PSI - Issue 28

A. Kostina et al. / Procedia Structural Integrity 28 (2020) 675–683 Author name / Structural Integrity Procedia 00 (2019) 000–000

678

4





 n ρ

where σ is the total stress tensor, is the effective density. Geometric relation for the total strain tensor ε in case of the small strains is defined as:   1 2 T    ε u u . , , i w o s  1   eff i i nS ρ r ρ 

(7)

where u is the displacement vector. The total strain rate tensor ε  is decomposed into elastic strain rate tensor e ε  , thermal strain rate tensor T ε  and structural strain rate tensor p  (Naimark (2003), Plekhov et al. (2009)):

e T    ε ε p ε     ,

(8)

where 0 d   p p p    , 0 p is the volumetric part of the structural strain tensor; d p is the deviatoric part of the structural strain tensor,   0 T T T T   ε E  is the thermal strain;  is the thermal expansion coefficient; E is the unit tensor; 0 T is the initial temperature. Hook’s law in the framework of effective stress concept proposed by Biot could be written as     0 : T B por α T T α p      σ C ε E p E , (9) 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. Structural strain p defines additional contribution to strain induced by the initiation and coalescence of the defects. Applying the standard thermodynamic formalism we can obtain constitutive equations for volumetric and deviatoric parts of the structural strain rate tensor in the form (Naimark (2003), Plekhov et al. (2009)):

 

0 F     p 

,

(10)

p 

σ - ρ

  

0

0 0

p

r

 

F ρ

d     p 

,

(11)

p 

σ

-

  

d

pd d r

where 0 p  , pd  are the kinetic coefficients, 0 σ is the mean stress, d σ is the deviatoric stress tensor, is the free energy of the reservoir in the presence of defects. To close the system of equations it is necessary to define approximations for the partial derivatives of the free energy function F which are included in equations (10)-(11). In case of the thermal oil recovery compressive stresses are small and the prevailing mechanism of the deformation is the shear dilation caused by the high temperature. According to the proposed model, this effect can be described as an accumulation of volumetric defects due to the increase in shear defects. Therefore, the approximation functions were assumed to depend on the first 1 I and the second invariant 2 I of the structural strain tensor. In the final form equations (10)-(11) can be written as ( , , ) e F F T  ε p





σ

1

  

  

  p

  d p

2

,

(12)

3 - bI

p 

σ σ 

c I

0





0

0

0

1

2

c

3 ' K

τ

0

p

   

   

σ

p p

( )

I

1

,

(13)

p 

σ σ 

2 2 d c G 



1

d

dc

  d p

τ

p

I

pd

2