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SPE-945228-G Arps Original 1945

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A new model for production decline, Heber Cinco-Ley
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  Soc ety of Petroleum  ngineers SPE 548 ANewModel  r ProductionDecline FernandoRodriguez,UNAM/Pemex,andHeberCinco-Ley,UNAM SP Members Copyright1993,SocietyofPetroleumEngineers,Inc.ThispaperwaspreparedforpresentationattheProductionOperationsSymposiumheld in OklahomaCity,OK,U.S.A.,March 21 23 1993.ThispaperwasselectedforpresentationbyanSPEProgramCommitteefollowingreviewofinformationcontained in anabstractsubmittedbytheauthor s .Contentsofthepaper,aspresented,havenotbeenreviewedbytheSocietyofPetroleumEngineersandaresubjecttocorrectionbytheauthor s .Thematerial,aspresented,doesnotnecessarilyreflectanypositionoftheSocietyofPetroleumEngineers,itsofficers,ormembers.PaperspresentedatSPEmeetingsaresUbject to publicationreviewbyEditorialCommitteesoftheSocietyofPetroleumEngineers.Permissiontocopyisrestrictedtoanabstractofnotmorethan300words.Illustrationsmaynotbecopied.Theabstractshouldcontainconspicuousacknowledgmentofwhereandbywhomthepaperispresented.WriteLibrarian,SPE,P.O.Box833836,Richardson,TX75083--3836,U.S.A.Telex,163245SPEUT. ABSTRACT Ananalyticalsolutionisdevelopedinthispaper to forecasttheproductionbehavior of two-dimensionalclosed-boundaryreservoirsproducingfrommultiplewellsunderconstantdifferentbottomholepressures.Wellsarerepresentedaslinesourcesarbitrarilylocatedinthedomain of thereservoir;damageisconsidered.Solutionisobtainedthroughcombination of LaplacetransformationandGreen sfunctionsMethods. The productiondecline of thewellsduringtheboundarydominated  ow periodisshown tobe ofa modified exponential nature: qw j  tD) = q ; ;;Dj   q~Dj - q ; ;;Dj)exp -DtD), j = 1 2 ..  w Whenallwellsproduce at thesamebottom-holepressure, it isshown that q ; ;;Dj = 0 andproductiondeclinesexponentially.   isalsoshown that the productiondeclineofthereservoirisinallcasesexponential: qD tD) = q exp -DtD), being o_  nw 0 q - L j l qwDj isfound that the declinecoefficient, D, is the samefor the reservoirandforeach ofthe wells.   dependsonthesize and shape ofthe reservoiraswellason the num ber ofwells,itslocation and damage.Parameters q~Dj q ; ;;Dj and q JJ, besides of dependingon the sameparametersas   alsodependonthebottomholepressures of thewells.Asexpected,productionunderunequalbottomholepressuresorunderirregularwells-reservoirpatterns,yieldunevenwellproductions.   isalsofound that stimulation,ordamage, of asinglewellmodifiestheentireReferencesandillustrations at the end of paperproductioncharacteristics ofthe wells-reservoirsystem.Asimplemethod of analysis that provides the productiondeclineparameters q~ q:0, j = 1,2,...,  w and D, isdeveloped. INTRODUCTION The searchfortools to forecast the productiondecline of wells-reservoirsystemshasbeen of greatinterest to manyresearchers.Arpsempiricalmodels 1 havebeenwidelyusedforthesepurposes.Analyticalmodelsforwellsproducing at constantwellborepressureincircularreservoirshavebeenpresentedsincelongag0 2  3 4 and include both transient and boundarydominatedflowperiods.Regarding the boundarydominatedflowperiod,Fetkovich 5 obtainedtheexponentialdeclinesolutionusingamaterialbalancedifferentialequation andthe pseudosteady state inflowequation.Lateron,EhlighEconomidesandRamey 6 obtained the samesolution;they started fromthesolutionfor the pressure of awellproducing at constant rate inacircularreservoirduringthepseudosteady state  ow period and used the Laplacespacerelationshipfor the pressure and flow rate givenbyvanEverdingenandHurst 4. They alsogeneralized the solution to  ow geometries otherthan thecircularusingthe shape factorconcept, C A  developedforconstant-rateproducingwellsinclosed-boundaryreservoirs. 7 Camacho and Raghavan 8 presented the theoreticalframeworkfor the productionperformance of awellin 639  2 ANewModel for ProductionDecline SPE 25480 and, iiw j = 1   qw j  tD exp -stD)dtD (8) The problem stated byEqs.1 to 6canbesolvedbytheLaplacetransformation andthe Green sfunctionsmethods,as it isnextpresented.Letusdefine the followingLaplacetransformsfordimensionlesspressureandflowrate: (5)  6) OP I- 0 OY YD:::::O -, and,solutiongas-drivereservoirsduringtheboundarydomi nated flow period and expressed the parametersinArpsequationsinterms of physicalproperties.ChenandPoston 9 presented the use of apseudotimefunctioninan attempt to improve the forecasting of productionperformance of constantpressurewellsinsolutiongas-drivereservoirs,during the boundarydominated flow period.Morerecently,Resurreit,;3.0andRodriguez lO developedasemianalyticalsolutionfor the productiondecline of finite-conductivityfracturedwellsandcharacterizedtheexponentialdeclinemodel.Correlationsfortheinitialdimensionless flow rateofthe well, q~ andthedeclinecoefficient,   asafunction of fractureconductivity and size of the reservoirwereprovidedforthecase of afracturedwellinthecenter of asquarereservoir.Solutionspresentedup to nowconsiderasinglewell.   isintendedinthis paper to studythe productionperformance of areservoirwithmultiplewellsproducing at constant arbitrary pressures and to provideamodelforinterpretacionandproductionforcastingpurposes. FORMUL TION  ND SOLUTIONOF THE PROBLEM Laplacetransformation of Eq.1,withinitialconditionconsidered,yields: (9) Weconsidera2Drectangularhomogeneousreservoirwithuniformthicknessandboundariesclosed to flow,seeFig.1. The reservoirisproducedthrough nw wells at constant but differentwellborepressures.Fluidsareslightlycompressible and haveconstantcompressibilityandviscosity.Pressureisuniform at initialconditions.Flowin the reservoirisdescribed,interms of dimensionlessparameters,seeAppendixA,by the followingpartialdifferentialequation:Transformedboundaryconditionsare, op OX D:::::O-, (10)  11) subject to the followinginitial andboundary conditions: _ OP - Ot 0< X < Xe 0< Y < Ye tD > 0  1) OP I - 0 OY YD:::::O -, and,  12) (13) PD XD,YD,O = 0, OP I- 0 OX D:::::O-, OP I- 0 OX D::::: eD -, (2)  3) (4) SolvingEq.9withboundaryconditions 10 to 13 according tothe Green sfunctionsmethodyields, nw PD XD,YD = -211 LG XD,YD,XDj,YDj;S) iiwDj  14 j:::::l 640   ~ 25480 Fernando Rodriguez and HeberCinco-Ley 3where G ZD, y/ , z ,  j s) istheGreen sfunctionassociated to the problem.  t is obtainedasthesolutionto the followingadjointproblem: 8 2 G8 2 G ,, 8z,2   81/2 - sG = 6 z- ZD,Y - YD),  15 Forareservoirproducingfrommultipledamagedwellsunderconstant, but arbitrary,wellborepressure,thefollowingconditionisset at thelocation of the wells: i = 1,2,... ,nw with,  I =0, 8z zo :O  I -0 8z ZO :ZO.D -,  16) 17) whereSiisthedamagefactorforwell i. Laplacetransform of conditionabovegives, - ) PwDi   s- PD ZDi, YDi =   iqwDi S  24)Equation 24 canbeusedinEq. 14 toyieldthefollowingsystemofEqs.in qwDj,j = 1,2, ... ,nw, and,  I  0  y y :O -,  I =0 8y ,, = .D  18) 19) nw I:F. - PwDi i q D =   3 w 3 211 s 3=1 where, i = 1,2, ... ,nw  25) SolvingtheadjointproblemprovidesthefollowingGreen sfunction: Fij = Gij   }:ij being,  26) G ZD, YD, z , y j s) = Si  :f : = J. }:ij = ãã 2 orzerootherwise  27)1 cosh[y S YD - YeD ] cosh[y Sy ]- y SZeD sinh[y SYeD] _2- t cosh[an YD - YeD ] cosh[any ] x ZeD n=1 an sinh[anYeD] COS[1Ul ZD/ZeD] cos[ml Z /ZeD]  20) and,Eq. 25 canbewritteninmatrixnotationas:1 FqwD =   PwD 1I S  28) 29) or, if G ZD,YD,Z ,y jS) = 0<:Y < YD where F isasquarematrix of order nw havingelements Fij givenbyEq. 26; qwD isthevectorofunknowns, qwDj j = 1,2,..., nw, and PwD isthevector of knownwellborepressures, PwDj, j = 1,2,..., nw. SolvingthesystemofequationsgivenbyEq, 25 leadsto,1cosh[y S YD)]cosh[y S y - YeD )] . - y SZeD sinh[y SYeD] _ 1 f cosh[an y - YeD ] cosh[anYD] x ZeD n=1 an sinh[anYeD] cos[ml ZD/ZeD] cos[ml Z /ZeD] if YD < 11 < YeD being, n1l an = S+(_)2 ZeD  n = 0,1,2,...  21) 22)  nw i+ _1 LA-1  -1) 3 MijPwDi  30) qwDj =- 2 S . I F I j=I 2 ... ,nw where M ij istheminor of Fij ofmatrix F. Up to ourknowledge,thereisnoananalyticLaplacespaceinversionof qwDj Eq, 30, Atlargetimesduringtheboundarydominated flow period, s -- 0, equation 30 reduces to alimitingformamenableforanalyticinversion,as it isnextpresented.641  4 ANewModelforProduction Decline SPE 25480 Production Behavior of Wells UnderBoundaryDominated FlowConditions. Inthissection we focusourattention to theproductionbehavior ofthe wellsduring the boundarydominated flow period.First,usingproperties of Laplacetransformation, we establish the limitingform of theGreen sfunction,Eqs. 20 and 21. At largetimes,when s tends to zero,Eqs. 20 and 21 canbewrittenas:lim G ZD,YD,Z ,y ;s) = _ - f3 ZD,YD,Z ,y ) 31)   0 s where the function f3 Z D,YD, z , y ) doesnotdependontheLaplaceparameter s. This is: where   is the matrix of  Yij elements; I   I isthedetermi nantof   and I   Ik is the determinant of that matrix   containing  Yij aselements,exceptforrow k whichhave ones, 1. The minor Mij of Eq.30,can also bewrittenas,(38)where mij isthe minor of  Yij inmatrix  Y, and  mij)k is obtainedbyreplacingrow k ofmatrix, mij, by ones. SubstitutingEqs. 37 and 38 inEq.30,andrearranging,leadsto:__ q :Dj   q~Dj/D S qwDj - s 1 + s/   j = 1,2,... ,nw  39 2   cosh mr YD - YeD )/ZeD] cosh (mry /ZeD]   mrsinh nll YeD/zeD] x cos nll ZD/ZeD] cos nu /ZeD] 32) where,  40) and, i f3 ZD,YD,Z ,y ) = 0< Y < YD and,  41) 2 f: cosh(nll (Y - YeD)/ZeD]cosh nll YD/zeD] n nll sinh nll YeD/zeD] x cos nll ZD/ZeD] cos(nll z /ZeD] i YD < y < YeD (33)  42) LaplaceinversionofEq.   yieldsthedimensionless flow rateofthewellsduring the boundarydominated flow period,this is: Usingapproximation 31 inEq. 26, it followsthat:1 Fij =   Yij  34) s The totaldimensionless flow rate of thereservoircannowbeobtainedasthesummation of thedimensionless flow ratesofthewells,where,and,  Yij = f3ij -  Xij (35)  36) or, nw qD tD) = LqwDj tD) j 44) Usingproperties of thedeterminants it canbedemonstrated that thedeterminant of matrix   withelements Fij givenbyEq.34,canbewrittenasfollows:where,  45) (37) nw q b =  q~ j j 46) 642
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