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DEVELOPMENTSINPETROLEUM SCIENCE75 AnIntroductiontoMultiphase, MulticomponentReservoirSimulation
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DEVELOPMENTSINPETROLEUM SCIENCE75 AnIntroductionto Multiphase, Multicomponent ReservoirSimulation MatthewBalhoff
Director,CenterforSubsurfaceEnergyandtheEnvironment; Professor,HildebrandDepartmentofPetroleumandGeosystems Engineering,TheUniversityofTexas,Austin,TX,UnitedStates; BankofAmericaProfessorshipinPetroleumEngineering
Elsevier
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1.Reviewofreservoirrockandfluidproperties
1.1Introduction 1
1.2Overviewofreservoirengineeringprinciples 1 1.3Definitions 2
1.3.1Phasesandcomponentsinsubsurfaceporousmedia2
1.3.2Porosity,saturation,density,andconcentrations3
1.4Phasebehavior 4
1.5RockandFluidProperties 6
1.5.1Formationproperties6
1.5.2Gaseousphaseproperties7
1.5.3Oleicphaseproperties9 1.5.4Aqueousphaseproperties12
1.6Petrophysicalproperties 14
1.6.1Darcy’slaw14
1.6.2Relativepermeability16
1.6.3Capillarypressure20
1.6.4Capillarypressurescanningcurves22
1.7Reservoirinitialization 23
1.8.1Relativepermeability29
2.Phasemassbalancesandthediffusivityequation
2.1Introduction 37 2.2Phasemassbalances 37
2.2.1MassbalanceofaphaseinCartesiancoordinates38
2.3Thecontinuityequation 40
2.4Thediffusivityequation 41
2.4.1Generalmultiphaseflow41
2.4.2Single-phaseflow42
2.5Analyticalsolutions 48
2.5.11Dheatequationinafinitemedium48
2.5.21Dheatequationinasemi-infinitemedium50
2.5.3Solutionincylindricalcoordinates(arounda wellbore)51
2.6Exercises 54 References 55
3.FinitedifferencesolutionstoPDEs 3.1Introduction 57
3.2Taylorseriesandfinitedifferences 57
3.2.1First-orderforwarddifferenceapproximation59
3.2.2First-orderbackwarddifferenceapproximation60
3.2.3Second-order,centereddifferenceapproximation61
3.2.4Approximationstothesecondderivative61
3.2.5Generalizationtohigher-orderapproximations64
3.3Discretizationoftheparabolicdiffusivity(heat)equation 68
3.4Boundaryandinitialconditions 70
3.4.1Dirichletboundarycondition71
3.4.2Neumannboundarycondition71
3.4.3Robinboundaryconditions72
3.5Solutionmethods 72
3.5.1Explicitsolutiontothediffusivityequation72
3.5.2Implicitsolutiontothediffusivityequation76
3.5.3MixedmethodsandCrank Nicolson77
3.5.4Linearsystemsofequations83
3.6Stabilityandconvergence 84
3.7Higher-orderapproximations 85
3.8Pseudocodefor1D,single-phaseflow 88
3.9Exercises 89 References 91
4.Multidimensionalreservoirdomains,thecontrol volumeapproach,andheterogeneities
4.1Introduction 93
4.2Griddingandblocknumberinginmultidimensions 93
4.2.1Gridblockindexingin2Dand3D94
4.2.2Griddimensions95
4.2.3Irregulargeometryandinactivegrids96
4.3Single-phaseflowinmultidimensionsandthecontrol volumeapproach 98
4.3.1Accumulation99
4.3.2Fluxterms100
4.3.3Sourcesandsinks(wells)101
4.3.4Single-phaseflow101
4.4Wells,boundaryconditions,andinitialconditions 102
4.4.1Constantratewells102
4.4.2Neumannboundaryconditions102
4.4.3Dirichletconditions103
4.4.4Cornerblocks104
4.4.5Initialconditions107
4.5Reservoirheterogeneities 107
4.5.1Fluidproperties109
4.5.2Geometricproperties109
4.5.3Accumulationterms113
4.6Matrixarrays 113
4.6.1Accumulationandcompressibility113
4.6.2Transmissibility114
4.6.3Sourceterms114
4.6.4Gravity115
4.7Pseudocodeforsingle-phaseflowinmultidimensions 120
4.7.1Preprocessing120
4.7.2Interblocktransmissibility120
4.7.3WellArrays121
4.7.4GridArrays121
4.7.5Maincode121
4.7.6Postprocessing122
4.8Exercises
5.Radialflow,wells,andwellmodels 5.1Introduction 127
5.2Radialflowequationsandanalyticalsolutions 127
5.3Numericalsolutionstotheradialdiffusivityequation 129
5.3.1Gridding129
5.3.2Discretization130
5.4WellsandwellmodelsinCartesiangrids 135
5.4.1Wellconstraints135
5.4.2Steady-stateradialflowaroundawell136
5.4.3Massbalanceonthewell-residinggridblock137
5.4.4Extensiontohorizontalwellsandanisotropy139
5.5Inclusionofthewellmodelintothematrixequations 142
5.6Practicalconsiderations 147
5.7Pseudocodeforsingle-phaseflowwithconstantBHPwells 147
5.8Exercises 148 References 150
Contents
6.Nonlinearitiesinsingle-phaseflowthroughsubsurface porousmedia
6.1Introduction 151
6.2Examplesofnonlinearitiesinsingle-phaseflowproblems 151
6.2.1Gasflow152
6.2.2Non-Newtonianflow153
6.2.3Forchheimerflow155
6.3Numericalmethodsfornonlinearproblems 156
6.3.1Explicitupdateoffluidandreservoirproperties157
6.3.2Picarditeration157
6.3.3Newton’smethod161
6.4PseudocodeforNewton’smethod 169
6.5Exercises 171 References 172
7.Componenttransportinporousmedia
7.1Introduction 175
7.2Transportmechanisms 175
7.2.1Advection175
7.2.2Hydrodynamicdispersion176
7.2.3Reactivetransportandothersourceterms182
7.3Componentmassbalanceequations 183
7.3.1Single-phaseflow184
7.3.2Overallcompositionalequations184
7.4Analyticalsolutions 186
7.4.11DCartesianADEinasemi-infinitedomain186
7.4.2Semianalyticalsolutiontotwo-phaseflow189
7.5Exercises 198 References 199
8.Numericalsolutiontosingle-phasecomponent transport
8.1Introduction 201
8.2FinitedifferencesolutiontotheADEin1Dforasingle component 201
8.3Discretizationofadvectiveterms 204
8.3.1Cell-centered204
8.3.2Upwinding205 8.3.3Matrices205
8.4Wellsandboundaryconditions 206
8.4.1Wells206
8.4.2Nofluxboundarycondition207
8.4.3Constantconcentration(Dirichlet)211
8.5Solutionmethods 212
8.5.1Implicitpressure,explicitconcentration(IMPEC)212
8.5.2Implicitpressure,implicitconcentration214
8.5.3Fullyimplicit218 8.6Stability
8.7Numericaldispersion
8.8Channelingandviscousfingering
8.9Multicomponents,multidimensions,andadditionalforms
8.10Pseudocodeforcomponenttransport
9.Numericalsolutiontotheblackoilmodel
9.1Introduction
9.3Finitedifferenceequationsformultiphaseflow
9.4.1Implicitpressure,explicitsaturation237
9.4.2Simultaneoussolutionmethod243
9.4.3Fullyimplicitmethod247
9.5Interblocktransmissibilitiesandupwinding
9.7.1Constantrateinjectorwells258
9.7.2Constantrateproducerwells259
9.7.3ConstantBHPinjectorwells261
9.7.4ConstantBHPproducerwells261
9.7.5Time-dependentwellconstraints261
9.8Pseudocodeformultiphaseflow 272
9.8.1Preprocessing272
9.8.2Blockproperties273
9.8.3Interblockproperties273
9.8.4Wellproductivityindex273
9.8.5Wellarrays273
9.8.6Gridarrays274
9.8.7Maincode274
9.8.8Postprocessing274
9.9Exercises 279 References 282
10.Numericalsolutiontomultiphase,multicomponent transport
10.1Introduction 283 10.2Compositionalequationsformultiphaseflow
10.4Solutionmethod 287
10.4.1Flashcalculations287
10.4.2Equationsofstate291
10.4.3Phasesaturation298
10.4.4Two-phasecompressibility299
10.4.5Phaseviscosity301
10.4.6Relativepermeabilityandtransmissibility304
10.4.7Wellsandsourceterms306
10.4.8Pressureandcompositionsolution308
10.5Oleic aqueousbipartitioningcomponents 311
Preface Theflowoffluidsinsubsurfaceporousmediaisimportantinmanyapplicationsincludingtheproductionofhydrocarbons,carbonstorage,hydrogen storage,aquiferremediation,andproductionofgeothermalenergy.Accurate modelingoftheseprocessesisofcriticalimportanceforpredictionsand decision-making.Forexample,inhydrocarbonproduction,modelscanbeused tomakebusinessdecisions,suchas:(1)Shouldafieldbeboughtorsold?(2) Where,when,howmany,andwhattypeofwellsshouldbedrilled?(3)What shouldbetheconstraint(wellrateorbottomholepressure)ofthewells?(4)If, when,andwhattypeofsecondary(andtertiary)recoveryshouldbepursued? (5)Whenshouldawellbeshut-inorconvertedtoaninjectorandwhatfluids shouldbeinjected?
Modelingofsubsurfacephenomenaischallengingformanyreasons.The subsurfacereservoiristhousandsoffeetbelowthesurfaceandcanbemassive (thousandsofacresinarea,orinthecaseintheGhawaroilfield,overa millionacres).Ourunderstandingofareservoir’ssize,lithology,permeability, porosity,fluidproperties,etc.,isanestimate.Reservoirsaregenerallyvery heterogeneousintheirpermeability,porosity,saturation,lithology,etc.,and canchangesignificantlyoversmallorlargelengthscales.Predictionsmaybe requiredforyearsordecadesintothefuture,orevenmillenniainthecaseof carbonstorage.
Subsurfacemodelsvaryincomplexityandcanbeassimpleasanalyticalor reduced-ordermodelssuchastankbalancesandthecapacitanceresistance model(Sayarpouretal.,2009).Suchmodelsaresimplificationsbutoften provideveryvaluableinformationandcanevenbepredictive.Thefundamentalequationsthatdescribeflowandtransportinsubsurfacemediaare multidimensional,multicomponent,multiphase,nonlinear,coupledpartial differentialequations(PDEs)withspatiallyheterogeneousandtime-dependent properties.Theseequations,withoutmajorsimplificationandassumptions,are notamenabletoanalyticalsolution.Numericalreservoirsimulatorsarethe mostadvancedtoolswehavetosolvethesePDEsandpredictflowand transportinsubsurfaceporousmedia.Thesesimulatorsaretheclosestthing wehavetoa crystalball.
Therearemanytypesofreservoirsimulators,withvaryingcomplexityand features,butgenerallytheyinvolvediscretizingthereservoirinto N grids, blocks,orelements.OnecanthinkofareservoirsimulatorasagiantRubik’s
Cube,witheachblockinthecubebeingagridinthemodel.Thesimulatorcan havethousands,millions,orevenbillionsofgridsandeachgridhasunique, constant(orsimplefunction)properties,suchaspermeability,porosity,saturation,composition,pressure,etc.Balance(mass,energy,momentum)equationsthatareimposedareoneachblockwhicharedependentonadjacent blockproperties.Asaresult,thecomplicatedPDEsreducetoasystemof N algebraicequationsand N unknowns.
Manycommercial(e.g.,CMG,ECLIPSE,INTERSECT,Nexus),academic, oropensource(BOAST,MRST,UTCHEM,UTCOMP,IPARS,TOUGH)and proprietary,in-housesimulatorshavebeendevelopedbyteamsofexpertsover decades.Thesesimulatorsvaryintheirapplicabilitybutarebasedonthesame basicfundamentals.Thesesimulatorsareoftenrelativelyeasyforthebeginnerto use,whichcanbeasmuchofaproblemasitisafeature.Failuretounderstandthe principlesandbasicequationsofnumericalsimulation(whatis underthehood) canleadonetonotrecognizethemodel’slimitationsandleadtocostlyoreven unsafedecisions.Themathematicsarecomplicatedandcanbedauntingforeven PhDscientistsandengineers.Manyoutstandingbookshavebeenwrittenonthe subject; AzizandSettari(1979), Ertekinetal.(2001), Chen(2007), Lie(2019), and Abou-Kassemetal.(2020) arejustafewofmyfavorites.Manyofthese booksarebestsuitedforadvancedgraduatestudentsorprofessionalswithsome experienceinsimulation.
Ihavetaughtthefundamentalsofreservoirsimulationfor15yearstoovera thousandundergraduatesandfirst-yeargraduatestudents.Breakingdownthe complexitiesofsimulationtostudentsnewtothesubjectischallenging,toputit mildly.Inthisbook,Ihaveattemptedtoorganizemynotes,teachingstyle,and “lessonslearned”inaconcisetextforthebeginner.Manyadvancedandmodern topicsareintentionallynotincluded,buttheinterestedreadershouldreadthe dozensofadvancedbooksandthousandsofpublicationsthatcoverthem.
Thisbookincludestwoimportantfeatures.Thefirstistheinclusionof dozensofsmall(e.g.,4 9block)exampleproblemsthataresolvedbyhand andcalculator,largelywithouttheuseofacomputer.ToquoteAlbertEinstein, “exampleisn’tanotherwaytoteach;itistheonlywaytoteach.”Ihavefound theseexamplesessentialforthebeginnertounderstandthebasicsofreservoir simulation.Inadditiontoexampleproblems,eachchapterincludesadditional exercisesforthereadertoattempt.
Thesecondfeatureofthebookistheemphasisonwritingcomputercode withtheend-goalofthereaderdevelopingtheirownmultiphase,multidimensional,andmulticomponentreservoirsimulator.Thefinalproductwillbe asimulatorthatwillproduceidentical(ornearlyidentical)resultsasthe aforementionedcommercial,academic,andin-housesimulators.Theuser’s codecanbeandshouldbevalidatedagainstthesesimulators,analyticalsolutions,orthesmallexampleproblemsprovidedinthetext.Thebookis organizedinsuchawaythatthecodestartsrelativelysimple(1D,single phase,homogeneous)andcomplexities(multidimensions,heterogeneities,
multiphase,etc.)areaddedalongtheway.Pseudocodeisprovidedineach chapter,withsomeexplanationanddiscussion,tohelptheuserdeveloptheir owncode.Themostcomputationallyefficient,vectorized,orelegantpseudocodesarenotalwaysprovided.Infact,thisisoftenintentional,assometimesthelesselegantcodesarebetterforunderstandingthelogicand mathematics.Thedeveloperofthesimulatorisencouragedtooptimizetheir codeoncetheyhaveaworkingcodethattheyunderstand.
Thesimulatordeveloperisencouragedtobepatientandavoidfrustration asbestaspossible.Ihavewrittenhundredsofsubroutinesandcodesformy reservoirsimulationcoursesovertheyearsandcansaywithconfidencethat everyoneofthemhaderrorsandbugsintheinitialversion.Theseerrorshave takenanywherefromminutestodays(orevenweeks)todebug.However, everysingletimeIhavefixedanerror,Ihavecomeawaywithabetterunderstandingofreservoirsimulationandreservoirengineeringingeneral.When thedeveloperobtainsresultsthatarenonphysicalordisagreewithanalytical solutions,exampleproblems,orcommercialsimulators,theyshouldaskwhat physicallyormathematicallycouldcausesuchadiscrepancy.Inmyexperience,99%ofthecodingerrorsareintheformationofthefewmatricesand vectorsthatareusedtosolvetheproblem.Theerror(s)canalmostalwaysbe identifiedbycomparisontothematrices/vectorscreatedbyhandintheexampleswithasmallnumberofgrids.
Yourfinalreservoirsimulator(albeitaccurateandflexible)willprobably notbeascomputationallyefficient,scalable,user-friendly,orhavenearlyas manyfeaturesasacommercialsimulator.However,youwilldevelopan excellentunderstandingofthedetailsandlimitationsofthesesimulators.And, justmaybe,youwilljoinateamorhaveacareerdevelopingthenextgenerationcommercial,in-house,oracademicsimulator.
References
Abou-Kassem,Hussein,J.,RafiqulIslam,M.,Farouq-Ali,S.M.,2020.PetroleumReservoir Simulation:TheEngineeringApproach.Elsevier.
Aziz,K.,Settari,A.,1979.PetroleumReservoirSimulation.1979.AppliedSciencePubl.Ltd., London,UK.
Chen,Z.,2007.Reservoirsimulation:mathematicaltechniquesinoilrecovery.SocietyforIndustrialandAppliedMathematics.
Ertekin,T.,Abou-Kassem,J.H.,King,G.R.,2001.BasicAppliedReservoirSimulation,7.Society ofPetroleumEngineers,Richardson.
Lie,K.-A.,2019.AnintroductiontoreservoirsimulationusingMATLAB/GNUOctave:User guidefortheMATLABReservoirSimulationToolbox(MRST).CambridgeUniversityPress. Sayarpour,M.,Zuluaga,E.,ShahKabir,C.,Lake,L.W.,2009.Theuseofcapacitance resistance modelsforrapidestimationofwaterfloodperformanceandoptimization.JournalofPetroleumScienceandEngineering69(3 4),227 238.
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Acknowledgments Firstandforemost,Iwouldliketothankthemanycurrentandformer,graduateandundergraduate,studentswhohelpedinthedevelopmentofthisbook. Althoughimpossibletolistthemall,Iwouldliketospecificallyrecognize NkemEgboga,YasharMehmani,HamzaSalimAlRawahi,TravisSalomaki, MoisesVelasco,JianpingXu,andSarahRazmara.IwouldliketothankMary Wheelerforintroducingmetothesubjectmatterofreservoirsimulationand themanycolleaguesforwhichIhavehaddiscussionsincludingLarryLake, KamySepehrnoori,GaryPope,RussJohns,DavidDiCarlo,andChengChen.I alsoacknowledgeCooperLink,JoannaCastillo,andJostineHoforhelping withthemanyillustrations.Iwouldliketothankmyfather,whotaughtmeto beanengineerandhelpedmenumericallysolvetheDiffusivityequationfor thefirsttime,mymother,whotaughtmetopersistentanddedicated,andmy sisters.Finally,thisbookwouldnotbepossiblewithouttheendlesssupport andloveofmywife,Julie.
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Nomenclature a cross-sectionalarea,ft2;empiricalcoefficientformechanical dispersion
A accumulationterm(Vif/Dt),ft3/day;parameterforcubicEOS
Ak ’ parameterforfugacitycoefficientofcomponent k incubicEOS
b empiricalexponentformechanicaldispersion
B parameterforcubicEOS
Ba formationvolumefactorforphase a,RB/STBorft3/scf
Bk ’ parameterforfugacitycoefficientofcomponent k incubicEOS
ca compressibilityofphase a,psi 1
cf formationcompressibility,psi 1
cp porecompressibility,psi 1
cr rockmatrixcompressibility,psi 1
cB bulkcompressibility,psi 1
ct totalcompressibility,psi 1
C constantforeffectiveshearrateinporousmedia fornon-Newtonianflow
Ck concentrationofcomponent k,lbm/ft3
Cj,i coefficientforblockiinIMPESmethod(j¼1,2,3)
dp graindiameter,ft
D depth,ft;hydrodynamicdispersioncoefficient,ft2/day
D1 capillarydiffusioncoefficient,ft2/day
Dm moleculardiffusioncoefficient,ft2/day
Dm,eff effectivediffusioncoefficientinporousmedium,ft2/day
DL longitudinalmechanicaldispersioncoefficient,ft2/day
Dr restricteddiffusioncoefficient(Dm,eff/Dm),ft2/day
DT transversemechanicaldispersioncoefficient,ft2/day
f weightingfactorforcapillarypressurescanningcurve
fa fractionalflowofphase a
fk,a fugacityofcomponent k ofphase a,psia
F residualofgridbalanceequation,ft3/day;formationresistivityfactor
g gravitationalconstant(32ft/s2)
G Gravityvector,ft3/day
h reservoirthickness,ft
i gridblockindex
j gridblockindex,x-direction
Ja productivityindexofphase a,ft3/psi-day
J totalproductivityindex,ft3/psi-day
Nomenclature
k permeability,mD;rateconstant,1/day;gridblockindexin y-direction
kapp apparentpermeability,mD
kB Boltzmannconstant(1.38 10 23 J/K)
kg Klinkenbergapparentpermeabilityforgasflow,mD
kH geometricmeanofpermeability,mD
kr,a relativepermeabilityofphase a
Kk K-values/equilibriumratioofcomponent k
l gridblockindex,z-direction
L reservoirlength,ft
m mass,lbm
_ mx massflux(inx-direction),lbm/ft2-day
M mobilityratio(krwmo/kromw)
Mk molecularweightofcomponent k
Ma molecularweightofphase a
n shear-thinningindexforpower-laworCarreaumodel
N numberofgridblocksinreservoirmodel
Nc numberofcomponentsinthereservoirmodel
Nk,a flux(advective,diffusive,ordispersive)ofcomponent k inphase a
Np cumulativeamountofoilproduced,bblorft3
Nx numberofgridblocksinx-directionofreservoirmodel
Ny numberofgridblocksiny-directionofreservoirmodel
Nz numberofgridblocksinz-directionofreservoirmodel
O scalingorder
p pressure(exact),psia
pb bubblepointpressure,psia
pc capillarypressure,psi
pc,k criticalpressureofcomponent k,psia
pe capillaryentrypressure,psia
pk sat vaporpressureofcomponent k,psia
P pressure(numericalapproximation),psia
Pwf bottomholepressureofwell,psia
Plim limitingpressureofproducingwell,psia
q sourceterm(1/time)
qwf wellflowingrate(ft3/day)
Q sourcevector,ft3/day
r radialdirectionincylindricalcoordinates,ft
re drainageradius,ft
rw wellboreradius,ft
req equivalentradiusforwellmodels,ft
rk effectiveradiusofcomponent k,ft
R idealgasconstant(10.73psi-ft3/lbmole-R)
Rs solutiongas-oilratio,scf/STB)
s skinfactor
Sa saturationofphase a
Sar residualsaturationofphase a
Swf watersaturationatshockfront
t time,day
Nomenclature xxi
T totaltransmissibility,ft3/psi-day;reservoirtemperature, For R
Ta phasetransmissibility,scf/psi-day
Tk componenttransmissibility,md-/cp-ft2
Tc criticaltemperature, R
Tr,k reducedtemperatureofcomponent k (T/Tc,k)
u Darcyvelocity,ft/day
U Dispersivetransmissibilityincomponentbalancematrixequations,ft3/day
v interstitial/frontalvelocity,ft/day
Vi volumeofgridblocki,ft3
Vp porevolume,ft3
Vm molarvolume,ft3/lbmole
w width,ft
Wk bulkconcentrationofcomponent k,lbm/ft3
x molefraction
X solutionvector(pressuresandsaturations)forSSmethod
z compressibilityfactor/z-factor;elevation,ft
Greeklettersandsymbols a phase(e.g.,oleic,gaseous,aqueous)
aL longitudinaldispersivity,ft
aT transversedispersivity,ft
b non-Darcycoefficient(ft 1);spatialdifferencingschemeforADE
ba capillarypressurecorrectedformationvolumefactor
g shearrate(1/s);Euler’sconstant(0.5772)
go specificgravityofoil
gg specificgravityofgas
gr coefficientforgeometricprogressionofgridsincylindrical coordinates
dk,l binaryinteractionparameterbetweencomponents k and l
dP changeinthepressurevectorbetweentwoiterations,psia
D discriminantincubicEOS
Dl tortuouslengthofporousmedium,ft
ε smallperturbationparameter
h dimensionlessdiffusivity/Fouriernumber(aDt/Dx 2 orDDt/Dx2)
q dipangle,degrees
J JacobianforNewton Raphsonmethods
k componentorpseudocomponent(e.g.,water,oil,gas)
l timeconstantforCarreaumodel,s
la mobilityofphase a (kkra/ma),mD/cp
m viscosity,cp
m0 zero-shearviscosity,cp
mN infinite-shearviscosity,cp
y molesgaseousphase/moleshydrocarbon
r density,lbm/ft3;variabletransformationforcylindrical coordinates(r2)
rr,a reduceddensityofphase a
sk parameterforcomponent k incubicEOS
sk,a volumefractionofcomponent k inphase a
s tortuosityinporousmedium
f porosity
fk,a fugacitycoefficientofcomponent k inphase a j pseudopressure,psi2/cp
u temporaldifferencingschemeforADE
uk weightfractionofcomponent k;acentricfactorofcomponent k
Superscriptsandsubscripts cen centered
GOC gas-oilcontactline,ft
g gaseousphase
o oleicphase
rc reservoirconditions
sc standardconditions(14.7psi,60F)
up upwinding
WOC water-oilcontactline,ft
x x-direction
y y-direction
z z-direction
y iterationnumberinnonlinearproblems
Dimensionlessvariables CD concentration
xD distance(x/L)
pD pressure((C-Cinit)/(Cinj-Cinit))
NPe,d Pecletnumberbasedondiffusioncoefficient(udp/Dm)
NPe Pecletnumberbasedondispersioncoefficient(uL/Df)
NRe Reynoldsnumber(rvdp/m)
Ng 0 gravitynumber(kkro 0 Drg/umo)
NpD oilrecovery(Np/Vp)
tD time/porevolumesinjected(ut/Lf)
h Fouriernumber(aDt/Dx 2 orDDt/Dx2)
p localPecletnumber(uDx/Df)
z Courantnumber(uDt/Dxf)
sD time/porevolumesinjected(ut/Lf) xxii Nomenclature
Chapter1 Reviewofreservoirrockand fluidproperties 1.1Introduction
Reservoirsimulationrequiresanaccuratedescriptionoffluidandreservoir propertiescombinedwiththeeffectpressure,temperature,andcomposition hasonthem.Manytexts(e.g.McCain,1991;Pedersenetal.,2006;Dandekar, 2013)provideanexcellent,thoroughdiscussionoffluidandrockpropertiesin thesubsurface;hereaconcisereviewisprovided.Thereservoirfluidsare composedofthousandsofuniquechemicalmolecules,areconfinedina complexporespace,andpartitionbetweenmultiplefluidphases,whichmake modelingofsubsurfacereservoirsverychallenging.
Inthischapter,anoverviewofbasicreservoirengineeringprinciplesisfirst presentedfollowedbyafewbasicdefinitions.Phasebehaviorprinciplesare introduced,followedbydefinitionsandequationsforfluid,rock,andpetrophysicalproperties.Finally,approachesfordeterminingtheinitialpressure andsaturationfieldsinthereservoirarediscussed.
1.2Overviewofreservoirengineeringprinciples
Petroleumreservoirsconsistofporousrockthousandsoffeetbelowtheearth’s surface.Thevoidspaceintherockcontainsfluids,includinghydrocarbons(oil andgas)andwaterthatcanbeextractedthroughwells.Uponexplorationand drilling,thepressureandtemperatureinthereservoirarerelativelyhigh(often thousandsofpsiaand >100 F,respectively)anddrillingawellwithalower pressureresultsinadrivingforce(drawdownpressure)forfluidproduction fromthereservoirtothewell.During primaryproduction thefluidsexpand; thereservoirpressure,drawdownpressure,andproductionratedecreasewith timeuntilproductionoffluidsisnolongereconomical.Attheendofprimary production,thereservoiriseitherabandonedor secondaryproduction methods areemployed.Duringsecondaryproduction,anaqueousphasecontainingsalts andotherdissolvedsolids(brine)orhydrocarbongasisinjectedthrough
2 AnIntroductiontoMultiphase,MulticomponentReservoirSimulation
injectorwellstorepressurizethereservoiranddrivemobilehydrocarbons towardtheproducerwells.Whentheinjectedfluidisanaqueousphase,itis oftenreferredtoas waterflooding.Secondaryrecoverycanoccurfordecades andcontinuesuntiltheprocessbecomesuneconomical,usuallyduetothe decreaseinproductionrateofhydrocarbonsandincreasein watercut (volume percentofproducedfluidsthatisbrine).Finally, tertiaryrecovery or enhanced oilrecovery (EOR)involvesinjectionofotherfluids(steam,carbondioxide, surfactants,polymers,microbes,etc.)notoriginallypresentinthereservoir. ThepurposeofEORistorecover unswept (bypassed)and/or residual (capillarytrapped)hydrocarbonsusuallybyreducingthe interfacialtension betweenphases,increasingthe viscosity ofthedisplacingfluid,ordecreasing theviscosityofthehydrocarbons.Enhancedoilrecoverymethodsareonly pursuedifeconomicallyviable.
1.3Definitions 1.3.1Phasesandcomponentsinsubsurfaceporousmedia
A phase (a)isaregionofspacewithuniformphysicalproperties.Subsurface reservoirsmayhaveseveralfluidphases;inthistextuptothreeareconsidered: liquidaqueousphase(w),liquidoleicphase(o),andgaseousphase(g).Other fluidphasessuchas microemulsions and supercriticalfluids arenotdiscussed inthistext.Asolidphase(s)consistsoftherockmatrix. Fig.1.1 showsa cartoonofareservoirwiththreefluidphasessealedbycaprock.
A component (k)isauniquechemicalspecies.Forexample,water(H2O), carbondioxide(CO2),methane(CH4),decane(C10H22),sodiumchloride (NaCl),andcalciumcarbonate(CaCO3)areallexamplesofcomponentsthat mayexistinareservoir.Aphasemaycontainmanycomponentsandthesame componentmaybepresentinmultiplephases. Compositionalreservoirsimulators (seeChapters7,8and10)areusedtomodelflowandtransportof individualcomponents.However,subsurfacereservoirsmaycontainthousands
FIGURE1.1 Threefluidphases,aqueous(blue),oleic(brown),andgaseous(yellow),andasolid phase(rockmatrix)inageologicaltrap.Importantlytheoleicandgaseouszonescontainconnate waterduetocapillaryforces.
ofuniquecomponents,modelingofwhichisneitherpracticalnorcomputationallyfeasible.Inpractice,componentsofsimilarsizeandphysicaland/or chemicalpropertiesarelumpedtogethertoform pseudocomponents.Most compositionalsimulatorsmodelthreetotensofpseudocomponents.
Aspecialcaseofthecompositionalmodelisthe blackoil (or b)model.A blackoilmodelcontainsthreepseudocomponents(water,oil,andgas)inupto threefluidphases(aqueous,oleic,andgaseous).Theoilpseudocomponent consistsofallcomponentspresentinaliquidhydrocarbonphasewhenbrought to standardconditions (sc)oftemperature(Tsc ¼ 60 F)andpressure (psc ¼ 14.7psia).Likewise,thegasandwaterpseudocomponentconsistsofall componentsinahydrocarbongaseousphaseandaqueousphase,respectively, atstandardconditions.Intheblackoilmodel,thegascomponentmaybe dissolvedintheoleicphaseatreservoirconditions(T > Tsc; p > psc)andoil mayormaynotbevolatilizedinthegaseousphase,buttheaqueousphaseis assumedtocontainnooilorgasandnowaterisinthegaseous/oleicphases.In reality,thereisaverysmallsolubility(< 100ppm)ofhydrocarbonsinthe aqueousphaseandremovalofthehydrocarbonsinshallowaquifersisoften thegoalin aquiferremediation strategies.
1.3.2Porosity,saturation,density,andconcentrations Porosity (f)isdefinedastheporevolumedividedbythebulkvolumeofthe reservoirorporousdomain, f ¼ Vp/Vb,andisreportedasafractionorpercentage.The saturation ofaphase(Sa ¼ Va/Vp)isthevolumefractionofthe voidspaceintheporousmediumoccupiedbythatphase.Phasesaturations mustsumtounity,
where Np isthetotalnumberoffluidphasesandsubscripts w, o,and g referto theaqueous,oleic,andgaseousphases,respectively.
Density (r)isamass(e.g.,lbm)perunitvolume(e.g.,ft3)andusuallyrefers toaphase(ra)butmayalsobeusedtodescribeacomponent(rk).Thedensity ofafluidisafunctionofpressure,temperature,andcomposition.Thedensity ofpurewater(H2Owithoutdissolvedsolids)atstandardconditionsis62.37 lbm/ft3,andthedensityofhydrocarbonliquidsisusually(butnotalways)less thanwater.
The poreconcentration ofacomponent(Ck)isdefinedastheamountof component k intheporevolume(Ck ¼ massof k/porevolume).Thus,theunits of Ck arelbm/ft3.Concentrationscanalsobeusedtodefinetheamountof componentwithinaphase, Ck,a ¼ amountof k/volumeofphase a.Finally,the
