatdownstream[6,12,19,20,22–28].However,fewexperimentaldataon󰀁pareavailablewithsufficientresolutiontoresolvepressuregradientasafunctionofpositionwithinthisregion.

MyersandRadke[26]contendthatmanyearlierfoam-flowexperimentswereconductedinshortcoresandsuspectedthatsuchexperimentswouldbedominatedbytheentranceeffect.Theyreportanentranceregionaslongas15cminsomeexperi-ments.Minssieux[22]alsoobservedasignificantwater-saturationentranceregionextendingforapproximately10cm.SomestudiesfindnoentranceregionobservableinSw(oronetooshorttomea-sure)[11,21].Incaseswherebotharemeasuredtheentranceregioncanbethesame[7]ordifferent[23,29]asreflectedin󰀁p,whichreflectsbubbletexture,andSw.FriedmannandJensen[20]reportthatthenatureoftheinjectedfoamaffectsfoampropertiesdown-streamoftheentranceregion.Thelengthoftheentranceregiondependsonmanyfactors,includinggastype,injectionvelocities,surfactanttypeandconcentration,andpresenceofoil,amongoth-ers.Duetotheexistenceofweakerfoaminthepresenceofmanyhydrocarbons,theentranceregionislongerwithoilpresentandtheremaybenosharpfoamfront,dependingonoilconcentration[26].Therearealsosomecontroversialresultsontheeffectofgastypeontheentranceregion.Farajzadehetal.[27]foundashorterentranceregionforCO2foam(around4cm)comparedwithN2foam(around6cm)forthesameexperimentalconditions,whileChou[24]reportfewerporevolumesoffoaminjectionrequiredtocreateN2foamratherthanCO2foam.

Inthisworkweassumethereisanentranceregionthroughwhichtheinjectedfluidscometolocalequilibriumbetweenfoamgenerationanddestruction.Inprincipleapregeneratedstrongfoamcouldcoarsenintheentranceregion;here,however,wefocusoncreationoffoamintheentranceregionfromgasandwater,withoutpre-generationoffoambeforeinjection.

Theissueofasteady-stateentranceregionisseparatefromthatofatimedelayinreachingsteadystate.Inmanyexperimentslargepressuredropsarenotobserveduntilmanyporevolumesoffluidhavebeeninjected[11,24,30–32]thisproblemcanbepronouncedforCO2foam[27].Chou[24]reportsfewerporevolumesofinjec-tionneededtocreatestrongfoamthanweakfoam.Hereweassumethatsteady-stateisachievedinthecore,andsolvefortheentrance

E.Ashoorietal./ColloidsandSurfacesA:Physicochem.Eng.Aspects377 (2011) 217–227219

regionatsteadystate.Whilethedelayinreachingsteadystatemaybelongonthelaboratoryscale,weexpectittoberelativelyshortonthefieldscale.

Wesolveforsteady-statewatersaturationandfoamtextureintheentranceregionaccountingforfoamkineticsemployingtwodifferentfoammodels:asimpleschematicfirst-order-kineticmodelandamore-realisticpopulation-balancemodelwithmulti-plesteadystates[5].Weshowhowconsiderationoftheentranceregioncanruleoutsomecandidatesforthestatedownstreamoftheentranceregion.TocomplementthetravelingwavesolutionworkdonebyAshoorietal.[18],wealsoexaminesensitivityoftheentranceregiontothekineticparametersinthemodelandfitthoseparameterstothelengthoftheentranceregiontypicalofcorefloodexperiments.

2.Entranceregioninfoamprocesses

Atsteadystate,nomobileoilispresentintheentranceregion,andsurfactantconcentrationisuniform.Animmiscibletwo-phase(gas–water),incompressibledisplacementinrectilinearflowthroughaporousmediumisgovernedbytheRapoport–Leasequa-tion[33,34]:∂S󰀆

󰀇

ϕw∂fw∂󰀃w󰀃gdPc∂t+u∂x+∂x

󰀃w+󰀃g∂x

=0(1)

whereSwandfwarewatersaturationandfractionalflow,respec-tively,ϕisporosity,utotalsuperficialvelocity,xposition,ttime,and󰀃wand󰀃garemobilitiesofwaterandgas.ThesecondandthirdtermsinEq.(1)representconvectiveandcapillarity-drivenwaterflow,respectively.Sincesurfactantispresentinuniformconcentra-tionintheaqueousphase,noseparatematerialbalanceisneededforsurfactant.

Infoamprocesses,propertiessuchas󰀃gandfwdependnotonlyonwatersaturationbutalsoonbubblesizeorfoamtexturenf,definedasthenumberoflamellaeperunitvolumeofgasphase.Modelingfoamprocessesthereforerequires,inadditiontothewatermass-balanceEq.(1),aso-calledpopulation-balanceequa-tionforfoamtexture:ϕ

∂󰀖∂tS󰀁∂󰀖󰀁ϕ

󰀂󰀃gnD+∂x

nDug=nSgrg−rc

(2)

maxwherenD≡nf/nmaxisthedimensionlessbubbletextureandnmaxis

theupperlimittofoamtexture(reflectingalowerlimittobubblesize)[35].Intherestofthispaperweworkwithdimensionlessfoamtextureratherthannf.

Becausefoamisnotpre-generated,butrathergasandwater(withdissolvedsurfactant)areco-injected,thereisanentranceregioninwhichfoamgainsitsultimateLEtexturecorrespondingtotheinjectedwaterfractionalflow.Inthisregion,watersat-urationisrelativelyhighandpressuregradientlow.Overtime,theentranceregionreachessteadystate,i.e.water-saturationandbubble-textureprofilesdonotchangefurther.

Fig.1showsaschematicdiagramofthesteady-stateentranceregionandofthedownstreamshockfrontforagivenfoamdis-placementandfoamfractional-flowcurve.Inthisfigure,thegraydashedcurverepresentsfoam-freewaterfractionalflowandthethickblackcurvewaterfractionalflowinthepresenceoffoamatlocalequilibrium(LE).Thethinblackcurvesarefractional-flowcurvesatconstantbubbletexture.Theentranceregion,inthegraysolidrectangleinthebottomfigure,startsatpointEatpositionx=0(topfigure),withzerobubbletexture;asxincreasesthefoamasymptoticallyapproacheslibriumandSw=SJ

stateJ,withbubbletextureatlocalequi-w.Becausefoamtextureiszeroattheinlet,pointEmustlieonthefoam-freefractionalflowcurvewheretotalwater

fractionalflow(includingthecapillarycontribution)equalsfJ

w.In

absenceofcapillarypressure,asshownbelowthewholesteady-stateentranceregionliesonthehorizontaldashedlinefw=fJ

wstartingfrompointE󰀈onfoam-freefractional-flowcurvetopointJonfoamfractional-flowcurve.InthisfigurestatesJandIdenotestatesupstreamanddownstreamoftheshockintroducedbyfoaminjection,respectively,shownbythegreendottedlineandgreenrectangleinFig.1.Generally,therecouldbeotherwavesbetweenJandtheshock;iftherewereaspreadingwaveatJitmustlieontheLEfoamfractional-flowcurve.WhetherthereisaconstantstateatJoraspreadingwave,thegradientofsaturationandotherprop-ertiesdownstreamoftheentranceregionarenegligiblecomparedtothosewithintheentranceregion.Therefore,forthepurposesoftheentranceregion,wecantreatJasaregionofconstantstate.Inthecasesweexamine,thereisashockdirectlyfromJtoIandJisindeedaregionofconstantstate.

ThesolutionfrompointEtostateJ,discussedinthispaper,representsthesteady-stateentranceregion.ThetravelingwaveorshockbetweenstatesJandIisthesubjectofAshoorietal.[18].

Atsteadystate,thetime-derivativetermdisappearsfromEq.(1):d󰀆

󰀇

uf󰀃w+

w󰀃gdPc

dx

󰀃w+󰀃g∂x

=0(3)

ThesumofthetwotermsintheparenthesisinEq.(3)representsthetotalwatersuperficialvelocityuw,includingbothconvectiveandcapillarity-drivenflow.Eq.(3)impliesthatintheentranceregionatsteadystate,waterandgassuperficialvelocitiesuwandugareeachuniformthroughouttheregion.ThetimederivativealsodisappearsfromEq.(2);usingtheuniformityofug,wecantakeugoutofthespatialderivative:udnDϕ

󰀂g

dx

=

nS󰀃grg−rc(4)

maxTheleft-handsideofEq.(4)arisesfromtransportoflamellae.AccordingtoEq.(4),ifugvanishes,rg−rc=0,soweareatsteadystate,evenattheinlet.Inotherwords,thereisnoentranceregionwithouttransportoflamellae.“BreakandReformTheory”ascur-rentlyimplemented[36,37],inwhichlamellaerepeatedlybreakandreforminplace,isnotconsistentwithexperimentalobserva-tionofanentranceregioninfoamcorefloods.

TheentranceregionmustsatisfyEqs.(3)and(4)andthefollow-ingtwoboundaryconditionsatstatesEandJ:

󰀋

B.C.Ex=0

nED

=0B.C.J

x→+∞SdSW→SJ

J

w

dnD

W,nD→nD,

dx

→0,dx

→0(5)

Thefirstboundaryconditionspecifieszerobubbletextureattheinletbecausethereisnopre-generationoffoam.Iffoamwerepre-generated,thenthegiveninjectedfoamtexturewouldbespecifiedinB.C.E.Thesecondboundaryconditionspecifiesthatwatersatura-tionandfoamtextureasymptoticallyapproachLEvaluesattheendofentranceregion.AccordingtoB.C.J,theentranceregionrequires“infinite”distancetoapproachstateJ,buttheobservableentranceregionisfinite.InthispaperwereporttheapparentwidthoftheentranceregioninSwasthedistanceoverwhichSwgoes95%ofthewayfromEtoJ.TheapparentwidthinnDisdefinedsimilarly.Thewidthoftheentranceregioncanbequitedifferentdependingonwhichmeasuredpropertyisusedinthedefinition.

Thecapillarity-drivenpartofwaterorgassuperficialveloc-itydisappearsatthedownstreamboundarycondition,because,accordingtoB.C.J,gradientsofsaturationapproachzeroasthisstateisapproached;therefore,attheendofaccordingtoEq.(3),uJJtheentranceJ

region,

w=ufw.ThedownstreamstateSwmustbeconsistentwiththewaterfractionintheinjectedfluids.Because

220E.Ashoorietal./ColloidsandSurfacesA:Physicochem.Eng.Aspects377 (2011) 217–227

Fig.1.Schematicoffoamsteady-stateentranceregion(graysolidrectangleinbottomfigure,lightgraycurveintopfigure)andthefollowingshockfront(dashedrectangleinbottomfigure,dashedlineintopfigure)correspondingstatestoJandIonthefoamfractionalflowcurve.Thin-blackfractional-flowcurvesinbetweenareatconstantbubbletexture;numbersoncurvesaredimensionlessbubble-texture.Solidthickblackcurveintopfigurerepresentssteady-stateorlocal-equilibriumfractional-flowcurve,wherebubbletextureisafunctionofwatersaturation.PointEliesonthe(dashed)foam-freefractionalflowcurve(representinginjectionofgasandsurfactant,with,asyet,

J

nofoam)wheretotalwaterfractionalflowincludingthecapillarycontributionequalsfw.Inabsenceofcapillarypressure,pointE󰀈isthestartingpointofentranceregionand

J

thewholesolutionliesonthehorizontaldashedlinefw=fwendingatstateJ.Thelightgraycolorinthebottomplotrepresentswatersaturationalongtheflowdirection.TheinitialstateIisnotimportantinthecontextofthispaper.

capillary-pressuregradientscanbesignificantattheinlet,however,andfwbydefinitionexcludescapillarity-drivenflow,ingeneralattheinletfwisnotequaltothefractionofwaterintheinjectedfluids.

Onasufficientlylargescale,oneneglectstheentranceregion,andtheconditionspecifiedinB.C.Jrepresentstheso-called“injec-tioncondition”Jforthedisplacement.Inanexperimentorfield

J

application,however,onecontrolsinjectionrate,uw(equivalentto

J

.Ifmultiplesteadystatesarepossible,onedoesfixingfw)andnED

notknowinadvancewhichLEstateJwouldformdownstreamoftheentranceregion.Mathematically,weassumeprovisionallya

J

particularLEstateJ(definedbySw)anddetermineifitisconsis-J

andfw.InsometentwithanentranceregionconstrainedbynED

casesthisanalysisrulesoutsomecandidatesteadystatesforJ.Onothercases,morethanonestateJmaybeconsistentwiththegiveninjectionconditions.

Eq.(3)statesthattotalwatersuperficialvelocity(includingthecapillarycontribution)doesnotvarywithxwithintheentranceregion.Consequently,totalsuperficialvelocityuwintheentrance

J

regionisconstant,equaltoufw:

thetotalgassuperficialvelocityug:

󰀂󰀃dnDϕ

S=r−rggcJdxnmaxu(1−fw)

(7)

Oneshouldkeepinmindthatnotonlydorgandrcdependon

SwandnD,butalsofwand󰀃g.SolvingEq.(7)alongwithEq.(6)requiresfoammodels,whicharediscussedbelow.Firstweshowtheprocedureusedtosolvefortheentranceregionforthetwocases,excludingandincludingcapillarypressure.2.1.Excludingcapillarypressure

Inabsenceofcapillarypressure,thediffusivepartinEq.(6)disappears:fw(Sw,nD)=fw

J

(8)

󰀆

󰀃w󰀃gdPcdSw

ufw+

󰀃w+󰀃gdSwdx

󰀇

=ufw

J

(6)

Asuwisconstantintheentranceregion,soisug.Thus,Eq.(4)canbesimplifiedbysubstitutingtheconstantvalueu(1−fw)for

J

TheentranceregionsatisfiesEq.(7),Eq.(8)andboundarycon-ditionsinEq.(5).Eq.(8)alongwithB.C.EmeansthatthestartingpointEoftheentranceregioncorrespondstothepointonthefoam-J

freefractionalflowcurvewherefw=fw.AccordingtoFig.1,intheentranceregionfwinthiscasefollowsthethinhorizontalgray

J

dashedlinecorrespondingtofwfromE󰀈toJ.AsnDvanishesatthe

EcanbeobtainedinletpointE,bysubstitutingnD=0intoEq.(8),Sw

E,0)=fJ.bysolvingfw(Sww

TheendpointJoftheentranceregionmustlieontheLEfractional-flowcurvesatisfying[rg−rc]=0withtheinjectedfrac-tionalflowgivenbyfw.Thisnonlinearsystemofequationscanhave

J

E.Ashoorietal./ColloidsandSurfacesA:Physicochem.Eng.Aspects377 (2011) 217–227221

multiplesolutionsforgivenfJ

w;see,e.g.,Fig.4below.Ifso,webeginbyassumingoneofthosestatesJ.

ThebeginningstateEandtheendstateJoftheentranceregionarenowknown.ForanyfixedvalueoffoamtexturewaterfractionalflowismonotonicallyincreasingwithSw(Fig.1).Eq.(8)thereforedeterminesnDuniquelyintermsofSw(callitnD=nD(Sw))whichsatisfies

fw(Sw,nD(Sw))−fJ

w=0

(9)

DifferentiatingEq.(9)withrespecttoSw,oneobtainsdSw/dnD

solelyasafunctionofSw:dS󰀆

w

∂fw∂f󰀇−1

w

dn(10)

D=−∂nD

∂Sw

wheretheright-handsideiscalculatedat(Sw,nD(Sw))inEq.(9).MultiplyingbothsidesofEq.(7)bydSw/dnDleadsto:

dSwdx=ϕdSw

un(1−S󰀂󰀃w)rg−rc(11)

max(1−fJw)dnD

wherethetermontherighthandsideisevaluatedat(Sw,nD(Sw))definedinEq.(9).

Thissingleordinarydifferentialequation(ODE)intheunknown

SwissolvedstartingfromSw=SEw

atx=0;theright-handsidemustbelessthanzero,anditcontinuestobelessthanzerountilstateJ,

where[rg−rc]vanishes.AtthispointSwisthewatersaturationSJ

wattheendofentranceregion.WewillsoonseethatJisthesolutionwithhighestwatersaturationamongallcandidatesteadystatesnomatterwhatthefoamkineticratesare.

Usingnumericalpackages,itiseasytosolveEq.(11).2.2.Includingcapillarypressure

Inthiscasethetotalwatersuperficialvelocityincludescon-vectiveandcapillarityterms,butitisstilluniformintheentranceregionaccordingtoEq.(6).RearrangingthisequationtransformsitintoanODEforwatersaturationasafunctionofposition:dSwu(fJ

w

−fwdx=)

f(12)

w󰀃g(dPc/dSw)

ThisequationhastobesolvedtogetherwithEq.(7)andbound-aryconditionsinEq.(5).

Contrarytothepreviouscase,theinletwatersaturationSEw

cannotbecalculateddirectlyfromfJ

w,becausewatersuperficialvelocityincludesacapillarycontribution.However,thiscontribu-tionvanishesatJfromB.C.JinEq.(5),socandidatesforstateJarecalculatedfromEq.(9)asbefore.

TheentranceregionmustsatisfyODEs(7)and(12),butthestateEisstillunknown.SincetheremaybemorethanonecandidateforJ,onemustintegratebackwardsinxfromeachcandidateforJ.SinceeachJisatequilibriumforbothequations(i.e.,theirrighthandsidesvanish),onedoesnotstartatJ,butatJ+󰀉JnearJ,asspecifiedinAppendixC.

Therearetwoissues:thefirstoneisthechoiceofpointJ+󰀉Jfromwhichtostartthebackwardintegration.Thischoicedependsonthebehavioroftheorbits(solutionpaths)oftheODEsneartheequilibriumpointJ.AsexplainedinAppendixC,suchanequilibriumcanbeofthreegenerictypes:saddle,attractorandrepeller.Inthecaseofasaddle,thereisanessentiallyunique,well-definedwaytoinitiatethebackwardintegration.Inthecaseofanattractor,onecannotintegratebackwards;oneimmediatelyconcludesthatthecandidateforJisnotacceptable.Inotherwords,itisimpossibletoachievethisstatefromfoam-freefluidsinjectedattheinlet.Inthecaseofarepeller,asolutionmayexistinprinciple,butdeterminingitismoredifficult;wedonotaddressthiscasehere.

ThesecondissueisthedeterminationofstateE,whereinte-grationstops.WeintegratebackwardsthesystemofODEsfromJ+󰀉J,startingatanarbitrarypositionx=xJ+󰀉Jthatwesetinitiallytozero.Asoneintegratesbackwardsinx,thevalueofnDdecreases,andtherearetwopossibilities:eithernDreacheszero(atthevalueofxthatwecallxE)atstateE,orelsenDstayspositive.IntheformercasewehavesatisfiedB.C.EinEq.(5)andreachedstateEandthewidthoftheentranceregionisthedifferencexE−xJ+󰀉J.Inthelattercase,thecandidateJisruledout.3.Foammodels

Wesolvetheentranceregionfortwofoammodels.Thefirstonecombinesfamiliarsteady-statefoambehavior(cf.[13,14])withafirst-order-kineticmodelfortheapproachoffoamtexturetoitsLEvalueatagivenwatersaturation.ThedetailsofthismodelaredescribedinAppendixA;Fig.1isbasedonthismodel.ThesecondmodelisthatofKam[5],describedinAppendixB,whichiscom-patiblewithmultiplefoamsteadystatesandfoamgenerationatathresholdpressuregradient,asseeninexperiments.AppendixCdescribesthemathematicalapproachforsolvingtheentrance-regionEqs.(7)and(12).4.Results

4.1.First-order-kineticmodel

WeinvestigatethecaseofinjectingfoamforfJ

w=0.268intoacorewithgasandsurfactantalreadypresent;weusethemodelandparametersthatareconsistentwiththetraveling-waveexampleinAshoorietal.[18].ForthesakeofcompletenessthefoammodelissummarizedinAppendixForknownfJA.

JmustsatisfyEq.(8):fw(SJJJ

wthestatew,nLE(Sw))=fw.Inthisequation,nLEisgivenbyEq.(A.1)inAppendixDA.Wefind(Sw,nD,fw)J=(0.372,D

0.664,0.268).Thereisonlyonefoamlocal

equilibriumforeachvalueoffwinthismodel;seeFig.1.

4.1.1.ExcludingForfJcapillarypressure

0.268,fromEq.(8)theequalityfw(SE,0)=fJSEw=givesw=0.082.SolvingEq.(9)withfww

wgivenin(A.6)wefindnD(Sw)asbelow:nk0gw

wD(Sw)=

r(S)󰀄krw(Sw)󰀄g(1/fJ

w−1)

(13)

HereweuseEqs.(A.4)and(A.5)forrelativepermeabilities.Sub-stituting[rg-rc]fromEq.(A.2),nD(Sw)anddSw/dnDfromEq.(13)intoEq.(11)producesthesoleODEgoverningtheentranceregioninabsenceofcapillarypressure.TheresultingODEcanbesolvedbytheRunge-KuttamethodusingMATLAB®(TheMathworks,Natick,

Massachusetts,USA)forwatersaturationbetweenSEwandSJw

,start-ingfromSEatx=0(B.C.EinEq.(5)).NotethatforagivenfJww,the

statesatthetwoendsoftheentranceregionarefoundindependentoffoamkineticrates.

Forfastkineticrates(e.g.Kc=200)theentranceregionhasverysmallwidth(oforderhalfofamicron)inSw.Forrelativelyslowerkineticrate(e.g.Kc=1)thewidthisoftheorderofone-tenthofamillimeter,andforKc=0.01,theentranceregioniswider.Forinstance,Fig.2showswatersaturationanddimensionlessbubbletexturevs.xintheentranceregionforKc=0.01.Theentranceregion

startsatx=0withzerobubbletextureandSEw

=0.0.82(thewatersaturationonthefoam-freefractionalflowcurveatfJ

w=0.approachestheLEstateJcorrespondingtothesamefJ

268)and

wonthefoamfractionalflowcurve(seethehorizontalgraydashedlineinFig.1).Foambubbledensityiszeroattheinlet,wheregasmobilityisthatintheabsenceoffoam,and,consequently,Swishighertherethan

222E.Ashoorietal./ColloidsandSurfacesA:Physicochem.Eng.Aspects377 (2011) 217–227

0.8wS0.6 0.4051015 x (mm)

202530351Dn 0.5005101520253035 x (mm)

Fig.2.Watersaturationanddimensionlessbubbletexturevs.xwithintheentrance

regionforfJ

w=0.268,applyingthefirst-order-kineticfoammodel(Kc=0.01)inabsenceofcapillarypressure.

intherestofthecore.Ourresultsshowthatentrance-regionwidthisinverselyproportionaltoKcbecausethederivativeofSwvs.xintheentranceregionisproportionaltothenetfoam-generationrate,accordingtoEq.(11).Weseethesametrendfortheotherfoammodel.

BycomparingtheentranceregioninSw,observedforinstancebyCTscanning,orinpressuregradient,whichreflectsnD,onecanroughlycalibratethefoamparameterKc.NotethatinthiscasetheapparententranceregioninSwwouldbeabout7mmandforpres-suregradient(ornD)abouttwotimeslonger,14mm.Iftheentranceregionisoforderofcm,asinmanyexperiments,thenclearlyavalueofKcgreaterthan1isnotconsistentwithexperimentaldata.Ashoorietal.[18]showthatfortherangeofKcbetween1and0.1,thetravelingwaveatthedownstreamshockfront(seeFig.1)oscillatesaroundtheinjectionpoint,butsuchalargevalueofKcisnotconsistentwiththeentranceregionobservableinexperi-ments.

4.1.2.Includingcapillarypressure

Inserting[rg-rc]fromEq.(A.2)intoEq.(7)andfwfromEq.(A.6)intoEq.(12)resultsintwoODEsthatshouldbesolvedtogetherwithboundaryconditionsspecifiedinEq.(5)ForthecasestudydefinedabovewithfJ

.

w=0.268,thepresenceofcapillarypressuredoesnotchangestateJ.Asdiscussedabove,stateEisnotknowninadvance.Thus,onemustsolvethesystemofODEsbackwardstartingneartheequilibriumstateJtothepointatwhichbubbletextureiszero,whichisstateE.

Table1showstheasymptoticbehaviorofthesolutionaroundJforsmallasmallvalueofKc,0.01.Inthistable󰀃1and󰀃2areeigenvaluesand󰀁1and󰀁2areeigenvectors.Theinjectionpointisasaddle.SeeAppendixCfortheprocedureofsolvingasystemofODEsstartingnearasaddlepoint.

Fig.3showstheentranceregionforthiscase.Theinletwater

saturation(SEw

)isabout0.55andthepractical(apparent)entranceregionwidthisaround1.4cmbothinSwandnD.ComparingFigs.2and3showsthatentranceregioninSwiswiderintheabsenceofcapillarypressure,inagreementwithKovsceketal.[6]foranotherfoammodel.ForlargervaluesofKc(notshown),theinletwatersaturationislowerandtheentranceregionisshorter.

Table1

EntranceregionbehavioraroundstateJ,(Sw,nD,fw)J=(0.372,0.664,0.268),applyingthefirst-kineticfoammodelinpresenceofcapillarypressure(Kc=0.01).󰀃1

󰀃2

󰀁1

󰀁2

−936.61033

21908.98651

(−0.05269,0.99861)

(0.80216,0.59710)

0.6w0.5S 0.40.30510 x (mm)1520251Dn 0.500510x 15(mm)2025Fig.3.Watersaturationanddimensionlessbubbletexturevs.xwithintheentrance

regionforfJ

w=0.268applyingthefirst-kineticfoammodel(Kc=0.01)inpresenceofcapillarypressure.

4.2.ModelofKam

Kam[5]proposeamodificationofthefoammodelofKamandRossen[38],whichfitsavarietyofsteady-stateexperimen-taldata:foamgenerationaboveathresholdpressuregradient[39],multiplefoamsteadystatesatgiveninjectionrates[40],andtwoflowregimesforstrongfoamdependingoninjectedwaterfrac-tion[35].DetailsanddefinitionsofallthevariablesandparametersofthismodelareavailableinAppendixB.Amongotherfactors,thelocal-equilibriumbehaviordependsontheratiooftwokineticparameters,CgandCc.Forgivensteady-statebehavior,withtheratiofixed,kineticratesaregovernedbythevalueofeitherparam-eter;belowweshowthebehaviorasafunctionoftheparameterCc.

4.2.1.Highvelocityco-injection

AccordingtoEqs.(B.3)and(B.5),thefractional-flowcurveforthisfoammodeldependsstronglyontotalsuperficialvelocityu.Initiallyweassumerelativelyhigh-velocityco-injectionofgasandwaterwithdissolvedsurfactant(u=5.29×10−5m/s,or4.57m/d).Fig.4showsthefoamfractional-flowcurvesatthishighveloc-ityandthefoam-freefractional-flowcurve,whichcorrespondsto

nD=0.WeassumefJ

w=0.05forthefirstexample,consistentwithacaseinAshoorietal.[18]andKam[5].Asaninitialcasewespec-

10.810.50.30.10.6w0.01f 0.0050.40.00100.2 LE foam fw foam-free fw fJw constant-n0D foam fw0.10.20.30.40.50.60.70.80.91 Sw

Fig.4.LEfractionalflowcurvesatu=5.29×10−5m/s(4.57m/d):foam(thicksolidcurve),withmultiplesteadystatespossible,andfoam-freefractional-flowcurve(thickdashedcurve).Thethinblackcurvesrepresentfractionalflowatconstantbubbletexture;numbersoncurvesaredimensionlessbubbletexture.TheLEfractional-flowcurveforfoamhasfoamtexturethatvarieswithwatersaturation.Horizontallineindicatesinjectionatwaterfractionalflowof0.05andthegreen,squarepointsareLEcandidatesforstateJ.

E.Ashoorietal./ColloidsandSurfacesA:Physicochem.Eng.Aspects377 (2011) 217–227223

0.7wS 0.50.301234 5678910x (mm)0.001Dn0.0005 0

012345678910x (mm)Fig.5.Watersaturationanddimensionlessbubbletexturevs.xinentranceregion

forfJ

w=0.05inabsenceofcapillarypressure(Cc=1).

ifycoalescenceparameterCc=1,whichmatchesthevalueofCcinsimulationsinKam[5].

4.2.1.1.Excludingcapillarypressure.ForfJ

w=0.05,wefindSEw

fromEq.(8),settingnD=0.TherearethreestateJatintersectionsoflinefw=fJ

possibleLEcandidatesfor

wandtheLEfoamfractional-flowcurve(seegreensquaresinFig.4).Eq.(8)indicatesthatfwisuniformthroughouttheentranceregion.Thereforethesolution

liesonthelinefw=fJ

wstartingfromtheno-foamfractionalflowcurvetothefirstequilibriumpointitencounters.Itisimpossibleforthesolutiontoextendbeyondthispoint,becausedSw/dx→0there(Eq.(11)).Inotherwords,intheabsenceofcapillarypres-sureandwithnopre-generationoffoam,foranyfoammodelwithmultiplesteadystates,thestatedownstreamfromentranceregion(J)istheonewiththehighestwatersaturationamongthepossi-blesteady-states.Intheabsenceofcapillary-pressuregradients,onecanidentifythesteady-stateinjectionpointJwithoutanysimulations,andindependentoffoam-kineticrate.Allthefinite-differencesimulationsofKametal.[4]intheabsenceofcapillary

pressureresultedinastateJthatobeysthisrule.ForknownfJ

w,statesEandJare(Sw,nD,fw)E=(0.62762,0,0.050000)and(Sw,nD,fw)J=(0.371342,0.00095022,0.050000).

WeconstructthesingleODEgoverningtheentranceregion:firstwedeterminedSw/dnDfromEq.(B.5)togetherwithEqs.(B.3)and(B.4)andthensubstitutergfromEq.(B.1),rcfromEq.(B.2)andthederiveddSw/dnDintoEq.(11).

Fig.5plotswatersaturationanddimensionlessbubblevs.positionxintheentranceregionforfJ

texture

w=0.05forCc=1.Accord-ingtothisfigure,theentranceregionstartsathighwatersaturationwithzerobubbletextureandasymptoticallyapproachesthefoam

LEstatewiththesamevalueoffJ

wandthegreatestvalueofSw.

Incontrasttothewidthoftravelingwave[18],theapparentwidthoftheentranceregionissensitivetofoamkineticrate.The

0.0592w0.059S 0.05880.058600.20.4 x 0.6(mm)0.811.21Dn 0.5000.20.4 x 0.6(mm)0.811.2Fig.6.Watersaturationanddimensionlessbubbletexturevs.positionxinentrance

regionforfJ

w=0.05andequilibriumJat(Sw,nD,fw)J=(0.058585,0.65218,0.05)inpresenceofcapillarypressure(Cc=0.001).

10.90.80.710.60.50.30.1wf 0.50.010.0050.40.001 fJ0w0.3 J0.2 LE foam fw0.1 foam-free fw constant-nD foam f0w0.10.20.30.40.50.60.70.80.91 Sw

Fig.7.LEFoamfractionalflowcurvesforu=2.798×10−5m/s(2.42m/d)(thicksolidcurve)andfoam-freefractional-flowcurve(thickdashedcurve);thethinblackcurvesinbetweenrepresentfractionalflowatfixedbubbletexture;numbersoncurvesaredimensionlessbubble-texture.Horizontaldashedlineindicatesinjectionatwaterfractionalflowof0.3.ThestateJcorrespondstotheLEinjectioncondition(seetheargumentinthetext).

entranceregionwidensalmostproportionallyto1/Cc:slowerfoamkineticratestendtoexpandtheentranceregion,andforfasterfoamkineticrates,theentranceregionshrinksalmostproportionally.Moreprecisely,theapparentwidthoftheentranceregionisabout0.15cm,0.16mand1.5mforthekineticparameterCc=1,0.01and0.001,respectively.

4.2.1.2.Includingcapillarypressure.Aswiththefirst-order-kineticfoammodel,thesystemofODEs(7)and(12)canberecastforthisfoammodelbysubstitutingtherespectiveexpressionsforrg,rcandfw(Eqs.(B.1)–(B.5)).

Asdiscussed,SEw

cannotbedeterminedbeforesolvingthesys-temofODEs.InthiscasetherearethreecandidatesforgivenfJ

stateJfor

w=0.05(seegreensquaresinFig.1).Exceptforthemid-dlestateatSw=0.1263,theothertwocandidatesaresaddles,asrevealedbytheasymptoticbehaviorofthesolutionnearthosepoints.(Wediscussthemiddlepointfurtherbelow.)WeassumeinitiallythatpointJis(Sw,nD,fw)J=(0.0585849073,0.65217855,0.0500000).Thisstateisconsistentwithfinite-differencesimula-tionforthecaseatfJ

w=0.05[5].

Theentranceregionisnarrowevenforslowfoamkineticrates.Forinstance,Fig.6showswatersaturationanddimensionlessbub-bletexturedistributionvs.xintheentranceregionatrelativelyslowkineticrates(Cc=0.001).NotethatalthoughthewidthoftheentranceregionissimilarforbothSwandnD,thechangeinSwissoslightthatitwouldnotbedetectableexperimentally.InsuchacaseonewouldobservenoentranceregioninSwand(depending

20000 Pc function (c=0.01) Pc function (c=0)1000cP 10000Pc 50000.70.8 S0.91w000.10.20.30.40.5 S0.60.70.80.91w

Fig.8.Capillary-pressurefunctionusedinfirst-order-kineticmodel(Eq.(A.7)).

224E.Ashoorietal./ColloidsandSurfacesA:Physicochem.Eng.Aspects377 (2011) 217–227

Table2

EntranceregioncorrespondingtotheJpoint,(Sw,nD,fw)J=(0.5339,0.0015,0.3),specifiedinFig.7:widthandinitialwatersaturationatEfordifferentfoamkineticrates.Anasterix*indicatesanentranceregionwithanundectablysmallchangeinSw.Cc

SEw

Apparent

Apparent

entrance-regionentrance-regionwidthinSw(cm)widthinnD(cm)NoPc10.8110.080.080.010.8117.57.50.0010.8118080WithPc10.534*0.030.010.544*30.001

0.63

50

50

ontheresolutionoftheexperimentalapparatus)anarrowoneinpressuregradient.

InsomecasestheasymptoticbehaviorofthesolutionaroundpossiblecandidatesforstateJcanruleoutoneormoreofthem.

Forexample,forfJ

w=0.05,theasymptoticbehaviornearstateJonthemiddlefoamfractionalflowcurve(Sw=0.1263,seeFig.4)showsthatforfoamkineticparameterCc=1,wehaveanattractorandforsmallerCc(0.001),arepellerwitheigenvalues󰀃1,2=8.48454±12.87156i.AsexplainedinAppendixC,arepellercannotbetheendpointofanentranceregion.Thus,onecanruleoutthiscandidateforstateJinthepresenceofcapillarypressureforsufficientlysmallvaluesofCc.

4.2.2.Lowvelocityco-injection

Fig.7showstheLEfractional-flowcurvesfortotalsuperficialvelocityu=2.798×10−5m/s,whichcorrespondsto2.42m/d.TherearethreepossiblestatesJforfw=0.3(seegreensquaresinFig.7).

Inabsenceofcapillarypressure,weshowabovethatstateJistheonewithgreatestwatersaturationamongthecandidates:(Sw,nD,fw)J=(0.5339,0.0015,0.3).StateEisfoundas(Sw,nD,fw)J=(0.811,0,0.3)onthecrossingpointoffw=0.3andno-foamfractional-flowcurve.Thewholeentranceregionliesonthelinefw=0.3betweenstatesEandJ.

Inthepresenceofcapillarypressure,theasymptoticbehaviorofthepossibleentranceregionaroundcandidatestatesJinFig.7showsthattheright-mostone(atSw=0.5339)isasaddlewhiletheothertwoareattractors.Theformerhasthepotentialofauniqueentrance-regionorbitendingatJ;theothertwomighthaveasolu-tionfortheentranceregionbutareruledoutbyconsiderationoftheshockaheadofstateJ.Ashoorietal.[18]reportthatforaninitialstateofhighwatersaturation,forCcbetween0.001and1,theright-mostJstate(Sw,nD,fw)J=(0.5339,0.0015,0.3)inFig.7istheonlystateforwhichtheycouldconstructashockdownstreamofthisstate.Thisfindingisconsistentwithfinite-differencesimulationsofKam[5].Asdiscussedbefore,stateEcanbedeterminedfromthe

Table3

Parametervaluesforfirst-order-kineticmodel.Swc0.2Sgr0.18

󰀄w0.001Pas󰀄0g0.00002Pask1×10−12m2ϕ0.25

u2.93×10−5m/snmax8×1013m−3S∗w0.37A

400

Table4

Relative-permeabilityandcapillary-pressurefunctions,modelparametersandsomepropertiesusedinfoammodelofKam[5].Relative-permeabilityfunctionskrw0.7888[(Sw−Swc)/(1−Swc−Sgr)]1.9575k0rg

[(1−Sw−Sgr)/(1−Swc−Sgr)]2.2868Capillary-pressurefunctionPc

󰀅(ϕ/k)0.5[(Sw−Swc)/(1−Swc−Sgr)]−0.2Foammodelparameters∇p04.2psi/ftn1

Cg/Cc3.6×1016

Cf1.535×10−16Cc

1

OtherparametersSwc0.04Sgr0.0

󰀄w0.001Pas󰀄0g0.00002Pask30.4×10−12m2ϕ0.31S∗w0.0585

nmax8×1013m−3󰀅

0.03N/m

backwardsolutionoftheentranceregionatthepointwherebubbletexturebecomeszero.

WesolvetheentranceregionforthelocalequilibriumJspecifiedinFig.7bothincludingandexcludingcapillarypressure,followingthemethodexplainedbefore.Table2summarizesthecalculatedwatersaturationattheinjectionface(E)andtheapparentwidthoftheentranceregionforeachcasefordifferentfoamkineticparam-eters.

Table2makesseveralpoints.First,thewidthoftheentranceregionpredictedforKam’sfoammodel,creatingeitherstrongfoam(section4.2.1.2)orweakfoam(Table2),ismuchshorterforCc=1thanisoftenreportedinexperimentalwork:fromaround10–15cm[6,12]tosmallervaluesofabout4–6cmdependingongastype[27].Inthepresenceofcapillarypressure,thevariationsinwatersaturationbetweenstatesEtoJaresosmallforCcbetween0.01and1thattheapparententranceregioninSwwouldbeunde-tectablebytypicalCTscanners.

Second,asthepreviouscase,theentranceregioniswider,withagreaterchangeinwatersaturation,intheabsenceofcapillarypres-sure.KovscekandRadke[41]reportthesametrendintheentranceregionfromsimulations.Oneexpectsmoreeffectofaccountingforcapillarypressurewhentheendstateoftheentranceregionisatlowwatersaturations,wherecapillarypressureasafunctionofwatersaturationisverysteep,orwhenfoamisweakerandviscouspressuregradientissmall.

Third,asinthecasewithhigherinjectionrate,accordingtoTable2,inabsenceofcapillarypressurethewidthoftheentranceregionisinverselyproportionaltoCc.Inpresenceofcapillarypres-surethisproportionalityislessquantitative.Thisisinagreementwiththesensitivityanalysisonfoamkineticrateparametersinthefinite-differencesimulationsofKam[5].5.Conclusions

WeassumethatonasufficientlylargescalefoamiseverywhereatLE,withtwoexceptions.Thefirstexceptionisshockfronts,seeAshoorietal.[18].Inthispaper,westudythesecondexception,whichistheentranceregionnearthewellorinjectionfaceinacoreflood.Wereachthefollowingconclusions.

•Weshowhowtoderiveandsolveequationsforwatersaturationandfoampropertiesalongtheentranceregionatsteadystate.Weshowthesolutionforsomecasesapplyingafirst-order-kineticfoammodelandafoammodelwithmultiplesteadystates.

E.Ashoorietal./ColloidsandSurfacesA:Physicochem.Eng.Aspects377 (2011) 217–227225

•Simplemathematicalconditionsontheentranceregionitselfcancontrolwhichoftheseveralpossiblesteadystatesfoamultimatelytakesdownstream.Forinstance,iffoamisnotpre-generated,andcapillary-pressuregradientsareneglected,asinmanypublishedsimulationstudies,thefinalsteadystatedown-streamistheonewithhighestwatersaturation–theweakestfoam.Inthepresenceofcapillarypressure,analysisoftheasymp-toticdynamicbehaviorinthevicinityofpossibledownstreamsteadystatesmayruleoutcandidatesinsomecases.

•Wefitfoamkineticparameterstothelengthoftheentranceregioninexperiments.Asinexperiments,theapparentwidthoftheentranceregioncandifferifonemeasureswatersaturationorpressuregradient.

•Inabsenceofcapillarypressure,thewidthoftheentranceregionisinverselyproportionaltofoamkineticrate;inpresenceofcap-illarypressurethisproportionalityisapproximatelyvalid.Acknowledgments

ThisworkwassupportedinpartbyagrantfromResearchCentreDelftEarth,aprogramofDelftUniversityofTechnology;CNPqunderGrants301564/2009-4,472923/2010-2,490707/2008-4;FAPERJunderGrantsE-26/110.972/2008,E-26/102.723/2008,E-26/112.220/2008,E-26/110.310/2007,E-26/112.112/2008.TheauthorsalsothankTUDelftforhospitalityduringthevisits.AppendixA.First-order-kineticmodel

Thisfoammodelreflectswell-knownsteady-statebehavioroffoaminporousmedia:specificallyalarge,nearlyconstantgasmobilityreductionathighwatersaturationsandanabruptweak-eningorcollapseoffoamatalimitingcapillarypressureorlimitingwatersaturation[13,14,42,43].Local-equilibriumtexturenLEisgiven󰀊

bythefollowingfunctionofwatersaturationSfw:

nLED(Sw)≡nLEf

/nmax=tanh(A(Sw−S∗w

))Sw>S∗wnLE(Sw)=0Sw≤S∗(A.1)

D

w

HerenLE

isdimensionless∗isthelimitingwatersaturationD

LEbubbletexture,Swatwhichfoamcollapses,nmaxistheupperlimittofoamtexture(reflectingalowerlimittobubblesize)[35],andAaconstant.Wetakenmax=8×1013m−3(cf.[4]).SeeTable3forotherparametervaluesandfluidpropertiesusedinthemodel.InthispaperweworkwithdimensionlessfoamtexturenD≡nf/nmaxratherthannf.

Fordynamicfoamtexturesweproposeafirst-orderapproachtolocal-equilibriumbubbletextureatanysaturation,withatimeconstant󰀂

󰀃

1/Kc:

rg−rc=Kcnmax(nLED(Sw)−nD)

(A.2)

Eq.(A.2)appliesfornD≤1;ifnD>1,nDisresetto1andthenEq.(A.2)isapplied.Thisassumesthatthemechanismsthatcontrolthelowerlimittobubblesize(interbubblediffusionanddifficultyincreatingbubblessmallerthanpores)arerapidcomparedtoothermechanismsoffoamgenerationandcollapse;seeChenetal.[7]foradifferentapproach.

Althoughgasrelativepermeabilityandgasviscosityareinsepa-rableinthepresenceoffoam,forsimplicity,werepresentthewholeeffectoffoamasareductioningasrelativepermeabilityandleavegasviscosityunaltered.Weassumealargemobilityreductionfac-torMRF(18500)forthestrongestfoam(cf.[14])andMRF=1fornofoam.Finally,weinterpolateMRFasalinearfunctionoffoamtexturebetweenthosetwofoamextremes:

krg(Sw,nD)=korg(Sw)/MRF(Sw

)MRF(Sw)=18500nD(Sw)+1

(A.3)

wherek0rg(Sw

)isgasrelativepermeabilityintheabsenceoffoam.Bythisdefinition,waterfractionalflowisnotsolelyafunctionofsaturation,becauseindynamicfoamflowfoamtexturecandeviatefromitslocal-equilibriumvalueforagivenwatersaturation.

Thefoam-freerelativepermeabilitiesarethoseofZhouandRossen[14],basedondataofPersoffetal.[11]fornitrogenandwaterflowinginBoisesandstone:

󰀆

kS󰀇4.2

rw=0.2

w−Swc1−S(A.4)

wc−SSgr

w>Swc

󰀆

󰀇1.3

ko−Sw−Sgr

rg=0.94

11−Swc−SS−Sgr

w<1gr

(A.5)

Obviously,krwiszeroforSw≤Swcandthesamefork0if1−Srg

Sw≥gr.Thefractional-flowforfoamisasfollows:

fw(Sw,nD)=

1

0(A.6)

1+

󰀄w

krg(Sw)krw(Sw)󰀄0MRF(nD)

Thelocal-equilibriumfoamfractionalflow,fw(Sw,nLE

),andfoam-freefractional-flowcurve,fSD

w(w,0),areshowninFig.1.

Thecapillary-pressurefunctionisgivenbyP0.022(1−Sc=15000×w−Sgr)c

(S,c=0.01

(A.7)

w−Swc)

wherethecapillarypressurePcisinunitsofPa.Incorporatedintothisfunctionisourassumptionofgas–watersurfacetension󰀅of0.03N/m(30dyne/cm).ThisfunctionisroughlyconsistentwiththatgivenbyKovsceketal.[6],thoughwithirreduciblesatura-tionsadjustedtothesamevaluesasintherelative-permeabilityfunctions.Thefactor(1−Swc−Sgr)cwithverysmallcguaranteesacontinuousfunctionatresidualgassaturation,whilehardlyaffect-ingcapillary-pressurevaluesattheotherwatersaturations(seeFig.8).Thisassumptiondoesnotaffectourresultsbecauseourexamplesoccurathighergassaturationsthanresidual.ThisfactorinEq.(A.7)doesnotaffectourresults.However,tobeconsis-tentwithtraveling-wavesolutionsforanalogouscasesinAshoorietal.[18]weapplythiscapillary-pressurefunction.Thiscapillary-pressurecurvecorrespondstodrainage,asdothedisplacementswemodelwithit.AppendixB.Kammodel

InthemodelofKam[5],lamellaecreation(rg)requiresexceed-ingaminimumpressuregradient;abovethethreshold,creationincreasesrapidlywithincreasingpressuregradient,reflectinglamellamobilizationanddivision,andathighpressuregradientsitreachesaplateau,ontheassumptionthatlamellacreationratemusthaveanupperbound.r=

C󰀄

󰀄gg

p−2

erf

∇√∇p0

󰀅

󰀄−2−erf

√∇p0

󰀅󰀅

2(B.1)

∇pmeans∂p/∂xinthispaper.Notethatthefoammodelequations

inthisAppendixrequirequantitiesinunitsspecifiedinTable3.Asinotherpopulation-balancemodels[6,41],therateoffilmbreakage(rc)isdescribedbyafunctionthatapproachesinfinityaseitherwatersaturationorcapillarypressureapproachalimit-ingvalue[42].WemodifythisfunctiontoincludedimensionlessfoamtexturenD≡nf/nmaxwherenmaxisfoamtextureatminimumbubblesizewhichisthoughttocorrespondtoaverageporesize:󰀄

rS󰀅n

c=CcnmaxnD

w

S(B.2)

w−S∗w

Asinthefirst-order-kineticmodel,Eqs.(B.1)and(B.2)applyfor

n≤nmax;ifn>nmax,nisresettonmax.

226E.Ashoorietal./ColloidsandSurfacesA:Physicochem.Eng.Aspects377 (2011) 217–227

InEqs.(B.1)and(B.2)∇pispressuregradientand∇p0,n,CgandCcaremodelparameters.Asinotherpopulation-balancemodels[41,44]effectivefoamviscosityisashear-thinningfunctionofgasvelocity:

󰀄fg=󰀄o

g+

󰀄Cfnmax󰀅nD

ug1/3

(B.3)

ϕSg

where󰀄0gisgasviscosityintheabsenceoffoamandCfisamodelparameter.TransportofliquidandgasisgovernedbyDarcy’slaw.Liquidandgasrelative-permeabilityfunctionsandliquid-phaseviscosityareassumedtobeunaffectedbyfoam.Hence,forhori-zontaldisplacements,neglectingcapillary-pressuregradients,gasvelocityandfractionalflowofwatercanbeevaluatedfromukk0g=

rg

󰀄f∇p

(B.4)

g

f1

w=

(B.5)

1󰀄k0+

wrgkrw󰀄f

Therelative-permeabilityandcapillary-pressurefunctionsandothermodelparametersandfluidpropertiesaregiveninTable4.FoamtextureatlocalequilibriumcanbeobtainedbyequatingEqs.(B.1)and(B.2).󰀄nLE=

Cg

Sw−S∗w

󰀅n󰀄󰀄∇p−√∇p0

󰀅

D

2Ccnmax

Serf

w

2−erf

󰀄−√∇p0

󰀅󰀅

2(B.6)

Atsteady-state,onecandetermine

󰀄fg,

ugandnDbysimultaneouslysolvingEqs.(B.3)–(B.6),andthenconstructthefractional-flowcurveusingEq.(B.5).AsshownbyKam[5]theresultingfractional-flowcurvescanbequitecomplex.AppendixC.Mathematicalapproachforsolvingentranceregion

TosolvefortheentranceregionatsteadystateinthepresenceofcapillarypressureonemustsolveEqs.(7)and(12)simultaneously.Forsimplicity,letFandGrepresenttheright-handsideofthesetwo⎧equationsrespectively:

⎪⎨dSwdx

=F(Sw,nD)⎪⎩

dn(C.1)

f

dx

=G(Sw,nD)whereSwandnDareevaluatedat(SJJ

w,nfindD).Eq.(C.1)representsasystemofnonlinearODEs.OurgoalistoSwandnfasfunctionsofxforanentranceregionwithknowndownstreamboundarycon-ditionJ.NotethatJisatequilibrium:i.e.SwandnDapproachtheirLEvaluesasymptotically.Inotherwords,dSw/dx=0anddnD/dx=0atJ.Asaresult,thissystemofnonlinearODEscanbelinearizedinthevicinityofJbyTaylorapproximationofFandG.Neglectingtermshigherthanfirstorder

⎡d󰀉S⎤⎡

∂F(Sw,nD)

∂F(Sw,nD)

⎤⎢w⎣

dx⎥∂Sw∂nf⎥󰀈

󰀉

d󰀉nf⎦=⎢⎣

∂G(Sw,nD)

∂G(Sw,nD)

⎦

󰀉Sw󰀉n(C.2)

D

dx

∂Sw

∂nf

J

ThebehaviorofthefinalorbitasthesolutionapproachesJcanbedeterminedbycalculatingtheeigenvalues(󰀃1and󰀃2)andeigen-vectors(󰀁1and󰀁2)oftheabovematrix.SolutionsofEq.(C.2)are

given󰀆

by

󰀉S󰀇

w

󰀃󰀉n󰀁2x

(C.3)

D

=c11e󰀃1x+c2󰀁2ewherec1andc2areconstants.Equilibriumpointsareclassifiedaccordingtotheireigenvaluesasfollows:anequilibriumwithtworealnegativeeigenvaluesiscalledanattractorpoint;anequilib-riumwithtwopositiveeigenvaluesiscalledarepellerpoint;anequilibriumwithtwoeigenvaluesofdifferentsigniscalledasaddlepoint;andonewithtwocomplexconjugateeigenvaluesiscalledanattractororrepeller(spiral)pointdependingonwhetherthesignoftherealpartoftheeigenvaluesisnegativeorpositive[45].

ConstructionofasolutionfortheentranceregionapproachingpointDasymptoticallyandsatisfyingthesystemofODEsisnotalwaysfeasible:forinstance,thereisnoorbitendingatarepellerpoint;therefore,anequilibriumpointwiththisbehaviorisruledout.

Oncethisnecessaryconditionissatisfied,weappliedtheODEsolverinMATLAB®(TheMathworks,Natick,Massachusetts,USA)tocalculatethesolutionofEq.(C.1).KnowingtheasymptoticbehaviorofthesolutionneartheendingequilibriumpointisneededtosolvetheODE.Forinstance,ifthedownstreamstateisasaddle,weknowthesolutionmustconvergetothesaddlealongtheeigenvector󰀁(1)correspondingtothenegativeeigenvalue󰀃1;thus,wesolvetheequationsinreversebymultiplyingourequationsby(−1),startingfromthepointJ+󰀉JnearthesaddleJ.inthiscaseeither󰀉J=ε󰀁(1)or󰀉J=−ε󰀁(1),whereεissmallnumberdeterminedbythedesiredaccuracyofthecalculation.Theentrance-regionsolutioncontinuestothepointatwhichbubbletexturebecomeszero.References

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