A continuum approach to high velocity flow in a porous medium

Abstract

A general macroscopic linear momentum balance equation is derived in the form of a constitutive relation for high velocity fluid flow in a porous medium. It shows the nonlinearity in Forchheimer's formula for nonDarcy flow arising primarily from microscopic inertial phenomena, and expresses the inertial force in terms of the macroscopic velocity in an anisotropic and nonlinear manner. The point of departure is Euler's first law of motion, valid at any point in the fluid phase which is assumed to completely occupy the void space. The geometry of the void space, i.e., of the solid matrix, is taken as arbitrary. By introducing an alternative description of the microscopic kinematic field, namely deviations of the local velocity magnitudes and directions from the macroscopic values of these quantities separately, a general macroscopic momentum equation for fluid flow in a porous medium is obtained after averaging over a REV. From the general equation, most of the established relations for nonDarcy flow can be recovered as special cases. Explicit analytic expressions are obtained for the involved inertial coefficients from which the origin and nature of nonlinear (inertial) effects for high velocity flow in a porous medium is clearly demonstrated. It is also shown that the coefficient associated with the quadratic term for nonDarcy flow is not material and is a function of the macroscopic flow. Finally, some previous results are discussed and an extension of the derived equation to include higher-order nonlinear effects, with regard to the resistivity force, is proposed.