An Improved SAGD analytical Simulator：Circular Steam Chambe(3)

Thosesliceswhoseoilsaturationreachestheresidualvaluewilljointhesteamchamberinthenexttimestep.Atanytimestepthe

Fig.8.De?nitionoflocalvelocityatdifferentpositionofsteamchamber(SC)andthelocationatwhich?owrateiscalculated.

32A.Azad,R.J.Chalaturnyk/JournalofPetroleumScienceandEngineering82-83(2012)27–37

ρwdensityofwater

QintheheatenergyinsidethesteamchamberQouttheheatlossaroundthesteamchamber.

Eq.(14)hastwomainterms;theleftsideistherateofinjectionofenergyintothereservoir,andtherightsidedeterminestheenergydistributionratethroughthesteamchamberintothereservoir.Forenergybalance,slicesarenotconsideredincalculation.TheenergyinsidethesteamchamberiscalculatedusingEq.(15)whereρisthedensityoftheformation,cisthespeci?cheatoftheformation,TsisthesteamtemperatureandTristhereservoirtemperature.Eachmovementfromoneslice(n)tothenext(n+1)wouldbeusedtode-terminethelastterm,dA/dt.dQin?ρceTdA

s?TrT:

e15T

IneachcasefromAtoC(seeFig.5),energylossfromthebodyofthesteamchamberiscalculatedusingEq.(16),ignoringtheexistenceoreffectofneighboringwells.ThederivationofEq.(16)isexpandedindetailsinAppendixB.dQ??αeT??

sidedρcs?TrTdt

?Dθ

dtaU1?cosθT:e16T

maxeIncasesBandCthatsteamchamberisnotafullcircle,andenergylossoccursthroughtheoverburdenattheseparationlinebetweenoilsandformationandotherlayersasderivedbyReis(1992):dQr????

topα

dt?2ρceTp??s?TrTUHte17T

where:UH

horizontalvelocityofthesteamchamberatseparationlinet

cumulativetimeaftertouchingthecaprock.

3.4.Initialconditions

Whenaclearcommunicationbetweeninjectorandproducerboreholesisidenti?ed,thesteamchambershapeappearsintheres-ervoiraroundtheinjector.Themaximumvelocityofthesteamcham-bergrowthisdeterminedbyEq.(18)(seeAppendixC).U2

8Koρoαg

max;initial?

μφΔS:

18T

osamoπ2DeStart

3.5.Calculationprocedure

Althoughtheformulationpresentedintheprevioussub-sectionslooksclosed-form,apartofthecalculationneedstobeperformednu-merically.Therefore,asimplecomputercodeisrequiredforimple-mentingthemodel.Thecalculationprocedurehasbeenlistedbelowinastep-by-stepalgorithm:

1Selectdiameterincrement.Thissinglevalueprovidestheincre-mentalincreaseindiameterandmathematicallydividesthereser-voirintoslices.Whileallthesteamchambercirclesareattachedtotheproducer,thesteamchambergrowsillustratedinFig.4.2CalculatetheinitialvelocityUmax,initialusingEq.(18).3Accumulatethetimeincrementsfromthetimezero.

4CalculatetheoilrateineachsliceaccordingtoitslocationusingEqs.(3)–(11).Iftherateislessthanaminimum,marktheslice

asinactive.Usuallytheslicesclosertothesteamchamberareactive

only.

5

Calculatetheincrementaltimerequiredforthe?rstsliceinfrontofthesteamchambertoreachtheresidualoilsaturationbasedonthedrainageratecalculatedinstep4.

6

Calculatetheaccumulatedoilproductionandupdatetheoilsatura-tioninactiveslicesfortheincrementaltimeperiodcalculatedinstep6usingEqs.(12)and(13).Now,thesteamchamberwilloccu-pyanothersliceandgetslarger.

7Calculatethevelocityofthesteamchamberfromthetimeandthediameterincrements.

8Calculatethesteaminjectionrateforthetotalincrementaloilpro-ductionrateusingEqs.(14)–(17).

9

Repeatallthestepsfrom3to8fortheperiodoftimeorproductionlimityouwanttoreach.

4.Modelvalidation:experimentaldata

ChungandButler(1988)conductedlaboratorytestsandcom-paredthedatatotheresultsoftheiranalyticalmodel.Thesmall

scalelaboratorymodelwas35cmwide,22cmhighand3cmthick.Theydesignedtwoinjectionstrategies:scheme‘A’and‘B’.Inscheme‘A’theinjectorwashorizontalandslightlyabovetheproducer.Inscheme‘B’,asinglehorizontalproducerandmultipleverticalcirculat-ingsteaminjectorswereinstalled.Scheme‘B’wasconsideredtomimictheconditioninwhichthesteamchambergrowslaterallyonlyandwasverysimilartothegeometryoftheButlermodel.TheyfoundthattheoriginalButlermodelisabletoreproducetheresultsofthetestwithscheme‘B’con?guration.OnlyTANDRAIN,amodi?edversioncouldbetterpredictthescheme‘A’con?gurationtest.Reis(1992)andAkin(2005)alsousedtheresultsofscheme‘B’tovalidatetheirmodels.

Fig.9showstheresultsofthemodelpresentedinthisstudycom-paredtothescheme‘A’data.TheparametersthathavebeenusedtorunthemodelarelistedinTable1.Closeagreementbetweenthelab-oratorytestdataandtheproposedmodel,revealstwonewaspectsofthismodel;(a)Unlikethepasttheories,thecurrentmodeldoesnotpredictaconstantvalueofoilrateanditisabletofollowthevariationoftheoilproduction.(b)Circulargeometrymodel,unlikeone-directionalmodels,isabletopredicttheresultsofarealSAGDgeom-etrytest.Howeverwemustemphasizethatonlytheinitialpartofthedataismatchedduetothefactthatboundaryeffectsarefeltintheex-perimentafter2handthecircularmodelthatisin?niteactingcannotproperlypredicttheprocess.

5.Modelvalidation:numericalsimulation

Experimentalvalidationinthelastsectioncon?rmedthatrunningtheproposedmodelwiththeexactvaluesreportedfromlaboratoryisabletomatchtheexperimentalmodel.Thismeansthatthemodeliscapableofworkingasa?owsimulatorforhistorymatchingpurposeswhileothermodelsmaynotbepowerfulenough.Toshowinghowthismodelcanbeutilizedasa?owsimulatortomatchthehistory,numericalanalysisresultshavebeencomparedtothecurrentmodel.

Anumericalsimulationisruntoproducesyntheticdataforcom-parison.MaterialpropertiesandotherrequireddataarelistedinTable2.Itisimportantthattheinformationinthetableisasmallpor-tionofthewholedataneededtorunanumericalsimulator.Inaddi-tion,Table2showsthenumberofparametersthatisrequiredforsimulationbythecurrentanalyticalmodel.

Forhistorymatching,relativepermeabilitycurveswereselectedastheunknownparameter.Thehistorymatchingwasthentrainedonthehistoryofoilproduction.Forthe?rstrun,arelativepermeabil-itycurve,commonforoilsands,asshowninFig.10wasselected.Aftereachrun,oilproductionhistorywascomparedtotheresultsofthenumericalmodelandamultiplierbetween0and1was

A.Azad,R.J.Chalaturnyk/JournalofPetroleumScienceandEngineering82-83(2012)27–3733

0200

0.20.40.60.811.21.41.61.82

Oil Rate, gr/hr

15010050

Butler-Like TheoriesThis Study

Chung and Butler Data

Cum. Oil Production, cc

300

Butler-Like TheoriesThis Study

Chung and Butler Data

200

100

Time, hr

Fig.9.ComparisonbetweentheexperimentaldatareportedbyChungandButler(1988).

selectedtomodifytherelativepermeability.Thetrialanderrorpro-cesswascontinueduntilcloseagreementwasfound.Sinceeachruntakes5sonly,thewholehistorymatchingprocesswasdonein5min.Althoughthe?nalrelativepermeabilitycurvewasnotquitethesameasthecurveusedinthenumericalanalysis,theaveragevaluewasthesame.ThisfeaturemayexplainwhyButlerchosetheaveragepermeabilitytobetheeffectiveparameterinhistheory.

Figs.11and12illustratethehistorymatchingprocess.AcloselookatFig.11clari?esthattheanalyticalmodelhasbeensuccessfulinpre-dictingthetrendofoilproductionfromtheverybeginningofthepro-cesswherethereisquickjumptothetimewhentheoilrateisdecreasing.Comparingthisabilityinthecurrentmodeltothecon-stantoilratepredictionbyButler-liketheoriescon?rmsthatthisnewmodelcanbeagoodtoolforfasthistorymatching.

ThenonlinearoilproductioncurveinFig.11hasbeencapturedbytheanalyticalsimulatorwhileButler-liketheoriesareincapableofpredictinganonlineartrend.Steaminjection,however,hasnotbeenpredictedwell.Theproblemismorelikelyduetothesimplicityofthetheoryusedforenergybalanceorthecomplexityoftheheatpropagationinthenumericalsimulator.

ThedashedhorizontallineinFig.11andthedashedstraightlineinFig.12havebeenincludedtobearepresentativeofButler-likemodels.AnyparallellinetothedashedlineinFig.11andanylinethatpassesthroughthecoordinateofthechartcanbematchedbychangingpermeability.Itmeansthatregardlessofthevalueofper-meability,thenatureofthesemodelsisnot?exibleenoughtobeusedasapracticalsimulator.Thesetwolinesshowtheinadequateca-pabilityofButlerorReismodelforhistorymatching.

Otherthanthemismatchbetweenthereporteddataandpre-dictedresultsforsteaminjection,steam/oilratio(SOR)plottedinFig.13isingoodagreement.ThedifferencebetweenSORsatthezonethathasthemostdivergence(from200to500days)isonly0.25inaverage.ThisdifferencewhenoriginalReismodelisemployedtopredictthesteaminjectionratecanvaryfrom1to10(dashedlineinFig.13).ThismeansthatthecirculargeometryandthemodelofsliceshaveimprovedtheanalyticalapproachtoSAGD.

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