John R. Fanchi - Principles of Applied Reservoir Simulation, Second Edition
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302
Principles
of
Applied
Reservoir Simulation
Once
Z is
known,
the gas
formation volume factor
is
easily determined
for a
given
temperature
and
pressure using
the
real
gas
law.
The
isothermal
gas
compressibility
c
is
obtained
from
Eq.
(28,2)
as
c
=
1
1
1
BZ
"a
P.
(28.3)
where
P
c
is the
critical
pressure (psia).
Real
gas
viscosities
are
computed using
the
method described
in
Govier
[1975].
This method
is a
computerized version
of the
Carr,
Kobayashi,
and
Burrows
[1954]
hydrocarbon
gas
viscosity determination procedure.
Pseudo-Pressure Calculations
Pseudo-pressures
are
defined
by
iK/0
=
2/
^-dP'
(28.4)
P
e
»
g
Z
where
P
1
-
dummy integration variable with pressure units (psia)
P
0
=
reference pressure
=
14.7 psia
P =
specified pressure (psia)
|l
g
=
gas
viscosity (cp)
Z
=
gas
compressibility
factor
The
pseudo-pressure
t|f
(P) is
often
written
as
m(P).
Since
\l
g
and
Z
depend
on
P',
evaluation
of Eq.
(28.4)
is
accomplished
by
numerical integration using
the
trapezoidal rule
and a
user-specified pressure increment
AP'
~
dP'.
Gas
Property Description
Four
different
gas
property descriptions
may be
specified.
Their
descrip-
tions
and
control parameter (KODEA) values follow:
KODEA
]
GAS
DESCRIPTION
Sweet gas:
0.0.0.
1.0
input
12
component mole
fractions
as
0. 0. 0. 0. 0. 0. 0.
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303
KODEA
2
3
4
GAS
DESCRIPTION
Sour gas: input
12
component mole fractions
in the
y\
y-i
y3
y*
0-
o. o. o. o. o. o. o.
where
j/j
=
mole
fraction
of
H
2
S,
y
2
=
mole fraction
of
CO
2
y
3
=
mole
fraction
of
N
2
,
and
Sweet
or
sour
gas
with
the
following
12
component
tions read
in the
following order:
order
mole
frac-
H
2
S,
CO
2
,
N
2
,
Cj,
C
2
,
C
3
,
iC
4
,
nC
4
,
iC
5
,
nC
5
,
C
6
,
C
7+
.
The sum of the
mole fractions should equal one.
Same
as
KODEA
= 3 but
also read critical pressure,
temperature,
and
molecular weight
of
C
7+
.
critical
Correlation
Range Limits
The
following range limits apply
to
correlations used
in
calculating
gas
Z-factors, compressibilities
and
viscosities:
1.05
< — < 3.0
T
c
0.01
< — <
15.0
PC
0.55
< SPG < 1.5
40 < T < 400
where
T
c
=
pseudo-critical temperature (°R)
P
c
=
pseudo-critical pressure (psia)
T
=
temperature (°R)
P
=
pressure (psia)
SPG
=
gas
specific gravity
No
values
of
r,
P, or
SPG
should
be
used
that exceed
the
above correlation
ranges.
If the
range limit
is
exceeded,
a
fatal
error will occur.
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Chapter
29
Initialization
It
is
important when making cross-section
or 3D
runs that
the
pressures
in
the
model
are
correctly initialized.
If
not, phase potential
differences
due to
gravity terms could cause
fluid
migration even though
no
wells
are
active.
Consequently,
a
simple pressure initialization algorithm
is
used
in
WINB4D.
It
is
reviewed below along with
an
option
to
correct pressures
to a
user-specified
datum
and an
option
to
initialize saturations using gravity segregation.
29.1
Pressure
Initialization
Consider
a
gridblock that
may
have
a
gas-oil contact
and a
water-oil
contact
as in
Figure
29-1.
Datum
X
y
\
>
„
GOC,
)
EL(+)
t
r
I
woe
*
w
\r
\
fjL.\
^
'
f
Figure 29-1. Depths
for
pressure initialization algorithm.
We
assume
the
pressure
in the
gridblock
at
model
location
(/,
j,
k)
is
dominated
by the
density
of the
phase
at the
block midpoint
and
that there
are
304
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305
no
transition zones between
different
phases initially.
The
pressure
and
depth
at
the
gas-oil contact
are
PGOC
and
GOC,
respectively.
Similarly,
for
the
water-
oil
contact
we
have PWOC
and
WOC.
The
initial pressure assigned
to the
gridblock
in
Figure
29-1
is
determined
by
the
depth
of the
node (midpoint) relative
to the
respective contact elevations.
Let
us
define
the
depth
of the
block midpoint
from
datum
as
EL
iJk
.
With
this definition,
the
pressure
in the
block
is
given
by the
following algorithm;
a.
IfEL,
yA
<GOCthen
p
g
=
p
g
JB
g
andP..
k
=
PGOC
+
p
g
(EL,
/t
-
GOC)/144
b.
IfEL,
7fc
>WOCthen
P
w
=
(P
w
*c
+
R
™
'
P
g
sc)
/5
w
and
P.
jk
=
PWOC
+
Pw
(EL,..,
-
GOC)/144
c.
If GOC
<;
EL
ljk
<
WOC
then
PO
=
(Post
+
R
so
'
P«c)^o
and
P
ijk
=
PWOC
+
p
0
(EL
iJk
-
GOC)/144
The
above algorithm should
be
reasonable
for
systems with initial transition
zones that
are
small relative
to the
total thickness
of the
formation.
Pressure
Corrected
to
Datum
Pressure P(I,
J, K) of
gridblock
I, J, K
with mid-point elevation EL(I,
J,
K)
may be
corrected
to a
datum depth PDATUM
by
specifying
a
pressure
gradient
GRAD.
The
pressure
at
datum
is
given
by
PDAT(I,
J, K) =
P(I,
J, K)
+
(PDATUM
-
EL(I,
J,
K))*GRAD.
29.2
Gravity-Segregated
Saturation
Initialization
A
simple model
of a
gravity-segregated saturation distribution
is
calculated when
KSI
=
1. For
depths increasing downward,
we
calculate
elevations
and
thicknesses using
the
geometry shown
in
Figure
29-1
as
follows:
Block
BOT = EL + 0.5 *DZ
Block
THICK
= DZ
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Principles
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Applied
Reservoir Simulation
Block
TOP = BOT -
THICK
Water
zone thickness
WTHICK
= BOT - WOC
Gas
zone thickness
GTHICK
= GOC - TOP
The
user must
specify
the
initial
oil
saturation (SOI)
for an
oil-water system
and
the
initial
gas
saturation (SGI)
for a
water-gas system. Given
the
initial
saturations
SOI and
SGI,
the
following algorithm
is
applied [Fanchi,
1986].
Case
1
Case
2
Case
3
Case
4
GOC
TOP
BOT
WOC
TOP
,
GOC
}ft
WOC
,
BOT
}
A
TOP
GOC
BOT
J-'
WOC
GOC
TOP
[/
WOC
;
BOT
S,
=
0
S
0
=
SOI
S
w
=l-SOI
f
GTHICK
g
THICK
f
WTHICK
THICK
C*
IOC*
o
w
- 1 -
5
0
-
bg
,
l
GTHICK
THICK
S
0
=\-
SOI*f
S
g
=
(l-f)*SGI
C
= 1
C
C
^w
L
-
*->o
"
^g
f
_
l
WTHICK
THICK
5=0
g
S
w
=
1 -
SOI*f
S
0
=
SOI*f
If
^^^^
then
5
0
-°0
s
g
=
fg
*
SGI
US
0
<
S
or
,
then
S
—
0
5
W
=
1 -
5G/
S
g
-
SGI
lfS
0
<
S
or
,
then
S
0
=
0
5
W
= 1
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307
Case
5
Case
6
TOP
EOT
GOC
woe
GOC
woe
TOP
BOT
S
0
=
0
S
w
=l-SGI
S
g
=
SGI
S
0
=
S
g
=
Q
S
w
=l
Water
saturation
is
calculated
as
S
w
= 1 -
S
0
-
S
g
in all
cases. Cases
2
through
4
require
the
user
to
enter residual
oil
saturation
S
or
.
29.3 Aquifer
Models
A
reservoir-aquifer system
can be
modeled using small
gridblocks
to
define
the
reservoir
and
increasingly larger gridblocks
to
define
the
aquifer. This
approach
has the
advantage
of
providing
a
numerically uniform analysis
of the
reservoir-aquifer
system,
but it has the
disadvantage
of
requiring more computer
storage
and
computing time because additional gridblocks
are
used
to
model
the
aquifer.
A
more time-
and
cost-effective
means
of
representing
an
aquifer
is to
represent aquifer influx with
an
analytic model. Three models
are
available
as
options
in
WINB4D.
Pot
Aquifer
Aquifer
influx
is
calculated assuming
the
aquifer
is
both small
and
bounded.
The pot
aquifer influx rate
q
wp
is
dependent
on the
pressure change
over
a
timestep
for a
specified
gridblock:
POT
(P
n
-
POT
*
0
(29.1)
where
P",
Af"
are
gridblock pressure
and
timestep
at the
present
time level
n\
P
n
+
\
Ar"
+
'
are
gridblock pressure
and
timestep
at the
future
time level
n
+
I;
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Principles
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Applied
Reservoir Simulation
and
POT is the pot
aquifer
coefficient.
The
minus sign preceding
the
bracketed
term
indicates water
is
entering
the
block when
P
n
>
P
n
+
l
.
Steady-State
Aquifer
The
steady-state aquifer model
is
based
on
Schilthuis's
assumption that
the
water
influx
rate
q
wss
is
proportional
to the
pressure difference between
the
aquifer
and the
hydrocarbon reservoir.
It is
further
assumed that
the
aquifer
is
sufficiently
large that
it
experiences
no net
pressure change throughout
the
producing
life
of the
reservoir. With these assumptions,
WINB4D
computes
steady-state
aquifer
influx
into
a
specified
gridblock
as
q
wss
=
-[SSAQ
(/>°
-
P
w
+
1
)];
SSAQ
*
0
(29.2)
where
P"
+l
is the
gridblock pressure
at the
future
time level
n
+
1
;
P°
is the
initial
gridblock
pressure;
and
SSAQ
is the
proportionality constant.
The
minus sign
preceding
the
bracketed term indicates water
is
entering
the
block when
we
have
the
inequality
p°>p
n
+
\
Carter-Tracy
Aquifer
The
Carter-Tracy
[
1
960]
modification
of
the
Hurst-
van
Everdingen
[
1
949]
unsteady-state
aquifer
influx
calculation
is
available
in
WINB4D.
The
Carter-
Tracy
aquifer
influx
rate
q
wa
for a
specified gridblock
is
*Wr
=
-M
-
B(P
n+l
-
P")]
(29.3)
where
P",P"
+l
are
gridblock
pressures
at
time levels
n and n +
1,
respectively,
The
coefficients
A and B are
given
by
A =
K
t
P(p°
-/>")-
DENOM
with
(29.4)
=
K
t
H
(29
5)
'
DENOM
l
J
DENOM
=
P"
D
+l
-
t£p'
t
£
+l
(29.6)
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309
dP.
tD
dt
D
(29.7)
K
t
=
0.00633
^--
=
AQPAR1
p
=
2n$hcrs
=
AQPAR2
(29.9)
and
c =
c
r
+
c
w
(29.10)
The
quantities
f
fl
and
P
tD
are
dimensionless time
and
pressure,
respectively,
with
t
D
=
K
t
t
and
P
tD
is the
Carter-Tracy influence function
for
the
constant
terminal
rate
case.
The
functions
P
lD
andP',
D
are
numerically represented
by
regression equations
[Fanchi,
1985].
All
remaining parameters
are
defined
as
follows:
c
r
-
rock compressibility
(psi'
1
)
c
w
=
water compressibility
(psi"
1
)
h
~
aquifer
net
thickness (ft)
k
~
aquifer permeability
(md)
r
e
=
external aquifer radius (ft)
r
w
=
external reservoir radius (ft)
s
=
0/360°
where
6 is the
angle
of
aquifer/reservoir interface
W
e
=
cumulative water
influx
at
time
level«,
SCF
[i
=
aquifer water viscosity (cp)
(f)
=
aquifer porosity
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Chapter
30
Well
Models
The
well models contained
in
WINB4D
are
described
in
this
chapter.
User-specified parameters
for
controlling these well models
are
defined
in
Chapter
25.
30.1 Rate Constraint
Representation
Case
1:
Oil
Production Rate
Q
0
Specified
In
this representation, rates
may be
specified
for
injectors
or
producers.
We
assume
the
well
may be
completed
in a
total
of
K
connections,
and the
production rates
for
each connection
k
for a
specified
oil
rate are:
Oil
(30.1)
Water
O
±
-
O
L
i
fin
9\
»w*
*£ok
i
/T,
PU./J
K
'
(PID)—
-
°J*
K
£
A=l
A
"
(PH))-±
o
310
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Gas
a
I.JB
g g
Qok
+
(
R
so\Qak
+
(
R
SW
\Q
W
k
(30,3)
0
°
>
k
where
A.
c
is the
fluid
mobility
of
phase
Q.
and PID is the
well
productivity index.
For
a
more detailed discussion
of
PID,
see
Chapter
31.
Notice that
a PID may
be
specified
for
each connection. This capability lets
the
WINB4D
user
take into
account permeability contrast.
Case
2:
Water Production Rate
Q
w
Specified
Assuming
the
well
may be
completed
in K
connections,
the
production
rates
of
connection
k for a
specified water rate are:
Water
Qwk
=
Qw
K
(30.4)
S
/"DTT\\
1 /
D
(
"LU
i
At
I Jo
,_.L
w
W
J
k
Oil
e
°*
=
^
w
*
T^/if
(30
-
5)
Gas
•^^
?
P
^-J
/T»
\
/-k
.
/•»>
\
/-V
/
,
+ (K
)L{S
,
+
(R
)
k
\s
.
/3Q
5)
?
g
MB.
Case
3: Gas
Production
Rate
Q
g
Specified
Assuming
the
well
may be
completed
in K
connections,
the
production
rates
of
connection
k for a
specified
gas
rate are:
Gas
Q = Q
*k
*
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