Page 70 - gas transport in porous media

Chapter 5: Two-Phase Gas Transport
63
While the van Genuchten and Brooks and Corey two-phase characteristic curves
maylooksimilar, thereareimportantdifferenceswhentheyareusedinflowsituations.
McWhorter and Sunada (1990) and Webb (1992) evaluated the differences between
the two sets of characteristic curves for analytical two-phase flow situations. The
differencesinthesaturationprofilesaresignificant, probablyduetothedifferentshape
of the gas-phase relative permeability expressions. In general, the van Genuchten (and
Parker) set of curves lead to a much sharper interface, while the Brooks and Corey
predictions are much more blunt.

5.1.3 Brinkman Extension
The Brinkman extension for two-phase (unsaturated) flow has been analyzed by
Whitaker (1994). The Brinkman correction terms are negligible for practical two-
phase flow problems in porous media and are generally ignored.

5.1.4 Forchheimer Extension

The Forchheimer correction would be important for two-phase flow when the
Reynolds number is sufficiently large compared to one. However, a formal anal-
ysis of the Forchheimer extension for two-phase (unsaturated) flow has not been
performed.

5.1.5 Low Permeability Effects
Section 2.1.5 presented the low permeability effects for all-gas conditions. For unsat-
urated flow in porous media, Reinecke and Sleep (2002) investigated the variation
of the Klinkenberg coefficient for air, b air , including summarizing previous investi-
gations. They developed a correlation for the Knudsen diffusion coefficient and the
Klinkenberg parameter for unsaturated flow conditions as a function of effective gas
permeability, which is simply the gas-saturated permeability times the gas relative
permeability. Their Klinkenberg parameter correlation for air is
b air = 5.57 k −24
g,e
2
where b air is in Pa and k g,e is in m . Note that the units for the above equation for k g,e
2
were specified as cm in the original reference (equation 30). The correct units for
2
k g,e are m (Sleep, personal communication, 2005) as given above.
2
At an effective gas permeability of about 10 −11 m , the above correlation is about
the same as Heid et al. (1950). As the effective permeability decreases, the above
correlation gives a smaller value of the Klinkenberg factor, or a smaller effect on the
permeability, than either the Heid et al. (1950) or Jones and Owens (1980) correla-
tions presented in Chapter 2. This difference may be due to the fact that liquid will
preferentially block smaller pores, which have the most influence on gas slippage.