Environmental Engineering Reference
In-Depth Information
(
)
∂
P
S
S
w
()
=
(
)
∫
(5.115)
PS
f
b
cow
b
d
b
c
w
w
∂
1
w
l
l
Where, the fractional-flow
function
f
w
measures the water frac-
w
=
c w cow
, then the total velocity can be
expressed as a function of the global pressure:
tion of the total flow. Since
∇= ∇
Pf
P
(
)
∇+
(
)
(5.116)
vv v
=+=−
K
ll
+
P
K
lrlr
+
gz
∇
wo
wo
wwo
o
(
)
+∇
k
e
Ek
+
k
w
erww
,
erow
,
Using the total mobility and global pressure, the following pressure
equation is obtained:
{
}
=
(
)
(
)
(5.117)
−∇
.
KK
l
∇
P
−
l r
+
l r
g
∇+ ∇
z
k
e
E
k
+
k
q
w
w
o
o
w
erww
,
erow
,
The pressure equation is complete for the problem at hand when no-
flow boundary conditions are imposed.
5.10 Numerical Implementation
Commonly used numerical methods for solving two-phase flow problems
include (i) finite difference method, in which functions are represented by
their values at certain grid points and derivatives are approximated through
differences in these values, (ii) method of lines, where all but one variable
is discretized and the result is a system of ordinary differential equations
(ODEs) in the remaining continuous variable, (iii) finite element method,
where functions are represented in terms of basic functions and the PDE
is solved in its integral form, and (iv) finite volume method, which divides
space into regions or volumes and computes the change within each vol-
ume by considering the flux (flow rate) across the surfaces of the volume
(Aziz and Settari, 1979; Allen et al., 1988; King, 1992; Chen et al., 2006).
When applied to reservoir simulation, finite difference method (FDM)
can be very easy to implement and in its basic form is restricted to handle
only rectangular shapes. The method introduces considerable geometrical
error and grid orientation effects. Finite element method (FEM), on the
other hand can handle complicated geometries, and reduces the grid ori-
entation effects. The quality of a FEM approximation is often higher than
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