Page 220 - gas transport in porous media

216
the interface between two finite difference grid blocks n and m.

Pruess
X κ X κ κ κ

g − g X − X
κ m n κ l m l n
κ
f =− − (12.4)
nm g l nm
nm D nm D nm
κ
Here we have introduced the as yet unknown diffusive strength coefficients ( ) nm
g
κ
and ( ) nm at the interface, which must be expressed in terms of the strength coeffi-
l
cients in the participating grid blocks. Invoking conservation of diffusive flux across
the interface between two grid blocks leads in the usual way to the requirement of
harmonic weighting of the diffusive strength coefficients. However, such weight-
ing may in general not be applied separately to the diffusive fluxes in gas and liquid
phases, because these may be strongly coupled by phase partitioning effects. This can
be seen by considering the extreme case of diffusion of a water-soluble and volatile
compound from a grid block in single-phase gas conditions to an adjacent grid block
which is in single-phase liquid conditions. Harmonic weighting applied separately
to liquid and gas diffusive fluxes would result in either of them being zero, because
for each phase effective diffusivity is zero on one side of the interface. Thus total
diffusive flux would vanish in this case, which is unphysical. In reality, tracer would
diffuse through the gas phase to the gas-liquid interface, would establish a certain
mass fraction in the aqueous phase by dissolution, and would then proceed to dif-
fuse away from the interface through the aqueous phase. Similar arguments can be
made in the less extreme situation where liquid saturation changes from a large to a
small value rather than from 1 to 0, as may be the case in the capillary fringe, during
infiltration events, or at fracture-matrix interfaces in variably saturated media.
A consistent space-discretized treatment of diffusion in multiphase conditions can
κ
κ
be achieved by introducing gas and liquid phase mass fractions (X ) i and (X ) i at the
g l
interface i between grid blocks n and m as additional unknowns. Diffusive fluxes to
and from the interface can be calculated by using finite difference expressions such

κ
κ
as (X ) i − (X ) n /D ni for mass fraction gradients, and applying the effective
g
g
diffusive strength coefficients in the individual grid blocks. Two additional constraints
are available to determine the unknown gas and liquid phase mass fractions at the

κ
κ
interface, namely, (1) the solubility relation C = X /X κ , and (2) the conservation
l g
of total diffusive flux at the interface.
12.2.2 Dusty Gas Model
The Dusty Gas Model represents the porous medium as a collection of giant spherical
molecules (dust particles) held in space by external force. The movement of gas
molecules in the spaces between dust particles is described by the kinetic theory of
gases. See Solcova and Schneider (this book) for more details.