Page 64 - gas transport in porous media

Chapter 5: Two-Phase Gas Transport
which can be rearranged as
1 −1/m 1/n 57
P c = S e − 1
α
S e is the effective saturation, or
S − S ,r
S e =
1 − S ,r
where α is the capillary pressure parameter, m and n are fitting parameters, and S ,r
is the residual liquid saturation.
The relative permeability of the liquid phase can be predicted from the capillary
pressure curve expression using the integral expressions of Mualem (1976) or Burdine
(1953). The resulting liquid phase relative permeability integral can be expressed
as a closed form expression for certain relationships between the m and n fitting
parameters as detailed by van Genuchten (1978, 1980). For Mualem (1976), the
relationship m = 1 − 1/n leads to a closed form solution for the liquid-phase relative
permeability, which is given by
m 2

k r, = S 1/2 1 − 1 − S 1/m (m = 1 − 1/n)
e e
For Burdine (1953), the relationship m = 1−2/n leads to a closed form expression, or
m

k r, = S e 2 1 − 1 − S e 1/m (m = 1 − 2/n)
In both cases, m must be between 0 and 1.
With these restricted forms, the limits on the parameter n are n ≥ 1 for Mualem and
n ≥ 2 for Burdine. As discussed by van Genuchten (1978, 1980) and van Genuchten
and Nielsen (1985), both forms give similar results, but the Mualem form is preferred
because of it has more general applicability due to the larger range for n, and the
Mualem restricted form with m = 1 − 1/n is generally recommended. The values
of α, n (or m), and S ,r are determined from fitting experimental data for capillary
pressure or head. A general program for fitting the experimental capillary pressure
data is RETC (van Genuchten et al., 1991), which is freely available on the web.
An important point for the present discussion is that van Genuchten (1978,
1980) did not address gas-phase relative permeability. Parker et al. (1987) extended
the van Genuchten–Mualem characteristic curves to include the gas-phase relative
permeability, or

2m
k r,g = (1 − S e ) 1/2 1 − S 1/m
e
Similarly, Luckner et al. (1989) present a gas-phase relative permeability
expression for the van Genuchten–Mualem characteristic curves.

2m
k r,g = (1 − S e ) 1/3 1 − S 1/m
e