Abstract

The problem of unsteady-state gas flow through porous media leads to a second-order non-linear partial differential equation for which no analytical solution has been found. In this paper a stable numerical procedure is developed for solving the equation for production of gas at constant rate from linear and radial systems. An electronic digital computer is used to perform the numerical integration using an implicit form of an approximating difference equation. Solutions are presented in graphical form for various values of dimensionless parameters. The solutions are compared with the laboratory study of gas depletion in a linear system.

Introduction

Production of fluids from porous rock reservoirs is essentially a transient process. Transient gradients develop as soon as production begins, and further withdrawals continue to cause disturbances which propagate throughout the reservoir, each adding in some way to the prior ones.

A correct mathematical analysis of this behavior is complicated by the fact that the transient or unsteady-state flow of compressible fluids must be described by difficult second-order partial differential equations. As a practical matter, three distinctly different cases arise:

  1. Flow of single-phase liquid

  2. Flow of gases

  3. Multiphase flow

The first of these has been found to give a linear second-order equation similar to the well-known heat flow equation.

Solutions of Equation (1) for both linear and radial flow are available in several forms.

Although a number of approximate solutions have, been proposed, each is limited in value by the associated simplifying assumptions.

Inasmuch as the analysis of transient flow is limited to liquid systems, a solution of the second case is necessary if further progress is to be made in studying underground fluid movement. For this reason a solution of Equation (2) was undertaken by means of numerical integration of approximating difference equations.

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