Pressure Transient Behavior of a Finite Conductivity Fracture in Infinite-Acting and Bounded Reservoirs

This papar pre8ente analytic solutions of the pre$sure trafwlent behavior of a well intarsaotad by a finite-conductivity fracture in an infinite-acting, or in cyfindric%lly or rectangularly Lmundsd finite reservoirs. Thsse solutions includa the practicat effects of reservoir psrmeabili~ anisotropy and dual porosity behavior. These solutions are analytic, and thus do not require dfscretization in epace. The analytical solutions of the fin~e-conductivity fracture Wnsient behavior presented in this papar efiminate me numerical difficulties associated with ofher mathematically rigorous finite-conductivity fracture solutions that have bean reported in fhe literature. Sofh fhe pressure and rate transient responses can be acarrately evafuat~ using the finitesonducfivity fracture solutions presentad in this paper. This is aepeciatly important for [ow-cenductivify fractures, for which tie pressure and rate transient behavior is often difficult to evaluab a~urately using file solutions available in the literature. INTRODUCTION The prassure transient behavior of finite-cmnducffvity verticaf fractures has bsen investigated extensively in the past few decades in order to better astfmate be propped fractura geometty and conductivity resulting from hydraulic fracture well stimulation treatments. The various types of models fhat have baen used in these investigations include both finite-cfifferenca and finite-element numerical models, 14 real and Laplace space anaiyfic solutions for the transient behavior of uniform flux and infinite+onductivity fractures,’a and raal end Laplaca space sern_@nalytfc solutions for the transient behatior of finita.conducfivify vertical fractures.5,8<’0”3 Of p~ular interest in this paper are the studiae pertaining to the evaluation of the transient behavior of finite-scmductivity fractures using tie Laplaca transformation tacluique and the @mdafy Elemsnt method. Two concurrenfty ad separatdy davelopecl solutions for tie fzansient behavior 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 end Meng” considered tie fracture storage effects to be nagifgible, whle tie model reported by van Kruysdijk’2 included the fracture storage effects. Bath mcdels were davelopad using fhe Bxrndary Element method and assumed that be vertical fracture was of uniform fracture width, conductivity, and height. The fracture height in sach of the models wae aeeumed to be aquel to me reservoir thickness. Later, mora general finiteccmducfiviiy fractufa models ware refxxtsd’s Mat permitted arbitrary ~~~~~s~:~aW and mnductivi~ distributions. AH of these semi-analytic for me transient behavior of a tinita-conductivity vertical fracture require dkcratizafion in space in order to sofve ma Fradholm integraf equafiens tiat comprise tie transient eelutions. This technique involves solving a system of equations numerically in order to detarmine the unknown flux distribution in the fracture and the wellbora preseure. Riley, et at.’” presented an anatytfc solution of me transient bahavia of an elfipticd finite-conductivity fracture which does not require discretizatfon of f3e fracture. The analytic solution presented by Riley et d.’” generaffy pfovides a somawhat more rapid evakfation procedure for the Wansient behavior of a finita-conductivity vertical fractura t?an de the semi-anal~c solutions reported in Refs, 11 13rough 13 for the same level of numerical accuracy. However, tie solution presented by Riley et al.’0 can still bs very time censuming to eva)uate due to the slow convergence of the series of fhe solution. The difficulty invoWd with evaluating tie transient behavfer of finiteconducfivity verfkel fractures using fhe solutions presented in Refs. 11 through 13 is primarily due to the singular nature of the integral aquations and tie numerical evaluation procedures required to avaluate tha unknown flux distribution and the wellbore pressure. At very early transient times, the principal component of the well production comes from tie rasewoir region nearest tie wellbore. This generstly requires the use of a large number of fracture elemenfe in the fracture nearest tie wellbore 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. This requires a large number of sfemente in ha fracture to accurately evaluate the flux disbibution in the fracture as well.