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RESEARCHPAPERPETROLEUMEXPLORATIONANDDEVELOPMENTVolume35,Issue1,February2008OnlineEnglisheditionoftheChineselanguagejournalCitethisarticleas:PETROL.EXPLOR.DEVELOP.,2008,35(1):85–91.Receiveddate:19June2006;Reviseddate:11October2007.*Correspondingauthor.Tel:+86-010-82324132;E-mail:lingzongfa@yahoo.com.cnCopyright©2008,ResearchInstituteofPetroleumExplorationandDevelopment,PetroChina.PublishedbyElsevierBV.Allrightsreserved.FloodpatternoptimizationofhorizontalwellinjectionLINGZong-fa1,*,WANGLi-juan2,HUYong-le2,LIBao-zhu21.SchoolofEnergyResources,ChinaUniversityofGeosciences,Beijing100083,China;2.ResearchInstituteofPetroleumExplorationandDevelopment,PetroChina,Beijing100083,ChinaAbstract:Tooptimizewellspacingandinjectionrateinhorizontalwellfloodpattern,formulaeofthereasonableinjectionrateandwellspacingarederivedunderfractionandno-fractionconditionsrespectivelyusingfiltrationtheory.Theformulaetakeintoaccountthefluidflowinlow-permeabilityandthinreservoirs,andassumethathorizontalwellinjectionisrigidwaterdriving.Accordingtotheseformulae,theinfluencingfactorsofpressurelossonthehorizontalsectionarediameterofcasingpipe,productionrate,andlengthofhorizontalsection.Forawellwithfixedcasingdiameterandhorizontalsectionlength,theinfluencingfactorsofpressurelossonhorizontalsectionisonlyproductionorinjectionrate;forareservoir,theinfluencingfactorsofmaximalwellspacearelengthofhorizontalsection,oilproductionrateandwaterinjectionrate.TheresearchresultsareappliedintheHade4thinlayersandstonereservoirofTarimOilfield.Thedesignedfloodpatterndecreasesinjectionpressure,augmentsinjection,andkeepsfieldpressure.Keywords:horizontalwell;horizontalwellinjection;wellspacingoptimization;low-permeabilityreservoir;thinlayerreservoirIntroductionHorizontalwelltechnologyhasbecomeausefulmeansindevelopingcomplexreservoirs[1-4],andthedevelopmentofhorizontalwelldrillingtechnologymakeshorizontalwaterfloodingtechnologycomeintoreality.Horizontalwaterfloodingtechnologybeganatthebeginningofthe1990s[5-7],andhasbeensuccessfullyutilizedinsomeoilfieldsallovertheworld[8-17].Theapplicationsofhorizontalinjector-producerpatterntodevelopreservoirswithlowpermeabilityandsmallthicknesscankeepreservoirpressure,increaserecoveryfactoranddecreaseinjectionpressure[17-27].Fromtheviewpointofeconomics,thelargertheareacontrolledbythehorizontalinjector-producerpatternandproduction,thebettertheeconomicresults.However,becauseofthelowflowcapabilityofthin,lowpermeabilityreservoirs,ifwellspacingistoolarge,thebottomholepressureofproducerwillbebelowthenormalworkpressureandthebottomholepressure(B.H.P.)ofinjectorwillincreaseandexceedthemaximallimitingpressure.Therefore,horizontalpatternhasmaximalcriticalspacingandreasonableinjectionrate.Usingporousmediaflowtheoryandconsideringthecharactersoffluidinlowpermeabilityandthinreservoirs,thispaperpresentsformulaeofthereasonableinjectionrateandwellspacingwithpressurelossandwithoutpressurelossonthehorizontalsectioninhorizontalwellsandanalyzestheirinfluentialfactors.1DerivationofreasonableWellSpacingThereispressurelossalongthehorizontalsectionofhorizontalwell,thereforetomakeiteasierforderivation,atfirst,formulaeofthereasonableinjectionrateandwellspacingarederivedwithoutpressureloss,andthenwecarryouttheworkconsideringpressurelossbasedontheseresults.1.1NotconsideringpressurelossTosimplifytheproblem,wemakethefollowingassumptions:thereservoirisisotropicandpermeabilityisK;pressurelossalongthehorizontalsectionofthehorizontalwellisneglected.piwfdenotestheuniformpressureoneachpointinthehorizontalsectionofinjector,andpwfdenotestheuniformpressureoneachpointinthehorizontalsectionofproducer;floodfrontaccordtoitsdrawdownfrontofinjector;gravitationandcapillarypressureareignored.Injectorisassumedtoberigidwaterdriving.Whenthewaterfrontofinjectormeetsthepressuredrawdownfront,thepressureatthelinewheretwofrontsmeet(Fig.1)isthereservoirinitialpressure,andisdenotedbypi.ThehorizontalsectionslengthofinjectorandproducerarebothL.dxrepresentstheinfinitesimalonthehorizontalsectionofinjector:xLqxqd)(dii(1)LINGZong-faetal./PetroleumExplorationandDevelopment,2008,35(1):85-91Fig.1Schematicrepresentationofwaterfrontmeetingwithpressuredecreasingfrontinhorizontalinjector-producerpatternDarcylawisusedforintervalbetweendxandwaterfront:ydpdKAxqdwiPD)((2)WhereA=hdx(3)FromEq.(1),Eq.(2)andEq.(3),weget:pdqLhKydwiPD(4)Eq.(4)isintegratedpqLhKyappdd0wiiiwf³³PD)(iiwfwippqLhKaPD(5)Eq.(5)canberewrittenasfollowsafterunitconversion)(088.0iiwfwippQLhKaP(6)ForproducerxLqxqd)(dpp(7)ypKAxqdd)(dopPD(8)SubstitutingEq.(3),Eq.(7)intoEq.(8)pqLhKyddopPD(9)TheintegralofEq.(9)intwosidescanbewrittenas:pqLhKyDbDppddwfiop³³PD)(wfiopppqLhKbPD(10)Eq.(10)canberewrittenasfollowsafterunitconversion)(088.0wfiopppQLhKbP(11)1.2Consideringpressureloss1.2.1ComputationofPressureLossSomeliteraturesgivemanymeanstocalculatepressurelossforhorizontalwell.Incomparison,thepressurelosscomputationmethodinliteratures[28]and[29]issimpleandtheresultscanfulfillthefidelityforestimation.Ifallfluidflowsintohorizontalwellfromoneside(toeorheel),wecandefinethisinflowmodelasOneTimeInflowModel.Thepressurelossissmallestwhenthefluidflowintohorizontalwellfromheel.Letusassumetheproducerisonehorizontalpipe,andfluidinthepipeisincompressiblesingle-phasefluid.Frictioncoefficientalongthehorizontalpipeisconstant.Thereby,usingmassconservationlaw,momentumconservationequationandenergyconservationlaw,pressurelossalongthepipecanbegivenbyddddhfappppLLLL(12)Consideredthatgravityandaccelerationpressurelosscanbeignored,thus,2fdd2pvLgdUO§·¨¸©¹(13)Thepressurelossforanypointalongthehorizontalpipecanbeobtained:gdxvxp2)(2UO'(14)ThetotalpressurelossalonghorizontalsectionisgdLvp22UO'(15)ThatisLgdqp5281.0UO'(16)Accordingtoliteratures[28]and[29],inflowprofilecanbedefinedasfivekindsofinflowmodels,including:uniforminflowmodel,lineardecreasinginflowmodel,linearincreasinginflowmodel,paraboladecreasinginflowmodelandparabolaincreasinginflowmodel(Fig.2).Takingtheuniforminflowmodel(modH,)asanexampletodeducethepressurelosspredictionformulae.FromtheillustrationofFig.2theinflowratealongdxcanbewrittenbyEq.(7),thustheflowratefromtoetoxisgivenbypp0()d()xqqxqxxL³(17)UsingEq.(14)andEq.(16)andthenthepressurelossalongLINGZong-faetal./PetroleumExplorationandDevelopment,2008,35(1):85-91Fig.2SchematicrepresentationoffivekindsofinflowprofilemodelsTable1PressurelossformulaeofkindsofinflowmodelInflowmodelFlowrateindxTotalpressurelossfromtoetoxTotalpressurelossIxLqdp381.03252poxLgdqUOp'31IIxLxLLqd2p5432452po2.03481.0xLxxLLgdqUOp'158IIIxLxLqd2p581.05452poxLgdqUOp'51IVxLxLqd32p781.07652poxLgdqUOp'71VxLxLLqd322p735.4381.076524334652poxLxxLxLxLLgdqUOp'149dxisasfollows:xxLgdqgdxpd81.02dd2252po2oUOQUO(18)Thepressurefromtoetoxisgivenby381.0d)(d3252po0IxLgdqpxpxUO³(19)UsingEq.(16)andEq.(19)andthenthetotalpressurelossalongthehorizontalsectionisasfollows:pLgdqxppL''³,313181.0)(d52po0IUO(20)Thedeductionofotherfourinflowmodelsissimilartotheabove-mentionedprocess,andtheflowrateindx,theinflowratealongdx,pressurelossfromtoetoxandtotalpressurelossareshowninTable1.Thecomputingmethodofpressurelossissameforinjectorandproducer.Theonlydifferenceisthedirectionofthepressureloss.Pressurelossofproducerdecreasesfromtoetoheel.Onthecontrary,pressurelossofinjectorincreasesfromtoetoheel.1.2.2Optimizationofwellspacing,injectionrateandproductionrateconsideringpressurelossThroughtheunitconversion,Eq.(16)canberewrittenbyLgdQp52po1010UO'(21)Assumepwfasheelpressureofproducer,andthentoepres-surecanbeobtained:LgdQpppp52po10wfwf10UO'(22)Eq.(11)canbewrittenas:)10(088.052po10wfiopLgdQppQLhKbUOP(23)WhenwaterfrontofinjectormeetspressurefrontofLINGZong-faetal./PetroleumExplorationandDevelopment,2008,35(1):85-91producer,thedistancebetweenwaterfrontoftoesectionandhorizontalsectionofinjectorcanbeobtained:)10(088.0i52iw10iwfwipLgdQpQLhKaUOP(24)Eq.(23)andEq.(24)areformulaecomputingthedistancesbetweenthepressurefrontoftoesectionandthehorizontalsectionforinjectorandproducerrespectivelywhenthewaterfrontofinjectormeetspressuredrawdownfront,whichareusedtocalculatetheeffectofpressurelossonthetoeandhorizontalsection.UsingthestudyofOneTimeInflowModelprofilepressurefrontandthefiveinflowmodelsofpressurelosscomputation,wecangivethepressurefrontcomputingformulae.Model,Eq.(19)istheformulatocomputethepressurelossforthepointonhorizontalsection,andthepressureofthepointcanbegivenby381.0)(33252powfIwfxLLgdqpxppp'UO(25)Thedistancebetweenthepressurefrontofproducerandthehorizontalsectioncanbewrittenby:»»¼º««¬ª310088.033252po10wfiopxLLgdQppQLhKbUOP(26)Thepressureatanypointoftheinjectorhorizontalsectionis3)(81.0)(3252iiwfIiwfxLLgdqpxppp'UO(27)Thedistancebetweenwaterfrontofinjectorandinjectorcanbewrittenby:»¼º«¬ªiiwiwfwipLgdxLQpQLhKa25321010088.0UOP(28)Inthesameway,Modelcanbegivenby>uOP10iwfwi10088.0pQLhKa»»¼ºi4554322iw)(2.0)()(3.1pLgdxLxLLxLLQU(29)>uOP10wfiop10088.0ppQLhKb»»¼º45543252po2.03.15.0LgdxLxxLLQU(30)Model:»¼º«¬ªi5452iw10iwfwi5)(10088.0pxLLgdQpQLhKaUOP(31)»»¼º««¬ª510088.055452po10wfiopxLLgdQppQLhKbUOP(32)ModelIV:»¼º«¬ªi7652iw10iwfwi7)(10088.0pxLLgdQpQLhKaUOP(33)»»¼º««¬ª710088.077652po10wfiopxLLgdQppQLhKbUOP(34)Model«¬ªu34652iw10iwfwi310088.0xLLLgdQpQLhKaUOP»»¼ºipxLxLLxLLxLL735.4765243(35)««¬ªu7652po10wfiop6.010088.0LLgdQppQLhKbUOP»»¼º¸¸¹·735.4376524334xLxxLxLxL(36)2DiscussionWecanseefromEq.(16)andEq.(21)thatthepressurelossismainlycausedbycasingdiameter,injectionrateandproductionrate.Forfixedhorizontalwell,pressurelossismainlyaffectedbyproductionrate.2.1ApplicationofformulaenotconsideringpressurelossIfDdenoteswellspacingbetweentheinjectorandtheproducer,andtheinjectiontoproductionratiois1(Qp=Qi=Q),thentheproductionrateandtheinjectionratecanbeoptimized.Whenthepressurefrontofproducermeetsthefloodfrontofinjector(seeFig.1):D=a+b(37)SubstitutingEq.(6)andEq.(11)intoEq.(37),weobtain)(088.0)(088.0wfioiiwfwppQLhKppQLhKDPPand¸¸¹·¨¨©§owfiwiiwfopt088.0PPppppDLhKQ(38)Ifpmin,pmax,QpandQioflimitlowpressure,limithighpressure,productionrateandinjectionrateareknownrespectively,thenthemaximumwellspacingcanbeLINGZong-faetal./PetroleumExplorationandDevelopment,2008,35(1):85-91calculatedin3conditions.7'>a+b.Wellspacingissolargethatpressurefrontsoftheinjectorcan’tmeettheproducer’sevenwhenB.H.P.oftheproducerdecreasestothelimitlowpressureandB.H.P.oftheinjectorincreasestothelimithighpressure,thereforewatercannotbeinjectedintheinjectorandtheproducerisnotaffected.8D=a+b.WhenB.H.P.oftheproducerdecreasestothelimitlowpressureandB.H.P.oftheinjectorincreasestothelimithighpressure,pressurefrontsoftheinjectorwillmeettheproducer’satthesametime.Thiswellspacingistheoptimizationwellspacing,namelymaximumwellspacing.Dmax=a+b(39)minmaxmax088.0088.0ppQLhkppQLhKDiopiwiPP(40)Ifinjectiontoproductionratiois1,namelyQp=Qi=Q¸¸¹·¨¨©§ominiwimaxmax088.0PPppppQLhKD(41)9D
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