Page 354 - gas transport in porous media

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356
water table, gas flows in the vadose zone are particularly sensitive to variations in
the porous media structure due to multiphase flow effects. As discussed in Chapter 2,
Darcy’s law for gas flow in the vadose zone includes a relative permeability term k rg
to account for pore blockage by liquid water:
kk rg
v g =− (∇p g + ρ g g∇z) (22.1)
µ g
and the relative permeability term is typically a nonlinear function of the water
saturation, such as (Fatt and Klickoff, 1959)
∗ 3
k rg = (1 − S ) (22.2)
where S = (S l − S lr )/(1 − S lr ) and S lr is the residual water saturation. Thus, the
∗
water content of the soil can have a large effect on the effective gas velocity, and
variations in the water content can have as much effect on the gas flow as variations
in the intrinsic permeability. Using the idea of Leverett scaling of capillary pressure
(Leverett, 1941), it is expected that capillary pressure should vary with the inverse
square root of the intrinsic permeability:

1
p c lg ∝ √ (22.3)
k
so layers or regions with low permeability also tend to have high capillary pres-
sure at a given water saturation. Therefore, under gravity-capillary equilibrium
conditions above the water table, low permeability zones tend to have high water
saturations, leading to very low gas phase relative permeabilities. This compound
effect on the effective gas phase permeability dominates the gas flow in heterogeneous
systems.
Consider, for example, the simplified layered vadose zone shown in Figure 22.3.
This relatively homogeneous formation consists of three different sands, with corre-
sponding capillary pressure curves, and an intrinsic permeability variation of a factor
of 30. By neglecting infiltration and assuming gravity-capillary equilibrium the indi-
vidual media capillary pressure curves can be used to construct the profile of water
saturation above the water table. This is shown as the dark line in the right hand part
of the figure. Due to it’s proximity to the water table, the bottom medium sand layer
has a very high water saturation, about 0.98 or so. Using Eq. (22.2) this would lead
to a gas phase relative permeability of only about 0.00001, so this unit would be
virtually impermeable to gas flow unless the water table dropped. The coarse sand
above the medium sand has a lower water content, roughly 0.37, giving a relative
permeability of about 0.4. Therefore, this coarse sand layer would have an effective
gas phase permeability of 4. The fine sand layer just above has a high water saturation
of about 0.75, and it’s gas phase relative permeability would only be about 0.025,
for an effective permeability of only 0.0075. Comparing these last two layers, the
contrast in intrinsic permeability is a factor of 30, while the contrast in effective gas
permeability is a factor of more than 500. The coarse sand layer just above the fine