A General Governing Equation for Flow of Fluids in Porous Media

Abstract

A general differential equation governing the motion of a fluid through porous media is formulated by applying the momentum balance to a formal control volume. The porous medium in the control volume is modeled as an assemblage bundle of hypothetical conduits through which fluid is transported across the medium. The momentum equation employed, will account for the rate of momentum influx and efflux by virtue of the bulk fluid motion in and out of control volume. This formulation will also take into consideration a net shear force acting on the surface of the control volume. The pressure forces acting on the fluid surfaces of the volume element and the resultant viscous force of the fluid acting on the interior wetted surfaces of hypothetical conduits are also included in the model. Furthermore, it is shown that equation of motion reduces to the familiar Darcy’s Law for slow flow rates; for higher velocities, it transforms into Forchheimer relationship. For non-uniform velocity fields, the governing equation yields additional cross-stream-inertia term and a drag term which do not appear in other formulations. In the absence of inertia effects, the derived equation of motion reduces to Brinkman’s findings. In addition, the individual coefficients of the expressions in the momentum equation are explicitly derived in terms of fluid properties, size characteristics of a given medium and some unknown coefficients. A procedure for extracting the unknown coefficients will also be discussed.