Environmental Engineering Reference
In-Depth Information
5.8.3
Contribution of Viscous Coupling
As discussed in section 5.8.2, Darcy's Law for single-phase flow has been
directly extended to two-phase flow using effective permeability coeffi-
cient (
k
α
):
−
1
(
)
v
=
k
∇− ∇
P
r
g
z
[α =
w,
for wetting phase;
=
n,
for non-wetting phase]
a
a
a
a
m
(5.65)
a
Where,
v
α
is the velocity,
m
α
is dynamic viscosity,
k
α
is effective permeabil-
ity,
P
α
is pressure gradient, and
r
α
is density of phase
α
. In reservoir simu-
lation, the relative permeability term (
k
rα
) is used widely which is related to
absolute and effective permeability through the following expression:
∇
(5.66)
k
=
K
.
k
r
a
a
Where,
K
is intrinsic permeability, and
k
rα
is relative permeability with
respect to phase α. Relative permeability coefficients are typically deter-
mined as a function of the wetting phase saturation using standard experi-
mental procedures (Brooks and Corey, 1964; Dake, 1978; Kalaydjian, 1991;
Bentsen, 1993; Corey, 1994).
The validity of the key assumption that Darcy's Law holds for two-
phase flow equations has been disputed since it implies that the flow of
two fluids are essentially uncoupled and each flow has its own channel
with own pressure or elevation gradients acting as the driving forces. The
generalized immiscible two-phase flow model states that the flow of each
fluid phase is a linear function of the gradients of both phases (Kalaydjian,
1991; Liang and Lohrenz, 1994; Avraam and Payatakes, 1995; Bentsen,
1993, 1998; Dullien and Dong 1996; Li et al., 2005). Therefore, fluid flow
of each phase depends not only on the gradient of that particular phase
but also on the corresponding gradient of the other phase. The gener-
alized two-phase flow models omit the viscous coupling between the
two immiscible fluids due to momentum transfer across the fluid-fluid
interface. In oil recovery applications, depending on the characteristics
of the formation rock, water and the residing oil, viscous drag could have
a significant contribution to the quantity of oil produced. Indeed, actual
production rates of heavy oil up to 100 times higher than those computed
by Darcy approach which omits the viscous coupling has been reported
in literature (Yeung, 1996; Tremblay, 1997; Ortiz-Arango and Kantza,
2008).
Search WWH ::
Custom Search