Analysis of Finite-Conductivity Fractures Intercepting Multilayer Commingled Reservoirs

Abstract

SUMMARY The response of a fractured well in a multilayered reservoir is the primary subject of this study. Both analytical approximations and numerical results are presented. The analytical solutions served three important functions:they enabled us to verify the numerical solutions used in this study;they provided information on the structure of the solution and thus increased physical understanding; andmost importantly, they suggested a method whereby we were able to correlate multilayer solutions with single-layer solutions. We introduce the concept of dimensionless reservoir conductivity and show that in most realistic cases this parameter can be used to correlate commingled-reservoir solutions with the single-layer solutions available in the literature throughout the infinite-acting period. This concept and its utility in correlating solutions are the main contributions of this work. We also consider the analysis of buildup data following a short producing time. As in the drawdown case, we show that the multilayer producing time. As in the drawdown case, we show that the multilayer buildup solutions can be correlated with single-layer buildup solutions through the concept of dimensionless reservoir conductivity. Introduction This work examines the performance of vertically fractured wells in reservoirs without interlayer communication. To the best of our knowledge, the results presented in this paper are not available in the literature. All results presented in this work were obtained by a finite- difference model. Numerical methods, however, are not useful in understanding the structure of the solutions. Analytical solutions, either exact or approximate, have an important role in this regard: they provide information on the structure of the solution. Thus they increase physical understanding and suggest procedures to combine results. The discovery of the bilinear-flow regime bears eloquent testimony to this observation. This flow regime was discovered mainly (perhaps only) because an approximate analytical solution was derived; the numerical solutions had preceded the analytical solutions by 2 to 3 years. preceded the analytical solutions by 2 to 3 years. New analytical solutions for a fractured well intercepting a layered reservoir were derived during the course of this study. The analytical solutions are summarized in the Appendix, and complete documentation is given in Ref 2. In addition to the advantages discussed previously, the analytical solutions served two important functions:they allowed us to verify the accuracy of the finite-difference solutions (no solutions are available in the literature); andthey suggested a method to correlate multilayer-reservoir solutions with the solutions for single-layer systems provided that boundary effects are negligible. From the viewpoint of provided that boundary effects are negligible. From the viewpoint of analyzing data, this result is the most important contribution of our study. Had the analytical solutions been unavailable, it is doubtful that we would have realized how to correlate the solutions for layered systems with the solutions for single-layer systems. Most of our results examine only two-layer reservoirs. We do demonstrate, however, the applicability of the results given here to systems containing more than two layers. The analytical solutions presented in Ref. 2 are valid for any number of layers. We examine the presented in Ref. 2 are valid for any number of layers. We examine the influence of the contrasts in layer properties (permeability, porosity, and compressibility) and show that the ratio of reservoir thickness to fracture height may be important under certain circumstances. Our primary aim is to show that multilayer solutions can be correlated with single-layer solutions by means of the dimensionless reservoir conductivity during the transient-flow period. It should be noted that numerical simulations of the pressure responses for commingled reservoirs producing from fractured wells are given in Refs. 3 and 4. Their objective was to examine the ability of an engineer to history match pressure data using a numerical model. Procedures to generalize the results given in these works are discussed by Procedures to generalize the results given in these works are discussed by Camacho et al. Mathematical Formulation We consider a two-layer reservoir in the form of a rectangular drainage region with the well located in the center of the drainage area (Fig. 1). However, all results given here assume that the influence of the boundaries is negligible. SPEFE p. 259