Environmental Engineering Reference
In-Depth Information
The advantage of this method is its stability and that it can take very large
time steps while stability is maintained. The sequential methods solve the
set of equations in an implicit fashion without developing a full coupling
between the equations (Chen et al., 2006; Aarnes et al., 2007). Compared to
IMPES and simultaneous solution schemes, the sequential schemes are less
stable but more computationally efficient. The adaptive implicit scheme
can be referred to as middle ground between the IMPES and sequential
solution schemes as at a given time step, the expensive simultaneous solu-
tion scheme is assigned to the blocks that require it, and the IMPES scheme
is implemented on the remaining blocks (Chen et al., 2006; Aarnes et al.,
2007).
The IMPES technique, developed by Sheldon and coworkers (1959) and
Stone and Grader (1970) is widely used in the petroleum industry. The
method separates the computation of pressure from saturation. The cou-
pled system is split into a pressure equation and a saturation equation and
the pressure and saturation equations are solved using implicit and explicit
time approximation approaches. The technique can be implemented for
solving the set of equations for two-phase immiscible flow under applied
pressure and electrical gradients. In the following section, the pressure
and saturation equations for the case of two-phase immiscible flow under
applied pressure and electrical gradients are derived to be implemented in
IMPES solution technique.
5.9.1
The Saturation Equation for Two-Phase Incompressible
Immiscible Flow
The continuity equations for the two phases are expressed as (Aziz and
Settari, 1979; Allen et al,, 1988; Ertekin et al., 2001):
∂
)
+∇
[
]
=
∂
(
(5.99)
nS
r
.
r
v
q
aa
aa
a
t
The water, oil, and total mobility (functions of water saturation) are
defined as (Aziz and Settari, 1979):
()
=
⎛
k
⎞
⎟
rw
,
(5.100)
l
s
⎜
ww
m
w
()
=
⎛
k
⎞
⎟
ro
,
(5.101)
l
s
⎜
ow
m
o
Search WWH ::
Custom Search