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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.

7/23/2019 Vogel Spe 1476 Pa


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

7/23/2019 Vogel Spe 1476 Pa


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

7/23/2019 Vogel Spe 1476 Pa


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.

7/23/2019 Vogel Spe 1476 Pa


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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OF PETROLEUM

TECHNOLOGY

7/23/2019 Vogel Spe 1476 Pa


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



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

91

7/23/2019 Vogel Spe 1476 Pa


1.oo....:----------------------

-

0,80

~ A Np/N'O, I , 2

'-

,

-

-

,

,

,60

-

a:

Ie.

~

6

3

e.

0.40

CASE

13

0,20


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


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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TECHNOLOGY

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