Pressure transient solutions to mixed boundary value problems for partially open wellbore geometries in porous media

Abstract

In this paper we present semi analytical pressure transient solutions for a limited-thickness open cylindrical production ring (often called limited entry or partially penetrated) on a non-permeable cylindrical wellbore in a porous medium which is used for fluid withdrawal (production) from the formation. Solutions are not readily available for the uniform-pressure boundary condition on the open cylindrical ring surface because such a condition creates mixed boundary value problem (MBVP) which is difficult to solve. It is well known that the approaches presented over the years for partially penetrated wells are based on the uniform-flux line source solutions for which the middle point, the equivalent-pressure point, or pressure averaging is used to approximate the uniform-wellbore-pressure boundary condition. Here we present exact pressure transient solutions obtained under assumption that the pressure on the open surface is uniform but a priori unknown and the rest of the wellbore surface is non-permeable (no-flow condition). Furthermore, we generalize our solution for the case of multiple production sections on the wellbore for both well testing and Wireline formation tester packer modules.To solve MBVP, we write it in the form of a dual-integral equation. Next, using the special set of basis functions for representing the flux density on the open surface, we reduce it to an infinite system of linear algebraic equations with a symmetric diagonally dominated matrix. These properties of the matrix together with the singular nature of basis functions ensure very fast convergence of the truncated solutions to their limit, thus only a few equations can be considered and the system can easily be solved numerically.The solutions presented in this paper are also compared with the middle point, equivalent-pressure point, and pressure-averaging solutions given in the petroleum literature.