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PRODUCTION
ESTIM TION
Inflow
Performance Relationships for Solution-Gas Drive
Wells
Abstract
J.
V
VOGEL
MEMBER A/ME
In calculating oilwell production, it has commonly been
assumed that producing rates are proportional to draw
downs. Using this assumption, a well s behavior can be
described by its productivity index (PI). This
PI
relation
ship was developed from Darcy s law for the steady-state
radial flow of
i
single, incompressible fluid. Although
Muskat pointed out that the relationship is not valid when
both oil and gas flow in a reservoir, its use has continued
for lack of better approximations. Gilbert proposed meth
ods
of
well analysis utilizing a curve
of
producing rates
plotted against bottom-hole well pressures; he termed this
complete graph the inflow performance relationship (IPR)
of a well.
The calculations necessary to compute IPR s from two
phase flow theory were extremely tedious before advent of
the computer. Using machine computations, IPR curves
were calculated for wells producing from several fictitious
solution-gas drive reservoirs that covered a wide range of
oil
PVT
properties and reservoir relative permeability char
acteristics. Wells with hydraulic fractures were also in
cluded. From these curves, a reference
IPR
curve was
developed that is simple to apply and, it is believed, can
be used for
most
solution-gas drive reservoirs to provide
more accurate calculations for oilwell productivity than
can be secured with PI methods. Field verification is
needed.
Introduction
In calculating the productivity of oil wells,
it
is com
monly assumed that inflow into a well is directly pro
portional to the pressure differential between the reservoir
and the wellbore - that production is directly propor
tional to drawdown. The constant of proportionality is
the PI, derived from Darcy's law for the steady-state rad
ial flow
of
a single, incompressible fluid.
For
cases in
which this relationship holds, a plot of the producing
rates vs the corresponding bottom-hole pressures results
in a straight line (Fig. 1). The PI
of
the well is the inverse
of
the slope
of
the straight line.
However, Muskat' pointed out that when two-phase
liquid
and
gas flow exists in a reservoir, this relationship
should not be expected to hold; he presented theoretical
calculations to show that graphs of producing rates
vs
bottom-hole pressures for two-phase flow resulted in
curved rather than straight lines. When curvature exists,
Original manuscript received in Society
of Petroleum Engineers
office
July
1966.
Revised manuscript received Dec.
6, 1967.
Paper (SPE
1476)
~ a s
presented at SPE
41st
Annual Fa l
e e t i n ~ held
in
I a las,
Tex., Oct.
2-5, 1966.
©Copyright
1968 American
InstItute
of Mmmg,
Metallurgical, and
Petroleum Engineers,
Inc.
lReferences given a t end of paper.
This paper will be printed in Transactions Volume 243, which will
cover 1968.
JANUARY. 968
SHELL
O L CO
BAKERSFIELD, CALIF.
a well cannot be said to have a single PI because the
value
of
the slope varies continuously with the variation
in drawdown.
For
this reason, Gilbere proposed methods
of well analysis that could utilize the whole curve of
producing rates plotted against intake pressures. He termed
this complete graph the inflow performan..:e relationship
(IPR) of a well.
Although the straight-line approximation is known to
have limitations when applied to two-phase flow in the
reservoir, it still is used primarily because no simple sub
stitutes have been available. The
calculations necessary to
compute
IPR's
from two-phase flow theory have been
extremely tedious. However, recently the approximations
of Weller" for a solution-gas drive reservoir were pro
grammed for computers. The solution involved the fol
lowing simplifying assumptions:
( l )
the reservoir is cir
cular and completely bounded with a completely pene
trating well at its center;
(2)
the porous medium is
uniform
and
isotropic with a constant water saturation
at all points; (3) gravity effects can be neglected; (4 )
compressibility of rock
and
water
can
be neglected; (5)
the composition
and
equilibrium
are
constant for oil and
gas;
(6)
the same pressure exists in both the oil
and
gas
phases; and (7) the semisteady-state assumption that the
tank-oil desaturation rate is the same at all points at a
given instant. Weller's solution did not require
the
con
stant-GOR assumption.
The resulting computer
program
proved convenient to
use and gave results closely approaching those furnished
by the more complicated method of West,
Garvin
and
Sheldon
The
program also includes the unique feature
...
a:
:>
a:
0..
..J
..J
..J
o
:
I
I:
I -
o
- - -" "CB"" - '''f'-'-'-''-----
DRAWDOWN il P
wf
PRODUCING RATE,
bopd
MAXIMUM
PRODUCING
RAT E.
(CIol",o.
Fig. l Straight line inflow performance relationship.
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o
making complete JPR predictions or a reservoir. Such
predictions for a typical solution-gas drive reservoir are
shown as a family
of IPR
curves on Fig.
2. Note
that
they confirm the existence
of
curvature.
I t appeared that if several solution-gas drive reservoirs
were examined with the
aid of
this program, empirical
relationships
might
be established that would apply to
solution-gas drive reservoirs in general. This
paper
sum
marizes the results
of
such a study that dealt with several
simulated reservoirs covering a wide range
of
conditions.
These conditions included differing
crude
oil character
istics
and
differing reservoir relative permeability charac
teristics, as well as the effects
of
well spacing, fracturing
and
skin restrictions.
The investigation sought relationships valid only below
2800r-----------------------------------------
2400
.;;
•
2000
'
I:
::0
CI)
CI)
'
1600
..J
..J
'
r
' 1200
..J
o
:I:
2
o 800
I:
o
400
40
RESERVOIR CONDITIONS:
ORIGINAL PRESSURE
2130
psi
BUBBLE POINT 2130 psi
CRUDE OIL PVT
CHARACTERISTICS
FROM FIG,A-IO
RELATIVE PERMEABILITY
CHAR
ACTERISTICS FROM FIG, A-20
WELL SPACING 20 ACRES
WELL RADIUS
0,33
FOOT
{
CUMULATIVE RECOVERY,
'\- _ PERCENT OF
ORIGINAL
1-"0 OIL IN PLACE
/.
80
120 160
200
PRODUCING
RATE. opd
Fig. 2--Computer-calculated inflow performance
relationships for a solution-gas drive reservoir.
1,0 r c - - - - - - - - - - - - - - -- - - - - - - - - - - -- - - - - - - - - - ,
W
0::
: ; )
0,8
en
0:
lI -
o:g
0. .0 :
oJ l 0,6
. . J i l l
1 10::
~ I L
o
I & I z
..J
o
o
i=
0.4
: 1 : 0
<I
::Eo::
O I L .
... 0::
'to
2
RESERVOIR
CONDITIONS
SAME
AS FIG,
2
O L -____ ______ ______ ______ ____
° 0,2
0.4
0,6 0,8 1.0
240
PRODUCING RATE (qo /(Qo)max), FRACTION OF MAXIMUM
Fig.
3 Dimensionless inflow performance relationships for
a solution-gas drive reservoir.
B4
the bubble point. Computations were made
or
reservoirs
initially above the bubble point, but only to ensure that
this initial condition
did
not cause a significant change in
behavior below the bubble point.
Shape of Inflow Performance Relationship
Curves with Normal
Deterioration
As depletion proceeds in a solution-gas drive reservoir,
the productivity
of
a typical well decreases, primarily
because the reservoir pressure is reduced
and
because
increasing gas saturation causes greater resistance to oil
flow.
The
result
is
a progressive deterioration
of
the IPR's,
typified by the IPR curves in Fig. 2. Examination of these
curves does
not
make
it
apparent whether they have any
properties in
common other than
that they
are
all con'
cave to the origin.
One useful operation is t o plo t all
the IPR's
as "di
mensionless
IPR's .
The pressure for each point on an
IPR
curve
is
divided by the
maximum or
shut-in pres
sure for
that
particular curve,
and
the corresponding pro
duction
rate is
divided by the maximum
(l00
percent draw
down)
producing rate for the same curve. When this
is
done, the curves from Fig. 2 can be replotted as shown in
Fig.
3.
I t is then readily
apparent
that with this construc
tion
the
curves
are
remarkably similar
throughout
most
of
the producing life
of
the reservoir
UJ
'
2 SOC
--
-----
2000
A-IPR FROM FIG 2 FOR
Np /N
1%
B-IPR
WITH A
DIFFERENT
CRUDE OIL
FLOWING, ALL
OTHER CONDITIONS
BEING THE SAME,
CRUDE
OIL PROP
ERTIES FROM FIG,
A-Ib,
( f )
UJ
' ..
oJ
oJ
UJ
1500
B
UJ 1000
.J
A
o
l :
5 0 0
O L - - L
_ _
__ L ~ __ __ _ ~ __
o
50
100 150
200
250
PRODUCING RATE, bopd
(a ) ACTUAL
IPR'S
1 0 ~ - - - - - - - - - - - - - - - - - - - - - - - - - - -
a::
ta.
~
r :;)
lOS
L&I
L I
0::
0::
: ; ) "-
(J)
(J)
L&I
0::
0..
~ 0 6
0::
L I
.J '
;j :: OA
IL.
L&I
0
..J
z
20,2
f-
o
X
::E
o
0::
::
IL.
o
O L -_ _ _ _ _ _ _ _ _ _
-L
_ _ _ __ -... l
<Xl
° 0,2 0.4
0,6
0,8 1,0
PRODUCING RATE (qo tqo) ma.l, FRACTION
OF MAXIMUM
(b)
DIMENSIONLESS
IPR'S
Fig. 4 Effect of
crude oil properties
on
IPR's.
300
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Effect
of
Crude Oil Characteristics
On
IPR
Curves
From
the foregoing results it appears that
IPR
curves
differing over the life of a given reservoir actually possess
a common relationship. To determine whether this same
relationship would be valid for other reservoirs,
IPR
cal
culations were made on the computer for different con
ditions.
The
first run utilized the same relative perme
abilities but a completely different crude oil.
The
new
characteristics included a viscosity
about
half that of the
first
and
a solution
GOR
about
twice as great.
Fig. 4a compares the initial IPR s
Np/N
= 0 1 per
cent) for the two cases. As would be expected, with a
less viscous crude (Curve B) the productivity was much
greater than in the first case Curve A). However, when
plotted on a dimensionless basis Fig.
4b)
the IPR s are
quite similar. As IPR s for the second case deteriorated
with depletion, no greater change of shape occurred than
was noted in the previous section. These two crude oils
IIJ
0.80
I:
::;
U
U
IIJ
tr:
a
II:
a
>
II:
IIJ
U
IIJ
II:
IL
0.60
0
z
0
t-
o
<I
II:
IL
Jc>.1I:
0.40
IIJ
II:
::;)
U
<I
IIJ
I
II:
0
...J
...J
IIJ
3t
IIJ
J
0.20
:t:
:E
0
t-
t-
o
m
o
had about the same bubble point.
lPR s
were then calcu
lated for a third crude oil with a higher bubble point.
Again, the characteristic shape was noted.
Two further runs were made to explore the relationship
under more
extreme conditions. One utilized a
more
vis
cous crude 3-cp minimum
compared
with I-cp minimum),
and the other used a crude with a low solution GOR
300 scf/STB). With the more viscous crude, some straight
ening of the IPR s was noted.
The
low-GOR
crude
ex
hibited the same curvatur e noted in previous cases.
Runs were also
made
with the initial reservoir pres
sure exceeding
the
bubble point.
During the
period while
the reservoir pressure was above the bubble point,
the
slopes of the IPR curves were discontinuous with the
upper
part
being a straight line until the well pressure
was reduced below the bubble point. Below this
point
the IPR showed curvature similar to that noted previous
ly. After the reservoir pressure went below the bubble
point, all the dimensionless IPR curves agreed well with
the previous curves.
-
o 0.20 0.40
0.60 0.80
1.00
PRODU ING
RATE (qo/(qo m a ~ I FRACTION
OF MAXIMUM
Fig. S-Inflow performance relationship for solution gas drive reservoirs.
JANUARY. 1968
85
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III
a:
;::)
1/1
l l
:f
0::
g
0::
III
~ I . O ~ __
0::
L
o
Z
0.8
u
<t
a:
IL
0.6
......
J
III
0::
::>
0.4
1/1
(/ )
w
a:
a
. J
- 0.2
w
W
.J
o
RESERVOIR CONDITIONS
SAME
AS FIG.2
REFERENCE CURVE
X
O L L ~
____
______
____
::;:
o
r-
r-
o
m
o 0.2 0.4 0.6
0.8 1.0
PRODUCING RATE (qo
I(qo)rnaxl.
FRACTION OF MAXIMUM
Fig. 6 Comparison
of
reference curve with computer
calculated
IPR
curves.
Effect of Relative Permeability
and
Other Conditions
The same basic shape
of the
curves was noted when
the study was extended to cover a much wider range of
conditions.
Runs
were
made
with three different sets of
relative permeability curves in various combinations with
the different
crude
oils.
The
results were in agreement
sufficient to indicate that
the
relationship might be valid
for most conditions.
0.8
a:
0.6
lel.
0.4
0.2
LIQUID FLOW
TWO-PHASE FLOW
(REFERENCE CURVU
0.2 0.4
0.8
1.0
Fig. S Comparison
of
IPR s for liquid flow, gas flow
and two-phase flow.
To explore further the generality of
the
relationship,
a
run
was
made
in which the crude oil
PVT
curves and
the relative permeability curves were roughly approxi
mated by st raight lines.
t
was surprising to find that, even
with no curvature in either the graphs
of
crude oil char
acteristics or the relative permeability
input
data, the out
put
IPR s exhibited about the same curvature as those
from previous computer runs.
Calculations also were made for different well spacings,
for
fractured wells and for wells with positive skins.
Good
agreement was noted in all cases except
for
the
well with a skin effect, in which case the IPR s more
nearly approached straight lines.
In
summary, calculations for 21 reservoir conditions
resulted
in IPR s
generally exhibiting a similar shape.
2 2 0 0 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - ~
RESERVOIR CONDITIONS SAME AS FIG. 2
POINT
OF
MATCH (WELL TEST)
'"
W
0::
::>
1/1
: .....
/1
W
..........
0::
a
~ t ~ S T R A I G H T L i N E
EXTRAPOLATION
. J
. J
III
w
\,: 0
/0
..........
J
0X
::;:
0
C O M P U T E R ~ A L C U L A T E D rPR
-
r-
0
m
400
..........
\
\.--r
PR EXTRAPOLATE;;
200
\
FROM REFERENCE
CURVE
\
......
60
80 100 120
140
160
180 200 220 240 260 280
300
320
340
PRODUCING RATE. bopd
Fig. 7 Deviations
when IPR s are predicted by reference curve from well tests at low drawdowns.
86
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Significant deviation was noted only for the
more
viscous
crude,
for
a reservoir initially above the bubble point,
and for a well producing through a restrictive skin. Even
in these cases, definite curvature was still apparent.
The curves
of
crude oil characteristics and of relative
permeability that furnished the input data for the various
conditions studied
are
given in Appendix A. Dimension
less IPR curves calculated for various conditions are
shown in Appendix B.
250
2000
5.0
2
4.0
N
[50
II B
g
0
3.0 -;;
0>
0>
'
V>
:t
0::
[00
2 0:
-
'
50
[,0
0
0
1000
2000
PRESSURE
psi
o)
Pb
2
[3
psi
250
2000
[/Bg
5.0
200
[600
4.0
N
0
[50
[200
3.0 -:
0>
0>
'
V>
0::
: t
[00
2.0 :to
;:,
m
50
1.0
0
0
0
0
[000
2000
PRESSURE, psi
c)
Pb
2
[30
psi
250
2
200
[/Bg
[50
0>
m
V>
0::
-
[00
50
0
[000 2000
PRESSURE, psi
e)
p b 3000
psi
250
200
[50
0>
'
[00
50
0
5,0
0>
: t
>
m
Proposed Reference PR Curve
I f the IPR curves for other solution-gas drive reser
voirs exhibit the same shape as those investigated
in
this
study, well productivities
can
be calculated more accur
ately with a simple reference
curve
than with the straight
line PI approximation method currently used.
Applying one reference curve to all solution-gas drive
reservoirs would not imply that all these reservoirs are
250 2000
5.0
200
[600
4.0
[/Bg
N
[50
3.0
w
0>
0>
'
V>
: t
0::
[00
0
2.0
: t
0
BO
'
50
1.0
0
0
[
2
PRESSURE,
psi
( b)
Pb
2[30
psi
2000
5.0 7.0
4.0
6.0
N
[200
[/Bg
3 0 ~
5.0
V>
0
0::
fLg
0>
: t
: t
800 2.0 ;:
4.0
'
Rs
400
BO
1.0 3.0
f O
0
0 2.0
0
[000
2000
PRESSURE,
psi .
d)
Pb
2[30
psi
250
2000
5,0
200
4.0
N
[50
[/B
g
3,0
x
0>
0>
: t
m
V>
0::
0
[00
2.0
::I..
0
m
Bo
50
f o
1.0
0
0
[000
2000
PRESSURE,
psi
f )
Pb
2[3
psi
Fig. 9 Input data, crude
oil
PVT characteristics co = 12 X
10-
in all cases).
l ANUARY,
1968 87
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identical
any
more
than
would the present use
of
straight
line PI's for all such reservoirs. Rather, the curve can be
regarded as a general solution
of
the solution-gas drive
reservoir flow equations with the constants for particular
solutions depending on the individual reservoir charac
teristics.
Although one
of
the dimensionless curves taken from
the computer calculations could probably be used as a
reference standard, it seems desirable to have a mathe
matical statement for the curve to insure reproducibility,
permanency
and
flexibility in operation.
0.45r--------------------------------------,
0 .40
0.35
Sgc
2.1 %
Sw
19.4 %
cf
13.9 %
0 .30
23.5 fl
k
20md
0.25
k
ro
(IOO SII) ,
0 .444
0.20
0.15
0.10
0.05
0
0.3 0.4 0.5 0,6
0.7 0.8 0.9 1.0
(0)
0.45
0.40
0.35
Sgc
10
Sw
19.4
cf
13.9
%
0.30
23.5 II
k
20
md
0 .25
k
ro
(100 SII ,
. l<
0.20
0.15
0.10
0.05
0
0.3
0.4
0.5
0.6
0.7 0.8 0.9
1.0
S (TOTAL LIQUID)
c)
The equation of a curve that gives a reasonable empirical
fit is
q = 1 - 0.20 ] J ; ~ - 0.80 ( J ; ~ ) '
, ( l)
(q.)max R R
where q
is
the producing rate corresponding to a
given
well intake pressure PU j,
p; is
the corresponding reservoir
pressure, and (q.) ,ax
is
the maximum (100 percent draw
down) producing rate. Fig.
5
is a graph
of
this curve.
For
comparison, the relationship for a straight-line
IPR
is
Sgc
8 .0
Sw
19,4
%
cf
13.9
23.5
II
k
20md
k
ro
(IOO S'I) ,
0.444
0.3
0.4 0.5
0.6 0.7 0.8 0.9 1.0
( b)
Sgc
5
Sw
19.4 %
cf
13 .9
23.5
II
k
20md
k
ro
(IOO SII) ' 0 , 425
krg
0.3
0.4 0.5
0,6 0.7
0.8
0.9
1 0
S (TOTAL
LIQUID)
(
d)
Fig
lO lnput
data relative permeability curves.
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qo _ 1 _ Pw
(qo)mRX - P ;;
(2)
When qo/(qo)max from Eq. 1
is
plotted vs Pwtl h. the
dimensionless
IPR
reference curve results.
On
the basis of
the cases studied,
it is
assumed that
about
the same curve
will result for all wells.
If qo is
plotted vs
pw ,
the actual
IPR
curve for a particular well should result.
A comparison of this curve with those calculated on
the computer
is
illustrated in Fig.
6. The
curve matches
more closely the
IPR
curves
for
early stages
of
depletion
than the
IPR
curves for later stages
of
depletion.
In
this
way, the percent
of
error is least when dealing with the
higher producing rates in the early stages of depletion.
The
percentage error becomes greater in the later stages
of depletion, but here production rates
are
low and, as
a consequence, numerical errors would be less in absolute
magnitude.
Use of Reference Curve
The
method
of
using the curve in Fig. 5 is best illu
strated by the following example problem. A well tests
65
BOPD with a flowing bottom-hole pressure
of
1,500
psi in a field where the average reservoir pressure is 2,000
cr
0.
--
i
0.
1 00
~ - - - - - - - - - - - - - - - - -
0.80
0.60
0.40
' - ;> ' - - -N p /N
-0.1 , 2 ,
4
1 2 % - - ~ >
10
- - - . : > . r ~ . _ _ _
CASE
I
CRUDE OIL PROPERTIES, FIG. A'lo
RELATIVE
PERMEABILITY, FIG A·20
0.20 WELL SPACING - 20 ACRES
WELL RADIUS = 0.33 FOOT
INITIAL
RESERVOIR PRESSURE=
2130psi
BUBBLE POINT - 2130psi
O L-
L
______
______
(0)
1.00
~ - - - - - - - - - - - - - - - - - - - - - - _ _ _ _ I
0.80
0.60
0.40
0.20
10
CASE
3
SAME
AS
CASE I, EXCEPT WITH
ABSOLUTE
PERMEABILITY
OF
200md
O L - - - - - ~ - -
__
L - - - - ~ ______
_
__
psi.
Find (1)
the maximum producing rate with 100 per
cent drawdown,
and (2)
the producing rate
if
artificial
lift were installed
to
reduce the producing bottom-hole
pressure to 500 psi.
The
solution is:
(1)
with PW =
1,500
psi,
PW;/PR =
1,500/2,000
=
0.75. From Fig. 5, when
PwJiR =
0.75,
qo/(q )n ,, = 0.40, 65/(qo)max = 0.40, (qo)max = 162
BOPD;
(2)
with pw = 500 psi, Pwtl i. = 500/2,000
0.25.
From
Fig. 5, qo/(qo)max = 0.90,
qo/162
= 0.90, q.
=
146 BOPD.
If
the same calculations had been made by straight-line
PI
extrapolation, the productivity with artificial lift would
have been estimated as 195 BOPD
rather than
145 BOPD,
This illustrates a significiant conclusion to be
drawn
for
cases in which such
IPR
curvature exists. Production in
creases resulting from pulling a well harder will be less
than
those calculated by the straight-line PI extrapolation;
conversely, production losses resulting from higher back
pressures will be less than those anticipated by straight
line methods.
I t
is
difficult to overstate the importance of using sta
bilized well tests in the calculations.
In
a low-permeability
10 - - - - < : - - ~
SAME AS CASE
I,
EXCEPT WITH
40-ACRE SPACING
(b)
CA
S E 4
SAME AS CASE I, EXCEPT THAT WELL
IS FRACTURED (PSEUDO WELL
RADIUS - 50 FEET)
o
0.20
0.40
0.60 0.80
1.00 0
0.20 0.40 0.60
O BO 1.00
JANUARY, 19611
qo/ Clolmax
(C)
(iig.
ll-Calculated dimensionless IPR curves.
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reservoir it frequently wiIl be found that significant changes
in
producing conditions should not be ma:de for several
days preceding
an important
test. This presents no prob
lem
if
a weIl
is
to be tested
at
its normal producing rate,
but
it becomes
more
difficult
if
multi-rate tests
are
required.
Accuracy of Reference Curve
I t is
anticipated
that
the most
common
use
of
the refer
ence
IPR
curve will be to predict producing rates at high
er
drawdowns from data measured at lower drawdowns.
For
example, from weIl tests taken
under
flowing condi
tions, predictions will be
made of
productivities to be
expected
upon
instaIlation of artificial lift. I t is necessary
to arrive
at
the approximate accuracy of such predictions.
Maximum
error
will occur when well tests
made at
very
low producing rates
and
correspondingly low drawdowns
are extrapolated with the aid
of
the reference curve to
estimate
maximum
productivities as the drawdown ap
proaches 100 percent of the reservoir pressure.
The error
that
would result
under
such conditions was investigated,
and typical results
are
shown in Fig. 7 In this figure the
dashed lines represent
IPR's
estimated
from
well tests
at
low
dra
wdowns (11 to
13 percent), and
the solid lines
represent the actual IPR's calculated by the computer.
1.oo .. .-------------------
0::
,0.
-
'i
0.
0.80
- . - - - . : ~ - N p / N ~ 0.1 .6 ,10
-.
-.
-.
0,60
'-.--
A
-
14
-
\
040
CASE
5
\
0,20
SAME
AS
CASE I, EXCEPT WELL HAS
PLUS 5 SKIN
\
\
\
\
\
O L - - - - - - L - - - - - - L - - - - - ~ - - - - - - ~ - - - - ~
(0)
1.00 I C - - - - - - - - - - - - - - - - - - - ,
0,80
~ ~ - - A . I O
0::
0,60
0. 14 6
-
0040
0,20
SAME AS CASE I,
EXCEPT
WITH
LESS VISCOUS CRUDE
OIL
FROM
FIG, A-I b
o
L - L _
o
0,20
0040
0.60 0,80
1.00
The
maximum
error for
the reservoir considered in Fig.
7
is
less than 5 percent throughout most
of
its producing
life, rising to 20 percent during final stages
of
depletion.
Although
the
20 percent
error may
seem high, the actual
magnitude of the
error is
less than
V BOPD.
I t is
obvious from Fig. 7
that
if weIl tests
are
made
at
higher drawdowns than the extreme cases illustrated,
the point of match of the estimated
and
actual
IPR
curves
is
shifted further
out
along the curves
and
better agree
ment will result.
Maximum-error calculations were
made
for all the res
ervoir conditions investigated. Except
for
those cases with
viscous crudes
and
with flow restr icted by skin effect,
it appears
that
a maximum
error
on the order of 20 per
cent should be expected if
all solution-gas drive IPR's
follow the reference curve as closely as have the several
cases investigated.
For
comparison, the maximum errors
for the straight-line
PI
extrapolation
method
were gen
eraIly between 70 and 80 percent, dropping to about
30 percent only during final stages
of
depletion.
The
figures cited above refer
to
the
maximum
errors
that
should be expected.
In
most applications
the
errors
should be much less
(on
the order of
10
percent) be-
o
BUBBLE POINT
CASE6
SAME
AS
CASE I,
EXCEPT THAT
RESERVOIR PRESSURE
IS INITIALLY
ABOVE THE BUBBLE POINT,BEING
3040 psi INSTEAD OF
2130
psi
(b)
___
--N
P IN
0.1 ,4
'-..
-.
-.
-
8 %
- - - - > . ~
_ '
-
~ A
' \
2%---- \
'SAME
AS
CASE
I, EXCEPT WITH MORE
VISCOUS CRUDE OIL FROM FIG.
A-Id
'\
0,20
0.40
0,60 0.80
\
\
\
\
1.00
Fig. 12 Calculated dimensionless IPR curves
90
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cause better agreement is noted between IPR s and refer
ence curve throughout
most
of the producing life of the
reservoirs and because well tests are ordinarily made at
greater drawdowns.
Application of Reference Curve:
Other
Types
of Reservoirs
The
proposed dimensionless
IPR
curve results from
computer analysis of the two-phase flow and depletion
equations for a solution-gas drive reservoir only and
would not be considered correct where
other
types of
drive exist.
In
a
major
field with partial water drive, how
ever, there
can
be large portions of the field
that are
ef
fectively isolated from the encroaching water by
barrier
rows of producing wells nearer
the
encroachment front.
It appears that the reference curve could be used for the
shielded wells for at least a portion of their producing
lives. Similarly, the reference curve might give reasonable
results for a portion of
the
wells producing from a res
ervoir
in
which expansion of a gas
cap
is a significant
factor.
Since
the
referel1ce curve is for
the
two-phase flow of
oil and gas only, it would not be considered valid when
three phases (oil, gas and water)
are
flowing. However,
1.00
~ - - - - - - - - - - - - - - - - - -
0.80
0:: 0.60
,, -
'
i
20
0.40
CA SE 9
0.20 SAME AS CASE I. EXCEPT WITH HIGHER
BUBBLE POINT CRUDE OIL FROM FIG.
A-Ie
INITIAL
PRESSURE
3000 psi
BUBBLE POINT 3000psi
o L L ~ ~ ~ ~
(a)
1.00 ------------------,
0.80
~ ~ : - - - 20
0.60
Np
/ N ~ O . I - . . - : : s ~ ~ - I O
28
- - - - - ~
0.40
CASE II
0.20 SAME AS CASE
I,
EXCEPT
WITH
PERMEABILITY CHARACTERISTICS
FROM
FlG.A-2c
it appears intuitively that some
curvature
should be ex
pected in the IPR s whenever free gas is flowing in a
reservoir.
For
radial flow, this curve should lie some
where between the straight line for a single-phase liquid
flow and the curve for single-phase gas flow. The dimen
sionless
IPR s
for the two types of single-phase flow
are
compared with the suggested reference curve
for
solution
gas drive reservoirs in Fig.
8.
Conclusions
IPR
curves calculated
both
for different reservoirs and
for the same reservoirs at different stages of depletion
varied several-fold in actual magnitude. Nevertheless, the
curves generally exhibited
about
the
same
shape.
This similarity should permit substitution of a simple
empirical curve
for
the straight-line
PI
approximations
commonly used. Maximum errors in calculated produc
tivities are expected to be on
the
order of 20 percent
compared with 80 percent with the PI method. Productiv
ity calculations
made
with the reference curve
method
rather
than
with the
PI
method will show smaller produc
tion increases for given increases in drawdowns and, con
versely, less lost production
for
given increases in back
pressures.
SAME AS CASE I, EXCEPT WITH
PERMEABILITY
CHARACTERISTICS
FROM FIG
A-2b
(b)
1
%
---- '<'-''''-
CA
S E
12
SAME AS CASE I, EXCEPT WITH
PERMEABILITY CHARACTERISTICS
FROM FIG A-2b AND CRUDE OIL PROP
ERTIES FROM
FIG.A Ib
0.20
0.40
0.60
0.80 1.00 0
0.20 0.40
0.60
0.80 1 00
Fig. 13 Calculated dimensionless IPR curves
J AN U AR Y, 19611
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1.oo....:----------------------
-
0,80
~ A Np/N'O, I , 2
'-
,
-
-
,
,
,60
-
a:
Ie.
~
6
3
e.
0.40
CASE
13
0,20
SAME AS CASE I, EXCEPT WITH
GOR
CRUDE FROM
FIG,A-If
0
(0)
1,00
O,BO
10 - - - - - ~ ~ : _ _ - - 20
0,60
a:
Ie.
-...
i
e.
0.40
CA SE
15
0,20 SAME AS CASE I, EXCEPT WITH
PERMEABILITY
CHARACTERISTICS
FROM
FIG,A-2e
AND CRUDE OIL PROP
ERTIES FROM FIG,
A-Ib
° 0 L - - - ~ 0 ~ 2 ~ 0 - - - 0 ~ L 4 ~ 0 - - ~ 0 ~ 6 - 0 - - - 0 ~ L B O ~ - ~
qo/(qol Max
(c)
o
10
%
CASE
14
SAME
AS
CASE
I,
EXCEPT
WITH
PERMEABILITY CHARACTERISTICS
FROM FIG,
A-2b
AND CRUDE OIL
PROPERTIES FROM FIG,
A-Ie
(b)
CASE
16
SAME AS CASE
I. EXCEPT
PERMEABILITY
APPROXIMATED FROM STRAIGHT LINES
OF
FIG,
A-2d
AND CRUDE OIL PROPERTIES
APPROXIMATED
PROM
STRAIGHT
LINES
OF
FIG,
A-Ie
0.20
0.40
0,60
qo (Qal
Max
(d)
O.BO
Fig.
14-Calculated
dimensionless IPR curves.
This technique needs to be verified
by
a comparison
with field results. As previously discussed, the conclusions
are
based only
on
computer solutions involving several
simplifying assumptions as listed in the Introduction.
References
1. Evinger, H. H. and Muskat, M.: Calculation of Theoretical
Productivity Factor ,
Trans.,
AI ME (1942) 146, 126-139.
2. Gilbert,
W.
E.: Flowing and Gas-Lift Well Performance ,
Drill. and Prod. Prac.,
API (1954)
126.
3.
Weller,
W.
T.: Reservoir Performance During Two-Phase
Flow ,
J Pet. Tech.
(Feb.,
1966) 240-246.
4.
West, W.
J.,
Garvin,
W. W.
and Sheldon,
J.
W.: Solution
of the Equations of Unsteady-State Two-Phase Flow in Oil
Reservoirs ,
Trans.,
AIME (1954)
201, 217-229.
APPENDIX A
Input Data
Figs. 9
and 10
illustrate graphically the input
data
(crude
oil
PVT
characteristics
and
relative permeability charac
teristics)
from
which
the
theoretical behavior
of
simulated
reservoirs was calculated by
the
computer.
92
APPENDIX B
Computer-Calculated IPR Curves
Dimensionless
IPR
Curves
Figs.
11
through
14
are graphs
of
the theoretical IPR's
calculated for various simulated reservoir conditions. So
that
the
IPR's
under various conditions
can
be compared
more easily, the initial
IPR
curve
Np/N =
0.1
percent)
from Fig. l1a
is
reproduced on all succeeding figures and
is
designated as Curve A
In
addition to the cases illustrated, five more calculations
were
made
in which individual curves of the crude
oil
properties
in
Fig. 9a were replaced one by one with the
curves
from
Fig. 9b.
The
results were
comparable to
those
shown, and, since the illustrations include
the
case in
which
the
curves
of
Fig.
9a
were completely replaced by
those
of
Fig. 9b, it was not considered necessary to repro
duce the cases in which the individual components were
replaced.
***
Editor s note: A picture and hiographical sketch of
J V. Vogel appear on page 60.
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