List of symbols

• g ₁ depolarization factor

• K intrinsic permeability, m²

• L length of the snow column, m

• n exponent

• S saturation (water volume/pore volume)

• S ₀ initial saturation

• S_i irreducible saturation

• S ₀ ^* effective saturation, S ₀ ^* = (S ₀−S_i )/(1−S_i )

• t time, s

• u water volume flux, m s⁻¹

• u^* relative volume flux

• W water content (water volume/total volume)

• α constant, m⁻¹ s⁻¹

• ε_∞ dielectric constant extrapolated for f → ∞

• ε_i dielectric constant of ice

• ε_w dielectric constant of water at 0°C

• ρ density of wet snow, kg m⁻³

• ρ_i density of ice, kg m⁻³

• ρw density of water, kg m⁻³

• Φ porosity (pore volume/total volume)

Introduction

The water-saturation of a natural snow cover varies, in general, from zero to approximately 20% of the pore volume. In this case two essentially different types of distribution of free water can be observed: the pendular mode and the funicular mode of distribution. Pendular distribution covers the range of very low saturations corresponding to a water film on an ice grain adsorbed to a degree of saturation where the various isolated water menisci surrounding the ice grains flow together into complicated structures. Funicular distribution covers the range from the upper limit of the pendular mode to the total saturation of the pore volume. The change from-the pendular to the funicular mode takes place at a saturation of between 10% and 20% of the pore volume. As there is a considerable change in the distribution of water during this transition from the pendular to the funicular mode, this also leads to a change in the physical characteristics of the snow-cover.

It is the purpose of this paper to determine the saturation regime in which the transition from the pendular to the funicular mode of water distribution in snow appears. For this purpose both the dielectric constant of wet snow and the drainage of water from snow columns were carefully analysed.

Dielectric measurements

A method particularly suitable for field measurements to find the distribution of free water in snow consists in measuring the dielectric constant. Theoretical studies have shown that the dielectric constant of heterogeneous mixtures is essentially influenced by the way the various components are distributed. Here the mixing formula of Reference Polder and SantenPolder and van Santen (1946) is particularly suitable for calculating the dielectric constant ε_∞ of the porous system which constitutes snow from the dielectric constant of the components ice, air, and water, from their parts per volume, and a depolarization factor (Reference Denoth and SchittelkopfDenoth and Schittelkopf, 1978). The mixing formula of Polder and van Santen in an adapted form for the system of snow reads as follows:

(1)

where

The geometrical configuration of water inclusions in the dry-snow matrix changes with increasing water content and so consequently does the depolarization factor g ₁. In Figure 1 the depolarization factor g ₁ calculated from the Polder and van Santen model (Equation (1)) is plotted against saturation S. The dielectric constant of snow samples with different liquid contents was measured by a capacitance bridge within a frequency range from 100 kHz up to 20 MHz, and the dielectric constant ε_∞ required for Equation (2) was obtained by-extrapolation for f→ ∞ according to the Cole-Cole model (Reference Ambach and DenothAmbach and Denoth, 1972). The free-water content W was obtained from a freezing calorimeter with an accuracy of ±0.5% by volume. The density ρ of wet snow was found by weighing; porosity Φ and saturation S were calculated from:

Fig. 1.

The depolarization factor g₁, as a function of saturation S. 1: pendular regime; 2: transitional zone; 3; funicular regime.

Figure 1 shows two significantly different arrangements of free water: within the saturation range 0 ≤ S ≤ 11% with a depolarization factor, g ₁ ≈ 0.053 an arrangement corresponding to the pendular mode of distribution, and within the saturation range of 15% ≤ S ≤ 25% with a depolarization-factor g ₁ ≈ 0.076 an arrangement corresponding to the funicular mode of distribution with complicated aggregates formed by the various water menisci flowing together. In between, i.e. within the saturation range from 11% ≤ S ≤ 15%, a transitional range from the pendular into the funicular distribution is formed. In the case of very low saturations (S < 4%) the depolarization factor decreases towards zero. This means, that in the case of such low saturations the shape of the water inclusions tends more and more to flat discs, forming a borderline case of oblate spheroids when g ₁ = 0.

It is obvious that a change in water distribution, as mentioned above, affects the flow characteristics of water in snow. Consequently, additional information as to the upper limit of the pendular distribution can be obtained by measuring the dependence on saturation of flux of the water volume from a snow column. Therefore, drainage experiments were carried out using old, natural snow, repacked in tubes, with saturations of up to 18% of the pore volume. The outflowing water volume was continuously recorded by a capaciative level pick-up.

Drainage measurements

According to the gravity-flow theory (Reference ColbeckColbeck, 1971) one obtains for the water volume flux u of a draining snow column with an initially constant saturation S₀ throughout the snow-column length (Reference DenothDenoth and others, 1979):

(2a)

where t ₀ is that time in which a constant flux u ₀ appears and is given by:

For comparison purposes, however, it is practical to rewrite Equation (2a) in dimensionless form:

(2b)

In Figure 2 the relative water-volume flux u* (represented by data points) from snow columnsFootnote ^* with saturations of S ₀ = 11.6% (symbol:•), S ₀ = 12.7% (symbol:∆), and S ₀ = 14.5% (symbol:○)respectively is plotted against time (in units of t ₀). As a change in flow characteristics is expected when water distribution changes, only drainage experiments with initial saturations corresponding to the transitional zone (cf. Fig. 1) are shown. In addition, the theoretical profile according to Equation (2b) is given (solid line). Satisfactory conformity with the experiments is obtained only for saturations less than approximately 13%. For saturations higher than approximately 14%, a substantially higher flux becomes apparent at the beginning than one would have expected according to the gravity-flow theory. This can be explained by the formation of an additional flux in an unstable channel system with total saturation, formed by coalescing water menisci during the transition from the pendular to the funicular regime (Reference DenothDenoth and others, 1979). Consequently, a saturation of about 14% of the pore volume can be assumed as the upper limit of pendular distribution.

Fig. 2.

Plot of relative water volume flux u⋆ against time (in units of t₀) according to the gravity-flow theory (Equation (2b)) compared with experimental data of relative volume flux from snow-columns with different initial saturations: S₀ = 11.6% (●), S₀ = 12.7% (Δ), and S =₀ 14.5% (●).

Discussion

By measuring the dielectric constant, a relationship between the distribution of water inclusions and the water saturation is obtained in the form of the depolarization factor g ₁. In Equation (1) the depolarization factor accounts for the shape of inclusions:

a₁, a₂ , and a₃ being the semi-axes of elliptic inclusions. Although the arrangement of free water around the contact points of ice grains (both in the pendular and in the funicular regime) does not appear in the form of ellipsoids or spheroids, it can be assumed that in the case of the low water saturations of a natural snow cover the arrangement of free water is somewhat disc-shaped. Thus the water menisci can approximately be described as oblate spheroids, so that the depolarization-factor g ₁ does not reflect quantitatively the geometrical shape of the inclusions, but it offers information as to the relationship between the geometry of distribution and the water saturation.

In the case of greater saturations, several water menisci flow together into complicated water structures (funicular regime), as has already been mentioned. This can be seen from a change in the depolarizing factor from g ₁ ≈ 0.053 to g ₁ ≈0.076 when the saturation increases from 11% to 15%. Here it must be noted, however, that with a growing percentage by volume of water, electric interactions between the various water aggregates can no longer be neglected. This limits the application of the basic mixing formula (Equation (1)) to saturations smaller than 20%, which is the case for a natural snow cover.

The drainage experiments show that the flow characteristics of water change substantially if the saturation changes from 11% to 15%: for saturations higher than approximately 14% of the pore volume an additional flow in saturated channels is formed. A saturation of 11% is at the lower end, and the saturation of 15% at the upper end, of the transitional range from the pendular to the funicular regime.

From both the dielectric measurements and the drainage experiments it results that a saturation of about 14% can be assumed as the upper limit of the pendular mode of distribution of water in snow of an advanced degree of metamorphism. This broadly corresponds to an estimation of the upper limit of the pendular regime in snow as 14% of the pore volume given by Reference ColbeckColbeck (1973) based on the theoretical work of Reference SmithSmith (1933). However it is likely that the value of the liquid content at the transition will vary with the degree of metamorphism of the snow.