Calculation of Unsteady-State Gas Flow Within a Square Drainage Area

Abstract

Published in Petroleum Transactions, AIME, Volume 204, 1955, pages 190–195. Abstract The problem of unsteady-state gas flow through porous media has been solved numerically only for the case of linear or radially symmetric reservoirs. A recently introduced numerical method for solving the unsteady-state heat flow equation in two dimensions is applied to the calculation of the depletion of a square region containing a perfect gas. Solutions are presented in graphical form for various values of dimensionless parameters. The solutions are compared with published solutions for radial reservoirs. Introduction The problem of unsteady-state flow of gas through porous media gives rise to a second-order non-linear partial differential equation for which no analytical solution has been found. Numerical approximations to solutions of the gas flow problem have been obtained by the stepwise solution of an associated difference equation. However, the methods so far developed have required that the reservoir be either linear or radially symmetric. This restriction in shape has been necessary so that only two independent variables be considered, namely, one distance variable and time. In order to deal with reservoirs having more realistic shapes, it is necessary to develop numerical procedures for the solution of the gas flow problem involving two distance variables. A numerical procedure, denoted as the alternating-direction implicit method, for the solution of the heat-flow problem in two dimensions has recently been introduced. By the use of this procedure, approximate solutions have been obtained for heat-flow problems in a square, and in regions having various non-rectangular boundaries. Because of the similarity between the equation for heat conduction in solids and the equation for gas flow in porous media, it is reasonable to expect that the alternating-direction implicit method should also be useful for solving the gas flow problem in various two-dimensional regions.