A Novel Uniform-Flux Solution Based on the Green's Function Method for Modeling the Pressure-Transient Behavior of a Restricted-Entry Well in Anisotropic Gas Reservoirs

Abstract

Summary Applied gas-well-testing data analysis considers that the nonlinear diffusivity equation becomes linearized by the pseudopressure function and by assuming the gas viscosity-compressibility product constant throughout the test history. This approach is justified for small pressure drawdowns only. This condition is seldom met in low-permeability gas reservoirs, whether the wellbore is fully perforated or not. This paper considers a restricted-entry vertical well in a homogeneous anisotropic infinite reservoir producing at constant-gas-flow rate, which, surprisingly, has not been the object of much research, either by numerical simulation or by analytical means. In this work, the Green's function (GF) method is used to rewrite the usual partial-differential formulation as a multidimensional integro-differential equation for the pseudopressure function that rigorously accounts the gas viscosity-compressibility product change. From there, an approximate explicit solution for a uniform-flux boundary condition is derived, which shows that the pseudopressure solution is given by the correspondent restricted-entry well slightly compressible (i.e., liquid) solution, plus a corrective term that handles the gas viscosity-compressibility product variation with pressure. Comparison with a commercial numerical simulator shows that the uniform-flux approximate solution is very accurate. Results are presented for a few perforation lengths and positions, with and without vertical anisotropy. It is shown that during the perforation-radial (early-time) and final-radial (late-time) flow regimes, the corrective term becomes constant. Their magnitudes, however, are quite different; during the perforation-radial flow regime, the corrective term is quite large compared with the one seen at long times. Thus, the pseudopressure derivative with respect to the natural log of time exhibits the same restricted-entry behavior observed in oil reservoirs. By averaging the uniform-flux solution over the open interval, an infinite-conductivity approximate solution for the wellbore pseudopressure is presented, which shows a very-good agreement with results obtained from a finite-difference simulator.