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

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.