TY - JOUR

T1 - Flows through porous media: A theoretical development at macroscale

AU - Wang, Liqiu

N1 - Funding Information:
The author is indebted to five anonymous referees for their critical reviews and constructive comments/suggestions on the original manuscript. This work was sponsored by the CRCG, the University of Hong Kong.

PY - 2000

Y1 - 2000

N2 - 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 trans-formation and second law of thermodynamics. This leads to a generalized Darcy's law, an algebraic nublap - v - L relation at macroscale with effects of G and M reflected in three material coefficients. Here nublap 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 nublap cannot depend on fluid velocity u itself. L affects nublap 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.

AB - 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 trans-formation and second law of thermodynamics. This leads to a generalized Darcy's law, an algebraic nublap - v - L relation at macroscale with effects of G and M reflected in three material coefficients. Here nublap 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 nublap cannot depend on fluid velocity u itself. L affects nublap 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.

KW - Convective inertia

KW - First principles

KW - Generalized Darcy's law

UR - http://www.scopus.com/inward/record.url?scp=0034003899&partnerID=8YFLogxK

U2 - 10.1023/A:1006647505709

DO - 10.1023/A:1006647505709

M3 - Journal article

AN - SCOPUS:0034003899

SN - 0169-3913

VL - 39

SP - 1

EP - 24

JO - Transport in Porous Media

JF - Transport in Porous Media

IS - 1

ER -