Performance-Prediction Procedure for Transient Flow of Fluids Through Pressure-Sensitive Formations
Authors: Samaniego V. F. , Brigham W.E. , Miller F.G. 
Published: December 2007

The concept of a continuous succession of steady states is applied to obtain a solution to the nonlinear partial differential equation describing the transient flow of a pressure-dependent fluid through a stress-sensitive formation. This equation also was solved numerically to check the validity of the succession of steady states solution.

Introduction
Porous media are not always rigid or nondeformable. Porous media are not always rigid or nondeformable. Usually, average values are used for both pressure-dependent rock and fluid properties. This method pressure-dependent rock and fluid properties. This method reduces the errors involved, but generally does not eliminate them.
In the diffusivity equation, the diffusivity is a constant independent of pressure. When both pressure changes and property changes are small, the constant property assumption is justified. If, however, rock and fluid property changes are important over the pressure range property changes are important over the pressure range of interest, then these changes cannot be neglected, and a variable property solution should be obtained.
Raghavan et al. derived a flow equation stating that rock and fluid properties vary with pressure. This equation, when expressed as a function of a pseudopressure, m(p), resembles the diffusivity equation. Samaniego et al. studied this problem for a greater variety of flow conditions. The diffusivity-like equation that describes this pressure-dependent flow is similar to the differential equation describing the flow of either an ideal or real gas through porous media. The concept of "a continuous succession of steady states" was applied successfully by several authors to obtain a solution for the nonlinear partial differential equation describing the transient flow of gas through porous media. This same method is used here to find an approximate solution for the transient-pressure dependent flow problem.
To date no general correlations have been available to predict reservoir performance when reservoir rock and predict reservoir performance when reservoir rock and fluid properties are general functions of pressure. This paper presents a performance-prediction procedure based paper presents a performance-prediction procedure based on the drainage radius concept and a material-balance equation. Radial- and linear-bounded systems are considered. Using this method, the sandface pressure and the average reservoir pressure are calculated easily. Results were obtained for five different sets of rock and fluid property data. Solutions of general utility can be obtained for engineering problems without using a digital computer.

Basic Assumptions
The method described here is based on the assumptions usually used in well testing theory. We assume horizontal fluid flow with no gravity effects, a fully penetrating well, an isothermal single-phase fluid obeying Darcy's law, an isotropic and homogeneous formation, and purely elastic rock properties (physical property changes purely elastic rock properties (physical property changes with stress changes are reversible).
The assumption of horizontal flow is not quite valid because of change in porosity and reservoir rock thickness with fluid pressure. However, Samaniego showed that the vertical component of flow is negligible for most rock and fluid properties of interest and can be neglected. Gavalas and Seinfel also assumed this.
The rock properties needed in the flow equation as functions of pressure are porosity, permeability, and pore compressibility. pore compressibility. JPT
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