Page 63 - gas transport in porous media

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permeability, k r , is the ratio of the permeability of the porous media to that particular
phase under unsaturated conditions divided by the permeability of the porous medium
for saturated conditions for that particular phase. Under all-gas conditions, the gas-
phase relative permeability is equal to 1.0. The liquid mass flux equation is similar
to the gas flux equation with liquid-phase parameters.
There are different pressures for the gas and liquid phases due to capillary forces
and interfacial curvature (see Ho (Chapter 3 of this book)). The difference between
the gas (nonwetting) and liquid (wetting) pressures is referred to as the capillary
pressure, or

P c = P g − P

For porous media where liquid is the wetting phase, the liquid phase pressure is
less than the gas phase pressure, and the capillary pressure is a positive quantity.
(Note that the liquid phase pressure can be greater than the gas phase as is the case
for droplets as discussed in Ho (Chapter 3 of this book)).
In order to close the equation set, relationships for the capillary pressure and rela-
tive permeability are needed. These relationships are often referred to as two-phase
characteristic curves. In general, the two-phase characteristic curves are a function
of the pore structure, phase saturation, surface tension, contact angle, and hysteresis,
(Dullien, 1992, pg. 338; Kaviany, 1995, pg. 428, 435). In addition, the phase relative
permeabilities may be coupled, which leads to a revised form of the two-phase Darcy
equations (Dullien, 1992, pg. 358). In practice, the two-phase characteristic curves
are usually posed as a function of the liquid saturation, S , in the porous medium, or
the fraction of pore space occupied by liquid (deMarsily, 1986).
There are numerous sets of two-phase characteristic curves. Relationships for the
capillary pressure as a function of saturation are usually based on experimental data.
Expressions for relative permeability are then derived using theoretical relationships
derived by Burdine (1953) or Mualem (1976) based on the capillary pressure expres-
sion. Two commonly used sets of characteristic curves are van Genuchten (1978,
1980) and Brooks and Corey (1964) as discussed below.

5.1.2 Two-Phase Characteristic Curves

5.1.2.1 van Genuchten
The van Genuchten curves (van Genuchten, 1978, 1980) are the most widely
used set of two-phase characteristic curves. The capillary pressure relationship is
given by

m
1
S e = n
1 + (αP c )