PSI - Issue 18

A. Kostina et al. / Procedia Structural Integrity 18 (2019) 301–308 Author name / Structural Integrity Procedia 00 (2019) 000–000

304

4

2.2. Energy balance equation Heat transfer in porous media is described by the energy balance law and accounts for the conduction and convection mechanisms as well as the latent heat associated with the phase transition process:

t      

    

  

  







 v





1 T n ρ c n   r r 

(

) ρ S c nT Q  i i i

i i i ρ S c

eff λ T    



,

(9)

i

, , i w o s 

, , i w o s 



 1  

 n λ

λ

i i nS λ



where subscript r stands for the reservoir properties, c is the heat capacity;

is the effective

eff

r

, , i w o s 

w Q Lq  is the heat source due to the phase transition; L is the latent heat.

thermal conductivity;

2.3. Momentum balance and constitutive equations Momentum balance equation, which accounts for the gravity force can be written as: eff ρ     σ g 0 ,

(10)



 1  

 n ρ

where σ is the Cauchy stress tensor;

ρ

i i nS ρ



is the effective density.

eff

r

, , i w o s 

In case of the small strains the total strain tensor is defined as   1 2 T     ε u u .

(11)

where ε is the total strain tensor; u is the displacement vector. The symmetrical tensor p which has a meaning of the additional strain induced by the initiation and coalescence of the defects is used as an internal variable describing structural evolution in the material. Therefore, the kinematic relation for the total strain rate has the form (Naimark (2003), Plekhov et al. (2009)):

0 T d     ε ε p p ε      , e

(12)

0 p is the volumetric part of the structural strain tensor; d p is the deviatoric part T T T   ε E  is the thermal strain;  is the thermal expansion coefficient; E is the

where e ε is the elastic strain tensor;



 0

of the structural strain tensor;

unit tensor; 0 T is the initial temperature. Effective stress concept is used to take into account the effect of pore pressure on the stress-strain state:     0 : T B α T T α p      σ C ε E p E , (13) where C is the stiffness tensor which has two components in case of an isotropic elasticity (Young’s modulus and Poisson’s ratio); B α is the Biot coefficient . Let us assume that the free energy F is a function of the elastic strain, structural strain and the temperature, i.e. ( , , ) e F F T  ε p . Taking into account this hypothesis, (12) and the standard thermodynamic formalism, we can obtain the constitutive equations for volumetric and deviatoric parts of the structural strain tensor in the form: