• / 10
  • 下载费用:5 下载币  

SPE-30710-MS

关 键 词:
SPE 30710 MS
资源描述:
ESPE30710SocietyofPetroleumEngineersOptimizingReservoirPerformanceUnderUncertaintywithApplicationtoWellLocationS.1.Aanonsen,SPE,NorskHydroa.s,A.L.Eide*,andL.Holden,SPE,NorwegianComputingCenter,andJ.D.Aasen,SPE,StatoilCopyright1995,SocietyofPetroleumEngineers,Inc.ThispaperwaspreparedforpresentationattheSPEAnnualTechnicalConference&ExhibitionheldinDallas,U.S.A.,22-25October,1995.ThispaperwasselectedforpresentationbyanSPEProgramCommitteefollowingreviewofinformationcontainedinanabstractsubmittedbytheauthor(s).Contentsofthepaper,aspresented,havenotbeenreviewedbytheSocietyofPetroleumEngineersandaresubjecttocorrectionbytheauthor(s).Thematerial,aspresented,doesnotnecessarilyreflectanypositionoftheSocietyofPetroleumEngineers,itsofficers,ormembers.PaperspresentedatSPEmeetingsaresubjecttopublicationreviewbyEditorialCommitteesoftheSocietyofPetroleumEngineers.Permissiontocopyisrestrictedtoanabstractofnomorethan300words.Illustrationsmaynotbecopied.Theabstractshouldcontainconspicuousacknowledgmentofwhereandbywhomthepaperispresented.WriteLibrarian,SPE,P.O.Box833836,Richardson,TX,75083-3836U.S.A.Telex,163245SPEDAL.AbstractStochasticparametersrepresentinggeologicaluncertaintiesinreservoirmodelingmaybeclassifiedin2types:1)Continuousstochasticvariables(e.g.,degreeofcommunicationthroughafault);and2)Discretestochasticvariablesrepresentingdifferentgeologicalinterpretations(e.g.,same/differentchannelobservedindifferentwells)eachwithagivenprobability.Amethodforoptimizingreservoirperformanceispresented,whichmaytakeintoaccountboththesetypesofuncertaintiesinaconsistentandsimplewaybyfindingvaluesofreservoirmanagementvariablesthatoptimizetheexpectedperformanceoverthepopulationofpossiblereservoirs.Themethodisbasedonresponsesurfacesandexperimentaldesignandisimplementedinauser-friendlycomputerprogramwhichmaybeusedtogetherwithanyreservoirsimulatororanalyticalflowmodel.Inthispaper,themethodisillustratedthrough3examplesofoptimizingwelllocationsunderuncertainty.*CurrentlywiththeNorwegianInstituteofTechnology67IntroductionTheoptimalfielddevelopmentstrategydependsonreservoirgeometryandpetrophysicalparameters.Oftenthereisconsiderableuncertaintyintheseparameters,andthisshouldbeaccountedforintheoptimization.Wepresentamethodforincorporatinguncertaintyviatheprobabilitydistributionofinputparametersorbytheuseofseveralrealizationsfromastochasticmodel.Themethodisimplementedinauserfriendlycomputerprogram,Decision,andmayinprinciplebeusedtooptimizeanyreservoirresponsevariable(oracombinationofvariables)withrespecttoanyreservoirmanagementparameter,suchaswelllocations,allocationofwellrates,etc.Thenumberofsimulationsneededisreducedusingmultipleregressionandkrigingtogetherwithmethodsforexperimentaldesign.DecisionmayalsobeusedtoperformuncertaintyanalysiswithoutoptimizationbyrunningMonteCarlosimulationsonreservoirresponsesurfacesgeneratedasfunctionsofstochasticvariables(seeRefs.1through3).Inthispaper,themethodisillustratedthrough3examplesofoptimizingwelllocationsunderuncertainty.Oneoftheexamplesisasyntheticmodelwithuncertaintiesrelatedtothelocationandtransmissibilityofsmall,subseismicfaults,whiletheothertwoarefromtworealNorthSeafluvialreservoirswithuncertaintiesrelatedtochanneldeposition.Oneexampleusesalargenumberofreservoirrealizationscoupledwithasimpleanalyticalflowmodel,whiletheotherreservoirismodelledwithafulldynamicreservoirsimulatorusing8reservoirrealizations.Well10cationscorrespondingtomaximumexpectedproductionaswellaslocationsthatminimizestheriskareconsidered.2OPTIMIZINGRESERVOIRPERFORMANCEUNDERUNCERTAINTYSPE30710whereYl,Y2,...,Ynaredrawnfromfey).Often,thestochasticvariables,y,areinputparameterstomodelsgeneratingstochasticrealizationsofthereservoir.Basedonthepriordistributions,anumberofreservoirrealizationsaregenerated,whichareuseda~inputtoreservoirsimulation.Inthosecases,theprobabilitydensityf(y)maynotbeexplicitlyknown,butexpectedresponseandFR(X;r)canstillbecalculatedfromEqs.(4)and(5)ifitispossibletosampl,efrornthedistributions.SeealsoExample2below.ExamplesThemethodisillustratedthrough3examplesofoptimizingwelllocationsunderuncertainty.Thefirstexampleisasyntheticmodelwithuncertaintiesrelatedtothelocationandtransmissibilityofsmall,subseismicfaults,whiletheothertwoexamples,aretakenfromtworealNorthSeafluvialreservoirswithuncertaintiesrelatedto~hanneldeposition.Inexample2,alargenumberofreservoirrealizationsissimulatedwithasimpleanalyticalflowmodel,'~hiletheexample3'reservoirismodelledwithafulldynamicreservoirsimulatorusing'8reservoirrealizations.Example1.Thisexampleconsidersthe'optimallocationsofoneinjectorandoneproducer.Theuncertaintyinthereservoirdescriptionisrelatedtothedensity,length,andsealingeffectofsmall,subseismicfaults.DiscountedoUproductionorNetPresentBarrels(NPB)isusedasameasureofreservoirresponse.'OptimalwelllocationsaredeterminedfromtheexpectedvalueforNPB(ENPB)ascalculatedfromEq.(4).Thesimulationmodelcontains13x25x11gridblocks.ThemainpartofthemodelisshowninFig.1.Increasingthedistancebetweentheproducerandtheinjectorwillincreasetheregionsweptbywater,andthusthetotalproduction.However,thereservoiristruncatedabovetheproducer,andmovingthewell'updipwilldecreasetheproductivityandthustheproductionrate.Consequently,weexpecttofindanoptimallocationoftheproducerformaximizingNPBawayfromthereservoirboundary.Thewellsareperforatedoverthewholereservoirthickness.Theproducerwasallowedtovarywithinarectangularregioninthexy-plane,whiletheinjectorvariedalongthexaxisonly,slightlybelowthewater-oilcontact(seeFig.1).Thus,thevectorxcontains3elements,x=(XOP,yOP,XWI),whereXOPandYOPdenotethexandycoordinatesoftheproducer,andXWIdenotesthexcoordinateoftheinjector.Uncertaintyinthereservoirmodel.AsimplemodelwasappliedassumingallthefaultstobealignedwithTheoryLetxdenotedeterministicreservoirmanagementparameters,andletydenotetheuncertainvariables.LetRbearepresentationofthereservoirresponse,whichwewanttooptimize(e.g.,NetPresentValue).Then:R=R(x,y)(1)Normally,theexpectedvalueforR,R(x),willbeoptimized.Iftheprobabilitydistributionforyisknown,R(x)isgivenby:R(x)=JR(x,y)f(y)dy(2)However,minimizingtheriskrepresentedbytheprobabilityofRbeinglessthanagivennumberr,maybeanotheralternative.DEmotethisprobabilitybyFR(X;r).Then,FR(x;r)=Prob(R(x,y)~.r)=[fey)dy,(3)J.nCT;X)whereD(r;x)={yIR(x,y)~r}.Ingeneral,ymayconsistofbothcontinuousstochasticvariables(e.g.,degreeofcommunicationthroughafault);anddiscretestochasticvariablesrepresentingdifferentgeologicalinterpretations(e.g.,sameldIfferentchannelobservedindifferentwells).Howeverbyinterpretingtheprobabilitydensityfunction,fey),asadistribution,bothtypesmaybeincludedinthenotationabove.Ourmethodisbasedonthegenerationofresponsesurfacesrepresenting'R(x,y),ortheintegrals,Eqs.(2)or(3).Themoststandardwayofestimatingaresponsesurfaceisregression.Forourapplicationswehaveusedbothregressionsurfacesandkriging4surfaces,whichareinterpolating.Theresponsesurfaceisestimatedfrom.alimitednumberofsimulations.TheinputvariablestothesesimulationsareselectedusingD-optimality.D-optimalityisamathematicalproceduretoselecttheoptimalrunsfroma'(large)setofpossibleruns(thecandidateset).Basedonasetofcandidateexperiments,thenumberofexperimentstobeselected,andanaprioriregressionequationdescribingtherelationsbetweeninputandresponsevariables,theoutputistheoptimaldesignwithrespecttoobtainingoptimallypreciseestimatesofthecoefficientsintheequationgiven.Fordetails,seeRefs.5or6.HavingobtainedtheresponsesurfaceforR(x,y),theintegrals,Eqs.(2)and(3)mayeasilybecalculatedusingforinstancetheMonteCarlomethod.Eq.(2)isthenapproximatedby.....1nR(x)=-2:R(x,Yi),(4)ni=lwhereYl,Y2,...,Ynaredrawnfromfey)(seeforinstanceRef.7).Eq.(3)isapproximatedby68F(.)_#(R(x,yi)0)LVi,j(x)(7)Z,]Vi,j(X»oThatis,Ui(X)isthemeanvolumefromchanneliifthewellinlocationxhitschanneli.Thisisaquantityweexpecttoincreasewithincreasingdistancefromxtothewellwherechanneliwasobserved.Theprobabilityofhittingthechannelsisestimatedfromthenrealizationsasnumberofhitsdividedbynumberofrealizations.TheprobabilityofhittingchanneliinwelllocationxisdenotedPi(x):#(Vi,j(x)>0)pi(X)=(8)nWeexpectthisprobabilitytodecreasewithincreasingdistancefromxtothewellwherethechannelwasobserved.TheexpectedvolumeVi(x)canbeestimatedbyVi(x)=Ui(X).Pi(X)(9)BysplittingVi(x)inthesetwofactorsweexpecttounderstandtheresponsesurfacesbetter.70Responsesurfaces.Inthisexample,thevolumecalculationsarecomputationallycheap,anditwaseasytoperformalargenumberofvolumecalculations.Hencewedidnotneedexperimentaldesignforsavingcomputingtime,butpreferredtodothecalculationsforalargenumberofwelllocationstoobtaingoodresponsesurfaces.So,foreachrealization,channelvolumebetweeninjectorandproducerwascomputedfor30differentlocationsofthenewwell(Fig.8).Weexpectedtheresponsesurfacestoberelativelycomplicated,andusedkrigingforinterpolatingallobservations.Theresponsesurfa-cesarethusbased.onupto36observations:30observationsofvolumeandprobabilityIfhittinginthe30differentlocationsoftheinputwellandobservationsfromthe6wellsalreadypresent.Theprobabilityofhittingis0inthewellswherethechannelwasnotobserved.Inwell3,wherechannellwasobserved,theprobabilityofhittingis1,butthevolumeisO.(Here,theobservationsareaveragesover50realizations.)Thewelllocationisx=(x,y),withxvaryingonascalefrom1to3.1equalswellxcoordinate1000mand3equalswellxcoordinate3000m.yvariesonascalefrom-1to1,where-1equalswellycoordinate350mand1equals1150m.Thetotalexpectedvolumeisgivenby:R(x)=Vi+V2•......••.••.•.••..•.••...•....••(10)RisshowninFig.9.Maximumexpectedvolumeisobtainedinwelllocationsontherighthandsideoftheplot.Theoptimalwelllocationisx=(3.0,0.39)innormalizedcoordinates.Thatis,betweentheexistingwells5and6,somewhatclosertowell5.Examplewithfaults.Inthissectionwebringanotheraspectintothemodeling:Barriersbetweeninjectorandproducer.Faultsbetweeninjectorandproducermaypreventcontactbetweenthesewells.Thefaultsmayeitherbesealingorjuxtaposethepartsofthechannel.Itisassumedthatthepositionofeachfaultisindependentofthepositionoftheotherfaults.Hence,thefaultsaremodeledasPoissondistributedalongthedirectionofthechannel.Thedistancefromawelltoafaultwillthenbeexponentiallydistributed.Theprobabilitythatthedistancefromawelltothenearestfaultislessthanalength1isthen1-e-)"I(11)Theprobabilityofcontactbetweenthewellswithadistance1betweenthemis1-(1-e-M)=e-)"I(12)Thiswillfavorwelllocationsnearthewellwherethechannelwasobserved.Theexpectedvolumewillthenbe-._""-)..1EVz-Uzpze".(13)SPE307105.1.AANONSEN,A.L.EIDErL.HOLDEN,J.O.AASEN5Theprobabilityofcontactislargestnearthewellwherethechannelswereobserved,anddecreaseswhenthedistancefromthewellsincreases.InthisexamplewehaveusedA=1andI=J(x-xd2+(y-YiP,where(Xi,Yi)arethelocationsofthewellswherechanneliwasobserved.Includingtheprobabilityofcontactchangesthetotalexpectedvolume(Fig.10).Theoptimalwelllocationsarenowmuchclosertothewellswherethechannelswereobserved.Sincechannel2iswiderthanchannell,itwillhaveagreatervolume,anditismoreimportanttohitchannel2.Globaloptimumisx=(2.65,-1.0)withasecondpeaklocatedatx=(1.86,-1.0).Thatis,theoptimaarelocatedonbothsidesofwell4,quiteclosetothewell.Example3.Theuncertaintiesassociatedwiththedepositionofchannelsandsareimportantfactorsregardingthedevelopmentofthisfield.Stochasticmodelinghasbeenusedtoevaluatetheseuncertainties,and8realizationsofthefieldhavebeengenerated.TherealizationsarebasedonaLowCase,aBaseCase,andaHighCasegeologicalinterpretation,withcorrespondingprobabilitiesestimatedto0.1,0.7,and0.2.TwoLowCase,fourBaseCase,andtwoHighCaserealizationsweregenerated.ThismayberepresentedinthenotationofEq.(1)bylettingybeascalar,discrete,stochasticvariablewith3possiblevalues,Yl,Y2,Y3;andwithcorrespondingprobabilitiesPl=0.1,P2=0,7,andP3=0.2,respectively.SimulationModel.Thesimulationsarebasedonamodeloriginallycontaining48576(46x88x12)blockswiththenumberofactiveblocksbeingoftheorder11000.Inthisstudy,thenumberofactiveblockswasreducedtoaround5500bysimulatingonlythecentralpartsofthemodelwithmostofthereductioningridblocksbeingintheaquifer.ThearealgridofthesimplifiedmodelisshowninFig.11.Themodelcontainsoneinjectorandoneproducer.Bothwellsareverticalandcompletedoverthewholereservoirthickness.Shownonthefigurearealsotwoexplorationwellspreviouslydrilledinthisarea,anditshouldbenotedthatthewellsproposedintheoriginalmodelarelocatedclosetothese.Bothexplorationwellspenetratedthechannels,soarelativelylowriskshouldbeconnectedwiththeoriginalwelllocations.Fortheoptimization,thewellswereallowedtomovewithintheareasmarkedonthefigure.Theregionofallowedvariationfortheproducercoversmostoftheoilzone,theEastboundarybeingclosetothewater-oilcontact.TheXcoordinatewaskeptfixedfortheinjector,givingatotalof3inputvariablestobedetermined.ExperimentalDesign.Experiencehasshownthatagoodresponsesurfaceiscriticalforobtainingagoodresult,andthatD-optimalityinsomecasestendstoproducetoo71manyexperimentsontheborderoftheinputvariablesdomain.Therefore8pre-definedpointswithvariationintheproducer'sXcoordinatewereincludedinaD-optimaldesignwith30experimentstotally.TheresultingdesignforthelocationoftheproducerisshowninFig.12.AD-optimaldesignwith30experimentsandnopre-specifiedcandidatesisshowninFig.13forcomparison.Simulationswithwelllocationsaccordingtothisdesignwererunforeachofthe8realizationsmakingatotalof240runs.Theprocessofgeneratingsimulatorinputdatafilesandstarttherunsinsequenceforeachrealizationwasautomated.Eachruntookapproximately1hrCPUontheSunSparcstationused,sothewholepr-o-oesstooksomewhatlessthat2weekstocompletemoreorlesswithoutanymanualinterference.Fig.14showstheproductionhistoryforallthesimulatedwelllocationsforoneoftherealizations.Thehistoryoftheoriginalmodelisshownwithsolidlinesforcomparison.Insomeofthecases,theproducerfailedtohitachannelandthesimulationterminated.Thusthenumberofcurvesarelessthan30.Notealsothelargespreadinthecurvesandthatforsomeoftheruns,thecumulativeproductionarehigherthanthecumulativeproductionusingtheoriginalmodel.Mostofthetime,theproductioniscontrolledbytubingheadpressure,andtheproductionratebeforewaterbreakthroughismainlydeterminedbytheproductivityoftheproducer.ResponsesurfaceModeling.LikeinExample1,discountedoilproduction,NetPresentBarrels(NPB),wasusedasresponsevariable.Foreachofthewelllocationsintheexperimentaldesign,expectedNPB,ENPB,andtheprobabilityforNPBtobelessthan1.106,3.106,and5.106Sm3(denotedFR1,FR2,FR3),wascalculatedfromEqs.(2)and(3)byassumingthatR(x,ydhasaGaussiandistributionforforeachofthe3valuesofYi.Results.Takingintoaccountthecomplexityofthisproblem,relativelycomplicatedresponsesurfacesshouldbeexpectedinthiscase.Thustheresponsesurfaceswerecalculatedusingvariousmethodsofkrigingtochecktherobustnessoftheresponsesurfaces.Inallcases,theresponsesurfaceforENPB(x)showed3distinctmaxima,whicharelistedinTable1.Thelocationscorrespondingtothe3maximainTable1areshownonFig.11.ExamplesofresponsesurfacesasfunctionsofthelocationoftheproducerareshowninFigs.15,16.Regardingflowpattern,theoptimalpatternistoinjectinthelowerrightpartandproduceintheupperleftpartofFig.11.BothFR2andFR3haveglobalminima(i.e.,leastprobabilityforalowNPB-value)atthislocation.RegardingFR1,whichistheprobabilityforareallylowvalue(NPBbelow1.106Sm3),thisfunctionhasitslowestvalueatalo
展开阅读全文
  石油文库所有资源均是用户自行上传分享,仅供网友学习交流,未经上传用户书面授权,请勿作他用。
0条评论

还可以输入200字符

暂无评论,赶快抢占沙发吧。

关于本文
本文标题:SPE-30710-MS
链接地址:http://www.oilwenku.com/p-70346.html

当前资源信息

吾王的呆毛

编号: 20180607205000268997

类型: 共享资源

格式: PDF

大小: 974.13KB

上传时间: 2018-06-08

相关搜索

广告招租-6
关于我们 - 网站声明 - 网站地图 - 资源地图 - 友情链接 - 网站客服客服 - 联系我们
copyright@ 2016-2020 石油文库网站版权所有
经营许可证编号:川B2-20120048,ICP备案号:蜀ICP备11026253号-10号
收起
展开