Transport in Porous Media

, Volume 92, Issue 3, pp 649–665 | Cite as

A Phenomenological Approach to Modeling Transport in Porous Media

Article

Abstract

The objective of this article is to make use of the phenomenological approach to construct models for the transport of extensive quantities, such as mass of a fluid phase, mass of a component of a fluid phase, momentum of a phase and energy, in porous medium domains. Special attention is devoted to express the fluxes of these extensive quantities, especially the non-advective ones, as functions of their relevant driving forces, obeying the principle of minimum entropy production. It is shown that for each extensive quantity, we have a linear diffusive flux term, a non-linear diffusive term, and a dispersive flux term. The latter is shown to be proportional to the velocity squared. In each case, the number of moduli that describe fluid and porous matrix properties is determined. The momentum balance equation for a porous medium domain, which is the “motion equation,” is analyzed and simplified for special cases, leading to Darcy’s law and to Brinkman’s equation.

Keywords

Phenomenological approach Transport in porous media Modeling Fluxes Entropy production 

List of symbols

cγ

Concentration of γ-chemical species

\({\mathcal{D}}\)

Molecular diffusivity

\({{\bf \mathcal{D}}^\ast }\)

Molecular diffusivity in a pm

E

Extensive quantity (e.g., m, mγ, M, \({\mathcal{E}}\))

\({{\mathcal{E}}}\)

Energy

e

Intensive quantity of E

F

Force

\({f^{E}_{\alpha \to \beta}}\)

Rate of transfer of E from α to β, per unit volume of pm

f

Subscript for fluid

I

Internal energy per unit mass

jE

Microscopic flux of E (...per unit flu. area)

JE

Macroscopic flux of E (...per unit flu. area)

\({{\bf J}_{{\rm pm}}^{E}}\)

Macroscopic flux of E (...per unit pm area)

k

Permeability

M

Momentum (=ρV)

m

Mass

T

Temperature

t

Time

W

\({= {\bf \nabla} {\bf V} + ({\bf \nabla}{\bf V})^{T}}\)

VE

Velocity of E

V

Velocity(\({\equiv{\bf V}^{\rm m}}\))

z

Vertical coordinate (pos. upward)

α

Subscript for α-phase

τ

Shear stress

ΓE

Source of E per unit mass

γ

Superscript for γ-species

Δ

Hydraulic radius

θα

Volumetric fraction of α-phase

λ

Thermal conductivity

λpm

Thermal conductivity of pm

μ

Fluid viscosity

ρ

Density

σ

Stress

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Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Department of Civil and Environmental EngineeringTechnion-IITHaifaIsrael
  2. 2.Kinneret College on the Sea of GalileeZemachIsrael

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