A Transform Approach to the Simulation of Transient Gas Flow in Porous Media

Abstract

Abstract The mathematical description of transient gas flow through porous media leads to a second-order nonlinear partial differential equation for which no analytical solution bas been found. For flow in one linear dimension, this paper outlines a procedure for using finite Fourier transforms to reduce the partial differential equations to a set of first-order, partial differential equations to a set of first-order, nonlinear ordinary differential equations with time as the independent variable. The dependent variables of these equations are the Fourier transform coefficients of the gas pressure in the porous media The procedure is applied to an porous media The procedure is applied to an example problem and calculated results are compared with other published results obtained by a numerical finite-difference technique. The good agreement shown between the two sets of results tends to confirm the validity of both solutions. Introduction The differential equation describing one-dimensional gas flow in porous media is a nonlinear equation with no known analytical solution. Numerical finite-difference methods are normally used to solve the flow equation. This paper presents a method which uses linear analysis to transform a second-order problem into a set of first-order problems. Basically the approach used is to assume an approximation of the solution consisting of known functions and unknown parameters and apply some error distribution parameters and apply some error distribution principles to determine the parameters of the principles to determine the parameters of the approximation. Although there are several error distribution principles, one of the most common is the orthogonal method. Recently special cases of the orthogonal method, the Galerkin or Bubnov-Galerkin have been applied by Price et al., to petroleum reservoir problems with considerable success. A classical orthogonal method is finite Fourier transforms. Recently several authors have demonstrated that this method can be used with considerable success in the simulation of systems described by a limited class of linear partial differential equations. This paper is intended to demonstrate that the finite Fourier transform method can be used to reduce the nonlinear partial differential equation describing gas flow in a linear porous media to a set of ordinary differential equations. THEORY The basic equation being studied is: ............(1) which is derived from the continuity equation ...............(2) the equation of state ..............(3) and Darcy's law ................(4) For the orthogonal method of error distribution, the procedure is to assume an approximate solution Pa, define an error L[pa] and require that this error Pa, define an error L[pa] and require that this error be orthogonal to N independent functions n(x) (n=1,2.... N) ...................(5) SPEJ P. 135