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gradientYaodevelopment,essuchof“drillingawell”detractsfromtheNPV,whenNPVismaximizedwithrespectinoilaresultioststhichmnrecovchwelJournalofPetroleumScienceandEngineering73(2010)220–226ContentslistsavailableatScienceDirectJournalofPetroleumScience.elseproblem.Forthisreason,mostoftheresearchonoptimalwellplacementhasbeenfocusingonnon-gradient-basedoptimizationalgorithms:simulatedannealing(BecknerandSong,1995;NorrenaandDeutsch,2002),geneticalgorithm(BittencourtandHorne,1997;OzdoganandHorne,2006;Yetenetal.,2002)andneural-networks(Yetenetal.,2002;Centilmenetal.,1999).Alltheabovediscreteoptimizationalgorithmsrequirealargenumberofreservoirsimulationrunsduetoslowconvergenceorthetrainingprocessfortheneural-Gradient-basedoptimizationalgorithms,withthegradientofafunctionalorobjectivefunctiontobeoptimizedmostcommonlycomputedbytheadjoint(optimalcontrol)method,havebeenusedinbothautomatichistorymatching(Chenetal.,1974;Wuetal.,1999;Lietal.,2003;GaoandReynolds,2006)andproductionoptimization(BrouwerandJansen,2004;deMontleauetal.,2006;Kraaijevangeretal.,2007).However,tothebestofourknowledge,thefirstpapersthatusethegradientdirectlytosolvetheoptimalwellplacementproblemnetwork.Bangerthetal.,(2006)comparedtheefsimulatedannealing(VFSA)algorithmwithalgorithms(finitedifferencegradientmethod-F⁎Correspondingauthor.E-mailaddresses:zhangkai@upc.edu.cn(K.Zhang),0920-4105/$–seefrontmatter©2010ElsevierB.V.Alldoi:10.1016/j.petrol.2010.07.002llocationsaresimulationdeterminingtheoptimaladiscreteoptimizationAremedytothisproblemistoconvertthediscreteoptimizationproblemintoacontinuousoptimizationproblemsothattraditionaloptimizationalgorithmscanbeappliedforfasterconvergence.gridblockindices.Therefore,theproblemofwelllocationswithareservoirsimulatoris1.IntroductionWellplacementisanimportantstepAgoodchoiceofwelllocationscanrecoveryatagivendrillingcost.Inmdefinedasamathematicalproblemwpresentvalue(NPV)orthehydrocarbolocationsinareservoirsimulator,inwhioptimizationprocessandthegradientprojectionmethodtoensurethattheconstraintsaresatisfied.Forthesynthetichomogeneousandheterogeneousreservoirexampleswheretheproblemistodeterminetheoptimallocationofwaterinjectionwells,thisproblemformulationyieldsgoodresultsaftertheoptimizationprocess.©2010ElsevierB.V.Allrightsreserved.ndgasfielddevelopment.nahigherhydrocarbonudies,wellplacementisaximizeseithertheneterybyadjustingthewellperturbationstochasticapproximation-SPSA).TheyshowthatbothSPSAandFDGalgorithmsaremoreefficientthanthenon-gradientVFSAalgorithm,butnoneofthesealgorithmscanensureanoptimalsolution.Themajorproblemoftheabovegradient-basedalgorithmsisthatthestepsizealongthesearchdirectionhastobechosensuchthateachfunctionevaluationpointcorrespondstothelatticepointsinthesimulationgridsystem.ficiencyofaveryfasttwogradient-basedDGandsimultaneousareHandelsetal.2008).Althoughthe(2008)canbeapplieditisexplainedmosttheoptimallocationwaterinjectioninjectionwells.Assumegaoming-li@utulsa.edu(G.Li).rightsreserved.continuous,weareabletouseadjointgradientforthetothewellrates,theratesofsomewellswillbedriventozeroandhencethesewellsareeliminatedfromthesystem.Asthecontrolvariables(wellrates)are“drillingawell."AsthecostOptimalwellplacementusinganadjointKaiZhanga,GaomingLib,⁎,AlbertC.Reynoldsb,JunaChinaUniversityofPetroleum(EastChina),Qingdao266555,ChinabUniversityofTulsa,800S.TuckerDr.Tulsa,Oklahoma74104,USAabstractarticleinfoArticlehistory:Received9January2009Accepted6July2010Keywords:AdjointgradientGradientprojectionWellplacementIntheprocessofreservoirhydrocarbonscanbeextractedtreatedasdiscretevariabloptimizationarenotapplicablenon-gradient-basedmethoddiscreteoptimizationprobleminitializetheproblembyputtingrespecttotheratesofthehypothesizedjournalhomepage:wwwa,LimingZhangaonewishestodrillwellsatoptimallocationssothatmoreatalowercost.Becausewelllocationsinareservoirsimulatorarecommonly(wellgridblockindices),standardimplementationsofgradient-basedforoptimalwellplacement,sooptimizationiscommonlydonewithaasthegeneticalgorithm.Thispaperpresentsanovelideatoconverttheintoanoptimizationproblemwithcontinuousvariables.Theideaistoawellineverygridblockandmaximizethenetpresentvalue(NPV)withwells.TheNPVincludesanadditionaltermtoaccountforthecostofandEngineeringvier.com/locate/petrol(2007),(Wangetal.2007)and(SarmaandChenmethodofHandelsetal.(2007)andZandvlietetal.tothesimultaneousplacementofseveralwells,simplybyconsideringtheproblemofdeterminingofasingleinfillwell,e.g.,thelocationofanewwellinareservoirthatalreadyhasproductionandflowisonlyinthex–yplanesoweusea2DK.Zhangetal./JournalofPetroleumSciencesimulationgrid.Inthiscase,giventhecurrentproposed(initialguess)placementoftheinjectionwell,whichisnotinagridblockadjacenttothereservoirboundary,a“pseudo-well”producedatasmallrateisplacedineachoftheeight“neighboring”gridblocks.Thenthegradientofthenetpresentvalue(NPV)overthereservoirlifewithrespecttotherateateachpseudo-welliscomputed.ThelargestvalueamongtheseeightgradientvaluesdeterminesthedirectioninwhichtheactualwellmovestoincreaseNPVthefastest.Specifically,theinjectionwellmovestothepseudo-wellgridblockthathasthelargestgradientvalue.TheworkbySarmaandChen(2008)isanextensionofthemethodinHandelsetal.(2007)andZandvlietetal.(2008).SarmaandChen(2008)distributeaspecifiedtotalwellrateamongtheactualwellandthepseudo-wellsaccordingtotheposition(xw,yw)oftheactualwellwithinitsgridblock.Intheimplementation,theyuseda2DGaussiandistributioncenteredattheactualwell(xw,yw)forratedistribution.TherateassignedtoeachwellisproportionaltotheintegraloftheGaussianfunctionovertheareaofthegridblockwherethewellresides.Therefore,theactualwellisalwaysassignedwiththelargestportionofthetotalrateandthentherateofeachpseudo-welldependsonitsdistancetotheactualwellposition(xw,yw).Withthisapproximation,theNPVoroilrecoveryisacontinuousfunctionofthewelllocationandthegradient-basedalgorithmscanbethenapplied.Wangetal.(2007)presentedtheunderlyingideaforthegradient-basedsolutionoftheoptimalwellplacementproblemconsideredinthispaper.Thesuppositionisthatonewishestoaddoneormorewaterinjectionwellstoa2Dreservoirthatcontainssomeproducingwells.Theoptimizationproblemisinitializedbyputtinganinjectionwellineverygridblockthatdoesnothaveawellandconstrainingtheproblembyspecifyingamaximumtotalinjectionratethatmustbeallocatedamongtheinjectorsremainingateachiterationoftheoptimizationalgorithm.Inthenetpresentvalue,adrillingcostisassignedforeachwellsothegreaterthenumberofinjectionwells,thegreaterthedrillingcost.DecreasingthenumberofinjectorsdecreasesthetotaldrillingcostwhichbyitselfresultsinanincreaseinNPVbutmayalsocauseadecreaseinNPVduetoadecreasedoilproduction.Iftherateofahypothesizedinjectorisreducedtozero,thewelliseliminatedfromthesystem.Initially,allinjectionwellsinjectatthesameratewhichisdeterminedbydividingthetotalallowableinjectionratebythenumberofinjectionwells.ThenasteepestascentalgorithmisappliedtoadjustratestomaximizeNPVoveraspecifiedreservoirlifetime.Astheoptimizationproceeds,somewellratesaredecreasedtozeroandareremovedfromthesystem.Astheinjectionrateshavetobenon-negative,itisaconstrainedoptimizationproblem.ThealgorithmofWangetal.(2007)usesarestrictedstepsizetosatisfysuchconstraints.Itcanonlyeliminateoneinjector(drivethewellinjectionratetotheboundof0)ateachstepandhenceisveryinefficientforalargescaleproblem.Inthispaper,theconstraintsareensuredwithagradientprojectionmethod,whichcaneliminatemorethanoneinjectoratatimeandhencemakethealgorithmpractical.UnliketheresultsinWangetal.(2007),thispapershowsthatitispossibletohavemorethanoneinjectorleftattheendoftheoptimizationprocess.Moregenerally,thenumberofinjectionwellstobedrilledisnotspecifiedapriori,butisdeterminedbythealgorithm.Inadditiontoitscomputationalefficiency,thisisanotherdistinctadvantageourprocedurehasovertheotherreferencescitedabove.2.WellplacementproblemformulationGivenareservoirwithexistingproductionandpossiblywaterinjectionwells,theproblemofdeterminingtheoptimallocationofnewwaterinjectionwellssubjecttoafixedtotalinjectionrate(qtSTB/D)isformulatedasanoptimizationproblem,whichmaximizesthenetpresentvalue(NPV)overaspecifiedreservoirproductionlifetime.TheNPVfunctionalincludingthedrillingcostsforeachnewwellisdefinedasJðqinjÞ=∑Ntk=1∑Nprodj=1roqko;j−rwqkw;jð1+bÞtk=365!"#Δtk−∑Ninji=1qkinj;iqkinj;i+β!Cinj"#;ð1ÞwhereNtisthenumberofreservoirsimulationtimesteps,Nprodisthetotalnumberofproductionwells,Ninjisthetotalnumberofwaterinjectionwellsspecifiedinitially,roin$/STBistheoilrevenueperunitvolume,rwin$/STBisthewaterdisposalcostperunitvolume,qo,jkandqw,jk,respectively,representtheaverageoilandwaterratesofthejthproduceroverthekthsimulationtimestep,Δtkrepresentsthesizeofthekthtimestepindays,tkrepresentsthetotalsimulationtimeindaysattheendofthekthtimestep,βisanadjustmentparameter,Cinjisthecostofdrillinganinjectionwellandbistheannualdiscountrate.InEq.(1),qinj,iistheinjectionrateoftheithinjectionwell,whichisfixedoverthetotalsimulationtime,whereasqinjdenotesthecolumnvectorwhichhasqinj,iasitsithcomponent.Notethatbecausethetotalwaterinjectionrateqtisfixed,thereisnoneedtoincludeatermintheNPVfunctionalforthecostofwaterinjection.TheoptimalwellplacementproblemismathematicallystatedasmaximizingthefunctionalJdefinedinEq.(1)subjecttotheconstraint,∑Ninji=1qinj;i=qtð2Þwherethetotalwaterinjectionrate,qt,isspecified.Asstatedintheintroduction,theoptimizationproblemisinitializedbyputtingawaterinjectionwellineverygridblockthatdoesnothaveawell.NotethatthefirstsumontherightsideofEq.(1)representsthetraditionaltermforNPVandthesecondsumrepresentsthetotalcostofallinjectionwellsdrilled.AlsonotethateachindividualterminthesecondsumontherightsideofEq.(1)isadifferentiablefunctionofqinj,iwhichdecaystozeroasqinj,i→0aslongasβ≠0.Thus,Jcanbemaximizedusingagradient-basedalgorithm.Whenapplyingagradient-basedalgorithmtomaximizeJ,someofthewellinjectionratesgotozero,effectivelyeliminatingtheassociatedtermsfromthesumthatrepresentsthecostofdrillingtheinjectionwellsinEq.(1).Atearlyiterations,oneexpectsthatthesumrepresentingdrillingcostswilldominatesothatmostoftheincreaseinNPVwillbeduetoeliminatingwells(settinginjectionratestozero).However,atlateriterationswhereonlyafewinjectionwellsareleft,thefirsttermmaydominateandifthisisthecase,itmaybepossibletoincreaseNPVbyredistributingthetotalwaterinjectionrateamonginjectionwells.Becausethetotalrateofwaterinjectionisfixed,theremustbeatleastoneinjectionwellleftattheendoftheiteration.ThenumberofremaininginjectorsattheendoftheoptimizationprocessdependsonthebalancebetweenthefirstandsecondtermsintheNPVofEq.(1).Asthecostofdrillingawellincreases,thenumberofremaininginjectorswilldecreaseasthesecondterminEq.(1)reducestheNPVvalue.3.AdjointgradientForthewellplacementproblem,onewishestomaximizetheNPV,J(Eq.(1))byadjustingthecontrolvectorqinjsubjecttotheconstraintsinEq.(2).Thecalculationofthegradientismostefficientlydoneusingtheadjointmethod.DetailsonthecalculationofthegradientofNPVwithrespecttowellcontrolsusingadjointgradientcanbefoundinBrouwerandJansen(2004)andSarmaetal.(2005)andthusareonlysummarizedhere.Thesimulatorequationscanbewrittenas,Fðx;qinjÞ=0;ð3Þ221andEngineering73(2010)220–226222K.Zhangetal./JournalofPetroleumScienceandEngineering73(2010)220–226wherexrepresentsthevectorofprimaryvariablesinthereservoirsimulator.Eq.(3)representsthediscretesetofequationsinthereservoirsimulator.TocalculatethegradientofNPV,wedefineβusingtheLagrangianformulationasβ=J½x;qinjC138+λTFFðx;qinjÞ;ð4ÞwhereλFistheadjointvectorfortheflowequations.Thetotaldifferentialofβcanbewrittenas,dβ=∂J∂xdx+∂J∂qinjdqinj+λTF∂F∂xdx+λTF∂F∂qinjdqinj:ð5ÞEq.(5)canberearrangedas,dβ=∂J∂x+λTF∂F∂xC18C19dx+∂J∂qinj+λTF∂F∂qinj!dqinj:ð6ÞThegradientofβwithrespecttoqinjcanbecalculatedasð∇qinjβÞT=∂J∂qinj+λTF∂F∂qinj;ð7Þor,∇qinjβ=∂J∂qinj!T+∂F∂qinj!TλF;ð8ÞprovidedthatλFsatisfiestheadjointsystemofequationsgivenby∂J∂x+λTF∂F∂x=0:ð9ÞThesystemofequationsinEq.(9)issolvedforλF.OnceλFiscalculated,itcanbesubstitutedintoEq.(8)tocalculatethegradientofthefunctionalβ,whichrepresentsthegradientofthefunctionalJ(NPV)withrespecttoqinj.4.ControlconstrainedoptimizationalgorithmAstheinjectionratesarerequiredtobepositiveandthetotalinjectionrateisconstant(Eq.(2)),theinjectionrateforeachinjectorshouldfallwithintheboundconstraints,0≤qinj;i≤qt:ð10ÞHere,agradientprojectionmethod(Luenberger,1984;NocedalandWright,1999)isappliedtosatisfyboththelineartotalinjectionrateconstraint(Eq.(2))andtheaboveboundconstraints(Eq.(10)).4.1.SearchdirectionThemathematicalproblemistomaximizetheNPV,i.e.maxJðqinjÞ;ð11Þsubjecttothefollowinglinearconstraints:aTiqinj=bi;i=1;⋯;M;ð12Þor,Aqinj=b;ð13ÞwhereMisthenumberofconstraintsincludingthelineartotalTinjectionrateandtheactiveboundconstraints,A=[a1,a2,⋯,aM]isamatrixofdimensionM×Ninjandb=[b1,b2,⋯,bM]TisanM-dimen-sionalcolumnvector.TomaximizetheNPV,J(qinj),weneedtofindanuphillsearchdirectiondℓattheℓthiteration,i.e.werequirethatðgℓÞTdℓN0;ð14ÞwheregℓisthegradientofJ(qinj)withrespecttothevectorofcontrolsqinj,i.e.,gℓ=∂J∂qinj;1;…;∂J∂qinj;k−1;∂J∂qinj;k;∂J∂qinj;k+1;…;∂J∂qinj;Ninj"#Tqℓinj;ð15Þwhichisobtainedthroughtheadjointformulation.Thecontrolsareupdatedasfollows,qℓ+1inj=qℓinj+αℓdℓ:ð16ÞAnotherrequirementonthesearchdirectiondℓisthatanypointalongthisdirectionsatisfiesthelinearconstraintsofEq.(13).AssumingthelinearconstraintsofEq.(13)aresatisfiedattheℓthiteration,i.e.,Aqℓinj=b;ð17Þwerequirethatqinjℓ+1alsosatisfiesthelinearconstraintssothatAqℓ+1inj=Aqℓinj+αℓAdℓ=b:ð18ÞComparingEqs.(17)and(18),itfollowsthatthelinearconstraintscanonlybesatisfiedbyrequiringAdℓ=0:ð19ÞAsearchdirectionthatsatisfiesbothconditions:uphill(Eq.(14))andlinearconstraints(Eq.(19))canbefoundthroughthegradientprojectionmethodandisgivenbydℓ=ðI−ATðAATÞ−1AÞgℓ:ð20ÞThedetailedderivationofEq.(20)isgivenin(Luenberger,1984).4.2.LinesearchandapplicationprocedureTwotypesoflinesearchmethodsusingtheprojectedgradientaretested.ThefirstmethodisessentiallythesameastheonefromWangetal.(2007).Inthismethod,thegradientisprojectedonlyontothetotalinjectionratelinearconstraint,soA=1;1;:::;1½C138.Aftergradientprojection,thesumofthecomponentsofthesearchdirectiondℓisequaltozeroaccordingtoEq.(19),i.e.∑i=1Ninjdiℓ=0.Thus,theupdatedinjectionrates(componentsofqinjℓ+1)automaticallysatisfythetotalinjectionrateconstraintaslongasthatconstraintissatisfiedforthepreviousiteration(componentsofqinjℓ).TheupperandlowerboundconstraintsofEq.(10)canbesatisfiedbylimitingthestepsizeαℓateachiterationsothatitisnotlargerthanαmaxℓdefinedbyαℓmax=minðαℓmax;iÞ;ð21Þwhereαℓmax;i=−qℓinj;idℓiifdℓib0;qt−qℓinj;iℓifdℓiN0:8>>>>>:ð22ÞdiFig.1.NPV($)contourmap,Example1.223K.Zhangetal./JournalofPetroleumScienceandEngineering73(2010)220–226Inthelinesearchalgorithm,αmaxℓisusedastheinitialguess.IfthisstepsizeresultsinanincreaseinNPV,itisaccepted.Otherwise,anewtrialstepisobtainedbycuttingthestepsizeinhalfuntilonefindsastepsizewhichresultsinanincreaseinNPV.However,astandardquadraticorcubicfit(NocedalandWright,1999)canalsobeusedinlinesearch.Unfortunately,theaboveprocedurecanonlyeliminateoneinjectoratatime,whichmakesthealgorithmveryinefficientifthealgorithmstartswithaninjectorineachgridblockthatisnotpenetratedbyaproducingwell,whichistheinitializationprocedureusedhere.Thefollowingproposesanim
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