Page 71 - gas transport in porous media

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5.2
Ordinary Diffusion
5.2.1 DIFFUSION Webb
As discussed in Section 2.2.1.1, Fick’s law is often modified for application to porous
media by the introduction of a porous media factor, β, which modifies the diffusion
coefficient.

D ∗
AB = β D AB
The term β is defined as

β = φ S g τ

where D ∗ is the effective diffusion coefficient for the AB gas system in a porous
AB,pm
media, D AB is the effective diffusion coefficient of theAB gas system in a clear fluid,
φ is the porosity, S g is the gas saturation, and τ is the tortuosity. The Millington and
Quirk (1961) correlation discussed in Chapter 2 is

τ = φ 1/3 7/3
S
g
which can be rewritten as
1/3 7/3
τ = τ o τ g = φ S
g
where τ o is the tortuosity due to the porosity and τ g is the tortuosity due to the partial
saturation of the porous medium. In Chapter 2, the gas saturation (S g ) was equal to
1.0. In this chapter, the gas saturation is generally not equal to 1.0 due to unsaturated
7/3
conditions, and the τ g factor (S g ) is retained.

5.2.2 Free-Molecule diffusion
Section 2.2.2 discusses the Knudsen diffusion coefficient, D iK , for gas-only
conditions. The gas-only equation based on the Klinkenberg coefficient is

k g b i
D iK =
µ g
For unsaturated conditions, the equation for the Knudsen diffusion coefficient
needs to be modified due to the presence of liquid. Sleep (1998) used the Millington
and Quirk (1961) tortuosity factor to evaluate the effect of unsaturated conditions,
although the author notes that the approach might not be appropriate because molec-
ular diffusion and Knudsen diffusion are two different processes. The porous media
porosity, τ o , is included twice with this approach, once in the tortuosity factor, τ, and
once in the Klinkenberg factor as discussed in Chapter 2, and the resulting equation
does not reduce to the proper form for all-gas conditions. Another approach for unsat-
urated flow was suggested by Webb (2001), who multiplied the Knudsen diffusion by