. # Pwr Correlations for Mkidle East crude oils Muhammad All Al=Mwhoun, SPE, King Fahd U. of Petroleum and Minerals /37/8 Summary. Empirical equations for estimating bubblepoint pressure, .. nil ~ ~$ p~~~~epo~ ~~~~, ~ ~ ~v~ for ~~e . .. ..- East crude oils were derived as a fimction of reservoir temperature, total surfkce gas dative density, solution GGR, and stock- tank oil relative density. These empirical equations should be valid for all types of oil and gas . muturea with properties failing within the range of the data used in this study. Introduction PVT correlations are important tools in reservoir-performance cal- culations. The major use of PVT data is in carrying out material- balance calculations. In 1947, Sttding ‘“~pubiished correlations for determining the bubblepoint pressure and FVF from known values of temperature, solution GOR, gas relative density, and oil API gravity. A total of 105 experimentally determined data points on 22 different crude oil and gas mixtures from California were used in deriving the corre- lations. Standing reported an average relative error of 4.8% for the bubblepoint pressure correlation and an average relative error of 1.17% for the FVF correlation. In 1980, Gla.w4 presented correlations for calculating bub- blepoint pressure, oil FVF, and total FVF from known values of temperature, solution GOR, gas relative densky, and oil API gravity. A total of 45 oil samples, mostly from the North Sea region, were used in obtaining the correlations. G1a.wrepcmed average relative errors of 1.28%, -0.43%, and -4.56% for the bubblepoint pres- sure, the bubblepoint oil FVF, and the total FVF correlations, re- spectively. Reviews of other empirical PVT correlations were presented by Sutton and Farshad5 in 1984. Standing used a graphic method and GlastDused both a graphic methcd and lir regression analysis in the development of their PVT correlations. The graphic estimation and curve-fitting, how- ever, do not lead to the best estimate. Therefore, this study devel- oped the correlations using only linear and nonlinear multiple regression analyses to obtain the highest accuracy. This paper (Ms with PVT COmlatiOIIS ~x~jU~~v~jy fcr am+jes of Middle East crude oils. However, they should be valid for all QW ef g~i~fl ~fi~~ums wft properties faii ing within the range of data used in this study. Moreover, this study evaluates the ac- curacy of Standing’s and Glaso’s PVT correlations, which are shown in Table 1. Error analyses were done for this study and also for Standing’s and Glasgis correlations to compare their degree of accuracy. Finally, nomography for bubblepoint pressure, bub- blepoint oil FVF, and two-phase total FVF wem conshucted on the basis of the developed empirical correlations. PVT Data ThePVTanalyses of 69 bottomhole fluid samples from 69 Middle East oil reservoirs were made available for this study. The ex- perimentally obtaind data points were 160 each for the bubblqmint pressure, pb, and bubblepoint oil FVF, Bd, correlations, and 1,556 for the total FVF, B,, correlation. The ranges of the data used are shown in Table 2. PVT Corrohtlons The correlations for bubblepoint pressure, bubblepoint oil FVF, and two-phase total FVF were developed by use of the linear and nonlinear multiple regression analyses shown in the Appendix. 650 Bubblepoint Pmasure. The following generaI relation of bub- blepoint pressure of an oil and gas mixture with its fluid and reser- voir properties was assumed *: Pb.=mss7gP70*n .......... . . . . . . . . . . . . . . . . . ...(1) Table 3 shows the 160 experimentally dckrmined bubblepoint pressures obtained from PVT analyses of 69 different Middle East oil/gas mixtures. The nonlinear multiple regression analysis was used to develop the following relatioa. pb =5.38088X 10 ‘3R$7t~~g- 1.577540y~.143700T-370, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...(2) where Pb = bubblepoint PWSSUE, R. = solution GGR, 78 = dissolved gas relative density (air= 1), 70 = stock-tank oil relative density (water= 1), and T = absolute temperature. Bubblepoint Oil FVF. Gil FVF at bubblepoint pressure can be de- rived as a function of solution GGR, average gas relative density, oil rehttive density, and temperature as follows: B& =j(R., Tg, -yo, T). . . . . . . . . . . . . . . . . . . . . . . . . . ...(3) The foUowing empirical equation was developed by use of the nonlinear multiple regression analysis and a trial-and-error method based on the 160 experimentally obtained data points shown in Ta- ble 3: B% =0.256805X 10-ZR~T4Z~7$323~T;l.= +1.63 x10-ST , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...(4) where B% is au intermedii oil FVF value. The bubblepoint oil FVF correlation @q. 4) was further retined by applying the lii regression analysis on the same data. This regression analysis yielded the following equation: B~ =0.497069+0.862963x 10 ‘3T +O. I132594X10-2F+0.318099 X1O-5F2, . . . . . ...(5) where F=R$T423~&232~y; 1.~~ . Journal of PetroleumTechnology,May 1988
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. #
Pwr Correlations for MkidleEast crude oilsMuhammad All Al=Mwhoun, SPE, King Fahd U. of Petroleum and Minerals
East crude oils were derived as a fimction of reservoir temperature, total surfkce gas dative density, solution GGR, and stock-tank oil relative density. These empirical equations should be valid for all types of oil and gas .muturea with properties failingwithin the range of the data used in this study.
IntroductionPVT correlations are important tools in reservoir-performance cal-culations. The major use of PVT data is in carrying out material-balance calculations.
In 1947, Sttding ‘“~pubiished correlations for determining thebubblepoint pressure and FVF from known values of temperature,solution GOR, gas relative density, and oil API gravity. A totalof 105 experimentally determined data points on 22 different crudeoil and gas mixtures from California were used in deriving the corre-lations. Standing reported an average relative error of 4.8% forthe bubblepoint pressure correlation and an average relative errorof 1.17% for the FVF correlation.
In 1980, Gla.w4 presented correlations for calculating bub-blepoint pressure, oil FVF, and total FVF from known values oftemperature, solution GOR, gas relative densky, and oil API gravity.A total of 45 oil samples, mostly from the North Sea region, wereused in obtaining the correlations. G1a.wrepcmed average relativeerrors of 1.28%, -0.43%, and -4.56% for the bubblepoint pres-sure, the bubblepoint oil FVF, and the total FVF correlations, re-spectively.
Reviews of other empirical PVT correlations were presented bySutton and Farshad5 in 1984.
Standing used a graphic method and GlastDused both a graphicmethcd and lir regression analysis in the development of theirPVT correlations. The graphic estimation and curve-fitting, how-ever, do not lead to the best estimate. Therefore, this study devel-oped the correlations using only linear and nonlinear multipleregression analyses to obtain the highest accuracy.
This paper (Ms with PVT COmlatiOIIS ~x~jU~~v~jy fcr am+jes
of Middle East crude oils. However, they should be valid for all
QW ef g~i~fl ~fi~~ums wft properties faiiing within the rangeof data used in this study. Moreover, this study evaluates the ac-curacy of Standing’s and Glaso’s PVT correlations, which areshown in Table 1. Error analyses were done for this study and alsofor Standing’s and Glasgis correlations to compare their degree ofaccuracy. Finally, nomography for bubblepoint pressure, bub-blepoint oil FVF, and two-phase total FVF wem conshucted onthe basis of the developed empirical correlations.
PVT DataThePVTanalyses of 69 bottomhole fluid samples from 69 MiddleEast oil reservoirs were made available for this study. The ex-perimentally obtaind data points were 160 each for the bubblqmintpressure, pb, and bubblepoint oil FVF, Bd, correlations, and1,556 for the total FVF, B,, correlation. The ranges of the dataused are shown in Table 2.
PVT CorrohtlonsThe correlations for bubblepoint pressure, bubblepoint oil FVF,and two-phase total FVF were developed by use of the linear andnonlinear multiple regression analyses shown in the Appendix.
650
Bubblepoint Pmasure. The following generaI relation of bub-blepoint pressure of an oil and gas mixture with its fluid and reser-voir properties was assumed *:
Table 3 shows the 160 experimentally dckrmined bubblepointpressures obtained from PVT analyses of 69 different Middle Eastoil/gas mixtures. The nonlinear multiple regression analysis wasused to develop the following relatioa.
Bubblepoint Oil FVF. Gil FVF at bubblepoint pressure can be de-rived as a function of solution GGR, average gas relative density,oil rehttive density, and temperature as follows:
The foUowing empirical equation was developed by use of thenonlinear multiple regression analysis and a trial-and-error methodbased on the 160 experimentally obtained data points shown in Ta-ble 3:
tooccur. The microprocesaor units also have logic that can detectwhen a well is logged up rather than pumped off and can preventprematw shutdown. The microprocessor unit can store data inmemolyandthemtransmitthe data bymdioorotherc0mawnicationlinks uponcommand toacentral hoatunnputer. TbiaallowarhemicqmXeWXunita to beusedwithout comrnunhtion, orthecul-tral communication can be added when justified.
Coneludons
1. Pumpoff control is a profitable technique to be used on rod-pumping wells except iu low-resemoir-pressure, high-PI wells.
2. Sevehl studies have shown that minimum crit&ia for justifi-cadonofpumpoffcontrol area20% reduotionin energy consumption, a2S%reduction inpullingjoba, anda lto4%increawinproduction. These are achievable when compamd with good oper-ation witbout pumpoff control. If poor qemtion currmtly exists,then greater benefits are available.
3. The anaiog and micqmceam r local logic type of pumpoffcontrollers are rwxxnmended. The decision to use central commu-nication to a host computer should be juatitkd individuaUy,~ on the existing conditions in a given field.
AdmowlodgmontWethankthemmagementof Shell Weaern E&PInc. forpemkionto publish this paper.
Roforoneos
1. Wcstemml,G.w.: “~ ~of ~~;”MM SPE 6853~ x the 1977SPE /innual Techakal Cuafer-ence arai Exhibhion, Denver, Get. 9-12.
3. Ghauri, W.K.: “Producdon Techao@y Ex#ence in a Large Car-bonate Waterflood, Deaver Unit, Waawn Sao Andres Field, WestTexas,” paper SPE 8406pmated at tk 1979SPE Anmd TechnicalCdererm and Exbibitioa, Las Vegas, Sept. 23-26.
4, HUa@r,I:Il., HUM!, M., d Reiter, c: “!kmr Uiiit%%!!SWVdhlWC and Pum@ffCoatrol S@tan,” JPT(Scpt. 1978)1319-26.
Nordii multiple regression analysis was applied to develop thefollowing dation, which is based on the 1,556 expmimmd y deter-mined two-phase total FVF:
Emor AMlyslsThe statisticaland graphic error analyses were used to check theperformance, as well as the accm’aey, of the PVT emrelations de-velopod in this study and by Standii and GIsw.
Statiatkal Error Analysis. The accuracy of eorrelations relativetotheexperkmd valueaia ~by Varioussmdstidmeana.The criteria used in this study were average percent relative error,average absolute percent relative error, minimum/maximum abao-lute percent relative error, standmd deviation, and the correlationcoefficient.
Journal of Petroleum Technology,May 1988
Avemge PmentRefative Envr. This is an indication of the rela-tive deviation in percent from the experimental values and is givenby
where Xem and Xew represent the estimated and experimentalvalues, respecdvely. The lower the value of E,, the more equallydistributed are the errors between positive and negative values.
Avemge Absolute Pement Relative Error. This is defined as
d indicates the relative absolute deviation in percent from the ex-YAEs. A lower value implies a better correlation.
aximum Absokte Parent Rdiziive Envr. After theabsolute percent relative error for each data point is calculated,iEii, i=i,2. . .nd, “ti the hum d _trm values wescanned to know the range of error for each correlation:
The accuracyof a correlation can be examined by maximum ab-solute percent relative error. The lower the value of maximum ab-solute percent relative error, the higher the accuracy of thecorrelation is.
Stam&udlbidion . StandaKJdeviation, s., is a measure of dk-persion and is expressed as
where (rid-n - 1) are the degrees of freedom in multiple regres-sion. The symbol x repmsem pb, Bd, or B,. A lower value ofstandard deviation means a smaller degree of scatter.
651
TABLE 3-SURFACE PROPERTIE8 AND EXPERIMENTAUYDETERMINED BUBBLEPOINT PRESSURE AND BUBBLEPOINT OIL FVF
::=3,2043,2013,1963,1803,1553,1553,1273,1013,0803,0663,0573,0573,0303,0032,8412,9252,8012,2002,6862,6712,8652,8452,8352,8312,8042,7892,7512,6672,6522,6392,6362,6172,6072,5862,5592,5582,5302,5212,5042,4452,4130 An4C,W I
The assumptionof normal distribution of errors allows establish- Table 4 presents the co-n of errors relative to the ex-ment of ~ntldence intervals for the estimated value. IfXm =.xe *z, then the confidence limits, z, in percent, areS==68.3, 2X=95.45, and 3s, =99.73.
Cbnddon Clr@cient. The correlation coefficient, r, representsthe degree of success in reducing the standad deviation by regression analysis. It is defined SS6
The correlation coefficient lies between Oand 1. A value of 1 indi-cates a perfect correlation, whereas a value of Oimplies no corre-lation at all among the given independent variables.
Graphic Error Analysis. Graphic means help in visualizing theaccuracy of a correlation. Two gfaphic analysis techniques wereused.
Cms$pfot. In this tedmique, all the dmated values are plottedvs. the experimental values, and thus a crossplot is formed. A450[0.79-rad] straight line is drawn on the crossplot on which estimat-ed value is equal to experimental value. The closer the plotted datapoints aii ‘b “&s iti, the ‘&tter the correlation is.
Erwr Distribution. Tbe deviations, Ei, for a good correlation
raree to be as close aspodble to the “normaldistriion.”Tbe uation of a norrnaldistribution curve to fit any data set canbe derived by use of the mean and standard deviation of that dataset. 7 This technique involvea pmenting relative _ of devi-~~~s ~mih~stomm. o-A ●L-- f-:- - - --
6*WU - ‘=11 ‘Lug u ‘@4~*tiu~m ~mc
to it. The accuracy of the correlation is then judged by matchhgthe error distribution with the normaldistribution curve.
Compwkon of CorrolatlonsStatisticalError Analysis. Average percent relative error, aver-age absolute percent relative error, rninirnumhaximum absolutepercent relative error, standard deviation, and correlation coeffi-cient were computed for each correlation.
perimentall~ determined bubblepoint pressure of 160 data points@imated from the three correlations. The cmrelation for bub-blepoint premure of this study achieved the lowest emors and stan-dard deviation, with the highest comletion coefficient accuracy of0.997, as prmented in Table 5. Standing’s correlation stood sec-ond in aceumcy, with a correlation coefficient of 0.979. Glas#scorrelation showed poor accuracy, with the highest errors and the10WCIW Correlation C4WffiCkDt of 0.891.
Wimated bubblepoint oil FVF end the experimentally obtainedbubblepoint oil FVF for the 160 data points are given in Table 6,akmgwhhtherelativedmhtiontbreachdatapoint.Forbubblepointoil FvFcormMon , this study again achieved the highest accura-~, foU~d by G~ @ Sti~~ M ti~ @Tabie7.
Atotalof l,556data points used indeveloping the total FVFcorrelation was sorted according to tbe percent relative error, and51&pointswemAectedby takingeveTyotller 3opointstorefkcttheaccumcy. Thesekcted51datapointsfort otalFVFwithreser-voir properties of the oil samples are given in Table 8. The rela-tive error for each data point is also presented in the same table.The statisdcal error analyses for the two correlations are presentedin Table 9. This study’s codauon. fortotal FVFoMained bigher~ than GIaso’s correlation.
G’OSSPIOtS.The crossplot of dmated vs. experimental valuesfor bubblepoint presswe codations are presented in Figs. 1tbrough 3. Mostoftheplotted pointaofthis study 'scomeWon“Wvery close to the perfect correlation of the 45” [o.79-rad] line. Thecorrelations of standing and Glas9reveal theii Ovemdmatl .OD.Ac-&fi#~g t-h~. 2 r.1.. +. --1A-.. “1.-.... -I —.- + —- - ---
,6.., .4. W.b=wuu ?Weu UIUGUmore overW-timation than Standi@s cordation.
Tile crossplot for bubblepoint oil FVF correlations are givenin Figs. 4 through 6. Moat of the plotted data points of this study’scomeladon fall on tbe 45° [0.%had] line, Mcating its high degreeof comlation, while the correlations of Glasoand StandingrevealMU OV~&~ ~ti~ ~ bJMi@fi CM ~ Of 1.5 P*!W.
[1.5 res mshtock-tank m3].The piotted i,556 data points of this study’s correlation for total
FvF fall very close to the perfect C43rrelationof the 45° [o.79-rad]line (see Fig. 7). Gntheotber hand, the plotted 1,556 data pointsof Glase’s codation scatted above or below the 45” [0.79-md]line, as prmentd in Fig. 8.
Error INstributimI. Error distribution histograms with overlaidnormal-distribution curve for tbe bubblepoint pressure correlations
Journal of PetroleumTechnology,May 1988654
,TABLE 4-COMPAR180N OP BUBBLEPOINT PREBSlfREB EBTIMATED
of this study, Standing, and Glsso are presentedin Figs. 9 throughii. Tine error ranges of ii5, f40. and *80% are used for thisstudy’s, Standing’s, and Giaso”s correlations, respectively. Thisstudy’s correlation has a mean almost equal to zero, while the peakiie@ of ‘h normai-distfloution curve for the Standing and Girt,wcorrelations are at about 7 and 18% error. indicating overestima-tion by positively skewed error distribution.
Error distribution for this study’s correlation for bubblepoint oilFVF is a normal distribution with a mean almost equal to zero (seeFig. 12). Norrnaldistribution curves for Standing and Glaso aregiven in Figs. 13 and 14. The mean distribution of Glas#s eorre-hkmyclwmtioftimml*m ofti*, ww-ing’s ccmdation indicated its overestimation by a positively skewednonnaldistribution curve with a mean of a60ut_2 %. -
The range of error dktribution for total FVF correlation for thisstudy is A 15% (see Fig. 15), while the error of Glas#s correla-tion ranges from -20 to +50%, as shown in Fig. 16. This study’scorrelation distributed the error evenly across the entire range witha mean of slmost 0.0%. Gn the other hand, the norrnaldistributioncurve for Glaso’s correlation shows a positively skewed error re-tribution with a mean of about 8%.
NomographyOn the basis of the mathematically developed PVT correlations,correlation charts have been devebped for bubblepoint pressure,bubblepoint OtiFVF, and tWC@SSe (oil/gas)totalFVF. The
chartsare presented in Figs. 17 through 19. The standatd proce-dures were followed in constructing nomography. 9*10
A nomogrsph is quite simple to use. The data points are con-nected from one scale to the other by a straight line. It is straight-
forward in determining pb, Bd, and B,. Engineering personnelwill find these charts very simple and useful tools in determiningthe reservoir performance or in designing production facilities.
conoluslon81. PVT cmrelations for Middle Bast oil and gas mixtums have
Mz?fi&V:iOped. E@. 2, 5, and 8 krrm the ‘basis for caicuisdngthe bubblepoint pressure, oil FVF at bubblepoint pressure, and ~td FVF below bubblepoint pressure. Moreover, the nomogrsphsemstructed in this study are so alternative solution without reduc-ing the accumey achievable by using Eqs. 2, 5, and 8 in a mucheasier manner.
2. Eqs. 2,5, and 8 were developed specifically for Middle Bastoilandmsm ixturesb utcanbeused foresdmabnrz“ thesame PVT*rsforrdltypesof oilsndgas “mmturea Wiipropeties fsu-ing within the range of data used ~Ath~s sttttd.
3. Deviations from eqerhenMy deterdd data, imiicated asaverage percent relative errors, average absolute percent relativeerrors, snd tlMstdd deviio@ We~ iowti fo~ t!& study Lhm
for estimations based on the correlations of Standing and Ghisg.4. The correlation ecreffiiients of the correlations of this study
tiw~Hontie Mtie Moti_lm~cl~rti lthsnthose of Standing and Glaso.
5. The PVT correkdions can be placed in the following orderwith respect to their accuracy: (1) for bubblepoint pmasure, thisstudy, Smnding, and Gw, (2) for bubblepoint oiI FVF, this study,Gw and Standing; and (3) for total FVF, this study and Glaso.
Nomondatum~= (n+l) v*ra = least-squares solution to tbe system Xz=y
B& = oil FVF at bubblepoint pressure, ItB/m[rea m3/stock-tsnk m3]
W* = intermediate value for B&B, = m Oti ~ bdoW bubtdepoint prSS.3~, RB/STB
[rea m3/stock-tank m3]~ = jnte~ v~ue for B,E . e~r
Ea = average absolute relative error, Eq. 11, %Ei = pxcent relativeerror,Eq. 10E, = average relative error, Bq. 9, %f = function
F = correlationparameter, Eq. 5
Journal of Petroleum Technology, May 1988 657
TABLE S-COMPARISON OF BUBBLEPOW/T OIL FVF’S ESTtMATEDSY CORRELATIONS PROM THIS STUDY, STANDING, AND GLASO
Experimental Estimated Bubblepoint Daviatbn in PercentBubblepoint Oil WF (RS/STS) of Estimated B ~
Oil FVF This misNumbar (FM/m) Study Standing Glaae Study Standing Giaao
6. Wdpole, R.E. d Myers, R.H.: Pru&bii@ and Srmistics@r kkgi-necrs and &icnrists, IvfcMillan PublishingCo. Inc., New York City(1985) 373.
7. Dixon, W.J. and Massey, F.J. Jr.: Intrddmn “ to Stati&d Analy-ses, KO@kusha CO. Ltd., Tokyo (1969) &i.
8. Leon, S.J.: LiworAlgebm with /4pplicadons, MacMilhn I%blishingCO. Inc., New York City (1980) 152.
9. Johnson, L.H.: Nomogmplryand EmpiridEqudons, fmutb edidon,John Wiley and Sons IDC., New York C@ (1966) 1S-57.
10. Davis, D.S.: Nomogqohy and Enqsirical Eqnatkw, second edition,Reinhold Publishing Coq)., New York City (1962) 137-210.
AP~-U-r and NonilnoarMuttiplo negroUbllLinear.Thebasicconceptof mrdtiple qression is to produce alinear combination of idepdent variables that will correlate asclosely as possiile with the depmdent variable.
Asample isofsize ndonwhich thepmpertiesy, Xl, xz. ..xnaremeaaumd. Thex'sare thei&pendent variables andtheyisthe depedent variable. The linear regression equation of y on x’sCai3hewriltenas
Fig. 13-Error didrlbutlon for bubblepoint dl FVF (Standingoorrddlon).
:
*QC1$lIW tRCOR (1)
FffI. 14-Ermrdis!rlbution for bubbfepdnt oil FVF (Gfaao cor-rdatlon).
664 Journal of Petvoleum Technology, May 1988
. . . .
UIAIIVC mtam (1)
‘lg. 15-Error distribution for totalFVF (this 8tudy’s corm.atlon).
R< Al AZ
3000 -
moo
11000 \,
I
800 ~,
60C+
400
300
200
100
ao
60
403
304
20
10I
I0.5
0.6 ‘b= 10000
FI!==&--.T 6000 To
..7 -- -, 41300 1.0----2000 ----
o.a 10003--V--- 0.9
‘ -+600
0.9o.a<
400
1.0 L5$ ‘m100
:.:
[.2
[.3
I..l
. 5
0.7
Ew!F!!sstimate bubble point pressure at 2400F of a reservoirluid with GOR 1203 SCF/ST8, average gas relative densityf 0.925 and stock tank oil relative density of 0.824.Memlined Pb jS 33~ ~sfa.
Ffg. 17-Correlation cfuti for bubblepoint preaure.
: I
@ELallVE EIW~[S)
‘lg. 16-Error di8trlWtin for total FVF (Gfno oorrofatfon).
20(
A3
u %,,~“? -
x,\ %‘,1 .5
$
l’!2\
1
Lo’
o.a
0.6
0.5~
L
20Estimsts fovmstion SOIUSS factor ●t 24@F of a bubble pointliquid with GOR 1203 SCF/SIB, average gas relatlve densityof 0.925 md stock tank oil relatiwe density of 0.824.kb~hd aob k 1.?7.
10
~!g. 4n.-e —la.. A.d L. L..idd.--, -. . . . --.—. m.mWII WIWL WE SSSSSJSJS8pUSfSS 4VSI rvr.
Journal of Petroleum Technology. May 1988665
\ M A?
moo P
1000
am
W
400
m
200
100
So
60
40
w
20
Qi?5L!S10
Estlmsts h phsse (oil -gss) totsl fomtion volums factorat lWF ●ti ●t ttsS PWSUW Of lS@ fjsia of . ~SWVOi rfluid with 60R 628 SCF/STB, wera~ ges relative densityof 0.a76 and stock tink oil relatlve &nsltY of o.a]b.OSterdned Bt IS 1.67.
where Xiaann~X(n+l) matrix, Z’iaan(n+l) vector, and ~is~ nd v-r.
Given an ndx(n+l) system of equations with n~>(n+l) asahown inllq. AA, a vector Zfor which XZequals ~cannotbefti. _,a~hfm av~or Zti~h XZkmcl~as possible to ~ia the maximum that can be achieved. Such a vec-tor is the least-aquarea solution. The unique kast-aquams solutiontotbesystemxir=yid
Nonlinear. Nonlinear multiple regression is achieved by reducingthe nonlinear relationship to a linear one by appropriate tranafor-rnation of variables.
Taking the total FVF correlation as an example, Eq. 7 can bewritten in a general form:
