The pressure transient behavior of a finite conductivity vertically fractured well is important in reservoir engineering because it provides valuable information concerning the well completion, and provides estimates of the in-situ reservoir properties and the propped fracture dimensions and conductivity. The pressure transient behavior of finite-conductivity wells has typically been analyzed using mathematical models that assume relatively simple fracture properties. Most of these models assume uniform fracture height, width, and proppant permeability distributions with respect to space, and many of these models also assume that fracture storage effects are negligible. This paper presents the pressure transient solution of a well intersected by a finite-conductivity vertical fracture with a lengthwise variation in the fracture geometric and material properties. The general solution of the fracture fluid-flow problem, and an approximate solution of the problem in which the effect of fracture storage has been assumed to be negligible, are presented. Comparisons of the pressure transient behavior predicted with the two finite-conductivity fracture solutions are given. The effects of spatially varying fracture height, width, and material properties are investigated with these pressure transient solutions. The new solutions provide more realistic pressure transient analysis models which may be used to better design, implement, and evaluate hydraulic fracture treatments. They will also provide better agreement between the prediction of the fracture geometry and its post-treatment evaluation. Introduction The pressure transient behavior of finite-conductivity vertical fractures has been studied extensively in recent years in an effort to obtain more reliable estimates of the geometry and the conductivity of the vertical fractures. Many of these studies have used semianalytic solutions to describe the pressure distributions in the reservoir and in the due to the production from the well. The semianalytic models that have been used in these studies have been commonly developed by the application of the Boundary Element Method (BEM). Gringarten et al. introduced the use of the Boundary Element Method for the development of pressure transient solutions of vertically fractured wells by an adaptation of the analogous solutions of heat conduction in solids. Gringarten and Ramey demonstrated that Source and Green's functions could be used to construct useful solutions for pressure transient problems in oil and gas reservoirs. These techniques have subsequently been used extensively in the analyses of the pressure transient behavior of vertically fractured wells. The application of the Boundary Element Method to the study of the pressure transient behavior of finite-conductivity vertical fractures was first made by Cinco-Ley et al. They developed a general semianalytic model in which the fracture geometry and material properties were assumed to be uniform. Their model also considered the fracture height to be equal to the reservoir thickness. This model has been commonly used for the analysis of the pressure transient behavior of wells with finite-conductivity fractures. It has also been a reference with which most of the later approximate models have been compared. P. 445^

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