PSI - Issue 13

A. Kostina et al. / Procedia Structural Integrity 13 (2018) 1159–1164 Author name / Structural Integrity Procedia 00 (2018) 000–000

1161

3

 

0 F      p    

,

(7)

p 

σ - ρ

  

0

0 0

p

 

,

(8)

p 

σ

d - F ρ p

  

d

pd d







where ' k - thermal conductivity coefficient,

0 22 33 σ σ σ σ / 3    - mean stress, K - bulk modulus, G - shear 11









modulus, 22 33 / 3 p p p p    - mean structural strain, Γ p 0 , Γ pd - kinetic coefficients. To close the system (1), (3)-(8) it is necessary to define approximations for 0 / F p   and / d F   p . Solution to the statistical problem of the defect distribution (Naimark (2001)) and hypothesis of coaxiality between σ and p allow us to rewrite the equations (7), (8) in the following non dimensionless form: 0 11 22 33 ε ε ε ε / 3 e e e e    - mean elastic strain, d σ - deviatoric stress tensor, 0 11

0 1 σ ( ) τ 3 p - f p K   1 0 0

  

,

(9)

p 

 

0

 

   

σ

p

1

,

(10)

p 

( )

- f p

 

d

d

2

d

τ 2 pd 

G

p

where τ p 0 - relaxation time for volumetric structural strain, τ pd - relaxation time for deviatoric structural strain, p - effective structural strain. Functions 1 0 ( ) f p , 2 ( ) f p are calibrated according to the experimental data and used for a description of the material hardening. The heat source related to the thermos elastic effect is small compare to the diffusion term. This source is neglected in this work. Therefore, the first law of thermodynamics (1) can be written as

T



   q

0

,

(11)

ρ c

p

t



where c p is specific heat capacity. Density, specific heat capacity under the constant pressure and thermal conductivity of the system, which can be in thermodynamic states “1” (fluid) or “2” (gas) are defined by the following set of expressions: 1 1 2 2 ρ θ ρ θ ρ   , (12)   1 1 ,1 2 2 ,2 1 m p p p α c θ ρ c θ ρ c L ρ T      , (13)

1 1 2 2 ' k θ k θ k   ,

(14)

where 1 (1 ) ρ n n ρ     ; c p, 1 = c p, w n w + c p, o n o + (1- n ) c p, r ; c p, 2 = c p, s n s + c p, o n o + (1− n ) c p, r ; s s ρ ρ n o o r 1 (1 ) w w o o r k k n k n n k     ; 1 θ θ  characterizes the fraction of the first phase in the material, θ 2 =1− θ characterizes the fraction of the second phase, L - is the latent heat of the phase change, n - the total porosity; subscripts w , o , s , r indicate water, oil, steam and 2 (1 ) n k     ; s s k k n k n o o r       θ ρ   2 1 1 2 0.5 1 1   m α θ ρ θρ θρ    ; (1 ) w w o o r ρ ρ n ρ n n ρ     ; 2

reservoir component respectively. 3. Results of numerical simulation

Identification of the material parameters for the constitutive equations (5)-(6), (9)-(10) was carried out based on the experimental data published in (Browning et al., 2017). For this purpose, a three-dimensional numerical simulation of the conventional triaxial loading of the cubic sandstone sample has been carried out. Results of the simulation is presented in figure 1.