Page 104 - gas transport in porous media

Chapter 6: Conservation Equations
problem is described in detail by Whitaker (1996). The analysis leads to the mappings
given by 97
γ
˜ v γ = M · v γ (6.144a)
γ
˜ p γ = µ γ m · v γ (6.144b)

The mapping tensor M and the mapping vector m are determined by a closure problem
that must be solved in some representative region which is necessarily a spatially
periodic model of a porous medium (Quintard and Whitaker, 1994a–e). Such a model
is illustrated in Figure 6.4. Use of Eqs. (6.144) in Eq. (6.142) and imposition of the
length scale constraints given by Eq. (6.117) leads to

γ
∂ v γ γ γ −1
γ T γ
ρ γ + ρ γ v γ ∇· v γ + ρ γ ε γ ∇· v γ · M M · v γ (6.145)
∂t
γ 2 γ
=− ∇ p γ + ρ γ g + µ γ ∇ v γ
⎧ ⎫
⎪ ⎪
1 γ
⎨ ⎬
+ µ γ n γκ · (− Im +∇M) dA · v γ
⎪V γ ⎪
⎩ ⎭
A γκ
It is shown elsewhere (Whitaker, 1996) that the solution for the closure variables is
analogous to the solution of the Navier-Stokes equations, thus the mapping variables
M and m can be determined by classical means. In general, the terms on the left hand
side of Eq. (6.145) are negligible and the volume averaged form of the Navier-Stokes

L

Representive
region
Figure 6.4. Spatially periodic model of a porous medium