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The only difference in the two expressions is the exponent on the first expression;
this exponent is referred to as the pore connectivity parameter. Webb
The pore connectivity parameter value was originally proposed by Mualem (1976),
who determined that a value of 1/2 minimized the data-model comparison error for
the liquid-phase relative permeability. The value of 1/2 has recently been questioned
by Schaap and Leij (2000), who found a better fit to the data using a value of −1.0
and a reduced value of the reference permeability. In any event, the pore connectivity
value is a fitting parameter that has only been determined for the liquid phase, not
the gas phase.
For gas-phase relative permeability, Parker et al. (1987) simply used the pore
connectivity parameter based on the Mualem (1976) investigation. The value of
1/3 proposed by Luckner et al. (1989) was not discussed in the reference. Nei-
ther extension compared the proposed gas-phase relative permeability expressions
to experimental data.
Figure 5.1 shows some generic van Genuchten–Mualem two-phase characteristic
curves for some typical parameters as noted on the figures. The van Genuchten
capillary pressure curve exhibits unphysical behavior as the liquid saturation is
reduced (gas saturation is increased). As the liquid residual value (0.2 in this case) is
approached, the value of capillary pressure goes to infinity; this behavior is discussed
in more detail below. The liquid relative permeability starts out low and increases
dramatically with increasing liquid saturation. The gas-phase relative permeability
decreases with increasing liquid saturation and is concave down. The difference
between the Parker et al. (1987) and Luckner et al. (1989) expressions is small for
these parameter values.
Typical values of the van Genuchten–Mualem parameters for 34 soils are tabulated
by Stephens (1996, pg. 186, Table 8), although the third column should be cm −1
instead of m −1 as can be verified by checking the original tabulations in Stephens
et al. (1987) and van Genuchten (1978, 1980). The parameter ranges are α from 0.004
to 0.12 cm −1 (1/α from 817 to 24,500 Pa), n from 1.17 to 7.62 (m from 0.15 to 0.87),
and S ,r from 0 to 0.4/φ (Stephens et al., 1987, used a water content form of the
effective saturation equation).
In order to alleviate the problem of an infinite capillary pressure as the
liquid residual saturation is approached, or as one approaches all-gas condi-
tions, a number of modifications to the van Genuchten capillary pressure curve
have been proposed that extend the curve to zero liquid saturation. Webb
(2000) presents a simplified approach to extending the van Genuchten capillary
pressure relationship; he also reviews other approaches. The approach simply
extrapolates the van Genuchten curve with a logarithmic extension (linear on
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a semi-log plot) to a capillary pressure value of 10 Pa at zero liquid satura-
tion imposing continuity of the capillary pressure derivative. As an example,
Figure 5.2 shows the result for Palouse soil (Webb, 2000). The approach is
simple to use and fits the limited existing low liquid saturation (high gas satu-
ration) capillary pressure data of Campbell and Shiozawa (1992) for a number
of soils.