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6. Dispersivity under Partially-Saturated Conditions
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In Figure (6.12) we can distinguish a trend in each k r −S w curve. Each k r −S w
curve is composed of two nearly linear parts (on semi-log scale). We have also
shown one of the k r −S w curves on linear scale, as a dotted line. The linear scale
plot shows that the slope of k r − S w is larger at higher saturation. This is due
to the fact that the non-wetting phase first occupies larger pores (which have
the most contribution to the conductivity of the phase), after which there is a
considerable drop in conductivity of the wetting phase. The substantial drop in
conductivity of the wetting phase will continue until the fraction of percolating
saturated pores is very low. After this point, further decrease in saturation will
cause only slight decrease in permeability of the wetting phase, since almost
all larger pores are already drained. Thus, the location of the change in slope
in the k r − S w curve, in the semi-log scale, can be related to the fraction of
percolating saturated pores. We may use the corresponding saturation as an
approximation for the critical saturation, S cr , at which maximum dispersivity
occurs. Figure (6.12) shows that the saturation at which change in slope of
the k r − S w curves occur is lower for networks with higher variance, in which
the α max also occurs at lower saturations. This observation is in agreement
with pore-scale observations such as change in fraction of percolating saturated
pores (Figure 6.8) and coefficient of variations of pore-scale velocities (Figure
6.7).
6.6 Conclusion
Multi-phase and unsaturated transport in porous media is characterized by
means of several macroscopic transport properties including relative permeability,
capillary pressure, and dispersivity. These properties have been found
to depend on macro-scale parameters, such as fluid saturation, but also on
pore-space morphology. Indeed, distribution of fluid phase within the pores is
important in determining transport properties.
In contrast to the relation between capillary pressure and saturation (P c − S w
curve), which has been investigated in many studies, there are only a few studies
on the relationship between dispersivity and saturation (i.e., α − S w curve).
Although, the dependency of dispersivity on saturation has been clearly shown
through experimental studies, almost all simulators of solute transport at the
macro scale, under unsaturated conditions, use a constant dispersivity (i.e., do
not include a dependance on saturation). This is mainly due to the unresulved
and complex functionality of dependence of dispersion on saturation.
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