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Cognition Marvin Chun
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DEDICATION
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PREFACE
Thisbookfocusesonthephysicsofmultiphase fluid flowanddisplacement inporousandfracturedmediaaswellasquantitativeapproachesand analysesfordescribingsuchphysicalprocessesinreservoirs.Thebookis intendedtocomplementtheexistingliteraturebypresentingnewadvances andupdateddevelopmentsinmultiphase fluid flowinporousmedia.The materialofthisbookisbasedprimarilyon(1)aseriesofpeer-reviewed papers,publishedbymeorwithco-authorsand(2)thecoursenotesthatI haveusedtoteachundergraduateandgraduatecourseson petroleumreservoir engineering and multiphase fluid flowinporousmedia attheColoradoSchoolof Mines.Thepublicationsthatthisbookisbasedonarerelatedtothe researchonthesubjectofmultiphase fluid flowsinporousandfractured media,whichIhavecarriedoutorbeeninvolvedwithsincethelate1980s attheUniversityofCalifornia,Berkeley,California(CA);HydroGeoLogic, Inc.,Reston,Virginia;theLawrenceBerkeleyNationalLaboratory, Berkeley,CA;andtheColoradoSchoolofMines,GoldenColorado.
Thebookcanbeusedasatextbookorreferenceforseniorundergraduateandgraduatestudentsinpetroleumengineering,hydrogeologyor groundwaterhydrology,soilsciences,andotherrelatedengineering fields, suchascivilandenvironmentalengineering.Itcanalsoserveasareference bookforhydrogeologists,petroleumreservoirengineers,andotherengineersandscientistsworkingintheareaof flowandtransportinporous media.
Thecontentofthebookisorganizedtocoverfundamentalsof multiphase fl uid fl owinporousmedia.Itdiscussesthephysicalprocesses andprinciplesgoverningmultiphaseporous-medium fl owusingDarcy’ s law,relativepermeability,andcapill ary-pressureconcepts.Thisbookuses theblack-oilmodelasanexampleofimmisciblemultiphase fl uid fl owto discuss fl ow-governingequationsandapproachesfortheirsolutionto quantity fl owanddisplacementprocessesinreservoirs.Specifi cally,this bookpresentstheextensionsoftheclassicalBuckley –Leverettfractional fl owtheorytoone-dimensionallinearandradialcompositesystems,to analysisofimmiscibledisplacementofnon-Newtonian fl uidsinporous media,andtonon-Darcydisplacementusing Forchheimer andBarreeand Conwaynon-Darcy fl owmodels.Inaddition,thebookreviewsthe concept,approach,anddevelopmentformodelingmultiphase fl owin
fracturedporousmediaandmultiphase fl uid fl owandheattransferin reservoirs.Inanefforttoincludethenewdevelopments,thebookalso presentsmathematicalformulations andnumericalmodelingapproaches formultiphase fl owcoupledwithgeomechanicsandfor fl owin unconventionalpetroleumreservoirs.
Yu-ShuWu ProfessorofPetroleumReservoirEngineering
DepartmentofPetroleumEngineering
ColoradoSchoolofMines Golden,Colorado,USA
FoundationCMGChairinReservoirModeling
1.2LITERATUREREVIEW,DEVELOPMENT,ANDADVANCE
Humankindmayhavenoticed flowinporousmediafromthebeginningof civilization,whentheystoodonbeachsandsfacingtides,lookedatrain percolatingintotheground,orbegantogrowplantsinsoils.
Thedevelopmentofthetheoryof flowthroughporousmedia,asa branchofappliedscience,beganwith HenryDarcy(1856) whodetermined experimentallytheproportionalityofpressuregradientand flux,now knownasDarcy’slaw.Darcy’slawwas firstdevelopedforwaterinsaturated flow,buthassincebeenextendedtomultiphaseandunsaturated flow,and toincorporateotherphenomena,suchassurfacetension,gravity,fracture flow,chemicalreaction,andchanging fluidproperties.Suchextensioninto variouscoupledprocesseshasbeenmadepossiblebytheuseofnumerical computation,whichinturndependsonpowerfulcomputertechnology andadvancesincomputationalmathematics.Darcy’slawanditsextensions havebeenappliedtoawiderangeofactivities,includingagriculture, environmentalmanagement,andmostnotably,petroleumreservoir engineering.
Darcy’slawhasbeenthefoundationforstudiesof flowandtransport phenomenathroughporousmedia.TheempiricalDarcy’slawcanalsobe derivedfromtheNavier–Stokesequationsviaavolumeaveragingmethod orhomogenizationtheory(e.g., Neuman,1977;Whitaker,1986).Even thoughoriginallyobtainedonlyfordescribing flowofasingle-phase fluid, Darcy’slawhasbeenextendedandgeneralizedtodescribethe flowof multiple,immiscible fluids(e.g., Scheidegger,1974;HassanizadehandGray, 1979a,b).ThemultiphaseextensionofDarcy’slawhasbeenusedexclusively asthebasisforquantitativestudiesofdynamicsofmultiphase fluid flowin porousandfracturedmedia.
Mostsignificantcontributionstounderstandingmultiphase flowin porousmediahavebeenmadesincethe1920s(Willhite,1986),1930s (Richards,1931),and1940s(BuckleyandLeverett,1942)withtherapid advancesinthepetroleumindustry,groundwaterhydrology,andsoilsciences.Thefundamentalunderstandingofimmiscible flowanddisplacementofNewtonian fluidsinporousmediawasinitiallycontributedby BuckleyandLeverett(1942) intheirclassicalstudyoffractional flowtheory.TheBuckley–Leverettsolutionprovidesinsightintoimmiscible-fluid displacementprocesses,describingasaturationprofile advancingwitha sharpfrontalongthe flowdirection,whilecapillarypressureandgravity effectsareignored.Effectsofgravityandcapillarypressureonalinearwater
Inaparalleldevelopment,soilandgroundwatercontaminationby Non-AqueousPhaseLiquids(NAPL),suchascontaminationfromoiland gasolineleakageorotherorganicchemicals,hasreceivedincreasing attention.Thesubsurfaceenvironmentalconcernhasmotivatedsignificant researchactivitiesindevelopingandapplyingmultiphase flowandtransport modelsforassessingNAPLcontaminationandassociatedcleanupoperations.Asaresult,manynumericalmodelsandcomputationalalgorithms havebeendevelopedandimproveduponforsolvingmultiphase fluid flow andorganicchemicaltransportproblemsthroughporousandfractured mediainthesubsurface(AbriolaandPinder,1985;Faust,1985;Forsyth, 1988,1991,1994;ForsythandShao,1991;KaluarachchiandParker,1989; Faltaetal.,1992a,b;Huyakornetal.,1994).
Recentdevelopmentinseveralfrontiersofenergyandnaturalresources andsubsurfacestoragehasrevitalizedtheinterestinandfurtherdriventhe researchof flowandtransportprocessesofmultiphase fluidsincomplicated reservoirsystems.Thelastfewdecadeshavewitnessedwide,diverseinterestsintheoryandapplicationofmultiphase fluid flowinporousmedia. Thesenewly,rapidlydevelopingandemerging fieldsincludethefollowing: CO2 geosequestrationinsubsurfaceformations: Toaddressthe increasingconcernsregardinggreenhousegasemissionanditsimpacton globalclimate,CO2 geologicsequestration,i.e.,injectinglargeamountsof CO2 intodeepsubsurfaceformationsforlong-termstorage,isconsideredto beaviableapproachfornear-termimplementation.Carbondioxidecanbe sequesteredindeepsalineaquifersaswellasindepletedoilandgasreservoirs,andcoalbedsthatareeitheruneconomicalorproducingcoalbed methane.ThemechanismsofCO2 storage(Metzetal.,2005)include structuraltrapping (trappingCO2 asamobile “ gas ”,i.e.,asupercritical fluid, informations); residualgastrapping (trappingCO2 asanimmobile “ gas ”); solubilitytrapping (trappingofCO2 asasolublecomponentinbrine);and mineraltrapping (conversionofCO2 intocarbonateminerals:calcite,dolomite,siderite,etc.).Thesemechanismsarecontrolledbyphysicaland chemicalprocessesofCO2 storage, flow,andtransportinreservoirs.To evaluatetheintegrityandproperdesignofaCO2 geologicstoragesystem andtoassesstheriskassociatedwithagivensubsurfacesystem,mathematicalmodelsandsimulationshavebeenusedasthemainapproachto studyandpredicttheperformanceofaCO2 storagesystem,becauseofthe largespaceandtimescales(PruessandGarcia,2002;Oldenburgand Lewicki,2006;PruessandSpycher,2007;BensonandCole,2008; Al-KhouryandBundschuh,2014;WinterfeldandWu,2014).
Insummary,ourunderstandingofmultiphase fluid flowanddisplacementprocessesinporousmediahasbeenenhancedsigni ficantlyinthepast fewdecades.Asapplicationexamples,manyphysics-basedmathematical modelsandassociatednumericalmodelingmethodshavebeendeveloped andusedasstandardtechniquesinsubsurfaceinvestigationsfrom developingenergyandothernaturalresourcesandcharacterizing contaminationsitestoengineeringdesignofsubsurfacestoragesystems.In thepastfewyears,thequantitativemodelingapproacheshavebeenused eventohelplaboratoryexperimentsfortestingdesignanddataandresult analysis.
1.3PURPOSEANDSCOPE
Theobjectivesofthisbookaretodiscussthephysicsandprinciplesgoverningmultiphase fluid flowinporousandfracturedmedia;toreview mathematicalmodels,quantitativeapproaches,andconstitutivecorrelations fordescribingmultiphase flowanddisplacementinporousmedia;andto discussandpresentanalyticalsolutionsandmathematicalformulationsfor evaluationofmultiphase flowanddisplacementprocessesinporousand fracturedmedia.Thisbookcomplementstheexistingliteraturebypresentingnewadvances,currentdevelopments,andupdatedquantitative approaches.Specifically,asignificantportionofitscontentisbasedonthe workandresearchoftheauthorandhiscolleaguesinthepastfewdecades. Inaddition,theappendicespresentaseriesofcomputationalprogramsfor calculatingtheanalyticalsolutionspresentedinthebook,makingitvery convenientforthereaderstousethesolutionsintheirownstudiesto analyzelaboratoryexperimentsortoverifynumericalmodelingresultsfor fieldapplications.
Theremainingchaptersofthebookareorganizedasfollows:Chapter2 presentsthefundamentalsofmultiphase fluid flowinporousmediawith basicscienceandengineeringconcepts.Itdiscussesphysicalprocessesand principlesforgoverningmultiphaseporousmedium flowandintroduces Darcy’slaw,relativepermeability,andcapillarypressureconcepts.Chapter3 describes flow-governingequationsandmathematicalmodelsformultiphase flowinporousmedia,usingtheblack-oilmodelasanexample.Italso discussesconstitutiverelationsandsolutionconditionsaswellashowto solvetheseequations.Chapter4discussesaunifiednumericalmodeland formulationforsolvingmultiphase flowanddisplacementinporousand fracturedmedia.
Forsyth,P.A.,Shao,B.Y.,1991.NumericalsimulationofgasventingforNAPLsite remediation.AdvancesinWaterResources14(6),354–367.
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Hassanizadeh,M.,Gray,W.G.,1979b.Generalconservationequationsformulti-phase systems:2.Mass,momenta,energy,andentropyequations.AdvancesinWater Resources2,191–203.
Helfferich,F.G.,1981.Theoryofmulticomponentmultiphasedisplacementinporous media.SocietyofPetroleumEngineersJournal21(01),51–62.
Honarpour,M.M.,Koederitz,F.,Herbert,A.,1986.RelativePermeabilityofPetroleum Reservoirs.CRCPressInc.,BocaRaton,FL.
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Huyakorn,P.S.,Panday,S.,Wu,Y.S.,1994.Athree-dimensionalmultiphase flowmodel forassessingNAPLcontaminationinporousandfracturedmedia,1.Formulation. JournalofContaminantHydrology16(2),109–130.
Huyakorn,P.S.,Pinder,G.F.,1983.ComputationalMethodsinSubsurfaceFlow.Academic Press.
Huyakorn,P.S.,Thomas,S.D.,Thompson,B.M.,1984.Techniquesformaking finite elementscompetitveinmodeling flowinvariablysaturatedporousMedia.Water ResourcesResearch20(8),1099–1115.
Istok,J.,1989.GroundwaterModelingbytheFiniteElementMethod.American GeophysicalUnion.
Kaluarachchi,J.J.,Parker,J.C.,1989.Anefficient finiteelementmethodformodeling multiphase flow.WaterResourcesResearch25(1),43–54.
Lake,L.W.,1989.EnhancedOilRecovery.PrenticeHallInc.,OldTappan,NJ.
Larson,R.G.,1978.Analysisofthephysicalmechanismsinsurfactant flooding.Societyof PetroleumEngineersJournal18(01),42–58.
Larson,R.G.,Davis,H.T.,Scriven,L.E.,1982.Elementarymechanismsofoilrecoveryby chemicalmethods.JournalofPetroleumTechnology34(02),243–258.
Lee,W.J.,Hopkins,C.W.,1994.Characterizationoftightreservoirs.JournalofPetroleum Technology46(11),956–964.
Looney,B.B.,Falta,R.W.,2000.VadoseZoneScienceandTechnologySolutions.Battelle PressColumbus,Ohio.
Mattax,C.C.,Dalton,R.L.,1990.Reservoirsimulation.JournalofPetroleumTechnology 42(6),692–695.
McWhorter,D.B.,Sunada,D.K.,1990.Exactintegralsolutionsfortwo-phase flow.Water ResourcesResearch26(3),399–413.
MercerJr.,J.W.,Faust,C.,Pinder,G.F.,1974.Geothermalreservoirsimulation.In: ResearchfortheDevelopmentofGeothermalEnergyResources,vol.1.Princeton University,Princeton,NewJersey,pp.256–267.
Metz,B.,Davidson,O.,DeConinck,H.,Loos,M.,Meyer,L.,2005.CarbonDioxide CaptureandStorage.IPCCGeneva,Switzerland.
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Whitaker,S.,1986.FlowinporousmediaI:atheoreticalderivationofDarcy’slaw. TransportinPorousMedia1(1),3–25.
Willhite,G.P.,1986.Water flooding.SocietyofPetroleumEngineers,Richardson,TX.
Winterfeld,P.H.,Wu,Y.-S.,2014.SimulationofCO2 sequestrationinbrineaquiferswith geomechanicalcoupling.In:Al-Khoury,R.,Bundschuh,J.(Eds.),Computational ModelsforCO2 Geo-sequestration&CompressedAirEnergyStorage.CRCPress, pp.275–304.
Wu,Y.-S.,1990.TheoreticalStudiesofNon-newtonianandNewtonianFluidFlow throughPorousMedia.LawrenceBerkeleyLaboratory,Berkeley,CA.
Wu,Y.-S.,2001.Non-Darcydisplacementofimmiscible fluidsinporousmedia.Water ResourcesResearch37(12),2943–2950.
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CHAPTER2
MultiphaseFluidsinPorous Media
2.1INTRODUCTION
Thephysicalprocessesassociatedwith flowandtransportofmultiphase fluidsinporousmediaaregovernedbythesamefundamentalconservation lawsasthoseusedinanybranchofthesciencesandengineering.Conservationofmass,momentum,andenergygovernsthebehaviorof multiphase fluid flow,chemicaltransport,andheattransferthroughporous andfracturedmedia.Thesephysicallawsinporousmediaarewellknown attheporelevel;however,inpracticeforaparticularstudyoflaboratoryor fieldapplication,onemaybeinterestedonlyinglobalbehaviororvolume averagingoftheporousmediumsystem.Becauseofthecomplexityofpore geometriesandtheheterogeneityofaporousmediumsystem,the macroscopicbehaviorisnoteasilydeducedfromthatontheporelevelor microscale.Forexample,anyattemptstodirectlyapplytheNavier–Stokes equationto flowproblemsthroughalotofporesinanactualreservoir porous-mediumsystemwillfacetremendousdifficulties.Theseinclude poorlydefinedorunknownporegeometriesor flowboundaries,complex dynamicphenomenaofphysicalandchemicalinteractionsbetweenpores and fluidsorbetween fluidsandsolids,andtoomanyunknownsor equations,withtoomanyundefinedparametersorcorrelations,which cannotbesolvedatpresentfor field-scaleapplications.
Themacroscopiccontinuumapproachismostcommonlyusedin studiesof flowanddisplacementprocessesinreservoirsfromlaboratoryto fieldscale.Forpracticalapplication,almostalltheorieson flowphenomena occurringinporousmedialeadtomacroscopiclawsapplicabletoa finite volumeorasubdomainofthesystemunderinvestigation,thedimensions ofwhicharelargecomparedwiththoseofpores.Consequently,theselaws leadtoequationsinwhichtheporousmediumand fluidsystemaretreated asiftheywerecontinuousandcharacterizedbythelocalvaluesofanumber ofthermodynamicvariablesandrockand fluidparameters,definedforall pointswithappropriateaveragingor representativeelementaryvolume (REV)
MultiphaseFluidFlowinPorousandFracturedReservoirs
ISBN978-0-12-803848-2
http://dx.doi.org/10.1016/B978-0-12-803848-2.00002-7
petroleumreservoirasanexample,primaryrecoveryisnaturallyoccurring flowtowardwells,butthiswillnotpersistwithoutenoughenergyeither storedwithinthereservoirorsuppliedfromoutside.Tomaintainlongtermproductivityofoilorgas,asecondaryrecoverymethod,suchas waterfloodingandgasinjection,ismostcommonlyusedtosupplyenergy toreservoirs.Inaddition,tertiaryoilrecovery(orEnhancedOilRecovery, EOR)hasbeenroutinelyusedtoinjectchemicalsorthermalenergyinto petroleumreservoirstoenhanceproductionorperformanceofthe reservoir.
A primaryrecovery stageofpetroleumreservoirsistheproductiontime periodwhen flowtowellsreliesonthenaturalenergyofthereservoiror beforeasecondaryortertiaryrecoveryapproachisimplemented.During thisstage,theoverallperformanceofreservoirsiscontrolledbythenature oftheenergy,i.e.,drivingmechanism(s),availableformobilizingtheoilto thewellbore.Thereareanumberofdrivingmechanismsthatprovidethe naturalenergynecessaryforoilto flowintowells(Craftetal.,1959; Willhite,1986;AhmedandMcKinney,2011): Rockandliquidexpansiondrive:Thisusuallyisthemainproductionmechanismforgroundwater flowtoawellfromaconfinedaquiferand forinitialoilproductionfroman “undersaturated” oilreservoir,i.e.,current reservoiroilpressureishigherthanitsbubblepointpressure.Then,the reservoirissaturatedonlywithliquidofoiland/orwater.Astheseliquids arewithdrawnfromwellsandthereservoirpressuredeclinesnearthewell, therockand fluidswillexpandaccordingtotheirindividualcompressibilities.Thereservoirrockcompressibilityiscontributedbytheexpansion oftheindividualrockgrainswithinreservoirandformationcompaction duetotherockmassoverlyingtheproductiveformations.Compaction occurswhenthereservoirformationis “compacted” asaresultoftheincreaseinthenetoverburdenstressforrocksolids,asthereservoirpore pressureisreduceddueto fluidwithdrawal.Thedegreeofconsolidationof bulkrockandadecreaseof fluidpressurewithintheporespacestendto reducetheporevolumeofthereservoirformationthroughthereductionof theporosity.Astheexpansionofthe fluidsandreductioninthepore volumeoccurwithdecreasingreservoirpressure,theoilandwaterwillbe forcedoutoftheporespacetothewellboreatlowpressure.
Becauseliquidsandrocksareonlyslightlycompressible,thereservoir willexperiencearapidpressuredeclinetomaintainagivenproductionrate. Theoilreservoirunderthisdrivingmechanismischaracterizedbyaconstantgas–oilratio,equaltothegassolubilityatthebubble-pointpressure.
Forconventionalpetroleumreservoirs,thisdrivingmechanismisconsideredaninefficientdrivingforceandmayresultintherecoveryofonlya smallpercentageofthetotaloriginaloil-in-place(OOIP).
Depletion-orsolution-gasdrive:Thisdrivingmechanismconsists mainlyofsolution-gasdriveandassociatedgasdisplacement.Adepletiontypepetroleumreservoirisnotincontactwithalargebodyofpermeable water-bearingformationoraquifer.Themainenergysourcefordriving flowinthiscaseisfromgasreleasedfromtheoilphaseandthesubsequent expansionofthesolutiongas.Asreservoirpressuredeclineswithcontinuousoilproductionandfallsbelowthebubble-pointpressure,gasbubbles areliberatedwithinthemicroscopicporesandthereservoirbecomes “saturated ” Thesebubblesexpandandforcethecrudeoiloutofthepore space.Oncethegassaturation,withpressurelowering,exceedsthecritical gassaturation,freegasforms flowchannelsatleastlocallyandbeginsto flowtowardwellsandthegas–oilratioofproduced fluidsincreases.
Adepletiondrivereservoirischaracterizedbyrapidandcontinuous reservoirpressuredeclineandgas–oilratiothatincreasestoamaximumand thendeclines(AhmedandMcKinney,2011).Oilproductionbydepletion driveisusuallynotefficient,becausegasismuchmoremobilethanoiland flowsmuchfasterthanoil,bypassingoilandleadingtoarapiddepletionof thereservoirenergy.
Gascapdrive:Agascap,ifexists,canprovideadditional fl ow-driving forcefromthetopofareservoir.Agascapcouldbenaturallyoccurring, secondarilyformedbyreleaseofsolutiongas,orgeneratedbygas(e.g., nitrogen)injectionforpressuremaintenance.Duetotheabilityofthegas captoexpand,accordingtoitssizeortotalenergystored,agas-capdrive reservoirischaracteristicofpressurefallingslowlyandcontinuously. Pressuretendstobemaintainedatahigherlevelthaninadepletiondrive reservoir.Thenaturalenergystoredinthegascap,availabletoproduce theoil,comesfromtheexpansionofthegas-capgasandexpansionofthe solutiongasasitisliberatedinoilzones.Oilrecoverybygas-cap expansionisactuallyatwo-phasedisplacingmechanism,whichisaided bygravitywhendenseroil,displacedbylightergas, fl owsdownwardand canyieldconsiderablylargerrecoveryef fi ciencythandepletion-drive reservoirs.
Waterdrive:Mostpetroleumreservoirsarepartlyorcompletely surroundedbyaquifersorlocatedaboveabottomwaterzone.Theseedge water,bottomwaterzonesoraquifersmaybelargecomparedtothe hydrocarbonreservoirvolume.Thesurroundingaquiferscansupply