John R. Fanchi - Principles of Applied Reservoir Simulation, Second Edition
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312
Principles
of
Applied
Reservoir Simulation
Oil
/""I
/"^
I
®
O
\
y
0
k
=
y
g
k\
.
,„
(30.8)
Water
a
, = O
J
——
—
flf)
Q\
wk
X'gkl
\
I
n
\J\J.y)
Solution
gas in
both
oil and
water
is
neglected when
a gas
production rate
is
specified.
This
is a
reasonable assumption
for
wells producing primarily
free
gas.
Case
4:
Total Production Rate Specified
When
the
total reservoir voidage rate
Q
T
is
specified,
we
first
compute
the
phase mobility ratio
for all
connections:
Oil
Mobility Ratio
*(
K
o
}
a
oT
=
SI
^
+
^
+
^
(30.10)
Water
Mobility Ratio
K
(
A...
1
Gas
Mobility Ratio
K
(
*
}
a
g
T
~
^
7
T
T~
(30.12)
*
=1
\
^o
+
^w
+
^gj
k
We
now
compute
the
total
oil
rate
(
\
x-v
T^^.
\^~
(30.13)
where
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*
0
=
7
£
(B
0
\
(30.14)
A
Jt=l
is
the
average
oil
formation volume
factor
for all
connections
in
which
the
well
is
completed.
Given
Eq.
(30.13),
we
simply proceed
as in
Eqs.
(30.1)
through
(30.3)
above.
Case
5:
Injection Rate Specified
If
the
well
is a
water
or gas
injector,
the
user must specify
the
total water
or
gas
injection rates
Q
w
or
Q
g
,
respectively,
and a
well
injectivity
index
(WI)
for
each connection.
The
injection rate
for
each connection
is
then allocated
as
follows:
Water
Injection Rate
(30.15)
Gas
Injection Rate
[WI(A,
0
+
X
w
+
A,p]
t
gk
~
g
~K~
~
(30.16)
k~
1
It
is
important
to
note that allocation
of
injection
fluids is
based
on
total
mobilities,
and not
just
injected
fluid
mobility. This
is
necessary
for the
following
reason:
If an
injector
is
placed
in
a
block where
the
relative permeabil-
ity
to the
injection
fluid is
zero, then
the
simulator using injection
fluid
mobility
only would prohibit
fluid
injection even though
a
real well would allow
fluid
injection.
A
common example would
be
water injection into
a
block containing
oil
and
irreducible water.
To
avoid
the
unrealistic result
of no fluid
injection,
we
assume
the
total mobility
of the
block should
be
used.
For
most
cases,
the
error
of
this method will only
persist
for a few
timesteps because,
in
time,
the
mobile
fluid
saturation
in the
block
will
be
dominated
by the
injected
fluid.
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314
Principles
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Applied
Reservoir Simulation
30.2 Explicit Pressure Constraint Representation
Case
\:
Oil
and/or
Water Production Wells
We
assume that flowing
bottomhole
pressures
(PWF)
and
well
PIDs
are
specified
for a
pressure-constrained well.
The oil and
water rates
in
STB/D
for
connection
k
are
given
by
V
Qnk
=
PID
(P
n
-
PWF),
(30.17)
and
A,
(P
n
-
PWF),
(30.18)
B
k
where
the
explicit pressure
P"
is
used.
If
P"
<
PWF,
the
well
is
shut
in.
When
P"
>
PWF,
Q
ok
and
Q
wk
are
calculated
and
then substituted into
Eq.
(30.3)
to
Case
2: Gas
Production Well
The
laminar-inertial-turbulent
(LIT) method
may be
used
to
represent
a
gas
production well.
The LIT
method entails
fitting gas
well test data
to the
equation
aQ
+
bQ
=
i|r
-
t|f
(30.19)
where
ty
R
=
pseudo-pressure corresponding
to
shut-in pressure
P
R
(psia
2
/
cp)
t|V
=
pseudo-pressure corresponding
to a
specified well
flowing
pressure
/^(psiaVcp)
ag
g
=
laminar
flow
^(?
g
2
~
inertial
and
turbulent
flow
WINB4D
employs user specified values
of
a, b,
P
wf
,
and a
table
of
pseudo-
pressure versus pressure values
to
compute total
gas
well production rate
as
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(3020)
where
ty
R
is the
pseudo-pressure corresponding
to the
nodal pressure
P".
Rates
for
each phase
in
connection
k are
computed
by
mobility allocation
as
shown
in
Eqs, (30.7) through (30.9).
Case
3:
Injection Wells
The
injection rate
for a
water
or gas
injection well
is
computed
from
(
A.
+
A,
+ V
I
B
,
where
the
subscript/?
denotes water
or
gas,
and
PID=WI.
Fluid injection occurs
when
P
n
<
PWF.
If P" >
PWF,
the
injection well
is
shut
in.
Also note that total
mobility
is
used
for the
injection
well rate calculation.
The
reason
for
this
was
discussed
in the first
section
of
this chapter.
30.3
GOR/WOR
Constraints
Maximum
gas-oil
and
water-oil ratios (GORMAX, WORMAX
respec-
tively)
are
input
by the
user
and
apply
to
every
oil
production well.
GOR for a
well
is
defined
as
total
gas
production divided
by
total
oil
production
for all
active well completion intervals.
If GOR
for
the
well
exceeds
GORMAX, then
the
completion interval (connection) with
the
highest
GOR
will
be
shut
in. If
more than
one
connection
has the
same maximum GOR,
the
shallowest
connection will
be
shut
in first. The
procedure
is
repeated until
GOR is
less than
GORMAX
or
until
the
well
is
shut
in.
The
ratio
WOR is
defined
as
total water production divided
by
total
oil
production
for all
active well completion intervals.
If
WOR
for
the
well exceeds
WORMAX, then
the
completion interval (connection) with
the
highest
WOR
will
be
shut
in. If
more than
one
connection
has the
same maximum
WOR,
the
deepest connection will
be
shut
in first. The
procedure
is
repeated
until
WOR
is
less than WORMAX
or
until
the
well
is
shut
in.
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Principles
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30.4 Fluid Withdrawal Constraints
Fluid
withdrawal
from
explicit pressure controlled production
wells
can
be
constrained
as
follows:
a.
A
minimum
oil
production rate
can be
specified;
b. A
maximum
oil
production rate
can be
specified;
and
c.
A
maximum
liquid
(water plus oil) withdrawal rate
can be
speci-
fied.
A
positive
value
of QO for a
pressure controlled production well
is
used
as
the
minimum allowed
oil
production rate.
If
the
calculated
oil
production rate
drops below
the
minimum allowed value,
the
well
is
shut
in.
A
positive value
of QW for a
pressure controlled production well
is
used
as the
maximum allowed
oil
production rate.
If
the
calculated
oil
production rate
exceeds
the
maximum allowed value, calculated production will
be
reduced
to
the
allowed value. Production
from
each connection
is
proportionally reduced
by
the
ratio
of
allowed
to
calculated
oil
production
rates.
A
positive value
of QT for a
pressure controlled production well
is
used
as the
maximum allowed liquid withdrawal rate.
If the sum of oil and
water
production exceeds
the
maximum allowed value, calculated production
is
reduced
to the
allowed value.
The
reduction
is
made
by
multiplying production
from
each
connection
by the
ratio
of
allowed-to-calculated
liquid withdrawal
rates.
IMPORTANT: When used
to
control total liquid withdrawal,
the
units
of
QT
are
STB/Day.
30.5 Fluid Injection Constraints
Fluid
injection
into explicit pressure controlled injection wells
can be
constrained
as
follows:
a.
A
maximum water
injection
rate
can be
specified;
and
b.
A
maximum
gas
injection rate
can be
specified.
A
negative value
of QW for a
pressure controlled water injection well
is
used
as the
maximum allowed water injection rate.
If the
calculated water
injection rate
exceeds
the
allowed value, calculated water injection will
be
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reduced
to the
allowed value. Water
injection
into each connection
is
proportion-
ally
reduced
by the
ratio
of
allowed
to
calculated water injection rates.
A
negative value
of QG
for
a
pressure controlled
gas
injection well
is
used
as the
maximum allowed
gas
injection rate
in
direct analogy
to the
water
injection
rate constraint described previously.
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Chapter
31
31.1 Productivity Index
Productivity index (PI)
is
defined
as the
ratio
of
rate
Q
to
pressure drop
AP, or PI =
£?/AP,
where
AP
~
P
e
-
P
w
,
P
e
=
average reservoir pressure,
and
P
w
=
wellbore
bottomhole
pressure BHP. From Darcy's
Law for
radial
oil
flow
we can
write
PI as
PI =
_
AP
"
Mo[H'A,)
+
S\
(31.1)
The
meaning
and
units
of all
terms
are
given
as
follows:
\i
0
= oil
viscosity (cp)
B
0
= oil FVF
(RB/STB)
r
e
=
drainage radius (ft)
r
w
=
wellbore radius (ft)
S
=
skin
K
e
=
effective
permeability (md)
=
k
ro
K
abs
k
ro
=
relative permeability
to oil
K
ahs
=
absolute
permeability
(md)
h
net
= net
thickness (ft)
Q
0
= oil
rate (STB/D)
318
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Some
of the
terms
in Eq.
(31.1)
depend
on
time-varying pressure
and
saturation, while other factors change
relatively
slowly
or are
constant with
respect
to
time.
We
separate these terms
to
obtain
PI
=
where
the
quasi-stationary
factors
are
collected
in the PID
term, that
is,
0.00708
J:
.
A
The
WINB4D
user
is
expected
to
provide
a PID for
each well connection.
A
connection
is a
gridblock
with
a
well perforation.
31.2
Vertical Wells
A
value
of the
connection
flow
index
PID for a
vertical well
can be
estimated
from a
formula
derived
by
Peaceman
[1978]:
0.007080
PID,
=
(31.2)
where
r
e
-
r
o
=
0.14(A*
2
+
Ay
2
)'
7
'
for
an
isotropic
system. With respect
to
permeability,
an
isotropic
system
is a
system
in
which
x
direction
and
y
direction
permeabilities
are
equal,
(K
x
=
K
y
).
For a
square well block
in an
isotropic system,
AJC
=
Ay
and
r
0
«
0.2
AJC.
The
subscript
k in Eq.
(31.2)
denotes
the
&th
connection.
For a
well
in a
rectangular
gridblock
and an
anisotropic
system (that
is,
K
x
*
K
y
),
well
PID is
estimated
using
an
effective
permeability
K
=
]/
K
X
K
y
and
an
equivalent well block radius
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Principles
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r
=
0.28
(K
y
/K
x
f<
+
(K
x
IK/<
The
remaining parameters
are
defined
as:
K
=
horizontal permeability
of
connection
k
(md)
h
-
thickness
of
connection
k
(ft)
r
w
=
wellbore radius (ft)
S
=
dimensionless skin
factor
In
principle,
the
well
flow
index
can be
related
to
measured values.
In
practice,
however,
the
terms
r
e
,
S, and
k
ro
/\1
0
B
0
are
seldom well known,
especially
for
a
multiphase
flowing
well.
As a
matter
of
expediency, therefore,
Eq.
(31.2)
is
often
used
to
compute
an
initial estimate
of
PID. This value
can
then
be
improved
by
adjusting
it
until
the
well rates computed
by the
simulator match
the
initial observed well rates.
31.3 Horizontal Wells
There
are
many ways
to
estimate connection
flow
index
PID for a
horizontal well
[Joshi,
1991
].
A PID
value
can
be
estimated
for
horizontal wells
in
a
manner similar
to
that
for
vertical wells
by
using
the
Joshi
formula
pro,
QMlOSKh
In
a+
.
\
2
_ I
L
I
I
2j
(
LI2
)
(
\
h
+5
J
^^
O
2r
wj
(31.3)
where
£
0.5
N
0.25
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The
subscript
k in Eq.
(31.3) denotes
the
&*
connection.
The
remaining
parameters
are
defined
as
follows:
K
=
horizontal permeability
of
connection
k
(md)
h
=
thickness
of
connection
&
(ft)
L
=
horizontal well length (ft)
r
w
=
wellbore radius (ft)
r
eh
=
drainage radius
of
horizontal well (ft)
S
=
dimensionless skin
factor
The
drainage
radius
r
eh
of the
horizontal well needs
to
account
for
elliptical
flow
into
the
wellbore. Babu
and
Odeh
[
1989]
present
a
procedure
for
estimating
r
eh
.
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