Page 96 - gas transport in porous media

Chapter 6: Conservation Equations
in which the column matrix on the right hand side of this result can be expressed as
⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 89
D Am ∇x A D Am 0 0 ...... 0 0 ∇x A
⎢ ⎥ ⎢ 0 0 ...... 0 0 ⎥
D Bm ∇x B D Bm ⎢ ∇x B ⎥
⎢ ⎥ ⎢ ⎥
⎢ ⎥
⎢ ⎥ ⎢ 0 0 ...... 0 0 ⎥ ⎢ ⎥
⎢ D Cm ∇x C ⎥ ⎢ D Cm ⎥ ∇x C
⎢ ⎥
⎢ ⎥ = ⎢ ⎥
... . . . ...... . . ⎢ .. ⎥
⎢ ⎥ ⎢ ⎥
⎢ ⎥
⎢ ⎥ ⎢ ⎥
0 0 0 ...... 0 ⎣ ⎦
⎣ D N−1m ∇x N−1 ⎦ ⎣ D N−1m ⎦ ∇x N−1
0 0 0 0 ...... 0 D Nm 0
(6.107)
The diffusivity matrix is now defined by
⎡ ⎤
D Am 0 0 ...... 0 0
⎢ 0 D Bm 0 ...... 0 0 ⎥
⎢ ⎥
⎢ 0 0 D Cm ...... 0 0 ⎥
[D]=[R] −1 ⎢ ⎥ (6.108)
⎢ . . . ...... . . ⎥
⎢ ⎥
⎣ 0 0 0 ...... D N−1m 0 ⎦
0 0 0 ...... 0 D Nm
so that Eq. (6.106) takes the form
⎡ ⎤ ⎡ ⎤
J A ∇x A
J B ∇x B
⎢ ⎥ ⎢ ⎥
⎢ ⎥ ⎢ ⎥
J C ∇x C
⎢ ⎥ ⎢ ⎥
⎢ ⎥ =−c [D] ⎢ ⎥ (6.109)
⎢ ... ⎥ ⎢ ... ⎥
⎢ ⎥ ⎢ ⎥
⎣ ... ⎦ ⎣ ∇x N−1 ⎦
J N 0
This result can be expressed in a form analogous to that given by Eq. (6.102) leading to
E=N−1

J A =−c D AE ∇x E , A = 1, 2, ... , N (6.110)
E=1
In the general case, the elements of the diffusivity matrix, D AE , will depend on the
mole fractions in a non-trivial manner. When this result is used in Eq. (6.101), we
obtain the non-linear, coupled governing differential equation for given by

E=N−1
∂c A
+∇ · (c A v) =∇ · c D AE ∇x E + R A , A = 1, 2, ... , N (6.111)
∂t
E=1
Use of the transport equations presented in this section to describe transport phenom-
ena in porous media requires that we upscale the equations from the so-called point
scale to an appropriate larger scale (Cushman, 1990, 1997). This analysis is described
in the following section.