where ifr and iunf are the temperature of the frozen and unfrozen zones. This expression is termed the Stefan condition at the moving phase boundary. This expression enables us to establish the law of phase boundary propagation. Thus in the case of constant negative temperature at the surface of the originally unfrozen ground, the freezing boundary £(t) propagates through the mass in accordance with the law £(t) = ol^Jt, where a is a constant coefficient depending on the thermal-physical earth material properties, the initial temperature of the ground and the temperature at its surface.

It is possible to approach the problem of ground freezing (thawing) in several ways depending on the degree of consideration of the character of physical processes occurring at the phase transition boundary and in the freezing and thawing zones. Selection of one or another way for approach-

ing a problem for practical calculations depends on the particular features of structure and properties of the ground unit under consideration. For the sake of simplicity we shall consider one-dimensional temperature fields in homogeneous ground masses.

The statement of the problem on ground freezing (thawing) with development of phase boundary (Stefan problem) The statement of the problem of freezing (thawing) with the formation of one phase boundary, i.e. a boundary at which the temperature of phase transition is maintained for the given situation, is the simplest (i.e. all water-phase transitions proceed on this boundary (Fig. 1.14)). Then the thermal physical properties of rocks on this boundary will undergo an abrupt change: Aunf + Afr, aunf + au and Cunf + Cfr where Aunf and Afr are heat conductivities of unfrozen and frozen ground, respectively, aunf and afr are their temperature conductivities [thermal diffusivity coefficients], Cunf and Cfr are the heat capacity coefficients. This statement describes very well the processes of freezing (thawing) in rudaceous materials and sands as well as in faulted (water-saturated) hard rocks. It is actually typical of these rocks that practically all the water phase transitions proceed at a freezing temperature ffre equal or close to 0°C. On further lowering of the temperature, phase transition is not observed and the thermal physical properties of the material remain constant.

Mathematical formulation of the problem includes two equations of heat conduction in this case (for unfrozen and frozen zones) and the Stefan condition for the moving phase boundary is written as: (1.22)

dz dz2

unf dz2


A, unf unf



where / is the position of the lower boundary of the region under consideration. The last two equations are the conditions for conjugate solutions to the equations of heat conduction in frozen and unfrozen zones with a moving phase boundary (Stefan condition). They couple the heat conduc-

X-const 1 jj, const! unf const fro C-const


X = const

0 0

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