Analytical study of fluid flow modeling by diffusivity equation including the quadratic pressure gradient term

Diffusivity equation which can provide us with the pressure distribution, is a Partial Differential Equation (PDE) describing fluid flow in porous media. The quadratic pressure gradient term in the diffusivity equation is nearly neglected in hydrology and petroleum engineering problems such as well test analysis. When a compressible liquid is injected into a well at high pressure gradient or when the reservoir possess a small permeability value, the effect of ignoring this term increases. In such cases, neglecting this parameter can result in high errors. Previous models basically focused on numerical and semi-analytical methods for semi-infinite domain. To the best of our knowledge, no analytical solution has yet been developed to consider the quadratic terms in diffusivity PDE of one-dimensional unsteady state fluid flow in rectangular coordinates and finite length. Due to the resulting errors, the nonlinear quadratic term should also be considered in the governing equations of fluid flow in porous media. In this study, the Fourier transform is used to model the one-dimensional fluid flow through porous media by considering the quadratic terms. Based on this assumption, a new analytical solution is presented for the nonlinear diffusivity equation. Moreover, the results of linear and nonlinear diffusivity equations are compared considering the quadratic term. Finally, a sensitivity analysis is conducted on the affecting parameters to ensure the validity of the proposed new solution. The results demonstrate that this nonlinear PDE is also applicable for hydraulic fractured wells, and well test analysis of fractured reservoirs.