PRODUCTIONESTIMATION

@ A

Inflow Performance Relationships for Solution-Gas Drive Wells

Jo V. VOGELMEMBER AIME

Abstract

Its calculating oilwell production, it has commonly beenassumed that producing rates are proportional to draw-downs. Using thi,s assumptimt, a well’s behavior can bedescribed by its productivity index (Pi). This PI relation-ship was developed from Darcy’s law for the steady-stateradial flow oj a single, incompressible fhid. AlthoughMuskat pointed out tha~ the relationship is not valid whenboth oil and gas jiow in a reservoir, its use has continuedfor lack of better approximations, Gilbert proposed meth-ods of well analysis utilizing a curve of producing ratesplotted against b ot!om-hole well pressures; he termed thiscomplete 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 ofthe computer. Using machine computations, IPR curveswere calculated for wells producing from several fictitioussolution-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 wasdeveloped that is simple to apply and, it is believed, canhe used for most solution-gas drive reservoirs to providemore accurate calculations for oil well productivity thancan be secured with PI methods. Field verification i.sneeded.

Introduction

In calculating the productivity of oil wells, it is cornmonly assumed that inflow into a well is directly pro-portional to the pressure differential, between the reservoirand the wellbore — that production is directly propor-tional to drawdown. The constant of proportionality isthe PI, derived from Darcy’s law for the steady-state rad-ial flow of a single, incompressible fluid. For cases inwhich this relationship holds, a plot of the producingrates vs the corresponding bottom-hole pressures resultsin a straight line (Fig. 1), The PI of the well is the inverseof the slope of the straight line.

However, Muskat’ pointed out that when two-phaseliquid and gas flow exists in a reservoir, this relationshipshould not be expected to hold; he presented theoreticalcalculations to show that graphs of producing rates vsbottom-hole pressures for two-phase flow resulted incurved rather than straight lines, When curvature exists,

Original matmacript received in Society of Petroleum Engln*rs o~@eJuly 11, 1966. Revised manuscript reaekd Dec. S, 1967. PaPer (SPE1476) wae presented at SPE 41st Annual Fall Meetbra held in Dallas,Tex,, Oot, 2.5, ISSC. @CbRyrlght 1S6S American Institute 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.

JANUARY, 19611

SHELL OIL CO.BAKERSFIELD, CALIF,

a well cannot be said to have a single PI because thevalue of the slope varies continuously with the variationin drawdown, For this reason, Gilbert’ proposed methodsof well analysis that could utilize the whole curve ofproducing rates plotted against intake pressures. He termedthis complete graph the inflow performarwe relationship(IPR) of a well.

Although the straight-line approximation. is known tohave limitations when applied to ttio-phase tlow in thereservoir, it still is used primarily because no simple sub-stitistes have been available: The calculations necessary tocompute IPR’s from two-phase flow theory have beenextremely tedious. However, recently the approximationsof 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 isuniform and isotropic with a constant water saturationat 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<oil and ,gas (6) the same pressure exists in both the oil and gasphasex and (7) the semisteady-state assumption that thetank-oil desaturation rate is the same at all points at agiven instant. Weller’s solution did not require the con-stant-GOR assumption.

The resulting computer program proved convenient touse and gave results closely approaching those furnishedby the more complicated method of West, Garvin andSheldon.’ The program also includes the unique feature

—

\ MA KIMUM

Ps4nwclns SATE, bwd

Fig. l—Straight-line inflow performance relationship.

of making complete IPR predictions for a reservoir. $mchpredictions for a typical solution-gas drive reservoir areshown as a family of IPR cur:’ s on Fig. 2. Note thatthey confirm the existence of curkdure.

U appeared that if several solution-gas “drive reservoirswere examined with the aid of this program, empiricalrelationships might be established that would apply tosolution-gas drive reservoirs in general. This p&per sum-marizes the results of such a study that dealt with severalsimulated 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, fractwingand skir~restrictions.

The i:westigation sought relationships valid only below

2800

k

RESERVOIR CONDITIONS:

ORIGINAL PRESSURE I 2130 Psi

2400 BuBBLE POINT , 2130 psiCRUDE OIL PVT CHARACTERISTICS

FROM FIG. A-10

~ RELATIvE PERMEABILITY CHAR-

- 2000 ACTERISTICS FROM FIG. A-20wa

k

WELL SPACING * 20 ACRES

JWEI L RAOIUS I 0.33 FOOT

g, 1600

-1

I

CUMULATIVE RECOVERV,, !,‘14+

dPERCENT 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~Np/N=OJ%,2eh,4”l.

6“le,8”h

~g ‘

.3b Io%

~z~og ~ 0.4 I 2 “A

v I 4“/.

ZSok1- --

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 jora solution-gasdrive reservoir.

n.s ,–– —

the bubbie point, Computations were made for reservoirsinitially above the bubble point, but only to ensure thatthis initial condition did not cause a significant change inbehavior below the bubble point.

Shape of Inflow Performance RelationshipCurves with Normal I’/eterioration

As depletion proceeds in a solutlon-gas drive reservoir,the productivity of a typical well decreases, primarilybecause the reservoir pressure is reduced and becauseincreasing gas saturation causes greater resistance to oilflow, The result is a progressive deterioration of the IPRs,typified by the IPR curves in Fig. 2. Exarr.ination of thesecurves does not make it apparent whether they have anyproperties 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 anIPR 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 isdone, the curves from Fig, 2 can bereplotted as shown inFig. 3. It is then readily ~ppaI’entthat with this construe.tion the curves are,, remmkably similar throughout mostof the producing life of the reservoir

2500 ———

F\ ~~

A+PR FROM FIG 2FCIR Npfld ’01%

BIPRw IT HP DIFFERENT CRUDE OIL

2000FLOWING, ALI. OTHER CON OIIIONS

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), fRACTIONOF MAXIM1’M

(b} DIMENSIONLESS IPR’S

300

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

JOURNAL OF PETROLEUM T13CHNO_&OCY

Effect of Crude Oil CharacteristicsOn IPR Curves

From the foregoing results it appears that IPR curvesdiffering over the life of a given reservoir actually possessa co,mmon rekdtionship. To determine whether this samerelationship 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 newcharacteristics included a viscosity about half that of thefirst 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 aless viscous crude (Curve B) the pr~ductivity was muchgreatec than in the first case (Curve A). However, whenplotted on a dimensionless basis (Fig. 4b) the IPR’s arequite sim!lar, As lPR.’s for the second case deterioratedwith depletion, no greater change of shape occurred thanwas noted in the previous section. These two crude oils

1,

wK 0.amu)waa

E5>awmwa

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 relationshipunder 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 whilethe reservoir pressure w-as above the bubble point, the”slopes of the IPR curves were discontinuous with theupper part being a straight line until the well pressurewas reduced below the bubble point. Below this pointthe IPR showed curvature similar to that noted previous-ly. After the reservoir pressure went below the bubblepoint, all the dimensionless IPR curves agreed well withthe previous curves.

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

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

1968 m

,, ,-

gu 1.0a

,REFERENCE CURVE

12%

I 4 “/.

RESERVOIR CONDITIONS

sAME AS FIC.2

0.2 0.4 0.6 0.8 LOot-1-

PROOUCIMG RATE (qo /@o)max), FRACTION oF MAXIMUM0m

Fig. 6—Con?pmison of reference curve with computer-ca[culated IPR curves.

Effect of Relative Permeability andOther Conditions

-. ‘l’he same basic shape of the curves was noted whenthe study was extended to cover a much wider range ofconditions, Runs were made with three different sets ofrelative permeability curves in various combinations withthe different crude oils. The results were in ‘agreementsufficient to indicate that the relationship might be validfor most conditions.

Lo

TWO-PHASE FLOW(REFERENCF CURVE I

0.8 1-

0.6 -!==>

.;LIQUIO FLOW

0.4 -

08 2-

0 ~—0 0.2 0,4

,,

. cl/qmQR

Fig. G-Comparison of IPR’$ for !iquid flow, gas flowand fwo-phase ffow.

To explore further the generality of the relationship,a run was made in which the crude oil PVT curves andthe relative permeability curves were roughly approxi-mated by straight linss. It was surprising to fi,ld that, evenwith 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 thosefrom 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 thewell with a skin effect, in which case the IPR’s morenearly approached straight lines.

in summary, calculations for 21 reservoir conditionsresulted in IPR’s generally exhibiting a similar shape.

\

“oo~ ‘-— RESERVOIR CONDITIONS SAME AS FIG. 2

2000~\poNToFMATcH(wELLTEsT)o1800

1600 - \\

\\

1400

~“ ‘

\\

—STRAIGHT. LINE4’\ 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 IIOK

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 JIO URNA?. OF PETROLEUM ‘CECEN(SLOGY

Significant deviation was no:ed only for the more viscouscrude, for a reservoir initially above the bubble point,and for a well producing through a restrictive skin. Evenin these cases, definite curvature was still apparent,

The curves of crude oil characteristics and of relativepermeability that furnished the input data for the variousconditions studied are given in Appendix A. Dimension.less IPR curves calculated for various conditions areshown in Appendix B.

““[’ooo~”o200 - [600

I50 . - 1200 .--w

x K*

- 100 - 800 -

50 -

0 - f)

- 4.0

No

- 3,0 ;

a-

- 2!0 *O

Lm

- 1.0

Jo. “-o 1000 Zooo

PRESSURE psi

(o) Pb : 2r30 psi

250 2000

r

I/Bg5.0

PRESSURE, psi

(c) Pb : 2130 psi

““[’ooo~”o

Proposed Reference IPR Curve

If the IPR curves for other solution-gas drive reser-voirs exhibit the same shape as those investigated in thkstudy, 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 drivereservoirs would not imply that all these reservoirs are,

:[:+—-----:

150

-1 .~f$

1200~w RS

\ K“’

I 00 800Pg

. B.

50

t

400

PO

nfl I

I I

1m“

1,0

-10“o 1000 2009

PREsSURE, psi

(b) : 2130 psi

25:[ 2000+5017”0200

-1. ~

1600

t

- 4,0 - 6.0

[50 I 200 - 3,0N: .. ~,f)

m l/Bg ~

e a’” ‘9 am2

100 800 - 2.0 “ - 440m“

I Y& - ‘:

%

50 400B. ,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~~’(oII

200

IL

[600

I50 1200l;!3g

m

: K-

100 s 00 Pg

B.

50 400 Po

- 4.0

N0

- 3,0 “m

a.

L

- 2,0 ?

II”

- 1.0

;1.OIAZI=Z!ZI=Z!Oo 1000 2000

PRESSURE, psi

~ (f) pb : 2130 psi

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

JANUARY, 196S ,.. . 87

,

identical any more than wou!d the preser.: use of straight-line PI’s for all such reservoirs. Rather. the curve can beregarded as a general solution af the solution-gas drivereservoir flow equaticms with the constants for particularsolutions depending on the individual reservoir charac-teristics,

Although one of the dimensionless curves taken fromthe computer calculations could probably be used as areference 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 kf9 \ /

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“” ..[

Sgc : 10%

Sw : 19,470

‘$ : 13.90/00.30 h : 23.5fi kro

k :20md

L1 25 kro (l OOO/OS+l ): 0.444

0.20 -

0.15 -

0.10 -

0.05 -

0’ b1 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 ‘1fit is

,qo _/. 1 – 0.20&- – 0,80 &M~\’ , , , (1)(q. )nmx Pn \ Pli /

where g. is the producing rate corresponding to a givenwell intake pressure p,,,, ~;, is the $orrespcnding reservoirpressure, 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 1PRis

,

Sgc : .s,0%

Sw : 19840/”

.# : 13,9%

h’ = 23.5ft

k : 20md

k~. (100% s,,1= 0.444 ‘

I k ro

krq

\

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

kr9

1 10,3 0,4 0,5 0,6 0,7 o#E 0.9

S(TOTAL LIOUIO]

(4)

Ffg. 10—Input data, relative permeability curves.

—

.

When q./(q.)~., from Eq, 1 is plotted vs p,,,/5,, thedimensionless IPRreference curve results, Onthe basis ofthe cases studied, it is assumed that about the same curvewill result for all wells, If q,, is plotted vs p~,, the actuallPR curve fora particular well should result.

A comparison of this curve with those calculated onthe computer is illustrated in Fig. 6. The curve matchesmore closely the IPR curves for early stages of depletionthan the IPR curves for later stages of depletion. In thisway, the percent of error is least when dealing with thekigher producing rates in the early stages of depletion.The percentage error becomes .greater in the later stagesof depletion, but here production rates are low and, asa consequence, numerical errors would be less in absolutemagnitude,

Use of Reference CurveThe method of using the curve in Fig. 5 is best illu-

strated by the following example problem. A well tests65 130PD with a flowing bottom-hole pressure of 1,500psi 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 artificiallift were installed to reduce the producing bottom-holepressure to 500 psi.

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. ),,,,. = 162BOPD; (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-linePI extrapolation, the productivity with artificial lift wouldhave been estimated as 195 BOPD rather than 145 IK)PD.

..This illustrates a significant conclusion to be drawn forcases in which such IPR r.,xvature-exists, Production in-.creases resulting from pulling a well harder will be lessthafi those calculated by the straight.line PI extrapolation;conversely, production losses resulting from higher backpressures 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-1oRELATIVE PERMEABILITY ,FIG, A-20WELL SPACING z 20 ACRESWELL RADIUS :0,33 FOOTINITIAL RESERVOIR PRESS URE:21300$IBuBBLE POINT= 2130psi

\

I 4 %

I 2 %

Io%

~

(o)

I 00

0.80 -

I 4 “/.

0.60 - 12”/e—

E I9 “A?

6 “/e, 8 %~:

0.40 -

CASE 3

0.20 - SAME AS CASE I, EXCEPT WITHABSOLUTE PERMEABILITY OF 200md

o1 ! 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. LIS FRACTURED (PSEUDO WELLRADIuS = 50 FEETI

1

0 0<20 040 0.60 0.80 1.00

qo/(%lmaa

[d]

.

ieservoir it frequently will be found that significant changesin 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 CurveIt 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 beexpected upon installation of artificial lift. It is necessaryto arrive at the approximate accuracy of such predictions,

Maximum error will occur when well tests made at very

low producing rates and correspcmdingly low drawdownsare extrapolated with the aid of the reference curve toestimate maximum productivities as [he drawdown ap-proaches 100 percent of the reservoir pressure. The errorthat would result under such conditions was investigated,and typical results are shown in Fig, 7. in this figure thedashed lines represent IPR’s estimated from well tests atlow drawdowns (1 I to 13 percent), and the solid linesrepresent the actual IPR’s calculated by the computer.

1.00

0.s0

0.50K

<

5la

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)

I 4 “/.

CASE 7

SAME AS CASE 1, EXCEPT WITH

LESS VISCOUS CRUOE OIL FROMFIG. A-lb

The maximum error for the reservoir considered in Fig.7 is less than 5 percent !hroughout most of its producingiife, rising to 20 percent during final stages of depiction.Although the 20 percent error may seem high, the actualmagnitude of the error is less than ?4 BOPD,

It is obvious from Fig. 7 that if well tests are madeat higher drawdowns than the extreme cases illustrated,the point of match of the estimated and actual IPR curvesis 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 withviscous 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’sfollow the reference curve as closely as have the severalcases investigated. For comparison, the maximum errorsfor the straight-line PI extrapolation ‘method were gen-erally between 70 and 80 percent, dropping to about30 percent only during final stages of depletion.

The figures cited above refer to the maximum errorsthat should be expected. In most applications the errorsshould 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%

I o “/.

CASE6

SAME AS CASE 1, EXCEPT THATRESERVOIR PRESSU2E IS INITIALLYABOVE THE BuB@LE PO IN T, BEING3040psi IN STEAO 0F2130Psi

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

I

cause better agreement is noted between IPR’s and refer-ence curve throughout most of the prodhcing life of thereservoirs and because well tests are ordinarily made atgreater drawdowns. ..=,,=—.

Application of Reference Curve:Other Types of Reservoirs

The proposed dimensionless IPR curve results fromcomputer analysis of the two-phase flo}” and depletionequations for a solution-gas drive reservoir only andwould not be considered correct where other types ofdrive 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 barrierrows of producing wells nearer the encroachment front,It appears that tht reference curve could be used for theshielded wells for at least a portion of their producinglives. Similarly, the reference curve might give reasonableresults for a portitin of the wells producing from a res-ervoir in which expansion of a gas cap is a significantfactm.

Since the referecce curve is for the two-phase flow ofoil and gas only, it would not be considered valid whenthree phases (oil, gas and water) are flowing. However,

[.00

0.80 -

= 060 -

c? o “/”

5n

0.40 -

CASE 9

02:t:,;:,E~,;,,,,,;,,?\SAME 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% I o %

\’<

~~ ~

28”/*— \

; \a \

040\

CASE [1 \,”

0.7.0 SAME AS CASE 1, ExCEPT WITHPERMEABILITY CHARACTERISTICSFROM 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 areservoir, For radial flow, this curve should lie some-where between the straight line for a single-phase liquidflow and the curve for single-phase gas flow, The dimril-sionless IPR’s for the two types of single-phase flov, arecompared with the suggested reference curve for mlution -gas drive reservoirs in Fig. 8.

Conclusions

IPR curves calculated both for differeut reservoirs andfor the same reservoirs at different stages of depletionvaried several-fold in actual magnitude, Nevertheless, thecurves generally exhibited about the same shape.

This similarity should permit substitution of a simpleempirical curve for the straight-line PI approximationscommonly used, Maximum errors in calculated produc.tivities are expected to be on the order of 20 percentcompared with 80 percent with the PI method, Productiv-ity calculations made with the reference curve methodrather 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

PERMEABILITY CHARACTERISTICS

FROM FIG. A-2b

Io %

18 “A

[

CASE 12

sAME AS CASE I, EXCEPT WITHPERMEABILITY CHARACTERISTICSFROM FIG. A-2b ANDCRUDEOIL PROP-

L~ERTIE!3 FROM F! G. A-lb

o 0!20 0.40 )

f.lo /(qol mol.(d)

1.00

0.80

0.4C

0.20

0

1,00

0.80

0,40

0.20

0

\

\\ , .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

SAME AS CASE 1, EXCEPT WITHPERMEABILITY CHARACTERISTICSFROM FIG, A-2c AN0 CRUDE OIL PROP-

ERTIES FROtd FIG, A-[b

~ 1 I 10.40 0.60 0.80 Lc

%A%)maa

(c)

Fig. 14--Calculated dim

This technique needs to be verified by a comparisonwith field results, As meviously discussed, the conclusions-L= UaWJ UIIIY UII WJUpULCJ >Ulutions involving severalsimplifying assumptions as listed in the Introduction.

..”,. L..”,.A -..1.. . . “1.”-.+- --1,,

I ReferencesI 1. Evineer. H. H, and Muskat, M,: “Calculation,of Theoretical

?actor”, Trans., AIME (1942) 146, 1;!6-139.e-., -—._-

%oductivity 1

I 2. Gilbert, W. E,: “Flowing and Gm-IJft Well Performance”,Drill. and Prod. Prac,, API ( 1954) 126,

;ng Two-Phase3. Weller, W. T,: “Reservoir Performance DuriFlow”, J. Pet, Tech. (Feb 1QK6~ ~Jn-~~~w., .,””, .-r”-& TV,

I

4. West, W, J,, Garvin, W. W. and Sheldon, J. W.: “Solutionof the Equations of Unsteady-State Two-Phase Flow in OilReservoirs”, Trans., AIME ( 1954) 201, 217-229.

I

IAPPENDIX A

Input Data

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

oil PV’T characteristics and relative permeability charac-teristics) from which the theoretical behavior of simulatedreservoirs was calculated by the computer,

\ \\

\\

\\

\

“\ \:’’N=O’’*’*Iov. ‘\

20 “i.

‘Y

T,A

26%\

\

CASE [4 \

\SAME AS CASE 1, EXCEPT WITH \PERMEABILITY CHARACTERISTICS

\FROM FIG, A-2b ANO CRUDE OILPROPERTIES FROM FIG. A-le

(b)

2% —

4 ‘(*

CASE 16

SAME 4.S CASE 1, EXCEPT PERMEABILITYAPPROXIMATELY FROM STRAIGHT LINES

OF FIG, A.2d AND CRUDE OIL PROPERTIES

APPROXIMATE FROM STRAIGHT LINES OFFIG. A-Ic I I 1

0.20 0.40 0.60 0,80 I )

(d)

m@lcss IPR clirves,

APPENDIX B IComputer-Calculated IPR Curves

Il)imensionless IPR Curves

Figs, 11 through 14 are graphs of the theoretical [PR’scalculated for various simulated reservoir conditions. Sothat the IPRs under various conditions can be comparedmore easily, the initial IPR curve (NP/N = 0,1 percent)from Fig 1la is reproduced on all succeeding figures andis designated as Curve A,

In addition KOthe cases illustrated, five more calculationswere made in which individual curves of the crude oilproperties in Fig, 9a were replaced one by one with thecurves from Fig, 9b, ‘~he results were comparable to thoseshown, and, since the illustrations include the case inwhich the curves of Fig. 9a were completely replaced bythose of Fig. 9b, it was not considered necessary to repro-duce the cases in which the individual components werereplaced. **

Editor’s no(e: A pictl(re and biograplzical sketch o/J. V, Vogel appear on page 60.

“1