..ml....I Society of Petroleum Engineers ISPE 28392Pressure Transient Behavior of a Finite ConductivityInfinite-Acting and Bounded ReservoirsB.D, Poe Jr. and J.L. Elbel, Schlumberger Dowell, and T.A. Blasingame,SPE MembersCopyright1994, Societyof PetroleumEngineers,Inc.IFracture inTexas A&M U.0This paperwas preparedforpresentationat the SPEWh AnnualTechnical~nfermce and Exhibitionheld In NewOrleans,LA,U.S.A.,25-2S September1994.‘rhis paperwas selecledtor presentationby an SPE ProgramCommitteefollowingre.lew of informationco.tained In 8nabstract submittedby the author(s).Contentsof the Psper,aspresented,havenotbeen reviewedbyiho .SoCletYof PefJoloumEngineersand are mbJectto wrreCtlOnbythe author(s).The material.as presented,doesnotnwessarily refltmtanYpmitionoftheSocietyof PetroleumEngineer?,IISoff!~rs, Ormem~rs. paPer8P<esenlti at SpE ?eetln9s araSUbM toPubkation revlswby EditorialCommitteesofthe SocietyofPetroleumEngineers.PermissiontoCOPYISreslnct,edtoanabstmtiofnotmmethan3C0words.lllUStt3t10nSrnaYnOlb+c%ded.Theabs!ractshouldcontainconspicuousacknowledgmentof where andby whomthe pacer is presented.WriteLibrarian,SPE, P.0. BOX833838. Richardson,TX 75083.3836, US.A. Telex, 163245 SPEUT.ABSTRACTThis papar pre8ente analytic solutions of the pre$sure trafwlent behaviorof a well intarsaotad by a finite-conductivity fracture in an infinite-acting, orin cyfindric%lly or rectangularly Lmundsd finite reservoirs. Thsse solutionsincluda the practicat effects of reservoir psrmeabili~ anisotropy and dualporosity behavior. These solutions are analytic, and thus do not requiredfscretization in epace.The analytical solutions of the fin~e-conductivity fracture Wnsientbehavior presented in this papar efiminate me numerical difficultiesassociated with ofher mathematically rigorous finite-conductivity fracturesolutions that have bean reported in fhe literature. Sofh fhe pressure andrate transient responses can be acarrately evafuat~ using the finite-sonducfivity fracture solutions presentad in this paper. This is aepeciatlyimportant for [ow-cenductivify fractures, for which tie pressure and ratetransient behavior is often difficult to evaluab a~urately using filesolutions available in the literature.INTRODUCTIONThe prassure transient behavior of finite-cmnducffvity verticaf fractures hasbsen investigated extensively in the past fewdecades in order to betterastfmate be propped fractura geometty and conductivity resulting fromhydraulic fracture well stimulation treatments. The various types of modelsfhat have baen used in these investigations include both finite-cfifferencaand finite-element numerical models, 14 real and Laplace space anaiyficsolutions for the transient behavior of uniform flux and infinite+onductivityfractures,’aand raal end Laplaca space sern_@nalytfc solutions for thetransient behatior of finita.conducfivify vertical fractures.5,8<’0”3 Ofp~ular interest in this paper are the studiae pertaining to the evaluationof the transient behavior of finite-scmductivity fractures using tie Laplacatransformation tacluique and the @mdafy Elemsnt method.Two concurrenfty ad separatdy davelopecl solutions for tie fzansientbehavior of finifa-senductivify fractures were reported by Cinco-Ley and...,: .—. ::,. –.- —- .-=References and”illustrafions atend of paper.Meng” and van Kruysdijk~2 The medel developed by Cincr+Ley endMeng” considered tie fracture storage effects to be nagifgible, whle tiemodel reported by van Kruysdijk’2 included the fracture storage effects.Bath mcdels were davelopad using fhe Bxrndary Element method andassumed that be vertical fracture was of uniform fracture width,conductivity, and height. The fracture height in sach of the models waeaeeumed to be aquel to me reservoir thickness. Later, mora general finite-ccmducfiviiy fractufa models ware refxxtsd’s Mat permitted arbitrary~~~~~s~:~aW and mnductivi~ distributions. AH of these semi-analyticfor me transient behaviorof a tinita-conductivity verticalfracture require dkcratizafion in space in order to sofve ma Fradholmintegraf equafiens tiat comprise tie transient eelutions. This techniqueinvolves solving a system of equations numerically in order to detarminethe unknown flux distribution in the fracture and the wellbora preseure.Riley, et at.’” presented an anatytfc solution of me transient bahavia of anelfipticd finite-conductivity fracture which does not require discretizatfon off3e fracture. The analytic solution presented by Riley et d.’” generaffypfovides a somawhat more rapid evakfation procedure for the Wansientbehavior of a finita-conductivity vertical fractura t?an de the semi-anal~csolutions reportedin Refs, 11 13rough 13 for the same level of numericalaccuracy. However, tie solution presented by Riley et al.’0 can still bsvery time censuming to eva)uate dueto the slow convergence of theseries of fhe solution.The difficulty invoWd with evaluating tie transient behavfer of finite-conducfivity verfkel fractures using fhe solutions presented in Refs. 11through 13 is primarily due to the singular nature of the integral aquationsand tie numerical evaluation procedures required to avaluate thaunknown flux distribution and the wellbore pressure. At very earlytransient times, the principal component of the well production comesfrom tie rasewoir region nearest tie wellbore. This generstly requires theuse of a large number of fracture elemenfe in the fracture nearest tiewellbore to accurately evakrata the unknown flux dkfribution.Simifarly, at vasy late transient times, tie principaf cemponent of tie well- production may be from tie resarvolr region beyond the fracture tip. Thisrequires a large number of sfemente in ha fracture to accurately evaluatethe flux disbibution in the fracture as well.305..2PRESSURE TRANSIENT BEHAVIOR OF A FINITE CONDUCTIVITY FRACTURE= SPE28392IN INFINITE-ACTING AND BOUNDED RESERVOIRSThe use of an increasing number of elements in the fracture to accuratelymodel the flux dfstribu&m introduces another numerical difficultly. @increasing tie number of fracture elements.used, thedistarmes betwaanconsecutive fracture nodes (midpoints of elements) decreases (tends fezero), resulting inanumerically singular coefficient matrix for tie system.An additional factor hat also must be considered Is that the use of anumerical Iaplace transform inversion procedure will tandtoma~ify tieerrors in the wellbore pressure and flux distribution values obtained.The use of higher order