Exact analytical solution of fluid flow equation in heterogeneous porous media

Investigation of reservoir or aquifer pressure response is mainly based on the analytical solution of the one-dimensional diffusivity equation, and it has limiting assumptions that deviates the analytical model from the reality of the reservoir. Based on the fact that in underground reservoirs the properties of the reservoir rock vary from point to point, by considering permeability as a function of the location, the general equation of fluid flow is expressed in the porous medium. The reservoir quality and permeability decrease from the crest of the reservoir to its flanks. To the best of the authors’ knowledge, there is no accurate analytical solution that could predict reservoir pressure behavior and well production rate in non-homogeneous reservoirs. Additionally, the position-dependent permeability could be obtained using the developed analytical solution, unlike the average permeability calculated by traditional well-testing methods. Previous models focused on numerical and semi-analytical solutions in a semi-infinite length. In this paper, by considering linear and quadratic functions in terms of location, the nonlinear diffusivity equation is defined, and then, an exact analytical solution of the diffusivity equation in a finite length and for different boundary conditions is developed. Also, in order to validate the proposed analytical model, the finite difference method is used, and the results of the analytical and numerical models are compared. Overlooking the position-dependent permeability changes could lead to error values up to 100% in predicting well production rate and additionally reservoir pressure at different points.