PSI - Issue 32

A. Kostina et al. / Procedia Structural Integrity 32 (2021) 101–108 A. Kostina/ Structural Integrity Procedia 00 (2021) 000 – 000

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3

for pore pressure, ice pressure and fluid pressure. More detailed description of the model can be found in [20]. The main equations of the model are presented below:       0 l l i i l l n ρ S nρ S ρ t t         v , (1)







,

(2)

λ T C T Q      v

C T

l

ph

t



   σ γ 0 ,

(3)

l l K p z ρ g          

v

,

(4)

l











,

(5)

e vol

3 p N n n b ε α b n T T      

0

0

0

T









ln / ρ ρ p ρ ρ L T T ρ p   

0

l

i

i l

ph

i l

,

(6)

p

 

i

ρ

l







 χ ρ ρ p       1



1

ln / χ ρ ρ L T T ρ p 

0

l

i

i l

ph

l

,

(7)

p



 1  

 χ ρ

l

χρ

l

i

 1 2 T     ε u u , 

(8)









0 : s α T T bp     σ C ε E E ,

(9)

where n is the porosity, l ρ is the water density, l S is the water saturation, t is the time, i ρ is the ice density, i S is the ice saturation,  is the divergence operator, l v is the water velocity, C is the volumetric heat capacity, T is the temperature, λ is the thermal conductivity of the porous media, ph Q is the heat source induced by phase transition, σ is the total stress tensor, γ is the unit weight of porous media, K is the hydraulic conductivity, l p is the pore water pressure, g is the gravity acceleration, z is the vertical coordinate, p is the pore pressure, N is the effective Biot tangent modulus, 0 n is the initial porosity, b is the Biot coefficient, e vol ε is the elastic volumetric strains, T α is the thermal expansion coefficient, 0 T is the initial temperature, i p is the pore ice pressure, L is the latent heat of phase transition, ph T is the phase transition temperature, χ is a parameter defined as   1.5 1 i S  , 0 p is the initial pressure, ε is the total strain tensor, u is the displacement vector, C is the stiffness tensor, E is the unit tensor. Nonlinear equations (1)-(9) were implemented in Comsol Multiphysics® software and solved numerically using finite-element method relative to porosity, displacement and temperature variables. 3. Results of numerical simulation Geometry and dimensions of the simulated domain are presented in Fig.1. Displacements in x and y directions were fixed on the upper edge of the unfixed section. Roller boundary condition was set on the right, left, front, rear boundaries as well as on the well boundary. Vertical displacements were constrained on the top and bottom boundaries of the domain. Zero heat flux was set on the boundary of the well and convective heat transfer with the environment was prescribed on the excavation boundaries. С hange in the stress-strain state of the soil due to its freezing was