An Improved SAGD analytical Simulator：Circular Steam Chambe(2)

Unlikepasttheories,thisapproachcanconsiderrelativeperme-abilityvariationwithoilsaturationchanges.Themethodologydividesthereservoirintoseveralslices.Subsequently,thematerialbalanceiscalculatedforeachsliceinfrontofthesteamchamber.Eventually,theoilsaturationisupdatedduetotheoilproducedfromeachsliceateachspeci?ctime(orsteamchamberposition).

Whenthesteamchamberisataspeci?clocation,foreachsliceinfrontoftheseparationline,oilrateiscalculated.Itisassumed

that

Fig.2.Modelofslices,fromAzadandChalaturnyk(2010).Oneslicehasbeenplottedononesideofthesteamchamberasanexample.

A.Azad,R.J.Chalaturnyk/JournalofPetroleumScienceandEngineering82-83(2012)27–3729

Fig.3.GrowthofsteamchamberobservedduringphaseAofUTFproject(adoptedfromItoandSuzuki,1996).

thesteamchambermovesfromoneslicetotheotheronlywhenthecurrentoilsaturationofthe?rstfrontslicedeclinestoresidualoilsat-uration.Therefore,theremainingoilvolumeinthe?rstslicecanbeusedtocalculaterelativepermeabilityandthetimeneededtoproducetheoilfromthatslice.Thisisalsotherequiredtimeforthesteamchambertomoveon.

ThemodelproposedbyAzadandChalaturnyk(2010)originallyusesalineargeometryforsteamchamberslicesthatgrowslaterallyonly,assumingthatthesteamchamberhasreachedthecaprock.Theyshowedthatthisthemeisinadequatetopredictthe?rstand?nalstagesoftheprocess.Fig.3hasplottedthegrowthofsteamchamberobservedinUTFphaseA(ItoandSuzuki,1996).FromthetemperatureisolinesillustratedinFig.3,onemightpickacircularge-ometryforthesteamchamberratherthanastraightline.Itisalsore-quiredthatthegeometrybe?exibletocapturethegrowthofthesteamchamberbeforeandaftertouchingthecaprock.

LineargeometryandcirculargeometryofthesteamchamberinSAGDhavebeencomparedinFig.4.The?gureshowsthatthecirculargeometrycanbeabetteroptioncomparedtothelinearshapeofthesteamchamberbecauseitcanmimictheinitialstagesoftheprocess.Inaddition,inlongtimeperspective,thesteamchamberdoesnotneedtogetunrealisticshapetocoverfurtherreservoirregions.3.Analyticalmodelforcirculargeometry

Inthissection,theformulationofthecircularmodelispresented.Themodelhastwomajorelements:(a)thedrainagemodeland(b)theinjectionmodel.ThedrainagemodelisconstructedusingDarcy'slawthatexplains?owof?uidinporousmediabasedonpressure

gradientoveradistance.Darcy'slawisalinearmodelthatisvalidforlaminar?ows.ForSAGDprocess,thepotentialgradientisas-sumedtobegravity-basedandhence,Darcy'slawdeterminesthe?owfromanylocationabovetheinjectortowardtheproducer.Tocalculatetheviscosityofoilwhichishighlysensitivetotemperature,thedrainagemodelwithsomesimpli?cationemploystheone-dimensionaltheoryofheattransferincontinuummediaforanad-vancingfront.Extendingthistheoryfortheedgesofthesteamcham-bergrowinginthereservoir,viscosityvariationinadvanceofthesteamchamberisestimated.BasedontheDarcy'sandheattransferlawsthedrainagemodelisdeveloped.

Theheatingenergyrequiredtokeepthegrowthofthesteamchamberproducingtheoilratepredictedbythedrainagemodeliscalculatedwithintheinjectionmodel.Theinjectionmodelassumesthatfortheentiresteamchambertobemaintainedatthesteamtemperature,therequiredenthalpyhavetobesuppliedaslatentheat.Thethermalheatingisdominatedbythelatentheatfortrans-formationofsteamtowater.AppendixAprovidesfundamentalas-sumptionsandthebasictheoryadaptedfromButlermodel(Butler,1997).

3.1.Drainagetheory

Foracirculargeometry,differentpossiblesteamchamberloca-tionshavebeenplottedinFig.5.ItisassumedthattheSAGDprocessstartswhenthesteamchamberreachestheproducerborehole.Thishappensaftertheprimaryperiodofheatingor‘startup’phaseinwhichsteamisinjectedthroughbothinjectorandproducertoestab-lishcommunicationinbetweenboreholes.Therefore,the?rststeamchamber(‘Start’inFig.5)isthecirclewhosecenteristheinjectorboreholeanditsradiusisthewellspacing.Thiscircleappearsashorttimeafterthe‘startup’phase.Thebottompointatwhichthesteamchambertouchestheproducerboreholeisnow?xedandthesteamchambergrowswhileitslowerpointisattachedtotheproduc-erborehole.Thisisanassumptionthatwasmadebasedonprevious?eldobservation(e.g.,seeFig.3,theUTFphasesteamchambergrowth).Case‘A’occurswhenthesteamchamberhasnotreachedthecap-rock.Assoonasthesteamchambertouchesthecap-rocktheshapeisnotacompletecircleanymore.Twopossiblepositionsofsteamchambermayoccur:case‘B’whenthecenterofthecircleislocatedinsidetheoilsandlayerandcase‘C’wherethe

steam

Fig.4.Comparisonbetweenlinearandcirculargeometryofthesteamchamber.

30A.Azad,R.J.Chalaturnyk/JournalofPetroleumScienceandEngineering82-83(2012)27–37

Fig.5.Possiblepositionsforacircularsteamchamber.

chamberextendstotheoverburdenandthecenterofthecircleis

shiftedfromthereservoirformationtotheoverburden.

ThefoundationofthetheoryproposedbyButlerisDarcy'slawfor?owofoilshowninEq.(3).ThiscanbeappliedtoonesliceasshowninFig.6tofollowthemodelofslices,whererelativepermeabilityisestimatedfromcurrentoilsaturationoftheslice.ForSAGDprocess,itiscommontoconsidertwophasesonly,oil–water,neglectinganygascomingoutofsolutionduringtheSAGDprocess.Therefore,rela-tivepermeabilityofoilisapplicable:dqKoρo

t?

Kro?ΦdAe3T

where:dqt

differential?owofoilattimetKro

relativeoilpermeability

Fig.6.Modelofslicesforcirculargeometryofsteamchamber.

Koabsoluteoilpermeabilityρooildensityμoilviscosity

?Φ?owpotentialfunctiondAdifferentialarea.

AllthematerialpropertiesinEq.(3)arede?nedforthesliceonwhichcalculationiscarriedout.Ifoildensityisassumedconstant,vis-cosityhastobecalculated.ToestimateviscosityatdistanceξfromsteamchamberthatmoveswithvelocityequaltoUinamediumwhosethermaldiffusivityisα,ButlerproposedtouseEq.(4)inwhichmisaconstantbetween3and5andμosistheoilviscosityatthesteamchambertemperature.MoreexplanationaboutthesetworecentequationsisincludedinAppendixA.μeξT?

μosexp?mUξ

:

e4T

CombiningEqs.(3)and(4)yieldstoEq.(5)whereaisanother

constant.Reis(1992)recommends0.4forathatadjuststhemaxi-mumlocalvelocityofthesteamchamber.??dqKamU??

roKoρo?localξt?exp

?ΦdA:e5T

os

Incase‘A’whenDbHt,the?owpotentialfunctionandtheeffec-tiveareacanbeexpressedusingEqs.(6)and(7).Intheseequations,Drepresentsthediameterofanycirclelargerthanthediameterofsteamchamber(DSC).?Φ?

Dg20:5πD?g

e6TdA?e1:0Tdξ?dD:

e7T

Inallothercasesbasedonthede?nitionofXandDinFig.7,steamchamberparametersarecalculatedasfollows:????D??2??

D??2

??0:5X?2?Ht?

2e8T

??

θ?Sin

?1

X???1??2X

??D=2?SinD

:e9T

IfHt>D/2,then?owpotentialfunctionisstatedasinEq.(10);

otherwiseEq.(11)isthe?owfunction.Forbothcasesdξisthediffer-entialareaandalltheremainingparametershavebeenplottedinFigs.7and8.?Φ?

Htg2Htg

0:5eπ?θTD?

e10T?Φ?

Htg0:5θD?2Htg

:e11T

3.2.Materialbalance

Materialbalanceisappliedtoeachsliceinfrontofthesteamchamber.Considerthatthesteamchamberisataspeci?clocationanditmoveswiththelocalvelocity,Ulocal.UsingEq.(5),eachsliceoutsideofthesteamchamberproducesoil.Itisassumedthatthesteamchamberjumpsfromitscurrentlocation(n)tothenextlocation(n+1)whenalloftherecoverableoil?owsoutofthecur-rentslice.Therefore,basedonthematerialbalanceinthe?rstslice,

A.Azad,R.J.Chalaturnyk/JournalofPetroleumScienceandEngineering82-83(2012)27–3731

Fig.7.SteamchamberparametersforcasesB(left)andC(right).

thetimethattakestogettheresidualoilsaturationvaluecanbecalculated:

??Δt?

So;1?So;Rt;1

??:

e12T

sliceswithresidualoilsaturationconditionorwithinsigni?cant?owratearenotconsideredincalculation.3.3.Energybalance

Reis(1992)utilizedthelawofconservationofenergytocalculatetherequiredsteaminjectionintothereservoir.Withthesameap-proach,theinjectionrateperunitlengthofthehorizontalboreholecanbederived.mLs?eqs:ρwTLs?

d

eQtQoutTdtin

e14T

InEq.(12),So,1anddqt,1arethecurrentoilsaturationandoilrateoftheadjacentslicetothesteamchamberandSo,Ristheresidualoilsaturation.BasedonthetimecalculatedinEq.(12),oilsaturationinotherslices(i=1,2,3…n)isupdatedasfollows:So;i?So;i?

qt;iΔt

:e13T

where:Lsqs

speci?clatentheatofsteam

steaminjectionrate(involumeofwater)

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