Development of a Generalized Darcy Equation

Abstract

Abstract General equations relating the pressure drop necessary to sustain the flow of a fluid through a porous matrix at a given rate have been developed. The results indicate that at high values of flow rate the pressure-flow behavior may not necessarily satisfy the usual Darcy equation. The mathematical analysis, carried through the micro-pore geometry and extended through the macro-reservoir scale, indicate that Darcy's law, of limited applicability to certain ranges of Reynolds numbers, can be generalized through the inclusion of some additional parameters. The "generalized Darcy equation" has also been formulated in dimensionless form permitting the evaluation of its predictive accuracy with regard to literature data. A comparison between predicted and experimental values indicates that the generalized Darcy equation predicts the pressure drops with good agreement over all possible ranges of Reynolds numbers. Introduction The limits and the nature of validity of Darcy's law has been a subject of every-day interest to the industry for many years. It is well known that as the Reynolds number, characteristic of the fluid flow through porous media, becomes large, Darcy's law gradually loses its predictive accuracy and ultimately becomes completely void. For the last 20 years much has been said and written on this subject. Unfortunately little has been accomplished to bring about a satisfactory agreement, at least on the nature of the threshold of validity of Darcy's law. Fluid dynamists, geo-physicists, and engineers all had their individual views, explanations, interpretations and concepts on the subject. To some, a mechanistic analogy with pipe-flow proved a satisfactory explanation. To others, turbulence, in its random character, was incompatible with the geometric structure of consolidated porous systems. To some, turbulence merely represented a factor influencing the permeability measurements and again to others, empirical or semi-empirical correlations proved satisfactory from an engineering viewpoint. Deviations from Darcy's law at high flow rates have been studied by systematic experiments by Fancher, Lewis, and Barnes. In an article on the flow of gases through porous metals, Green and Duwez conclude that the onset of turbulence within the pores appears unsatisfactory to explain deviations from Darcy's law. This view is held by many others.