Analytical solutions for radial pressure distribution including the effects of the quadratic‐gradient term

Abstract

This study provides a quantitative analysis of the effects of neglecting the quadratic‐gradient term in solving the diffusion equation governing the transient pressure distribution during high pressure‐gradient injection of compressible liquids in porous media. Mathematical solutions of the two‐dimensional cylindrical‐symmetry nonlinear diffusion equation are derived by using the Laplace transform. A fully penetrating well bore in a homogeneous and isotropic porous medium is considered. The analysis accounts for well bore storage and incorporates a wide range of boundary conditions. Analytical early‐ and late‐time solutions are also presented for some cases. Quantitative deviations from existing linear solutions are related to a dimensionless group, α, which is proportional to the fluid compressibility; the higher the magnitude of α, greater is the deviation of the nonlinear solutions from the linear ones. The linear pressure and rate solutions are generally within 0.5% of the corresponding nonlinear solutions for the constant pressure inner boundary. However, for the constant discharge‐ rate condition, the error may be as high as 10% (within the ranges of a and dimensionless radius and time considered). The error may be even higher for higher injection rates in flow systems with smaller transmissivity.