ON THE ALGEBRAIC AND DIFFERENTIAL FORMS OF DARCY'S EQUATION

Darcy's law is a cornerstone of the quantitative analysis of flow in porous medium systems. A considerable amount of work subsequent to that performed by Darcy has extended the original algebraic form of Darcy's law to a variety of commonly used forms, including a form posed in terms of the differential of the macroscale pressure of the fluid. Results from both a megascopic and macroscopic analysis of single-fluid-phase flow are used to show that a standard form of Darcy's law posed in terms of a gradient of macroscale fluid pressures neglects a term that is related to the gradient in porosity and subscale pressure variations.We examine several approaches for proper inclusion of this term. The analysis shows that posing the macroscale Darcy equation in terms of gravitational and chemical potentials is the natural form and does not require the artificial redefinition of fundamental thermodynamic quantities. Finally, we provide a simple example of a physical system that illustrates the completed form of a Darcy equation expressed in standard pressure form and comment on the importance of this development for more complicated multiphase flow problems.