Oil Reservoir Production Forecastingwith Uncertainty EstimationUsingGeneticAlgorithms

Harald H. SolengNorwegianComputingCenter

P.O.Box 114BlindernN-0314Oslo,NorwayHarald.Soleng@nr.no

Abstract: A geneticalgorithm is applied to the problemofconditioning the petrophysical rock propertiesof a reser-voir model on historic production data. This is a diffi-cult optimization problem where each evaluation of theobjective function implies a flow simulation of the wholereservoir. Dueto the high computing costof this function,it is imperative to make use of an efficient optimizationmethod to find a near optimal solution using asfew itera-tions aspossible. In this study we have applied a geneticalgorithm to this problem. Tenindependentruns areusedto give a prediction with an uncertainty estimatefor thetotal futur eoil production using two differ ent productionstrategies.

1 Oil production history matching

In orderto beableto forecasttheoil productionusingdiffer-entproductionstrategies,oneneedsrealisticgeologicalmod-elsof theoil reservoir. Thegeologicalmodelsareformulatedon grids with thousandsor millions of grid cells. Eachgridcell hasseveralphysicalvariables,andhence,thenumberofunknown parametresin a realisticreservoir characterizationis formidable.

The history matching problem is the problem of find-ing geologicalmodelswhich areconsistentwith both staticdata—suchas permeabilitiesand porositiesas measuredinwellboreplugs—andwith dynamicdatasuchasproductionrates,bottomholepressures,andgasoil ratiosthroughouttheproductionhistoryof thefield.

In general,thehistorymatchingproblemis a non-uniqueinverseproblem; several combinationsof parametrevaluesrepresentingthegeologycouldgivethesameproductionper-formance.In afull scaleheterogeneousreservoir, thenumberof unknown parametresis often ashigh asseveral millionswhile the numberof observablesis muchsmaller. The taskis to find a setof parametresso that the differencebetweentheresultsof flow simulationsandthetrueproductionhistoryis assmallaspossible.This is a hardoptimizationproblem.Sincethecomputationalcostof differentiationwithin a flowsimulatoris very high, we have chosento experimentwith ageneticalgorithm(GA) [1, 2] asanoptimizationtool.

For any realisticcaseoneexpectthat therearemany lo-cal optimain the history matchingproblem. Sinceproduc-tion parametresto a largeextentaredeterminedby thegeol-ogy nearthe wells, regionsfar from wells arenot very well

determinedby this kind of inversemodelling. This induceslarge uncertaintiesin the predictedproduction,especiallyifnew wells aredrilled. In orderto estimatethispredictionun-certainty, wehavegenerateda populationof historymatchedmodelswhereeachindividual is generatedby an indepen-dentGA optimization. In this way it is hopedthat we spana significantpartof theparametrespacecompatiblewith theknown productionhistoryandthus,thatwe canestimatethepredictionuncertainty.

2 The PUNQ S3case

Themethodis testedon a syntheticoil field preparedaspartof thePUNQ(productionforecastingwith uncertaintyquan-tification)projectsponsoredby theEuropeanCommunity. InthePUNQprojecttenpartnersfrom industry, researchinsti-tutesanduniversitiesarecollaboratingon researchon uncer-tainty quantificationmethodsfor oil productionforecasting[3]. The‘historic data’of thecasestudiedin this paperweregeneratedusingtheEclipseoil simulator. Gaussiannoisewasaddedto both thehistoricdataandthewell observationsbe-fore the datasetswith uncertaintieswere presentedto thepartners.At thetimethehistorymatchingwascarriedout,thetrue reservoir wasnot known to the partners.In the presentwork weusedtheMoresimulatorfor historymatching.

Figure1: A reservoir modeloptimizedwith GA.

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Figure2: Wellboreplugdata:logarithmof verticalandhorizontalpermeability���������

v and��������

h versusporosity � , respectively.

The productionhistory for the oil field wasknown for aperiodof 8 years.Thefield has6 wells. In thepresentcasestudywework onarelatively smallblock-centeredCartesiangrid with dimensions�� ���������� . Thegeologicalparame-tresarethehorizontalandverticalpermeabilities,� h and � v,andtheporosities,� . About onethird of thegrid blocksareinactive. Figure1 shows the porosityfield of an optimizedmodel. The horizontalsize of the reservoir is much largerthanits verticalsize,andhence,thegrid blocksarevery thinslivers.Consequently, thethird dimensionis hardlyresolvedin Fig. 1.

Therearesix wells with well boreplug observationsforall five layers.Thewell dataindicatesa 95%correlationbe-tweenhorizontalandverticalpermeabilitiesandtheporosity,cf. Fig. 2. For this reasonwe usedthreehighly correlatedGaussianrandomfields conditionedon well observationstointialize thepopulationbeforestartingtheoptimization.

Thepressureof thefield wasmaintainedby a numberofaquifers.It wasthereforenotnecessaryto useinjectionwells.In thetestcase,ananalyticaquifermodelof theCarter–Tracytype was specifiedas part of the Eclipsemodelfile. Sincethereis no suchaquifermodel in the currentversionof theMore simulator, we had to constructone by using setsofisolatedinactive blocksto representhugewatersourcesandnon-neighbourconnectionsto modelwaterflow into thegridblocksspecifiedby theEclipseaquifermodel.

After 8 yearsof production,two differentrecoverystrate-giesshouldbe considered.The first alternative wasto con-tinue productionfor another8.5 yearsusing the original 6wells. Thesecondalternative wasto add5 new wells for thenext 8.5years.Theaimwasto giveforecastswith uncertaintyestimatesfor the total oil productionfor eachof theproduc-tion strategies.

3 The geneticalgorithm

The objective of history matching is to find a geologicalmodelgivingsimulationresultsascloseaspossibleto therealproductionhistory. Themainideaof evolutionprogrammingis to searchfor the optimumfrom a populationof possiblesolutions.New statesareproposedby recombinationof ge-neticmaterialin thepopulation.By applicationof Darwin’s

survivalof thefittestthepopulationincreasesin fitnessmak-ing efficient useof gradientinformationimplicitly encodedin the population. The geneticalgorithmis basedon a ge-netic representationof possiblesolutions,a matingoperatorfor producingoffspring,anda mutationrule.

3.1 History mismatchand fitness

In historymatchingwetry to minimizethedeviationfrom thetrue productionhistory. Thusthe mostfit modelsarethosewith aminimaldeviationfrom thehistoricdata.Hence,let usquantify this deviation by a weightedsumof squareddevia-tions ��������� � "! #%$& ' (*) � ' #�+&, (*)

- ' , �/. ' , �����10�2 ' , �435 3' , (1)

where�

is thereservoir characteristicsvector, ! is thenum-berof time series,

'is numberof elementsin time series6 ,. ' , are the differentsimulatoroutput resultsfor the bottom

holepressure,thegas/oilratio, andthewatercut,2 ' , arethe

correspondingobservations,-' , aretheweights,and 5 3' , are

the variances.The weightsandvarianceswerespecifiedinthePUNQS3case.

Thenwe definethefitnessof a particularsetof reservoircharacteristics

�as 7 ������� ��98;: �<�=��� (2)

wherethe history mismatch�<�����

is given in Eq. (1). If forsomereason,the flow simulatorcrashedduring the simula-tion, the fitnesswassetequalzero. Thusthe fitnessis nor-malizedto theinterval > ?�@A�CB .3.2 Geneticrepresentation

In order to preserve the high degreeof correlationbetweenthepetrophysicalparametresasseenin thewells (cf. Fig. 2),the reservoir genomewas codedas an array of active gridblocks. Eachgrid block carriesa setof threepetrophysicalblockvalues,namelythehorizontalpermeability, theverticalpermeability, andtheporosity. Thusin this modeleachgenecodesthe completesetof petrophysicaldatafor a reservoirblock.

3.2.1Geneticoperators

Theinitial populationwascreatedby usinghighly correlatedtransformedGaussianrandomfields(conditionedonthewellobservations)for thegeologicalparametres.Thetransforma-tion mapsthe Gaussianrandomfield to porosityvaluesbe-tween0 and0.3. For the permabilitieswe usedexponentialtransformsof theGaussianrandomfieldsto getonly positivevalues.

Thecrossingoperator is a simpleonepoint crossover onaone-dimensionallist of blocks.Suchasimplecut-and-pasteoperatorin statespacecould leadto somerealizationswithvery high densitycontrasts.Onecouldavoid this by imple-mentinga smootherinterpolationbetweengeneticmaterialfrom the two parents,but we have chosento let the objec-tive function take careof it; realizationsthat make the flowsimulator crashare given zero fitness. However, using athree-dimensionalcrossover methodwith smoothinterpola-tionswould probablyimprovethemethod.Themutationop-erator is a simpleswap operatorinterchangingpairsof gridblocks.Theprobabilityof swappingthegrid blocksobservedin thewells is zero. Thus,theconditioningon well observa-tionsis preservedby thealgorithm.

3.2.2GA algorithm and parametres

We useda steadystategeneticalgorithm with overlappingpopulations[4, p. 32]. We useda populationsizeof D � ��? ,amutationprobabilityof 1%,acrossoverprobabilityof 90%,andareplacementpercentageof 25%in eachgeneration.

Thus,in eachgeneration12 childrenareborn,and12 in-dividualspassaway. We let 90% of the childrenbe formedby crossoverfollowedby mutation.Theremaining10%werecreatedasmutatedclonesof existing individuals.Themuta-tion probabilityfor eachgenein a chromosomeis 1%. Witharound1700genesin eachchromosome,thegenomeof atyp-ical child hasundergone17mutations.

The evolution was stoppedwhen the meanfitnesswaswithin 98% of the fitnessof the best individual. Childrenreplacedtheworstindividualsof thepopulation.For mating,a roulettewheelselectorwasused. Let

7 'be the fitnessof

individualnumber6 . If thetotal fitnessof thepopulationis

EGFIH& ' (*)7 '@ (3)

thentheprobabilityfor theindividual J to bechosenfor mat-ing is . , �

7 ,ELK (4)

A totalof tenindependentrunsweremadein orderto beableto giveanestimateof theforecastuncertainty.

4 Results

The objective of this studywasnot to find a singlesolutionmatchingtheobservedhistory. Ratherit wasto give a good

forecastof thetotal oil productionandto estimatetheuncer-tainty of this forecast. Thus, it wasnecessaryto produceapopulationof historymatchedreservoir modelswherethein-dividualsmay representdifferentlocal optimaof the fitnesslandscapeof historymatching.Thiswasobtainedby pickingthe bestindividual of the last generationof ten independentGA runs. In this sectionwe reporton the resultsfor historymatchingandforecastsusingthis setof optimizedreservoirrealizations.In theoptimizedpopulation,themeanvalueoftheweightedsquareddeviation

�, asdefinedin Eq. (1), was�%KM�%� (rangingfrom �NKO��? – �%K PNQ ). The meanof meansof the�A?R�S��? realizationsdrawn from theprior was T�KM��T . Themin-

imalandmaximalvaluesof thehistorymismatchof theunop-timized realizationswere �%KM��� and ��K TN? , respectively. Thus,a significantimprovementof

�is observedin theoptimized

populationcomparedto theprior distribution.

4.1 History match

Sincethe stoppingcriterium of the GA was that the meanfitnessis with 98%of thebest,thenumberof flow simulationrunsbeforeconvergencewasreachedvariedamongthe GAruns. The meannumberof reservoir simulationrunsin thegeneticalgorithmwas �%?U� spanningthe rangeof ��QNQ – �%����P .Thetwo runswith lowestandhighestfinal valuefor

�������are

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Figure3: Evolutionof V �XW of thebest(black)andworst(red)indi-vidualsof two independentGA runsasfunctionsof numberof flowsimulations.

shown in Fig. 3. Observe that the run with thehighestfinalhistorymismatchstoppedearly. Thereis a generaltendencyof earlystoppersbeinglesswell adaptedthanthoserunninglonger. This maybetakenassignof prematureconvergenceof theGA.

Detailsaboutthehistorymatchfor theoptimizedrealiza-tionsareshown in Figs.4–6.

Figure 4 shows the simulatedbottomhole pressurever-sustime for thetenoptimizedrealizations.Thefigureshowsthehistoricdatawith � 5 errorbars.Theinitial step-like be-haviour is dueto well testscarriedout in thefirst partof thesimulation. The long plateaurepresentsa shut-down periodof 1461days. In this periodthereis a slow increasein thepressure.In the productionprofilesof the optimizedreser-voirs this increaseis slightly smallerthanin thehistoricdata.Thus,thesimulatedbottomholepressures(Fig. 4) at theendof the shut-down periodfall outsidethe � 5 errorbar at this

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Figure4: Bottomholepressureversusdaysfor six productionwells. Thespikesaredueto periodictestingandmaintainance.Shuttingtheproducingwellscausesa rapidincreasein theirpressuresfollowedby a largedropwhenproductionis restarted.

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Figure5: Gasoil ratiosversusdaysfor six productionwells.

datapoint. The periodicspikesin the plots aredueto welltests;at certaintimesthewellsareshutin for testsandmain-tainance.During shut-inthepressureincreases,followedbya quickdropwhentheproductionis resumed.

Figure 5 shows the simulatedgasoil ratios versustimetogetherwith observedvalueswith � 5 errorbars.Thefluctu-ationsin thegasoil ratioof wellsPro-1andPro-4arehardtomatch.Thesimulationresultsfor thegasoil ratio (Fig. 5) areoutsidethe � 5 errorbarfor severaldatapoints.

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Figure6: Wateroil ratio versusdaysfor the only productionwellwith anonvanishingwaterproduction.

In all wells exceptone, the watercontentis almostzerothroughoutthe whole productionhistory. This is the caseboth in the ‘true’ history andthe simulationdata. The sud-denappearanceof water in well Pro-11is well matchedbythesimulationsfor all tenrealizations(cf. Figure6).

4.2 Forecasts

Oneof theobjectivesof historymatchingis to beableto pre-dict theproductionof a field. After 16.5yearsof productionwith the 6 original wells, we founda meantotal productionof T�K ���[�\��?%] m oil ( 5 � ?_K ?%P��`�A?N] m ). To studythe

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Figure7: Estimatedprobabilitydistributioncurvesfor total oil pro-ductionin unitsof acb�d me with theoriginalwells.

effectof incrementaldrilling, fivenew productionwellswereadded. Their productionstartedat the end of the “historymatchingperiod”. With theseaddionalwells, the forecastedtotal productionis PfK QN <�g�A?N] m oil ( 5 � ?_K ?N <�g�A?N] m ).Theestimatedprobabilitydistribution curvesfor thetotal oilproductionandtheincreasein theoil productionwith 5 incre-mentalwells aredepictedin Figs.7 and8, respectively. Thenumericlabelson thedatapointsarethecorresponding

�<�����values.

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Figure8: Estimatedprobabilitydistribution curve for increasedoilproductionwith fiveadditionalwells in unitsof ahb d me .

Thepredictionsof Figs.7 and8 shouldbecomparedwiththe ‘true total production’obtainedby runningMore on the‘true reservoir’. With theoriginalwells the‘true’ productionwas T_K �U�<�i��? ] m . This agreeswell with thepredictionofT�K ���kj�?_K ?%Pl�m��?%] m . With five infill wells,thetrueproduc-tion increasedto �Kn���[�g��?%] m comparedto a predictionofP_K Q% RjL?_K ?N <�g�A?N] m . In this casethepredictedrangefallsfarbelow thetrueanswer.

4.3 History mismatchand production

Onecouldaskif thereis a correlationbetweentotal oil pro-ductionandvaluesof

�<�����. As seenin Figs.9 and10 this

seemsnot to bethecase.Hence,theprematureconvergenceof thegeneticalgorithmhasnotbiasedthepredictions.

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Figure9: The total oil productionwithout incrementalwells as afunctionof V �XW .

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Figure10: Theincrementof thetotal oil productionwith additionalwellsasfunctionof V �XW .

Figure11 shows a crossplot of total oil productionwithandwithout incrementalwells. As oneshouldexpect,there

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Figure11: Total oil productionwith incrementalwells versustotaloil productionwith theoriginalwells.

seemsto be a weaklinearcorrelationbetweentotal produc-tion with thetwo productionstrategies.A goodreservoir with6 wells is still goodwith 11.

4.4 Discussion

The true value for the casethat no new wells aredrilled iswithin the � 5 uncertaintyof theforecast.However with fivenew productionwells in additionto theoriginalsix, thefore-castis significantlylower thanthetrueproduction.Thismis-matchcanbeexplainedfrom thefollowing arguments:

a) Sincethe‘true reservoir’ wasnon-Gaussian,initializationusingGaussianfields may be too restrictive, becauseofthefinite correlationlengthof theGaussianfields.

b) Thetrueproductionhistorywasgeneratedwith adifferentflow simulatorthantheoneusedfor historymatching.

c) Theregionsfar from wellswith known productionhistoryarenot muchconstrainedby inversemodelling,andthus,theproductionuncertaintyof new wells in suchregionsismuchhigherthantheproductionuncertaintyof thehistorymatchedwells.

Argument(a) is strengthenedby a study using an adaptiveMetropolis–Hastingsalgorithm confinedto Gaussianfields[5] wherethe predicteduncertaintyis even smallerthan inthe presentcase.Thus,the useof Gaussianfields focusthesearchon a too smallpartof theparametrespaceandforcesthe geologicalmodelsinto a small subspacewhich may notcover that of the true reservoir. We now know that the syn-theticreservoir is non-Gaussian.

(b) Usinga differentsimulatorfor matchingthantheoneusedto generatethe ‘true’ history may alsoproducea bias.For regionsnearwells with a known productionhistory, thehistorymatchingwill compensatefor thesimualtorbias,butnot in otherregions.

The point raisedin argument(c) comesinto play whennew wells aredrilled. Genesrepresentingpetrophysicaldataof grid blocksfar from wells with known productionhistoryaresubjectto a very weakselective pressure.Consequently,geneticoperatorsleadto mediocrepermeabilitypropertiesinsuchregions. If high-permeablegrid cells arerequirednear

the existing wells, thenthe grid cell swappingof the muta-tion operatorwould tendto put low-permeablecells in lessimportantregions. Furthermore,thegrid cell swappingpro-duceswhitenoisein thepermablityfields.Dueto thelack ofcorrelationin white noise,suchfieldshave lower long rangepermeabilitythansmootherGaussianfields. Thus, the pre-dictedproductionof new wells arebiasedtowardsmediocreor low yield.

Althoughthebiasingof predictionscanbereduced,it cannevervanish.Thestartingpoint for optimizationmustalwaysbea classof geologicalmodels.Theadvantageof narrowingthesearchby a restrictive modelhasto betradedagainsttherisk of missingimportantpropertiesof the true geology. Inaddition,theflow simulatorhaslots of simplifying assump-tions.As aconsequence,it will neverreproducethetrueflowof a real reservoir even if we hadexactknowledgeaboutitspetrophysicalrock properties.However, by makingsurethatour history matchingmethodsrespectthe prior geologicalknowledgeaboutthereservoir, i.e., by doingbetterthanpro-ducingwhite noisein regionswith weakselective pressure,thepredictabilityof theyield of new wellsshouldimprove.

5 Conclusions

A steadystategeneticalgorithmhasbeenappliedto ahistorymatchingproblemin oil reservoir exploration. The methodhasproven to be reasonablyfast in obtainingnearoptimalsolutions,i.e., geologicalmodelsgiving flow simulationre-sultscloseto theobservedproductionhistory. To beabletoquantify the uncertaintyof the predictedoil production,wecreatedapopulationof historymatchedrealizationsusingthebestindividualsof tenindependentGA runs.Usingthesetengeologicalmodelsasinput to theflow simulator, productionforecastsweremadefor two differentproductionstrategies.Theseforecastsaresimilarenoughto indicatethatall theendrealizationsof the reservoir aregeologicallysimilar. Albeitthis methodis not a statisticallycorrectmethodfor drawingfrom theposterior, it givesavaluableestimateof theforecast-ing uncertainty.

This work hasshown thatgeneticalgorithmsarepromis-ing in historymatchingfor realpetroleumreservoirs. In thisconnectiontheinherentparallelismof thealgorithm—andthehugepotential for speedingup the calculationson parallelmachines—willbe essentialwhen attackinglarger models.It is worth mentioningthat very powerful parallelmachinescan be built as Linux clustersusing off-the-shelfPC hard-warewhich to a largeextentalreadyis part of the inventoryof mostoil companies.

However, therearestill a few problemswith themethodasimplementedhere.Most of thesearesmallandsolvable;wehave, for example,not experimentedmuchwith parametresof thegeneticalgorithmitself. For instance,it is believedthatprematureconvergencecanbeavoidedby usingseveralsep-aratepopulationswith migrationinsteadof thesinglesteadystatepopulationusedin this paper. Also, asmentionedear-

lier, thedisruptiveeffectof crossovercanbereducedby usinga 3D crossoveroperator. It canbefurtherrefinedby interpo-latingbetweengeneticmaterialfrom thetwo parentsnearthecrossoversurfaces.This would reducetherisk for endingupwith superficialhighcontrastsurfacesin thechildren.

Otherproblemsaremorefundamentalandharderto solve.For example,we needto find a geneticalgorithmrespectingprior geologicalknowledgeaboutthe reservoir suchas thecorrelationstructureof thepetrophysicalfields. In this studyweusedapopulationof 50transformedGaussianfieldsasthestartingpoint for ourgeneticalgorithm.Thesefieldshadcer-tain trendsrepresentingprior informationaboutthegeology.However, by breakingup thecorrelationstructurethegeneticoperators(especiallytheblock swappingmutationoperator)tendto biastheproductionfrom new wells towardslow yield.Similar andmoresevereproblemsoccurif onetries to com-bine geneticalgorithmswith moreadvancedreservoir char-acterizationmethodsmakinguseof moredetailedgeologicalknowledgeaboutthereservoir. For example,a so-calledflu-vial reservoir [6] consistsof a network of channelsin a lowpermeablebackground(afluvial reservoir is theresultof riverdepositsbuilt upovermillions of yearsby thechangingposi-tion of ariver-bed).Suchreservoirscanberepresentedby ob-jectmodels.In orderto applyageneticalgorithmto reservoirmodelsof this type, oneneedsto find suitableparametriza-tionsandgeneticrepresentationsof objectmodelssothatonecandefinemeaningfulcrossingandmutationoperatorswith-out loosingtheprior geologicalknowledgeinherentin thesemodels.

Ideally, onewould like to find notonly geologicalrealiza-tions that matchthe history well, but their statisticallycor-rectdistribution. To meetthis demand,thegeneticalgorithmcouldbecombinedwith aMarkov chainMonteCarlosampler[7, 5].

Acknowledgements

Theauthorwould like to thankAnneRandiSyversveenandLars Holdenfor commentsandsuggestions,Hakon Tjelme-land for permissionto usehis codefor Gaussianfields,andSmedvigTechnologiesfor supportandfor donatinga lisence

for theModularOil Reservoir Evaluationprogramme(More).Theauthorespeciallywantsto thankDavid Pontingof Smed-vig Technologiesfor his kind assistancein setting up themodelfiles for More. The software for this work usedtheGAlib geneticalgorithmpackage,written by Matthew Wallat the MassachusettsInstituteof Technology. The authorisgratefulfor financialsupportfrom theEuropeanCommunitythroughtheJoule-IIInon-nuclearenergy programme.

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