Solutions to the Diffusion Equation for a Region Bounded By a Circular Discontinuity

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Abstract
Solutions to the diffusion equation for a line source located anywhere in a region bounded by a circular discontinuity are presented. Such solutions follow automatically from the Green's function presented by Jaeger. Petroleum industry applications of this problem would include transient compressible fluid flow in porous media bounded by a discontinuity. Since several recent papers have treated the subject in less general form, it would appear that many workers in this area of the petroleum industry are not aware of Jaeger's work. STATEMENT OF THE PROBLEM In several treatments of the transient flow of compressible fluids in porous media it was convenient to consider a region to be bounded by a circular discontinuity. In such a region (Fig. 1), the properties in Region 1, r less than a, are constant but different from the properties in Region 2, r greater than a. Hurst, using the Laplace transform, obtained solutions for a line source located in the center of such a porous medium. The purpose of this work is to point out that a much earlier paper by Jaeger presents the solution for a line source located anywhere inside the discontinuity. The assumption is made that a line source may be used to approximate a well. In the following material it is convenient to use the terms source and well as being equivalent; however, the reader should remember the distinction in applications. SOLUTIONS Jaeger presented the Green's function for a line source, or well, in the composite region shown in Fig. 1. In the most general form, the solution for a system of j wells in Region 1, each producing at a time varying rate, qi(t), is ..................(1) The Green's function vi corresponds to the solution for an instantaneous line source at position ri, 0i (Fig. 1). Integration in time, previously shown, gives the contribution from a single well, and summing over all j wells in the reservoir gives the total solution. Equations defining v are presented by Jaeger as well as an outline of the method of solution.From Jaeger's work it can be shown that ....(2) for r less than a, ri less than a (3) SPEJ P. 113^

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