Environmental Engineering Reference
In-Depth Information
instance, on the reservoir boundary one can impose the boundary condi-
tion as,
v
w
.j
= 0, where
j
is the normal vector pointing out of the boundary.
This boundary condition results in an isolated flow system where water
cannot enter into or exit from the reservoir.
5.8.2
Two-Phase Immiscible Flow Under Applied Pressure
Gradient
To derive the two-phase flow equations for water and oil under applied
pressure gradient, several simplifying assumptions are made. These
assumptions include (i) the two fluids are immiscible and incompress-
ible and there is no exchange of chemical species among the fluid phases,
(ii) the rock matrix is incompressible, (iii) the total flow of oil (as displaced
fluid) and water (as displacing fluid) remains constant, (iv) the gradient in
phase pressure of the individual phases is the driving force of that specific
phase, and (v) the effective permeability for the individual phases is a func-
tion of the saturation of each phase.
Assuming Darcy's law holds for flow of both phases (water and oil) and
considering the mass conservation in the system, the simplified two-phase
flow equations are (Aziz and Settari, 1979; Allen et al., 1985; Ertekin et al.,
2001; Chen et al., 2006):
∂
∂
(
)
=−∇
[
]
+
(5.59)
nS
r
r
v
q
ww
ww
w
t
−
k
k
(
)
rw
(5.60)
v
=
∇− ∇
P
r
g
z
w
w
w
m
w
∂
∂
[
]
=−∇
[
]
+
(5.61)
Φ
S
r
r
v
q
oo
oo
o
t
−
k
k
(
)
(5.62)
v
=
ro
∇− ∇
Pg
r
z
o
o
o
m
o
1
(5.63)
S
w
+=
S
o
(5.64)
PPP
C
=−
o
w
Table 5.1 summarizes all the parameters used in the simplified two-
phase flow equations. Capillary pressure used in the equations is due to the
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