On the derivation of the Forchheimer equation by means of the averaging theorem

The averaging theorem is applied to the microscopic momentum equation to obtain the macroscopic flow equation. By examining some very simple tube models of flow in porous media, it is demonstrated that the averaged microscopic inertial terms cannot lead to a meaningful representation of non-Darcian (Forchheimer) effects. These effects are shown to be due to microscopic inertial effects distorting the velocity and pressure fields, hence leading to changes in the area integrals that result from the averaging process. It is recommended that the non-Darcian flow regime be described by a Forchheimer number, not a Reynolds number, and that the Forchheimer coefficient be more closely examined as it may contain information on tortuosity.English a _i gravitational acceleration (m/s²) - A _fs interfacial area between the fluid and solid phases (m²) - Fo Forchheimer number - k permeability (m²) - k ₀ permeability at zero velocity (m²) - p thermodynamic pressure (Pa) - r _i coordinate on the microscopic scale (m) - Re Reynolds number - t time (s) - u _i ,u bulk velocity (m/s) - V volume (m³) - V _f fluid volume (m³) - w _i ,w microscopic velocity (m/s) - x _i ,x coordinate on the macroscopic scale (m) Greek the Forchheimer coefficient (1/m) - _ij extra (viscous) stress tensor (Pa) - _ij stress tensor (Pa) - Viscosity (Pa. s) - density (kg/m³) - porosity - a general variable Symbols < > phase average - < > ^f intrinsic phase average - the fluctuating part of a variable