Introduction

Previous work on the deformation and consolidation of snow has concentrated on the macroscopic rheological behavior and has resulted in the introduction of macroscopic rheological parameters that are snow-type, temperature and density dependent.

Extensive experimental work has been done to investigate small deformations under low-stress conditions. Reference LandauerLandauer (1955[b]) and Reference Ramseier and PavlakRamseier and Pavlak (1964) empirically expressed the time-dependent component of the deformation, for a given constant stress, as a simple power law in terms of time. Other researchers (including Ramseier and Pavlak) compared experimental results with the equations of certain rheological models: Reference BucherBucher (1948) discussed the use of the Maxwell model; and Reference Quervainde Quervain (1946), Reference YosidaYosida and others (1956) and Reference Ramseier and PavlakRamseier and Pavlak (1964) employed the Maxwell-Voigt model. It was found that the power law and the rheological equations agreed quite well with experimental data; however, these investigators showed that the macroscopic elasticity and viscosity coefficients as well as the experimental parameters in the power law are definite functions of at least temperature and density. Application of these equations to a consolidation process in which there is a significant density change necessitates knowing the functions relating the viscosity coefficient or the parameters in the power law to density and temperature.

The following investigators have advanced simple, yet very different, empirical equations for the macroscopic coefficient of viscosity of snow as a function of density: Reference LandauerLandauer (1955[b]), Reference KojimaKojima (1956, Reference Kojima1957, Reference Kojima1958), Reference Bader and KingeryBader (1963), and Reference Mellor and HendricksonMellor and Hendrickson (1965).

Examining recent densification theories one finds that Reference Costes and KingeryCostes (1963) circumvented the problem of an explicit viscosity-density-temperature relationship by developing a rather complicated empirical equation, involving several parameters, for the rate of densification. Reference Bader and KingeryBader (1963) used the empirical viscosity-density equation of Kojima, and modified it by arbitrarily introducing a function which satisfied certain boundary conditions. Reference Kojima and MellorKojima (1964) showed that Bader’s modified equation was approximately of the same form as his original equation for a large density range, and proceeded to use the original in the analysis of his densification studies in the Antarctic.

In order to gain further insight into the process of snow consolidation and to allow formulation of a theoretical equation describing the density-dependent viscous behavior of snow, it is suggested that the theory should originate on the microscopic level by considering the actual phenomena occurring in the grains and grain bonds.

In this way the properties and behavior of ice are immediately incorporated into the theory. By considering the snow mass to be composed of a finite number of ice particles joined together by a finite number of ice bonds one can introduce parameters related to grain and pore-space geometry. Then introducing the macroscopic variable porosity, which is a measure of the mass of ice in an elemental volume, the density dependence is incorporated and a transition to the phenomenalistic level of interest is achieved.

Consolidation Theory

The theory of consolidation is developed for a laterally confined cylindrical snow mass subjected to an axial load. The lateral stresses which develop are considered negligible.Footnote * Shear components of this axial force in the grain bonds produce a viscous flow of the ice composing the bonds, and pore space is decreased by the relative movement of the individual snow particles as they slide along intergranular boundaries into a closer configuration. Consolidation is considered to proceed with little change in the shape and size of the snow grains until the closest possible arrangement of the particles is reached. This closest configuration corresponds to a bulk porosity somewhere in the range of 36 to 43 per cent (Reference BensonBenson, 1962) and further consolidation can only continue by mutual intrusion of the particles, which results in a marked transition in the rate of consolidation.

It is assumed that the given snow mass is statistically homogeneous, and contains a large number of particles joined together by k bonds. The total volume of the voids, V _v, in the mass is imagined to be partitioned into k equal volume elements with a common dimension, r, i.e. the elemental volume is represented by

where the proportionality constant α may be considered to be a volume shape factor.

Let δϵ _i denote the small shear strains which occur in the bonds when V _v is decreased by a small amount δV _v. It is assumed that there exist k geometrical parameters λ _i which enable one to approximate the change in the geometrically complicated pore volume as

then from equations (1) and (2) the unit change in pore volume is

It has been found experimentally that the Theological behavior of ice is simple Newtonian at low stresses, i.e. the relationship between shear stress and strain-rate is linear. Reference Jellinek and BrillJellinek and Brill (1956) report this linearity to exist for the stress range of 3.4 × 10⁵ to 2.3 × 10⁶ dyn./cm.² and Reference Butkovich and LandauerButkovich and Landauer (1960) found the linear relationship in the stress range of approximately 10⁴ to 10⁵ dyn./cm.². In this paper the forces of consolidation are considered small enough that the grain bond in the snow mass will exhibit this linear Newtonian behavior.

Let τ _i represent the shearing stresses acting in the grain bonds which produce the strains δϵ _i in time δt. Then

where η is the coefficient of viscosity for ice. Equation (3) becomes,

For the confined snow mass let P be the axial force and define a surface S, in general perpendicular to P, which intersects only grain bonds in the planes of τ _i . Defining the number of bonds intersected by S to be l, the area of each bond cross-section to be A _i , and the angle between the normal to A _i and the direction of P to be θ _i and assuming that the vertical force on A _i is

, then

It is now assumed that the “two-dimensional porosity” of the surface S projected on a plane perpendicular to P is identical with the effective porosity, n _f, of a potential failure surface as is defined by Reference Ballard and McGawBallard and McCaw (1965). If A is the right cross-sectional area of the mass then

and

where σ is the axial external stress P/A. Substituting equation (5) in (4)

Allowing δt to approach zero

If V represents the bulk volume of the snow mass with bulk porosity n, then the insignificant compressibility of ice allows the substitution of dV for dV _v, nV for V _v, and dn/(1−n) for dV/V. Substituting these relationships in equation (6)

The dependence of n _f and

on n will now be discussed. According to Ballard and McGaw (1965) n _f for an age-hardened snow can be represented by an where a is the reciprocal of the limiting porosity n ₁. In the summation k increases as consolidation progresses and the quantities λ _i and θ _i will in general change as n decreases; however, if the separate distributions of the values of λ _i and θ _i are invariant with porosity for k infinite, then when k is large will not change appreciably with porosity.

There is very little evidence in the literature to substantiate the invariance of the distributions of θ _i and λ _i . Certainly a porosity-dependent distribution of θ _i would require a preferred orientation of the bonds in a naturally consolidating snow mass, a condition which to date has not been observed. The exact physical significance of the geometrical parameters λ _i is not yet clearly visualized; however, if the geometry of pore space remains similar for some range of n as n decreases, then it seems reasonable to assume that the distribution of the values of λ _i is independent of n for this range of n. Replacing

by a constant ν and n _f by an in equation (7) produces the equation

which is now restricted to age-hardened snow.

Representing the height of the snow mass by z, then since

equation (8) may be rewritten as

to give the expression for the rate of consolidation.

The Coefficient of Viscosity

The term

in equation (10) is the macroscopic coefficient of compressive viscosity for the conditions specified in the development: the confining stresses are negligible; the macroscopic stress condition is uniaxial; and the porosity is such that consolidation can occur through particle rearrangement, but is less than 1/a. Denoting this macroscopic coefficient of compressive viscosity as η _c, then

The dependence of η _c, on temperature, snow-type, and porosity is immediately apparent: temperature dependence is inherent in the temperature-dependent coefficient of viscosity for ice, η, and variation with different snow-types is introduced by the parameters ν and a.

Equation (11) was compared with the viscosity data of Reference Mellor and HendricksonMellor and Hendrickson (1965) from confined creep tests at “Byrd” station, Antarctica. The porosity range of these data, 0.35 < n < 0.55, is within the range of applicability. Making the substitution

reduces equation (11) to a linearized equation through the origin,

A series of regression analyses of η _c on X for 1.70 ≦ a ≦ 1.80 in increments of 0.01 showed a maximum correlation coefficient of 0.940 for an a of 1 78. The reciprocal of this value of a predicts a limiting porosity, n ₁, of 0.561, which agrees very well with the values found by Ballard and McGaw. The corresponding value of the slope, η/ν, was 1.23× 10¹⁵ poise. Figure 1 shows a comparison of the theoretical equation developed here with the empirical equation of Mellor and Hendrickson for their “Byrd” station data.

Fig. 1.

Comparison of theoretical compressive viscosity

, for a = 1.78, with empirical relationship,

for “Byrd” station data (Reference Mellor and HendricksonMellor and Hendrickson, 1965)

Reference HaefeliHaefeli (1939) showed that the effect of lateral confinement is small for unit deformations up to several per cent; hence equation (11) should be applicable to unconfined consolidation for small deformations. From viscosity experiments (unconfined case) Reference LandauerLandauer (1955[a], p. 24–26) found that the compressive viscosity of snow varied as exp (−4.4e), where e is the void ratio. From equation (11)

The function f(e) is well represented by an equation of the form exp (−4.4e) for a = 1.82 (or n ₁ = 0.566) in the porosity range of 38 to 53 per cent (Fig. 2).

Fig. 2.

Comparison of theoretical compressive viscosity

, for a = 1.82 with empirical relationship, η_c ∝ exp (−4.4 e) of Reference LandauerLandauer (1955[a])

Equation (11) is also compared in Figure 3 with the experimentally determined viscosity curve of Reference Ramseier and PavlakRamseier and Pavlak (1964) and the empirical relationship of Reference Kojima and MellorKojima (1964), η _c ∝ exp [20(1−n)] which was derived from depth-density data. In general dη _c/dn for the theoretical curve is inconsistent with the slopes of these other curves.

Fig. 3.

Comparison of the theoretical compressive viscosity

for a = 1.8, with the experimentally determined viscosity curve of Reference Ramseier and PavlakRamseier and Pavlak (1964) and the empirical relationship of Reference Kojima and MellorKojima (1964) η _c ∝ exp[20(1−n)]

Porosity—Time Relationship

Equation (8) can be integrated for a constant value of σ to give

where n ₀ is the value of n at t = 0. Figure 4 is a graph of equation (14) in units of νσ/η for a = 1.8 and n ₀ = 0.555. To investigate the validity of equation (14) porosity-time data may be analyzed statistically by using the linearized form

Fig. 4.

Theoretical porosity—time curve

for a = 1.80 and n₀ = 0.555

where

Data from a U.S. Cold Regions Research and Engineering Laboratory Research Report in preparation by Ballard and Feldt were analyzed according to equation (15) by regressing ϒ on X. The results are summarized in Table I and a typical example is shown in Figure 5. Ballard and Feldt reported data for a large series of tests, but only the data for the lower axial loads appeared to be explained by equation (14).

Fig. 5.

Linearized time—porosity equation compared with data from a U.S. Cold Regions Research and Engineering Laboratory Report in preparation by Ballard and Feldt

Table I.

Summation of Regression Analysis of Porosity—Time Data According to Equation (15)

The consistently high correlation coefficients and the narrow range of values of the limiting porosity with a mean of 0.564 support the validity of equation (14). Values of η/ν are plotted against the reciprocal of absolute temperature in Figure 6. An Arrhenius-type relationship for an activation energy of 16 kcal./mole appears to fit the data very well. This agrees with Reference Jellinek and BrillJellinek and Brill (1956), who found that the variation of the coefficient of viscosity of polycrystalline ice with temperature predicted an activation energy for creep of 16.1 kcal./mole. It therefore appears that ν is indeed constant for the particular type of snow used by Ballard and Feldt.

Fig. 6.

Arrhenius-type relationship, for an activation energy of 16 kcal./mole compared with data of Ballard and Feldt

Unit Deformation

Representing the height of the snow mass at time t = 0 and t by z ₀ and z ₀−Δz respectively, and making the substitutions

and

then equation (14) may be written in terms of the unit deformation d Δz/z ₀ as

Equation 16 was checked against the creep curves given by Reference LandauerLandauer (1955[b]). Choosing a = 1.80 (a value which is consistent with previous predictions) and using n ₀ = 054.2 (Landauer’s initial porosity), a logarithmic plot was made in Figure 7 of the left side of equation (16), designated as F(Δz/z ₀), against Δz/z ₀. Landauer’s creep curves (Reference LandauerLandauer, 1955[b], Fig. 6, p. 7) can be represented by an equation of the form t = b(Δz/z ₀)^1.25 where b varies with σ. This empirical relationship is a very good approximation to equation (16) in the range 0.001 < Δz/z ₀ < 0.1 as can be seen in Figure 7 where the straight line is the power function 0.179(Δz/z ₀)^1.25. Replacing F(Δz/z ₀) by this approximation one has

Fig. 7.

Comparison of theoretical unit deformation equation

, for a = 1.80 and n₀ = 0.542, with power function approximation F(Δz/z₀) = 0.179^{(Δz/z₀)1.25}

which requires

Values of b from Landauer’s curves and the calculated values of η/ν are shown in Table II.

Table II.

Values of η/ν Predicted from Landauer’s Creep Curves

Equation 16 was also compared with the data from Reference Ramseier and PavlakRamseier and Pavlak (1964, Fig. 1, p. 326) by using the Iinearized form of the equation,

where

A regression of ϒ on X produced the results shown in Table III.

Table III.

Summation of Regression Analysis on Creep Data of Ramseier and Pavlak According to Equation (18)

Although widely separated areas are represented the values of n ₁ are considerably lower than previously predicted.

Densification of Natural Snow Cover

Finally, application of the result of this theory to densification of a natural snow cover should indicate whether the imposed conditions used in the development of the theory are applicable to in situ consolidation.

Assuming that the snow is accumulating at a constant rate A, then a snow layer which fell at time t = 0 will be subject to a vertical stress σ = At at time t, and for the given snow layer equation (8) becomes

Integrating gives

where n ₀ is now the surface snow porosity.

Denoting the depth as h, the density of the snow as γ and the density of ice as γ _i then from Sorge’s Law (Reference BaderBader, 1954.)

or in terms of n

Substituting from equations (19) and (20) in (21) and integrating from the surface to depth h produces the depth–porosity curve for a constant temperature,

A depth-porosity curve from Reference Kojima and MellorKojima (1964, p. 186, Fig. 54, BH58) was selected as representative for snow with a low surface porosity for comparison with equation (22). Estimating n ₀ = 0.525 and choosing a = 1.80, the integral in equation (22) was evaluated numerically by changing the lower limit to 0.5249 to avoid the point n = n ₀ where the integrand becomes infinite. Then a choice of

produced the depthporosity curve shown in Figure 8. The curve expresses the data quite well down to a depth of about 450 cm., where the porosity is just within the range indicated by Reference BensonBenson (1962), for close packing. A value of η/ν = 4.25 × 10¹³ poise was calculated from Kojima’s estimated accumulation rate.

Fig. 8.

Depth-porosity curve predicted in this paper compared with data and depth–porosity curve of Reference Kojima and MellorKojima (1964)

Discussion of Parameters

Two parameters, a and ν, were introduced in the development of the consolidation theory. The reciprocal of the first parameter, 1/a, was assumed to be equivalent to the limiting porosity, n ₁, as defined by Ballard and McGaw. Comparison of the theory with consolidation data from several different sources has predicted values of 1/a that agree very closely with the values of n ₁, predicted by Ballard and McGaw from strength data.

The parameter, ν, defined by

is not directly related to any quantity that has been previously defined in snow mechanics. It was introduced to relate the combined individual infinitesimal displacements in the structure of the snow mass to the total change in pore volume. Its value cannot be predicted precisely from consolidation data, but rather the quotient η/ν. A knowledge of the value of η for a particular temperature allows an estimation of ν. Jellinek and Brill found that η for polycrystalline ice could be represented by

where R is the ideal gas constant in kcal./mole °K. and T is the absolute temperature. Using equation (24) and the values of η/ν predicted in this paper, the values of ν shown in Table IV were calculated. It is seen that the order of magnitude of ν ranges from 10⁻² to 10². This inconsistency in the values of ν as predicted from the data of different investigators may indicate that the value of ν is greatly affected by the diagenetic history of the snow and the specific conditions of the experiment. The magnitude of ν (equation (23)) is equal to the mean value of the term λ _i sin θ _i cos θ _i From equation (2) the magnitude of λ _i should be of the order of 1; and if θ _i can assume all possible values between 0 and π/2, then the average value of sin θ _i cos θ _i is 1/π. Therefore the magnitude of ν should be of the order of 1. From Table IV it is seen that only the data of Mellor and Hendrickson predict a value of ν of this magnitude. It may be rioted, however, that their experiment was the only one which actually satisfied the conditions of the theory in that the samples were fully age-hardened snow and were confined during the tests.

Table IV.

Estimated Values of the Parameter ν