Unsaturated or twophase flow

In unsaturated wet snow, both air and water occupy the pore space and, strictly speaking, flow of both phases should be considered. However, since the volume of water in freely draining snow is usually less than 10% (Colbeck, 1978), air is not confined by the snow and a modified one-phase treatment may be used (Scheidegger, 1974). Replacing water pressure in Equation (2.19) with capillary pressure (pat = pa -pt) and the intrinsic permeability K with the liquid permeability Kt, the downward water velocity vt becomes

The equation for water flow through snow then derives from liquid continuity as

d x rate of change in net water flux compaction melt or water saturation condensation where s is the liquid saturation, vi is the velocity of the compacting ice matrix, and S is a phase change source term. The compaction rate is determined from the snow viscosity in Equation (2.11). Otherwise, Equation (2.23) contains three unknowns —s, pat, and Kt - which are related through the s - pat and s-Krt constitutive functions. In the s-Krt function, Krt is the relative permeability, expressed in terms of the saturated permeability as Krt = Kt / K.

Figure 2.14. (a) Measured saturation-pressure curves from Wankiewicz (1979), Colbeck (1974), and Colbeck (1975). Lines show curves fit with the van Genuchten (1980) function. (b) Saturation-relative permeability curve computed from the Brooks-Corey function with e = 3.

Capillary forces and the pressure-saturation (or s-pat) curve Capillary forces result from pressure drops across the concave water menisci in wet snow (see Figs. 2.4 and 2.6). Such forces suck water from wetter to drier regions of the snow cover and oppose or augment the gravitational forces. Recalling that Zae is the radius of curvature of the meniscus, the pore-scale pressure drop or capillary pressure, pai, computes from the Laplace equation (2.5) as 2aai/Zai. Although known as capillary "pressure," pai is actually a capillary suction or tension. Because Zae relates directly to the pore diameter, capillary tension is highest for small pores and water entering dry snow will fill the smallest pores first. A plot of water saturation versus water tension thus approximates a cumulative size distribution of the pore space. Such plots are referred to as water retention or s-pat curves and are standardly used to characterize soils.

The fragility of snow and its loss of cohesion when wet make it very difficult to measure water retention curves. Figure 2.14a shows the few published s-pai drying curves for snow (Colbeck, 1974,1975;Wankiewicz, 1979), which resemble those for coarse sand. Water suction in snow typically ranges from 0.1 to 1 kPa -several orders of magnitude less than in fine-grained soils. Water retention curves exhibit a directional hysteresis, with suctions in draining snow (drying phase) being about one-half that in infiltrated snow (wetting phase). Water pathways disconnect

Table 2.5 Constitutive relationships between water saturation and capillary pressure, and between water saturation and relative permeability in partially wetted porous media.

Brooks and Corey (1964)

van Genuchten (1980)

s-Pat s-Kt

pal pe

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Renewable Energy 101

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