PSI - Issue 18

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

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2.1. Mass balance equations and Darcy’s law Mass balance equations for steam, water and oil components have the following form:     s s s s s n ρ S ρ q t     v ,

(1)





n ρ S







w w

v

ρ

q





,

(2)

w w

w

t







n ρ S







o o

v

0

ρ

 



,

(3)

o o

t



where subscripts s , w and o stand for the values related to the steam, water and oil, respectively; n is the porosity; ρ is the density; t is the time; S is the saturation; v is the fluid velocity; q is the mass source. The mass sources s q and w q in equations (1)-(2) arise due to the phase change process, which converts water into the steam. These sources have the following form (Lee et al. (2015)):

T T 



,

s s rnS ρ

T T 

sat

     

sat

T

sat

,

(4)

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 the phase change temperature. Darcy’s law is used to describe filtration of the each component:

Kk





rs μ     v g , s s p ρ

(5)

s

Kk





rw μ     v g , w w p ρ

(6)

w

Kk





ro μ     v g , o o p ρ

(7)

o

where K is the absolute permeability; r k is the relative phase permeability; μ is the viscosity; p is the pressure; g is the gravity. Equations (1)-(7) are supplemented with the condition of the full saturation:

1 s w o S S S    .

(8)