Environmental Engineering Reference
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exhibit significant hysteresis, and the verification of an appropriate model in the
presence of multiphase fluids (with three or more phases) (Stone
1973
) or compo-
sitional effects (Bardon and Longeron
1980
; Amaefule and Handy
1982
) are still
not clear. According to Eq. (
10
), each fluid phase has its own pressure at any point
in the reservoir. At the microscopic scale, the effects of interfacial tension and pore
geometry on the curvatures of fluid-fluid interfaces lead to capillary effects, which
at the macroscopic scale can be quantified in terms of the capillary pressure, defined
as the difference between the pressures of two adjacent phases
ʱ
and
ʲ
in a porous
medium
p
c
ʱʲ
=
p
ʱ
−
p
ʲ
.
(20)
In simple models, the capillary pressure is also assumed to be a function of saturation.
However, in general it is a function of the pore geometry, the physical properties of
the fluids, and the phase saturations (Parker
1989
), i.e.,
p
c
ʱʲ
=
f
(ˆ, ˃
ʱʲ
,
S
1
,
S
2
,...,
S
P
),
(21)
where
interface andwhere it has been assumed
the presence of
P
phases. In actual flows the capillary pressure is generally multi-
valued, exhibits hysteretic behaviour (Hoa et al.
1977
), and depends on fluid compo-
sition (Coats
1980
). If
P
phases coexist, then
P
˃
ʱʲ
is the interfacial tension at the
ʱ
-
ʲ
−
1 independent capillary pressure
functions will appear in the system.
The most restrictive hypothesis concerning Eq. (
10
) is the one that considers the
flow laminar and the fluid movement as dominated by viscous forces. It is therefore
valid when the velocities of the fluids are small. As the flow rate increases, deviations
from Darcy's law occur due to inertia, turbulence, and other high-velocity effects
(Chen et al.
2006
; Mei and Auriault
1991
). Although Darcy's law is valid for Re
<
1,
its upper limit of validity can be extended to Re
10 (Bear
1988
), approximately at
the border between the linear and nonlinear laminar flow regimes. For Re
=
100, in
the turbulent regime, a correction to Darcy's law can be described by the quadratic
relation (Forchheimer
1901
)
>
(μ
ʱ
I
+
ʲ
ʱ
ˁ
ʱ
|
v
ʱ
|
K
ʱ
)
·
v
ʱ
=−
K
ʱ
·
(
∇
p
ʱ
−
ˁ
ʱ
g
∇
z
) ,
(22)
where
ʲ
ʱ
is a factor including the inertial or turbulence effects and
|
v
ʱ
|
is the modulus
of the velocity of phase
. This relation is commonly known as Forchheimer's law
and incorporates laminar, inertial, and turbulence effects (Chen et al.
2006
; Amiri
and Vafai
1994
). A formal derivation of the Forchheimer's equation from volume-
averaging the microscopic momentum balance equation is given byWhitaker (
1996
).
However, much more fundamental research on multiphase flow in porous media is
needed to rigorously include these non-Darcian effects into the model.
If the flow is nonisothermal, we must add a balance equation for the energy,
which introduces the temperature as an additional dependent variable to the system.
In a reservoir, the average temperature of the solid matrix and the fluids in a porous
mediummay not be the same and so heat conductionmay occur between the solid and
ʱ
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