Pressure Response of a Reservoir With Spherically Discontinuous Properties
Authors: Onyekonwu Michael O. , Horne Roland N. 
Published: December 2007

Summary
This paper addresses the problem of pressure transient analysis of a reservoir with spherically discontinuous mobility and hydraulic diffusivity. Such a situation may exist as a result of formation of a burned zone in in-situ combustion, by phase change in a steam flood or geothermal operation, or simply by the formation of a bank of injected fluid with properties different from those of the reservoir fluid. The spherical configuration is expected to represent geothermal applications more closely than does a radial configuration, and is also of interest in hydrocarbon wells with limited entry or partial penetration, or those wells in the vicinity of a pinchout. The problem is solved analytically by using the Laplace transform, which is then inverted using the Stehfest algorithm. The solution is different in detail from the radial case but still shows the important pseudosteady-state behavior at the end of the inner region response (in some cases). It is this pseudosteady-state behavior that permits determination of the distance to the discontinuity (or, in fact, the "burned volume"). Other features of the solution are that the mobility ratio and diffusivity ratio may often be determined from pressure drawdown behavior. The mobility ratio can be calculated from the intercept of a pressure-vs.-reciprocal-square-root-time plot. The diffusivity ratio also may be determined from the ratio of the two straight lines, which appear on the same plot. These methods are not possible in the case of large mobility contrasts in which the outer region is the more permeable. This work was motivated by the need in the geothermal industry to interpret well tests in reservoirs of very large production "thickness," especially in cases where the fluid close to the well may be in a phase different from that further away. Such a configuration occurs frequently since the most commonly performed completion tests involve injection of cold water. However, the application of methods derived in this paper are not restricted to the geothermal area and have applications in in-situ combustion as well.

Introduction
Discontinuities around a well can occur in a wide variety of reservoir situations. In hydrocarbon reservoirs the inflow or outflow of a fluid bank will result in a compressibility discontinuity, or an in-situ combustion or stimulation program may result in a permeability discontinuity. In a gas condensate or geothermal reservoir, reduction of reservoir pressure in the vicinity of the well will give rise to changes in relative permeabilities as the fluid changes phase, and, in the case of water, enormous changes in compressibility. The same would be true in the case of a steamflood. Recent work by Grant and Sorey indicates that the compressibility of a two-phase steam/water mixture at the boiling point has an effective compressibility four orders of magnitude larger than that of water and two orders of magnitude larger than that of steam. Clearly, the appearance of a flash front in a water region or the start of condensation in a steam region would result in a sharp discontinuity of considerable order.
JPT
P. 2127^