Flows Through Porous Media: A Theoretical Development at Macroscale

Abstract

Good separation of microscale with macroscale leads to the existence of a macroscale description of flows through porous media. Such a macroscale description is developed in a systematic and rigorous way through exploiting necessary and sufficient condition for three fundamental principles regarding physical relations: principle of frame-indifference, principle of observer transformation and second law of thermodynamics. This leads to a generalized Darcy's law, an algebraic ∇p−v−L relation at macroscale with effects of G and M reflected in three material coefficients. Here ∇p is piezometric pressure gradient. G denotes macroscale geometric properties of the medium. M stands for thermophysical (material) properties of the medium and fluids. v is the fluid velocity vector relative to the solid. L is the velocity gradient tensor of the fluid velocity u. Such a generalized relation can be used for both low and high flow rates. Also developed in the present work is a linear theory to simplify the works of determining effects of G and M.

It is found that ∇p cannot depend on fluid velocity u itself. L affects ∇p only through its symmetric part (velocity strain tensor D). The symmetry and positive-definiteness of H, the inverse of permeability tensor, follow logically from the three fundamental principles. Eigenvectors of H are the same as those of D with corresponding eigenvalues related to those of D through a quadratic relation. Six scalars formed by v and D (rather than the Reynolds number) are found to be scalars characterizing convective inertia effects. The incompressibility is found to be responsible for the vanishing of the first correction term to the classical Darcy's law as the Reynolds number tends to zero. The vanishing of D forms the applicability condition of classical Darcy's law. This requires u to be vanished, uniform, or in rigid body rotation.