Page 218 - gas transport in porous media

Pruess
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12.2.1 Advective-Diffusive Model
We begin with the formulation of an “advective-diffusive” model (ADM), in which
advective and diffusive contributions to gas transport are evaluated separately, using
phenomenological flux laws, and then added.
AdvectivegasfluxiscommonlydescribedwithamultiphaseversionofDarcy’slaw,

k rg ρ g
F g = ρ g u g =− k (∇P g − ρ g g) (12.1)
µ g
where u g is the volume flux, often referred to as “Darcy velocity.” Whereas for liquids
absolute permeability k is a material constant, this parameter becomes dependent on
pressure for gases, according to the Klinkenberg (1941) relationship

b
k = k ∞ 1 + (12.2)
P g

The physical effect that is causing an apparent increase in permeability for low
gas pressures is “slip flow:” at low pressures, the mean free path of gas molecules
is no longer negligibly small in comparison to pore sizes, so that the assumption of
advective velocity at the pore walls approaching zero is no longer valid.
Eq. (12.1) gives the total flux of gas phase. The flux for an individual mass com-
ponent κ is obtained by multiplying with the mass fraction of that component in
κ
κ
the gas phase, F = X F g . Under general multi-phase conditions, an expression like
g
g
Eq. (12.1) is also written for advective transport in the liquid phase.
Diffusive transport in the gas phase is modeled as Fickian in the ADM; that is,
diffusive flux is proportional to the concentration gradient, usually expressed as mass
fractions or mole fractions. In the TOUGH2 code, the diffusive flux of component κ
in the gas phase is written as
κ
κ
j =− φτ 0 τ g ρ g D ∇X κ (12.3)
g g g
Here φ is porosity, τ 0 τ g is the tortuosity which includes a porous medium dependent
factor τ 0 and a coefficient that depends on gas phase saturation S g , τ g = τ g (S g ), ρ g
κ
is gas density, D is the diffusion coefficient of component κ in bulk gas, and
g
κ
X is the mass fraction of component κ in the gas. For a binary system with
g
equal diffusion coefficients, Eq. (12.3) yields a net mass flux of zero, so that
total mass flux in the gas phase is given just by the advective term, Eq. (12.1).
Formulations similar to Eq. (12.3) are employed in other continuum codes (see
Table 12.1).
In some cases the mole fraction gradient rather than the mass fraction gradient is
used as driving force, giving equimolar counterdiffusion in a binary system with equal
diffusion coefficients (Bird et al., 1960). The TETRAD code (Vinsome and Shook,