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USAhydrocarbonnearNorthdirectiondeveAnincreasingnumberofhydrocarbonreservoirsmaturing.Morethan70%oftheoilandgasproducedsecondaryortertiaryproduction(MengandFuh,productionfromthesefieldshasresultedindepletion,causingtheeffectivestressinthereservoirsorfailure(e.g.,Dusseaultstatevariablesarescalarandhenceyieldanisotropicchangeinperme-ability,i.e.assumethechangeinpermeabilityisequalinalldirections,yindepletedres-dformationscanthatthedegreeofcase,increasesasintheradialdi-Theseeffectsmaystillneedtobeconsideredwhendesigningadrillingfluidschemeforwellsindepletedformations.*Correspondingauthor.DepartmentofPetroleumandGeosystemsEngineering,UniversityofTexasatAustin,200EDeanKeetonSt.,StopC0300,Austin,TX78712-1585,USA.E-mailaddress:daigle@austin.utexas.edu(H.Daigle).ContentslistsavailableatScienceDirectJournalofPetroleumScienceandEngineeringjournalhomepage:www.elsevier.com/locate/petrolhttp://dx.doi.org/10.1016/j.petrol.2017.07.046Received13April2017;Receivedinrevisedform13July2017;Accepted14July2017Availableonline17July20170920-4105/©2017ElsevierB.V.Allrightsreserved.JournalofPetroleumScienceandEngineering157(2017)302–311ofthereservoirandaccuratelyforecastproduction,modelsthatcouplethestress,deformationandpermeabilitymustbeincorporatedinthedrillingplanningprogramsandreservoirsimulators.Thepermeabilitychangesofaporousmediumsubjecttodeformationareusuallydeterminedthroughastatevariablethataccountsforthevolumetricchange,suchasporosity,voidratioorvolumetricstrain.Suchtheimportanceofconsideringpermeabilityanisotropervoirs.Weshowthatoverbalanceddrillingindepletemitigatenear-wellborepermeabilityreduction,andpermeabilityreduction,particularlyintheanisotropicthewellborepressuredecreases,butthepermeabilityrectionstillremainsaround90%oftheoriginalvalue.anddrillingfluidinvasioninthesubsequentwellsthataredrilledthroughoutthedevelopmentprogram.Toproperlyplanthedevelopmentanisotropicpermeabilityalterationmodels,weassessthepotentialerrorassociatedwithassuminganisotropicpermeabilitychangeandhighlightphenomenoncanresultinconsolidationsubsidence,lostcirculation,pressuremanagemefluidproductionissuesandcasingfailure1988;Aadnoy,1991;Alietal.,1994a;Hillis,2002;vanOortetal.,2003).Aparticularproductionistheassociateddepletion-inducedthereservoir,asthiswillinfluencetheproductiointheworldaretodaycomesfrom2010).Hydrocarbonsignificantporepressuretoincrease.Thisoftherock,surfacentissuesduringdrilling,SulakandDanielsen,etal.,1998;Streitandconcernfordrillingandpermeabilityalterationinnratesofexistingwellsbothcross-sectionalareaavailabletoflowandtortuosity(Rasromani,2016).DaigleandDugan(2011)developedamodelforchangesinpermeabilitybothasafunctionofgrainorientationandporosity.Theirmodelallowsdeterminationofthepermeabilitytensorfromtheprincipalstrainsratherthanrelatingabulkpermeabilitytovolumetricstrain.Thisthereforeallowsforanisotropicchangesinpermeabilityandhencepro-ducesamorerealisticrepresentationofdeformation-inducedperme-abilityalteration.Asanexample,weevaluatedthechangesinnear-wellboreandfar-fieldpermeabilityofaverticalproducingwellintheTorformationoftheValhallfieldintheNorthSeaduringporepressuredepletion.Incomparingthepredictedpermeabilitychangesthroughisotropicand1.Introductioneventhoughthedeformationsaredifferentineachdirection.Straincanresultinbothvolumechangeandgrainrearrangement,whichaffectsNear-wellborepermeabilityalterationindepleted,HughDaigle*,EbrahimRasromani,K.E.GrayDepartmentofPetroleumandGeosystemsEngineering,UniversityofTexasatAustin,Austin,TX,ARTICLEINFOKeywords:DepletedreservoirsPermeabilityHoopstressAnisotropyNorthSeachalkABSTRACTPorepressuredepletionfromcanresultinsignificantcompaction.bothinthefar-fieldandthecrucialinforecastinghydrocarbonmodelsaretypicallyusedinwhichanisotropic,andthestresschangescomparedisotropicandanisotropicmationinValhallfieldinthematethedegreeofpermeabilitythepermeabilityintheradialconsideredwhendesigningaanisotropicreservoirsproductioncausesanincreaseineffectivestressinthereservoirandThechangeinstressstatewithdepletioninducesachangeinpermeabilitywellboreregion.Modelingthedepletioninducedpermeabilityalterationisproductionanddesigningadrillingstrategy.Porosity-basedpermeabilitythepermeabilitychangesareassumedtobeisotropic.However,mostrocksareduetodepletionanddrillingawellarealsoanisotropic.Asanexample,wepermeabilitypredictionsaroundahypotheticalwelldrilledintheTorfor-Sea.Theresultsshowthatisotropicpermeabilitymodelstendtounderesti-reductioninthenear-wellboreregion,butevenwhenanisotropyisconsidered,stillonlydecreasesbyabout10%.Theseresultsmaystillneedtobelopmentstrategyindepletedreservoirs.equationsforaverticalwellboreareH.Daigleetal.JournalofPetroleumScienceandEngineering157(2017)302–3112.Stressesandpermeabilitiesindepletedreservoirs2.1.StressesAhydrocarbonreservoirexperienceslargestressesduetooverlyingrock,surroundingrock,nearbyfaultsandtectonicforces.Aportionoftheexternalloadofthetotalstressissupportedbytheporepressureofthefluidinthereservoir.Asporepressuredecreasesduetohydrocarbonproduction,theloadcarriedbytherockitselfincreases.Suchloadistermedtheeffectivestress.Terzaghi(1924)andlaterBiot(1962)definedtheeffectivestressofafullysaturatedrockasσ'ij¼σijC0αδijPp;(1)whereσij’istheeffectivestress,σijisthetotalstress,δijistheKroneckerdelta(δij¼1ifi¼j,δij¼0otherwise),Ppistheporepressure,andαistheBiotcoefficient.Deformationsinarockareevaluatedthroughaconsti-tutiverelationthatrelatestheeffectivestresstothestrains.ForalinearelasticmaterialthestraintensorεrelatedtothechangeintheeffectivestresstensorΔσ′throughafourth-orderinvariantcompliancetensorD:ε¼D⋅Δσ0:(2)Notethatboldvariableshererefertotensors.ItisoftenusefultoexpressEq.(2)inmatrixnotation,alsocalledVoigtnotation.Voigtno-tationtakesadvantageofthesymmetryofthestressandstraintensorandexpressesthestressandstraintensorasasix-dimensionalvector.Eq.(2)inVoigtnotationforisotropicmaterialsisexpressedas2666664ε11ε22ε33ε23ε13ε123777775¼26666666666666666666641EC0νEC0νE000C0νE1EC0νE000C0νEC0νE1E0000001þνE0000001þνE0000001þνE377777777777777777777526666664Δσ011Δσ022Δσ033Δσ023Δσ012Δσ01337777775;(3)whereνandEarePoisson'sratioandYoung'smodulus.Notethatlinearelasticityassumesinfinitesimalstrainsorsmalldeformations.Ifstrainsobservedinthereservoirarelargeandfinite,thenonlineartheoryofelasticitymustbeused.Inthisstudy,positivestressesareassumedtobecompressive,sopositivestrainscorrespondtovolumeloss.Variousmodelsexistforcouplingthestress,strainandpermeability.Thesimplestmodelsrelatechangesineffectivestressrelativetosomereferencevaluedirectlytochangesinpermeability(e.g.,ZhaiandSharma,2005).Morecomplexmodelsinvoketheporosity-permeabilityrelationshipinaccountingforchangesinporosity,tortuosityandsur-faceareathroughevaluationofthevolumetricstrain.Notethatinthelineartheoryofelasticity,thevolumetricstrainεvolisexpressedfromthenormalstrainsasεvol¼C0ΔVV¼1C0ð1C0ε11Þð1C0ε22Þð1C0ε33Þ:(4)Ifthesolidgrainsareassumedtobeincompressible,thevolumetricstrainisrelatedtothechangeinporositythroughεvol¼ϕ0C0ϕ1C0ϕ;(5)whereϕandϕ0arethefinalandinitialporosityrespectively.Eq.(5)isderivedintheAppendix.303σr¼12ðσhmaxþσhminÞC181C0R2r2C19þ12ðσhmaxC0σhminÞC181þ3R4r4C04R2r2C19cos2θþR2r2pw;(6a)σθ¼12ðσhmaxþσhminÞC181þR2r2C19C012ðσhmaxC0σhminÞC181þ3R4r4C19cos2θC0R2r2pw;(6b)σz¼σvC02νðσhmaxC0σhminÞR2r2cos2θ;(6c)σrθ¼12ðσhmaxC0σhminÞsin2θC181C03R4r4þ2R2r2C19;(6d)σrz¼σθz¼0;(6e)whereσijisthetotalstressincylindricalcoordinates,Ristheradiusofthewellbore,σhmaxandσhminarethemaximumandminimumhorizontalstressesrespectively,σvistheoverburdenstress,νisPoisson’sratio,randθaretheradialandazimuthalcoordinatesofpointofinterest,respec-tively,andpwisthewellborepressure.Fig.1illustratesthelocalstressfieldaroundthewellinrelationtothefar-fieldstresses.Porepressuredepletiondoesnotonlyincreasetheeffectivestressinthereservoir,butitalsochangesthefar-fieldstresses.Althoughtheoverburdenstressiscommonlyassumedtobeconstant,thehorizontalstresseschangesignificantlywithdepletion.ThestresspathAisdefinedastheratioofthechangeinhorizontalstressestothechangeinporepressureandisexpressedasA¼ΔσhΔPp¼α1C02ν1C0ν;(7)forlinear,isotropichomogenousreservoirofinfiniteextent(Addis,1997;Hillis,2001;EngelderandFischer,1994).NotethatEq.(7)assumesthatthemagnitudeofthechangesintheminimumandmaximumhorizontalstresseswithdepletionarethesame.AccordingtoSegallandFitzgerald(1998),Eq.(7)isapplicablewhentheratioofthereservoirlateralextenttoitsthicknessis10:1.2.2.PermeabilitiesPredictinghowpermeabilityanisotropychangesduetodepletionandassociatedstrainsrequiressomesortofphysicalmodel.Here,wecombinetwoexistingmodelsintheliteraturetoaccountforpermeabilityanisotropydevelopmentduetoacombinationofverticalconsolidationandhorizontalstrain.DaigleandDugan(2011)developedamodeltoexplainpermeabilityanisotropyinsedimentaryrocksasafunctionofconsolidationandshearing.Inthismodel,directionvariationsinpermeabilityareassumedtobedueonlytodirectionalvariationsinTheKirschequationsdescribethestressaroundacircularcavityofalinearisotropicmaterialsubjecttoexternalloading(Kirsch,1898).Thediscontinuityinmaterialpropertiesandstressesatthewallofthecavitycausesalocalperturbationinthestressfieldwithastresscomponentactingtangenttotheboreholewall,perpendiculartotheboreholeaxis,whichmaybetensileorcompressivedependingonthefar-fieldstressesandthepressurewithinthecavity.AadnoyandChenevert(1987)suc-cessfullyappliedtheKirschequationstocalculatethestressesaroundadeviatedwellboreandassessedthefracturepressureandcollapsepres-surethroughincorporatingvariousrockfailurecriteria.TheKirschsesstressesH.Daigleetal.JournalofPetroleumScienceandEngineering157(2017)302–311tortuosity(e.g.,WittandBrauns,1983;Rasromani,2016).Theyconsideredaporousmediumcomposedofcylindricalgrainswithaspectratio(ratioofdiametertothickness)m(Fig.2).Ifthelongaxesofthegrainsareinclinedatanangleψwithrespecttothehorizontal,theratioofpermeabilityinthehorizontaldirectionkhtothatintheverticaldi-rectionkvisgivenbykhkv¼"1þC2089mcosψþ2πsinψC21C30C203π8ð1C0ϕÞC012C211þC2089msinψþ2πcosψC21C30C203π8ð1C0ϕÞC012C21#2:(8)ThetheoryofMarch(1932)statesthatanelongategrainwithaninitialinclinationψ0ofitslongaxiswithrespecttothehorizontalwillrotatetoanewinclinationψ(εz)whensubjectedtoauniaxialstraininthez-direction(i.e.,vertical;Fig.2bandc).Thenewinclinationisgivenbyψ(εz)¼tanC01[(1–εz)tanψ0](DaigleandDugan,2011).Eq.(8)wasshownbyDaigleandDugan(2011)toaccuratelyreproduceresultsfromlattice-Boltzmannsimulationsofflowthroughidealizedporousmedia.DaigleFig.1.(a)Far-fieldstressdistributionaroundaverticalwellboredrilledthroughahomogeneousdistributionaroundthewellbore.Thecoordinatesystemisillustratedinred,whilethestresθ-components,aswellasthetractionσrθ,changetheirorientationwithrespecttothefar-fieldlegend,thereaderisreferredtothewebversionofthisarticle.)andScreaton(2015)latershowedthatthemodelaccuratelydescribespermeabilityanisotropydevelopmentinsedimentsthathaveundergoneburialinanormalfaultingstressregime(σ1vertical).WhileEq.(8)describespermeabilityanisotropyduetoburialanduniaxialstrain,amoresophisticatedmodelisnecessarytodescribechangestothepermeabilitytensorduetoarbitrary,anisotropicstrains.Wong(2003)developedaframeworkfordescribingsuchpermeabilitychanges.Inhismodel,changesinpermeabilityarelinkedtostrainFig.2.(a)Cross-sectionrepresentationofsedimentgrainsintheDaigle-Duganmodel.GrainsareDepictionofporousmediumpriortodeformation.(c)Depictionofporousmediumaftervertical304throughaconstitutivemodelsimilartoEqs.(2)and(3).Foranisotropicmedium,thechangeinpermeabilityisexpressedas24k11C14k11;0k22C14k11;0k33C14k11;035¼241k22;0C14k11;0k33;0C14k11;035þ24abbbabbba3524ε11ε22ε3335;(9)wherek11,k22,andk33arethepermeabilitiesinthethreeprincipaldi-rectionsandthesubscript0referstotheirinitialvalues.Eq.(9)islinearandisthereforeonlyvalidforsmallstrains.ValuesfortheconstantsaandbmaybedeterminedbycomparisonwithEq.(8).Duringburialandconsolidation,ε33isequaltothevolumetricstrainsinceε11¼ε22¼0.Takingk33¼kvandk11¼k22¼kh,Eq.(9)canberewrittenaskhC14kh;0¼1þbεvol¼1þbϕ0C0ϕ1C0ϕ;(10a)kvC14kh;0¼kv;0C14kh;0þaεvol¼kv;0C14kh;0þaϕ0C0ϕ1C0ϕ:(10b),linearlyelastic,isotropicmaterial.VariablesrefertotermsinEq.(6).(b)Localstressactingonanelementofformationnearthewellareshowninblack.Notethatther-andasθvariesfrom0to2π.(ForinterpretationofthereferencestocolourinthisfigureSomealgebraicmanipulationyieldskvC14kh;0¼ðkv=khÞðkhC14kh;0Þ¼ðkv=khÞC181þbϕ0C0ϕ1C0ϕC19¼kv;0C14kh;0þaϕ0C0ϕ1C0ϕ;(11a)orientedatangleψwithrespecttothehorizontal,havethicknessδ,andaspectratiom.(b)strainhascausedgrainrotation(i.e.,reductionintheangleψ)andporosityloss.kv=kh¼kv;0C14kh;0þaϕ0C0ϕ1C0ϕ1þbϕ0C0ϕ1C0ϕ:(11b)Porosity-horizontalpermeabilitydatathereforemaybeusedtodefinebthroughEq.(10a),andEq.(11b)maythenbecombinedwithEq.(8)todeterminea.ThisthenallowspermeabilitychangesfromarbitrarystrainstobedefinedthroughEq.(9).3.Example:Torformation,ValhallfieldTheValhallfieldisachalkreservoirlocatedintheNorthSea.ItconsistsoftwolateCretaceousoil-bearingformations:theTorformationandtheunderlyingHodformation.Inthispaper,wewillusetheTorformationasanexampletoevaluatethedepletion-inducedpermeabilityalterationthroughtheWongmodel.Fig.3showsalocationmapandstructuremaponthetopoftheTorformation.TheaveragethicknessoftheToris30m(Yorketal.,1992).TheTorformationisprimarilychalkwith2.3.co;2.Fabricius,I.L.,Røgen,B.,Gommesen,L.,2007.HowdepositionaltextureanddiagenesiscontrolpetrophysicalandelasticpropertiesofsamplesfromfiveNorthSeachalkfields.Pet.Geosci.13(1),81–95.http://dx.doi.org/10.1144/1354-079306-707.Gommesen,L.,Fabricius,I.L.,2001.DynamicandstaticelasticmoduliofNorthSeaandRøgen,B.,Fabricius,I.L.,Japsen,P.,Høier,C.,Mavko,G.,Pedersen,J.M.,2005.UltrasonicvelocitiesofNorthSeachalksamples:influenceofporosity,fluidcontentandtexture.Geophys.Prospect53(4),481–496.http://dx.doi.org/10.1111/j.1365-2478.2005.00485.x.Segall,P.,Fitzgerald,S.D.,1998.Anoteoninducedstresschangesinhydrocarbonand22(10),949–952.http://dx.doi.org/10.1130/0091-7613(1994)022<0949:PaperPresentedattheInternationalOilandGasConferenceandExhibition,Paper48864.SocietyofPetroleumEngineers,Beijing,China.Engelder,T.,Fischer,M.P.,1994.Influenceofporoelasticbehavioronthemagnitudeofminimumhorizontalstress,Shinoverpressuredpartsofsedimentarybasins.Geologyzones:avoidinglosses,formationdamageandstuckpipe.In:PaperPresentedattheMiddleEastDrillingTechnologyConferenceandExhibition,PaperSPE-85326.SocietyofPetroleumEngineers,AbuDhabi,UAE.Røgen,B.,Fabricius,I.L.,2002.InfluenceofclayandsilicaonpermeabilityandcapillaryentrypressureofchalkreservoirsintheNorthSea.Pet.Geosci.8(3),287–293.10.1016/S0012-821X(03)00425-4.Dusseault,M.B.,Bruno,M.S.,Barrera,J.,1998.Casingshear:causes,cases,cures.In:Darley,H.C.H.,Gray,G.,1988.CompositionandPropertiesofDrillingandCompletionFluids.GulfPublishingCompany,Houston.Dugan,B.,Flemings,P.B.,Olgaard,D.L.,Gooch,M.J.,2003.Consolidation,effectivestress,andfluidpressureofsedimentsfromODPSite1073,USmid-Atlanticcontinentalslope.EarthPlanet.Sci.Lett.215(1–2),13–26.http://dx.doi.org/PetroleumEngineering,PaperSPE-47274.SocietyofPetroleumEngineers,Trondheim,Norway.Rasromani,E.,2016.ExploringAnisotropyinRockFluidFlowandElasticBehavior.M.S.thesis.DepartmentofPetroleumandGeosys
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