Approximate Solutions for Fractured Wells Producing Layered Reservoirs

Abstract

Abstract New analytical solutions for the response at a well intercepting a layered reservoir are derived. The well is assumed to produce at a constant rate or a constant pressure. We examine reservoir systems without pressure. We examine reservoir systems without interlayer communication and document the usefulness of these solutions, which enable us to obtain increased physical understanding of the performance of fractured physical understanding of the performance of fractured wells in layered reservoirs. The influence of vertical variations in fracture conductivity is also considered. Example-applications of the approximations derived here are also presented. Introduction We recently examined the response of vertically fractured wells in layered reservoirs. The conductivity of the fracture was assumed to be finite. The solutions presented in Refs. 1 and 2 were obtained by solving the partial differential equations and the associated conditions by standard finite-difference methods. During the course of that study, several analytical solutions were derived. Although approximate, the analytical solutions serve important functions. First, they provide information on the structure of the solution; thus they increase physical understanding. Second, they suggest procedures to correlate results obtained by finite-difference methods. (If the analytical solutions had been unavailable, it is doubtful that the correlations in Refs. 1 and 2 could have been obtained.) Third, they allow us to verify the accuracy of the finite-difference solutions when no solutions are available in the literature or when the solutions in the literature are not in agreement, which is important. We had difficulty validating our finite-difference model because solutions in the literature are not in agreement for all times of interest. To our knowledge, the analytical solutions presented here are not available in the literature. Comparisons with the numerical solutions are presented and the advantages of the analytical solutions are documented in this study. In addition to being used to examine fractured wells intercepting layered systems, the analytical solutions presented here also can be used to study cases where the presented here also can be used to study cases where the fracture extends above and/or below the productive interval and cases where the conductivity of the fracture is a function of depth. In this work we refer to a fracture with fracture height greater than the formation thickness as an "extended fracture." It is emphasized that the objective of this paper is to discuss the analytical aspects of the problems stated above. Practical considerations of these results are not germane to this paper. This aspect is discussed in more detail in Refs. 1 through 4. However, we show that the analytical solutions presented here lead to new methods of analysis that are not given in the literature. Example applications of the new methods of analysis are presented. Physical Model Physical Model The physical system discussed in this work is a multilayer, rectangular reservoir that is being drained by a hydraulically fractured well. The well is located at the center of the reservoir and the fracture. The fracture is parallel to two sides of the reservoir. Thus, the parallel to two sides of the reservoir. Thus, the reservoir-fracture system is symmetric about the well. For this reason, only a quarter of the physical system is modeled (Fig. 1). (For simplicity, only two layers are shown; all derivations given here are applicable to systems with more than two layers.) The top, bottom, and outer boundaries of the reservoir are assumed to be impermeable. Each layer of the reservoir is assumed to be a uniform and homogeneous porous medium. The properties of any given layer (porosity, compressibility, permeability, and thickness), however, may be different from that of another layer. There is no vertical communication between the layers, except by the vertical fracture. The fracture is considered to be a layered porous medium. This idealization is used to model variations in fracture conductivity with depth. Each layer of the fracture is assumed to be a homogeneous porous medium. The width and length of the fracture are independent of the vertical coordinate. The permeability of the fracture in a given layer is independent of the distance from the well. The most important feature of the mathematical model used in this study concerns the fracture height and the thickness of the layers in the fracture and reservoir. The model assumes that the fracture height, hf, can be equal to or greater than the formation thickness, h. More importantly, the thicknesses of the fracture layers are independent of the thicknesses of the layers in the reservoir. This feature of the model is in accordance with the conditions that should exist in practice. Fig. 2 depicts some of the conditions that can be simulated by this model. For convenience of notation, z=0 represents the top of the reservoir and the top of the fracture. SPEJ P. 729