Application of the Line Source Solution To Flow in Porous Media--A Review
Authors: Ramey H. J. 
Published: December 2007

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This paper is to be presented at the SPE-AIChe Symposium to be held in Dallas, Tex., Feb. 7–10, 1966, in conjunction with the 58th Annual meeting of the American Institute of Chemical Engineers, and is considered the property of the Society of Petroleum Engineers. Permission to publish is hereby restricted to an abstract of not more than 300 words, with no illustrations, unless the paper is specifically released to the press by the Editor of the Journal of Petroleum Technology or the Executive Secretary. Such abstract should contain conspicuous acknowledgment of where and by whom the paper is presented. Publication elsewhere after publication in the Journal of Petroleum Technology or the Society of Petroleum Engineers Journal is granted on request, providing proper credit is given that publication and the original presentation of the paper.

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Ideal flow of slightly compressible single phase fluids from horizontal, isotropic reservoirs of linear and radial shapes has been thoroughly discussed in the literature, although specific information for closed radial reservoirs at large re/rw ratios is not readily accessible. Superposition of the continuous line source solution can be used to generate solutions for large drainage area-well radius ratio cases for a variety of reservoir shapes. Application of this method to pressure build-up testing of wells in a variety of reservoir shapes was made by Matthews, Brons and Hazebroek. Application of their results to the general problem of fluid flow from odd reservoir shapes is demonstrated in this paper. Results can also be used to generate Miller-Dyes-Hutchinson type pressure build-up curve for a variety of reservoir shapes, or to produce dimensionless pressure-drop data from Miller-Dyes-Hutchinson type data. Other applications include generation of flow equations for steady-state injection in developed well patterns, and heat conduction shape factors for circular sources and sinks in bounded shapes.

Introduction

Transient flow of a slightly compressible fluid from horizontal isotropic reservoirs of constant thickness has been thoroughly studied for radial and linear systems. Flow in other reservoir geometries [e.g., square, rectangular, triangular, etc.] has been considered thoroughly in connection with pressure build-up but only sporadically for depletion type of, flow. Although it is not often necessary to consider production from oddly shaped reservoirs, there are cases where such information is useful. It is the purpose of this paper to organize available information in the literature on flow from oddly shaped reservoirs, to develop pertinent relationships and to demonstrate the utility of the information.

THEORY

The laminar flow of a slightly compressible fluid in a radial, horizontal, isotropic reservoir under the, condition of small pressure gradients in the reservoir may be described by the diffusivity equation in radial coordinates:

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