Krt [1 1 S1m

at lower moisture contents and, thus, all four curves in Fig. 2.14a reach an immobile or irreducible saturation limit, si. Isolated liquid inclusions cannot be reduced below this limit except through freezing or evaporation. Water flow parameters thus scale to an effective saturation, s* = (s - si)/(1 - si), so that capillary pressure tends to infinity and relative permeability to 0 as s approaches si. While the retained immobile saturation after drainage is typically around 0.07, the minimum saturation requirement during infiltration can be much lower (see Denoth et al., 1979; Kattel-mann, 1986; Dullien, 1992). Jordan et al. (1999a) suggests an "average" value of 0.04 for general modeling purposes, but values as low as 0.01 may be appropriate when modeling infiltration.

Experimental s-pal data is typically represented by inverse power functions between s and pal. The simplest of these is the Brooks-Corey (1964) function, while the somewhat more complicated van Genuchten (1980) function provides a closer representation of the "knee" that occurs near full saturation. These functions are summarized in Table 2.5. Both functions have two fitting parameters - an air entry or bubbling pressure pe and an exponent Ap, termed the pore-size distribution index. Figure 2.14a includes curve fits of the van Genuchten function to the four sets of experimental data. Air-entry pressures ranged between 240 and 440 Pa and pore-size distribution indices ranged between 2.6 and 3.8. Jordan (1983) obtained a value near 3.0 for Ap using in situ tensionmeter and lysimeter measurements. These values for Ap are within the range suggested by Brooks and Corey (1964) for sand, while the values for pe are somewhat lower.

The wicking height of dye water into a snow sample provides a rough estimate of its entry pressure. Water in snow is under tension and, thus, rises spontaneously to a height where its weight balances the pressure drop across the liquid meniscus. Conceptualizing the pore space as bundles of capillary tubes, the capillary rise, h, relates inversely to the pore size as

2aaf cos 6

Here, 9 is the contact angle between the ice grain and water meniscus, which is usually taken as 0. Rise in natural snow ranges from about 0.5 cm for old, coarse snow to 5.0 cm for ice crusts (see Wakahama, 1968; Wankiewicz, 1979; Coleou et al., 1999; and Jordan et al., 1999b), which corresponds to a range of entry pressures (= Ptgh) between 49 and 490 Pa.

Liquid and relative permeability

In wet snow, air and water share the pore space, leaving less cross-sectional area available for flow of either phase. As with saturated permeability, liquid permeability Kt relates to the square of the pore size - but the size distribution is now limited to water-filled pores. Since water flow is usually restricted to smaller pores in the snowpack, liquid permeability is typically several orders less than saturated permeability. Kt depends as well on tortuosity and interconnectivity among the water-filled pores. Because of the implicit pore-size relationship in the s-pat curve, liquid permeability is parameterized as a power function of s. Table 2.5 presents the s-Krt functions of Brooks and Corey (1964) and van Genuchten (1980). The simple Brooks and Corey formula is more commonly used for snow modeling and has only the exponent e as a fitting parameter.

Figure 2.14b depicts the Brooks and Corey permeability function for e = 3. While no laboratory measurements have been made for the s-Krt curve in snow, Denoth et al. (1979) determined values of e between about 2 for new snow and 5 for old, clustered snow by observing the discharge rate from cylindrical tubes filled with sieved snow. Colbeck and Anderson (1982) found that a value of 3.3 predicts flow rates for well-metamorphosed snow that are in close agreement with field data.

Flow with a uniform wetting front

When a steady rainfall infiltrates a homogeneous snowpack, the solution to the flow equation (2.23) is a traveling wave of fixed shape as shown in Fig. 2.15a (Illan-gasekare et al., 1990; Gray, 1996; Albert and Krajeski, 1998). Because capillary forces in snow are small, Colbeck (1972,1978) neglected the pressure term in Equation (2.22) and derived a gravitational solution. For gravitational flow, the front has the shape of a shock wave or step function and the transition between wetted and dry snow regions is abrupt. The downward wavefront velocity through homogeneous snow is easily approximated by balancing the interior flow rate with the surface influx (rain or meltwater) and then computing the time it takes to warm the snow to 0 °C and to satisfy the immobile and equilibrium saturation deficits (Colbeck, 1976; Jordan, 1991; Albert and Krajeski, 1998). Because capillary suction accelerates water movement into the snowpack, the gravitational approximation will somewhat underpredict the wavefront velocity, but the effect is minor. Capillary forces do, however, play a vital role in the formation of capillary barriers and flow fingers and in the wicking of ponded water.

Liquid Water Saturation, s Capillary Tension, p3t

Figure 2.15. Simulated profiles of (a) liquid water saturation and (b) capillary tension at 5 min intervals for an irrigation rate of 5 cm h-1. The snowpack consists of fine and coarse layers with respective entry pressures of 262 and 118 Pa. Depth is with respect to the snow surface (after Jordan 1996, with modifications).

Liquid Water Saturation, s Capillary Tension, p3t

Figure 2.15. Simulated profiles of (a) liquid water saturation and (b) capillary tension at 5 min intervals for an irrigation rate of 5 cm h-1. The snowpack consists of fine and coarse layers with respective entry pressures of 262 and 118 Pa. Depth is with respect to the snow surface (after Jordan 1996, with modifications).

Figure 2.16 compares outflow from lysimeters at two snow research sites (Davis et al., 2001) with simulations from the SNTHERM snow model (Jordan, 1991). SNTHERM's assumptions of gravitational flow and a uniform wetting front closely represent outflow in the full melt season, once the snowpack is ripe or totally wetted. Heterogeneity in the snowpack in the early melt season may explain why SNTHERM predicts outflow not captured by the lysimeters. In heterogeneously stratified snow covers, flow is proportionally slower in fine-textured, dense snow or ice crusts, and water saturation is therefore higher to maintain flux continuity. Water levels must also adjust to maintain pressure continuity across textural discontinuities. So-called capillary barriers arise between fine and coarse layers because suction is higher in the finer layer (Jordan, 1996). Infiltrating water thus accumulates above the fine-coarse horizon until it fills pores of the same size (and hence the same tension or pressure) as the underlying layer (see Fig. 2.15). In cold snow, backed-up water refreezes to form ice crusts or lenses, which can further impede the downward flow of water. In sloped terrain, capillary barriers and ice layers can direct rain or meltwater to the bottom of a hill before it infiltrates to the bottom of the snowpack. For very deep snowpacks, this can accelerate the arrival of hydraulic pulses by hours or even by days.

Unstable flow

In addition to the effects of layering, the flow field in snow is complicated by the development of vertical channels or fingers, which concentrate flow ahead of the

Removable Wall Theather Pit

Figure 2.16. Observed outflux(cmh-1) from the snow cover and predictions from the SNTHERM snow model: (a) between days 100 and 160, 1994, at Mammoth Mountain, CA and (b) between days 80 and 120,1997, at Sleepers River Research Watershed, VT. The solid line shows the measurements from lysimeters, while the dotted line shows the simulation results (after Davis et al, 2001, copyright 1999; copyright John Wiley & Sons Limited. Reproduced with permission).

Figure 2.16. Observed outflux(cmh-1) from the snow cover and predictions from the SNTHERM snow model: (a) between days 100 and 160, 1994, at Mammoth Mountain, CA and (b) between days 80 and 120,1997, at Sleepers River Research Watershed, VT. The solid line shows the measurements from lysimeters, while the dotted line shows the simulation results (after Davis et al, 2001, copyright 1999; copyright John Wiley & Sons Limited. Reproduced with permission).

background wetting front. Raats (1973) and Philip (1975) established that flow instabilities develop in soil when the suction gradient is in opposition to the flow, which causes the flow velocity to accelerate with depth. Selker et al. (1992) later confirmed their criterion with measurements of the capillary tension within growing instabilities in unsaturated soil. Such conditions occur for flow through fine-over-coarse layers and in homogeneous media when the surface flux decelerates.

Marsh and Woo (1984a) observed flow fingers in snow similar to those reported for soils, which are characteristically narrow and uniform in width and randomly spaced at frequent intervals. In snow at subzero temperatures, fingers often freeze to form ice columns (see Fig. 2.17a). Marsh and Woo (1984a) and Marsh (1988, 1991) reported finger widths in cold arctic snow of from 3.5 to 5 cm thick, with mean spacings between the fingers of 13 cm and areal coverages of 22-27%. About

Figure 2.17. (a) Refrozen flow finger (from Albert et al., 1999, copyright 1999; copyright John Wiley & Sons Limited. Reproduced with permission). (b) Dye studies showing flow fingers and pre-melt horizons in layered snow (after Marsh and Woo, 1984a, photo by Philip Marsh, with permission. Published 1984 American Geophysical Union. Reproduced/modified by permission of American Geophysical Union).

Figure 2.17. (a) Refrozen flow finger (from Albert et al., 1999, copyright 1999; copyright John Wiley & Sons Limited. Reproduced with permission). (b) Dye studies showing flow fingers and pre-melt horizons in layered snow (after Marsh and Woo, 1984a, photo by Philip Marsh, with permission. Published 1984 American Geophysical Union. Reproduced/modified by permission of American Geophysical Union).

one-half of the total flow was carried by the fingers. Using a novel thick-section cutter, McGurk and Marsh (1995) measured mean finger diameters of 1.8-3.0 cm in warm snow, with mean spacings of 1.9-4.6 cm, and a mean wetted area of 4-6%. Using a high-frequency FMCW radar to detect flow fingers in a seasonal snowpack, Albert etal. (1999) estimate an areal density of about 3 fingers m-2 from rain falling on temperate snow in the early melt season and a diameter range from 2 to 5 cm. Clearly, the surface flux intensity as well as snow temperature and grain size will impact the size and spacing of the fingers.

Marsh andWoo (1984a) and Marsh (1988,1991) observed that fingers often originate at horizon interfaces (see Fig. 2.17b), regardless of the snow properties above and below the interface. In layered studies of sifted and poured snow, Jordan (1996) found that finger formation was limited to fine-coarse transitions. This inconsistency suggests that natural snow layers develop surface characteristics distinct from their bulk properties (e.g. surface melt crusts or surface hoar) that can affect the flow pattern. Using a multiple dye tracer application, Schneebeli (1995) demonstrated that the position of the finger paths usually changed between subsequent melt-freeze events.

Marsh and Woo (1984b) developed a multiple path simulation model that separates water flow into background and finger front components. Their model also accounts for ice-layer growth at strata horizons and demonstrates that ice-layer growth in subfreezing snow can be sufficiently large to interrupt finger-flow advance for short periods of time. Because much of the finger flow is laterally diverted to growing ice horizons, their simulations predict that the finger front in cold snow is never more than 10-15 cm below the background front. Predicted finger flow is more rapid through warm snow, however, and reaches the base of the pack well ahead of the background wetting front.

Flow fingers become more permeable than the surrounding dry snow because snow coarsens faster in the presence of water. Grain bond weakening within fingers can also cause snow to collapse, leaving "bowl-shaped melt depressions" or dimples in the snow surface (McGurk and Kattelmann, 1988; Marsh, 1991).

Despite the omission of fingering, even-wetting front theory does a reasonable job predicting the timing of basal outflow once the snowpack is ripe (Tuteja and Cunnane, 1997; Davis et al., 2001). There is a certain logic to this, given the trend over time towards homogeneity in well-wetted snowpacks. Thus, while finger flow plays an important role at the beginning of the melt season or for mid-winter rain-on-snow events, the even wetting front approach may be adequate once late season melting is well underway.

Renewable Energy 101

Renewable Energy 101

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