Pressure Falloff Behavior in Vertically Fractured Wells: Non-Newtonian Power-Law Fluids

Abstract

Summary. This paper examines pressure falloff behavior in fractured wells after the injection of a non-Newtonian power-law fluid. Results are presented in a form suitable for field application. Responses at wells presented in a form suitable for field application. Responses at wells intercepting infinite-conductivity and uniform-flux fractures are considered. Procedures to identify flow regimes are discussed. The solutions presented here are new and to the best of our knowledge are not available in the literature. The consequences of neglecting the non- Newtonian characteristics of the injected fluid are examined. Although the main objective of our work is to examine pressure falloff behavior at fractured wells, we also examine responses at unfractured wells. This part of our study examines the validity of using the superposition principle to analyze pressure falloff data. (The pressure distribution for this problem is governed by a nonlinear partial- differential equation.) If the solutions given in the literature are used, then correction factors are needed to analyze pressure falloff data. The results of this phase of our work can also be used to analyze data in fractured wells provided that pseudoradial flow conditions exist. Introduction This paper examines pressure falloff behavior in fractured wells after the injection of non-Newtonian power-law fluids. This problem was considered recently by Murtha and Ertekin, who problem was considered recently by Murtha and Ertekin, who developed a simulator that uses polar coordinates to model the flow of a non-Newtonian fluid injected by a vertically fractured well; however, no procedure was presented to analyze pressure falloff data and thereby determine formation parameters or fracture half-length. The solutions presented in this study were obtained numerically. The numerical scheme used here has proved robust enough to apply to a wide range of parameters. Procedures to determine fracture half-length are presented. Although the primary objective of this study is to analyze data in fractured wells, we also examine falloff behavior in unfractured wells. Analysis procedures for pressure falloff behavior in fractured wells should be similar to that pressure falloff behavior in fractured wells should be similar to that for unfractured wells provided that pseudoradial flow conditions prevail. prevail. The partial-differential equation that governs the flow of a slightly compressible power-law fluid in porous media is a nonlinear equation. Odeh and Yang and Ikoku and Ramey (see also Ref. 5) linearized this equation, obtained a closed-form solution, established the validity of the closed-form solution for the flowing period, and then invoked the principle of superposition to analyze pressure falloff data. This approach was also taken in Ref. 6, where field examples were analyzed. No justification, however, for the use of the principle of superposition to the nonlinear problem of interest was principle of superposition to the nonlinear problem of interest was presented. A rigorous and complete examination of the falloff presented. A rigorous and complete examination of the falloff response following the injection of a non-Newtonian power-law fluid has not been considered until now. We examine flowing and shut-in pressure responses at unfractured and fractured wells for the power-law index, n, in the range 0.1 less than or equal to n less than or equal to 0.9, respectively. We show, that significant corrections to the expressions given in Refs. 3 and 6 are needed if we are to examine falloff behavior when n less than 0.6. The value of n is less than 0.6 for all field examples presented in Refs. 3 and 6. We present a correction factor correlation to analyze falloff data in unfractured wells. In summary, the objectives of this paper are (1) to examine the consequences of using linearized solutions to analyze pressure data (injection and falloff), (2) to examine the validity of the superposition principle to analyze falloff data and to present a correction factor correlation to analyze test data when the linearized solution will yield incorrect results, and (3) to present new solutions to examine falloff behavior in wells intercepting vertical fractures and discuss procedures to estimate fracture half-length. Modifications to the procedures to estimate fracture half-length. Modifications to the example applications considered in the literatures are also discussed. Mathematical Model The mathematical model considered in this study is similar to the models considered in Refs. 1 and 3 through 7 in many respects. We examine the flow of a non-Newtonian and slightly compressible fluid in a uniform porous medium. The thickness of the reservoir is assumed constant and gravitational effects are assumed negligible. The well is assumed to penetrate the reservoir completely. Fluid is injected at a constant rate. Falloff responses are simulated by closing the well and then noting the change in pressure with time. Wellbore storage effects are assumed negligible (see Ref. 2). Unless it is explicitly stated, boundary effects are assumed negligible. When we examine the influence of boundaries, we assume that the reservoir boundary is at a constant pressure. Our prime objective is to analyze pressure falloff data; for this situation, the constant-pressure outer-boundary condition is the appropriate boundary condition. The-features discussed above are identical for both unfractured and fractured wells. To model the vertical fracture, we impose two different boundary conditions on the fracture surface. infinite conductivity and uniform flux. Both solutions are useful for falloff-data analysis. Justification for using these idealizations is discussed later. Fig. 1 is a schematic of the fractured-well model used in this study. The vertical fracture is assumed to extend over the entire vertical extent of the reservoir (fracture height is equal to formation thickness), and fluid enters the reservoir only through the vertical fracture. Although a square drainage region is shown in Fig. 1 and the fracture plane is parallel to two of the boundaries, we will assume that boundary effects are negligible for the most part. The performance of well-intercepting fractures of finite width performance of well-intercepting fractures of finite width (finite-conductivity fractures) is examined in Ref. 2. In the unfractured-well case, we incorporate a skin region by use of the thick skin concept. The procedure to determine the skin zone permeability for a given value of the skin factor and skin radius is discussed in Appendix A. SPEFE P. 456