Pressure Behavior of a Limited Circular Composite Reservoir
Authors: Carter R.D. 
Published: December 2007

Abstract
The analytical solution is presented to the problem of flow of a slightly compressible fluid in a limited, composite reservoir with radial symmetry which is produced by a well at the center. Numerical results are given for a specific case. It is believed that this type of heterogeneity can account for some actually observed pressure behavior and should be of special value in the interpretation of reservoir limit tests.
The system of interest is composed of two zones of different permeability in concentric series. There is no flow across the outer boundary of the outer zone, and fluid is withdrawn from the system at a well represented by a point sink located at the center of the inner zone. The solution to this problem is useful in the fundamental study of the behavior of reservoirs having a low permeability "rim" and in the pressure transient behavior of some wells having a large horizontal fracture or a large fractured area such as would be created by a nuclear explosion. The analytical solution to this particular problem has apparently not been previously published.
This paper containsa mathematical statement of the problem,the analytical solution,numerical results for a specific problem, anddiscussion of the physical interpretation of these results.
The Appendix contains descriptions of the procedures used to obtain the analytical solution and the tabulated results for a specific problem. The specific numerical results given show that reservoir fluid from a very low permeability rim can contribute to production from a well located in the high permeability area. Predicted pressure drawdown and build-up behavior for the system is given.

INTRODUCTION
The present work was undertaken to develop a basis for interpretation of some observed well pressure transient behavior that did not appear to be otherwise explainable. The solution described herein has special significance in the interpretation of well pressure transient tests designed to indicate reservoir limits.
The partial differential equations describing transient heat conduction and the transient flow of fluids having small and constant compressibility in porous media are mathematically identical. Analytical solutions to this equation for systems involving media of different conductivities in concentric series have appeared in connection with both types of problems. In the heat conduction literature, solutions have been published by Jaeger1 and Carslaw and Jaeger.2 In reservoir engineering literature, solutions have been published by Hazebroek, Matthews and Rainbow,3 Hurst4 and Loucks and Guerrero.5 Additional published solutions are cited in Ref. 2. In all of these references the Laplace transform method equivalent to that introduced to the petroleum literature by van Everdingen and Hurst1 has been utilized to obtain solutions. This method was also employed in the present work.
Although radial symmetry is specifically assumed in this treatment, (he numerical results should give some qualitative insight into the behavior of many reservoirs having a roughly circular area of commercial pay surrounded by a hydrocarbon-containing region in which wells would be noncommercial because of low permeability.
Hopkinson et al.8 gave an expression for the linear asymptote portion of the solution for the linear zone which is equivalent to the one given in this paper. Ref. 8 considers a ratio of diffusivities between the two zones which may be independent of the ratio of permeabilities. In this paper, a difference in permeabilities only is considered and the ratio of the hydraulic diffusivities in the two zones is equal to the ratio of the permeabilities.