Compositional Flow in Fractured Porous Media: Mathematical Background and Basic Physics - Selected Topics of Computational and Experimental Fluid Mechanics - page 11

with the constraint

J ʱ =

,

0 (30)

ʱ

which can be used to decompose the phase velocity in Eq. ( 28 ). Summing up Eq. ( 28 )
for all fluid phases and solid matrix and using the constraint ( 27 ) along with Eq. ( 29 ),
we finally obtain

T

− ˆ)ˁ R C R ] ∂

T
∂

[ ˆˁ C + ( 1

t + ʳ C ˁ

C v ·∇ T + C I

ʱ +

C R I R

ʱ

ʱ

ʱ ) +∇· k T , eff ∇

T −

=−∇· ( p v C J ·∇ T

ʱ ʱ ʱ

ʱ

− ʵ r ˃ SB T 4 + Q , (31)

for the energy balance equation of the multiphase mixture, where ˁ C is the heat

capacity of the multiphase fluid mixture defined as

ˁ C = ˁ ʱ S C

ʱ ,

(32)

ʱ

ʱ

k T , eff is an estimation of the effective thermal conductivity of the composite system
given by

k T , eff = ( 1 − ˆ) k T , R + ˆ S k T ,ʱ , (33)

ʱ

ʱ

and ʳ C is the correction factor for energy advection defined as

ˁ ʱ ʻ ʱ

C

ʱ

ʳ C = ʱ ˁ ʱ . (34)

S C

ʱ ʱ

To complete the mathematical description of multiphase flow and heat transfer in
non-deformable porous media, the above equations need to be supplemented with a
number of constitutive equations. As expressed by relations ( 18 ) and ( 21 ), the relative
permeabilities, capillary pressures, and thermal conductivity in most applications are
assumed to be functions of the fluid saturations, while the phase density and dynamic
viscosity are treated as functions of the pressure and temperature (Wu and Qin 2009 ).
Equations ( 7 ), ( 10 ), and ( 31 ) provide 2 P

+ 1 differential equations, while there are

3 P , and T . The additional P relations to

determine a solution of the system are provided by the constraint ( 6 ) and the P

+ 1 independent variables: S , v , p

ʱ ʱ ʱ

− 1

independent capillary pressure functions.