Flow Mechanism of Forchheimer's Cubic Equation in High-Velocity Radial Gas Flow Through Porous Media

Abstract

ABSTRACT Until recently, the visco-inertial flow equation, which is an adaptation of Forchheimer's quadratic equation, has been used to describe gas flow behavior at higher flow rates and pressures. The inability of this equation, in some cases, to fully describe high-velocity, high-pressure gas flow behavior, especially around the wellbore, led to the consideration of other empirical equations. In this paper, formal derivation of Forchheimer's cubic equation is made by considering the kinetic energy equation of mean flow and dimensional relations for one-dimensional, linear, incompressible fluid flow. By the addition of the cubic term, this equation is regarded as a modified Forchheimer's quadratic equation which accounts for the flow rates obtained beyond the laminar flow condition. The cubic equation spans a wide range of flow rates and regimes, i.e. Darcy type, inertial type, and turbulent. For suitable use in gas flow studies, this equation has been adapted, modified, and corrected for the gas slippage effect. The physical basis of the cubic term has been established by using boundary layer theory to explain the high-velocity, high-pressure flow behavior through a porous path. Gamma, the main parameter in the cubic term, is directly related to a characteristic, dimensionless shape factor which is significant at higher flow rates. It is inversely related to viscosity, but has no dependence on the gas slippage coefficient in the higher flow regime.