Compositional Flow in Fractured Porous Media: Mathematical Background and Basic Physics - Selected Topics of Computational and Experimental Fluid Mechanics - page 8

ʽ ʻ ʱ

where is the mobility of each

phase in the multiphase mixture, which is commonly referred to as the fractional
flow function. It must obey the constraint

is the kinematic viscosity of the mixture and

ʻ ʱ = 1 . (14)

ʱ

Using the definitions ( 12 )-( 14 ) into Eq. ( 10 ) and summing up over all phases, we
obtain the momentum conservation equation of the multiphase mixture

· ∇ z

k
ʽ

ˁ v = ˁ ʱ v

ʱ =−

p − ʳ ˁ ˁ g ∇ (15)

ʱ

where

∇ p = ʻ ʱ ∇ p

ʱ ,

(16)

ʱ

and

ʱ ʻ ʱ ˁ ʱ

ˆ ʱ

ʳ ˁ =

S ʱ ˁ ʱ ,

(17)

are, respectively, the mixture gradient pressure and the density correction factor
(Starikovicius 2003 ). The definition of the mixture pressure according to Eq. ( 16 )
is somewhat non-conventional. For instance, it is satisfied only for homogeneous
cases, while for heterogeneous, multidimensional, multiphase systems it is not always
possible to define such function.
In multiphase flow the function k r ʱ to wet the

porous medium. In practical applications, the relative permeabilities are assumed to
be known functions of the phase saturations, which must be empirically determined
(Morel-Seytoux 1969 ). The simplest correlations used for the relative permeabilities
are power functions of the phase saturations, which for a gas-liquid system are

indicates the tendency of phase ʱ

s l ,

n

k rl = k rg = ( 1 − s l ) , (18)

where s l is a normalized liquid saturation defined by

S lr
S lm −

S l −

s l =

S lr ,

(19)

with S lr being the residual or irreducible liquid saturation and S lm the maximum
achievable liquid saturation, which in many cases is less than unity. At the irre-
ducible saturation, the liquid becomes immobile since no interpore connections of
liquid exist. Identical forms to Eq. ( 18 ) with n

3 are widely used in petroleum and
nuclear safety engineering, porous heat pipes (Wyllie 1962 ). However, this picture
of relative permeabilities is quite simplistic. In nature relative permeabilities often

=