Fractional differentiation approach to modeling of fluid filtration processes in complex heterogeneous porous media

Abstract

Fractional differential mathematical models of diffusion type for description of filtration processes in complex fractured porous media are proposed.  A nonlinear pressure equation with the Riemann–Liouville time-fractional derivatives is derived for a single phase filtration of nonNewtonian fluid with a fractional equation of state in naturally fractured porous media. A space-fractional generalization of the Darcy’s law is proposed for a single phase filtration modeling in porous media with fracture network, and corresponding fractional differential equation of anisotropic filtration is obtained. Also, a fractional modification of the Barenblatt-Gilman model of nonequilibrium two-phase capillary counter-current imbibition is derived. The model takes into account a power-law memory effects which can accompany the process of system relaxation to local equilibrium state.