PSI - Issue 28

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

677

3

2.1. Three-phase flow Three-phase flow of the pore fluid which consists of steam, water and oil was described by mass balance equations and Darcy’s laws of filtration. The mass balance equation has the following form:     i i i i i n ρ S ρ q t       v , (1) where n is the porosity of the reservoir, i ρ is the density of steam ( i s  ), water ( i w  ) and oil ( i o  ), i S ( , , i s w o  ) is the saturation of each phase, t is the time,  is the divergence operator, i v is the phase velocity, i q ( , i s w  ) is the mass source induced by phase transition. It should be noted that mass sources are considered only for steam and water mass balance equations because phase transitions in oil are beyond the scope of this article. Phase velocities i v are described by Darcy’s law:   ri i por i i Kk p ρ μ     v g , (2) 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. Mass sources i q in (1) were defined according to the Lee et al. (2015):

T T 



,

s s rnS ρ

T T 

sat

     

sat

T

sat

,

(3)

q q

s

w

T T 



,

rnS ρ

T T 

sat



w w

sat

T

sat

where r is the mass transfer intensity factor, T is the temperature, sat T is phase transition temperature. Equations (1)-(3) were supplemented by the condition of a fully saturated media:

1 s w o S S S    .

(4)

2.2. Energy balance equation Energy balance equation for the three-phase flow in porous media with accounting for the convective heat transfer, conductive heat transfer and latent heat source induced by the phase transition can be written in the following form:

t      

    

  

  







 v





1 T n ρ c n   r r 

i i i ρ S c

eff λ T    

i i i i ρ S c T Q 

,

(5)

, , i w o s 

, , i w o s 



 1  

 n λ

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

λ

i i nS λ

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



eff

r

, , i w o s  w Q Lq  is the latent heat source due to the phase transition, L is the latent heat.

is the effective thermal conductivity,

2.3. Momentum balance equation and constitutive equations The equilibrium equation with accounting for the gravity has the form: eff ρ     σ g 0 ,

(6)