An exact analytical model for fluid flow through finite rock matrix block with special saturation function

Abstract

An exact analytical solution for one-dimensional fluid flow through rock matrix block is presented. The nonlinearity induced from flow functions makes the governing equations describing this mechanism difficult to be analytically solved. In this paper, an analytical solution to the infiltration problems considering non-linear relative permeability functions is presented for finite depth, despite its profound and fundamental importance. Elimination of the nonlinear terms in the equation, as a complex and tedious task, is done by applying several successive mathematical manipulations including: Hopf-Cole transformation to obtain a diffusive type PDE; an exponential type transformation to get a convective-diffusive type PDE with suitable boundary conditions; Laplace transformation; Integral transformation to homogenize the boundary condition in the Laplace domain; and finally Laplace inversion method to find the time domain solution. The obtained solution is used for developing a new matrix-fracture transfer function. The developed analytical equations are used for prediction of drying front in the case of evaporation from deep water table. The model results are in close agreement with the experimental data and numerical simulation of the process. Results of sensitivity analysis of different parameters on recovery rate and ultimate production based on design of experiment method are also discussed.