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6. Dispersivity under Partially-Saturated Conditions
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through all pore throats at the inlet or outlet boundary of the network), L is
the network length in the flow direction, V f is total fluid volume, θ is porosity,
and A is cross-sectional area of the network perpendicular to the overall flow
direction.
Knowing the total discharge through the network, we can also calculate the
relative permeability. The relative permeability of the network to water at a
given saturation and capillary pressure is calculated from Darcy’s law
k rw =
µ wQ t
k A ∆P/L
(6.14)
where µ is the wetting phase viscosity, k is the intrinsic permeability, and ∆P
is the pressure difference between inflow and outflow reservoirs.
Repetition of this process at consecutively larger imposed capillary pressures
results in a graph of capillary pressure versus wetting saturation and relative
permeability versus wetting saturation.
6.4.2 Simulating solute transport through the network
Commonly, under saturated conditions, one average concentration is assigned
to each pore body or throat [Raoof et al., 2010, Li et al., 2006a, Acharya et al.,
2005a, Sugita et al., 1995a]. This is done assuming that each pore space is a
connected well-mixed domain and that diffusion is fast enough compared to the
fluid flow within individual pores (i.e., gradients in concentration are negligible
within a single pore). This assumption may be reasonable under saturated
conditions, but, under partially saturated conditions the situation is different.
Under partially saturated conditions, in the case of cubic pore body, we have
eight corner units, each comprised of a corner domain together with half of
the three neighboring edges, as shown in Figure (6.3). Thus, we assign eight
different concentrations to a drained pore body, one for each corner unit. In
the case of drained pore throats, we assign different concentrations to each pore
throat edge.
Thus, in our formulation, the unknowns will be either concentrations of saturated
pore bodies, c i , and saturated pore throats, c ij , or concentrations of
edges of drained pore throats, c ij,k , and corner units of drained pore bodies,
c CU,i . We assume that each corner unit is a fully mixed domain. To show the
formulation, we introduce mass balance equations for a system of two drained
pores connected using a drained pore throat, as the most general case. We
assume that flow is from corner unit j towards corner unit i through corners of
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