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'1 '
APJ BUL DH First Edition Odoher 19fi7
A STATISTICAL STUDY OF RECOVERY EFFICIENCY
AMERICAN PETROLEUM INSTITUTE New York, N. ·y. 10020
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1976
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Issued by AMEHJCAN PETROLEUM INSTl'l'lJ'l'E
Division of Produdion Dallas. Texas 75201
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Copyright© 1967 hy American Petroleum Institute
A STATISTICAL STUDY OF RECOVERY EFFICIENCY A Report By The API Subcommittee on Recovery Efficiency Chairman
J. J. Arps Petroleum Consultant Dallas, 'l'exas Currenl Memf¡ership
F. H. Brinkman Esso Production Research Co.
Houston, Texas *Prof. Folkert Brons U niversity of Texas
of
the S11l,r,om111if./ec
T. M. Geffen Pan American Petroleum Corp. Tulsa, Oklahoma
Austin, Texas
R. K. Guthrie R. K. Guthrie & Assoc. San Antonio, Texas
R. W. Buchwald Sun Oil Company Dallas, Texas
Gulf Oil Corp. Houston, Texas
R. C. Craze Esso Production Ilesearch Co. Jlouston, Texas
G. L. Hancock
G. K Hendrickson
W. J. Rog-ers Sinclair Oil & Gas Co. Tulsa, Oklahoma
A. E. Smith
Bartlesville, Oklahoma
Petroleum Consultant Dallas, Texas
T. S. Hutchinson
E~. F. Egan Texaco lnc.
Prof. H. T. Kennedy Texas A&M Universi~y College Station, Texas
*L. F. Elkins Sohio Petroleum Co. Oklahoma City, Oklahoma
J.M. Miller Standard Oil Co. of Calif., Western Operations, lm~. San Francisco, California
Cities Service Oil Co.
J. N. Dew Continental Oil Co. Panca City, Oklahoma
Houston, Texas
*V. T. McGhee Phillips Petroleurn Co. Bartlesville, Oklahoma
Atlantic Richfield Co. Dallas, Texas
L. C. McColloch Sun Oil Co. Dal1as, Texas
*A. F. van Everdingen DeGolyer & MacNaughton Dallas, Texas ,!,O. E. Van Meter, .J r. Mohil Oil Col'p. Shreveport, Louisiana
,J. W. Wilson Union Oil Co. of California Brea, California
Requests for,· permi.
'
"
TABLE OF CONTENTS
l'age I
SUMMARY AND CONCLUS10NS .. .
1
II
INTRODUCTION ................. .
4
JU
PRELIMINARY ANALYSIS OF CASE HISTORIES ...
5
A. Number and Distribution of Case Histories by Groups ..
5
B. Range of Basic Parameters ........ . C. Rating and Weighting of Case Histories ..... .
6
D. E. F. G.
!)
9
Predominant Drive Mechanisms ....... . Rccovcry Efficiencies of thc Various Mechanisms ... Reservoir Rock .. . Well Spacing .......... .
13
IV
CORRELATION METHODS A. Recovery Parameters or Dependent Variables ......... . B. Reservoir Parameters and Independent Variables .. . C. Regression Equations
13 14 16 16
V
EARLY CORRELATION EFFORTS .... A. Primary Reéovery from Water Drive Reservoirs. B. Primary Recovery from Solution Gas Drive Reservoirs ... C. Secondary Recovery by Fluid Injection ............ .
9 .13
17 17 18 . ... 18 18
D. Inter-Correlation of P. V. T. Data ...... . E. Inter-Correlation of Rock Parameters. . . . . . . . . . . . . . . . . . . . 19 F. Overall Results ................................................................... '19 VI
FINAL CORRELATIONS ... . A. Results .............................................................. . l. Water Drive ................................................................... 2. Solution Gas Drive . . . . . . . . . . . B. Numerical Examples C. Discussion of Results . .....................................
?II
NOMENCLATURE ..... .
. ............................................ 0
VIII
REFERENCES
. ..........................................
.......... .
'APPENDIX: REGRESSION ANALYSIS AND TESTS OF SIGNIFICANCE ....
~{2):U3996
_\;,
19 19 19 20 20 22
~
28
.Figures Fig. Fig. F'ig.
1-Cumparison uf Olmcrvcd an
3
2-Comparison of Observed and Computed Values of Recovery Factors in Barreis· per Acre Foot from 80 Solution Gas Drive Reservoirs below Bubble Point .......... .
4
3-Cumulative Frequency Distribution of Permeabilities for Water Drivc and Solution Gas Drivc Cases-Sandstone Reservoirs ................... ·............ , ... .
JO
Fig.
4-Cumulative Frequency Distribution ·of Permeabilities for Water Drive and Solution Gas Drive Cases-Limestone Reservoirs ................................... . 11
Fig.
5-Cumulative Frequency Distribution of Recovery Efficiencies for Four Drive Mechanisms ........................................................... . 12 G-Cumulative Frequency Distribution of Recovery Factors for J<'our Drive Meclianisms .. H
Fig. Fig. Fig.
7-Cumulative Frequency Distribution of Residual Oil Saturation for Water Drive ami Residual Gas Saturation for Solution Gas Drive Reservoirs .......................... . 15 8-Correlation of Observed and Computed Dependent Variables ......................... . 29
Fig.
9-" stu.dent" t Distribution ........................................................... . 31
Fig. 10-Significant Coefficient of Correlation ............................................... . 32 Fig. 11-Significant Increase in Coefficient of Correlation ..... .
32
Tables Table Table
1-Distribution of Case Histories by Rcservoir Mcchanism and Rock Typc .. 2-Range of Parameters for Water Drive Reservoirs .......................... :.
Table
3-Range of Parameters for Solution Gas Drive Reservoirs without Supplemental Drives ............................................................ .
Table Table Table Table Table Table Table Table Table
IUUf
:5 6
7
4-Range of Parameters for Solution Gas Drive Reservoirs with Supplemental Drives ............................................................. . 7 5-Range of Parameters for Gas Cap Drive Reservoirs ............................... . 8 6-Range of Parameters for Gravity Drainage Reservoirs ............................. . 8 7-Median Val ues of Recovery Efficiency by Recovery Mechanisms .................... . 12 8-Summary of Exponents of Terms in Regrcssion Equations, Rccovcry Factor for Water Drive Reservoirs ..................................... . 9-Summary of Exponents of Terms in Rcgrcssion Equations, Rccovcry Factor for Solution Gas Drive Rcservoirs ............................. . 21 10-Effect of the Addition of Various Tcrms in the Regression Equations on Computed Recovery Factors ........................................ . 2~ 11-Effect of Variation in Independent Variables on ltecovcry Factor from '1 Water Drivc Reservoirs ................................................... . 24 12-Effect of Variation in Independent Variables on llecovery Factor from Solution Gas Drivc Reservoirs ......................... ·......... '................. . 25
¡,_~ = = = - -
2)
l
1
AStatistical Study of Recovery Efficiency J. J. ARPS, FOLKERT BRONS, A. F. VAN EVERDINGEN, R. W. BUCHWALD, AND A. E. SMITI! I-SUMMARY AND CONCLUSIONS 'fhe cquations ami conclusions prcsentc
lahoratory data. The study began as an effort to apply machine methods of analysis to updated data from the 103 reservoirs investigatcd by Cr~ze and Buckley in 1945,t using multiple correlation analysis procedurcs dcscribed by Guthrie and Grcenberger in 1!)55.2 ·This initial effort was latcr expanded to include collection and analysis of reservoir and recoYery data on additional reservoirs. A vailability of advanced computers matle it ¡iossible to apply more sophisticated techniques to arrive at statistically significant correlations betwccn indicated recovenr factors and the reservoir parametcrs. The subcommittee collected data on 312 reserYoirs ,vhich were weighted 1, 2, ami 3 according to thcir rcliahility. This rcport deals with: a. Oil reservoirs under_ a water drivc mcchani_sm, where invading bottom or edge water is the dominant
Not covered iú this report are combinations of water drive and solution gas drive, gas cap drive, or gravity drainage which tog·ether comprised 30 to '10 pen.'ent of the case histories. Another 10 percent of ali cases received cóul
For Water Driv~ Reservoirs (Sands and Sandstones) NA/•'= (J,259) •
~(l {.
v·
-.
f1m
sw )it( +1.0422 ( ¡. . )+o.orro ( • ¡.µ,m • ,),,, ,
/foi
-0.1!1().7( p· )-0.2J,5.'J )
•
--¡l;;
b;incl:; per acre fout
( 1)
For units iiscd, sce VII-NOMJ::NCLATURK* Thc· cocfficicnt of mulliplc corrclation r for this c11uatio11 i8 0.!~58. This coeflicienl iudicatcs the "goodness of fit" ami mea·ns that thc ·regression was successful in rcmoving (0.958)2 or 91.8 percent of the original variancc in thc data. The standard error of estimate of this equation is 17.G percent. Aftcr dividing Equation (1) by 77.58 times the oíl-in-place term tp(1 -Su,) the cofrcs¡1oncling equation for the Recovery Efficiency (in pcrcent of initial stock ' B;ii ' tank oil in place) was found to be:
,
_
!.!,
f ~(1-Sw) l¡· +0.0422 )-/-o.ono =-(.,4.s.,s) ·, IJ . ·. (''''"'' --. · ( Sw, . . \ "' ,µ{JI 0
See p. 27 for references. *Symbols used ar.e those adopted.as standard -by Society of Petroleum Engineers of A IME.
' 1
pcrcent
(2)
J. J.
2
ARl'S, FOLKI·:RT BRONS, A. F. VAN EVERDlNGEN, R. W. RUCHWALD, ANDA. E. SMITH
Vor Solution Gas Drivc Rcscrvoirs (Sands, Sandstones, and Carbonate Rocks) Thc hcst equation for thc Rccovcry Factor bclow the hulihlc point ,vas found tu he:
l!AF
( 1(1-S,,,) ¡-f-J,JGll (
.,
= (.,24;,)
·, l
fl
(I"
f
·
k )-f-0,0!17,? ( -· µub
, )-/-0,.7722 (
8,,,
•
¡n, )-f-0,1141 -Prr.
barreis pcr acre Ioot (:J)
Thc cocflicieut of rnultiplc corrclation r for this cquatioH is 0.!>:3:.:! and thc standard error of cstimatc is 22.9 pcrccnt. Jn applying this correlalion to under-saturated oil rcscrvoirs, (bubblc point prcssure p11 below initial prcssurc p;) thc additional recovery due to expansion of the rcservoir and its fluids ovcr thc prcssure diffcrence (p¡ -
Ph) should
be added to thc compute
B,,
covery Efficiency (in pcrcent of stock-tank oil in place at huhblc point) is ohtaine
,
,
(
0 hL= (J,t.;t.,) ··¡
\
~(1-S,,,)}-/-0.1611 ( 1.- )+o.0.?79 ( , )+o.:n22 ( J)I, )-1-0.n.;1 B • -• S,, • --·91,
/!u{,
JICl"CCJ!\,
(-1)
1hi
A comparison of thc obscrved Recovcry Factors ami those eomputed by mea ns of Equations ( 1) and (3) is 5hown on Fig. 1 and 2. The .relative ,veighting of the data points is indicated by the size of the circlcs. It may be noted that in Equations (1) ancl (3) for the Recovery Factor, in barreis per acre foot, íour oí thc signif1cant indcvendent variables are not only similar hut their exponents are also of the same arder of magnitude (although oí opposite sign in thc case oí the water saturation and pressure ratio). 'l'hesc findings may point to a fruitful field for further research. Independent variables other than those Usted in these equations may have a significant ctfect on recovery, but the number of case histories of acceptable accuracy was insufficient to support inclusion of additional parameters under the requircments of statistical vali
IIIHf
.,
· BULLETIN Dl4: A STATISTICAL STUDY OF REC0\'1-:RY EFFICIENCY
o
1
1000 '
'
'
'
'
500
200
100
50 2000 o
I
WATER ORIVE
."
2000
'
:1 / 1, .
/
,
1000
•
• '¡ /
.
_I000
,,/.
._
e
,
i
500
~
º/
ONE STANDARD ERROR OF ESTIMATE
.. ~ • ~- . v¿· 0,._" .. • /,
.
/
•, /
/,
/ :L. • 1 •
/
..
.
t
•
500
/
/
f-
/~
0
.e
LL
o
<(
en
0
.e
·:xf_,v .. /. ' '//+. ,
LL
<(
en
/;:,·· /
200
200
100
WEIGHTING OF DATA • -3x • -2x . - , ¡x
f-
'
50
100
200
500
1000
00
1
50 2000
8 AFcomp. --1~~ In
o
~ ( 1-S.) ? ki., p. BAF-_•1n4259+1.04221n - --+.07 01n-,-.,- -.190310 S,-.215910-P-' 8 oi a
Fig. l-Comparison of' Observed · and Computed Values of Recovery Factors in Barreis per Acre Foot from 70 Water Orive Reservoirs Coefficient of Multiple Correlation, r = ·o.958 Standard Error of Estímate S, 0.162 ar 17.6 percent
=
·¡
J. J. ARPS,
4
FOLKERT BRONS, A. F. VAN EVERDINGEN,R W. BUC!IIVALD, ANO A. E. S~IIT!I
---------------------------------
15 600
30
100
50
'
'
200
'
'
I
I
600
t
• I/
1
1SOLUTION GAS ORIVE
I
/ /
.
V
7
•,
.
200 .
-. ,, " '.
'~• .
/
'~
i 100
LLº
I'-~
<(
\
en
/. /
50 .
/
.
/ / "
/
~ ,
/
/
;,•
/
/;
.
... ..
1,'
·./
100
•V,
50
•
/ ..
-·
,
30
····· .
/
WEIGHTING OF DATA
/
.
• --3x ' 2x • -· · - . lx '
In
"o
.e
en
.
•
/
t LL. <(
.
V
30
BAF,omp.'
200
,
l,l •
/
'
In
'
/
/
15
/
V "•/
/
30
/
\
''
1/
. ,¡·): . • /',.'·
ONE STANDARD ERROR OF ESTIMATE
1
/
'/
;.-/ .,,. . V ,
/
/
'
3244+1.1611
200
BAFcomp. In
5
1
'
'
100
50
-
\! ( I-S.) +.0979 In
B,,
k ~ob
+.37221~ s.+:1741
600
in.~ P0
Fig. 2 - Comparison of. Ob.served and Computed Va!ues of -Recovery Factors. in Barreis per Acre Foot
from 80 Solution Gas Orive Reservoirs below Bubble Point
Coefficient of Multiple ·Correlation, r ·Standard Error of Estímate, Sy
=
=
0.932
0.207 or 22.9 percent
11 - INTRODUCTJON Tn HJ/15 Craze anrl Ruckleyt pnhlished their now classic investigation on the eífect of well spacing on ultimate recovery efficiency using data from 10:1 reservoirs. In 1955 Guthrie and Grcenberger!! used the same data to illustrate thc use of multiplc correlation analysis in obtaining the best relation between the recovery cfficiency of water clri,·e reservoirs and various combinaiions of parameters which were thought to control it. During lf>fi6 a group of engineers consisting of l\Iessrs. R. .T. Dobson, .J. H. Sullivan, and S. C. Oliphant recommended the formation of an .API Subcommittee on Recovery Efficiency. It was approved hy the F.xecutive Committee
5
íllJLLETIN 1)]~: A STATISTICAL STUIJY Ol•' RECO\'EHY Et•'FICIENCY
on Drilling and Production Practice in June of thc samc year. The assignment to the Subcommittee, as indicated in Mr. A. W. Thompson's report to the Board of Directors in the 1956 Proceedings, was "to evaluate the relative impóÚance .of thé various types of data which are used or can be u;ed in the estimation of reserves." ·· · . As·ª first step, the Sub.committee under its energetic Chairman, Mr. E. Kraus of The Atlantic Réfining Company, designed"a ·comprehensive q~esti~nnaire covcring over 100 items of pertinent 'informatiÓn on .each. reservoir:. . . · · It was intended to use ~lectronic computers and statistical techniques, to determine from the Ínfo~ation obtaincd to what extent various factors and parameters a·ppear to affect ultimate Tec¿very froÍn oil reservoirs. The illduStry
would thus be provided with recovery factors based on actual performance of producing oil fields inste.at of.,theory or laboratory studies, .. To,,,ard- the end of 1959 the Subcommittee had .rcceived in excess of 300 case histories on oil res~rvoirs in. the United States and abroad, and by the middle of 1960 the time-con_suming job of checking the individual ques~ionnaires was well on its way. WÍth Mr. KTaus' re'tirément in Oc'tober 1960, the Subcommittee lost its inSpiring· ChaiI'~án whO h8.d provided much capa.ble· guidance through the organizational and data-gathering· phasés of the ·assig'nment; · Mr. J. J. Arps took over as Chairman a~Hl directcd thc Subcornmittee through the corrclation and computational work to thc cornpletion of thc present report. He divided the work by gcographical grouping of the membcrship between: 1, a Houston task force (Chairman, Prof. Folkert Brons, University of Texas, succceding Mr. E. K. Schluntz of Shell Developrnent Cornpany) to handle the correlations for water drive reservoirs; 2, a Dallas task force (Chairman, Mr. A. F. van Everdingen, DeGolyer and MacNaughton, succeeding its original chairman, Mr. E. G. Trostel), to study possible correlations for solution gas drive resenoirs; 3, an Oklahoma City-Tulsa task force (Chairman, Mr. L. F. Elkins, Sohio Petroleum Company) t9 sh~dy _the recoveries from reservoi_rs under fluid injection; .L a BartlesvillePonca City task force (Chairman, 1\'lr. V. T. l\kGhee, Phillips Pctroleum Company) to assist in checking pararneters of the reservoi.r fluids; and 5, a.,California task force (Chairman, Mr. O. E. Van Meter, Jr., Mobil Oil Corporation) to assist in checking the parameters of the reservoir rocks. ··: The pÍ-esent rei:,ort is the result ·of the combined effort of the entire Subcommittee membership represented by thcse different task forces. III-PRELIMINARY ANALYSIS OF CASE HISTORIES A. Number nnd DiSfribution of Casé Histories by Groups By" the end oí 1960 the nurnber of questionnaires receiYed by the Subcommittee had risen to 312, reporting on many different drive mechanisms all(l rock types. The number ,vas considered somewhat small for statistical validity, nevertheless it was decided to proceed with the correlation work rather than attcmpt to secure a larger sarnple. Thc tedious and detailed work of reviewing and checking for obvious crrors, duplication, or missing data and rating' thc qucstionnaire data was essentially finishe
Table 1 Distribution of Cose Histories by Reservoir Mechanism and Roc.k Type Rescrvoir Ho,ck Typc l'redominanl Hescrvoir l\fcchanism Water Drivc .............................. . : Solution Gaf-1 Drive (,.,·ithout supplemental (lrivc) ............. . Solution Gas Drive (\vith supplemental
Sand and Sandstone
Limestone, Dolomite, Othcr
Total
72
~rn
111
77 60
21
B8
21
.. 81· ·
11
3
H
2
8
86
312 ·
G
Total. ....................................... · 22G
J. J.
6
ARrs, FOLKERT BRONS, A. F. l'AN En;RDINGEN,
R. w.
BUCH\VALD, ANDA. E. S~!!Tll
B, Range of Basic Parameters The range oí thc various paramctcrs for cach o_f thc prcdominant mechanisms is listed 01l Tables 2 through 6.0n these tables thc_indcpendcnt variables are sub
fluids in the reservoir; ande, the environment under which these fluids are being produced. For each parameter thc mínimum, median, and maximum values are listed; and, when sufficient cases werc available, scparate data are shown for sancls and sandstones and for limestone, dolomite, and other.
as
The variables uSed rccovery parameters in thc corrclation studies are: barrcls per acre foot (BAF) and recovery efficiency (RE) for all cases; residual oil saturation (S(lr) for water drive rcscrvoirs; residual gas saturation (S0 .) for solution gas drive rcservoirs. The minirnum, median, and maximum valucs are shown for cach scparalc parameter; thc values shown do not neccssarily belong to the same case history.
Many of the contributors requested that all data submitted he treated as confidential. For that .reaso_n identification of field and rcservoir names and their gcographic locations werc deletc
Table 2
Range of Parameters of Reservoirs with Water Orive as Predominan! Orive Mechanism Sand and Sandstone
Limestone, Dolomite and Other
------'-
Minimum
Median
darcys fraction fraction
0.011 0.111 0.052
0.568 0.256 0.250
dég API cp cp ratio ratio ratio ,g/cc
15.G 0.2 0.24 0.997 1.008 1.004
Parameters
------'-
Maximum
Minimum
Median
4.000 0.350
0.010 0.022
0.470
0.033
0.127 0.154 0.180
Maximum
Rock k ~
Sw
1.600 0.3000.500
Fluids
g, µr,i
µw Boif
Bobf Boof
pw-po
0.054
35.3 1.0 0.46 1.238 1.259 1.223 0.241
50 500 0.95 2.950 2.950 1.970 0.490
1G 0.2
:J7 0.7
G4 142
l.110
1.321
1.933
fJnvironmcnt
«
ft dcg
Db,
ft
T
deg F psig psig psig
hn
J)i
ps
p,
G.G 0-5 1,400 84 . 450 52 100
17.G
0-ií G,260
163 2,77~ 1,81G 1,!)70
160 15-45 12,400 270 6,788 G,'100 G,010
!l
2¡210 90 700 :10
G0.2
· 6,790 182 3,200 1,80G
185 J :J,100 226 5,668 3,821
Ultimatc Recovcry
BAF
; bbl/AF
RE
¡ percent
s,,
fraction
155 27.8 .0.114
571 51.1 0.327
1,641 86.7 0.635
6 6.3 0.247
172 43.6 0.421
1,422 80.5. 0.908
BULU~TIN D14: A STATISTICAL STUDY
OF RF;COVERY
7
EFFICIENCY
Table 3 Range of Parameters of Reservoirs with Solution Gas Orive as Predominan! Orive Mechanism without Supplemental Orives Sand and Sandstone Limestone, Dolomite and Other ~
-----'-
Rock k ~
Sw Fluids g, µ,b
R,, Bobd
Bobt BomJ. 1
Parameters
Minimum
Median
Maximum
Minimum
Median
Maximum
; darcys
0.006 0.115
0.051 0.188
0.940 0.299
0.001 0.042
0.016 0.135
0.252 0.200
0.150
0.300
0.500
0.163
0.250
0.350
49 6 1,680 1.900 1.740 1.400
32 0.2 302 1.200 1.200 1.060
40 0.4 640 1.346 1.402 1.120
f>0.2 1.5 1,867 2.067 2.350 1.420
¡ fraction fraction deg APl ; cp cf/bbl ; ratio ; ratio ; ratio
20
:t5 0.8
0.3 60 1.050 1.050 1.000
5G5 1.310 1.297 1.090
Em,ironment hn
ft
a
deg-
I.h~ T
deg F
1}1, ]Ju
ft ; psig· ; psig
Ultinwle Reco-very BAF bbl/AF percent RE ,S'g, fraction
32.2 5-15
3.4 0-G 1,GOO 79 6:39 10
fi,380
772 >45 11,!:iOO
150 1,750 150
4,403 1,000
2GO
fi34 46.0 0.382
154 21.3 0.229
47 9.5 0.130
3.9 0-5 3,100 107 1,280 50
27 0-5 6,:100 174 2,383 200
20 15.5 0.169
88 17.6 0.267
425 5-1.fí 10,fiOO 209 3,G78 1,800 187 20.7 0.447
Table 4 Range of Parameters of Res ervoirs with Solution Gas Orive as Predominant Orive Mechanism and with Supplemental Orives 0
Sand and Sandstone
Limestone, Dolomite and Other
~
~
Mínimum
Median
Maximum
Minimum
Median
Maximum
darcys fractioh fl"actioh
0.010 0.120 0.100
0.216 0.210 0.310
2.500 0.359 0.579
0.002 0.033 0.0:35
0.019 0.133 0.250
0.867 0.248 0.600
deg- A PI cp
15.5 0.4 10 1.010 1.015 1.000
36 0.9 390 1.230 1.230 l.050
46 20 1,010 1.580 1.845 1.220
22 0.3 60 1.050 1.050 1.020
38 0.8 615 1.328 1.310 1.110
46 2 1,325 1.682 1.680 1.500
Parameters Rock
k ~
Sw Fluids g, µ,b
R,, BoM
Bvbt Buo,I
cf/bbl ratio ratio ratio
]<,,'n1Jironment h,, ft
•D,, T
p, p,
; deg ; ft
; degF ; psig ; psig
Ultimate Recovery bbl/AF BAF percent RE
s,,
fraction
4
0-5 300 77 5 5
109 13.1 0.077
:30 0-5 4,237 146 1,360 100 227 28.4 0.255
714 >45 10,280 260 4,275 800
820 57.9 0.435
8 0-5 2,800 88 530 40 32 9.0 0.112
31 0-5 (i,000 128 1,830 200 120 21.8 0.260
154 5-15 10,530 225 2,935 1,550 464 48.1 0.426
s
J. J. ARrs, FOLKl,RT BRONS, A. F. YAN
En:RDINGf:N,
R. \V. RUCHIVALD, ANDA. E. SM!T!I
Table 5 · Rang·e·
of
Parameters _of ReservofrS With GaS Cap
f'arameters Rock k ,¡,. ". Sm ·¡:
fraction
0.047 0.086
fraction
O.l 50
darcys
.
Minimum
DriV'e
as· PredO~ir\anf :[)íiVe A 11 Rock Types Combined
M'eCha·nis"m ·:
Median·
l\faximum
0.600 .
1.966 0.358 0.4:lO
0.225 0.2112
Pllúd.i:: llo /folt
Na,, /J.,1,.l B,.1,1
; deg A Jll ; cp cí/hhl
ratio ratio ratio
Boml
:M
4:J
40 0.fi 70:¡ l.:J74·. 1.:)50 l.1fí9• ·
0.3
~:w l.lHi
l.l ](i 1.040
...... 9 •)
t",:1:1lí 1.fi75
l.63.t_' 1.490::
Environment ft deg ft
hn a
/)¡,,
dcg F psig
1' p¡, p,
psig
3G
15
7
lf,-,tlí
0-G !J,500 17G
0-fi :i,:JOO
7,ti7G
854 o
2,213
200 !1,S8a
fiOO
2,900
289
8G4
108
Ulti111atc Recovery
BAF
bbl/AF
(i8
RE
percent
Sur
fraction
15.8 0.223
m.o
:12.G
0.Gl?,
0.271
Table 6 Rari9é of PCu~meters of Reservoirs With Groviiy _Dr~inaQ·e: as 'pí~do,,,"ir'\cir'lf Ori~e· Mechan is~ Sa1ld ·and Sandstone Only ---'-'-----"--
Parameters
Minimnm
Maximum
Medi.~m.
Rock
" ,¡,
Sw
; darcys
fraction ; fraction
0.305 0.194 0.0:10
2.000 · 0.350 0.400
1.285 0.329 0.29:l
FI11jd~
deg A l'J cp cf/bhl
g,
µ,,
Ri/¡¡I
/1,,1,,¡ Ro1,1
H,,,,,i
ratio ; ratio ; ratio
15 0.7 96 1.070 · 1.070 1.0:m
38.fí 8
22.G 4 ioo 1.106
n5 u8:J . l.:l4Q 1.1 ooc:
l.10fi 1.0·10
/<.,'1111·iron 111ent
"·D,, "
; ft ; deg·
T p, p,
; degF
; ft ; psig
; psig
40 0-fi '1,170 100 497 o
71 5-Hi '.:2,400 100 1,044 20
200 15-4!i -~ G,!iOO 132 2,670 50
U ltfrna te Recoveri.J
RAF
hhl/A l'
RE
percent fraction
Sur
250 . 16 0.151
096 57.2 0.377
. 1,124 '· 63.8 ·,. 0.fi!í4'
.. , ....
mm
BuLLETIN D14:
A SrATISTICAL
Sruoy OF RECOVERY EFFICIENCY
9
C. Rating and Weighting of Case H.istories The Subcommittee members reviewed and rated all questionnaires with respect to reliability of the volumetric data and degree of performance control on which estimates of recovery were based. From these two ratings the reviewers furnished a combined rating of each questionnaire as either "good," "adequate," or "poor." On this basis 47 percent of the questionnaires were rated as good, 34 percent as·adequate and the remaining 19 percent as poor. In la ter regression analysis these ratings were "weighted" as follows:
Combined Rating of Volmnetric Data and Performance Control
Weiaht
Good Adequat.1!
Pool' In the final regression equations the weighted t.lata conslstently showed better correlations than the unweig·hted. Consequently, the correlations presented in this report are based on dat.a weighted as shown. D. Predominant Orive Mechanisms In 81 case histories (25.9 percent) solution gas driYe was listed as the pre
It is realized that classification into wáter drive or solution gas drive used is still open to criticism. This classification has been adopted even though it is nearly impossible to conceive of any solution gas ·drive reservoir in ,vhich gravity would not be operative once free gas exists, nor of a water driye reservoir where free gas coming out of solution does not help recovery. Sorting the questionnaires by predominant recovery .mechanisms revealed sorne interesting relationships. For example, the two curves on the log probability charts of Fig. 3 and 4 may indicate that the probability of natural water influx into a sandstone or limestone reservoir increases as the permeability increases. The top curve oll each of these charts relates the frcquency distribution of the nnmber of cases with water drive as a predominant mechanism to the logarithm of their average permeability, while the bottom curve shows the same distribution for those cases where no water influx is reported either as a predominant or supplemental mechanism. The distribution of this last curve also includes those cases which show gas cap drive or gravity as the predominant· mechanisrn, without supplemental w'ater influx. It may be noted from t~e curves on Fig. 3 and 4 that the permeabilities for the sandstone reservoirs in both cases exceed severalfold those for the limestone reservoirs. Combination of the ·data from these two curves may be useful, when information other than permeability is lacking, in estimating the probability of a natural water drive occurring. This, in turn, may influence the choice of recovery factor for a given case between the extremes of water
10
,J. ,J. ÁRPS,
FOLKERT BRONS, A. F. VAN E\'ERDINGEN, R. \V. BUCIIWALD, ANDA. E. S~IITH
-
PERGENTAGE OF GASES LARGER THAN 20
10
5
. 2 10.000
30
40
T
' 5.000
.,
\
"
1
'
" Q
1
z
~
¿¡.100
1
t: _,
1
1
1
' .
[\
"' \
(/)
w
o
1
a: o
"
'
1\~ 1
'' '\
T
'
NON-WATER ORIVE WlTH k >.001 O I 103 SANDSTONE RESERVOIRS Í
..> "
~. ó,
100
' ' 1 1 1
'
1 .050
500
1 1
\
' '
>-
·ººº
1
\
~
ü
n:
l
WATER ORIVE WITH k> .010 D 72 SANDSTONE RESERVOIRS
./
1 1 \..
o..
5,000
1
' \
---1
10.000
.
1
w
w
98
'1
~
(/)
a:
95
90
1
:500
":,e
80
.
-
'
w
70
.
\
¡¡;
60
SANDSTONE PERMEABILITIES
1
1.000
50
~
\
_, .
'1'\
1
'"'-
¡¡; 050
a:
1
1
.010
w o..
\
\
\.
1
j
:,e
'"'\
'
f.:,,:.~
\
.
'\
010 \
"
.
005
. 005 1
1
'
+o-__!_J_
L-o-
'
' 1
11
.1
z
~
ow
,. 1
.001 2
5
10
1
' 20
30
40
50
60
70
80
001 90
95
98
-+ PERCENTAGE OF CASES LARGER THAN Fig .. 3 - Cu_mulative Frequency Distribution of Permeabilities for. Water Orive
and Solution Gas Orive Cases - Sandstone Reservoirs
BULLETIN
D1 ,¡: A
11
STATISTICAL STUDY OF RECOVERY EFFICIENCY
-
PERCENTAGE OF CASES LARGER THAN 2
10
5
20
10.000
30
40
50
l
'
60
70
80
90
95
98
1 1
LI MESTONE PERMEABILITIES
5,000
10.000
5,000
'1 '
---
!.000
1
1 ' 1
1
' 1
1
1
.ooo
1
'1 \
.500
500
\
1
\
\
.
\
/'
l
\_
.Jj.100
WATER ORIVE WITH k > ,010 D 23 LIMESTONE RESERVOlRS
1/
1
\
' '
-'
f-
-'
050
'\ \._
1. rNON-WATER
\. .
ORIVE WITH k>.001 D LIMES TONE RESERVOIRS
w n.
~
1[>
\
1
r\
' '
1
' '
"
010
1
1
.005
.005
,:
1
1
.
+CT_!__J
~-(T
l
'
' z
1
~
a
w
".
1
.001 2
5
10
20
30
40
50
1
60
70
60
"
w
:, o:
1
\
1
.010
>-
\
1
33
'
\.
1
""' !ti
100
\
1
.050
t !_ ~
1
'
' ,-
f-
t-
'
--- i\
1
>~ w
\"-,__
\
1\ 90
95
98
ºº'
PERCENTAGE OF CASES LARGER THAN Fig. 4 - Cumulative frequency Distribution of Permeabilities for Water Orive and Solutión Gas Orive Cases - Limestone Reservoirs
12
J. J. ARPS,
FOLKERT BRONS, A. F. VAN EVF.RD1NGEN,
R
W. BUCH\VALD, ANDA. E. SMITH
Table 7 Median Volues of Recovery Efficiency by Recovery Mechonisms Predominant Recovery Mechanism Water Drive ............................. Water Drive ............................. Gas Cap Orive ..................... , ..... Solution Gas Drive .... ·................... Solution Gas Drive ....................... Solution Gas Drive .......................
Whether
Median Value of Recovery Efficiency (RE), Percent 51.1 43.6 32.5 28.4 21.8 21.3 17.6
Róck Type*
Supplemented
SS LS
. . . . . .
Solution Gas Drive ...................... , .
Yes Yes Yes No
SS & LS SS LS
No
LS
SS
*SS= Sand and Sandstone. LS = Limestone, Dolomite and other. Table 7 shows: a, that dissolved gas asan expulsive force is the least efficient.rccovery mechanism; arnl b, that it works somewhat better in reservoir rock with intcrgranular porosity, sueh as sa1Íd and sandstone, than in reservoirs with the intermediate typc of porosity ( carbonate rocks)., The table further underscores the much higher efficiency of water drive. This mechanism also seems to work better in sands and sa11dstones than in carbonate rocks. The number of gravity cases (8) was too small to permit a well-founded comparison. However, under ideal con
2
5
10
1
1
20
30
40
50
'
1
1
1
1
60
70
80
90
95
'
1
1
1
1
RECOVERY EFFICIENCIES
l
100
"RE
1 1
72 WATER ORIVE
-
I
1/
', 1
50
w
-" w
-
_a. >- z z
ºw -' S! o ¡e o ~~
'' I'
---..
1
t
:---,._
RE
1
Q.
,;
1 ·
ºº
1 1 1
1/
-
-- 4--
_,:_
---- '-1-,' ------ [:7--..-r---------
',,
"1
1/ 1RE 80 SOLUTION GAS ORIVE WITHOUT SUPPL. ORIVES
"
1
1
1
5
'
1 1
13 GAS CAP ORIVE ._
1
10
o oc o lú
lli
='--...._-
SOLUTION GAS ORIVE" ti" WITH SUPPL. ORIVES
>- ,_
ww
,_ __ ,_ ·- -,
1RE-75
oc z
>O
1
.......
98
5
'
1
1
h-----CT 1
+CT~. z
~~
1
o
w
~
1
1 2
5
'º
-
1
1
20
30
40
50
60
70
80
1 90
95
98
PERCENTAGE OF CASES LARGER THAN
fig. 5 - Cumu1otive .Distribution of Recovery Efficiencies for four Orive Mechanisms
lllUI
BULLETIN
Dl4: A STATISTICAI, STUDY OJ<' RECOVERY E1''FICIENCY
13
The distribution of recovery efficiency values for the four main drive mechanisms is illustratc
lt may be noted on Fig. 5 that: a. Within the range of one standard deviation on either side of the median value, the curves for thc four :·: ·mechani.Sms·follów' the typicál-:Straightlinc i.'clatiónships of a lognormal distribution. :r·
:12 Th~
~ario&s trends a~e · és~entially · parállel except far divergence in the trend for supplemented solution gaS dl'iVC cáseS: · · · ·
c. The hig·h~st ,;~coven; éiÍiciency 'unde~: water ~lrive c~nditions reaches 84 percent of the oil in place, or aboul 3 times the lowcst rccovcry shown for this mechanism. Two-thirds of the water drive cases fall in thc 35 to 71 percent range with 51.1 percent as the median value . . d. Iri:'gen'eral,
,~atC~ ·driv~
sc~Íns to· be· aroun~l 2.6 times more efficient than unsuppÍémented ·s·o~ution gas drivc .
.e" ·-Th~· 11i;l1.~st r_eco~ery ~IÍ~ic~~Y- under ~n unsupplemented solution gas drive m~ci;;nism reaches 3·7· ;Jercent, or ábout 3 times.,t_h~)owest rccover.Y shown.for this mechanism. Two~thirds of the cascsfall i,n tJ10 1G ~o 27 pcrccnt ·:range, ,Vitlí á medi"an vallle of 19.G percent. ' · ·
-.·!' '; '.. .:i, -; . . . ::·.... f. Thc distribution curves for the supplemented and unsupplemcnted solution gas tlrive mechanisms appear to mergc at the low end of the rangc and show a diverging trcnd toward the high end of the range. This clearly indicates that the effect of supplcmentation is much more pronounced when the rcservoir characteristics themselves become more favorable. At the median value the improvement due to such supplementation by partial water drive, gas cap drive, or gravity is about one-third.
g. For those cases where a gas cap ·tlriv/ Ís 'reported as the · predominant mechanism the average recovcry efficiency improve:3._by ab9ut two•thirds ov~r th~t oJ soJution gas drive alone.
F. Reservoir Rock For complete statistical analysis it would ha,y~ beén dÜ-¡:;iráhle to separ~te the case histories according to the typc of drivc mcchanism, with furthcr suhdivisions according to thc type of rcservoir rock. However, becausc thc numhcr uf case histories was limited, scparate correlations could not he made for the different rock types. Of the 80 cases ~ith' 3.~CcúlU_l)Ic accuracy in th~ solution gas group, which wcre used by the b~J)as task force in their final corre_lations, G7 wcre obtairi.ed fro~ san_d anwd -~andstonc reserVoirs and 13 from carbonatq rock reservoirs.
Most of the .case histories .qf. water drh:es in lim~st;~e ;reservoirs -carne from· the Middle East where limited depletion made it diffcult-to·confirm recovery estimates-by·pcrformance. The rcmaining water drive case histories covering carboriate rock l'eservÓirs ·Ú1cke
1
G. Well Spacin~ ;. The inclusion ·of ,vell spacing as an .independent variable affecting recovery received considerable attention in thc study of the_;{Íata. It is believ~d that the choice of well spacing normally is strongly affected by :·eci:momic factors, such as ultima te iecovery .and well cost. This "·as confirmed by an analysis of the data obtained from the available : questionnaires On- the domestic fields, which showcd a consistent incrcase of. both thc ~verage-U1t;mate rccovcry ( per wcll and the average wcll spacing with increasing well depth.
1
1
•
; In vicw of this observation .it may be ,expectcd that in those cases which, for reasons other than well spacing, · exhibit a recovery factor apprCClibIY higlici- or lowcr than average, a bias toward closer or wider well spacing may also be present. BeCause Of· thiS-:- the Subc01~m.lltee :-f~it t}:i_at,.the inclusion of wcll spacing as an independent '. variable in the analysis,o:f recoy~ry)!~·dencY/:oúiftl!ad-_t'?.~~ro~Cou_s concl~_sions, and it w~s, therefore, omitted from . .•.\ ·. the regression analysis·.
1V -
cu111mLAT!ON METHODS
In an analytical COrrelalio1i··stirdy variables ,are _separat9d into dependent variables which are known and · independent variables whÍch are thouiht to a'trect them. Thc connection bctween them is called a regression equation. 'l'his Rection will discuss in succcssio11 n, dependent .variables;, l,, independent variables; and e, the various types of rcgression equa~i_o!_l~_.usr~_:,.
14
J. J. ARI'S, FOLK~~RT BRONS, A. F.
VAN
En:1mnrnEN, R.
w.
RUCHWALD, AND "A.
E. SMITH
A. Recovery l'arameters or Dependen( Variables· ScYeral rccovery paramelers wcrc considcrcd as dcpcndcnt Yariablcs.
a. /úJcuuay Factor (RAF) The commonly uscd Recovcry Factor in barreis pcr acre foot of nct pay becamc the first choice. In thc study of solution gas drive reservoirs thc observcd Recovcry Fadors werc correctcd by the. Rartlesville-Ponca City task force for expansion abovc thc bubble point. The distribution of Recovcry Factor valucs for various drive mechanísms shows essentially straight-linc relationships on thc lag probability chart of Fig. 6, paralleling cach othcr. b. Rccovcry Bjficicncy (RE) Rccovery Efliciency in perccnt of sioc~ tank oil initially in place was thc second choice. Rccausc of th_e. logarithmic format selecled for the regression equation, Rccovery Efliciency is relatcd to Recovcry Factor so tha~ esscntially the same equation expresses the correlation for both. Distribution of the Reéovery Efficicncy values for different drivc mcchanisms also shows a parallel straight-line relationship on the log ¡)robability chart of Fig. 5. It may be notcd that thc respective c1:1rvcs show a ~nuch lcsscr slopc, and t.hercfore smaller variance, than the Uccovery Factor curves of Fig. (l.
PERGENTAGE OF CASES LARGER THAN
-+ 2
10
5
20
30
40
'
'
'
'
50
60
70
80
90
95
'
'
'
'
JRECOVERY FACTORS ( BAF)
98
1
?,
?,
n..
...w
n. 2,000
2,000
z
"'-- I'---
~
o
...
o o
1,000
1
' '1
;;: w
.. a:
"' ,¡
a: a: o w ... n. o .J tt 6
,..a: "'z
·'
'
500
w ..
~J-
º"' w o a: o
.
...
-1-- '·
~
o
.J
."'
' '
"':
"-
'
'-
.;:,-...
y
1-..... ..__,__ .....
' ,J '---.._ "'-- .......
"-"'-
1
¡._:_ _,,.
20 2
5
'º
' '
.
'
'1
,. '
-·
WITHOUT SUPPL. ORIVES
50
"
_
' ¡BAF-80 SOLUTION GAS ORIVE
a:
m
1--' ...::::::
;§5. _
BAF-13 GAS CAP ORIVE 1--
-- -- -- f< -b
1 1
'ºº
'
+a--___.¡
'
z
1
•o
1
;' 30
40
50
"'.J
w
.
a: a: 50
"'\'
m "':
\
w
20
8
;;: w
1 1
1\
IBAF-75 SOLUTION GAS ORIVE WITH SUPPL. ORIVES
"'
"'w
-
r---
'~
1
l,000
~
·-- '
' -...
...o
1
~
t
~
1
IBAF-72 WATER ORIVE ,1
~
u,
z
,
1
. 1--
1
60
70
80
20 90
95
98
-+ PERCENTAGE OF CASES LARGER THAN
Fig. 6 - Cumulative Distribution of Recovery Factors for Four Orive Mechanisms
: BULLETIN Dl1: A STATÍSTIC,11, STUDY o;,. fü:cOVERY EF1"IC1"NCY
15
c. Residual Saturalions (Sor u.nd Si;r) 'fhe actual physical
the 1945 Crazc-Buckley study of thc ·effcct of wcll· spacing on rccovcry.1 'fhc Guthrie-Grcenberger study of 1955' used Recovery Efficiency in rcgression analysis Of data on !he water drive rcservoirs reported in the Craze-Buckley study. ~esidual Oil Saturation in water drive reservoirs at abandonment time is expressed as a fraction of the total
pore space. Distribution of Residual Oil Saturation valucs computcd for the water drive cases.in this study is shown
as a dashed curve on the !og probability chart of Fig. 7. Residual Gas Saturation of solution gas clrive reservoirs at abandonmcnt time is also expressed as a íraction of total pore space. Thc distribution of Residual Gas Saturation values for solution gas cases is shown as a solicl curve on thc log probability chart of Fig. 7. Both Residual Oil Saturation and.Rcsidual Gas Saturatjon are computcd indircctly from Rccovcry Fa~tor and the parameters of rock and fluids. Many· attempts were madc to. find dircct correlations between Residual OH Saturation and Residual Gas Saturatfon and the various indcpendent variables. However, the results were generally unsatisfactory since the coefficients of multiplc corrclation wcre consistently lowcr t~an thosc for t.he rec<,)Very factors.
-
PERCENTAGE OF CASES LARGER THAN 2
5
10
'
'
20
1
30
40
'
'
50
60
70
80
90
95
'
'
'
'
'
'
RESIDUAL SATURATIONS
98
1
¿ 1.00
'
-
,50
s,,
/
'-
1
>-
72 WATER ORIVE RESERVOIRS
- --...... - ""-
...... ¡
-
-
"
::,
1
1
-
o iñ w ·"' o
/
1
-1'-
-
10
Sz ~Q z >-
5
ii ~ f' "
'
1
1
1
1
'
'
r-i-----o-
_,
2
5
'º
ow• ~ 20
30
40
50
U>
6
_,
"::,o
¡;; w
z
1
.1
"
<(
" ' <(
+o--+j
1
ºº >-
w
;/. >o >-
z~ i ;:_o
~
1
~¡;;
~ g; _,
r--.1
"
_,
"'
"o ::, a.
1
6
"w U> o w
.10
'.5
t--- .50
1
~
1S9r - 80 SOLUTION GAS ORIYE .RESERVOIRS 1
_,
z 2
>-
1 1 60
70
80
90
95
1
98
-'+ PERCENTAGE OF CASES LARGER THAN
Fig. 7 - Cumulative Frequency Distribution of Resi_dual Oil .Saturation for_ Water Orive and
Residual Gas Saturation for Solution Gas Orive Reservoirs
J. J.
16
ARPS, FOLKERT BRONS,
A.
F •. \'AN En;RDINGt:x,-R.
,v. BUCHWALD, ANO A. E. Sl\lITH
B. lleserroir Parameters and Independent Variables In thc prcliniinary correlations, practically all of the availllblc rescrvoir ami fluid characteristies (pnramüters) wére·,used as independcnt variables. From the. rcsults it. was poSsiblc .to pinpoint those parameters -which sccmed· consistcntly· to affect· the ultimatc TCcovcry. The Suhcominittce was· fortunatc. in having out.standing rcservoir-eng'ineering talent ari1óng its memhcr~hip, -which -provcd· most hclpful in sdecting aq0 -grouping- thcsc param~te;r_s.,: Thc in
n. For tite ffn.la Driue Stmly 'P (l-Sw)
, oil in place _(initially).
.
.
Boi
2. ~µw!_, rnobilit.y ratio (under initial resel'\'(;~~ c~·ndit.i~ns). µoi_
Permca!Jility k was.tcstcd ii1 thrce différcnt ways agáiilst thc Rccov'ery Factor "(IJAF): as arithi11élic average
· permcability; as median pcrnicahilit\', and as log meal1 of pcnncabllit)'. It ·was found that thc strongcsVcorrcJa'.· tion was ohtaincd with thc arithmetic average ·pcrmciahilily so this paramctcr was u sed .throughout. .•.'
:J. S10:, fnter~liÜal water Saturat.ion. ,t.
_PJ_, prcssure ratio.
P• !).
f)w -
6.
~
po, density difference hctwccn rcservoir \\'a ter and oil.
;
, "hydra11lic radins" function.1
In correlation work with thc logarithmic model the regression process indcpendcntly determines thc appropriatc exponen\ for each parameter arnl thc /;/
1.
rf,
(1-Sw), oil in place (at huhhle ,puint.). Ilo1,
2. _!!:_, mohilily ralio (al Uuhblc 1wint).
µ,,
Actual
,')'m,
interstitial ,valer saturation.
,L R~, dissolved· gaswoil ratio.
5.
_P.!!_ ,pressurc ratio.
P• G.
{_!':._, "hydraulic radius'' function: 1 . . . .
~ f
In lhc correlat.ion wor~ the squarc .l'Oot. sig-n \vits
cll"OJ)pcd.
Othcr indcpcndcnt pa,:amctcrs may 11rovc to havc a measurcahlc cíTcct ·un t.hc ltccovcry Efficicncy. In particular those rclated to rclati,·e pcrmeability of thc rock, ,·ariancc of thc pcrméahility distributi~n, a1HI intcrfadal tcn,::ion would be of intcrest for íurthcr study. Such a study must, howcvcr, await t_hc collcclion of more dclailed data. IH!
C. Hegrcssion Equations Severa! typcs of regrcssion cquations werc considcrcd in t.hc carly
currclation' work
with the following- three rnain
types actually bcing used. o. Luyarithmic Modcl
h, y ~ · rt 11
+ l,
= (c)i'
ln
Xi!
+ e ln
XJ
+ ..
. (:rc!)b • (r.1Y .. ..
The nolne1\datui:e of these symbols is Sf~t out scparately in Sec. VII.
( G)
( 6)
BULLETIN D14: A STATISTICAL STUDY OF RECO\'ERY El'FICIENCY
17
This logarithmic model was selected as most applicable for the final correlations because:
l. Both the dependen\ variables and most oí the indei1endent variables indicate a. good lognorrnal sampling distribution (see Fig. 3, 4, 5, 6, and 7). 2. The indicated Recovery_ Factor (BAF) showed a strong and alrnost linear córrclation with the· oil-in-place term so that after dividing by this factor ami changing thc constan! (a) essentially the same correlation applies to Recovery Factor an
b. Algebraic Model (7) i
Thc algcbraic model was used largely in sorne of the early correlation work, but was dropped in thc final correlation work in favor of the more suitable logarithmic model. e,. Logis'tic M odel
C,
Y= C, + c-Ca+o
,,+clnx, + ..
.i
(8)
'J'his model ,.,;m; used eX:tem,ively during the eárly correlation efforts bccause it has the rathcr unusual advantagc of allowing preset limits in the correlation rclationship. For example, the range of the depcndcnt variable would . be betwE:en a mínimum value of zcro, whcn thc tc!m (a+ b ln x 2 + e ln XJ + .. .) reached minus infinity anda maximum of ~: when this term reachcd plus infmity. In this manner tl~e correlation could be prevented.from assum.in~ physically impossible values. However, the log-istic model consistently yielded lo,ver correlation coefficients than those ob~ained With the other models, in addition to being somewhat umvieldy. It was, t_h_erefore, dropped in favor oí tl"!e logarithm~C model. V-EARLY CORHELATION EFFOHTS Com.plet~ sets of elata printouts on all 312 case histories wer~ prepared by Mr. R. K. Guthrie and distributed .fo merilbcrs of the Subcommittec ata meeting in Dallas.in August 1962. At that meeting, prcferred rccovcrY. yardstic1cs and thc" bcst mathematical corrclation models. wcrc also agrecd upon. Corrclation ,vork was thcn assignc,l to each .uf thc Subcommittcc's ti.ve task forces. Rcsults of thcsc early cfforts are summarizcd· herc. A. Pri.mary RccO\·cry from Water Drivc Rescnoirs l. In August 1962, Mr. J. C. Burkc, British-Amcrican Oíl l'roducing Company, ¡,rescnted a correlation of unweighted dá.ta for 52 sandstone reservoirs with water dri,·e as the predominant drive mechanism. Using the "logistic model", l\fr. Burke developcd ti.ve rcgression equations with recovery efficiency as the dependent variable. His work demonstratecl that the coefficient of multiple correlation improved steadily as additional independent variables were included in the regrcs·sion equation. His best cquation for recovery cfficiency included permeability, viscosity ratio, and interstitial water saturation as independent variables ancl yielded a multiple correlation coefficient of 0.691. 2. In February 1964, l\fr. R. E. Carlile, rcprescnting Prof. H. T. Kennedy of Texas A&M University, presented correlations used for his graduate Ph.D. thesis, "Recovery, Rccovery Fraction, and Residual Oil Saturation Correlations for Sandstone ,vater Drive Reservoirs." Mr. Carlile, with data for 57 sandstone water drive reservoirs, developed 29 equations with recÓvcry factor, rec_overy efficiency, and residual oil saturation as dependent variables. Indepen.dent variables werc investigated as follows: IloCk paramcters of pcrmeability, porosity, and water saturation_; fluid parameters of oil-water viscosity ratio, solution gas-oil ratio, formation volumc factor, and API gravity of. the oil; and environmcntal paramctcrs of net pay thickness, reservoir temp.eraturc, depth, and bubble point pressure. Mr. Carlile's best equation was obtained by relating residual oil saturation to viscosity ratio~ porosity, bubble point pressure, rescrvoir tcmpcrature, and solution gas-oil ratio. 3. · In Februá.ry 1966, Mr. G. L. ·Hancock, .Jr., Gulf Oil_ Corporation; presented correlations developed by 1\.'lessrs. \V.· R Cóok and D. G... Stl'ittm"ater of Gulf Research and Development Company, using 71 sandstone reservoirs with water· dri\'e·as· the prcdominánt drive mechanism. The Gulf group dcveloped two sets of 14 regression equations;. onc with unwcighted data and a second one with weig·hted data. Algebraic, logistic, aml, for the first time, logarithmic models werc used. Dcpendent variables conclated were: recovcry factor, recovery efficiency, and · residual oil satq.ration. Indepcndent variables which provcd to be significant were: in the rock parameter catcgory, perrneability, pÓrosif:y, ancl interstitial water saturation; in thc fluid paramcter grou_p, oil-water viscosity ratio, solution gas-oil ratio, and initial formation volumc factor; and in the enviromental category, initial pressul'e, a;nd bubble point pressure. The highest coefficients of multiple correlation obtained were: for recovery factor, 0.886; for recovery efficiency, 0.708; and for residual oil saturation, 0.715. Gulf's correlation study showed thc superiority of the logarithmic model and focussed attention-.upon the comhinatíon of parameters into gToupings ::;ig·niticant from a reser\·oii· enginee·ring standpoint. The most important groupings wcre the "oil-in-place tcrm", ~nd the "mobility ratio".
18
J. J. ÁRPS,
FüLKI-mT IlRONS, A. F. VAN Ev,moINm:N, R. W. IlUCIIWALD, ANDA.
E. SMITH
B. Primary Rcco,·ery from Solution Gas Dri,·e Reservoirs l. In May 1963, l\.fr. W. Sturm, undcr thc able dircction of 1fr. KG. 'l'rostel, DeGolycr ancl MacNaughlon, rcportcd on his corrclations for data from 95 sandstone ami limcstonc rcscrvoirs with solution gas drive as thc predominant drivc mechanism. Using a logistic~type modcl with 18 inclepcndcnt variables, his best correlation coefficicnt íor recovery efüciency below thc liubblc point rcached 0.706, ,-..·hile-- his best correlation for residual gas saturation showed a maximum of 0.802. · 2. In Fcbruary 1964, Messrs. J. K. Latimer, ,Jr. aml V,,l. C: -~turdivant, Jr. with Sun Oil Company in Dallas, presented their best corrclations on 183 solution gas drive reservoirs, including those with ami without other supplcmental mechanisms. Thc algebraic mo
meters, net pay thickness, bubble point pressurc, reservoir tempcrature, productive area, and depth. subsea .. Thc best coefficients of multiple correlation obtained wcrc: for rccovery factor 0.918, for recovcry efficiency 0.517, ami for residual oil saturation 0.973. 3. Also in February l!J64, Mr. F. H. Hunter, with Jersey Production Rcsearch, presented an equation for recovery effi.ciency below the bubhle point for
G8
sandstone solution gas drive reservoirs without
supplemontal m(lc_hanisms.
Indepcndent variables were mobility ratio, porosity, interstilial water saturation, depth, ratio of wcll drainage ,radius squared to oil pay thickness, and ratio of solution gas-oil rntio to formation volume íaclor at the bubble point. Mr. Hunter reported that his cquation providccl a calculated recovery cíliciency of within 25 percent for ovcr twó-thirds oí the reservoirs and within 10 perccnt for one-thircl of the reservoirs. 4. During 1964, Mr. R C. LeRoy, with The Atlantic Refining Company, prepared· ancl presented to the Dallas Task Force a large numher of corrclaÜons on solution gas drive reservoirs. Dependent variables Correlated were:
recovery factor, recovery efficicncy, and residual gas saturation. As independent variables he used rock parametérs of permeability, porosity water saturation, and critica} gas saturation. For fluid parameters he used solution gas-oil ratio and_API gravity of the oil. As significant parameters to describe environment he used reservoir temp~rature, bubble: point pres~u1·e, a'nd ratio between bubble point anQ. adbandonment pressures. In his aliebraic. e(]Uations, different functions of"thesc individual parameters were used, such as theír linear, square root, squarc, ancl loga~ithmic form. The best coefficients of multiple corrclation obtained in this work for various recovery parameters and for· different groupings of solution gas drive case histories are enumerated helow. Solution Gas Drive Rcservoirs Correlated 70 without supplemental drive ........................... 55 with supplemental drive .............................. 125 ,vith and without supplemental drive ................. 21 with and without supplemental drive .... . 13_ with supplemental drive .............. , ............... 39 with k 9 /ko data,. . , ..... , , , ...................... .
. . . .
Rock
Highest Correlation Coefficient for:
TypC
BAF
RE
s,,,,
SS SS SS LS LS
0.885 0.885 0.892
0.719 0.702 0.709 0.865 0.884 0.894
0.860. 0.762 , 0.794.
0.%8
Both the algebraic and t:µe Iogistic mathe.m3.tical models were used in the foregoing work. In an_ effort to improve correlations.further Severa! additional _runs were máde, using inter-related funciions of the indepen\'.lent param.eters (x) of the type n(xct + C1)B and Mn (ex';_+ C'2); htit the additional ·comPicxity did· ~o~ ~ield any further in~1uovement in thc correlatiofl coefficients. · C: Secondáry Re'éO\·ery by Fluid Jnjection In February 1964, l\fr. F. H. Hunter, with ,Jersey Production Research, presented a correlation for recovery· elficiency of secondary recovery by water injection based on data from 17 sandston'e réservoirs. Mr. Hunter also presented a correlation for recovery efficiency of secondary recovery by gaS .injection based On data from 15 sandstone reservoirs. Because of the limited availability of rclative perrneability data, such data ,Vere not includecl in his correlations. ' Mr. Lincoln Elkins, Sohio Petroleum, Chairman of ihe Oklahoma City - Tulsá Task Force, recommended,'' and the Subcommittee agrecd, that no furthcr cffort should be madc to correlaic fil.lid injection rccoverics against avaiiablc paranieters becausc of thc srnall number of available cases. ·
D.,.l!]ter-Correlation of P.V.T. Data .Under the chairman:ship of .21,fr .. V .. T. l\IcGhee, ,vith Phillips Petrolcum Company, the Rartlesville_.- Ponca City Task .Force rendcred valuablc seniee to the· othci-, ta:-k force::- in :-olut.ion of prob\{m1s 1·clating to P.V.'T'. data. In 196!5, this group provided the Dallas Task Force witl_~.a ('.Orrela.tion of _oil corn1iressibilit~, and oil formati_on Yolumc
BULJ.t;TJN Dl.4: A STATISTICAL
Snnn: OF Til•'.CÓ\'F.RY
EFFTCIENCY
19
· factors so that correlations for residual gas saturations at abandonincnt and recovery factors below the bubble point for solution gas drive reservoirs could be developed. In 1967 the P.V.T. group provided the Houston· Task Force with reservoir water and oil density values for water drive reservoirs so that the density difference between water and oi) could be tested asan independent variable in the regression equations. E. Inter-Correlation of Rock Parameters In February 1964, Mr. J. M. Mil!er, with Standard Oil Company of California, reported that his group ha
F. o'verall Resulte Early corrclation efforts, as summarized hére, provirovement in the multiple correlatíon coefficient was no longer statistically significant. Whether to accept each improvement as significant was decided by the so-called "F-test" (see A ppendix). It was found in both cases that the probability was Ve:Í-y small that a_d~ition of the Jirst four independent variables was not statistically significant; in no case more than 3.0 percent and in most cases 0.02 percent-or 1ess. The probabilitY that the fifth and sixth variables are not statistically significant increased sharply to around 9 and 50 percent, respectively. At the same time, the addition of these two variables showed only a minor improvement in the standard error of estima te. The Subcommittee clecided, therefore, that an equation comprising on]y the first four variab_les oil in place, mobility ratio, ,vater saturation, and pressure ratio~ be. accepted as statistically significant.
1. Water Dl"ive Ilased on figures shown on Table 8, the fo~lowing logarithmic reg;ression equation íor the Recovery Factor under a water drive mechanism with the four signifieant independent variables is recommended: -t-1.0422
RAF
= (4259)
1 { f(
-)-0.07'70
;;,i~"') } {-":-::'-}
{
-0.1.10,'J
S," }
{
-0.21fi.1
;:
}
· barrcls per acre foot (9)
, where 4259:::: e+s. 3569 The multiple correlation coefficient. for t.his equation is 0.958 and the standard· error of estimate is 17.6 percent. Fig. 1 shows observed Recovery Factors plotted against those computed by means of Equation (9). The two dashéd lines on this chart indica te the range of values between the computed amount plu~ .or minus the standard error of estimate of 17.6 percent. It should be recognized that this standard error of estimate strictly applies only at the mean of the distrihution.
J ..J. ARPS, FOLKF.RT BRONS,
20
A. F. \'A1' El'RRDING!:N; R.
w. BUCHWAI.D, ANDA. E, s,mn
By dividing both sides of Equation (9) by 77.58 times the oil-in-place term, the. following equation giving the . Recovery Efficiency for a water drive reservoir is obtained: IIE
,
= (54,898)
8w
)-0.190.? ( ¡J; )-0.215.9
'
percent (10)
-Po
2. Solution Gas Dri-ve Based on figures shown on Table 9, the following logarithmic regression equation for Recovery Factor under a solution gas drive rnechanism below the._bubble point ,vit11 fo_ur significant indepenrlent vari~bles is recomménded:
, · /,A F
= (!1244)
•
+0 ..1722 ( p,, )+0.1741 · { ,¡,(1-S,,,)) +1.1a11 ( k )+0.0979 ( , · --;-hancl~ pcr ncrc foot ·(11) R,,,, ( · • --. • 8w ) . ¡t,.f¡ p,.
, whérc .'1244 =-e+s.01?-1r.
The eoeflicient of multiple corrclation for lhis cquation is O.D32 and lhe standard error of cstimate is 22.9 p_crcent. Fig. 2 shows observed Recovery Factors plotted against those compuled by means oí Equation (11). The two dashed Jines on this chart indicate the range of values between the computed amount plus or minus the standard error oí estima te of 22.9 percent. lt should be recognized that this standard error ~í estima te strictly applies only at the mean
of the distribution. Ry dividing both sides of Equation (U) by 77.58 time~ the oil-in-place term, the following eqnation giving the t!ecovery Efficiency betow the bubble point in solution gas drive 1·ese1•1•oirs is obtained: . , { ,¡,(1-Sw) }+0.1611 (. k )+0.0.979 ( . )+0.-1722 ( P• )-j-0,1741 RE= (41,815} • • -- . • Sw • . Bob /lob Pn
(12)
per~ent
B. Numerical Examples 'l'ó illustrate the effect of ad
Table 8 Summary of Exponents of Terms in Regression Equations Recovery Factor (BAF) for Water Orive Reservoirs, Sonds,.and Sandstones - 70 Case Histories (1)
(3)
(2)
Mobility Ratio
Conslant Oil in Place ~
(1-S.,,)
---Boi
9.0Gfií 8.íGGf,
+ l.3871 +1.1414 + 0.9897 + 1.0422
8.1585 8.:;569
8.fi079 8.6421
-
-
+ 1.1027 + 1.l!"'l:H
,va ter Saturrition
kµwi
-
Sw
lloi
+ 0.10:!7 + 0.1140 + 0.0770
-0.2280
-
-
+
- 0.0940
+ 0.0701
~0.2040 -0.19G9
Pressure Ratio - P!..... P11
(7)
Hydraulic Ra
Density Differenee
q,
pw-po
/,,
-0.21G9
-0.1903
-
( G)
(5)
( 4)
-
-
-0.201'1 -·0.1969
-
-0.0444 - 0.0243
-0.0901
(8)
(9)
Standard Coefficient gl'l'Ol' OÍ of Multiple Estima te, Correlá tion Percent
r
s,
0.8650 0.9268 0.9409. 0.9575
32.2 2a.2 20.8 17.6
0.9594 0.9597
17.4 17.4
The probability of improvement in the conelation due to the addition of the Jast parameter in the regression equation being insignifü-ant· is less than 0.02 percent when the variables representing oil in place, mobility ratio, water saturation, and pressure ratio are successively inclu
lll!U
. BULLETIN D14: A STATISTICAL STUDY OF RECOVERY EFFICIENCY
21
· Table 9 Summary af Exponents of Terms in Regressions Equotions· Reco.very Factor (BAFJ for Solution Gas Orive Reservoirs, Sands, Sandstones, Carbonates and Other 80 Case Histories ·
(3)
(2)
(1)
Constan\
Oil in Place
(1-S,,.)
Mobility
Préssure
Ratio
Ratio
·:.~;º
µ,,
JJ,i
k
(6)
Solution Gas-Oil Ratio
R,
p,_
(8)
(7)
. .¡. I.:l0·1 !)
+ 1.221G
-1- 1.1 f,Jl -· - · -1- l.OG19
8.fiG97. ··
. + l.OG94
-1- 0.2:199
+ 0.3128 -f- 0.3722
-
-
+ 0.3586 + o:3800
+ 0.0771; -
-1- 0.0979 - -1- 0.1162
-
+ 0.0881
+ 0.17•11 -
Hydraulic
Coefficient Error oí of Multiple Estimate, Conelatiim Pe1·cent
Radius
k
r
s.,
0.8899 0.89G8 0.9090 o.n17
·29.a · 28A 2fi.8 22.9
0.9345 0.9349
22.5 22.5
-
-1- 0.1723 -f- 0.1728
-0.0828 -0.0567
(9)
Standard
+ 1.2()9:1
8.o:wo
8.4007
(5)
Water Saturation
----B~ 8Alfi/¡ 8.G~OO· 8.084G
(4)
-0.0351
The Pi-_obability of the impi:ove_ment in the correlation due to the addition of the last p;rameter · in the regresRion equatidn' being insignificant is n~t more than 3.0 percent when the variables represcnting oil in place, water satu'ratión, mobiÚtf. ratio, and pressuré·'ratio are sucl'essively included. lnclusion of the solution g·as-oil ratio ami hydraulic radius has an 8.G and 52.0 percent chance, respectively, of being insignÚicant. As the standard error of estimate is not appreci_ably improve
S,,,
•
-
p,, p"
)-j-1).J 7.',J
Table 10
Effect of the Addition of Various Terms in the Regression Equations on Computed Recovery Fq:ctors Barrels per Acre Foot Computed
WATER DRJVE A constant ph~s oil-in-place term ... .
A.hove terms plus mobility ratio .. . Ahove tenns plus water saturation .. ;·A ~o~e. terrris pl.us ·prcssure_. 1·atio .... ·........ . ·Abo\·e·. terms.,plus hydraulic radins· .. . Ah°Ove· terms plus density ·
------A,---
Low
Average·
High
301 243
714
655
1,135 1,085
225
581
1,231
244
554
1,326
232 231
575 585
...J,3G2 1,349
245
570
1,250
129
2í7
133 131
279
SOLUTION GAS DRIVE A constant plus oil-in-place terrn ......... . Above terrns plus water saturation /bo\·~· ·t~~·ms' r·,u_s ~ ~o.billty ·r;tio ...... ·.. ·.... . _.AbQyeJerrñ~< plus pressure ratio .. ·........· .. ·Alio\·e.terms·plus gas-oil ratio ....... ·.................................. . Above terms plus hydraulic radius .......... . -'Fhesc·ngtlt"e~ mú"s(be· ~om.pared to ohserved values of: ·.............. ·.... . ,., .
0
\
5G 53 51 48 50 51 55
244
119
2n2~
11 (l
271
115·
270
112
276
· .The .results obtained from the recommen'ded regressioÍl EquationS (9) and (11) are ·underlined in Table 10. ~--. Jt ma), ..he·.Jloted l:.hat; for Ü1e .wafer driv"e ñtechanism, the inclusion .of m_ore terms res~lts in a computed recoYerv nearer to the observ~ value. Adding, term~ in th~ solution gas rlrive m~chanism appears to he less effective. · ·
J..r. ARrs, FoLK>:R'f BRONS, A; F. v.1N EVERDIKr.EN, R. W. BUCHWALD, ANDA. E. S,mn
22
The íollowing examples show, in detail, how the Recovery Factor rnay be cornputed for the forementioned two average cases by means of the prefened regression equations:
Constant ....... : Natural log of constant ...... . Porosity ...
óll-i\lled ¡,oré gpnrQ. Shrinkage factor .
• , , • , , i 11111 • ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' • • • • • • • • • •
....................... ' ......... ·............ ; ..
Oil-in-place facf.or ........................................ , , , , , , , , , , ·
-Water Drive 4,259 8.3569 0.282 (1-0.350)
Solution Gas Drive
3;244 8.0845
0.174 (1-0.340)
1.100
1.402
0.16664 -1.79194 0.249
0.08191 -2.50212
Natural log of oil in place .......................................... , .. Perrneability ....................................................... , Oil viscosity at bubhle point ..................... , , , , , , , , , ... , , . , ·•· .. . Oil viscosity at initial pressure ............................. . Water viscosity at initial pressure .................................... . Mobility frictor ...................... , ...... · .. · .... · .... · .... · .... · · Natural log of mohility factor ........................... : ........... .
0.540 · 0.10264 -2.27652
-3.22389
Water sal111'ation .................... , .............................. , .
0.350
0.340
Natural log of water saturation ....................................... .
-1.04982 1,986
-1.07881
Initital pressure ........................... , ..... : . , . , , , , , , , , , , , , , .. .
Buhhle point pressure . . . . . . . . . . . . . . . . . . . . . . . . . Pressure at abandonment ............ , ....... .
r.310
0.03980
3,660
. ......... ,
Pre,1;sure ratio .......................... . Natural log of pressure ratio .................. ; ........ .
0.0199 0.500
800 2.48250
580
0.90927
1.84219
6.31034
The regression cquation can now he written as follows:
\Valer Drive Case
= =
+
In BAF 8 ..'/5/W 1.0422 (-1.79194) -1- 0.0770 (-2.27U52) fnl]AF=fi ..117.52 BA F 554 compuled vs. 570 ol;scrvcd.
0.190.1 (-1.0J,982) -
0.2159 (0.90927)
Solution Gas D1'i\;e Case· In BAF In BAF BAF
= 8.0845 + 1.1611 (-2.0,5212) + 0.0979 (-3.2288.9) -1- 0.3722 (-1.07881) -t- 0.1741 (1.84219) = 4.7828G = 119·computed vs. 112 obscr11ed.
The reader will recognize the· figures outside the parcnthescs as thc exponenb, appeariúg in Er¡uations (9) and (11).
e: Discussion of Results Equations (9) and (11) ,vill )rield ihe estimated values of the Recovery Factor for water drive and solU.tion gas drive reservoirs, respectively. In the event the bubhle point pressure of a solution gas drive reservoir, 7>1,, is less than its initial pressure, p¡, the amount of oil expected during the time the pressures drop from p¡ to p,, must be added to the values computed from Ec¡uation (11). The variables appearing _in these equations are arranged approximately in order of importance. ··
Oil-in-Place Ter-m In the accepted Equation (9) for water drivc, the amount to be recovered, in barrels pcr acre foot, is equal to a constant, 4529, modified by the influence exerted by the succeeding four independent variables. A recove"ry proportional to the amount of oil in place needs no further explanation. The exponent of the oil-in-place factor, .+1.0422, being slightly higher than unity, indicates that the recovery factor for water drive increases somewhat faster. than the amount of oil in place. In the accepted Equation (11) for solution gas drive reservoirs, the constant term, 3244, is about 25 percent less than the constant appearing in the water drive equation. It would be interesting to explore the reasons íor the decrease; howcve1\ it is felt that an analysis.in depth of both figur~s should be left until data on a greater-number of sample reservoirs are available. The exponent of the oil-in-place variable, +1.1611, for the so.lution gas drive case, iS somewhat larger than the· one fo:t' ·water ·arive,· which indica tes thát· the recovery factor will increáse moré progressively with higher oil-in-pláce contént; '
IIIHf
RULLlmN D14 ,. A STATl8TICAL SnmY or R,;covi;ny 1,n·1C11::
The first two columns of Tables 11 and 12 Jist the oil content of the forrnation and thc uncorrec\ed recovery efficiency fa.ctl>r bas_cd ~hereon._ To show the improvemcn_t in rcc?very cfficiency w_hen the oil content of ·th~. rescrvoir rock increasés,.tlie uncofrected rccoVcry cfficiency, RE\ is giYén in percent of Oil originally in place. This recovery efficiency for water drivc rcscrvoirs improves from 17.3 pcrccnt to 52.2 pcrcent for a 10-fold incrcase in oil·content. It increases from 23,8 percent to 34.4 percent for thc same 10-fold incrcase in oil contcnt of solution .gas drive reservoirs. Also shown on these tables is the cffect of each of the three values observed in the reservoirs studied.
additional independent
variahles over the range of
M obility T.<;nn
The mobility ter.m appe_ars in· thc water dri\'C case.to. the powcr +ü.0770 ai~d in thc solution gas drive case to the power +o.0979. Tables 11 ·and 12 show that a maximum increase in the mobility term will approximately double thc reco\'ery from the solution gas drive case and increase thc recovery from the water drive case hy about 50 percent. The significance of this variable in the recovery process is wcll-recognizcd in thc litcrature, 1 ami thc results of thc present study confirm and give quantitativc mcaning to its cffcct.;
Water SatHration Terni The interstitial water saturation pararneter appears in the water driYe equation to the power -0.1903 and in the solution gas drive case to the power +ü.3722. From a reservoir cngineering standpoini these opposite elfects are as expected. Low water saturation in water clrive reservoirs generally means a low surface arca of the pores per unit of pore-space and, therefore, larger pores ami a more efficient displaccmept process by ·water In solutiOl1 gas drive reservoirs low wate1; saturation means more oil in .place initially present' in th~ pores; Apparently the void sp_ace created by shrinkage of this }arger· .amount' of ·oil u pon préss'Ure reduction causes gas to flów without moving much oil, thercby impairing the efficiency of thc recoven, process. These effects have bcen noted previously by s~veral authors, in~luding P. A. J?ic!'--ey and R. B ...Bosslcr in 19446 and M. Muskat and M. O. Taylor in 1946.' According to the Tables 11 and 12, a 10-fold increase in the water saturation will decrease the contrihution of ihc water saturation term in ,vater drive rcscrvoirs by about one-third. At the same time, such an increasc will more than double the effect of the water saturation tenn in solution gas drive reservoirs.
Preisuri Ratio Tenn; .
.
..
'
..
.
.
.
.
The pressure ratio can be considered as an indication of the effectiveness \~'ith ..wllich the available water dri\'e meehanism is being utilized. A value elose to one mcans that the operator is using all of the water drive. According to Table 11, an increase in the pressure ratio from 1 to 20 decreases the contribution of ,this term t? the wate; drive recovery by al~ost 50 p_ercent. In solution.gas drive-reser~'oirs the cxpansion of the gas co~·ing ou_t of\olution is· the solc cxpulsive force,.and it is to b~ exp~cted that a,ni~{X:imuffi ¡;ressu_re.dr;wdown iri thc r.eser\,ai·r is .rcquircd to make full ,Use of the availahle en~rgy cont"clined.in the dis~olved gas. T~ble 12.shows that an ·e,;cntu_al _drawdO\vn of the reservoir p.ressure to l/30th of the bubble point pressure will increase the· contrihution of tl~is terlll tO the recovenr by around 75 percent. In addition to the foregoing significant independcnt variables, the following tcrms wcre also invcstigatcd, but not inc~uded in the final correlatif;ms. (;~,s iit .'-Í~ÍÍltim¡fén11 ,Ji;r ',)'-1~'tution- Gú.S Ih/uc· Rcscrvoirs
Accor
ii
J. J, Alll'S 1 FoLKERT Bnoxs, A. F'. VAN
24
W. BUCHWALD, ANO A. E.
E%'RD1NGEN, Jt.
SMITH
Table 11 Effect of Variation in lndependent Variables. on Recovery F~ctor fr~m Water o"rjve ReServoirS. Ilecovcry Efficiency(RE)
=
_ f ~(l -Sw) } +0.0422 B,;
.,4.00 (
.
( kµ"'.' ) +o.ono
X
X
Pm
( Sw )-0.1903
X
( p, )-0.2159
· . pcrccnt
--
P,
1
WATER SATURATION TERM
1
OIL-IN-PLACE TERM Then the uncorrected If
OIP
equals:
MORILITY TERM
recovery
If mobility
RE',is:
µ()¡.
kµwi ls: efficiency, --equa
47.a% 48A% 49.8% 50.7% 51.3% 52.2%
3% 5%
10% 15% 20% 30%
0.01
0.03 0.10 0.30 1.00 3.00.
Then the multiplier
is:
0.701
0.763 0.838 0.911 1.000 1.088
PRESSURE RATIO TERM
If water saturation equals:
multiplier
If
is:.
P,/ P, equals.:
0.05
1.77
Then the
is:
l.O
l.00 0;92 ·
0.10
l.55
1.5
O.lG 0.20 0.30 0.40 0.50
1.1:1
2,0 3.0 5.0 10.0 20.0
1.36 1.26 1.19 1.14
Then the multiplier
0.86
0.79 0.71 0.61 0.52
Example:
,¡, = 0.282) S..,= 0.350) OIP = 16.7%
Br,i
=1.10 )
RE'= 50.9 1%
k
=
µwi.=
poi=
Rccováy Facto}· (flAi)
0.249 d ) 0.540 cp ) 1.310 cp )
kµ,,¡ JLoi
= 0.103
=
Multiplier 0.839 ·
Sw = 0.350
Multiplier = 1.22
P, = 1986#) p, P, 800 #) p;
=
Multiplicr
=2.48
= 0.82
= 0.509 X 0.8S9 X 1.22 X 0.82 X 0.167 X 7758 =·554 Barreis pcr Ac1·e Fo"ot ( vs. 570 observcd)
Hydraulic Radius Tcnn in Both Tupes of Mcchanisms According to the results shown on Tables 8 and 9, the effect of the "hydraulic radius" term in both equations appears to be small, probably because both permcability aml porosity are already included in other variables and the_addition of thiS term is, iherefore, redundant. Dcnsity-Diffcrcnce Tcrm in lf'ater Drivc llcscrvoirs ' According to the data of Table 8, the effect of a dcnsity-diffcrcncc term in the water drivc C(lUatiun appears to he small. It should also be pointcd out that the addition of thi.s ·term ha
IH!
25
BULLETIN Dl4: A STATISTICAL STUUY O!' RECOVF.RY ~;FFICIENCY
-Table 12 Effect of Variation in lndependent Variables on Recovery Factor from Solution Gas .Drive Reservoirs .
. 1/ecovuy Efficiency (RE) = ¡ ~(t _;__ Sw) \ +0.1011 41.SB . { ·-
B,,
~--¡·
j
.
-
OIL-IN-PLACETEfilr Jf
OIP eqnnls:
x
( Y-)·+0.0979 µ,b
X
~l
MOBILITY TERM
Then the uncorrected lf mobility recovery efficiency, -k- equals: µ,, RE', is:
+n. ?722 ( Sw ) . WATER SATURATION TERM
is:
If water saturation SU) equals:
Then the multip1ier
X
PRESSURE RATIO TERM
Then the multiplier If P11/Pri equals: is:
Then the multiplier is:
0.001
0.509
0.05
0.328
1.50
1.07
"3%
23.8%
0.01
0.637
0.10
0.42_4
2.00
1.13
!;%
25.8%.
0.03
Q.709
0.15
0.494
3.00
1.21
10%
28.9%
0.10
0.798
0.20
0.549
5.00
1.32
'15%·
. 30.8%
0.30
0.889
0:30
0.639
10.00
1.49
20%
32.3%
1.00
1.000
0.40
0.711
20.00
l.GS
34.4%
3.00
1.114 ·
0.50
0.77_3_
30.00
1.81
30%
-~---··-----·
Example: ~
= 0.174)
Sw = 0.340) 1.40 )
llob
=
k = 0.020 ·d ) __le_ µ,,~ RE'::::: 27.9% µ01, 0.50 cp ) Multi-
OIP = 8.20%
Ueeoucry Ji'actor (flAJi')
=
= 0.279
=
= 0.040 Sw 0.340 Multi- plier.::::: 0.729 plier = 0.669
X 0.72.CJ X O.tHi!J X J ••18 X 0.082 X 7758
A== 3660# ) A P«=:580#) P, = 6.31 Multiplier= L38
= 119 Earrcls per Ac1·c F'~ót ·' ('1Js.112 ol>ser11ed)
variables, viz., ir.creased mobility, lower water saturation, and a pressure drawdown ratio closer to one. According to the data on Table 11, each of these three additional factors under optimum conditions rnay .be responsible for an improvement of around 1.5 times as cornpared to minimum conditions. The cornbined effect of theSe four factors, whiéh are a1I in the same direction when the reservoir rock and its fluids approach ideal conditions; ~~;~\lid cause a maximuril improvement oí better than four times over the poorest conditions'. · In the solution gas driYe case a maximum variation of the oil-in-place variable may result i,1 an irnprovement in recovery efficiency of 1.5 times. This improvement expresses essentially the beneficia! erre~t of bette/ reserVoir rock. When the permeability and, therefore, the.mobility·increases, the water saturation usu·ally·decreases. According to Table 12, the results of such changes in the mobility factor and the water saturation factor are in opposite directions and approximately cancel each other. Table 12 also shows that a maximum pressure drawdown may cause an improvement in recovery efficiency of around 75 percent. The combined effec_t of the oil-in-place factor and this pressure ratio factor, together with the cancellation of the mohility and water Rnturation factorR, wou1rl cause ·a total improvement oí slightly less than three times over the minimum Conditions.
QG
J. J. ARfl~, FOLK!:RT RRONs,:A. F. VAN EY1,RD1NGE1" 1 R, W,
BUCU)VALD1 ANDA. E. SMITH
VII -NOMENCLATURE Petrolt>um F.ngineering Symbols a
BAF'
= angle of formation dip; degrees = recovery factor; ultmate recovery in barreis of stock-tank oil per acre foot of ne~ pay = oil formation volume factor;
a dimensionlcss factor repregenting the volume of oil under reservoir conditions per unit volume of stock-tank oil
D,,
= depth below the surface; feet = gravity.of stock-tank oil; degrees API
= net pay thickness; feet
=arithmetic averag~ of absolute permeahility; daJ'Cys ''º JJ
RE
= viscosity of reservoir oil; cenÚp~ises
:::: visc6sity of resei;voir water; centipoises
:::: pressure; psig = recovery efficiency.; recoverable oil as a pe1·centage oí stock-tank oil initinlly in place in the reservo ir
RE'
= recovery efficiency yet to be corrected for the effect of mobility, water saturation,
R,
=
p.,
=density of reservoir oil; grams per cu~ic centimeter
pw
= density of 1·eservoir water; grams per cubic centimeter
and pressure ratio terms gas solubility factor; the number of standard cubic feet oí gas liberated under specific separator condjtions which, under reservoir conditions, are in solution in one stock-t:_:rnk harrel of oil
s,,
=residual gas saturation under reservoir conditions; a fraction of the total pore space
Sw f
.=.interstitial water content; a fraction of the total pore space_ =· reservo ir ·tempera tu re; 5Iegrees Fahrenheit effective porosity; a fraction of rock hulk volume
"1,
= initial conditions = abandonment conditions = buhble point conditions
= residual oil T
,¡
f o
w
saturation under reservoir conditions; a fraction of the total pore space
=
=differential separation = flash separation =·oil ~ wafer
,.
= gas = i·eSirlual at ahafüion.ment e~JHÜti_onR
N
= mimber of elements in a· sample from
[/
·1n,;mt,- ••• mr1 'x
y y y X
a, b, e, ... o.,fi,"C' .•.
n. population
= elements ·or:sets
of ·
= = = y value eomputed by regression analysis. = mean of y values = mean of x values = eoefficients in 1·egression equation = exponents in reg·ression equation
ltlítt
BULLETIN Dl4: A STATISTICAL STUDY
O}'
27
RECOVERY EFFICIENCY
=constants in logistic equation =variancc of y with rcspcct to Ü befare regreso ion =variance of x with rcspcct to ;; beforc rcgrcssion
G,, C1,
si s_/
= covariance of y and x
=standard deviation of y =squarc root of variancc s,,'
n
= standard deviation of x =square root of variance si = number of degrees of freedom = N-1, N-2, cte.
w
= Weight factor; w, mean of w values
s,
f·
1
...
•. ?
~ - - ·_ _
,..:=: ¡
·.r: t~ ~·'.:;:.·0NP~ nt;~ r;. .
~ ':"!J...,_I; :;' ~
.. .., . ..,!..,~ .. '
=variance remaining after regression
Si
= number of independent variables = summation symbol = coefficicnt oí multiplc correlation = "F" test probability function (Rcf. !), page 240) = "studcnt" t proLability function (Reí. 9, pagc 225) = standard error oí estima te in y = standard deviation of a normal distriLution = natural logarithm to the base e = base oí natural logarithm
p
,.
fn e
Sub,cript,
= any number = significant = first regression = second regression
i 8
o
1
~
•
significance level
= observed = computed
obs
comp
VIII 1 Craze,
REFERENCES
R. C. and Buckley, S. E: A Factual Analysis oí the Effect of ,vell Spacing on Oil Recovery, API Drilling
and Production Practice, 144 (1945). 2 Guthrie,
R. K. and Greenberger, M. H: The Use of Multiple Correlation Analysis for Interpreting Petroleum Engineering Data, API Drilling and I'roduction Practice, 130 (1955). 3 Arps, J. J. and Roberts, T. G: The Effect of Relative Permeability Ratio, the Oil-Gravity and the Solution GasOíl Ratio on thc Primary Recovery from a Depletion Type Reservoir, Trans. Am. lnst. Mining Met. Engrs. (Petroleum Development and Technology, 204, 120 (1955).
4Leverett, M. C: Capillary Behaviour in Porous Solids, Trans. Am. lnst. Mining Mcl. Enyrs. (Pct·ro_leun1, Dcvelop,nent and Tcchnology) 142, 152 (1941). SEzekiel, M: Methods of Correlntion Annlysis, 211, John Wiley & Sons, !ne., New York, 1941, 2nd Ed. 6Dickey, P. A. and Rossler, R. B: Role oí Connatc Water in Sccondary Recovcry oí Oil, Trans. Am. Inst. Mining
Met. Engrs. (Petrolcum Dcvelo¡nncnt and Technology) 155,175 (1944). 7Muskat, M:. and Taylor, M. O: Effect oí Rescrvoir Fluid and Rock Characteristics on Production Histories of ·Ga:1 Orive Reservoirs, Trans. Am. Inst. il!ining Mct. Engrs. (I'etrolcmn !Jcvclopmcnt and Tcchnology) 165, 78 (1946). BArps, J. J: Estimation of Primary Oil and Gas Reserves, Frick fetrolemn Produ..ction Handbook, II, 37, McGraw, Hill Book Company, New York, 1962. ºBeyer, W. H: Jfondbook of Ta!Jles for Pruhability rmd Statistics, The Chemical RubLer Company, 1966.
J. J.
28
Al(l'S, Ful,KEHT lliWNS, A.[<', VAK E1'81UJI1'GEN, R.
w. RIJCill\'Al,U, ANUA. B. s,m11
Appcndix REGRESSION ANALYSIS AND TESTS OF SIGNIFI<.:ANCE Consider a populalion of N elements m1, ·111-:t, ••• nis, (which may be N cores, oilfields, etc.) ancl let 1/i and :r.¡ be properties measured on clcmcnt mi. The purpose of regression analysis is to cstablish if thcre is á significant relatioi~ between y and x. This relation is cxpressed hy the rcg¡·cssion cqua.lion. More specific, a linear rclation of y to :r. ;'; expressed by y= n equation:
+ l,x.
This mcans that for every ol;served value
Xi
a calculatcd value
Y is 1
dctermined by the
yi;::: a+ ÚX¡
(13)
Ilefore carrying out thc process of dctermining a arnl .b the following statislical propcrties are dcHncd:
.\lean of ¡¡:
1Iean of x:
:z (y,-y)2
Variance of ¡¡ with respect to y:
N-1
Variance of x with respect to x: si::: CoYariancc of // a111l x:
,.,-=
s1
::,; (x,-x) 2
N-1
: ,; <11,-u)(x,- xJ N-i
Thc standard dcviatio'ns s11 and' .'l.r are the squarc roots of thc varianccs s¡/! and
SJ: 2 ,
The factor N l/1 appearing above signifies the number of indepcndcnt observations (or degrccs of frcedom n) which is one less than thc number of clements N, hcr::ause one indepen
Y
lw¡y¡ y=~
si=
::,; w,(y,
-y)'
~W;-W
X:
etc.
Hénvever, ·the number of independent observations remains N. Tn Equation (13) the constants
)
''=sx2
t
n=y-l1x
f
( 1,1)
At this p¿int it maY\e ·desirable to note that the correlation between y and x may be either an "npriori" or a "nonapriori" corrclation. 1f the corrclation is "apriori", then the physical relation between y and x ·and thc mathcmatical cquation cxpressing this rclation is largcly known. The purpose of the regrcssion analysis is here only to determine the constants in the equation which give the best fit. If, howevcr, the correlation is "nonapriori", then the physical relation iS as yct unknown; 'and, · thereforc, thc 1iurpose oí the regression is to fincl the hcst form of hoth thc equation and the sizc of thc constants in it, which gives the best statistical fit. The physical relation not heing known, there is a dang"er oí conStructing by regreSsion analysis a curve which wavés through all observed points causing overjitting.2 Ü\·crfitting can largely be an1ided first, by using the simplest equation possiblc; if the; num~ her of observations is small; aml second, by using any a priori information available to select the equation which hest
BULJ.l:TIN DH: A STATISTICAL STUIJY 01" Rl•:co\'1']ty EFncrnKCY
agrccs with this information. For instance, if it is known that y incrcascs with incrcasing x; thcn the cquation expresH: ing this relation must be unidircctional. An equation y==: a+ bx + cxi whcre sorne of the constants are of oppositc sign would not mcet this rcquircrncnt.
+ ...
Coming back to thc rcgression in Equation (1:J), thc observcd values 1/i are
J¡. Only
if this distribution is independent of x,, i.e., the position of ;, on thc rcgression linc, will thc values of a, ami b computed (14) givc the best estímate of the linear regres·sion. This is clarified in the following example. Let l/E be the recovery efficiency of a field and_Jc_-the mobility ratio of its oíl. In Fig. 8a the result is shown of a rcgrcssion 0 f f k A . A Y= RE on x =- by y,= a+ bx;. The observed values y, are plotted vs. the computcd values y,. It shows that thc µ
spread of y¡ around
Y, increases from left to right. Now instead the logarithms of RE' and ..!.. are regressed by taking µ
y==: log RE and x ==: lag~. Fig. Sb shows that the spread of y¡ around y~ in this case is independent of the position µ
the regression linc. 'l'hercfore, this last rcgrcssion is prefcrred ovcr thc first. 'l'hc average of the fluctuations of the o\Jserved y Yalucs around thc calculatcd ?/ valucs can he cxprcsscd by thc rv1wtining v,iriunr;c:
011
. S" :! --
" (y·
'
y·)'
~---~·-'-
(N-2)
( 15)
·
where N - 2 are thc rcmaining dcgrees of frccdom, becam;e two indcpendent oliscrvations are lost in detcrmining n. and b. More general, if a multiple rcgression is carried out with 11 indcpendent variuhlcs (p ==: 1 for single rcgres•
sion) then .(15) becomes:
·
s
2- ~
' (y¡-y¡)2
( 1G')
"-(N-p-1)
To investigate ho\v significant the regression is, the variance rcrnaining ufter regression, S/, must be compared with either the variance befare regression sy2 or with the part removed by regression s1/-Sy2. This is achieved as follows: Rccause n
= -y - bx- thc rcgrcssion equation J/i' ==:a+ bx¡ may also be written as: · y¡-'Y¡==: y;-1}-b (x¡-X)
. which givcs ~
A (y¡-y;}:J ==:
~
(y¡-y)2-j-b2~ (:r: 1-x),Z-fJb
or (N-2) S,, 2
Y¡ (obs)
=(N-1)
~
(y¡-y)(x¡-,r,)
[s,'+ ú's,'-2bs,-,]
Y¡ (obs)
"Y¡ y= RE,
X
(comp)
k = ¡;.
/
"Y¡ y = In RE,
(cornp) X
( a) Fig. 8 - Correlation of Observed and Computed Dependen! VariablH
k = In -¡;.
( b)
J. J. Anrs, FOLK!c'RT BRONS, A. F. VAN EYERDING~N, R. w. BUCHWALD, ANDA. E. S~!ITH
30
For large cnough valucs of N only a ncgligible error is made by writing instead:
or,
.
USIII~
Sur
b =-,s.
Introducing lhc cocfficienL uf corrclation Syx
r=-s11sx
gives:
(16)
dcgrces uf írcetlom: 'I'he coefficient of corrclation r rangcs from zero (no correlation) to plus or minus one (pcrfcct correlation).
To investigate-the meaning- of an r between zero and plus or minus one, the variance removed by regression r2sy2 (with one degree of freeclom) is compared with the part rernaining after regression S,2 = s/ {1-rii) (with N-2 degrees oí íreeclom). For that purpose the F test can be used, where F is defined as:
v1wiance removed/degrees of freedom ·uariance remaining/degrees of freedom ,,.2
r2 811 2¡1
F,,n
=(1- r') s,2/(n) -
n 1 - r' · (n
(17)
·
= N -2 = number o/ dcgrces of frccdom)
The subscript 1, n of F signifies the numbcr of degrees of freedom in the numerator and denominator and is used íor reading F• values from tables. f The F function has a known probability function which can be rcad from tables.9 Howcver, it is usually more convenient to use the "sludent" l function whose aLsolutc value is related to F by: or 1
t,
1
- = \jI _nr2 =vF,,n 1
,-2
(18)
where nis the number of degrees of freedom of the remaining variance. Note that for multiple regression with p independent variables, the F function becomes
F,,.,
=
where n N t-function.
=
r2s,l/p (1-r") s:,2 /(n)
=
n • r2
>1 (19)
¡, (1- r')
p -1, thc degrees of frccdom of thc remaining variance. In this case F cannot be rcplacc
The reasoning by which the significance of r is rletermined gocs as follows: r was determined on a sample of limited size N. To draw conclusions from this sample valid for a much larger population from which N was taken, we should considcr that another samplc taken from the large population might havc givcn a somewhat differcnt value of r. Now suppose that in the large population y aml x are unrelated. Thc expected value of 1· in N woul 1 ta I even if no correlation exists. ¡ tn,o: 1 = 2.0. If from regrcssion aualysis
not related there is still 1/2 a.S{, chance that tas calculated from the regresa.% chance that it is smallcr than - ta. Therefore, there is a.% chance that Suppose we take a= 5% arnl Jet n he GO. Then from tables9 it is read that a valuc of r is found corresponding to a value of I t 1 3, and ,ve accept this
=
IIIHf
)
31
BULJ.ETIN 1)1,t: A STATISTICAL STUDY OF RECOVERY EI•'FICIENCY
r as a signifieant correlation coefficient, then there is less than 5% chance that we are wrong in our decision. Actually the value of a corresponding to I t 1 =: 3 is less than 1%. Therefore, we will accept the correlation between y and x to be significant with the chance being
wlt) 1
•
-2 (} º''º
lcss than 1% that our decision was wrong. If, as is
1
2
conventional in certain industries, the significant levcl a is taken as 5%, with a corresponding value of I t0 1,
-ta
then we_ will only accept those r values as being significant, for which the corresponding I t I value is larger than J ta/. The minimum value of r which is significant is 1·, which is related to Ita! as found from (18):
r,
== 1V -
with ta= tn, 5% read from tables and n
o,
- a 'º
Fig. 9 -
11
student" t Distribution
= ~ n ;2t.'-·
(20)
2 for single regTession (p;::: 1).
For multiple regression with p independent variables the F function must be used, giving-
(21) with
Fa= Fp,
n(.'í%)
ruul n=N-p-1 In Fig·. 10 valucs of ,,., al'e shown for
= 1 and /1 = 2 hasecl
J)
011
(20) and (21 ).
a+
For example, on a sample N = (iO, the i· of the equation y= bx (JJ = l) is only significant if it is larger than 0.2fi and for the equation y= n l>x f'Z, ,. must be larger than 0.32. For N == 10 these values become 0.65 and 0.75 respectively, Next considera regression analysis giving·
+
+
y=a+l>x+cz+ ...
(}} independent variables)
Let the remaining varianee he Suo2 an
y= Let the rernaining variance he
ri
+ bx +e::+ en+...
(p
+ .1 indepenrlent variables)
ancl the coefiieient of correlation
8y12
r 1,
which is larger than
To investigate if the second correlation is significanlly better than the first, compare Sy1::
variance a f l e r /
== Syo2- (S!102 -Sy1':!)
variai1 afler
scco11d regression
first regression
\
1 = (N -··J)--1) -
tkATN:s of fre~dom! (N - v - 2 )
\variance 1·emo,·ecl betwccn /first nnil seeond Tegrcssin11 (1)
:\gain t.lw /,' lei'lt is 11scd, deíine
mirimu;c ,;·cmoi•1uljdef¡r
01'
S, 0·! -8,¡1 2
1 !1'1.11 := (11) ---'-~',--!! • - . '-J/1
But
which gives
with n
=N -
p-
2
= degrees of freedom after second regression.
Sy12
?\¡.
with Syo?:
J. J.
~2
ARrS, FoLKERT BRONS, A. F. VAN EvERDINGEN,
1.0 f- -·
0.9
-
\ \
- - -
1 \
- -
\
- -
-
~ - -'-- -
- -
!';:
.
' ,:- '
04
~
- - -
,_ -
---
1
-1-
- - - -
-
::::I_ - --
- - - -
-
- - -
1
o
-
-
o
,_ -
1
~
, 20
10
-
40
+
..
1
- - -
- -
-
~- -
-
-
1
-
-
- -
1
-
1
¡_
1
1 - f-
1
,_ -
-
30
'
-
1
1 ~
+ _L
1
-
0.1
[
1
1
1
1
'
-f1
-·
-1
-
---
-
- - - - - -
,_
AT et= 5% 1
-· -
- - - - -
·p - 1. --..;"-'
-
f-- f--f-
-
""' -.;: ._!:=2 -
1 [
p = NUMBER OF INDEPENDENT VARIABLES r 5= MINIMUM COEFFICIE/\JT OF CORRELATIQN-
1
'
-
0.2
N, NUMBER OF OBSERVATIONS l
- -
'-
0.5
0.3
- -
-f- f--
- '-- - \
07
06
1-
w. BUCHWALD, ANDA. E. SMITH
,_
\
0.8
•
f--
R.
70
60
50
60
100
90
N
Fig. 1O - Significan! Coefficient of Correlation .28
-f- _¡_ i~-f- -i- -f- ,_ -1- -/- -f- _L -- -- -
-
-, -
- - -
1
- -
.24
- -
\~- 20
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-
-
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f--
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r,,_
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.os
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F/API/BUL D-14/1967 703996
"" '
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r--
Fig. 11 - Sipnificant lncrease in Coefficient of Correlation
11111111~
1-
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r,.;
~
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1
'40
i'
\
-----~ :¡=
0.3
= 5%
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n::120
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30
1
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1
.
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\;
C(
-
-
n =10 '\
-
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\
1
1
n = N-P-1 = DEGREES OF FREEDOM ór 5 = SIGNIFICANT INCREASE IN r 1 FOR A SIGNIFICANCE LEVEL
~- -
1'\
- -
1
- - - - - -
0.8
0.9
10
33
BULLETIN D14: A STATISTICAL STUDY OF RECOVERY EFFICIENCY
Again as before (22)
The minimum value of t" needed to be a significant improvement is le;:. The minimum value .,. which is significantly higher than 1·0 is found from (22):
anti a significanl increase over 1·u is
6-r,
= r,-ro = ~ nra2+ + ta. n t 0
2
2
(23)
-'to
Equation (23) is graphically represented in Fig. 11 for a= 51¡;1, As an example oí the use of Fig. 11 consider a sample of N 80. First a single regression analysis is made (1> 1) idving ro o.so. Then a variable is added (p 2) giving ri 0.82. Is this a significant improvement?
=
=
=
=
=
=
=
Solution: n N -p - 2 76. 'The minimum increase in ro to be significant as read from Fig. 11 is ahout 0.01. 'l'herefore this is a significant improYement. The ¡,rohlem can also be considered in a 1·everse manner:
r1
= 0.82,
re= 0.80,
n
= 7r,
From Equation ('.::!'.!) it follows that
The corresponding value of a is found to be less than 1 percent, therefore, there is less than 1 percent chance that the increase in r from 0.80 to 0.82 is insignificant.
UNAM FECHA DE DEVDLUCIDN
El lector se obliga a devolver este libro antes del vencimiento de préstamo señalado por el último sello,
/ /
FliPI
BUL D-1'1
J?t1 /
