Compositional Flow in Fractured Porous Media: Mathematical Background and Basic Physics - Selected Topics of Computational and Experimental Fluid Mechanics - page 7

ʱ

The momentum conservation equation for any phase obeys the form

ˁ ʱ

∂ (ˁ ʱ v

ʱ )

v v

ʱ ʱ

+∇· =−∇ p

ʱ +∇·

T

ʱ + ˆ ʱ ˁ ʱ

F

ʱ +

M

ʱ ,

(9)

∂ t ˆ ʱ

where p is the mechanical pressure in fluid ʱ , T is the viscous stress tensor, F

ʱ ʱ ʱ

is a body force, and M is the rate of momentum exchange from all other phases to

ʱ

phase T represents the viscous transfer per unit volume from the pore

walls to the pore spaces. In most reservoir applications, the flow in porous media is
characterized by low Reynolds numbers (Re

ʱ .Theterm ∇·

1), i.e., the viscous forces dominate
over the inertial ones causing the fluids to move slowly. Under these conditions the
inertial terms on the left-hand side of Eq. ( 9 ) can be neglected. In addition, taking
the average of the right-hand side of Eq. ( 9 ) over the volume of fluid phase

<

within
a representative elementary volume and assuming that gravity is the only body force
acting on fluid

ʱ

z , where g is the magnitude of the gravitational
acceleration and z is a reference depth), it can be demonstrated that the momentum
balance reduces to the well-known Darcy's law (Bear 1988 )

ʱ (i.e., ˆ ʱ F ʱ = g ∇

v ʱ =− K ʱ · ( ∇ p ʱ − ˁ ʱ g ∇ z ) , (10)

where the tensor

k k r ʱ K

ʱ =

μ ʱ ,

(11)

is the mobility of fluid phase . In this expression k is the absolute permeability

tensor of the porous medium, which measures its ability to transmit the fluid, k r ʱ

ʱ

is

the relative permeability of phase , which describes the effects of the other fluid

phases in obstructing the flow of fluid

ʱ

ʱ , and μ ʱ is the dynamic viscosity of phase

ʱ . In any particular rock-fluid system, the mobility K ʱ accounts for much of the

predictive power of Darcy's law. In general, constitutive relations for K ʱ are largely
phenomenological and an expression of common use in many applications is just that
given by Eq. ( 11 ), where the mobility is proportional to the product of the absolute
and relative permeabilities. If the medium is isotropic, the absolute permeability is
diagonal, i.e., k

=

k I , where I is the identity tensor. Otherwise, we say that the
porous medium is anisotropic.
Equation ( 10 ) can be modified into a momentum conservation for the multiphase
mixture. To do so we first define the kinematic viscosity of phase ʱ as ʽ ʱ = μ ʱ /ˁ ʱ

and introduce the following definitions

− 1

k r ʱ
ʽ ʱ

ʽ = , (12)

ʱ

k r ʱ
ʽ ʱ ʽ,

ʻ ʱ = (13)