Environmental Engineering Reference
In-Depth Information
Analytical expressions that correlate EO relative permeability coeffi-
cients to the wetting phase saturation for the porous medium of interest
can be developed also, as discussed in section 5.8.3.
Assuming that the capillary pressure has a unique inverse function as:
−
1
(5.96)
S
w
=
P
C
Equations 5.87 to 5.90 can be combined as follows:
⎡
⎛
⎜
⎞
⎠
k
⎛
⎜
⎞
⎟
+
k
∂
dP
dS
∂
(
)
=∇
rww
,
rwo
,
(5.97)
Φ
S
r
r
k
∇
P
−
c
∇
S
k
∇
P
⎢
⎢
⎟
ww
w
o
w
0
t
m
m
⎣
w
w
o
⎤
+
k
e
k
∇
Eq
+
⎥
⎥
er ww
,
w
w
⎦
⎡
⎤
⎛
⎜
k
k
⎞
⎟
+
∂
∂
(
)
=∇
(
)
+
row
,
roo
,
Φ
(
1
−
S
)
r
r
k
∇
P
k
∇
P
k
e
k
∇
Eq
o
⎥
+
⎢
wO
O
o
0
er ow
,
o
t
m
m
⎣
⎦
w
o
(5.98)
To solve the above system of equations, appropriate initial and bound-
ary conditions are needed in the space and time domains. For initial con-
ditions
,
usually the main unknowns (phase pressures) are specified over
the entire space domain at t = 0. For boundary conditions, the possible
cases include (i) phase pressure specified as a function of position and
time (Dirichlet), (ii) pressure gradient is known (Neumann), (iii) phase
pressure and pressure gradient in an impervious boundary are specified
(Mixed). In reservoir simulation, usually the no flow boundary condition
is used (Aziz and Settari, 1979; King, 1992; Chen et al., 2006). Also, the
boundary conditions for applied electric field need to be specified on the
reservoir boundaries.
5.9 Solution Strategy
There are multiple solution schemes for the system of equations used in
reservoir simulation (Aziz and Settari, 1979; Eymard, 2003; Chen et al.,
2006). Among these schemes, IMPES (Implicit pressure, explicit satura-
tion), simultaneous solution, sequential, and adaptive implicit methods are
widely used in the petroleum industry. The simultaneous solution method
solves the coupled non-linear equations simultaneously and implicitly.
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