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SPE-20739-PA

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SPE 20739 PA
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ConsiderationsAffectingtheScalingofDisplacementsinHeterogeneousPermeabilityDistributionsThomasA.Hewett,SPE,StanfordU.,andRonaldA.Behrens,SPE,ChevronCanadaResourcesSummary.DispersionandconvectionscalingcanbeusedtomodelIDmiscibleandimmiscibleflows,respectively,butthesescalingtechniquesfailinmultidimensionalheterogeneouspermeabilitydistributions.Thelengthdependenceobservedfordispersioninducedbycorrelatedheterogeneityinmiscibledisplacementprocessesprecludesthedefinitionofaneffectivedispersioncoefficient.Effectiverelativepermeabilitiescanbedefinedthatwillreproducetheresultsofthecross-sectionalsimulationsfromwhichtheywerederived,butcomplicationsarisewhentheyareusedinarealmodels.IntroductionThestudyofreservoirheterogeneityanditsinfluenceonoilrecoveryprocesseshasreceivedmoreattentionrecently.Theissuesthatariseinmodelingflowinheterogeneousreservoirscanbeplacedinfourcategories.I.Geologicquantification:quantitativespecificationoffluidflowpropertiesincludesstatisticalmeasurementsofvariationandspatialcorrelation.2.Dataintegration:datafromdifferentsourcesandmeasuredatdifferentscalesarecombinedatdifferentdegreesofreliability,resolution,andspatialsamplingintoareservoirmodelconsistentwithalltheavailabledataandtheirspatialstructures.3.Numericalrocks:simulationsofgeologicpropertydistributionsatarbitraryresolutionsthatreproducemeasureddataandstatisticalcorrelationmeasurements.4.Scale-up:definitionofeffectiveflowpropertiesforcoarsegrid,finite-differencesimulationsorstreamtubescalingmethodsforfield-scaleperformancepredictions.Thispaperfocusesonscale-upbyreviewingthescalingbehaviorofmiscibleandimmiscibledisplacementstodeterminethevalidityofdefiningeffectiveflowpropertiestorepresenttheeffectsofpermeabilityheterogeneityonfrontaldisplacements.Weshowthatheterogeneityeffectsonmiscibleprocessescannotbereducedtoaneffectivedispersioncoefficientorbemodeledwithanappropriatelevelofnumericaldispersion.Heterogeneityeffectsonimmiscibledisplacementscanbecapturedinpseudorelative-permeabilityfunctions,buttheiruseentailsmanycomplicationsandrestrictionsthathavenotbeendiscussedpreviously.Theserestrictions,whichincludemaintainingthesamecellsizeandstreamwisesequenceandusingthesameinitialsaturationdistributionasthecrosssectionfromwhichtheywerederived,willbeshown.First,itisnecessarytodevelopscalinglawsforIDdisplacementprocesses.ThesescalinglawsformthebasisforscalingtheequivalentIDformoftheheterogeneouscross-sectionalresults.ScalingLawsinMultiphaseDisplacementProcessesTheIDtransportequationformulticomponent,multiphaseflowinaporousmediumis!aCiaFiaCiaDi;;+vaCi--;;z=aL'...........................(I)NpwhereCi=EcijSj,...............................(2)j=!NpFi=Ecijij,.......................:...........(3)j=!Copyright1993SocietyofPetroleumEngineers258kNpkFJ.=~I~rjJli..J.............••..................(4)I'-jj=!I'-jNp"uCijDi=EKijSj-,..............................(5)j=!aLNpESj=I,........................:.............(6)j=!andv=ql¢A........................................(7)Fi=fractionalfluxofComponenti,whichdependsontherelativepermeabilitiesandviscositiesofPhasej,which,intum,dependonlyonthelocalfluidcompositions,Ci.Di=thedispersivefluxofComponenti,whichdependsonthedispersioncoefficientofComponentiinPhasej,Kij.Thisequationincludesthesimplifyingassumptionsthatthefluidsandrockareincompressibleandisothermalandthatthefluidsflowingthroughthemediumareinlocalthermodynamicequilibriumanddonotgothroughchemicalreactions.InEq.I,thefirsttermrepresentsthelocalaccumulationofComponenti,thesecondtermrepresentsthenetfluxoutofalocalvolumeelementbyconvectivefluidmotion,andtherightsiderepresentsthenetfluxintoavolumeelementbydispersioninthephases.Thescalingbehaviorofdisplacementfrontsdependsontheformsandrelativemagnitudesofthefluxterms,whichcontainthetransportproperties,krjandKrj.Ifweconsiderdisplacementsinasemi-infinitemedium(i.e.,thedownstreamboundaryisfarenoughawaythattheboundaryconditiondoesnotinfluencelocaldisplacementbehavior),thentheonlylengthscaleisthemeandisplacementdistanceafteratimeto,Lo=vto.TherelativemagnitudesofthetwofluxtermsthenaregivenbythePecletnumber,definedasNpeLo=vLOIKij'...................................(8)Whendiffusionisnegligiblecomparedwithmechanicalmixinginthemicroscopicporespace,Kij=av,wherea=amicroscopicmixinglengthandNpeLo=Lola......................................(9)Measurementsofafromoutcropsandstonesshowvalues2typically200m),NPe>10,000.Insimulationswithfinite-differencemodels,thetruncationerroreffectsintroduceanumericaldispersionthat,forsmalltimesteps,scalesasNPe==2N,whereN=numberofgridblocksalongtheflowdirection.4Thismeansthatinmostflowsimulations,numericaldispersioneffectswillbelargerthanproperlyscaledphysicaldispersion.Indefiningeffectiveflowpropertiestorepresentpermeabilityheterogeneityeffects,itmustbeunderstoodthattheycanreproduceonlythekindsofscalingbehaviorproducedbythefluxtermsinEq.1.IfthescalingbehaviorobservedindetailedheterogeneousSPEFormationEvaluation,December19931.1..........,.........,........,.~,..........,.........,........,.~,.........,.........,.........,.~,.........,.........,.........,.......,1.01--------_0.90.80.7~0.61-Swi0.50.40.30.20.1\.~4'~~-~3~~-~2~~-~1~~~0~~~~~~~~3~~~4~.=112Np.,"'(ULo·1),K=1Fig.1-Solventconcentrationprofilesat50differenttimesasafunctionof~.fromasimulatedsolventfloodinahomogeneouspermeabilitydistribution.1~r--1----1-------100.-100Slope=0.50V10¥'"-"""--*"'""'"./Lc/K./././././10./1.0~./1-------+--+-+++1111111111111111111.1.0L-____---'________--'--_____~0.11.0101001000MeanDisplacementDistance,LOFig.2-Concentrationprofilelengthandeffectivedispersioncoefficientasafunctionofmeandisplacementdistance.flowsimulationsdoesnotconformtothescalingbehaviorgivenbythesefluxterms,thennodefinitionofeffectiveflowpropertiesforuseinEq.1willbeadequateandtheproblemmustbereformulatedtoaccountforthephysicalprocessesthatproducedifferentscalingbehaviors.ToseethetypesofscalingbehaviorsthatcanbemodeledwithEq.1,wewillconsidertwolimitingcasesthatisolatethescalingbehaviorsoftheindividualfluxterms.Thefirstcaseisthatofamiscibledisplacementinwhichtherelativepermeabilitiesareunity,sotheconvectivefluxtermdoesnotintroduceanydisplacementfrontspreadingandonlydispersionchangestheshapeofthefront.Thesecondcaseisanimmiscibledisplacementofpurephasesinwhichthereisnodispersion,andonlythenonlinearitiesintheconvectivefluxtermchangetheshapeofthefront.DispersionScaling.Forthemiscibledisplacementofonefluidbyanother,thefractionalfluxofacomponentisalinearfunctionofitsconcentrationandthecoefficientmUltiplyingtheconvectivefluxtermisconstant.ThisreducesEq.1tothewell-knownconvectivedispersionequation.Thesolutionoftheconvective-dispersionequationforastepchangeinsolventconcentrationfromzerotooneisgivenas!11C=-erfc[(L-vt)/2v'Kt]=-erfc(O,................(10)22SPEFormationEvaluation,December19931.8QCt)/Vp.btack-4-2BBsui6.4.2BB123ScaledDistance,liLa1.8Q(t)/Vp,block-56-2BBSui6r~.4.2B_123ScaledDistance,LIlaFig.3-Watersaturationprofilesfromawaterfloodasafunc-tionofthelinearscalingvariable,LlLo:(a)all50profilesand(b)late-timeprofiles.where~=(L-vt)/2v'Kt.Inthepresenceofimmobilewater,asinthecasesconsideredhere,C=Ss/(l-Sw;)andv=q/cf>A(l-Sw;)'Forafixedtime,to,wecandefineLo=vtoand1y,~=-Np~(LlLo-l),..............................(11)2whereNPe=Lov/KisthePecletnumber.Thisformforthesimilarityvariableshowsthattheprofilewidthscaleswiththesquarerootofthemeandisplacementdistance,Lo.Toshowtheself-similarformoftheconcentrationprofileswhenthedispersioncoefficientisnotknown,Ccanbeplottedasafunctionof~*=V2N{~1(LlLo-l),whereNpCl=vLo/l.Bynotingthevalueof~*correspondingtoC=0.9,denotedby~~,andthevalueof~*correspondingtoC=0.1,denotedbyH0'thevalueofthedispersioncoefficientcanbedeterminedfrom2K=(~~0.~1;0)2................................(12)TheequalityofphysicaldispersionandnumericaldispersioncanbeseenbysettingthephysicaldispersioncoefficientinEq.1equaltozeroandsolvingthemiscible-displacementproblemwithafinitedifferencesimulator.Fig.1showstheconcentrationprofilesat50differenttimestepsfromaIDsolventfloodsimulationwith200gridblocks.Alltheprofileshaveasimilarshapewhenplottedasfunctionsofthedispersionscalingvariable,~*.ApplicationofEq.12toanyoneoftheprofilesresultsinaneffectivedispersioncoefficientofK=0.15711L2/d.Forthe5-ftgridblocksusedinthissimulation,thiscorrespondstoaphysicaldispersioncoefficientofK=0.37m2/dandadispersivityofa=K/v=0.94m.Thisismuchlargerthandispersivitiesmeasuredinthelaboratoryandiscomparableinmagnitudetothesizeofthegridblocksusedinthesimulation.Lantz4hasgiventheinfluenceofgridblockandtimestepsizesontheeffectivedispersionintroducedbytruncationerrorsinfinitedifferencesolutionsoftheconvective-dispersionequation.Theeffectivedispersioncoefficientfortheerrorsintroducedwhenanimplicitbackward-differencingschemeisusedis1K=-(vI1L+v2I1t)................................(13)2Inthissimulation,thevelocitywas0.25l1L1dandthetimestepmultipliedbyvelocitywas0.2511L.ThisgivesanumericaldispersioncoefficientofK=0.15611L2/d.ThevaluegivenbyLantz'sequationcomparesfavorablywiththevaluecalculatedwithEq.12.AccordingtoEq.11,thethicknessofthemixingzoneproducedbydispersionshouldincreaseproportionallyasthesquarerootof259SOLVENTSATURATIONMAP--100DAYSoABOVED."o0.63-Q.tI?r&30.40-0.53_0.27-0.40_0.1'3-0..27_0,00-0.13CJBELOW0,00,..-SOLVENTSATURATIONMAP--200DAYSoABOVE0.•7Do.'"-0.•70.40-a.!!3_0.2:7-0-40-._0.13-Q..270.00-0.13~-,er:t.ow0.00SOLVENTSATURATIONMAP--300DAYSoABOVE0.•7D{L53-0.61_0.40-0.'"_0.27-0.40_0.13-0.27_0.00-0.13oBELOW0.00.-~~-'~---~--""""'"-.SOLVENTSATURATIONMAP--400DAYSoABOVE0.•7o0...-0.•7_0..0-D.."_Q27-0.40_0."-0.27_0.00-0.13DBtt..o"0.00SOLVENTSATURATIONMAP--500DAYSoABOVE0.•7DO~-OJ!70.40-0,153_0.27-0..0_0.13-O.Z7_0.00-0.13oBELOW0.00HORIZONTALPERMEABILITYMAPoABO""=.0GJ10Q.0-~.D_33.3-100.0_10.0-33.3_'.3-10.0_0.0-3.3DSJ;:f..DWD.O3003504004505005506006507007508008509009501000INJECTOR-PRODUCERDISTANCEFT.Fig.4-Verticalcrosssectionwithfractalpermeabilityvariationsandsolventsaturationsatselectedtimesfromasimulatedfirst-contact-mlsclble,equal-density,constant-mobility,solventflood.thetimeormeandistancetraveledincreases.UsingLc=LlO-L9(j,whereLIOandL90aresitesofthe0.10and0.90concentrations,asameasureofthemixing-zonewidth,weseeinFig.2thatthesquare-rootspreadingbehaviorfortheeffectsofnumericaldispersionisobeyed,andnumericaldispersionhasallthecharacteristicsofphysicaldispersion.Fig.2alsoshowstheeffectivedispersioncoefficientforeachoftheprofilesinFig.l.Theeffectivedispersioncoefficientforthissimulationisconstantthroughout.ConvectionScaling.Fortheimmiscibledisplacementofonefluidbyanother,thefractionalfluxofacomponentisanonlinearfunctionofitsconcentration,sothecoefficientmultiplyingtheconvec-260tivefluxtermisafunctionoftheconcentration.Thisnonlinearityintheconvectivefluxtermcausestheresultingdisplacementfrontstochangeshapeastheypropagate.Forpureimmisciblephases,thecompositionofeachphaseremainsconstantandthedispersiontermontherightsideofEq.1iszero,sothescalingofthedisplacementiscausedbyonlytheconvectivefluxterm.ThesesimplificationsreduceEq.1towell-knownfractionalflowequationswiththesolutiongivenbytheBuckley-Leveretttheory.!Inthissolution,eachcompositionmoveswithaconstantvelocity,sotheshapeofthedisplacementfrontscaleslinearlywithtimeormeandisplacementdistance.Theself-similarprofileshapesmaybegivenasafunctionofthelinearscalingvariableLlvt,oratanyinstantto,byLILo,whereLO=vtO·5SPEFonnationEvaluation,December19931.00.9080.7~0.61-Swi0.50.40.30.20.1-3-2-1~.=1/2Np.,"2(ULo-1),K=1Fig.5-Solventconcentrationprofilesat50differenttimesasfunctionsof~.fromasimulatedsolventfloodinaheterogeneouspermeabilitydistribution.Infinite-differencesolutionsofthefractionalflowequations,numericaldispersionagainsmoothscompositiondiscontinuities,butfarfromtheinletwherethenumericalPeeletnumberislarge,theinfluenceofnumericaldispersionissmallandthecalculatedprofilesarefunctionsofthelinearscalingvariableLfLo.Thiscanbeseeninawaterfloodsimulationwithafinite-differencesimulatorthathas200gridblocks(Fig.3).Thewatersaturationprofilesat50differenttimestepsareplottedinFig.3aasafunctionofthelinear-scalingvariable;numericaldispersioncausestheearlyprofilestodeviatefromthelinear-scalingbehaviorpredictedbytheBuckley-Leveretttheory.InFig.3b,onlytheprofilesfromlatetimesareshown;thelinear-scalingsolutioniswell-approximated.EffectofPermeabilityHeterogeneityonScalingInMultiphaseFlowsForconstant-ratemiscibledisplacementswithaunitmobilityratio,statisticaltheoriesbasedonmeasuresofthespatialcorrelationsofpermeabilityfieldshavebeenusedtoderivethestatisticsofdisplacementfronts.6,7Thesetheoriesshowthat,whentherangeofcorrelationsinapermeabilityfieldissmall,thespreadingofmisciblefrontscanbemodeledbyaddinganeffectivedispersioncoefficienttothemacroscopicdispersioncoefficientinthetransportequation.Iftheconditionofsmallcorrelationlengthcomparedwithflowpathlengthisnotsatisfied,thecalculatedeffectivedispersioncoefficienthasascaledependencethatalsohasbeenobservedin1ooor----------,----------.-----~~---,100Slope~0.99V/,('-/10010l.////)/Slope~0.982/;Y//.;t//C'//~~-//+///"f++-/-V101.01.0'-----------'/'-------'---------------'----------------'0.11.0101001000MeanDisplacementDistance,LOKFig.7-Characteristicconcentrationprofilelengthandeffectivedispersioncoefficientsasafunctionofmeandisplacementdistance.Slopeonlogarithmiccoordinatesindicatesnearlylinearspreading.SPEFormationEvaluation,December19931.0K=4.877l>L'IDay0.90.80.70.6~0.51-Swl-J0.40.30.20.10-4-3-2-101~.=1/2Np.l'12(ULo-1),K=1Fig.6-0neoftheprofilesfromFig.5showingthecharacteristicprofilelengthusedforcalculatingtheeffectivedispersioncoefficient.fieldmeasurementsoftracerdispersionoverdistancesgreaterthanseveralkilometers.1,8Fortypicaloilfieldproblems,wherecorrelationsbetweenwellscommonlyareobserved,thesesolutionswillbeoflittlevaluefordescribinginterwellflow.Intheapproachtoscale-upadoptedhere,wefirstconstructaconditionalsimulationofinterwellpermeabilityvariationsinarepresentativeverticalcrosssectionbetweentwowellsandrunasimulationoftheprocessofinterestinthiscrosssection.Theresultsofthissimulationthenarereducedtoanequivalentsingle-layersolutionthatdependsonlyonstreamwisedistanceandtime.Thisdimensionreductionisaccomplishedbyverticallyintegratingthesolutionfromthecrosssectiontoobtainverticallyaveragedsaturations,fractionalflows,andmobilities.5Havingreducedtheheterogeneityeffectsintheverticalcrosssectiontoanequivalentsingle-layersolution,itthenispossibletodeterminewhethereffectiveflowpropertiescanbedefinedthatwillreproducethesamebehaviorwhenuseddirectlyinaIDsimulation.Whenthisispossible,theseeffectivepropertiescanbeusedinafinite-differencearealmodeltopredicttheperformanceoftheprocessinthreedimensions.Wheneffectiveflowpropertiesthatg~>.~coQ.'"i510~~ooo§o00o0IP§B0=B•x•u•XElxxxElElx.,f.~.El~xxLallemand-BarresandPeaduecerf(1978)8•PickensandGrlsak(1981)'ElGelharatal.('985)'5oLabDate(Arya,1986)7--DataFromFigure7--DataFromFigure2Distance(m)Fig.8-FieldandlaboratorymeasureddispersivitiescomparedwiththeeffectivedispersivitiescalculatedfromFig.7.261Fig.9-Fiveof200effectiverelativepermeabilitiesatequallyspacedstreamwisedistanceincrementsderivedfromawat
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