An Investigation by Numerical Methods of the Effect of Pressure-Dependent Rock and Fluid Properties on Well Flow Tests
Authors: Raghavan R. , Scorer J.D.T. , Miller F.G. 
Published: December 2007

Abstract
Well test analyses of unsteady-state liquid flow have been based primarily on the linearized diffusivity equation for idealized reservoirs. Studies of pressure behavior of heterogeneous reservoirs have been highly restricted, and no general correlations have been developed for systems in which reservoir porosity, permeability and compressibility, together with fluid density and viscosity, are treated as functions of pressure. A second-order, nonlinear, partial-differential equation results when variations of the above parameters are considered. in the present study, this equation was reduced by a change of variables to a form similar to the diffusivity equation, but with a pressure- (or potential-) dependent diffusivity. pressure- (or potential-) dependent diffusivity. By making this transformation, the solutions to the linear diffusivity equation may be used to obtain solutions to nonlinear flow equations in which reservoir and fluid properties are pressure dependent. This paper provides correlations in terms of dimensionless potential and dimensionless time for a closed radial-flow system producing at a constant rate. The solutions obtained have been correlated with the conventional van Everdingen and Hurst solutions. It also has been shown that the solutions can be correlated with the transient drainage concept introduced by Aronofsky and Jenkins, even though no theoretical basis exists whereby their validity can be proved. In fact, the latter correlation provides a better approximation to the nonlinear provides a better approximation to the nonlinear equation than the van Everdingen and Hurst solutions for large values of dimensionless time. Substitution of the potential described has many important consequences in addition to those already mentioned. Usually, the second-degree pressure gradient term is neglected by assuming that pressure gradients in the reservoir are small. in the present study, these gradients are handled rigorously. Moreover, the selection of parameters such as "average reservoir compressibility" is avoided.

Introduction
The concept that the porous medium is absolutely rigid and nondeformable is a valid assumption for a wide range of problems of practical interest. It has been long realized that in many problems this assumption leads to certain discrepancies, however, and that the use of "average" properties of the medium would reduce these errors. Considerable research effort has been made to study the effect of pressure-dependent rock characteristics (compressibility, pressure-dependent rock characteristics (compressibility, porosity, permeability) and fluid properties porosity, permeability) and fluid properties using analytical and /or numerical techniques. As a result, numerous methods of solution have been outlined in principle, and a larger number of particular problems have been solved by means of particular problems have been solved by means of high-speed digital computers. Rowan and Clegg give a thorough review of the basic equations governing fluid flow in porous media, showing how the form of the equation changes depending on which of the parameters are taken as functions of pressure of space variables. They also discuss the implications of linearizing the basic equations. Bixel et al. have treated problems involving a single linear and a single problems involving a single linear and a single radial discontinuity. Mueller has considered the transient response of nonhomogeneous aquifers in which permeability and other properties vary as functions of space coordinates. Carter and Closmann and Ratliff have considered the problem of composite reservoirs and studied pressure response and oil production.
SPEJ
p. 267