Issue 49

M. Semin et alii, Frattura ed Integrità Strutturale, 49 (2019) 167-176; DOI: 10.3221/IGF-ESIS.49.18

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 k k p      v l

(4)

l

l

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.

H on the temperature T of the water-saturated rock mass is expressed by

The dependence of the total specific enthalpy tot

the formula:

2 2 c T T nw T L T T               1 1 ( c T T nL T T ) , ( ) ( ) , sc l sс l

sc

tot H T

( )

(5)

sc

where 1  is the density of the frozen rock mass, kg/m 3 ; 2 specific heat capacity of the frozen rock mass, J/(kg  °C); 2

 is the density of the unfrozen rock mass, kg/m 3 ; 1 c is the specific heat capacity of the unfrozen rock mass, c is the

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 :

( c n T T nL T T      ) ,

  

l l

sc

l

sc

H T

( )

(6)

l

( ) , nw T L T T  



l

sc

As follows from (6), when the unfrozen water content w is equal to zero, the specific enthalpy l 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: H of water in the pores is

1,

  

T T T T  

sc

w T

( )

(7)





  

B T T

exp   

,



sc

sc

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,

  

T T T T 

sc

k T

( )

(8)





r

   

M T T

exp   

,



sc

sc

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:

(9)

out T T  

0

 fr T T T t T n       ( )

  



 

(10)





0

 



fr

170