A Variable Rate Solution to the Nonlinear Diffusivity Gas Equation

Abstract

AbstractThe hydraulic diffusivity equation that governs the flow of compressible fluids in porous media is nonlinear. Although gas well test analysis by means of the pseudo pressure function has become a standard field practice, the effect of viscosity and gas compressibility variation with pressure is often neglected. Moreover, in field operation, the gas well is submitted to a variable rate production in order to determine well/reservoir properties and an estimation of the absolute open flow (AOF). For slightly compressible fluids, variable rate can be properly handled by superposition in time. Unfortunately superposition cannot be offhand justified to gas reservoir due to its non-linear behavior. In this work a general solution which properly accounts for both fluid property behavior and variable rate is presented. The proposed solution, which is based on Green's Functions method by considering the viscosity-compressibility product change as a non-linear source term, can handle variable gas rate for several well and reservoir geometries of practical interest. From the general solution, an analytical expression for variable rate tests of a fully penetrating vertical well in an infinite gas reservoir is derived. This expression is applied to a synthetic data set to calculate the pressure response for a buildup test in an infinite homogeneous reservoir. The results compared to a finite difference numerical simulator shows close agreement. It is also shown that the dimensionless pseudo pressure converges to the slightly compressible fluid solution for long shut-in times. Thus, at such long times, Horner analysis and log-log derivative plot can be applied to obtain good estimative of reservoir parameters as already discussed previously in literature.