PSI - Issue 17

M. Zhelnin et al. / Procedia Structural Integrity 17 (2019) 316–323 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

318

3

2. Theoretical model

Let us consider a thermo-hydro-mechanical model of artificial freezing process of rock mass. It supposed that the rock mass is three-phase material that consists of a drained rock skeleton, water and ice in pore space. During the freezing water in pores is converted to ice at temperature of the phase transition. Rock skeleton is supposed to be a multi-layered poro-elasto-plastic medium that undergoes small deformation. The model includes the energy conservation equation, the mass balance equation and the equilibrium equation. The phase transition is described by Stefan’s model with using the apparent heat capacity coefficient. Water flow is governed by Darcy’s law. The interaction between a fluid and a rock skeleton is described on the basis of Biot theory. Plastic deformation is determined by Drucker-Prager criterion. A system of governing equations of the thermo-hydro-mechanical model has the following form:



,

(1)

v

, p f p f c T c   +

( T  −  = T 

) 0

t



p





t 

( ) f  v

f

,

(2)

S

+

= −

f 

 

vol

f

B

t





k p

(  = −  + v g , ) f f 

(3)

(

) ) 0 − = I

(

 − σ

g ,

B f p p 



(4)

(

)

: T h pl = − − − σ C ε ε ε ε ,

(5)

1 2

( )   =   +   ε u u , T

(6)

0 T s T T  = − ε I , ( )

(7)

ε

I .

0.09 (1 ) n  −

=

(8)

h

p c – effective heat capacity, [J/(kg·K)], T – absolute temperature, [K], t

In (1)-(8):  – effective density [kg/m 3 ],

– time, [s], , p f c – specific heat capacity of fluid [J/(kg·K)],  – the Nabla operator, v – Darcy’s velocity, [m/s],  f p – pore pressure of fluid, [Pa], S – the fluid loss coefficient, vol  – volumetric part of the full strain tensor, g – acceleration of gravity, [m/s 2 ], k – permeability coefficient, [m 2 ],  – dynamic viscosity of the fluid, [Pa·s], σ – Cauchy stress tensor, [Pa], C – stiffness tensor, [Pa], which has two components bulk modulus K , [Pa], and shear modulus G , [Pa], B  – Biot coefficient, f p – initial pore pressure of fluid, [Pa], I – unity tensor, ε – full strain tensor, u – displacement vector, [m], T ε – thermal strain, s  – the thermal expansion coefficient, 0 T – initial temperature of the rock mass, h ε – strain induced by frost heave or thawing, n – porosity,  – phase indicator function, pl ε – plastic strain. Expressions for the thermophysical, hydrodynamic and mechanical parameters are same as in (Panteleev et al. (2017)). According to Drucker-Prager criterion plastic strain is determined by the following way: – effective thermal conductivity coefficient, [W/(m·K)], f  – fluid density, [kg/m 3 ],