Application of convolution theory for solving non-linear flow problems: gas flow systems

Abstract

The accurate description of fluid flow through porous media using physically representative analytical or numerical models greatly improves an engineer's ability to correctly analyze and predict the flowrate and pressure performance in a reservoir. It is well known that fluid flow through porous media can be described mathematically with a partial differential equation of the diffusion type, and that both analytical and numerical solutions of this "diffusivity" equation differ widely depending on flow conditions, spatial geometry, and fluid type. The diffusivity equation for the case of single-phase flow of a slightly compressible liquid yields analytical solutions in the Laplace domain for a variety of well geometries. However, the analytical solution of the gas diffusivity equation remains unconquered-except for approximate solutions given by ideal gas behavior, by perturbation, and by linearization. The difficulty in solving this problem stems from the highly compressible nature of real gases which results in a non-linear partial differential equation (i.e., pressuredependent coefficients in the differential equation). Our objective is to develop a se@-analytical solution specific to the case of compressible gas flow that will eliminate the use of limiting assumptions and/or pseudotime. This thesis proposes a new approach that uses pseudopressure to linearize the spatial portion of the differential equation and uses convolution to account for the pressure-dependent non-linear term in the time portion of the gas differential equation. This solution is "semi-analytical" because although the approach is rigorous, we evaluate the non-linear term based on the average reservoir pressure predicted from material balance.