Simulation of Non-Darcy Flow in Porous Media Including Viscous, Inertial and Fractional Effects

In this paper a new diffusivity flow equation has been derived to describe fluid flow in porous media including both Darcian and non-Darcian behaviors. This equation is based on the fundamental Darcy's equation, Forchheimer's equation and Brinkman's equation. The pressure gradient as predicted by the new diffusivity equation includes both viscous terms in Darcy's and Brinkman's equations and the inertial forces term in Forchheimer's equation. Both Forchheimer's inertial and Brinkman's viscous effects are expected to become significant at high flow velocity due to the interactions between fluid layers among themselves and with the media. A numerically simulated model has been specifically developed based on the newly derived partial differential equation. The Crank-Nicholson approximation technique was successfully used to model the newly derived diffusivity equation using suitable boundary conditions. A wide range of fluid flow and porous media characteristics has been tested, and predictions of the numerical model showed very consistent results in all ranges. An experimental laboratory program was designed to verify the numerical model predictions. Comparison showed excellent agreement between experimental data and the numerical model predictions. The flow velocity versus pressure gradient profiles resulting from both the numerical and the experimental programs depicted a great deal of compatibility. At the non-Darcy region "high velocity", the inertial forces tend to predict non-linear higher pressure drop than the Darcian linear prediction, while the frictional non linear effect holds the pressure gradient closer to the Darcian trend. The effect of the shift upward from the Darcian linear trend is much more significant than the shift downward caused by the frictional effects. The findings of this study are expected to be applicable to both gas and oil reservoirs with no scale-up effort.