vogel 1968 para el tema 2

8/9/2019 Vogel 1968 Para El Tema 2

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PRODUCTION

ESTIMATION

@

A

Inflow Performance Relationships for Solution-Gas Drive Wells

Jo V. VOGEL

MEMBER AIME

Abstract

Its calculating oilwell production, it has commonly been

assumed that producing rates are proportional to draw-

downs. Using thi,s assumptimt, 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 oj a single, incompressible fhid. Although

Muskat pointed out tha~ the relationship is not valid when

both oil and gas jiow 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 bot om-hole well pressures; he termed this

complete graph rhe infiow performance relatiorsship (IPR)

of a well.

The calculations necessary to compute IPR’s from two-

phme fiow theory were extremely tedious bejore advent of

the computer. Using machine computations, IPR curves

were calculated for wells producing from several fictitious

solution-gas drive reservoirs that co>ered a wide range O

oil PVT properties and reservoir relative permeability char-

acteristics. Wels 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

he used for most solution-gas drive reservoirs to provide

more accurate calculations for oilwell productivity than

can be secured with PI methods. Field verification i.s

needed.

Introduction

In calculating the productivity of oil wells, it is corn

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 matmacript rece ived in Society of Petroleum Engln*rs o~@e

July 11, 1966. Revised manuscript reaekd Dec. S, 1967. PaPer (SPE

1476) wae presented at SPE 41st Annual Fall Meetbra held in Dallas,

Tex, , Oot, 2.5, ISSC. @CbRyrlght 1S6S American Inst itute of Minims ,

Metallurgical, and Petroleum Errsineers, k,

preferences given at end of papsr.

This paper will be printed hr Transactions Volume 24L3,which will

cover 196S.

J ANU ARY, 19611

SHELL OIL 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, Gilbert’ 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 performarwe relationship

(IPR) of a well.

Although the straight-line approximation. is known to

have limitations when applied to ttio-phase tlow in the

reservoir, it still is used primarily because no simple sub-

stitistes have been available: The calculations necessary to

compute IPR’s from two-phase flow theory have been

extremely tedious. However, recently the approximations

of Welled for a solution-gas drive r~.servoir were pro-

grammed for computers. The solutior, invoived the fol-

lowing simplifying assumptions: (1) ihe reservoir is cir-

cular and completely bounded with t. 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 writer can be neglected (5)

the composition and equilibrium are constant for


2/10

of making complete IPR predictions for a reservoir. mch

predictions for a typical solution-gas drive reservoir are

shown as a family of IPR cur:’ s on Fig. 2. Note that

they confirm the existence of curkdure.

U 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 p&per 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 permwbiiity charac-

teristics, as well as the effects of well spacing, fractwing

and skir~restrictions.

The i:westigation sought relationships valid only below

2800

k

RESERVOIR CONDITIONS:

ORIGINAL PRESSURE I 2130 Psi

2400

BuBBLE POINT , 2130

psi

CRUDE OIL PVT CHARACTERISTICS

FROM FIG.A-10

~

RELATIvE PERMEABILITY CHAR-

- 2000

ACTERISTICS FROM FIG. A-20

w

a

k

WELL SPACING * 20 ACRES

J

WEI L RAOIUS I 0.33 FOOT

g, 1600

-1

I

CUMULATIVE RECOVERV,

, ,

‘14+

d

PERCENT OF ORIGINAL

. \

“o. OIL IN PLACE

.—

.0

PRODUCING RATE , bo$~

Fhr. 2—Contrx4ter-culculatetl inflow performance

,

relationships for a solution-gas drive reservoir.

I.0

k

a

g g 0,8

:go,

r~

p/N=OJ%,2eh,4”l.

6“le,8”h

~g ‘

.3b

Io%

~z

~o

g ~

0.4

I2 “A

v

I4“/.

ZS

ok

1- --

g go.?

RESERVOIR CONDITIONS

SAME AS FIG.2

.J.

“..

o 0.2

0.4

0,6

0.8

1.0

PRODUCING”RATE”fqo /~o)mox), FRACTION OF MAXIMUM

NE. 3—Dimensionless injlo’w performance relationships jor

a solution-gasdrive reservoir.

n.s

,–– —

the bubbie point, Computations were made for 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 I’/eterioration

As depletion proceeds in a

solutlon-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 IPRs,

typified by the IPR curves in Fig. 2. Exarr.ination of these

curves does not make it apparent whether they have any

properties in common other ‘:han that they are all con.

cave to the origin.

One useful operation is to plot all the IPR’s as “di-

mensionless IPR’s”. The prissure 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 isdivided bytt.e maximum (l OOpercentdrtw-

down) producing rate for the same curve. When this is

done, the curves from Fig, 2 can bereplotted as shown in

Fig. 3. It is then readily ~ppaI’entthat with this construe.

tion the curves are,, remmkably similar throughout most

of the producing life of the reservoir

2500

———

F\ ~~

A+PR FROM FIG 2FCIR Npfld ’01%

BIPRw IT HP DIFFERENT CRUDE OIL

2000

FLOWING, AL I. OTHER CON OII IONS

BEING THE SAltE. CRUOE OIL PROP-

~TIE”

FRO M)”l G, A-lb,

1500 -

\

B

1000

@

500

\

o

l—_L_L~

50

100 50

200 250

PRODUCING RATE , bopd

(al ACTUAL

IPR’S

0.2

0.4

0.6 0.8 1.0

ROOUCIN5

RATE klD~%)maa), fRACTION

OF MAXIM1’M

(b} DIMENSIONLESS IPR’S

300

Fig. 4-Eflect of crude oil properties on IPR*s.

J OU RNAL OF P E TROLE U M T13C HNO_&OCY


3/10

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 co,mmon rekdtionship. 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 complete] y different crude oil. The new

characteristics included a viscosity about half that of the

first and a sohrtion GOR about twice as great.

Fig. 4a compares the initial IPR’s (~{P/N = 0.1 per-

cent) for the two cases. As would be expected, with a

less viscous crude (Curve B) the pr~ductivity was much

greatec than in the first case (Curve A). However, when

plotted on a dimensionless basis (Fig. 4b) the IPR’s are

quite sim lar, As lPR.’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

1,

w

K 0

a

m

u)

w

a

a

E

5

>

a

w

m

w

a

0

0

JANUARY,

had about the same bubble point, IPR’s were then calcu-

lated for a third crude oil with a higher bubble poirit.

Again, the characteristic shape was noted,

Two further” runs were made to explore the relationship

under more extreme conditions. C)ne utilized a more vis-

cous crude (3-cp minimum compared with 1-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 W?.Snoted. The low-GOR crude ex-

hibited the same curvature noted in previous cases.

Runs were a[so made with the initial reservoir pres-

sure exceeding the bubble point, During the period while

the reservoir pressure w-as 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.

PRODUCIN9 SATE (qo/(qo )ma*), FfSACTION OF

MAXIMUM

Fig. 5-Irrf70w performance relationship for solution-gas drive reservoirs.

1968

m


4/10

,, ,-

g

u

1.0

a

,REFERENCE CURVE

12%

I4 “ /.

RESERVOIR CONDITIONS

sAME AS FIC.2

0.2

0.4

0.6 0.8

LO

o

t-

1

PROOUCIMG RATE (qo /@o)max), FRACTION oF MAXIMUM

m

Fig. 6—Con?pmison of reference curve with computer-

ca[culated IPR curves.

Effect of Relative Permeability and

Other Conditions

-.

‘l’he 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.

Lo

TWO-PHASE FLOW

(REFERENCF CURVE I

0 .8 1-

0.6 -

==

>

.;

LIQUIO FLOW

0.4 -

082-

0 ~—

0

0.2

0,4

,,

. cl/qmQR

Fig. G-Comparison of IPR’ for iquid flow, gas flow

and fwo-phase ffow.

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 straight linss. It was surprising to fi,ld 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 curvttturc as those

from previous computer runs.

Calculations also were made for different wdl spacings.

for fractured wells and for WCIISwith positive skhs.

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.

\

“oo~ ‘-—

RESERVOIR CONDITIONS SAME AS FIG. 2

2000~\poNToFMATcH(

ELLTEsT)

800

1600 -

\

\

\

\

1400

~“ ‘

\

\

—STRAIGHT. LINE

4’\

EXTRAPOLATION

I 200

t~+ \

Y? ~

[000

\ { ‘~,

800

k+

COMPUT*FR -< AL CULATED IPR

\ \f2 \

\

600

\-\ \@ \ i,

\

i\

\

\\. ?

400

\~o’f \\\

~IPREXTRApOLAJF~\

20 0

~\;\

\ :tiv:EFERENcE \

so

\\ i, \ ,

\ \

\

.-

1

lU

1

Ii

1

II

OK

1

I,i

\\r l\

I

1 1

1

1

1

\

0

20 40 60 60

100 120 f40

[60 180

200 220 240 260 280 300 320 :

PRODUCING RATE , b@Pd

I

Fig.

7—Deviations when IPRs are predicted by reference curve from well tests at low drawdowns.

86

J

I O U RNA?. OF P E TROLE U M ‘CE CE N(S LOG Y


5/10

Significant deviation was no:ed 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.

““[’ooo~”o

200 - [600

I 50 . - 1200

.-

-w

x

K*

- 100 -

800 -

50 -

0 - f)

-

4.0

N

o

- 3,0 ;

a-

- 2 0 *O

L

m

- 1.0

o

“-o 1000

Zooo

PRESSURE

psi

o Pb :

2r30 psi

250 2000

r

I/Bg

5.0

PRESSURE, psi

(c) Pb : 2130 psi

““[’oo~”o

Proposed Reference IPR Curve

If the IPR curves for other solution-gas drive reser-

voirs exhibit the same shape as those investigated in thk

study, well prodl.activities can be calculated more accur-

ately witha simple reference curve than with the straight-

Iine PI approximation method currently used.

Applying one reference curve to all solution. gas drive

reservoirs would not imply that all these reservoirs are,

:[:+—-----:

150

-1 .~f$

1200

~w

RS

\

K“’

I 00

800

Pg

.

B.

50

t

400

PO

nfl

I

1

m“

1,0

-10

“o

1000

2009

PREsSURE, psi

(b)

: 2130 psi

25:[ 2000+5017”0

200

-1. ~

1600

t

- 4,0 - 6.0

[50

I 200

- 3,0N: .. ~,f)

m

l/Bg ~

e

a’”

‘9

am

2

100

800

- 2.0 “ - 440

m“

I Y& - ‘:

50 400

B. ,0

3.0

~o

on

o 2,0

-o

1000 2000

PRESSURE, psi.

(d] pb : 2130 psi

PRESSURE, psi

(e) pb :3000 psi

“or ‘o”o~~’ o

II

200

IL

[600

I5 0

1200

l; 3g

m

:

K-

100 s 00

Pg

B.

50 400

Po

- 4.0

N

0

- 3,0 “m

a.

L

- 2,0 ?

II”

- 1.0

;1

.OIAZI=Z ZI=Z

1000

2000

PRESSURE, psi

~

(f)

pb : 2130

psi

Fig, 9-Input data, crude oil PVTchoracteristics (c,= 12 XIO-gin all cases .

J ANU ARY, 196S

,.. .

87


6/10

,

identical any more than wou d the preser.: use of straight-

line PI’s for all such reservoirs. Rather. the curve can be

regarded as a general solution af the solution-gas drive

reservoir flow equaticms 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 flexibilityy in operation.

0.40

t

0.35

[

Sgc : 2.1 %

Sw

: 19.49/0

+ : 13.90/.

0.30

h : 23.5fl

k

: 20md

0.25

k

~.

(l OO?/ s, ,): 0,444

I

k

f9

\

/

k

ro

-0,3 0,4 0.5 0(6 0.7 0,8 0.9 1,0

0

0.45

0,40

0835

r“” ..[

Sg c

: 10%

Sw

: 19,470

‘$

: 1 3. 90 /0

0.30

h : 23.5fi

k

ro

k

:20md

L125

kro (l OOO/OS+ l ): 0.444

0.20 -

0.15 -

0.10 -

0.05 -

0’ b

1

L

1

0.3 0,4 0,5 0.6 0,7 08

0 9 1,0

S( TOTAL LIOUID]

(cl

The equation of a curve that gives a reasonable empirical

fit is

,qo _

/

1 – 0.20&- – 0,80 &M~\’ , , ,

(1)

(q.

mx Pn \ Pli /

where g. is the producing rate corresponding to a given

well intake pressure

p,,,, ~;, is

the orrespcnding reservoir

pressure, and (qO),,,, ,. 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 1PR

is

,

Sgc

: .s,0%

Sw

: 19840/”

. : 13,9%

h’

= 23.5ft

k

:

20md

k~.

(100% s,,1= 0.444 ‘

I

k

ro

k

rq

\

0,3 ‘ 0,4 0,5 C ,6

0,’7 OS

0.9

1.0

(b)

Sgc , ‘5%

/

Sw

:

19.4°0

+ : 13,970

h

k : 20md

k

r9

1 1

0,3

0,4 0,5 0,6

0,7 o E 0.9

S(TOTAL LIOUIO]

4

Ffg. 10—Input data, relative permeability curves.

—


7/10

.

When q./(q.)~., from Eq, 1 is plotted vs p,,,/5,, the

dimensionless IPRreference curve results, Onthe basis of

the cases studied,

it is

assumed that about the same curve

will result for all wells, If q,, is plotted vs p~,, the actual

lPR curve fora 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

kigher 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 130PD with a flowing bottom-hole pressure of 1,500

psi in afield where rhe average reservoir pressureis2 ,000

psi, Find (~) 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 500psi.

The solution is: (1) with p., = 1,500 psi, p.,/~R=

1,530/2,000=0,75, From Fig, 5, when p,,j/~k =0.75,

q.~(q,)t,,ii. = 0.40> 65/(qr>)[tlrix= 0,4j, (q. ),,,,. = 162

BOPD; (2) with p., = 500 psi,

p,.J /p,, =

500/2,000 =

0,25, From Fig. 5,q./(q,, ),,,,,, = 0.90, q./l62 =

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 IK)PD.

..This illustrates a significant conclusion to be drawn for

cases in which such IPR r.,xvature-exists, Production in-.

creases resulting from pulling a well harder will be less

thafi those calculated by the straight.line PI extrapolation;

conversely, production losses resulting from higher back

pressures will be less than those anticipated by straight-

Iine methods,

It is ditlicult to overstate the importance of using sta-

bilized well tests in the calculations. In a low-permeability

0.20

0

—

0.80 -

0.60 -

0,4( -

CRUOE OIL PROPERTIES, FIG A-1o

RELATIVE PERMEABILITY ,FIG, A-20

WELL SPACING z 20 ACRES

WELL RADIUS :0,33 FOOT

INITIAL RESERVOIR PRESS URE:21300$I

BuBBLE POINT= 2130psi

\

I 4 %

I 2

Io%

~

o)

I

00

0.80 -

I 4 “ /.

0.60 -

12”/e—

E

I 9“A

?

6“ /e, 8 %

~:

0.40 -

CASE 3

0.20 -

SAME AS CASE I, EXCEPT WITH

ABSOLUTE PERMEABILITY OF 200md

o

1

1

I

o

020

0.+ 0 0.60 0.80

I 4 % ——

~

SAME AS CASE 1 ,ExCEPT WITH

.40-ACRE SPACING

1 1

1 1

(b)

14%

SAME AS CASE I, EXCEPT TIIAT WEI. L

IS FRACTURED (PSEUDO WELL

RADIuS = 50 FEETI

1

0

0


8/10

.

ieservoir it frequently will be found that significant changes

in producing conditions should not be made for several

“days preceding an important test. This presents no prob-

lem if a well is to be tested at its normal producing rate,

but it becomes more difficult if multi-rate tests are required

Accuracy of Reference Curve

It is anticipated that the most common use of the refer-

ence IPR curve will be ‘to predict producing rates at high-

cr drawdowns from data measured at lower drawdowns.

For example, from well tests taken under flowing condi-

tions, predictions will be made of productivities to be

expected upon installation of artificial l ift. It 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 correspcmdingly low drawdowns

are extrapolated with the aid of the reference curve to

estimate maximum productivities as [he 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 drawdowns (1 I to 13 percent), and the solid lines

represent the actual IPR’s calculated by the computer.

1.00

0.s0

0.50

K

<

5

la

0.40

020

0

1.00

0.s0

0.40

0,20

—

\-

‘\

\

\

\

\

=.

~Np/N: 0,1’/.,6”/,,10”/.

\

\

\

\

yA

14%

\

\

\

\

CASE 5

\

\

SAME AS CASE 1, ExCEPT WELL HAS

\

PLUS 5 SKIN

\

,

1 1

[

o)

I4

“/.

CASE 7

SAME AS CASE 1, EXCEPT WITH

LESS VISCOUS CRUOE OIL FROM

FIG. A-lb

The maximum error for the reservoir considered in Fig.

7 is less than 5 percent hroughout most of its producing

iife, rising to 20 percent during final stages of depiction.

Although the 20 percent error may seem high, the actual

magnitude of the error is less than ?4 BOPD,

It is obvious from Fig. 7 that if well 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-erro: calculations were made for all the res-

ervoir conditions investigated. Except for those cases with

viscous crudes and with flow restricted by skin effect,

it appears that a maximum error on the order of 20 per-

cent should be expected if al 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-

erally 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 0.20

0.40 0.60 0,60 1,00

(cl

BuBBLE POINT

16%

Io “/.

CASE6

SAME AS CASE 1, EXCEPT THAT

RESERVOIR PRESSU2E IS IN ITIALLY

ABOVE THE BuB@LE PO INT, BEING

3040ps i IN STEAO 0F2130Ps i

8

12

*

C=

*SAME AS CASE 1. EXCEPT WITH MORE

VISCOUS CRUQE OIL FROM FIG. A-id

1

I I

I

1

0

0,20

0.40 0460

0.80 1, )

qo/%)mOx

(d)

Fig. 12-Calculated dimensionless IPR curves.

I


9/10

cause better agreement is noted between IPR’s and refer-

ence curve throughout most of the prodhcing 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 flo}” 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 tht 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 portitin of the wells producing from a res-

ervoir in which expansion of a gas cap is a significant

factm.

Since the referecce 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,

[.00

0.80 -

= 060 -

c

? o “/

5

n

0.40 -

CASE 9

02:t:,;:,E~,;,,,,,;,,?

AME AS CASE I, ExCEPT WITH HIGHER

BUBBLE POINT CRUDE OIL FROM FIG. A-le

(a)

Loo

\

0.80 -

\

yA

\

20 %

\

= 0.60

-

N ~ IN =0.1%

Io %

\’

<

~~ ~

28”/*—

\

; \

a

\

040

\

CASE [1

\,”

0.7.0

SAME AS CASE 1, ExCEPT WITH

PERMEABILITY CHARACTERISTICS

FROM FIG. A-2c

qo/(qOl mox

(c)

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 dimril-

sionless IPR’s for the two types of single-phase flov, are

compared with the suggested reference curve for mlution -

gas drive reservoirs in Fig. 8.

Conclusions

IPR curves calculated both for differeut 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-

)

pres?urcs.

10“/0

16%

cASE 10

SAME AS CASE 1, EXCEPT wITH


FROM FIG. A-2b

Io %

18“A

[

CASE 12

sAME AS CASE I, EXCEPT WITH


FROM FIG. A-2b ANDCRUDEOIL PROP-

L~

RTIE 3

FROM F G. A-lb

o

0 20

0.40

)

f . lo / (qol mol.

(d)


10/10

1.00

0.80

0.4C

0.20

1,00

8

0,40

0.20

\

\\

, .A, NP/kz O.l”/e, 2%

\“’..\

“\\

N,

\

\

\

10% 6 v,

\

\

\

\

CA SE13

\

.—

\

\

SAME AS CASE 1, EXCEPT WITH LOW-

\

GOR CRUDE FROM FIG. A-if

\

IO*A—

CASE [5



FROM FIG, A-2c AN0 CRUDE OIL PROP-

ERTIES FROtd FIG, A- [b

~

1 I

1

0.40 0.60 0.80

Lc

A )maa

c

Fig 14--Calculated dim

This technique needs to be verified by a comparison

with field results, As meviously discussed, the conclusions

-L= UaWJ UIIIY UII WJUpULCJ >Ulutions involving several

simplifying assumptions as listed in the Introduction.

. L..”,.A -..1..

. .

“1.”-.+- --1,,

References

I 1. Evineer. H. H, and

Muskat, M,:

“Calculation,of Theoretical

?actor”,

Trans

AIME (1942) 146, 1; 6-139.

e-. -—._-

%oductivity 1

2. Gilbert, W. E,:

“Flowing and Gm-IJft Well Performance”,

Drill. and Prod. Prac,, API (1954) 126,

;ng Two-Phase

. Weller, W. T,: “Reservoir Performance Duri

Flow”, J. Pet, Tech. (Feb 1QK6~ ~Jn-~~~., .,””, .-r”- TV,

I

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.

I

I

APPENDIX A

Input Data

Figs. 9 and 10 illustrate graphically the input data (crude

oil PV’T characteristics and relative permeability charac-

teristics) from which the theoretical behavior of simulated

reservoirs was calculated by the computer,

\

\

\

\

\

\

\

\

“\

\:’’N=O’’*’*

Iov.

‘\

20 “i.

‘Y

T,A

26%

\

\

CASE [4

\

\


\


\

FROM FIG, A-2b ANO CRUDE OIL

PROPERTIES FROM FIG. A-le

(b)

2% —

4 ‘(*

CASE 16

SAME 4.S CASE 1 , EXCEPT PERMEABILITY

APPROXIMATELY FROM STRAIGHT LINES

OF FIG, A.2d AND CRUDE OIL PROPERTIES

APPROXIMATE FROM STRAIGHT LINES OF

FIG. A-Ic

I

I 1

0.20

0.40

0.60 0,80 I

)

(d)

m@lcss

IPR clirves,

APPENDIX B

Computer-Calculated IPR Curves

l)imensionless IPR Curves

Figs, 11 through 14 are graphs of the theoretical [PR’s

calculated for various simulated reservoir conditions. So

that the IPRs under various conditions can be compared

more easily, the initial IPR curve (NP/N = 0,1 percent)

from Fig 1la is reproduced on all succeeding figures and

is designated as Curve A,

In addition KOhe 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, ‘~he 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 no(e: A pictl(re and biograplzical sketch o/

J. V, Vogel appear on page 60.

vogel 1968 para el tema 2

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