Pressure Transient Analysis in the Presence Of Two Intersecting Boundaries
Authors: Prasad Raj K. 
Published: December 2007

Prasad, Raj K., SPE-AIME, H. J. Gruy and Prasad, Raj K., SPE-AIME, H. J. Gruy and Associates, Inc.
The paper presents an analytical solution for the transient pressure behavior of a well located near two intersecting boundaries in an otherwise infinite system. A least-squares method is used to solve for the pertinent reservoir parameters, providing a rational method for selecting the parameters in any equation that result in a "best fit" of the observed data.

Introduction
Pressure-transient testing has been extensively Pressure-transient testing has been extensively applied to detect anomalies in a reservoir system. These anomalies may be present in the form of faults or change in rock or fluid properties. Horner presented a technique based on the method of images presented a technique based on the method of images to analyze the transient data in the presence of a single fault. van Poollen utilized the method of images to generate pressure drawdown curves for a well located near two intersecting faults in an otherwise infinite system. The image method can be used when angles of intersection are /n, where n is a positive integer for cases where the wells are not located on the bisector of the angle. If the well is located on the bisector, the method is applicable for angles equal to 2 /n in which n is again a positive integer greater than unity. The image technique fails when one or more of the image wells fall in the real plane. This situation prevails when the angle of intersection is n /m, where m and n are both positive integers, prime to each other. For example, when the angle between two intersecting boundaries is 2 /3 and a well is located asymmetrically at P, as in Fig. 1, the image P3 falls in the real plane, and the image technique fails. This is well described by Carslaw and Jaeger. In this paper an analytical solution for the transient pressure behavior for a well located near two intersecting boundaries in an otherwise infinite system is presented. This solution is valid for all angles of intersection and well locations. A least-squares method is used to solve for the pertinent reservoir parameters such as distance from the well to the parameters such as distance from the well to the boundaries, angle between the two boundaries, flow capacity, and the initial reservoir pressure best describing the observed pressure transient data. The least-squares technique provides a rational method for selecting the parameters in any equation that result in a "best fit" of the observed data.

Theory Mathematical Treatment
The physical model considered in this analysis consists of a well located near two intersecting faults in an otherwise infinite system. The fluid flowing into the wellbore is slightly compressible and of constant compressibility and viscosity; and the reservoir system is homogeneous, isotropic and of constant thickness. A well is approximated by a line source. We consider a source at (r', 0') in a radial coordinate system with the origin at the intersection of the two boundaries, as in Fig. 2. Carslaw and Jaeger present the Green's function for the temperature present the Green's function for the temperature distribution due to a unit instantaneous point source at (r', 0') with no flow of heat across the two boundaries in such a wedge system. Their solution is integrated along the coordinate axis, z, and in time to obtain a continuous line source. The integrated solution in terms of fluid flow is presented below.(1)r 2 x - +1 2 e r' 2 pD (r, 0, tD) = g(x)dx, pD (r, 0, tD) = g(x)dx, 0o rr' 2- 2tD d1
JPT
P. 89