PSI - Issue 37

A. Kostina et al. / Procedia Structural Integrity 37 (2022) 431–438 A. Kostina/ Structural Integrity Procedia 00 (2019) 000 – 000

433 3

Kk

(

)

ri μ = −  − v g , i por i p ρ

(2)

i

where K is the absolute permeability, ri k is the relative permeability of each phase, i μ is the dynamic viscosity, por p is the pore pressure, g is the gravity acceleration. Oil dynamic viscosity o μ is assumed to be function of temperature, while water and steam dynamic viscosities are considered as constant. To describe steam condensation we have utilized mass sources i q similar to Lee et al. (2015):

T T −

, sat

,

(3)

q q rnS ρ = − =

T T 

s

w

s s

sat

T

sat

where r is the mass transfer intensity factor, T is the temperature, sat T is the phase transition temperature. Condition of a fully saturated media is applied to close equations (1)-(3):

1 s w o S S S + + = .

(4)

2.2. Energy balance equation Energy conservation law which takes into account convective heat transfer, conductive heat transfer and latent heat induced by steam condensation is given by the equation:

t       

    

  

  

(

)



 v

(

)

1 T n ρ c n ρ S c − +

eff λ T +   −  +  

i i i ρ c T Q =

,

(5)

r r

i i i

, , i w o s =

, , i w o s =



( 1 + −

) n λ

λ

i i nS λ

, , , i r s w o = ) is the heat capacity,

=

where subscript r stands for the reservoir properties, i c (

eff

r

, , i w o s = w Q Lq = is heat source due to the steam condensation, L is the latent heat.

is the effective thermal conductivity,

2.3. Momentum balance equation and constitutive equations The equilibrium equation with gravitational acceleration is expressed by: eff ρ  + = σ g 0 ,

(6)



( 1 + −

) n ρ

ρ

i i nS ρ

=

where σ is the total stress tensor,

is the effective density.

eff

r

, , i w o s = Linear geometric relation for the total strain tensor ε is applied: ( ) 1 2 T =  + ε u u .

(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 plastic strain rate tensor pl ε :

e T pl = + + ε ε ε ε ,

(8)

(

) 0

where 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.