A Variable-Rate Solution to the Nonlinear Diffusivity Gas Equation by Use of Green's-Function Method

Abstract

Summary The hydraulic diffusivity equation that governs the flow of compressible fluids in porous media is nonlinear. Although the gas-well test analysis by means of the pseudopressure function has become a standard field practice, the effect of viscosity and gas-compressibility variation with pressure is often neglected. Moreover, in field operations, the gas well is submitted to a variable rate production 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 casually justified for gas reservoirs because of its nonlinear behavior. In this paper, a general solution that properly accounts for both fluid property behavior and variable rate is presented. The proposed solution, which is derived from the Green's-function method by recasting the effect of the viscosity-compressibility product variation as a nonlinear source term, can handle variable gas rate for several well/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 with a commercial finite-difference numerical simulator show close agreement for both drawdown and buildup periods. It is also shown that the dimensionless pseudopressure converges to the slightly compressible fluid solution for long shut-in times. Thus, during those long times, Horner analysis and log-log derivative plot can be applied to obtain good estimation of reservoir parameters, as discussed previously in literature.