Page 105 - gas transport in porous media

98
equations simplifies to
γ 2 γ Whitaker
0 =− ∇ p γ + ρ γ g + µ γ ∇ v γ (6.146)
⎧ ⎫
⎪ ⎪
1 γ
⎨ ⎬
+ µ γ n γκ · (− Im +∇M) dA · v γ
⎪V γ ⎪
⎩ ⎭
A γκ
Rather than attack the general closure problem for m and M, it is convenient to decom-
pose the problem into two parts. The first part will produce the Darcy’s law (Darcy,
1856) permeability tensor that depends only on the geometry of the porous medium
under consideration, and the second part will lead to an inertial correction, i.e., the
Forchheimer equation (Forchheimer, 1901). To accomplish this decomposition, we
represent m and M as
m = b + c, M = B + C (6.147)

The details of the closure problems for b and B and for c and C are given elsewhere
(Whitaker, 1996), and here we only note that the use of Eq. (6.147) leads to
γ 2 γ
0 =− ∇ p γ + ρ γ g + µ γ ∇ v γ (6.148)
⎧ ⎫
⎪ ⎪
1 γ
⎨ ⎬
+ µ γ n γκ · (− Ib +∇B) dA · v γ
⎪V γ ⎪
⎩ ⎭
A γκ
⎧ ⎫
⎪ ⎪
1 γ
⎨ ⎬
+ µ γ n γκ · (− Ic +∇C) dA · v γ
⎪V γ ⎪
⎩ ⎭
A γκ
in which the Darcy’s law permeability tensor is defined by

1
n γκ · [− Ib +∇B] dA =− ε γ K −1 (6.149)
V γ
A γκ
and the Forchheimer correction tensor is defined by
1
n γκ · [− Ic +∇C] dA =− ε γ K −1 · F (6.150)
V γ
A γκ
Here we note that the definitions of K and F have been deliberately chosen to produce
a momentum equation containing the superficial average velocity rather than the
intrinsic average velocity that appears in Eq. (6.148). Use of these two definitions in
Eq. (6.148) leads to a result
γ
γ
0 =− ∇ p γ + ρ γ g + µ γ ∇ v γ − µ γ K −1 · v γ − µ γ K −1 · F · v γ (6.151)
2