John R. Fanchi - Principles of Applied Reservoir Simulation, Second Edition
Подождите немного. Документ загружается.
Chapter
32
The
IMPES
Formulation
The
following
section
from
Fanchi,
et
al.
[1982] shows
how the flow
equations
for a
black
oil
simulator
can be
recast
in a
form
that
is
suitable
for
solution
by a
numerical technique.
The
numerical technique
is
based
on the
formulation
originally presented
by
Sawyer
and
Mercer
[1978].
32.1 Flow
Equations
and
Phase Potentials
The
form
of the
Darcy velocities (Eqs.
(4.10)
through
(4.12))
may be
simplified
by
defining
the
potential
€>
p
of
phase
p as
$
=
p
-
_Z
(
32>
i)
P
P 144
and
we
have used
the
assumption that
g=g
c
.
In
this notation, including
x, y, and
z
directional permeabilities
and
unit vectors
i, j, k, the
Darcy velocities
may
be
written
as
v
= -K
dx
y
By
z
dz
(32.2)
K
•
A...VO...
=
-A.
322
(32.3)
TEAM LinG - Live, Informative, Non-cost and Genuine!
Part
V:
Technical
Supplements
323
a$
°
" -
---
'•
"--
(32,4)
iK
—-
+
jK
—
+
kK
By
z
dz
have
used
the
dyadic notation
K
to
signify
that permeability
is a
tensor
of
rank
two.
The
expanded
form
of
Eqs. (32.2) through (32.4) employs
the
common
assumption that
the
coordinate axes
of our
reference system
are
aligned along
the
principal axes
of
K.
As
discussed
in
Chapters
7 and 8, and
associated
references,
this assumption impacts
the
ability
of the
simulator
to
accurately
model fluid flow.
Combining
Eqs. (4.27) through (4.29) with Eqs. (32.2) through (32.4)
gives
(32
.
5)
St
B
(32.6)
and
(32.7)
Equations
(32.5) through (32.7)
are
equivalent
to
Peaceman's [1977] Eqs.
(1-
105)
through (1-107)
for a
three-dimensional system, except
we
have also
allowed
gas to
dissolve
in the
water phase.
Our
rate
and
coordinate system sign
conventions also
differ.
If
these
differences
are
taken into consideration,
the
formulations
are
seen
to be
equivalent.
32.2 Introduction
of the
Capillary
Pressure Concept
The
presence
of
oil-,
water-,
and
gas-phase
pressures
in
Eqs. (32.5)
through
(32.7) complicates
the
problem.
We
simplify
the
handling
of the
phase
TEAM LinG - Live, Informative, Non-cost and Genuine!
324
Principles
of
Applied
Reservoir Simulation
pressures
and
potentials
in the flow
equations
by
using
the
capillary pressure
concept.
Let us
define
the
difference
in
phase
pressures
as
p
cow
=
P
o
-
P
w
(32.8)
and
P
=
P
~
P
-
(32 9)
ego
g o
\~>^.J/
The
differences
P
cow
and
P
cgo
are the
capillary pressures
for
oil-water
and
gas-
water systems, respectively. Experimentally
P
cow
and
P
cgo
have been observed
to
be
principally
functions
of
water
and
gas
saturations,
respectively.
Using Eqs.
(32.8)
and
(32.9) lets
us
write
the
water
and gas
phase potentials
as
Pw*
$
= P ~ P -
-2—
(32
10)
*w
J
o
cow
~
AA
\J±.iv)
144
and
®
8
=
p
o
+
p
cgo
-
~7
(32.11)
*
* 144
Combining
Eqs. (32.5) through (32.7) with Eqs. (32.10)
and
(32.11)
and
rearranging yields
Oil
—
~.
t
%•„}
_
_
?„
d
.
S
}
~
(32-12)
«
I
A,
1 a
V
• K • —
VP
+
CGL
-
'
v^J
Po,c
5r
Water
^
L
=
~~
U~
(32.13)
f
—
\
w
i
TEAM LinG - Live, Informative, Non-cost and Genuine!
Part
V:
Technical
Supplements
325
Gas
(32.14)
d_'
dt
B^
B,
The
gravity
and
capillary contributions
to the
phase
pressures
have been
collected
in the
terms
CG
0
,
CG
W
,
and
CG
g
:
CG„
= -V • K
•Z
V
=
-V • K
(32.15)
(32
-
16)
and
,
CG„
= V • K
-
-*
-,
,
144
- V
ego
JLtt
(32.17)
>
V
\P.
1
A
i
\
n
I COW
t
A A
144
/
S
I
144
Essentially
our
task
is to
solve Eqs.
(32.12)
through
(32.14)
and
saturation
constraint
Eq.
(4.20)
for the
four
unknowns
P
0
,S
0
,S
W
,
and
S
g
.
All
other physical
properties
in the
equations
are
known,
in
principle,
as
functions
of the
four
unknowns,
or from field and
laboratory data.
32,3
The
Pressure Equation
The
procedure used
in
WINB4D
to
solve
the flow
equations requires that
we
first
combine Eqs. (4.20)
and
(32.12)
through
(32.14)
such that
we
have
only
TEAM LinG - Live, Informative, Non-cost and Genuine!
326
Principles
of
Applied
Reservoir Simulation
one
equation remaining
for the
unknown pressure
P
0
.
We
proceed
by
using
the
following
shorthand
for
Eqs.
(32.12)
through
(32.14):
Oil
Water
r
d
f
.
O
L
= —
4>-Jt
(32.19)
a*
I
*J
Gas
I.
=
a
SL
R.-S^
RS
"g
"so~o
(32.20)
where
L=V-K-
—
VP+CG-
—,
(32
21)
/)
__
rt
/)
?
f
*»!
JM'
*
JU
J.
I
«
A
r
= v • K •
-
and
L
0
- V •
g
K
•
g
+
so
°
+
R R
L
\
"g
"°
*«A
)
w
/
CG
-
-^-
(32.23)
Recognizing that formation volume factors,
gas
solubilities,
and
porosity
are
functions
of
pressure,
we use the
chain rule
to
expand
the
accumulation terms
(time derivatives)
of
Eqs.
(32.18)
through
(32.20):
Oil
Q\
O
i3
n
O
CD
/i
T
^
"*"^
TEAM LinG - Live, Informative, Non-cost and Genuine!
Part
V:
Technical
Supplements
327
Water
L
- A
^
"
Gas
L. =
B_
dt
B.
a/>
R
2
a
p.
o
o
dP,
dB^
(32.26)
dt
B...
dt
B
.
o
w
o
dt
The
saturation constraint
S
o
+
S
*
+
S
g
=
1
(32.27)
is
now
used
to
remove
dS^/d/
from Eq.
(32.26). Differentiation
of Eq.
(32.27)
by
t and
rearranging gives
as
as
as
—I
=
_—°-
-
—in.
(32.28)
a?
a/
ar
Substituting
Eq.
(32.28) into
Eq.
(32.26)
and
simplifying yields
TEAM LinG - Live, Informative, Non-cost and Genuine!
328
Principles
of
Applied
Reservoir Simulation
JLt
^
"°~
A
B
sl
4>
dS..
dt
\
B.
Bl
dt
as
2
^o
^
3P
0
a/
*
(32.29)
Equations
(32.24), (32.25),
and
(32.29)
are
three equations
for the
three
unknowns
P
0
,
S
0
,
S
w
.
Multiplying
Eq.
(32.24)
by
(B
0
-
R
so
B
g
),
Eq.
(32.25)
by
(B
w
-
R
sw
B
g
\
Eq.
(32.29)
by
B
g
,
and
adding
the
results gives
TEAM LinG - Live, Informative, Non-cost and Genuine!
Part
V:
Technical
Supplements
329
B
dt
?
0
*J
dt
BP
n
o o
dt
-
E
dt
!4.
s
»*
a
^
,
J
«
S
°
J?
"'
a
*
I
8P
°
8P
0
B
g
dP,
B
0
dP
0
j
dt
dP
dt
a*
,
(32.30)
*„
a/>.
B,
dp
0
B
*
d
where some simplification
has
been performed. This mess
can be
greatly
simplified
by
multiplying
the
bracketed terms
and
then combining with appro-
priate terms
in the
curly brackets.
We
also notice
the
terms involving time
derivatives
of
S
0
and
S
w
vanish identically.
The
result
is
+
s
S\
d
^
-
o)
dp
g
°.
dt
(32.31)
£,
B
TEAM LinG - Live, Informative, Non-cost and Genuine!
330
Principles
of
Applied
Reservoir Simulation
Oil
and
water
compressibilities
include
the
effect
of
gas:
B
0
dP
o
(32,32)
W
^
Gas
compressibility
is
B
4-
£
(32.34)
while rock compressibility
has the
form
1
ad:
r*
(32.35)
0
Total compressibility
for the
system
is the sum
c
t
=
c
r
+
c
o
s
o
+
c
w
s
»
+
c
g
s
g
(32.36)
Employing these definitions, Eqs. (32.21) through
(32.23)
and
(32.27)
in Eq.
(32.31)
gives
(*o
-
R
s
«
A a
+
CG - —
V
•
K
-
—VP
+
CG
+
CG -
*
(32.37)
a/>
^
t
-%
dt
Equation (32.37)
is
called
the
pressure equation.
The
pressure
equation
does
not
contain
any
time derivatives
of
saturations.
WINB4D
is
coded
to
solve
the
three-
TEAM LinG - Live, Informative, Non-cost and Genuine!
Part
V:
Technical
Supplements
331
dimensional,
three-phase
flow
equations
by
first
numerically solving
the
pressure
equation
for
P
0
,
then using
the
results
in
Eqs.
(32.18),
(32.19)
and
(32.27)
to find
the
phase saturations. This procedure
is an
example
of a
numerical method
known
as the
IMplicit Pressure/Explicit Saturation
(IMPES)
procedure.
TEAM LinG - Live, Informative, Non-cost and Genuine!