Unsteady Flow to a Well Produced at a Constant Pressure in a Fractured Reservoir

Abstract

Summary. The objective of this theoretical study is to delineate the characteristics of a well producing at a constant pressure in a naturally fractured reservoir. We consider flow to a well located at the center of a closed circular reservoir. New solutions not documented previously are presented. Five flow regimes are identified and information that can be extracted from each regime is documented. Methods to determine formation parameters and to predict well deliverability are discussed. Introduction This paper is a theoretical study of the performance of a well located in a fractured reservoir and produced at a constant pressure. As is well known, a fractured reservoir is usually considered to be a system in which the conducting properties of the rock are a result of the fracture system and the storage capacity of the reservoir is due to the matrix system. All mathematical models that describe the performance of fractured reservoirs assume that the well response is a combination of a rapid response that is a result of the fracture system and a slower induced response that reflects the contribution of the matrix system. The principal difference between the models suggested in the literature involves the interaction of the matrix system and the fracture system. Warren and Root and Odeh assume quasisteady- or pseudosteady-state flow in the matrix system, whereas Kazemi, deSwaan- O., Najurieta, and Kucuk and Sawyer assume unsteady-state flow in the matrix system. Mavor and Cinco-Ley and DaPrat et al. used the Warren and Root model to examine the response of a well flowing at a constant pressure. Their objective was to predict well deliverability when the reservoir parameters were determined independently. In this work, we assume that the fractured reservoir may be represented by the rectangular slabmodel idealization used in Refs. 3 and 4 (see Fig. 1). Our first objective is to discuss procedures whereby early-time rate data may be analyzed at a well to determine the properties of the fracture and the matrix systems. The second objective is to present methods to predict future performance or well deliverability. In this work, we identify five possible flow regimes. Three of these flow regimes may exist if the well response is unaffected by boundary effects. Following the terminology of Ref. 9, we identify the flow regimes as Flow Regimes 1 through 3. Two other flow regimes may be evident after the outer boundary begins to dominate the well response. We refer to these flow regimes as Flow Regimes 4 and 5. In the following, we delineate conditions under which Flow Regimes 1 through 5 will exist and discuss the information that can be extracted from each. In particular, we show that for specific values of reservoir properties, not all flow regimes will be evident. We also show that reservoir size determines the flow regime that will govern the well response during boundary-dominated flow. Mathematical Model We consider the flow of a slightly compressible fluid of constant viscosity in a cylindrical reservoir in which the outer boundary is closed. The well is located at the center of the cylinder and fluid is produced at a constant pressure. Initially, the pressure is uniform throughout the reservoir. Gravitational effects are assumed to be negligible. We consider a skin region and assume that it is "infinitesimally thin." The wellbore storage effect is not considered. As already mentioned, we assume that the naturally fractured reservoir be described by the slab model suggested by Kazemi and deSwaan-O. (Fig. 1). Individually, the matrix system and the fracture system are assumed to be uniform, isotropic porous media. All production is by way of the fracture system, and flow from the matrix system to the fracture system is onedimensional (1D) (normal to the fracture system). For modeling purposes, the mathematical problem may be formulated by examination of flow in only one of the repetitive elements shown in Fig. 1. The mathematical formulation is discussed in the Appendix. The solution in Laplace space (for infinite-acting and bounded systems) and the relevant asymptotic approximations are given in the Appendix. All results presented in this study were obtained by inversion of the rigorous analytic solution given in the Appendix numerically, and not from the asymptotic expansions. The asymptotic expansions are useful primarily in identifying the structure of the solutions. SPEFE P. 186^