An Improved SAGD analytical Simulator：Circular Steam Chambe(4)

Table2

Parametersofthenumericalmodel.Property

Permeability

AveragerelativeoilpermeabilityOildensity

GravityaccelerationHt

DynamicviscosityatsteamtemperaturePorosity

ReservoirthermaldiffusivityReservoirtemperatureSteamtemperature

FormationheatcapacityLatentheatofsteamSteamqualityWaterdensity

Coef?cientofviscositychange,mCoef?cientofaveragevelocity,aInitialoilsaturationInitialwatersaturation

Value25000.4810089.81200.00732

0.000000610240

2,100,0001,740,0009510003.90.48515

UnitmD–

kg/m3m/s2mPa.s%m2/s°C°C

J/(m3°C)J/kg%

kg/m3––%%

Table1

Experimentalparameters,ChungandButler(1988).Properties

Porosity

InitialoilsaturationResidualoilsaturationAbsolutepermeabilityRelativepermeabilityOildensityModelheight

ThermaldiffusivityConstant‘a’Constant‘m’OilviscosityModelthickness

Value39100529300.480.980.210.05070.43.690.03

Unit%%%

Darcy–

gm/cm3m

m2/day––

m2/daym

34A.Azad,R.J.Chalaturnyk/JournalofPetroleumScienceandEngineering82-83(2012)27–37

Relative Permeability

1

Kro0.750.50.2500

KroKroAfter History Matching

0.20.40.60.81

Fig.10.Relativepermeabilitycurvesresultedfromhistorymatching.

Oil or Steam Rate, m3/day/m

0.750.60.45

Circular Model

STEAM

0.30.150

Numerical Model

OIL

Butler-Like Models

02004006008001000

Time, day

Fig.11.Historymatchingofoilproductionandsteaminjectionratebythecurrentanalyticalmodel.

Cumulative Oil or Steam, m3/m

500400300

Butler-Like Models

Numerical ModelCircular Model

2001000200

600

Time, day

Fig.12.Cumulativeoilproductionandsteaminjectionpredictedbythecurrentnumericalmodelcomparedtothenumericalanalysisresults.

Thetransitionzoneoftemperaturehasbeenshownforthreemo-mentsofrecoveryprocessinFig.14.SimilartoButlertheory,thismodelalsoconsidersasharpseparationbetweenthesteamchamberandotherpartsofthereservoirasisshowninFig.14bydashedcurves.However,thelocationofthesteamchamberisfairlypredicted.

6.Conclusions

Inconclusionitcanbesaidthattheproposedmodelwhichbene-?tsfromacirculargeometryofsteamchamberandmethodofslices,canbeaveryhelpfulmethodtobeusedasafastsimulatorinSAGDprojects.Theadvantageofthemodelisthatthefastspeedruntime

Steam/Oil Ratio, m3/m3

3.632.41.81.20.600

200

400

Reis Model

Circular Model

Numerical Model

6008001000

Time, day

Fig.13.Steam/oilratiopredictedduringhistorymatchingcomparedtotheresultsofanumericalsimulator.

A.Azad,R.J.Chalaturnyk/JournalofPetroleumScienceandEngineering82-83(2012)27–3735

Fig.14.Oilsaturationdistributionatthreemomentsoftheanalysis.

isservedbyaphysics-basedtheory,nota‘black-box’algorithm.Thelongtermgoalofusingsuchasimulatoristojoinwithanoptimiza-tionmethodologyforfastdecisionmakingpurposes.However,itshouldalsobenotedthatatthisstagetheproposedmodelhasitslim-itationsanditneedsmoreimprovementforpracticalcases.Inrealprojectswherehorizontalboreholepairsaredrilledclosetoeachothertooptimizetheoilproduction,boundaryeffectsofneighboringwellsshouldbeconsideredinwhichthecircularmodelsisnotcapa-bleofperformingsuchanalysesyet.

NomenclatureLatinacoef?cientofvelocitycspeci?cheatoftheformationdqtdifferential?owofoilattimet(alsoseeFig.6)dAdifferentialareaDDiameter(seeFig.7)DSCDiameterofthesteamchamber(seeFig.8)DStartDiameterofthesteamchamberatthebeginningofthe

process

ggravityHreservoirheightHtreservoirheightkpermeabilityKoabsoluteoilpermeabilityKrorelativeoilpermeabilitylactuallengthLlengthofboreholesLsspeci?clatentheatofsteammcoef?cientofviscosityntheindexofeachsliceqoilproductionrateqssteaminjectionrate(involumeofwater)QintheheatenergyinsidethesteamchamberQouttheheatlossaroundthesteamchamberrradiusSo,icurrentoilsaturationinithsliceSo,Rresidualoilsaturationinithslicetcumulativetimeaftertouchingthecaprock(inEq.(17))TssteamtemperatureTrreservoirtemperatureUnormalvelocitytothesteamchamberedge

Ulocal

UmaxUHXlocalnormalvelocitytothesteamchamberedge

maximumnormalvelocitytothesteamchamberedge

horizontalvelocityofthesteamchamberatseparationlineEq.(8)(seeFig.7)

GreekαβΔSo?ρρoμμos?Φξρwρθ

thermaldiffusivityofthereservoirseeFig.8

(initial?residual)oilsaturationporosityoildensityoildensity

dynamicoilviscosity

dynamicoilviscosityatsteamtemperature?owpotentialfunctionSeeFig.8

densityofwater

densityoftheformationseeFig.7

Acknowledgments

AuthorswouldliketoacknowledgeNathanDeismanforhiscon-tributiononreviewingthispaper.Histechnicalsupportandhelpfulcommentsarehighlyappreciated.

AppendixA.Fundamentalsofthedrainagemodel

Fig.A.1showstheparametersofthedrainagemodelonasmallsegmentofthesteamchamberedgethathasbeenadaptedfromBut-lermodel.Themodelisderivedforaunitlengthoftheboreholeper-pendiculartothepage.BasedonDarcy'slaw:dq?

kΔρ

?ΦdAeA:1T

where:dqk

differential?owofoilpermeability

36A.Azad,R.J.Chalaturnyk/JournalofPetroleumScienceandEngineering82-83(2012)27–37

Fig.A.1.Asmallsegmentofthesteamchambercrosssection.

Δρρo?ρg

μoilviscosity

dAdξ×1.0(unitlength)=dξ?Φ

?owpotentialfunction

Sinceρo?ρg,then:dq?

kρo

?Φdξ:eA:2T

Forthespeci?csituationillustratedinFig.A.1thatthesteam

chamberedgemakesanangleofθtothehorizon,theDarcy'sequa-tionbecomes:

?Φ?ΔΦ

?gsinθeA:3T

dq?kρo

egsinθTdξ:eA:4T

Eq.(A.4)canberecalculatedfordifferentshapesofthesteamchamber.Inthisequation,viscositycanbealsosubstitutedbythefunctionproposedbyButlerthatconsidersthevariationofviscosityinfrontofthesteamchamber.Thisfunctioncomesfromthecombina-tionofEqs.(A.5)and(A.6).T?Tr?Uξ=T?e

α

eA:5T

sr??

T?T??rm

?

μos

sr

:eA:6T

Eq.(A.5)isthesolutionoftheonedimensionalheattransferforan

advancingfrontwiththevelocityofU.However,Eq.(A.6)determinestheviscosityofoilatthedistanceξfromthesteamchamberedge.The

resultofsubstitutingEq.(A.5)inEq.(A.6)isEq.(A.7)whichisiden-ticaltoEq.(4)inthetext.μeξT?

μos:

eA:7T

exp?

mUξ

AppendixB.Energylossaroundthesteamchamber

Thetotalthermalenergylossoutofthegrowingsteamchamberiscalculatedrelatedtothetemperaturedifferencebeforeandaftertheinjectionofsteam:

θ∞

Qside?∫∫ρcΔTdξerdθT:

eB:1T

00

TemperaturedifferencecanbesubstitutedusingEq.(A.5):θ∞??Q?ρceTU??

localξ

side

s?TrT∫∫exp?dξerdθTeB:2T

00

??

Qr

??

θ

side?ρcαeTs?TrT

Udθ

eB:3T

local

∫0

QρcαeTs?TrTrθ

side?

U:

eB:4T

local

Forafullcircle,therateofenergylossoutsideofthesteamcham-beriscalculatedasfollows:dQ????

sidedρcαeTs??TrTDθ

:eB:5T

maxAppendixC.Initialcondition

Whensteamchamber'sdiameterislessthanDStart,Eq.(5)canberewrittenforanysizeofsteamchamber(D)asfollows.Theshapeofthesteamchamberisnoneofthecases‘A’to‘C’(seeFig.A.1).Inthiscase,thecenterofthecircleisontheinjectoranditgrowsbeforetouchingtheproducer.

????0ooghDiStartD1dqKρ?amUmaxξexp

@tt?DdξeC:1T

osStartD?tD

dqK??????D??

oρo?amUmaxξg?StarttD??t?exp

osDdξeC:2T

Start∞

q?∫

K????????

oρo?amUmaxξg?DStarttD??

texpDdξC:3T

Starte0

os

??qKρ????????

D??ooαg?StarttD??

t?

:eC:4T

osmaxStartMaterialbalanceforthesamecasewouldbe:

qd????12????1dDt?dteφΔSoT8πD?8eφΔSoTe2πDT

dt?1

4

eφΔSoTeπDUmaxT:eC:5

T

A.Azad,R.J.Chalaturnyk/JournalofPetroleumScienceandEngineering82-83(2012)27–3737

Therefore,themaximumvelocityatthebeginningoftheproduc-tionisformulated:

??qKρ????????

tD????ooαg?DStartD?1t?eφΔSoTeπDUmaxTeC:osamUStart46T

max???

Kρ??

??D??????U2

oo??α??g?StarttD??4max;initial

eC:7T

osamDStartoU2

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