Page 15 - gas transport in porous media

Webb
8
smaller than the radius, and continuum flow occurs. As the capillary tubes get smaller
and smaller, the gas molecular mean free path becomes of the same order, and free-
molecule, or Knudsen, diffusion becomes important.
Low permeability effects were experimentally investigated by Knudsen in 1909
for gas flow in capillaries as discussed by Cunningham and Williams (1980, pg. 49).
Based on Darcy’s law, the mass flux for a given pressure drop should decrease as
the average pressure is reduced due to the change in gas density. However, Knudsen
found that at low pressures, the mass flux reaches a minimum value and then increases
with decreasing pressure, which is due to slip, or the fact that the fluid velocity at the
wall is not zero due to free-molecule flow.
Klinkenberg (1941) derived an expression for the effective gas permeability, k g ,
of a single gas in the Knudsen diffusion regime, which is a function of the liquid
permeability, k λ , the average pressure, P, and the Klinkenberg coefficient of gas
i, b i ,or

b i
k g,i = k 1 +
P g
For a large average pressure, the correction factor in parentheses goes to zero, and the
gas and liquid permeabilities tend to become equal.As the average pressure decreases,
the two permeabilities can deviate significantly from each other. This behavior is
confirmed by data presented by Klinkenberg (1941) for glass filters and core samples
and by Reda (1987) for tuff. The Klinkenberg parameter for a given porous medium
can be derived by plotting the effective gas permeability as a function of the inverse
of the average pressure. The slope of the line is related to the Klinkenberg parameter,
and the intercept at zero inverse average pressure is the liquid permeability.
The Klinkenberg coefficient, b i , is a function of the porous medium, the gas, and
the temperature. The Klinkenberg coefficient for air can be estimated from the Heid
◦
et al. (1950) correlation for air at 25 C as a function of permeability (Thorstenson
and Pollock, 1989a, Figure 3), or
b air = 0.11 k −0.39

2
where b air is in Pa and k is the liquid permeability in m . The data used in this
correlation are from oil-field cores with permeability values between about 10 −12
2
and 10 −17 m .
Another expression for the Klinkenberg coefficient is from Jones and Owens
(1980), who performed similar measurements for low-permeability gas sands
2
with permeabilities between 10 −14 and 10 −19 m . They developed the following
correlation for air (presumably at 25 C)
◦
−0.33
b air = 0.98 k

where the units are the same as for the Heid et al. (1950) correlation. Between
2
10 −14 and 10 −17 m , which is where the permeabilities for the data sets overlap, the
Klinkenberg factors from both correlations are quite similar.