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6. Dispersivity under Partially-Saturated Conditions
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invades the pore body and occupies the bulk space of the pore, we consider
each corner of the pore as a separate element with its own pressure and concentration.
Thus, for a cubic pore body, 8 different corner elements exist with
8 different pressure and concentration values assigned to them. Fluid flow and
solute mass fluxes between these elements occurs through the 12 edges of the
pore body (see Figure 6.3(a)).
In the same manner, we assign only one concentration to a saturated pore
throat. However, after invasion of the pore throat by the non-wetting, each
edge of the pore throat will have its own conductance (and flow rate), and we
assign separate concentrations to each edge.
The conductance of each edge needs to be determined as a function of the thickness
of water film residing in the edge. This thickness depends on the radius
of curvature of the fluid-fluid interface, which in turn depends on the capillary
pressure. Corner elements of a given drained pore body are connected to the
neighboring pore body corners via pore throats. Therefore, we need to specify
connections of pore throats to the corners of pore bodies. The algorithm
used to associate different pore throats to different corners of neighboring pore
bodies is described in Appendix B.
6.3 Unsaturated flow modeling
We wish to simulate drainage in a strongly water wet porous media saturated
with water. The non-wetting phase is assumed to be air, which can flow under
negligibly small pressure gradient. To simulate drainage in our network, the
displacing air is considered to be injected through an external reservoir which is
connected to every pore-body on the inlet side of the network. Displaced water
escapes through the outlet face on the opposite side. Impermeable (no flow)
boundary conditions are imposed along the sides parallel to the main direction
of flow.
6.3.1 Drainage simulation
Initially, the network is fully saturated with water. At low flow rates, the
progress of the displacement is controlled by capillary forces. This forms the
basis for the invasion percolation algorithm used to model drainage [Wilkinson
and Willemsen, 1983, Chandler et al., 1982]. At every stage of the process, air
invades all accessible pore bodies and throats with the lowest entry capillary
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