SPE-96260-PA-P

AlongaHorizontalWellDuring

InjectionPeriod

GuohuaGao,SPE,ChevronCorp.;andYounesJalali,SPE,Schlumberger

Summary

Thispaperpresentsamathematicalmodeldescribingthevariationoftemperaturealongthelengthofahorizontalwellduringtheprocessofwaterinjection.Themodelisobtainedfromatheoret-icaltreatmentaccountingforbothmasstransferandheattransferbetweenahorizontalwellandareservoir.Thetreatmentis1Dlinearinthewellboreand1Dradialinthereservoir.Anumericalalgorithmforreservoirtemperaturecalculationisproposedandananalyticalsolutionisderivedonthebasisofsomerealisticas-sumptions.Theanalyticalsolutioncanbeusedtogeneratethetemperatureprofileinahorizontalinjectionwellforanyassumeddistributionofinjectionprofilealongthelengthofthewell,in-cludinginjectionprofilethatisuniform,skewedtowardtheheelorthetoe,orexhibitssomediscontinuity(e.g.,leakoffintoahighpermeabilitystreakorfracture).Thispaperalsopresentscompari-sonoftemperatureprofilesobtainedwiththeanalyticalsolutiongiveninthispaperandthoseobtainedwithanumericalreservoirsimulatorwithtemperatureoption(ECLIPSE),whichshowsthattheanalyticalsolutionyieldsreasonabletemperaturepropagationprofilesalongthewellbore.Theeffectsofinjectionrateandtheinjectionprofileareanalyzed,andaquickin-situinjectionpattern-recognitionmethodisproposed.Finally,examplesaregiventoshowthepracticalapplicationofthetheoreticalmodel.

Introduction

Useofhorizontalwellsforinjectionpurposesisnowcommon-place.Thisisbecausematuringfieldsproduceincreasingamountsofwater,andhorizontalwellsenabledisposaloflargevolumesoffluid.Therefore,disposalobjectivescanbemetwithafewernum-berofwells.Thisisimportantinoffshoreoperations,inwhichwellnumbersarelimitedbecauseofslotconstraintsontheplatform.Intermsofpressuremaintenance,horizontalwellsarealsoattractive.Insaturatedreservoirs,inwhichvoidagereplacementisfundamentaltoachieveagoodrecoveryfactor,horizontalwellshelpachieveinjectionvolumescommensuratewithproductionvolumesinthefield.Therefore,theyareinstrumentalinreservoirmanagement.Finallyintermsofrecovery,horizontalwellsareeffectiveinachievingagoodvolumetricsweep,particularlyinthickreservoirsinwhichgravitationalforcesandstratificationtendtounderminesweep.

Measurementofinjectionprofileinhorizontalwellsisalsoacommonrequirementforassessingtheefficiencyofdrillingorcompletionprocess,cleanuporstimulationprocess,andinjectionorrecoveryprocess.

Injectionprofileinhorizontalwellscanbeestimatedwith“pro-ductionlogging”technology,butthismayrequirecoiledtubingtoaccessthefulllengthofthewell.Evenwithcoiledtubing,therearepracticallimitsofhowfarthewellcanbelogged(e.g.,currentlyoftheorderof1to2km).“Interventionless”meansofdeterminingtheinjectionprofileinhorizontalwellsarethereforeofinteresttooperators.Afeasibleinterventionlessapproachinhorizontalwellsisefficientcompletiondesignsfordeploymentofthefiberoptic

Copyright©2008SocietyofPetroleumEngineers

Thispaper(SPE96260)wasacceptedforpresentationatthe2005SPEAnnualTechnicalConferenceandExhibition,Dallas,9–12October,andrevisedforpublication.Originalmanuscriptreceivedforreview15July2005.Revisedmanuscriptreceivedforreview28March2007.Paperpeerapproved20May2007.

February2008SPEReservoirEvaluation&Engineering

line.Therearepublishedcasesintheliteraturewithfiberoptic-distributedtemperaturesensorsdeployedinhorizontalwells.Thesearedeployedeitherthroughanextendedtailpipe(i.e.,“stinger”completion)(Al-Asimietal.2002/2003),orthroughagrooveinthesandscreen.

Thetechnologyformeasurementofdistributedtemperatureonaperiodicorcontinuousbasisisawell-establishedtechnologyandhasbeenwidelyappliedtoreservoirandproductionmonitoring(Brownetal.2000,2003,2006;LaurenceandBrown2000;Ou-yangandBelanger2006;Nathetal.2007;Lee2006;Johnsonetal.2006).

Asthereisongoingeffortinimprovingthedeploymenttech-nologyofsuchsensorsinhorizontalwells,itisthereforeappro-priatetoinvestigatetheinterpretationmethodologythatisrequiredtoinferthecharacteroftheinjectionprofilefrommeasurementsoftemperatureprofile.

Inferringinjectionprofilebasedonmeasuredtemperaturedataisaninversionproblem(Tarantola1982,2004).Onemethodofestimatinginjectionprofileandrockpropertiesonthebasisofmeasuredtemperaturedataistofollowtheproceduresimilartohistory-matchingproductiondatawithapplicationsofnumericalreservoirsimulationwitheithertherandomizedmaximumlikeli-hood(RML)method(Oliver1996;Reynoldsetal.2004;Gaoetal.2006)ortheensembleKalmanfilter(EnKF)method(Zafarietal.2005;GuandOliver2006;Wenetal.2006;Evensen2007).Al-thoughnumericalthermalreservoirsimulationisavailablenowa-days,onelimitationofapplyingthermalreservoirsimulationisthatitisveryexpensivetoobtainsatisfactorynumericalresults,whichrequireveryfinegrids,especiallynearthewellbore.TheLBFGSalgorithm,aquasi-Newtonalgorithm,hasbeenprovedanefficientandrobustoptimizationalgorithmforlarge-scalehistory-matchingproblems(Zhang2002;Gao2005;GaoandReynolds2006).TheLBFGSalgorithmandotherquasi-Newtonalgorithmsrequirecalculationofthegradientinformationofhistory-matcheddatawithrespecttomodelparameterstobeadjustedduringtheprocessofhistorymatching.Thegradientinformationcanbeef-ficientlycalculatedwiththeadjoiningmethod(Lietal.2000;Li2001;Zhang2002).However,developmentoftheadjoiningalgo-rithmsforthermalreservoirsimulatorsisquitetedious.

Anothermethodofestimatinginjectionprofileonthebasisofmeasuredtemperaturedataistodevelopanalyticalorsemi-analyticalmodelsoftemperaturepropagationduringwater-injectionperiodortemperaturerecoveryduringshut-inperiod.Goodworkhasbeendonebyauthorswhohaveconsideredtheinterpretationofmeasuredtemperaturedatafromtheshut-inpe-riodtoestimatetheinjectionprofile.Thismethodoriginatedinworkdoneonverticalwellsandisbasedontheconceptthatthedegreeoftemperaturerecovery(duringshut-in)observedacrossvarious“sandface”intervalsisinverselyrelatedtotheefficiencyofinjectionintothatinterval(Nowak1953).Theseworkshavebeenadaptedtohorizontalwells(BuiandJalali2004).Thereisalsoaninnovativemethodofestimatingtheinjectionprofilefromtem-peraturedataacquiredimmediatelyafterashut-inperiod,inventedandreportedbyBrownetal.(2003).However,shut-inisnotadesirableeventininjectionoperations,especiallyinloosesands,becauseoftheproblemofbackflowoffinesandpluggingofthecompletion.

Severalmodelsforpredictingtemperaturedistributionalongverticalwellbores(Ramey1962;Alvesetal.1992;Izgecetal.

131

2007;Mendesetal.2005)andhorizontalwellbores(Yoshiokaetal.2005a,2005b;Romeroetal.2005;OuyangandBelanger2006)havebeeninvestigated.Thesemodelseitherconsideronlyheatorenergyexchanges(heatconduction)betweenwellboreandreservoirbutnomassexchangesexceptatthebottomofthewell-bore,orassumethatthereservoirtemperaturedistributionisinasteadystateandsingle-phaseflowinthereservoir.Suchtreatmentsarevalidforwellsperforatedinaveryshortintervalatthebottomorvalidforproductionwells.However,theyarenotvalidforinclinedorhorizontalinjectorsbecausethetemperatureofinjectedfluidisquitedifferentfromthereservoirtemperature,andtheinjectedfluidwilldisturbthetemperaturefieldinareservoirsig-nificantly.Moreimportantly,theinvadedfluidhasdifferentther-malpropertiesthantheformationfluid,andtheshockwave(dis-continuity)ofwaterandoilflowrateatthewaterfrontmakethemodelmorecomplicated.

Ontheotherhand,manyauthorsinvestigated1Dtemperaturemodelinareservoirintowhichhotwaterorcoldwaterisinjected(Sumnu-DindorukandDindoruk2006;Kocabas2006;Nasrabadi,etal.2006;Platenkamp1985;Rouxetal.1980).Inthesemodels,thewellboretemperatureisassumedknownanddoesnotchangewithtime.Thisisnotthecaseofhorizontalinjectorsinwhichwellboretemperatureisafunctionofbothtimeandlocation(dis-tancefromtheheel).

Theobjectiveofthispaperistodevelopamathematicalmodelofpredictingwellboretemperaturethatispracticalforrealfieldin-situapplication.Oneadvantageofthemodelisthattheinjectionprofilecanbeestimatedbeforeanyshut-in.Moreimportantly,thetemperaturedistributioninthereservoiratthetimeofshut-incanbepredictedwiththismodel,whichprovidestheinitialconditionfortemperaturerecoveryanalysisforthesubsequentshut-inpe-riod.Althoughthemodelcanbealsousedastheforwardmodelforwellboretemperatureinversiontodeterminehowinformationaboutinjectionprofilemaybeobtainedfromanalysisoftempera-turedataduringtheinjectionperioditself,weonlyfocusonthedevelopmentoftheforwardmodelinthispaper.

PhysicalDescription

Letusassumewehaveabox-shapedreservoirwithahorizontalwellplacedinthemiddle.Also,letusassumetherearelayersonthetopandbottomofthisbox,whichcanconductheatbutnotfluid.Thereservoiranditsboundinglayersareinthermalequi-libriumconformingtoastaticgeothermalgradient.(Typicalvaluesare1to2°C/55m)Atinitialcondition,therefore,thereisacon-stanttemperaturealongthelengthofthehorizontalwell.Now,letussupposethiswellisfullysealedfromthereservoir.Ifweexposetheheelofthewelltoafluidstreamthatiscoolerthanthereser-voir,thisstreamwillflushthewell,andatemperatureprofilewillgetestablishedinthewellthatrisesfromthetemperatureattheheeltotheinitialreservoirtemperaturesomewherebetweentheheelandthetoe.Thistemperaturechangeissensedatthewell-bore-reservoirboundary,resultinginheatfluxfromthereservoirtowardthewellbore.Thisradialheatfluxwillgraduallypropagateintheaxialdirectionfromtheheeltowardthetoeofthewell.Asimilarphenomenonoccursifweallowfluidinfluxfromthewell-boretothereservoir,exceptthattheheatfluxintheoppositesensewillbeattenuated,comparedwiththecasewhenthereisnofluidinflux.Forthegeneralproblem,therefore,thereisalinearflushingofacoldfluidalongthewell,triggeringheatfluxfromthereser-voirtothewellandanattenuationofheatfluxthatiscausedbydistributionoffluidinfluxfromthewellboreintothereservoir(masstransfer).Theconditionsunderwhichthisscenarioholdsarebroadlythefollowing:theendsofthewellarefarfromthereser-voirboundaries;andtheperiodduringwhichthesolutionholdsisequaltothetimeittakesforthelocationofthetemperaturefrontofinjectedfluidtoapproachthebedboundaries.

Tosimplifythemathematicalproblem,wemakethefollow-ingassumptions:

•Massandheattransferinreservoirinthedirectionparalleltothewellboreisneglected.

•Injectionprofileisassumedconstantwithtime.•Injectionispiston-like,andformationisisotropic.

132

•Injectedfluidisincompressible,andthereservoirtemperat-urefrontiswithintheupperandlowerboundarybedsofthereservoir.ReservoirTemperatureModel

EnergyAnalysisforTwo-PhaseRadialFlow.Letusconsideratinyradialelementbetweenrandr+drofunitlength,asshowninFig.1.

Thetotalheatstoredinthiselementiscomposedofthreeterms:theheatstoredinthewaterphaseQt),theheatstoredintheoilphaseQwס2␲rdr␾sw(x,r,t)cwT(x,r,t),andtheheatstoredintherockoסQ2␲rdr␾[1−sw(x,r,t)]coT(x,r,rס2␲rdr(1−␾)crT(x,r,t),thatis,

Qres==Q2␲wr+drTQo͑x+,Qr,r

t͓͒sw͑x,r,t͒...........................␾͑cw−co͒+co␾+͑1−␾͒cr͔,

(1)inwhichT(x,r,t)isthetemperatureinthereservoir;␾isporosity;co,cw,and3crare,respectively,theheatcapacityofoil,water,androck(J/(m.K));andsw(x,r,t)iswatersaturation.

Letqw(x,r,t)andqo(x,r,t)denote,respectively,thewateroilvolumeflowrateinreservoirandq␲␩rѨT

and

T͑x,r,t͒ס2denotetheconductiveheatinwardflux.TheenergyconservationѨrequationforthisradialelementis:

ѨQres

Ѩt

=qrin−qrout,....................................(2)inwhichqrinandqroutdenotethetotalheatfluxintoandoutofthetinyradialelementdr.qheatflux,carriedbyriniscomposedoftwoterms:theconvec-tivetheoilandwaterflowingintotheelementthroughtheinnersurfaceatr,[ctheconductiveheatfluxintowqthew(x,r,t)+ctinyelementoqo(x,r,t)]T(x,r,t),andthroughtheoutersurfaceatr+dr,qT(x,r+dr,t).qtheconvectiveheatfluxcarriedroutisalsocomposedoftwoterms:bytheoilandwaterflowingoutoftheradialelementthroughtheoutersurfaceatr+dr,[cwqw(x,r+dr,t)+coqo(x,r+dr,t)]T(x,r+dr,t),andtheconductiveheatfluxoutofthetinyelementthroughtheinnersurfaceatr,qT(x,r,t).qrin=͓cwqw͑x,r,t͒+coqo͑x,r,t͔͒T͑x,r,t͒+qT͑x,r+dr,t͒,qrout=͓cwqw͑x,r+dr,t͒+coqo͑x,r+dr,t͔͒T͑x,r+dr,t͒

+qT͑x,r,t͒,

qѨT

T͑x,r,t͒ס2␲␩r

,inwhich␩isthethermalconductivityformation(J/(m.K.h)),Ѩrofandweassumethatitisnotaffectedbythepresenceofoilorwaterinporevolume.

SubstitutingthepreviousequationsandEq.1intoEq.2yields:Fig.1—Massandheatflowinradialdirection.

February2008SPEReservoirEvaluation&Engineering

͓sѨT

w␾͑cw−co͒+co␾+͑1−␾͒cr͔

Ѩt+␾͑c͒TѨsww−coѨt=

1

2␲rͫ2␲␩ѨѨrͩrѨTѨrͪ−cѨ͑qwT͒Ѩ͑qoT͒wѨr−co

Ѩrͬ.....(3)Forpiston-likeinjection,thereservoircanbedividedintotwo

mainflowregimes:waterflowregime(rwͱr2qwi͑x͒tw+

␲␾͑1−ss,....................(4)

or−wi͒representsthepositionofwaterfrontatlocationxandtimet.Inthewaterflowregime(r)סqwѨt=ѨrѨrͩrѨrͪ−2␲rѨr,....................(5)

inwhichawססIn0.theqoilflow(1−sregimeor)␾(cw−c(ro)+co␾+cr(1−␾).

(x,r,t)decreaseswf(x,t)aѨT␩Ѩ

ͩѨTͪcoѨ͑qoT͒oѨt=rѨrrѨr−2␲rѨr,.....................(6)inwhichaoסswi␾(cw−co)+co␾+cr(1−␾).Theinitialconditionisgivenby

T͑x,r,0͒=TR........................................(7)Theboundaryconditionsarespecifiedas

T͑x,rw,t͒=Tw͑x,t͒,..................................(8)and

limr→ϱ

T͑x,r,t͒=TR.....................................(9)

Letrrx␩t

Dסr,xDס,andtDס2denote,respectively,thedimen-sionlessradius,wLdimensionlesscwrdistancew

fromtheheel,anddimen-sionlesstime.BothEqs.5and6canbenormalizedas

␣ѨTѨ2T1ѨT2Ѩ͑␯T͒Ѩt=2+−.......................(10)DѨrDrDѨrDrDѨrDInthewaterflowregime,␣ס␣awcwqwi

wס,and␯ס␯;whereasintheoilflowregime,␣ס␣סacwwס

4␲␩ocoqo

owaterinjectionrate.

c,and␯oס␲␩.␯wisthedimen-sionlessw4DiscontinuityattheWaterFront.Becausewatersaturationswaterflowrateqw,spatialdomainatw,andoilflowrateqthewaterfront,(i.e.,o,arediscontinuousinthe␣w󰀎␣o,and␯w󰀎␯o),such

discontinuityinthoseparametersmaycausediscontinuityofT,ѨT

Ѩr,

andѨ2T

2.TisacontinuousfunctionofrbecausetermѨtheconductive

isr

includedinthemodel.Otherwise,thecontinuityofTinthespatialdomaincannotbeguaranteed.FromEqs.10and11,itis

obviousthatѨ2T

2isnotcontinuousatthewaterfront.continuityofѨHowever,the

ther

temperaturegradientisnotclear.Tosolveequations,itiscriticaltoknowwhetherѨT

the

Ѩr

iscontinuousorthedifferenceofѨT

acrossthewaterfrontifitisLetusconsiderѨrnotcontinuous.

atinyradialrѨrelementbetweenrwf(x,t)and

wf

wf(x,t+⌬t)forasmall⌬t,⌬r≈⌬t.Theheatsstoredinthistinyelementattimet(sѨtwסswi)andtimet+⌬t(swס1−sor)are,re-February2008SPEReservoirEvaluation&Engineering

spectively,Qt͒ס2␲arѨrwf

res͑owf͑x,t͒ѨT⌬t,andQres͑t+⌬t͒ס2␲awrwf

͑x,t͒Ѩrtwf

TheѨtT⌬t.heatfluxesintoandoutofthetinyelementintervalof⌬tare,respectively,qqѨT

duringthetime

rinסcwwT+2␲␩r

Ѩr⌬t,rסrwf

͑x,t+⌬t͒

andqqѨT

routסcooT+2␲␩ͩrѨrͪͬͩͪͬ⌬t.

Theenergyͫͫconservationrסlawrwf

͑x,tgives

͒

Qres͑t+⌬t͒−Qres͑t͒=qrin−qrout,......................(11)whenѨT−

ͩͪ⌬t→0,and⌬r→ͩ0,

ͩͪrѨTѨrͪ→,andrѨTrסrwf͑x,t͒

ͩrwfѨrnote,Ѩr→r͑x,t͒ѨT+,inwhichѨT−andѨT+de-respectively,rסrwf͑x,t+⌬t͒wfthetemperatureѨrgradientbehindѨrͪ͑x,t͒

ͩͩͪandinѨrfrontͪofthewaterfront.ApplyingQ11yields:

res(t),Qres(t+⌬t),qrin,andqoutintoEq.͑c+

−

wqw−coqo͒T+2␲␩ͫͩrѨTѨrͪ−ͩr

ѨTѨrͪͬ=2␲͑aѨrwf

w−ao͒rwf͑x,t͒

Ѩt..............(12)FromEq.4,wehaverѨrwfqw

wf

Ѩt=

␲␾͑1−s............................(13)or−swi͒UseͩofEq.13ѨTͪ+

−

ͩinEq.12yields:

ѨT+

͑qo−qw͒

ѨrѨrͪ=

co2␲␩rT.......................(14)

wf

Notethatqsoweprovedthat

ѨT

oסqwiatrסrwf,ousatthewaterfront.

Ѩrisalsocontinu-Withtheproperlydefinedinitialcondition,boundarycondi-tionsgivenbyEqs.7through9,andthecontinuousconditionoftemperatureanditsgradientatthewaterfront,wecanobtainthereservoirtemperaturedistributiongivenoilflowrate,qitisdifficulttogettheo(x,r,t),beyondthewaterfront.However,exactanalyticalsolutionofthereservoirtemperaturemodelbecauseboththelocationofthewaterfront,rwf(x,t),andtheoilflowratebeyondthewaterfront,qo(x,r,t),aretimedependent.

NumericalSolution.Inthissection,anewfinitedifferenceequa-tionisdevelopedtosolvethereservoirtemperaturemodel,

problemofdiscontinuityofѨ2especiallytoaccountfortheT

thewaterfront.

Ѩr

2at

Letusconsidertworadialgridblockswithdifferentgridsizesof⌬rD,i−1סrD,i−rD,i−1and⌬rD,iסrD,i+1−rD,i.Let␣␯intheithgridblockbetweeniand␯rirepre-sentthevaluesof␣andD,iandr␣D,i+1␣.Iftheithgridblockiswithinthewaterflowregime,theniסw,and␯iס␯w.Otherwise,␣iס␣o,and␯i=␯o.Attimetj,oneofthenodesischosensuchthatitisexactlyonthewaterfront.LetTi,j,Ti−1,jandTi+1,jdenote,ͩrespectively,ͪ−ͩthetemperatureͪat

rѨ2TѨ2T+

D,i,rD,i−1,andrD,i+1attimetj;

anddenotethesecond-orderѨr2Ѩr2Di,jDi,␧→0,respectively,derivativesinwhichoftemperature␧>0.FromtheatTaylor’srj

D,i−␧andseries,rD,i+we␧when,j+ͩѨT⌬r1D,i+

2ͩѨ2T+Ti+1,j=Ti2⌬r2

D,i.............(15)i,j

−

Ti−1,j=Ti,j−

ͩѨrDͪѨTѨrD

ͪ⌬r1

ͩѨrD

ͪhave

Ѩ2TD,i−1+

ͪi,j

⌬r2D,i−1..........(16)i,j

2Ѩr2D

i,j

133

Fig.2—Reservoirtemperaturedistribution,numericalsolution.

SubstitutingͩѨTͪEq.10intoEqs.15andͩ16gives

=Ti+1,j−Ti,j␣i⌬rD,iѨT

ͩѨrDi,j

⌬rD,i͑1−ci͒−

2͑1−ci͒ѨtD

ͪ..............(17)ѨTTi,j−Ti−1,j␣i−1⌬ѨrD

ͪ=

i,j

⌬rD,i−1͑1+ci−1͒+

rD,i−1ѨT

2͑1+ci−1͒ͩi,j

ѨtD

ͪ,.........(18)

i,j

inwhichc͑1−2␯i͒⌬rD,i͑1−2␯i−1͒⌬rD,i−1

iס,ci−1ס,ͩ2rand

D,i2rD,i

ѨT

ѨtD

ͪ=

Ti,j−Ti,j−1

⌬t.................................(19)

i,j

D,j

ByuseofEq.19inEqs.17and18,wecanderivethefollowingfinitedifferentialequation:

−Ti−1,j+fi,jTi,j−giTi+1,j=hi,jTi,j−1,.......................(20)foriס2,...,Nr−2andjס1,g⌬r2,...Nt,inwhichfi,jס1+gi+hi,j,

D,i−1͑1+ci−1͒⌬rD,iס

⌬r,andhi−11+ci−1

D,i͑1−ci͒i,jס2⌬tD,jͫ1−ci␣i⌬ri+␣i−1⌬ri−1

ͬ.UseoftheboundaryconditionT0,jסTw(xD,tj)foriס1inEq.20yields:

f1,jT1,j−g1T2,j=T0,j+h1,jT1,j−1..........................(21)Similarly,useoftheboundaryconditionTNr,jסTRforiסNr−1inEq.20yields:

−TNr−2,j+fNr−1,jTNr−1,j=hNr−1,jTNr−1,j−1+gNr−1TR............(22)GivenTi,j−1(iס1,2,...,Nbysolvingr−1),T0,thej,andTlinearR,wecandetermineTi,j(iס1,2,...,Nr−1)equationsgivenbyEqs.20through22.

Figs.2and3showthenumericalresultsofreservoirtempera-turedistributionandtemperaturegradientdistributionforthree

Fig.3—Reservoirtemperaturegradient,numericalsolution.134

differentcasesattס12(hr).TheparametersusedforthenumericalsolutionarelistedinTable1.Thewellboretemperatureisafunc-tionoftimegivenby:

Tw0͑t͒=50+48exp͑−0.3964t͒........................(23)Forcaseone,wesetqoסqinwiandcwhichoסthecwbeyondthewaterfront,whichrepresentsthecasewaterfrontisartificiallymovedtoinfinity.Forcasetwo,wesetq6beyondthewaterfront.Forcasethree,weoסsetqwiandcqoס2.13×10beyondthewaterfront,whichrepresentstheextremeoסqwiandccaseofoסno0flowbeyondwaterfront.Fig.2showsthattemperaturedistribu-tionsfordifferentcaseshavenosignificantdifference.Thetem-peraturebeyondthewaterfrontisveryclosetotheinitialreservoirtemperature,soassumingqo(x,r,t)סqwiandcoסcwbeyondthewaterfrontwillnotsignificantlyaffecttheenergyexchangebe-tweenthesetwoflowregimes,(i.e.,agoodapproximationofthereservoirtemperaturedistributioncanbeobtainedbyartificiallymovingthewaterfronttoinfinitybecausethefrontpartoftheinjectedfluidhasgivenallitsheattothematrix).Fig.3showsthatthetemperaturegradientspredictedforthreedifferentcasesarealmostidenticalexceptaslightvariationatthewaterfrontforbothcasetwoandcasethree.

AnalyticalSolution.Ananalyticalsolutionofthereservoirtem-peraturemodelcanbeobtainedbyartificiallymovingthewaterfronttoinfinity.Let⌬T(x,r,t)סTR−T(x,r,t)denotethetempera-turedropinthereservoir,andEq.10becomes

␣Ѩ⌬TѨ2⌬T1−2␯͑xD͒Ѩ⌬TѨtD=Ѩr2+,.....................(24)DrDѨrDinwhich␣ס␣aw

wס.Let⌬T(x,rD,tD)סr␯Dw(x,rD,tD),andEq.24becomes

cw

ѨwѨ2␣Ѩt=w1ѨwwѨr2+−␯22..........................(25)DDrDѨrDrDTheboundaryconditionsandinitialconditioncanberewrittenasw͑xD,1,tD͒=⌬Tw͑xD,tD͒.............................(26)

rlim,rD→ϱ

w͑xDD,tD͒=0.................................(27)

w͑xD,rD,0͒=0......................................(28)ApplyingLaplacetransformtoEqs.25through28gives:Ѩ2w1ѨwѨr2rѨr␣pͩ1−␯2+−␣pr2w=0...................(29)DDDD

ͪw͑xD,1,p͒=⌬Tw͑xD,p͒,rlimw͑xD→ϱ

D,rD,p͒=0..........(30)

Eq.29isamodifiedBesselequationwithageneralsolutionofw͑xD,rD,p͒=c1I␯͑͌␣prD͒+c2K␯͑͌␣prD͒............(31)

⌬T͑xD,rD,p͒=c1r␯DI␯͑͌␣prD͒+c2r␯DK␯͑͌␣prD͒,......(32)

February2008SPEReservoirEvaluation&Engineering

inwhichIv(z)andKv(z)arethesecondkindmodifiedBesselfunctions,andctheboundaryconditions.1andc2aretwoconstantsthataredeterminedbyWiththeboundaryconditionsatinfinity,rlim⌬T͑xD,rD,tD͒ס0,weknowc1ס0.AndwiththeboundaryconditionD→ϱ

atwellbore(rDס1),⌬T(xD,1,p)ס⌬Tw(xD,p),wehave

c⌬Tw͑xD,p͒

2ס

.Thetemperaturedistributioninreservoiriscom-pletelyK␯͑͌determined␣p͒

bythewellboretemperature⌬Tdimensionlessinjectionrate␯,andthedimensionlessw(xthermalD,p),thepa-rameter␣.⌬T͑xTw͑xD,p͒D,rD,p͒=

KDK␯͑͌␣prD͒...............(33)

␯͑͌␣p͒

r␯TheBesselfunctionK␯(z)hasthepropertyof

dK␯͑͌␣prD͒

drD

=−͌␣p͑K␯−1͑͌␣prD͒

+␯͌␣prK␯͑͌␣prD͒͒.................(34)DThus,wehave:

Ѩ⌬TdK␯͑͌␣prD͒ѨrD=⌬Tw͑xD,p͒Kͫ␯r␯D−1K␯͑͌␣prD͒+r␯D␯͑͌␣p͒drD

..........................ͬͫͬ(35)rѨ⌬TD

Ѩrp,␯͒,....................(36)

D

=−⌬Trw͑xD,p͒f͑D=1

inwhichf͑p,␯͒ס͌␣p␶K␯−1͑͌␣p͒

.Fig.4showsthenumericalK␯͑͌(solid␣p͒

curve)andanalytical(dottedcurve)resultsofreservoirtemperaturegradientatthewellbore.Bothcurvesarealmostidenticalexceptforveryearlytime.WellboreTemperatureModel

LetQinjectionwi(x)denotethewellboreflowrate(m3/h)andqrateperunitlengthalongwellbore(mwi2(x)denotethe/h).Ifweignorethecompressibilityofinjectedfluidinthewellbore,themassconservationequationgives

ѨQwi

Ѩx=−qwi͑x͒.......................................(37)Integratingtheaboveequationfromheel(xס0)totoe(xסL)yieldsthetotalinjectionrateattheheel:QLinj=

͐0

qwi

͑x͒dx,

...................................(38)

Fig.4—Comparisonbetweennumericalandanalyticalsolutions.February2008SPEReservoirEvaluation&Engineering

inwhichLdenotesthelengthofthehorizontalsectionofwellbore,

andQwi(x)isgivenbythefollowingintegration:Qxwi͑x͒=Qinj−

͐0qwi

͑␵͒d␵.

...........................(39)

LetTandqw(x,t)denotetemperaturedistributionalongthewellboreTw(x,t)denotetheheatfluxperunitlengthfromformationintothewellborecausedbyheatconduction,thatis,qѨTTw͑x,t͒=2␲␩ͩr

Ѩrͪ.............................(40)

r=rw

Thetotalheatstoredinthetinyelementdxis

Qwell=cwAwTw͑x,t͒dx,...............................(41)

inwhichA22wס2␲rwdenotesthewellborecross-sectionalarea(m).Theenergyconversationequationforelementdxis

ѨQwell

Ѩt=qwin−qwout..................................(42)Here,qwinandqwoutdenotetheheatrateflowingintoandoutofthetinywellboreelementdx.qflowingwiniscomposedoftwoterms:theheatcarriedbythefluidintotheelementthroughthewellborecross-sectionalareaatx,cwQwi(x,t)Tw(x,t),andtheheatflowingfromtheformationtothewellboreelementthroughthewellboresurfacecausedbyheatconduction,qTw(x,t)dx.qheatcarriedbythefluidflowingwoutisalsocom-posedoftwoterms:theoutoftheelementthroughthewellborecross-sectionalareaatx+dx,cwQwi(x+dx,t)Tw(x+dx,t),andtheheatcarriedbythefluidflowingoutoftheelementthroughthewellboresurface,cwqwi(x,t)Tw(x,t)dx.qwin=cwQwi͑x,t͒Tw͑x,t͒+qTw͑x,t͒dx...................(43)qwout=cwQwi͑x+dx,t͒Tw͑x+d..........................x,t͒+cwqwi͑x,t͒Tw͑x,t͒dx

(44)Substitutingqwin,qout,andEqs.40and41intoEq.42givescwwAѨTѨt+cѨTw

ѨTw

wQwiѨx−2␲␩ͩrѨrͪ=0.............(45)

r=rw

Theinitialconditionis

Tw͑x,0͒=TR........................................(46)Theboundaryconditionattheheelis

Tw͑0,t͒=Tw0͑t͒......................................(47)AndTw0(t)canbemeasuredwiththeDTSsystem.

Similarly,let␤͑xcwQwi͑xD͒1

D͒ס=͐␯͑␵͒d␵denotethedimension-lesswaterflowrate,and⌬4␲␩TLxD

w(xD,t)סTR−Tw(xD,t).ThenEq.45canbenormalizedas

Ѩ⌬Tw

Ѩ⌬TwѨ⌬TѨtD+2␤͑xD͒ѨxD−ͩrD

ѨrD

ͪ=0...............(48)

rD=1

Theinitialconditionandboundaryconditionbecome⌬Tw͑xD,0͒=0.......................................(49)⌬Tw͑0,tD͒=⌬Tw0͑tD͒.................................(50)TakingtheLaplacetransformofEq.48incombinationwiththeinitialconditiongivenbyEq.49yields:⌬Tw+2␤͑xD͒

Ѩ⌬TwѨ⌬TѨxD−ͩrD

ѨrD

ͪrD=1

=0................(51)

UseofEq.36inEq.51yields:

Ѩ⌬Tw

1+f͑p,␯͑xѨx=−D͒͒2␤͑x⌬Tw..........................(52)DD͒135

Fig.5—Comparisonoftemperatureprofilesobtainedwiththeanalyticalsolution(symbols)andanumericalreservoirsimula-tor(curves).

TheanalyticalsolutionofEq.52isgivenby

⌬Tw͑xD,p͒=⌬Tw0͑p͒F͑xD,p͒..........................(53)F͑x͑p,␯͑␵͒͒

D,p͒=expͫ−

͐xD1+f0

2␤͑␵͒

d␵ͬ................(54)

TakingtheinverseLaplacetransformof⌬Ttemperatureprofile:

w(xD,p)givesthewellbore⌬Tw͑xD,tD͒=⌬Tw0͑tD͒*F͑xD,tD͒......................(55)Herethesymbol“*”representstheconvolutionoperator.TheanalyticalsolutiongivenbyEq.55isasolutionvalidforvariablewelltemperature.

ResultsandApplications

ComparisonofAnalyticalandNumericalSolutions.BuiandJalali(2004)providedadetailedanalysisoftemperaturebehavior

Fig.6—Effectsofinjectionrateonwellboretemperature.136

inhorizontalwaterinjectors,inwhichacommercialreservoirsimulator(ECLIPSE)wasusedasthenumericalanalysistool.Fig.5showsthecomparisonofthetemperatureprofilesobtainedwiththeanalyticalsolutiondiscussedabove(representedbysymbolsinFig.5)andthenumericalsimulationresults(representedbycurvesinFig.5)atdifferenttimes(1.2hr,2.4hr,4.8hrand12hr)foracasewhentheinjectionprofileisperfectlyuniformalongthehori-zontalwell.TheparametersusedforthissyntheticcasearelistedinTable1.Thetemperatureattheheel(xס0)isgivenbyEq.23.AsshowninFig.5,fortheveryearlytime(t<2.4hours),thewellboretemperaturepropagationpredictedwiththeanalyticalso-lution(opencirclesorsquares)issomewhatslowerthanthatpre-dictedwiththenumericalsolution(dashedordottedcurves).Theassumptionsofpiston-likeinjectionandincompressiblefluidandthetreatmentofartificiallymovingthewaterfronttoinfinitymayintroducesomeerrorstoourmodelandtheanalyticalsolution.Atveryearlytime,thewaterfrontislesslikeapiston-likeinjection,andmoreerrorisintroduced.Whiletimeincreases,sucherrorreduces,andthedifferencebetweentheanalytical(solidtrianglesandsolidcircles)andnumericalsolutions(dashed,dotted,andsolidcurves)vanishes.Thegoodagreementbetweentheanalyticalsolutionandnumericalsolutionindicatesthattheassumptionsofpiston-likeinjection,incompressiblefluid,andthetreatmentofartificiallyremovingthelocationofwaterfronttoinfinityarere-alistictosimplifythereservoirtemperaturemodelduringwaterinjectionthroughahorizontalwell.

EffectsofInjectionRate.Fig.6illustratestheeffectsofinjectionrateonwellboretemperatureprofile.InFig.6,theinjection-rateprofilealongthewellboreisuniform,(i.e.,injectionrateisacon-stantalongthewellbore).ThetemperatureattheheelisalsogivenbyEq.23.Thetimeissettot0.25,0.5,1.0,and2.0representDסdifferent5.Differentinjectionvaluesratesofalong␯ס0.15,thewellbore.Thewellboretemperaturedecreasesfasterwhenmorewaterisinjectedorthevalueof␯becomesgreater.Forextremecases,ifnowaterisflushedthroughthewellboreandinjectedintothereservoir(␯ס0),thewellboretemperatureisthesameasthatofthereservoir,(i.e.,Tlargeinjectionrate(␯→wϱס),TtheRס98temperature°C).Onthealongotherthehand,wholeforwell-veryboredecreasesinstantaneouslyfromthereservoirtemperaturetothetemperatureofinjectedfluid.

EffectsofInjection-RateProfile.Fig.7showsthreedifferentpatternsofinjection-rateprofiles,decreasingratepattern(blacklinewithopencircles),uniformratepattern(blacklinewithnosymbol),andincreasingpattern(blacklinewithsolidcircles).Fig.8illustratesthewellboretemperatureprofilespredictedwiththethreedifferentpatternsofinjection-rateprofiles.Similarly,curveswithopencircles,solidcircles,andcurveswithnosymbolrepresent,respectively,temperatureprofilespredictedwithde-creasingratepattern,increasingratepattern,anduniformratepat-tern.Solidcurvesrepresenttemperatureprofilespredictedattimetס2hranddashedcurvesrepresentthosepredictedattimetס10hr.

Fig.7—Differentpatternsofinjection-rateprofile.

February2008SPEReservoirEvaluation&Engineering

Fig.8—Effectsofinjection-rateprofileonwellboretemperature.

FromFig.8,weseethatwellboretemperaturepredictedwithincreasinginjection-ratepatternpropagatesfasterthanthatpre-dictedwiththeuniforminjection-ratepattern.Thisisreasonablebecauselesswaterisinjectedneartheheel,morevolumeofwaterflowsinthewellbore,andthusitcoolsthewellborefaster.Ontheotherhand,wellboretemperaturepredictedwithdecreasinginjec-tion-ratepatternpropagatesslowerthantheuniforminjection-ratepattern.AsshowninFig.8,curveswithsolidcircles(increasinginjection-ratepattern)arebelowcurveswithnosymbols(uniforminjection-ratepattern).Curveswithopencircles(decreasinginjec-tion-ratepattern)areaboveothercurves.Thesefeaturesofwellboretemperaturebehaviorsfordifferentinjection-ratepatternscanbeusedtoidentifyinjection-ratepatternswithoutapply-ingcomplicatedtechniquessuchastemperatureinversionorhis-torymatching.

Becausewellboretemperatureprofilesarechangingwithtime,itisinconvenienttodirectlyusethetemperatureprofilesatdiffer-enttimestoidentifytheinjection-ratepattern.Tocapturethemajorfeaturesofthetemperaturebehaviorsfordifferentinjectionpat-terns,weproposeamethodof“isotherm”plot.

Fig.10—FieldDTSdata.

February2008SPEReservoirEvaluation&Engineering

Fig.9—Isothermplotsfordifferentinjectionpatterns.

Foragiveninjectionprofile,thewellboretemperatureatanypointxD(orx)andatanytimetmodeldiscussedD(ort),Tabove.w(xD,tWeD),canbesolvedfromthetemperatureselectsuch(t)thatTaconstant,andthecurveofsuch(tD,xDw(xanD,tisothermD)isplotforthegiventemperature.Fig.9D,xshowsD)representssomepredictedisothermplotsfordifferentinjectionprofilesataconstanttemperatureof80°C.

InFig.9,theabscissaisln(twithnosymbolrepresentsD)andtheordinateisln(xtheisothermplotforauniformD).Thecurveinjection-rateprofile.Similarly,thecurveswithopencirclesandsolidcirclesare,respectively,theisothermplotsforadecreasinginjection-ratepatternandincreasinginjection-ratepattern.Asdis-cussedabove,foranincreasinginjection-rateprofile,wellboretemperaturepropagatesfaster.Thus,forthesameinjectiontimeinterval,(i.e.,sametD),thedistanceoftemperaturepropagatingalongthehorizontalwellbore—hererepresentedbythex—wouldbelargerforanincreasinginjection-rateDintheisothermplotprofilethanforauniforminjectionprofile.So,theisothermplotforanincreasinginjectionprofileisalwaysabovetheisothermcurveforuniforminjection.Ontheotherhand,theisothermplotforadecreasinginjectionprofileisbelowtheisothermcurveforuniforminjection.So,fromtherelativepositionoftheisothermplot,wecaneasilyrecognizethepatternofaninjectionprofile.InjectionPatternRecognition.Fig.10showstemperaturepro-filesrecordedinahorizontalinjectionwell,whichhasbeenre-portedbyBrownetal.(2003).TheparameterscharacterizingheattransferinthereservoirarelistedinTable2.ThelengthofthehorizontalsectionisLס1760(m).Fromthemeasuredtemperatureprofile,wegetthereservoirtemperatureTlibriuminjectiontemperatureattheheelisRסT98(°C)andtheequi-liststhemeasuredbottomholeinjectiontemperaturesweס50(°atC).theTableheelat3differenttimes.ThefittedtemperaturefunctionattheheelisgivenbyEq.23.

137

ThecurvewithopencirclesinFig.10showsthetemperatureprofilealongthewellborebeforewaterinjection.Fig.10clearlyshowsthatthewellboretemperaturedecreaseswheninjectiontimeincreasesbecauseofthecoolingeffectofinjectedwater.Wheninjectiontimeincreases,thetemperatureattheheelreachesitsequilibriumtemperatureofapproximately50°Cforthegivenin-jectionrate.Thehorizontalsectionofthewellboregraduallycoolsoff(i.e.,lowtemperatureisgraduallypropagatedtothetoeofthewellwhilemorewaterisinjectedintothereservoir).

Forexample,letustakethetemperatureof80°C,readthedistancesfromtheheeltothepointsinwhichthemeasuredwell-boretemperatureequals80°Cforaseriesofinjectionintervals,andputthesereadingsintotheisothermplot.TheisothermplotonthebasisofthemeasuredDTSdatainFig.10isshownasthecurvewithopentrianglesinFig.9,anditisfarbelowtheisothermoftheuniforminjectioncase.Therefore,wecanconcludethattheinjec-tion-rateprofileforthefieldcaseisfarfromuniform.Specifically,theinjectionratemustdecreasedrasticallyfromtheheeltothetoe.ThisconclusionisconsistentwiththefindingsofBuiandJalali(2004).

FractureDetection.AnotherpossibleapplicationofthewellboretemperaturemodelandDTSmeasurementsisfracturedetectioninhorizontalinjectionwells.Detectingandanalyzingfracturesisacriticalstepinmodelingfracturedcarbonatereservoirs(Elsaidetal.2007).Newtechnologieshaveevolvedinthelastseveralyearsfordetectingfractures,includingimagelogs(Bartonetal.1997),3DseismicP-waveanalysis(Wangetal.2006),anddy-namicdataanalysis(Ozkaya,2006,2007;GangandKelkar2006).Ifthehorizontalwellintersectsafractureplane,theinjectionrateatthelocationofthefractureplanecanbemuchgreaterthanthatatotherlocations.Here,weconsiderasyntheticcase.ThethermalparametersofthereservoirarethesameasthoselistedinTable2.Theinjection-rateprofileisshowninFig.11.Theinjec-tionrateatthelocationoffractureisapproximately100timesgreaterthanthatatotherlocations.Fig.12isthetemperature

Fig.12—Temperatureprofilewithfracture.

138

Fig.11—Injectionprofilewithfracture.

profilegeneratedonthebasisoftheinjectionrateillustratedinFig.11.Becausethevolumeflowrateofwaterinthewellboreintervalbetweentheheelandthefractureismuchgreaterthanthatbeyondthefracture,theconductiveheatthatflowsfromtheformationintothewellboremaywarmupthewaterflowinginthisintervalverylittle.Thechangeoftemperature,orthetemperaturegradientbe-tweentheheelandthefracture,ismuchlowerthanthatintherestofthewell.Therefore,onthebasisoftemperaturegradientinfor-mation,wecanidentifyafractureanditslocation.Fig.13illus-tratesthetemperaturegradientprofile.Temperaturegradienthasabigjumpatthelocationofthefracturebecausethevolumeflowrateofwaterinthewellboreatthispointisdiscontinuous.Discussion

Becausethewellboretemperaturemodelproposedinthispaperwasbasedonseveralassumptions,theapplicationofthissimpli-fiedmodelisofcourselimitedbysuchassumptions.

Althoughnotmathematicallyproveninthispaper,theeffectsofmassandheattransferinthedirectionparalleltotheaxisofthehorizontalwellbore,orthedirectionalongx,isprettysmallandcanbeneglected.First,thepressuregradientinthereservoiralongxdirectionismuchsmallerthanthatintheradialdirectionduringtheperiodofwaterinjection.So,masstransferinthereservoiralongthedirectionofxcanbeignored.Second,theconvectiontermofheattransferinthereservoirismuchmoresignificantthanthatofheatconduction,andthus,masstransferintheradialdi-rectionmakestheheattransferinthexdirectionalsonegligible.

Fig.13—Temperaturegradientwithfracture.

February2008SPEReservoirEvaluation&Engineering

Injectionratevaryingwithrespecttotimewillmakethemodelverycomplicated,becausebothdimensionlessparameters␯and␤becometimedependent.Ifthetotalinjectionratedoesnotchangewithtime,theinjection-rateprofilealongthewellborewouldnotchangewithtimesignificantly.Foranisotropicformation,thewa-terfrontwillbeanellipseinsteadofacircle.Inthiscase,weneedtoinvestigatethepossibilityofapplyingthetransformfromanellipsetoacircle.

Piston-likeinjectionisareasonableassumptiontosimplifythemodel.Forrealcases,thisassumptioncanslightlychangethelocationandtheshapeofthewaterfrontonlyandwillnotchangeverymuchthetemperaturedistributioninthereservoir,especiallytheheatexchangebetweenwellboreandreservoir.Withtheas-sumptionofpiston-likeinjection,wecanconvenientlygetridofcomplexitiescausedbyfractionalflow,relativepermeabilitycurves,pressurechanges,andchangesofoilandwaterviscositywithrespecttothechangeoftemperature(i.e.,wecaneasilydecouplethereservoirtemperatureequationfromthereservoirpressureandsaturationequations).Therequirementisthatthetotalinjectionratebekeptconstant,whichcanbeeasilyachievedforin-situinjectionoperations.Thetreatmentofartificiallymovingthewaterfronttoinfinitymakesitpossibletoobtainananalyticalsolutionforbothreservoirtemperaturemodelandwellboretem-peraturemodel.Thegoodagreementbetweenanalyticalsolutionandnumericalsolutionalsoprovesthatsuchassumptionsandtreatmentsarereasonable.

The1Dradialtemperaturepropagationmodelofinjectingwa-terintoareservoironlyholdsfortheso-called“earlytime”injec-tion,whichmeansthatthetemperaturefrontdoesnotreachthetoporbottomboundarybedsoftheformation.Theintervalofthis“earlytime”periodmayvaryfromseveraldaystomonths,de-pendingupontheinjectionrate,thethicknessoftheformation,thethermalparametersofrockandinjectedfluid,andthelocationofthewellbore.Ifthethicknessoftheformationisbigenoughsothatthewellboretemperaturecanreachthesteadytemperatureofin-jectedfluidattheheelbeforethetemperaturefrontinthereservoirreachesthetoporbottombeds,thenwecanignorethelimitationof“earlytime.”So,weneedtocheckwhetherthereservoirtem-peraturefrontremainswithinthetopandbottomboundariesbeforethewellboretemperaturereachesthesteadytemperatureofin-jectedfluidattheheelandmakesurethatthemodelproposedisvalidforagivenproblem.

Asimplifiedwellboretemperaturemodelisproposedinthispaper.Theauthorshopethatthisworktriggersmoreextensiveinvestigationsonhowtoinferinjectionprofilesandrockproper-tiesfrommeasuredtemperaturedatawithDTS,includingrigorousmethodsofinversionandhistory-matchingtechniques.Conclusions

Onthebasisofthediscussionabove,wecandrawthefollowingconclusions:

1.Atheoryisdevelopedtodeterminethedistributionoftempera-tureprofileinahorizontalwellwhena“cold”fluidflushesthewellborefromtheheeltothetoeandwhenthefluidisinjectedintothereservoir.

2.Thetheorycanaccountforbothavariableinjectiontemperatureattheheelandaninjectionprofilethatvariesalongthelengthofthewellbutdoesnotchangewithtime.However,itholdsfor“early-time”(i.e.,beforethetemperaturepropagationinares-ervoirisaffectedbybedboundaries).

3.Ananalyticalsolutionisobtainedwiththetreatmentofartifi-ciallymovingthewaterfronttoinfinity.Comparisonoftheanalyticalsolutionandnumericalsolutionindicatesthattheas-sumptionsarereasonable.Themodelcanbeusedtogeneratethetemperatureprofileexpectedfromanarbitraryinjectionprofile.4.Injection-rateprofilehassignificantimpactonthebehaviorofwellboretemperaturepropagationduringtheperiodofinjection.Thisrelationshipcanbeusedtoinferthepatternofinjectionprofileandidentifysharpfeatures,suchashigh-permeabilitystreaksandfractures.

February2008SPEReservoirEvaluation&Engineering

5.Thepreviousworkcomplementsexistingsolutionsinthelitera-turethatusetemperaturedataduringshut-inorpost-shut-inperiodstoinfertheinjectionprofile.

Nomenclature

Aסarea,m2cסheatcapacity,J/(m3.K)Lסlength,m

qסvolumeinjectionrateperlength(m2/h)orheatflow

rateperlength,J/(m.h)

Qסvolumeflowrate(m3/h)ortotalheat,Jrסradius,msסsaturation

Tסtemperature,K

␩␾ססthermalporosityconductivityofformation,J/(m.K.h)Subscripts

Dסdimensionlessvariableoסoilphaserסrock

wסwaterphaseorwellbore

Acknowledgments

WethankthemanagementofSchlumbergerforsponsoringthiswork,GeorgeBrownforprovidingdatatotestthemethodology,andThangBuifornumericalcomparisons.References

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February2008SPEReservoirEvaluation&Engineering