Page 23 - gas transport in porous media

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Knudsen diffusion as experimentally discovered by Graham (1846), or
N 1 K m 2 1/2 Webb
− K =
N m 1
2
For ordinary diffusion (no advection), Graham’s law of diffusion applies as
experimentally discovered by Graham (1833) or

N D m 2 1/2
− 1 D =
N 2 m 1

Even though the ratios are identical, each equation applies to a different diffusion
regime. In terms of a mass flux ratio, these equations become
F K m 1 1/2
1
− K =
F m 2
2
F D m 1 1/2
− 1 D =
F 2 m 2
The above relationships are significant. Graham’s laws were “lost” and were inde-
pendently rediscovered in the 1950s and 1960s (Mason and Malinauskas (1983, pg.
3)). Graham’s laws are fundamental relationships for gas diffusion in porous media
and were used in the development of the Dusty Gas Model. These relationships will
be used in the data-model comparisons that follow.
The data used in these comparisons were obtained by Evans, Watson, and Truitt
(1962b, 1963). They performed these experiments to support development of the
Dusty Gas Model. The experiments consisted of flow and diffusion of helium and
2
argon across a low-porosity (0.11) and low-permeability (2.13×10 −18 m ) graphite.
At this low permeability, Knudsen diffusion plays a significant role.
In addition to the data and model predictions, Mason and Malinauskas (1983,
pg. 91) provide closed-form solutions for the DGM for special cases, which are also
plotted as a continuous line labeled DGM in Figure 2.1 and Figure 2.2. In the data-
model comparisons, the DGM is compared to the data as is a model labeled ADM.
The ADM Model, which stands for advective-diffusive (or dispersive) model, is sim-
ply a linear addition of Darcy’s law and ordinary diffusion using Fick’s law, which is
incorrectly applied to stationary coordinates as discussed above. This simple linear
addition is commonly used as exemplified by TOUGH2 (Pruess, 1991) and Abriola
and Pinder (1985). Slip effects, or Knudsen diffusion, are included through a Klinken-
berg parameter to define an effective permeability for the advective flux as discussed
earlier. Porous medium effects for ordinary diffusion are included through a porosity-
tortuosity-gas saturation factor applied to the diffusive flux in free space. This simple
additive approach, while intuitively appealing, ignores coupling between advective
and diffusive mechanisms. The expression for the ADM used in this comparison is