Numerical Modeling of Forchheimer's Equation to Describe Darcy and Non-Darcy Flow in Porous Media

Abstract

Abstract This paper adds a new dimension to fluid flow in porous media by replacing Darcy's equation with Forchheimer's equation, which in fact takes inconsideration both Darcy and non-Darcy flow behaviors. Mathematical derivation of the diffusivity equation based on the Forchheimer equation has been accomplished for linear flow in both one- and two- dimensional cases. Numerical simulation of the new diffusivity equation has been achieved in both cases of Darcy and non-Darcy (Forchheimer) domains. Both numerical model shave been tested and verified giving very reasonable accuracy in describing flow characteristics of Darcy and non-Darcy behaviors. Interestingly, a new dimensionless number "Be" relating ß (mainly a function of permeability and porosity), velocity, density and viscosity has been introduced to differentiate between Darcy and non-Darcy flow in porous medium for any types of rock and flowing fluid. This new dimensionless number is far from being considered declaration of turbulence flow in porous medium rather the energy loss is contributed to the nature of both flowing fluid and the porous medium. Theoretically, the break point of non-Darcy behavior from Darcy's has been found at (Be = 0), for practical use it has been determined that non-Darcy flow would start at the point when "Be" become greater than zero. A practical range of permeability with porosity changing accordingly, fluid velocity, density, viscosity and the non-Darcy coefficient () estimated from different types of available correlations in the literature have been examined using the numerical models obtained. In all cases "Be" remains the same at the point of flow behavior change to non-Darcy. Introduction The first equation introduced to describe fluid flow in a porous medium was the one presented by Darcy1 in 1856. Muskat in 1937 was the first to utilize Darcy's law in deriving fluid flow equations in oil reservoirs. Linear, radial and spherical flow of compressible and incompressible fluids for single and multi-phase flow patterns have been derived and employed at later times. In1901, when Forchheimer2 modified Darcy's equation by adding a newterm to account for inertia caused by high velocity flow in gas reservoirs. It has become evident lately that non-Darcy flow occurs not only in gas reservoirs, but in fractured reservoirs and multi-permeability systems within oil reservoirs also. These reservoirs manifest non-linearity due to non-Darcyflow behavior. Most currently used reservoir simulators give inaccurate predictions due to dependency on the traditionally used flow equations, that are simply extended versions of Darcy's equation.