Page 27 - gas transport in porous media

Webb
20
2.5 TRACE GAS LIMIT
The above data-model comparison was for gas diffusion where the different gas mass
fractions are significant.As shown byWebb and Pruess (2003), in the limit of trace gas
diffusion in a binary mixture where one gas has a vanishingly small mass fraction, the
DGM and ADM reduce to similar equations. Two “correction” factors are needed to
bring the ADM in line with the DGM. The first correction factor is an additional
tortuosity term on the diffusion coefficient. The second correction factor is on the
Klinkenberg coefficient, b.
The correction factors can also be viewed as ratio of the mass flux predicted by the
DGM to that predicted by the ADM. As will be seen, the tortuosity correction factor
is always 1 or less, which indicates that ordinary diffusion is always overpredicted by
the ADM, in some cases by orders of magnitude. The magnitude of the Klinkenberg
correction factor is much smaller and may be less than or greater than 1.0 depending
on the molecular weight ratio of the trace gas to the bulk species.
The standardADM equation incorporating the Klinkenberg coefficient is as follows

k b
∗
F g =− 1 + ρ g (∇P g − ρ g g) − ρ g D ∇x
12
µ g P g
Introducing the correction factors gives

k l b DGM b
∗
F g =− 1 + ρ g (∇P g − ρ g g) − τ DGM ρ g D ∇x
12
µ g P g
The first term on the RHS is simply the convective flux including the Klinkenberg
correction factor. The second term on the RHS is ordinary diffusion with a tortuosity
correction factor. For trace gas diffusion, these factors are given by (see Webb and
Pruess, 2003)
1/2
1 + m rat D rat
b DGM =
1 + D rat
and
1
τ DGM =
1 + D rat
where D rat is the ratio of the effective ordinary diffusion coefficient to the Knudsen
diffusion coefficient, or
∗
D 12
D rat =
D 1K
and m rat is the ratio of molecule weights
m 2
m rat =
m 1
where gas 1 is the trace diffusing species and gas 2 is the bulk species.