Issue 49

M. Semin et alii, Frattura ed Integrità Strutturale, 49 (2019) 167-176; DOI: 10.3221/IGF-ESIS.49.18 170 where 1  is the thermal conductivity of the frozen rock mass, W/(m  °С); 2  is the thermal conductivity of the unfrozen rock mass, W/(m  °С). The groundwater velocity is calculated from the Darcy’s law: r l l l k k p      v (4) where r k is the water relative permeability; k is the absolute permeability of rock mass, m 2 ; l  is the dynamic viscosity of groundwater, Pa  s; l p is the hydrostatic pressure in the pore space, Pa. The dependence of the total specific enthalpy tot H on the temperature T of the water-saturated rock mass is expressed by the formula: 2 2 1 1 ( ) , ( ) ( ) ( ) , sc l sc tot sс l sc c T T nL T T H T c T T nw T L T T               (5) where 1  is the density of the frozen rock mass, kg/m 3 ; 2  is the density of the unfrozen rock mass, kg/m 3 ; 1 c is the specific heat capacity of the frozen rock mass, J/(kg  °C); 2 c is the specific heat capacity of the unfrozen rock mass, J/(kg  °C); L is the specific heat of groundwater phase transition, J/kg; sc T is the temperature when the groundwater freezing begins, °C. The specific enthalpy l H can be represented as the function of temperature T : ( ) , ( ) ( ) , l l sc l sc l l sc c n T T nL T T H T nw T L T T            (6) As follows from (6), when the unfrozen water content w is equal to zero, the specific enthalpy l H of water in the pores is also assumed to be zero, since there is no unfrozen water in the pores at such temperatures. The dependence of the unfrozen groundwater content w on the temperature T (or the soil freezing characteristic curve) can be written in the form:   1, ( ) exp , sc sc sc T T w T T T B T T             (7) where B is the empirical parameter, which characterizes the reduction of the water content with decreasing temperature. It is assumed that the relative permeability of the water in pores is a temperature-dependent parameter. Hence we can write   1, ( ) exp , sc r sc sc T T k T T T M T T             (8) where M is the empirical parameter, which characterizes the reduction of the water relative permeability with decreasing temperature. The problem (1) — (8) is supplemented with boundary and initial conditions: 0 out T T   (9)     ( ) 0 fr fr T T T t T n               (10)