Computing Pressure Distributions in Wedges
- Chih-Cheng Chen | Rajagopal Raghavan
- Document ID
- Society of Petroleum Engineers
- SPE Journal
- Publication Date
- March 1997
- Document Type
- Journal Paper
- 24 - 32
- 1997. Society of Petroleum Engineers
- 5.11 Fundamental Research in Reservoir Description and Dynamics
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Pressure distributions for wells in wedge-type systems are derived in terms of the Laplace transformation Characteristics of responses are discussed and computational issues are addressed. The algorithm given here is a practical tool for analyzing flows in wedge-type systems and may be incorporated immediately into existing software packages. Existing solutions, in the petroleum-engineering literature, for pressure distributions in wedges are a subset of the solution given here. Introduction Conventional application of the method of images for a system with intersecting faults in terms of the Laplace transformation is limited to situations wherein the angle of the wedge is equal to p/m, where m is a positive integer.1 This limitation is not an issue if we simply wish to understand pressure responses for various well locations and wedge-angles. From the viewpoint of analyzing tests, however, this restriction may be a serious limitation, particularly so, if techniques such as nonlinear regression or 'history matching' are used. The purpose of this work is to present a tractable solution for the case wherein the wedge-angle is np/m, where n and m are positive integers that are prime to each other. The contributions of this work are: 1) A general algorithm is outlined to compute pressure distributions in wedge-type systems in terms of the Laplace transformation; thus the limitations of existing solutions are overcome, 2) Schemes to compute well responses for vertical, vertically-fractured and horizontal wells are outlined, and 3) Computational issues are discussed; specifically, we address computations that involve products of the Bessel functions of the form I(a)K(a), where is a fraction.
Consider flow in an uniform and isotropic porous medium with the axis of the wedge coincident with the z axis, and with the two bounding surfaces of the wedge located at q=0 and q=q0. The top and bottom boundaries are given by the planes z=0 and z=h, respectively. Each bounding surface may be assumed to be impermeable or at a pressure equal to the initial pressure.
For each combination of boundary conditions, the instantaneous point-source solutions of Carslaw and Jaeger1 may be used along with procedures of Ozkan and Raghavan2 to obtain pressure distributions for the various circumstances noted in the Introduction. The pertinent expressions and procedures are documented in the Appendix.
As is evident from the solutions given in the Appendix, all solutions involve computing the product I(a)K(a) where is a fraction. Thus, in computational terms, the main issue is to compute the product I(a)K(a). Our experience suggests that values beyond the range ~10-306 to ~10+306 are needed because I(a) increases without bound even though K(a) approaches 0 for large . Unfortunately, a limiting form for this product when is large for all values of the argument a is unavailable. Consequently, it is difficult to terminate computations and ensure that convergence is attained. To compute such products, we used Microsoft Visual C++ with IEEE, long double-data-type that supports values of variables to »10-4932 and »10++4932. This step avoids convergence problems when a large number terms (greater than 30) is needed to compute the relevant summation terms. Additional details are given in Ref. 3. The Stehfest4 algorithm is used to obtain well responses.
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