PSI - Issue 18

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

305 5

 

0 F     p 

,

p 

σ - ρ

  

0

0 0

p

 

   

,

p 

σ

d - F ρ p

  

d

pd d



where 0 p  , pd  are the kinetic coefficients; 0 σ is the mean stress, d σ is the deviatoric stress tensor. To close the system of equations it is necessary to define approximations for the functions 0 F / p   and d F /   p . In case of the thermal oil recovery compressive stresses are small and the prevailing mechanism of the deformation is the dilation caused by the high temperature. According to the proposed model, this effect can be described as an accumulation of volumetric structural 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 (14)-(15) can be written as





σ

1

  

  

  p

  d p

2

,

(14)

3 - bI

p 

σ σ 

c I





0

0

0

0

1

2

c

3 ' K

τ

0

p

   

   

p p

σ

1 I ( )

1 σ σ

,

(15)

p 

c

d







d

dc

  p

τ

2G

p

2 I

pd

2 d

where τ p 0 is the relaxation time for volumetric structural strain;  denotes Macaulay brackets; 0 c σ , dc σ , b , c are the material parameters; ' K is the bulk modulus; G is the shear modulus; τ pd is the relaxation time for deviatoric structural strain; σ :  σ σ ; : p  p p . 2.4. Coupling of structural changes with formation reservoir properties As it was mentioned above, the prevailing mechanism of deformation during viscous oil production is the shear dilation. Shear dilation arises near the thermal front and induces increase in porosity by 4-6% (Shafiei and Dusseault (2013)). The following equation was used to describe the effect related to the rise in porosity due to the changes in the volumetric strain (Rahmati et al. (2017)):   0 1 vol vol n n      , (16)

where 0 n is the initial porosity; 33 vol        . Permeability is related to the porosity according to the relation (Hu et al. (2013)): 11 22

3

.

(17)

 1 K n  



2

n