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ComputGeosci(2009)13:235–244DOI10.
1007/s10596-008-9087-9ORIGINALPAPERAniterativeensembleKalmanlterforreservoirengineeringapplicationsM.
V.
Krymskaya·R.
G.
Hanea·M.
VerlaanReceived:29November2007/Accepted:14April2008/Publishedonline:14June2008TheAuthor(s)2008AbstractThestudyhasbeenfocusedonexaminingtheusageandtheapplicabilityofensembleKalmanlteringtechniquestothehistorymatchingprocedures.
TheensembleKalmanlter(EnKF)isoftenappliednowadaystosolvingsuchaproblem.
Meanwhile,tradi-tionalEnKFrequiresassumptionofthedistribution'snormality.
Besides,itisbasedonthelinearupdateoftheanalysisequations.
Thesefactsmaycauseprob-lemswhenlterisusedinreservoirapplicationsandresultinsamplingerror.
Thesituationbecomesmoreproblematiciftheaprioriinformationonthereservoirstructureispoorandinitialguessaboutthe,e.
g.
,per-meabilityeldisfarfromtheactualone.
TheabovecircumstanceexplainsareasontoperformsomefurtherresearchconcernedwithanalyzingspecicmodicationoftheEnKF-basedapproach,namely,theiterativeEnKF(IEnKF)scheme,whichallowsrestartingtheM.
V.
Krymskaya(B)·M.
VerlaanFacultyofElectricalEngineering,MathematicsandComputerScience,DelftUniversityofTechnology,Mekelweg4,2628CDDelft,TheNetherlandse-mail:m.
v.
krymskaya@tudelft.
nlM.
Verlaane-mail:verlaan@dutita2.
twi.
tudelft.
nlR.
G.
HaneaTNOBuiltEnvironmentandGeosciences,BusinessUnitGeoEnergyandGeoInformation,TNO,Princetonlaan6,3584CBUtrecht,TheNetherlandse-mail:remus.
hanea@tno.
nlR.
G.
HaneaFacultyofCivilEngineeringandGeosciences,DelftUniversityofTechnology,Stevinweg1,2628CNDelft,TheNetherlandsprocedurewithanewinitialguessthatisclosertotheactualsolutionand,hence,requireslessimprovementbythealgorithmwhileprovidingbetterestimationoftheparameters.
ThepaperpresentssomeexamplesforwhichtheIEnKFalgorithmworksbetterthantra-ditionalEnKF.
Thealgorithmsarecomparedwhileestimatingthepermeabilityeldinrelationtothetwo-phase,two-dimensionaluidowmodel.
KeywordsReservoirengineering·Historymatching·Permeability·EnsembleKalmantler·IterativeensembleKalmanlter1IntroductionThemathematicalmodelingapproachtotheanalysisofreservoirperformancehasgainedpopularitythrough-outtheyears.
However,themodelcanbeusedtoforecastreservoirbehavioronlyifithasbeencali-bratedbeforehand.
Thecalibrationstage,called"his-torymatching"inthereservoirengineeringcontext,aimsatadjustingtheparametersofthereservoirsimu-lationmodelinsuchawaythatthecomputedvaluesofobservablevariablesatindividualwellsareconsistentwithavailablemeasurementsofthosequantities.
Asthemodelsbecomemorecomplicatedandlargerscaled,thereincreasesaneedofautomatichistorymatchingtechniques.
Themainproblemthathastobesolvedviaauto-matichistorymatchingissearchingforthecombina-tionofreservoirparametersforwhichanerrorfunction(objectivefunction)attainsitsminimum.
Thisfunc-tionrepresentsasumofsquareddifferencesbetween236ComputGeosci(2009)13:235–244theobservedreservoirperformanceandtheresultsofsimulationduringthehistoricalperiod[2]:fE=npari=1wi(xioxis)2,(1)wherefEdenoteserrorfunction,nparisthenumberofreservoirparameters,wiistheweightingcoefcient,andxioandxiscorrespondinglyrefertoobservedandsimulateddatathathavetobematched.
Historymatch-ingisusuallyanill-posedproblemsincetherearemoreunknownsthanconstraintstoresolvealltheunde-terminedquantities.
Then,theGauss–NewtonmethodappliedtominimizecostfunctionfailsbecausetheHessianisill-conditioned.
Suchaproblemcanbeover-comebyapplyingsomeregularizationstrategy(e.
g.
,byuseofthepriorgeostatisticalmodel[8]).
Thereareseveralapproachestoautomatichistorymatching,whichdifferinthewaytheyobtainthesetofparametervaluesminimizinganobjectivefunction.
Thechoiceofminimizationtechniqueismainlybasedonwhethertheerrorfunctionhaslinearornonlinearform.
However,mostofthesetraditionalhistorymatchingapproachesareeitherlimitedtothesmall-scaledandsimplereservoirmodelsorinefcientintermsofcom-putationalcosts.
Ingeneral,thesemethodsalsoper-formthetreatmentofuncertaintyviarepeatedhistorymatchingprocessesforvariousinitialmodels,whichre-sultsinevengreatercomputationalefforts.
Moreover,traditionalhistorymatchingdoesnotallowcontinu-ousmodelupdating.
Namely,asthenewdatabecomeavailableforbeingincludedintothematch,thewholehistorymatchingprocesshastoberepeatedusingallobserveddata.
Atthesametime,theamountofde-ployedsensorsforpermanentmonitoringofpressure,temperatureorowratesincreases.
Thisfactyieldstheincreaseofdataoutputfrequencyandlightensupaproblemofincorporatingobtaineddatainthemodelassoonasitbecomeavailablesothatthemodelisalwaysup-to-date.
TheKalmanlteringtechniquesareknownasthemostpopularmethodologyforassimilatingthenewmeasurementstocontinuouslyupdatethestateofthesystem.
Originally,theKalmanlterwasdevelopedforoperatingonthelinearmodels,whilenonlinearityrequiresusingsomefurthermodications,e.
g.
,theextendedKalmanlter.
However,whenthemodelishighlynonlinearorthescaleofthespacevectoristoolarge,applicationoftheextendedKalmanlteralsomeetsdifculties.
ThesedifcultiesareovercomebyapplyingtheensembleKalmanltering(EnKF)algo-rithmbasedontheMonte-Carloapproach.
Thegreatmajorityoftheproblemsinreservoiren-gineeringarehighlynonlinearandcharacterizedbyalargenumberofvariables;thus,theideatouseEnKFinreservoirsimulationseemstobenatural.
Inparticular,itispresentedinthepublications[5,10].
Otherpapers[4,7]reporttheresultsofusingtheEnKFapproachinhistorymatchingprocesses.
Theyconsidertheapplica-tionofEnKFtoaPUNQ-S3model.
AlthoughthesestudiesclearlyshowthatEnKFissuccessfulinassim-ilatingproductiondatatoupdatetheinitialreservoirmodelanditsapplicationallowsreducingcomputa-tionalcostsforhistorymatching,thereisstillenoughspaceforfurtherinvestigationandimprovement.
Specically,theresearchdescribedin[11]hasshownthat,forsomenonlinearmodels,theEnKFdoesnotprovidecompletelyacceptablecharacterizationsoftheuncertainties.
Thesituationbecomesmoreproblematiciftheaprioriinformationonthereservoirstructureispoorandtheinitialguessaboutthesystemstateisfarfromtheactualone.
ThisleadstotheideaofusingimprovedEnKFmodications,namely,iterativeEnKF(IEnKF)schemes,whichallowimprovingtheinitialensembleusedforsimulationand,hence,theresultingestimatedstatevector.
Inthispaper,weapplyaniterativealgorithmbasedonasequentialdataassimilationschemetoestimatingthepermeabilityeldinreservoirmodels.
Thisalgo-rithmisinspiredbyinvestigationsofA.
H.
JazwinskionaniterativeextendedKalmanlteringapproach[6].
TheimplementationandtheimprovementofthismethodagainsttheclassicalEnKFarepresentedwitha2Dreservoirsimulationorientedtowardsestimatingthepermeabilityeld.
2TheensembleKalmanlterKalmanlteringisapowerfultechniquedesignedforsolvingdataassimilationproblems.
ThissectionpresentsthegeneralideaofKalmanlteringinaman-nersimilarto[9]andofensembleKalmanlteringasgivenin[3].
Letusrestrictourselvestothecaseofthefollowinglinearsystem:xk+1=Fkxk+Bkuk+Gkwk,(2)zk=Mkxk+vk,(3)whereFk,Bk,Gk,Mkarematrices,kisthetimeindex,xkdenotesthestateofthesystem,ukisthesysteminput,zkisthevectorofmeasurements,wkisGaussianwhitesystemnoiseprocesswithzeromeanandcovari-ancematrixQk,andvkisGaussianwhitemeasurementnoiseprocesswithzeromeanandcovariancematrixComputGeosci(2009)13:235–244237Rk.
Theinitialstatex0isassumedtobeGaussianwithmeanx0andcovariancematrixP0.
Moreover,processesx0,wk,andvkareassumedtobeindependentfromeachother.
Vectorxk,whichcontainsinformationonthecurrentsystemstate,cannotbedirectlyobserved.
However,itispossibletomeasurezk,whichissomefunctionofxkaffectedbynoiseprocessvk.
Theideaistousetheavailablemeasurementszkforestimatingthestateofthesystemxk.
Tosolvethelteringproblem,Eqs.
2and3,wehavetodeterminetheprobabilitydensityofthestatexkconditionedonthehistoryofavailablemeasurementsz1,zl.
ItturnsoutthatthisconditionaldensityfunctionisGaussian;hence,itcanbecharacterizedbymeansofacovariancematrix.
However,fornonlinearmodeloperatorFk(whichispreciselythecaseofreser-voirengineeringapplications),suchconditionaldensityfunctioncanberepresentedbythersttwomomentsonlyapproximately.
TheEnKFhasbeenexaminedandappliedinanumberofstudiessinceitwasrstintroducedbyGeirEvensenin1994.
Thislteringapproachisrelativelyeasytoimplementandhasaffordablecomputationalcosts.
TheEnKFisbasedonarepresentationoftheprobabilitydensityofthestateestimateattimekbyanitenumberNofrandomlygeneratedsystemstatesxk,i,i=1,N.
EquationstoobtainthemeanxkandcovariancematrixPkofprobabilitydensityofstatexkattimekconditionedonthehistoryofthemeasure-mentsz1,zlviaEnKFalgorithmcanbeformulatedasfollows[3](wheresuperscriptspandustandforpredictedandupdatedsystemstates):Initializationstep:xu0,iN(x0,P0),i=1,N.
(4)Forwardstep:xpk,i=Fkxuk1,i+Bkuk+Gkwk,i,i=1,N,(5)xpk=1NNi=1xpk,i,(6)Lpk=xpk,1xpkxpk,NxpkT,(7)whereLpkdenesanapproximationofthecovari-ancematrixPpkwithrankN:Ppk=1N1LpkLpTk.
(8)Assimilationstep:Kk=LpkLpTkMTkxMkLpkLpTkMTk+(N1)Rk1,(9)whereKkistheso-called"Kalmangain"matrixdeterminingtheweightswithwhichthemeasure-mentshavetobeincorporatedintothemodelupdateoutcome,xuk,i=xpk,i+KkzkMkxpk,i+vk,i,i=1,N,(10)xk=1NNi=1xuk,i.
(11)NotethatEq.
10involvesgeneratingadditionalnoisevk,iwhileconstructingthemeasurementsetcorre-spondingtotheensemble.
Thisnoisevk,ihasthesamestatisticsasassumedfortheobservationerrors.
Theperturbedmeasurementsarenecessaryduetothefactthattheabsenceofperturbationleadstotheupdatedensemble,whichhastoolowvarianceandcausesthedivergenceofthealgorithm[1].
Actually,theforwardmodelintegrationstepwithinreservoirengineeringframeworkcanbeperformedbymakingaforwardrunofthereservoirsimulatorxpk,i=Fxuk1,i,(12)whichcanbedevelopedseparatelyandusedasablackboxinEnKFanalysis.
ItturnsoutthatparameterestimationviaEnKFisalsopossible.
Thiscanbedonebyconstructingthefollowingstatevector:x=my,whereyconsistsofdynamicvariableschangingwithtimeandmisavectorofstaticmodelparameters,whichareconstantintimeandhavetobeestimated.
Now,theKalmanlteranalysisisperformedontheaugmentedstatevector.
Theforwardstepofthealgo-rithmresultsinupdatingonlythedynamicvariableswithtimeandconservingthevaluesofstaticparame-ters.
However,attheassimilationstep,thevariablesofbothtypesaresimultaneouslyupdated,providingcorrectedestimationsofthestatevectorand,hence,modelparameters.
Themodeldescribingmultiphaseuidowinreser-voirishighlynonlinearandthenumberofvariablesincludedintostatespacevectorisverylarge,normallyatleasttwopergridblock.
AlthoughEnKFperformsfairlygoodforthiskindofproblem,itsometimesfails238ComputGeosci(2009)13:235–244toprovideappropriatecharacterizationofuncertainty.
Anexampleisgivenin[11]inrelationtothecasewhentheconditionalpdfforthereservoirmodelismulti-modal.
SuchaphenomenonresultsfromthefactthatmodelnonlinearitydestroysthenormalityofapriorandaposteriordistributionswithinKalmanlteringanalysis.
Moreover,thetaskofobtainingaccurateestimationofthestatevectorbecomesharderiftheaprioriin-formationaboutthereservoirstructureispoorandtheinitialguessabout,e.
g.
,permeabilityeldisfarfromactualone.
TheKalmanltertechniquesaredesignedinsuchawaythatinitialconditionstendtobeforgottenbythelteralgorithmasmoredataareassimilated[9].
However,itisimportantforreservoirengineeringcasesthatreasonableestimationsofthemodelparametersareobtainedbasedonthedatacollectedbeforethewa-terbreakthroughevents.
Inturn,thesedatamaynotbeenoughtoimproveapoorinitialguessabouttheeldstructure.
Insuchasituation,forsomeapplications,theideaofiteratingltergloballycanbeofhelpsinceitallowsrestartingtheprocedurewithanewinitialguessthatisclosertotheactualsolutionand,hence,requireslessimprovementbythealgorithm.
WearegoingtoconsiderthehistorymatchingviaEnKFalgorithmasthestartingpointforfurtherin-vestigations.
WecontinuewithsomeintroductionintoalternativeEnKFtechniques.
3IterativeKalmanlteringThecurrentsectionpresentstheideasofKalmanl-teringalgorithmsthat,inouropinion,canbealterna-tivelyappliedtosolvingthehistorymatchingproblem.
IterativeformsoftheKalmanlterarenotcompletelynewwithinthescopeofreservoirengineeringapplica-tions.
Thesemethodsaimatobtaininganyensemblethatprovidesimprovingtherepresentationofthestatedistribution.
Thereexistseveralapproachesinpetro-leumengineeringliterature,e.
g.
,thead-hocconrmingEnKFmethodproposedby[10].
WewouldliketoexploittheideaofJazwinskitoiteratethelterglobally[6].
AlthoughitwasoriginallysuggestedtoiteratetheextendedKalmanlter,wemodifytheapproachforthecaseofparameterestima-tionviaEnKFtechnique.
Thealgorithmlooksasfollows.
IncorporatingallavailabledataviaEnKFstartingwithx0andP0,weobtaintheestimatedvaluesofxtendandPtend,wheretenddenotestheendtimepointofdataassimilationperiod.
Ifthenumberofavailablemeasurementsissuf-cientlylarge,wecanexpectthattheestimatedmodelparametervaluemtendisclosertothe"true"onethaninitialm0.
Theestimatedmodelparametermtendnowreplacesm0andbecomesanewinitialguessforthenextglobaliteration,whichisdonebyrerunningtheEnKFbasedonthesamebunchofobservations.
Af-terwards,thisprocedurecanberepeateduntilnosuf-cientchangeinestimatedmodelparameterisobtained.
Notethat,whenrerunningthelter,wechangeonlythemeanestimatorofinitialguessaboutthemodelparameterandnotthestatisticsy0andP0.
Thisyieldsthat,inthecaseofGaussianinitialensemble,thenewinitialdistributioncanbegeneratedbyresamplingwithupdatedmeanmtend,y0Tandthesameinitialcovari-anceP0.
TheowchartofsuchanIEnKFispresentedinFig.
1,wherethedashedblockscorrespondtothestepsthatactuallyarethepartsoftheEnKFalgorithm.
Althoughthemoreeducatedchoiceofinitialguessnaturallyshouldresultinbetterestimation,thereisnoguaranteethatiterationwillconverge.
Thus,theplanistoinvestigatethefeaturesoftheaboveIEnKFtech-niqueandtocheckwhetheritindeedallowsimprovingthestatevectorestimations.
ThenextsectionoutlinesthesettingsoftheexperimentusedtotestEnKFandIEnKFperformances.
Fig.
1IEnKFalgorithmowchart00000ComputGeosci(2009)13:235–2442394ExperimentalenvironmentThestudyisaccomplishedonthebasisofatwo-dimensional,two-phaseuidowmodel.
Themodelimplementationisprovidedbytheforwardreservoirsimulatorusedasablackboxtoperformthetimeupdateinlteringalgorithm.
Themodelisappliedtoatwo-dimensionalsquaredpetroleumreservoirwithasizeof700*700mequippedwithuniformcartesiangridconsistingof21gridcellsineachdirection.
Thereservoiristakentobe2minheight;however,weassumethatallquantitiesareverticallyhomogeneous,whichallowsconsideringuidowprocessesonlyintwodimensions.
Weconsiderthewateroodingstageoftherecoveryprocess,whichisperformedthroughtheexploitationoftheinjectionwelllocatedatthecenterofthereservoirandfourproductionwellsestablishedatthecornersoftheeld.
Theinjectionwellisconstrainedbyaprescribedinjectionrateof0.
002m3/sandproductionwells—bybottomholepressureof2.
5*107Pa.
4.
1StatespacerepresentationInthecaseofreservoirengineeringapplications,thestatevectornormallyconsistsofvectorsofpressures(p)andsaturations(s)correspondingtoeachgridblock.
Toperformparameterestimation,wehavetoincludetheparameterofinterestintothespacevector.
Thestudyisfocusedonestimatingthepermeabilityeld.
Itturnsoutthatthenaturallogarithmofpermeabilityisnormallydistributed,hence,wewouldliketoaug-mentthestatevectorbythevectoroflog-permeability(logk)correspondingtoeachgridcell.
Moreover,whileoperatingonaeld,onecanmeasurethefollowingparametersatthewells:bottomholepressures(pwell),oil(qwell,o),andwater(qwell,w)owrates.
Wecanalsoincludeintoavailablemeasurementsthepressureandsaturationquantitiesatthewells.
Finally,themegastatespacevectortakesthefollowingform:x=logkpspwellqwell,oqwell,w.
Themodelparameterisconsideredasstatic,i.
e.
,time-invariant.
Meanwhile,thevalueofthestaticparameteriscorrectedwithinthedataassimilationstep.
Notethatthestatevectorconsistsof441permeabil-ityvalues,441pressurevalues,441watersaturationvalues,1observedbottomholepressureattheinjectionwell,4observedoilowrates,and4observedwaterowratesattheproductionwells,orsimply,x∈R1332.
TherelationEq.
3betweenthemodelvariablesandthemeasurementsforourexampleisassumedtobegiventhroughtrivialmeasurementmatrixM∈R19*1332withonlyzerosandonesasitselements,arrangedintheblockform:M=0M10000M10000M2withblocksM1∈R5*441andM2∈R9*9ofthefollow-ingform:ElementsofmatrixM1correspondingtotheob-servationsatthewellgridblocks[i.
e.
,elementsindexedas(1,1),(2,21),(3,221),(4,421),and(5,421)aresettoone,therestofthematrixiflledinwithzeros].
M2is,infact,anidentitymatrix.
Letusnotethat,althoughthemeasurementoperatorhasalinearform,theactualrelationbetweenthemodelstatevectorandobservablevariablesisnonlinear.
Thederivednotationonlyshiftsthesourceofnonlinearityanddoesnotvanishitseffects.
WeconsiderthemodelEq.
12asbeingperfect,whichmightseemtobenotveryrealistic.
However,suchanassumptionspeciesbetterenvironmentforinvestigatingaparticularIEnKFmethod.
Weexpectthat,inthecasewhentheensemblespreadisnotin-uencedbymodelnoise,theiterativetechniqueshavetodemonstratetheirspecicfeatures.
Onthecontrary,thevaluesofobservablevariablesareassumedtobeimprecise.
Toinitializethelter,oneneedsgeneratinginitialensemblesofonlystaticanddynamicvariablesbecausethereisnoproductiondataavailableatthestartingtime.
Sincethereservoiristypicallyinastateofequi-libriumatthetimewhenproductionstarts,theinitialdynamicvariables(i.
e.
,initialpressuresandwatersatu-rationscorrespondingtoeachgridblock)areassumedtobeperfectlyknown(withoutuncertainty).
Therefore,theyarethesameforeachensemblememberandequaltotheinitialconditionofthe"true"model(i.
e.
,p=3107PaandSw=0.
2).
Thus,attheinitialmoment,theonlypermutationscontainedintheinitialensemblearecausedbyinitialensembleofpermeabilitymodels.
Withinthestudy,wearegoingtousetheinitialpermeabilityensembleconsistingof999members.
Theensemblemeanand240ComputGeosci(2009)13:235–244Meanofpermeabilityfieldsensemble(log(m2))5101520246810121416182029.
12928.
928.
828.
728.
628.
5(a)Varianceofpermeabilityfieldsensemble((log(m2))2)5101520246810121416182000.
20.
40.
60.
81(b)Fig.
2Mean(a)andvariance(b)ofpermeabilityeldsensemblevariancearevisualizedinFig.
2,wherethetoppicturecorrespondstotheensemblemeanandthebottomimagetothevariance,respectively.
4.
2SyntheticmeasurementsgenerationTotesttheperformanceofEnKFalgorithms,wearegoingtodoaso-calledtwinexperiment.
Itrequiresthatthe"true"valuesofobservablevariablesaregen-eratedsyntheticallybyapreliminaryrunofthemodelitselfandthenoisyobservationsarethencreatedbypermutatingthetruevalueswiththemeasurementer-rornoise.
Thisprocedureensuresthatthemodelisindeedabletomatchthedata.
Thereafter,thesyntheticdataareusedintheassimilationexperiments.
Theimplementationofourin-housesimulatorprovidesthe"true"permeabilityeld(seeFig.
3),whichoriginatesfromthetrainingimageofmeanderingchannels.
Nowitispossibletogeneratesyntheticdatainitial-izingthesimulatorwithtruepermeabilityeld,gridblockpressuresp=3107Pa,andwatersaturationsSw=0.
2.
Theerrorineachobservablevariableistakentobe5%ofitsactualscale.
Thesamecovariancematrixisthenusedtorepresentthemeasurementsnoisewithindataassimilationanalysis.
4.
3MeasuresoflterperformanceWequantifythequalityofestimatingatruepermeabil-ityeldintermsofthefollowingspaceaveragedrootmeansquare(RMS)errorsattimek:RMS(logk)k=(logk)k(logk)true22dim(logk),(13)orRMS(logk)k,i=(logk)k,i(logk)true22dim(logk),(14)wheredim(logk)statesforthesizeofvectorofesti-matedparameters(i.
e.
,dim(logk)=441inourstudy),(logk)kistheestimatedvectoroflog-permeabilityaf-terthekthassimilationstep,vector(logk)k,idenotestheithupdatedensemblememberafterthekthassim-ilationstep,andvector(logk)truerepresentsthetruepermeabilityeld.
Truepermeabilityfield(log(m2))510152024681012141618203231.
53130.
53029.
52928.
528Fig.
3"True"permeabilityeldComputGeosci(2009)13:235–2442415ResultsanddiscussionAsequenceofsimulationshasbeenaccomplishedtotesttheperformanceofEnKFandIEnKFassimilationalgorithmsintheframeworkofestimatingthemodelparametersforatwo-phasetwo-dimensionaluidowmodel.
Weproceedbydescribingparticularinstancesanddiscussingtheobtainedresults.
Thestudyofeachparticularalgorithmincludessolv-ingthehistorymatchingproblemandobtainingtheestimateofmodelstaticparameter(i.
e.
,permeability).
Thelteranalysisisdonefromtimet0=0daystilltend=510days,whichensuresthatthewaterbreak-througheventoccursinnoneoftheproductionwells.
TheuseofclassicalEnKFinreservoirengineeringframeworkmeetsanimportantobstacleconcernedwithobtainingphysicallyunreasonablevaluesofthestatevariables.
Itoriginatesfromperformingdataassimila-tiononthestatevectorwithoutanyconstraintcomingfromthephysicalnatureoftheparameters.
Hence,theupdateddynamicvariablesmaybecomeunfeasibleandinconsistentwithestimatedstaticvariables.
Theauthorsof[10]proposedtoincludeoneadditionalso-called"conrming"stepintotheEnKFalgorithminordertoensurethattheupdatedstateisphysicallyplau-sible.
Theideaoftheconrmationstepisthefollowing:Startingattimemomentk1,we,atrst,performaforwardsimulatorrunuptotimekandthenadataassimilationstep.
Then,wetakeonlyrecentlyupdatedstaticmodelparametersandrunagaintheowsimu-latorfromcurrenttimektothenexttimemomentk.
ThedynamicvariablesobtainedreplacethosegotafterthemeasurementupdatestageofEnKFandbecomeaninitialguessforthenexttimeupdatestep.
Thisprocedureavoidsnonphysicalresultsofthemodeling.
Theinclusionoftheconrmationstepintotheal-gorithmresultsinalmostdoublingthecomputationaltimeduetoadditionalforwardmodelrunperensemblememberateachtimestep.
Infact,weusethecon-rmingEnKFtechniqueinsteadofclassicalensembleKalmanlteringforourinvestigations.
So,fromnowon,wemeanconrmingEnKFtechniqueundertheabbreviationofEnKF.
5.
1HistorymatchingviaEnKFItturnsoutthatEnKFfacesanimportantpracticalproblem,namely,standarddeviationoftheerrorsinthestateestimateconvergesveryslowlywiththenumberofensembles.
ThismakestheEnKFquitesensitivetothenumberofensemblemembersusedforsimulation.
Thus,itseemsreasonabletostartbyinvestigatinghowlterperformancedependsonanumberofensemble01002003004005006000.
40.
60.
811.
21.
41.
61.
82Time(days)RMSerror(log(m2))EnsemblemeanEnsemblemembersFig.
4EnKF:RMSerrorinestimatedpermeabilityvstimemembersandndsomeoptimumensemblesize.
Pre-liminaryanalysisshowsthat,inourcase,perform-ingfurtherEnKFrunsonN=60ensemblemembersseemstobeoptimal.
SinceiterativemodicationofEnKFactuallyhasthesameorigin,wenditappro-priatetousethesameensemblesizeasbeingoptimalalsoforIEnKFruns.
Letusnowpresenttheoutcomeofthedataassimi-lationprocedureaccomplishedviatheEnKFalgorithmwithrespecttotheoptimalnumberofensemblemem-bers.
Weconsiderthequalityofestimatingthemodelparameter.
Forthatpurpose,space-averagedRMSer-rorsEqs.
13and14areplottedintime(seeFig.
4).
Thesequantitiesarerelatedtothepartofensemblemeanandensemblemembers'valuescorrespondingtoevaluatedpermeability.
Thegraphdemonstratesim-provementoftheparameterestimationintherstfewdataassimilationstepsfollowedbystabilizationoftheerror,andreductionoftheuncertaintyforestimatedvalue(sincetheensemblespreadbecomesnarrower).
Thismeansthat,atthelatertimes,assimilateddatacarrylessusefulinformationonreservoirstructurethanattheearlytimes.
Indeed,weobtainapermeabilityeldresemblingthetrueone,althoughsomeunderes-timationofthevaluesintheupperrightandoveresti-mationofthevaluesinthebottomleftcornerareaoftheeldoccur(compareFig.
3andleftbottomchartonFig.
5).
Thevarianceeldisactuallyobtainedasthediagonaltermsofcovariancematrixcomputedfromthestatisticalpropertiesoftheensemble.
Thediffer-encebetweenthetoprightandthebottomrightsub-plotsinFig.
5indicatesreductionofthevarianceand,therefore,uncertaintyintheestimation.
242ComputGeosci(2009)13:235–244Meanofinitialensemble(log(m2))510152051015203231302928Varianceofinitialensemble((log(m2))2)5101520510152000.
20.
40.
60.
81Varianceofestimatedensemble((log(m2))2)5101520510152000.
20.
40.
60.
81Estimatedpermeabilityfield(log(m2))510152051015203231302928Fig.
5EnKF:Initialandestimatedpermeabilityeldsandcorre-spondingvariancesAlthoughhistorymatchingonthebasisofEnKFtechniquehasdemonstrateditsefciencyforproperestimatingmodelparameters,thereisstillspaceforimprovement.
Wemayaimatobtainingbetterrepre-sentationofreservoirheterogeneousstructures,whichinturnwillresultinincreasingthequalityofforecasts.
5.
2HistorymatchingviaIEnKFWeproceedbyrunningtheIEnKFalgorithmforthetrialexample.
Infact,weaccomplishthesecondglobaliterationoftheEnKFmethod.
SpaceaveragedRMSerrorsEqs.
13and14areplottedintime(seeFig.
6)toevaluatethequalityofestimatingthemodelparame-ter.
Thegraphdemonstratesimprovementforneitherparameterestimationnoruncertaintycharacterization,01002003004005006000.
40.
60.
811.
21.
41.
61.
82Time(days)RMSerror(log(m2))EnsemblemeanEnsemblemembersFig.
6IEnKF:RMSerrorinestimatedpermeabilityvstimewhichcanbeexpectedsincetherstEnKFiterationdoesnotprovidereducingtheparameterestimationerrorinlatertimesandactuallygivesusarelativelyaccurateestimate.
Indeed,thereisalmostnovisualdifferencebetweenpermeabilityeldsobtainedwithEnKFandIEnKFalgorithms(compareFigs.
5and7).
Considernowasituationwhenaprioriinforma-tiononthevaluesofmodelparametersisfarfromreal.
Forthatpurpose,wetaketheinitialensem-bleoflog-permeabilityeldsandshifteachmemberofitbyaddingthevector0.
5Ishift,whereshiftingvectorIshiftconsistsofonesandIshift∈R1*441.
Notethatsuchashiftdoesnotaffectthevariancestatis-tics,hence,thestructureoftheinitialensembleiskept.
Thedataassimilationisperformedfromtimet0=0daysuptotimemomenttend=510days,andthecovarianceofmeasurementerrorisscaledbythefactorof102topreventlterdivergence.
Suchpara-metersfordataassimilationallowsomereducingtheerrorinestimationofpermeabilityvaluesperformedviaIEnKF(seeRMSerrorsEqs.
13and14plottedintimeonFig.
8).
Indeed,weobtainapermeabilityeldwithastructureresemblingthetrueone,al-thoughsomeoverestimationofthevaluescorrespond-ingtolow-permeabilityareasoftheeldoccurs(seeFig.
9).
Theparametervaluescorrespondingtotheseareasare,inparticular,improvedafterglobaliteration(compareFig.
9aandb).
ThedifferencebetweenthebottomrightchartsinFig.
9aandbindicatesreduc-tionofthevarianceand,therefore,uncertaintyintheestimation.
Notethat,althoughregularlyprovidingoveresti-matedvalues,theltertendstocapturethestructureMeanofinitialensemble(log(m2))510152051015203231302928Varianceofinitialensemble((log(m2))2)5101520510152000.
20.
40.
60.
81Varianceofestimatedensemble((log(m2))2)5101520510152000.
20.
40.
60.
81Estimatedpermeabilityfield(log(m2))510152051015203231302928Fig.
7IEnKF:Initialandestimatedpermeabilityeldsandcor-respondingvariancesComputGeosci(2009)13:235–24424301002003004005000.
60.
811.
21.
41.
61.
822.
2Time(days)RMSerror(log(m2))EnsemblemeanEnsemblemembers(a)01002003004005000.
60.
811.
21.
41.
61.
822.
2Time(days)RMSerror(log(m2))EnsemblemeanEnsemblemembers(b)Fig.
8IEnKF:RMSerrorforestimatedpermeabilityvstime(shiftedinitialensemblewith0.
5Ishiftandmeasurementerrorcovariancematrix102Rareusedinexperiment).
(a)Firstiteration.
(b)Seconditerationoftruepermeabilityeld.
Thishappensbecausetheensembleofpermeabilityelds,usedforthecurrenttest,isonlytheshiftedversionofanalogousensemblepreviouslyusedforinvestigations.
Suchanensem-blecontainssomeinformationontheeldstructure,whichisnotinuencedbysimpleshiftingsinceashiftchangestheensemblemeanandnotthecovariance.
Thegiveninitialstatisticscannotbechangedbecauseitcomesfromthestatisticsofensemblepopulation.
Thus,thepossibilitiesofimprovingparameterestimationbyvaryingstatisticsofinitiallyguessedvaluesofmodelparameterare,inacertainsense,restricted.
AlthoughdemonstratingausageofIEnKFapproachforestimatingpermeabilityvalues,theperformedtestsraiseadditionalproblemstobesolved.
Thelistofsuchproblemsincludesndingcriteriatoevaluatewhethergloballteriterationisneededintherealcasewhennotruepermeabilityvaluesareavailable.
WesupposethatonemayconsidertheRMSdifferencesbetweentheparametervaluesobtainedattwosequentialdataassimilationsteps.
Anotherissueisconcernedwithde-terminingappropriateerrorstatisticsthatcanhaveagreatimpactonimprovementoftheestimationsandthenumberofglobalEnKFiterationsrequiredforthatpurpose.
Summarizing,wemayassertthathistoryMeanofinitialensemble(log(m2))510152051015203231302928Varianceofinitialensemble((log(m2))2)5101520510152000.
20.
40.
60.
81Varianceofestimatedensemble((log(m2))2)5101520510152000.
20.
40.
60.
81Estimatedpermeabilityfield(log(m2))510152051015203231302928(a)Meanofinitialensemble(log(m2))510152051015203231302928Varianceofinitialensemble((log(m2))2)5101520510152000.
20.
40.
60.
81Varianceofestimatedensemble((log(m2))2)5101520510152000.
20.
40.
60.
81Estimatedpermeabilityfield(log(m2))510152051015203231302928(b)Fig.
9IEnKF:Initialandestimatedpermeabilityeldsandcorrespondingvariances(shiftedinitialensemblewith5Ishiftandmeasurementerrorcovariancematrix102Rareusedinexperiment).
(a)Firstiteration.
(b)Seconditeration244ComputGeosci(2009)13:235–244matchingonthebasisoftheIEnKFtechniquehasdemonstrateditsefciencyforimprovingmodelpara-meterestimation.
6ConclusionThestudyhasbeenfocusedontheanalysisoftheusageandtheapplicabilityofensembleKalmanlter-ingtechniqueswithrespecttohistorymatchingprob-lems.
Followingtheideapresentedin[6],aniterativemodicationofEnKFisproposed.
TheaccomplishedcasestudyhasconrmedtheusefulnessoftheEnKFtechniqueforsolvingthehistorymatchingproblemandestimatingreservoirmodelparameter.
TheremightoccurproblemsatwhichtheEnKFalgorithmdoesnotprovideresultsofsufcientaccuracy.
AnappropriateuseoftheIEnKFmethodinsuchacasecanimprovetheestimations.
AcknowledgementsTheauthorswouldliketothankProf.
J.
D.
Jansenforprovidingthereservoirsimulatorandsyntheticreservoirstructure.
WearegratefultoProf.
A.
W.
Heeminkforusefulremarks.
Thecommentsofthetwoanonymousreviewershelpedtoclarifytheobjectiveofthepaper.
OpenAccessThisarticleisdistributedunderthetermsoftheCreativeCommonsAttributionNoncommercialLicensewhichpermitsanynoncommercialuse,distribution,andreproductioninanymedium,providedtheoriginalauthor(s)andsourcearecredited.
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