## Fro t ayx

"< , * , , >boundary conditions iunf(l,T) = 02(T)j

This statement holds good for the process of soil freezing as well as thawing while the direction of the process (freezing or thawing) is determined in the course of the problem solution.

In this mathematical formulation of the problem the boundary conditions are of type I, taken as the upper (z = 0) and lower (z = /) boundaries, i.e. the change of temperature with time at these boundaries is given. Other types of specification of boundary conditions are also possible:

- of type II, when the value of heat flux or temperature gradient is specified on the boundary:

-of type III, when the conditions on the boundary are specified as a combination of temperature and temperature gradient values on this surface, with the values of both the temperature and the gradient being unknown individually:

where a and are the constant coefficients depending on the character of heat exchange on the surface.

It should be stressed that 4>1(z),4>2(z), O^t), <I>2(t),/(t), i/Kt), are the known functions of their arguments and are specified either in the form of specific formulae with the help of which we can calculate their values at any point or at any moment of time, or in the form of tables for a particular digital set of arguments. The boundary conditions in the statements of the Stefan problem most commonly encountered in geocryology are mixed: the conditions of type I are specified on the upper boundary as a rule, while the conditions of type II are specified on the lower one, i.e. thermal flux from the Earth's interior is given.

It is rather difficult to solve the Stefan problem even in the most simple form. The exact analytical solution, i.e. the solution in the form of formulae for calculating the position of the freezing (thawing) front and the temperature field for the particular moment of time, has been obtained up to now in individual cases only. In general the problem is solved numerically using rather powerful computers.

In the case when the temperature field does not vary with time, i.e. when the field is permanent, dt/dx = 0, the heat conduction equation 'regenerates' into the Laplace equation:

or in a multidimensional case:

dh dx2 dy2 dz2

If the conditions of type I are specified on the boundary of the region under investigation, it is termed a Dirichlet problem. If the conditions of type II are specified, such a problem is termed a Neumann problem.

Statement of the problem of soil freezing (thawing) over a range of temperatures (with the development of a freezing zone) Investigations of the freezing (thawing) processes in soils (dusty sands, sandy silts, silty-clays, clays, etc.) show that the above mathematical model turns out to be too rough. Actually, a great amount of bound water is typical of such soils. As was shown above only free water freezes at temperatures close to 0°C, while the bound water transforms into ice over a range of negative temperatures and these phase transitions turn out to be fundamental, practically down to temperatures of —5 to — 7°C. Thus freezing soils can be divided by convention not into two zones (unfrozen and frozen) with a clear dividing line as in the previous case, but into three zones: unfrozen, freezing and frozen (see Fig. 1.14b). The interface between the unfrozen and freezing zones is the zero isotherm (strictly speaking, the temperature of the beginning of free water freezing ffre, which turns out to be slightly below zero). All the free water freezes on this boundary. The freezing/frozen zones interface is less distinctive and corresponds to the temperature at which less-pronounced phase transitions come to a close. This temperature for various soils is in the range from — 3 to — 7°C. Within the freezing zone the phase transitions proceed with an intensity which falls with decreasing negative temperature and which can be determined from the curve of the unfrozen water content (see Fig. 1.8). This fact can be regarded as the presence of uninterruptedly distributed heat sources (or sinks):

or dt or

Phase transition of free water proceeds at the phase interface (between unfrozen and freezing zones) where the temperature is constant and equal to that of the beginning of phase transition. Therefore in this case the specific heat of phase transitions on this boundary should be understood as the value 0ph = 334[W/vol— Wuni(ttlJ] where Wvol is the initial (volumetric) soil moisture content; ffre is the temperature of the beginning of freezing (or the temperature of the beginning of unfrozen water crystallization fcr).

The essential dependence of thermal physical characteristics of soils in the freezing zone on the temperature, Afr =/(f), Cfr = </>(f), is of great importance when approaching a problem, because the relationship between ice and unfrozen water in this zone is not constant and depends on temperature. This situation complicates the problem because the parameters which are responsible for the temperature field themselves depend on the required temperatures. So, mathematical formulation of the problem for the case of soil freezing in the range of negative temperatures takes the form:

0 0