Page 68 - gas transport in porous media

Chapter 5: Two-Phase Gas Transport
Gas relative permeability

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k r,g = (1 − S e ) 2 1 − S e (2+λ)/λ
van Genuchten (1976, 1980) presents Mualem-based expressions for the Brooks
and Corey liquid-phase relative permeability. A gas-phase expression could be sim-
ilarly derived. However, because these expressions have not been compared with
experimental data, they will not be presented here.
Figure 5.3 shows some generic Brooks and Corey two-phase characteristic curves.
Note that the Brooks and Corey capillary pressure does not go to zero as the liquid
saturation goes to 1.0. This behavior is expected because the straight-line fit does not
fit the data in this region (P c < P b ). In contrast, the van Genuchten capillary pressure
curve in Figure 5.1 does go to zero as the liquid saturation goes to 1.0. This behavior
at high liquid saturations is one reason why the van Genuchten characteristic curves
are much more popular than the Brooks and Corey formulation. The behavior of the
Brooks and Corey capillary pressure curve is similar to van Genuchten as the satura-
tion approaches the residual value. Note that Webb (2000) only discussed extension
of the van Genuchten capillary pressure curve, although the same procedure could
be applied to Brooks and Corey. The liquid-phase relative permeability increases
significantly as liquid saturation is increased, similar to van Genuchten–Mualem
shown earlier. The gas-phase relative permeability decreases with increasing liquid
saturation and is concave up. The shape of the curve is opposite of the van Genuchten–
Mualem curves (Parker et al., 1987; Luckner et al, 1989 extensions) shown earlier.
Note that Brooks and Corey (1964, 1966) present data-model comparisons for the
capillary pressure as well as the wetting (liquid) and non-wetting (gas) relative perme-
abilities, which show good results for all relationships including non-wetting relative
permeability. The two van Genuchten–Mualem relationships have not been compared
to data.
The range of the Brooks and Corey parameters for the nine soils investigated by
Brooks and Corey (1966) are P b from 14 to 75 cm (1400 to 7500 Pa), λ from 1.8 to
7.3, and S ,r from 0.085 to 0.577.
The Brooks and Corey characteristic curves can be shown to be a limiting case of
van Genuchten (van Genuchten (1978, 1980). For large values of capillary pressure
n
((αP c ) 1), the following relationships can be derived
1 n
P b = (αP c ) 1
α
and

n
λ = mn (αP c ) 1
Using the Mualem relationship, λ = m/(1−m); for Burdine, λ = 2m/(1−m). These
relationships were used for the Brooks and Corey capillary pressure curves presented
in Figure 5.3 based on the van Genuchten parameters in Figure 5.1.