Generalized Darcy's Law by Control Volume Analysis Including Capillary and Orifice Effects
Authors: Civan F. 
Published: March 2010

Abstract
The equation of motion for single-phase fluid flow through porous media is derived by rigorous analysis of forces acting upon fluid based on a control volume momentum balance approach. Resistive forces associated with pore surface and pore throat are characterized by the capillary-orifice model of porous media. This analysis reveals deficiencies of previous efforts in deriving the generalized momentum equation. The present approach provides resolution to several issues, including porous media averaging of the pressure and shear-stress terms, effect of porous media heterogeneity and anisotropy, and threshold pressure gradient that must be overcome for fluid to flow through porous media.

Introduction
Derivation of generalized equation of momentum for fluids flowing through porous media has occupied many researchers and is not by any means a trivial task. Since its inception, the fundamental law of flow through porous media given by Darcy(1) has been scrutinized greatly and been the subject for many modifications to take into account the effect of other factors which are not considered by this law. In spite of an exhausting amount of effort, and prolonged occupation and obsession with derivation of an improved formulation, a satisfactory generalized equation of fluid motion in porous media still does not exist. In an effort to shed some light into the adequate treatment of the control volume balance approach, this paper attempts to revisit, modify and improve the generalized formulation presented by Das(2).
Many researchers attempted to improve Darcy's law by various approaches to take into account other conditions of practical importance. The outstanding approaches have been developed primarily on the bases of dimensional analysis(3), control volume balance(1, 2) and porous media averaging(4–12). There are some discrepancies, however, between the results presented by various researchers. As pointed out by Civan(13), porous media averaging does not only require extremely complicated and intricate algebraic manipulation procedures, but also has been implemented inadequately in many studies. Consequently, debate about proper handling of various issues dealing with porous media averaging is still continuing, judging by the high number of papers being published. The results presented by various studies do not agree whether the porous media averaged pressure gradient term should include the porosity. The same concern can be raised about shear-stress. Liu and Masliyah(12) emphasize that averaging of pressure and viscosity terms in the microscopic equation of motion should be treated differently than the other terms. Also, the mathematical definition of partial derivatives assumes a limit as control volume approaches zero, i.e., collapses to a point of zero volume. By and large, discussion of differences introduced by the application of such a condition in deriving porous media equations of mass, momentum and energy conservation have been omitted, whereas, representative elementary bulk volume is a finite quantity in definition of volume averaged quantities used for deriving the porous media (macroscopic) equation of motion.