Derivation of basic equations of mass transport in porous media, Part 2. Generalized Darcy's and Fick's laws

Abstract

This is the second part of a series of two papers on the development of a general thermodynamic basis for the study of transport phenomena in porous media. The porous medium is modelled as a superposition of one solid continuum coexiting and interacting with an N-component fluid-phase continuum. Macroscopic balance laws derived in Part 1 provide the equations of mass and momentum for the mean motion of the fluid phase and diffusive motions of individual components. The Coleman and Noll's method of exploitation of the entropy inequality is applied to restrict a rather general set of constitutive equations. Gradients of the fluid-phase density and concentrations of N-1 components are included among independent variables to account for buoyancy and cross-coupling effects properly. Extensions of certain classical relations for fluid-phase pressure, solid stress tensor, and components' chemical potentials are obtained as results of the constitutive theory. Further simplifications and linearizations of constitutive equations and balance laws yield a general extension of Darcy's and Fick's laws, applicable to cases where the fluid has more than one main component. It is found out that both relations have to be modified to account for the effect of high concentrations. It is shown that classical forms of those laws are valid only if fluid components exists at low concentrations. All assumptions are carefully and explicitly stated during the course of development. As an illustration of the theory, proper forms of Darcy's and Fick's laws for the flow and transport of concentrated brine in porous media are given. The development here also provides a fundamental basis to equations used in the description of chemico-osmosis effects.