Page 72 - gas transport in porous media

Chapter 5: Two-Phase Gas Transport
the partial saturation tortuosity coefficient, τ g , which is consistent with the arguments
above and reduces to the proper expression for all-gas conditions. 65
In more recent work, Reinecke and Sleep (2002) have developed expressions for
the Knudsen diffusion coefficient of air using the above Knudsen diffusion expres-
sion discussed in Section 5.1.5. Correction factors for the Klinkenberg coefficient for
different gases and different conditions are discussed in Section 2.1.4. The gas per-
meability in the above equation is simply the effective value, or the gas-only value
times the relative permeability. Any tortuosity factor is implicitly included in the
Klinkenberg parameter.
Obviously this expression for the Knudsen diffusion coefficient is preferred to
those proposed by Sleep (1998) and by Webb (2001), which were only estimates of
the effect of unsaturated conditions.

5.3 COMBINED EFFECTS

Section 2.3 discusses combined effects for gas-only conditions. A number of param-
eters in the Dusty Gas Model need to be modified for application to unsaturated
conditions as discussed above. The relative permeability should be used for the
advection part of the transport, while tortuosity factors and a modified Klinkenberg
parameter should be used for the ordinary (molecular) and free-molecule (Knudsen)
diffusion components, respectively.

5.4 DISSOLVED GAS
In addition to the added complexity of unsaturated conditions on gas transport in the
gas phase, the addition of a liquid phase adds another pathway for advection and
diffusion of the gas. Gas will not only be transported in the gas phase; gas will also be
transported as a dissolved species in the liquid phase. Due to the low concentration
of dissolved gas in the liquid as discussed by Bird et al. (1960, page 538), Fick’s
law is adequate for dissolved gas diffusion. Note that advection of liquid will also
transport dissolved gas. A simple addition of the two fluxes, similar to the ADM
discussed in by Webb (Chapter 2 of this book), is probably adequate for dissolved
gas flux in the liquid phase. In general, diffusion coefficients for liquids are about
4
a factor of 10 lower than for gases at low pressure (Reid et al., 1987, Tables 11.2
and 11.5). However, the diffusion flux of gas in the liquid phase can be significant
because of the concentration gradient in liquid phase, which can be larger than in the
gas phase. The ratio of the gas concentration in the liquid and gas phases is given by
the dimensionless Henry’s constant as discussed by Ho (Chapter 3 of this book), or

c g
K H =
c
where c g is the concentration in the gas phase, and c is the concentration in the
liquid phase.Alow value of the dimensionless Henry’s constant indicates a significant
concentration of dissolved gas in the liquid phase compared to the gas phase. Values