because tunf(z,q) = 0 while

so substituting the calculated gradients into the Stefan condition (1.21) we obtain the standard first differential equation, Q^ = kUotJ£, the solution of which is the following function:

2kiro\t0\x

The formula for calculation of the thawing depth is derived in a similar manner; however, Aunf should be substituted for Af in this formula. As mentioned above this formula can be used for an estimate of the freezing (thawing) depth. It gives too large a value as a rule. At the same time, the closer the freezing (thawing) conditions are to the model used as a basis for deriving this formula, the more accurately it describes the process.

The value |i0| t = Q(t) is termed the sum of freezing degree-hours (in the case of soil freezing) or of heating degree-hours (in the case of thawing). The Stefan formula has been derived from the assumption of the temperature constancy (f0 = constant) at the ground surface; however, it can be used when the values t0 are variable also. To do this it is essential to determine the value of freezing degree-hours or heating degree-hours (Qfr, Qth) in a time interval during which freezing or thawing proceeds from the seasonal temperature change on the surface. Then the expression (1.30) will take the following form:

V 2ph

Given the depth of freezing caused by the known sum of freezing degree-hours on the ground surface Q, the calculation from the Stefan formula can be improved. Then for the same soil the depth of freezing £2 as a result of the effect of another sum of freezing degree-hours Q2 wiU be:

L.S. Leybenzon's formula for determining the depths of seasonal and perennial ground freezing (thawing)

As noted above, the essential error of the Stefan formula is associated among other things with neglect of the heat flux flowing from the unfrozen zone in the upward direction because of the difference of the temperature of this zone from 0°C. The model accepted in Leybenzon's method takes account of this by entering the initial conditions in the form t(z,0) = t1 where t1 is the constant ground temperature (across the section) before the beginning of the freezing (thawing) process. In the case of freezing > 0. In addition it is suggested that in the course of freezing (thawing) the temperature changes with depth in the unfrozen zone in accordance with the law erfl —--¡= I where erf x = —— e s2 ds is the error integral, with

\2flunfx/t/ ^jnjo erf(0) = 0 and x—> + oo erf (x) —» 1. Otherwise the statement of the problem is in line with that in the course of Stefan formula derivation (see Fig. 1.16 b). Thus, the temperature distribution in both zones is taken as:

where aunf = /lunf/Cunf is the diffusivity of unfrozen soil. The problem is solved in much the same way as in the previous case by substituting the expressions for heat fluxes in unfrozen and frozen zones into the Stefan condition at the moving freezing boundary and integrating the standard differential equation obtained.

The solution for evaluating the freezing depth m(q) is of the form:

In the case of thawing Aunf and /lfr change places and Cfr is substituted for Cunf in the formula (1.34).

When this formula is compared with the Stefan formula, it is apparent that the values of freezing (thawing) depth calculated from the Leybenzon formula will be lower than those calculated from the Stefan formula for the same period of time. This is to be expected because the Leybenzon formula takes into consideration the non-zero temperature distribution in the unfrozen (frozen) zone before its freezing (thawing). This slows down the front propagation as some portion of heat energy is spent on the temperature lowering (raising) in this zone up to the value at the beginning of phase transition. As one would expect, the Leybenzon formula transforms to the Stefan formula, if the initial temperature in the unfrozen (frozen) zone is 0°C.

Among the formulae for approximate calculations, that put forward by V.S. Luk'yanov and M.D. Golovko in 1957 taking into consideration the heat capacity of freezing (unfrozen) ground as well as the presence of insulating covers on the surface is also worth noting. These researchers had derived a transcendental equation with reference to the solution of which presents calculation difficulties. Therefore Golovko developed a nomogram allowing the solution to be obtained easily to an accuracy sufficient for practical purposes. In spite of the fact that Luk'yanov and Golovko's formula is very useful for engineering calculations, the essential weakness of this formula is the uncertainty in assigning some parameters.

By and large there exist many (more than a hundred) approximate, often semi-empirical formulae put forward by various researchers for calculating the freezing (thawing) depth. However all of them have been obtained assuming the constancy of temperature on the ground surface with time and, what is more important, they do not make allowance for the clear-cut association of the thermal-physical aspects of ground freezing (thawing) and the geologic-geographical nature of this phenomenon. The approximate formulae put forward by V.A. Koudryavtsev and published in the General Geocryology textbooks in 1967, 1978 and 1981 do not have these disadvantages.

V.A. Kudryavtsev's approximate formula for determining the heat turnover and seasonal ground freezing (thawing) depth

The attributes (used for classification) of the freezing (thawing) processes put forward by P.I. Koloskov and developed by V.A. Kudryav-tsev are the required parameters for these approximate formulae. There are four input parameters: annual temperature amplitude at the ground surface A0, the mean annual temperature at the base of the seasonal freezing (thawing) layer f„ean, ground composition and moisture content defining all thermal-physical characteristics such as 1{T and lun[, Cvol, Qph. Kudryavtsev's formulae have been derived for the periodic steady-state temperature regime at the ground surface and represent generalized Fourier's laws. They allow consideration of temperature waves to be extended to media with phase transitions.

The formulae for determining the freezing (thawing) depth, m, and heat storage through the surface for the layer of seasonal freezing (thawing), Qsurf, and the heat storage in the ground underlying this layer, Q , are written as:

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