~ society of Petroleum Engineers

SPE 51086

CORRELATION OF BUBBLEPOINTCOMPARATIVE STUDY

PRESSURES FOR RESERVOIR OILS--A

W. D. McCain Jr., R. B. Soto, P.P. Valko, and T. A. Blasingame

Copyrkght1998, Society of Petre!eurrr*EnglneeraInc.

This paper was prepared for pressmtatfonat the 1998 SPE Eastern Regional Conference andExhibition held in Pittsburgh, PA, 8-11 November 198S.

This paper was selected for presentation by an SPE Pregram Comminee followicg review ofinformation contained in an abstract submitted by the author(s). Contente of the paper, aspresentad, have not bean reviewad by the Society of Petroleum Engineers and are subject tocorrection by the author(s). The material, ae presented, does not necessarily reflect any positionof the Society of Petroleum Engineers, its officere, or members. Papers presentad at SPEmeetings are subject to publication review by Editorial Committees of the Society of PetroleumEnginaers. Electronic reproduction, distrikwtion, or storage of any part of this paper forcommercial purposes without the written consent of the Society of Petroleum Engineers isprohibited. Permission to reproduce in print is restricted to an abstract of not more than 300words; illustrations may not be copied. The abstract must centain conspicuous acknowledgmentof where and by whom the paper was presentad. Write Librarian, SPE, P.O. Sox S33S38,Richardson,TX 75083-3S36, U.S.A., fax 01-972-952-9435.

Abstract

None of the currently proposed correlations for bubblepointpressure am particularly accurate.

Knowledge of bubblepoint pressure is one of the importantfactors in the primary and subsequent developments of an oilfield. Bubblepoint pressure is required for material balancecalculations, analysis of well performance, reservoir simulation,and production engineering calculations.

In addition, bubblepoint pressure is an ingredient, eitherdirectly or indirectly, in every oil properly correlation. Thus anerror in bubblepoint pressure will cause errors in estimates of alloil properties. These will propagate additional errors throughoutall reservoir and production engineering calculations.

Bubblepoint pressure correlations use data which aretypically available in the field initial ploducing gas-oil ratio,separator gas specific gravity, stock-tank oil gravity, andreservoir temperature. The lack of accuracy of currentbubblepoint pressure correlations seems to be due to aninadequate description of the process – in short, one or morerelevant variables are missing in these correlations.

We considered three independent means for developingbubblepoint pressure correlations. These are (1) non-linearregression of a model (traditional approach), (2) neural networkmodels, and (3) non-parametric regression (a statisticalapproach which constructs the functional relationship betweendependent and independent variabies, without bias towards aparticular model).

The resuIts, using a variety of techniques (and models),establish a clear bound on the accuracy of bubblepoint pressurecoyelations. Thus, we have a validation of error bounds onbubblepoint pressure correlations.

Introduction

Correlations of the bubblepoint pressures of reservoir oilcommonly use reservoir temperature, stock tank oil gravity,separator gas specific gravity, and solution gas/oil ratio at thebubblepoint as independent variables. These are the only datacommonly available from field observations.

This paper presents a comparative analyses of the correlationof bubble-point pressures for reservoir oils using:

1. Standing-type models1-3

fitted using non-linearregression.

2. Non-parametric regression.4,5

This approach yields a“non-parametric” (point-to-point) regression of thedata, but can also be adapted to provide functionalapproximations for the transforms of the individualvariables.

3. Neural network modeling.A data set consisting of data from 728 laborato~ reservoir

fluid studies (PVT studies) was used to create these threemodels for prediction of reservoir bubblepoint pressure. Theranges of data in this data set maybe found in Table 1.

Standing-Type Models:

A Standing-type model was used to correlate this database. In

particular,’ we present the work of Velarde,3 where the

modification of Standing’s model given by Petrosky2 was used.This modified Standing-type relation is given by

kPb ‘al ;; y~ 10x r

5– ab

...................................... (1)

where,

X=(%T”7)-L3‘4PI”’)........................................... (2)

267

2 W. D. MCCAIN JR., R. B. SOTO, P. P. VALKO, AND T. A. BLASINGAME SPE 51086

This model was fitted to the data set with non-linearregression, the following result was obtained:

Pb = 1091.47 [R$O*1465 i5.354891

&161488 ~@ . ().74()152 .

.........................................................................................(3)

where,

.X= (0.013098 T02!23.72)– $.2 x 10-6 AFY2”176124).(4)

The Velarde relation (Eqs. 3 and 4) reproduced thebubblepoint pressures from the data set used in its creation to anaverage error of 0.6% and an average absolute error of 11.50A.Fig. 1 is a comparison of bubbiepoint pressures predicted by theVelarde relation with the experimental data.

Non-Parametric Regression: (GRACE algorithm4’5)

Parametric regression--that is, regression where a prescribedmodel is fitted to data, is a robust and effective mechanism forrepresenting a data function. However, it provides little insightinto the interrelation of the independent variables, nor does itprovide a “global” minimum expected error of the dependentand independent variables.

The non-parametric regression approach proposed by

Breiman and Friedman,4 and refined by Xue, et al,5 providesexactly such a “non-biased” mechanism for the purpose ofestablishing the minimum error relationship between thedependent - and independent variables. -The method of

Alternating Conditional Expectations (ACE)4 is based on theconcept of developing an optimal transformation of eachvariable--both the dependent variable, as well as theindependent variable(s).

The ACE approach finds individual transformations of theindependent variables (xIJ2,.. .,xn) which are of the form:

-zI=fl (q] z?- = fz (X2), . .Zn = fn (an) ......................

In addition, the ACE approach also finds a transformationfor the dependent variable, y, of the form

20 =fo (y) .....................................................................(6)

The development of these data transforms uses an intuitivelystraightforward requirement--maximize the correlation of thetransformed dependent variabie zo with the transformedindependent variables 21,22,...,zn. The most important aspect ofthe ACE4 (or GRACE5) algorithm is that the datatransformations are constructed pointwise, rather thanfunctional--therefore there is no need to associate a specificalgebraic form with the individual data transformations.

The application of the trivial restrictions of zero mean andunit variance for the individual transformations results inessentially unique solutions. A final requirement is a

“smoothness” for the individual transformations. Although thisoperation is somewhat hidden in the algorithms (where a “datasmoother” code is used), we believe that the ACE algorithm (aswell as the slightly different GRACE algorithm) does in factprovide a unique and robust minimization of the expected error--in a non-parametric sense.

The procedure for this approach is given by1. Calculate the data transforms:

Z1 = fl(xl), 22 = fz(x~, .... Zn =fn(x~ and zo =fo(y).2. Calculate the transform sum z~ = z] + Z2 + . . . + Zn.

3. Calculate the inverse transform: y = f~l (zo).

While the ACE and GRACE algorithms do provide a non-parametric optimization of the dependent and independentvariables, this approach does not provide a computation (i.e.,predictive) model. Figs. 2 through 5 show the optimaltransformations of the individual data for the four independentvariables. The optimal transform of the data for the dependentvariable, bubblepoint pressure, is given in Fig. 6. This “non-parametric” regression results in a reasonable match ofcalculated and measured bubblepoint pressures. Fig, 7 showsthis comparison of calculated and measured bubblepointpressures for the non-parametric optimization.

The ACE and GRACE algorithms do not provide apredictive model, i.e., they do not result in equations. Howeversimple quadratic polynomials can be fitted to the optimal datatmnsforms, resulting in:

Z~ =1.2206 – 1.8029 X10-2 (MY)-3.8682 X10q (API)2 ......

......................................................................................... (7)

22 =3.8649- 6.5080 (yg )+ 1.9670x (yg~ .................. (8)

Z3 = -7.7161 + 2.3604 ~n (T,,,)]- 1.6678x 10-1 [h (T,e,)~ ..

......................................................................................... (9)

24 = –1.0056x l(!l + 1.6664 [in (R.b )].......................(10)

-1.7682 X 10-2 [In (R~b)]2

The inverse transform is defined as:

ZO=ZI+Z2+Z3 +Z4 ...................................................(n)

And the inverse transform, in terms of In@b), is given by

in (pb )= 7.8421 + 4.9506 x 10-1 [h (20 )]

..................(12)-2.5726 x 10-3 ~n (z. )~

The application of these quadratic polynomials to theoptimal transforms determined by non-parametric regressiondoes not seriously degrade the quality of the fit to thebubblepoint pressures. Eqs. 7 through 12 reproduce thebubblepoint pressures in the data set to an average error of 3.1percent and an average absolute error of 11.8 percerit. Fig. 8gives a comparison of bubblepoint pressures calculated with

268

SPE 51086 CORRELATION OF BUBBLEPOINT PRESSURES FOR RESERVOIR OILS--A COMPARATIVE STUDY 3

equations 7 through 12 with the measured bubblepointpressures.

Note that it is not necessary to fit the optimal data transformswith quadratic polynomials -- equations of any fictional formmay be used. However, in this case these simple functions areadequate.

Neural Network Modeling:

MatlabTMwas used to build an “intelligent” software packagehfor data characterization. The software has three integratedmodules: a) a preprocessing module for multivariate statisticalanalysis; b) a neural network module; in this case thebackpropagation algorithm with the Levenberg-Marquardtprocedure as an optimization method for convergence, and c) apost-training analysis module for evaluation of the performanceof the trained network by calculation of the errors for thetraining, validation, and testing data sets.The 728-sample data set was divided into three subsets fortraining, validation and testing. Half the data was used for thetraining subset and one-quarter each for the validation andtesting subsets. The training and validation data subsets areused iteratively to develop the neural network model. Themodel is “trained” with the training data and the validation dataare used to determine when the “training” is optimal. Thequality of model is then tested with the testing data subset.

The final neural network model has four input nodes and thetwo hidden layers with 5 and 5 nodes respectively. A completedescription of the trained neural network model is given inTable 2.

The neural network model showed very good performancefor prediction of bubblepoint pressure. The bubblepoint

pressures predicted by the trained model agreed with the 728sample data set to within 0.3 percent average error and 6.0percent average absolute error. The comparison of calculatedand measured bubblepoint pressures is in Fig. 9.

After the neural network was “trained,” the weight and biasvectors were incorporated into Fortran-90 and Visual Basicinterfaces so that the results could be used in a practical manner.

Table 3 shows a comparison of the quality of fit of the threemodels with the bubblepoint pressures used in their

development.

Validation of the Predictive Models

An independent data set consisting of data from 547 PVTstudies was used to test the three models. None of these datawere included in the 728 sample data set used in developing themodels. Table 4 gives the ranges of data in the independent dataset. Notice that the maxima and minima in the several variablesare very similar in the two data sets (Table 1, Table 4).

The predictive abilities of the three models were tested usingthe independent variables from the independent data set andcomparing the calculated values of bubblepoint pressure withthe measured values. Table 5 gives the results. The non-linearregression and non-parametric models worked reasonably wellon the independent data set--retaintig their accuracy of

approximately 13 percent average absolute error. The averageabsolute error of the neural network model increased from 6percent (for the “original” data) to 25 percent for theindependent data set. Figs. 10 through 12 show the calculated-measured comparison. Especially noteworthy is the comparisonbetween Figs. 9 and 12 which illustrates that a trained neuralnetwork will not necessarily provide accurate predictions fordata not involved in the “training.”

Several other bubblepoint pressure correlations from theliterature were tested with the independent data base. Table ~and Figs. 13 through 15 show how predictions of the Standing,Vasquez and Beggs,7 and Kartoatmodjo and Schmidts equations(all non-linear regression type models) compare with themeasured bubblepoint pressures from the independent data base.

Other proposed bubblepoint pressure correlations2’9’10’1*werealso tested with the independent data set. In each case theaverage absolute error was larger than those given in Table 6.

Unfortunately, the best results for any correlation, whetherprepared by non-linear regression, non-parametric regression, orneural network, can only predict bubblepoint pressure (given thefour commonly available independent variables) to an averageabsolute error of about 13 percent. This means that errors of 25percent or greater are possible for a given situation.

Bubblepoint pressure is used, either directly or indirectly, inall oil property correlations. 12’3Thus the errors in estimates ofbubblepoint pressure will propagate throughout all estimates ofother fluid properties such as oil formation volume factor, oilviscosity, oil density, etc.

Correlations for these other oil fluid properties arereasonably accurate given accurate values of bubblepointpressure. It appears that an accurate bubblepoint pressurecorrelation is not possible (given the usually available inputdata). Thus, two options are available: regular measurement ofaverage reservoir pressure (pressure buildup tests, etc.) orobtaining a representative sample of the original reservok fluidand measuring bubblepoint pressure and other properties in thelaborato~.

Conclusions

1.

2,

3.

269

The best possible correlations of bubblepoint pressure;given the usual input data; solution gas-oil ratio atbubblepoint, stock-tank oil gravity, separator gas specificgravity, and reservoir temperature; are accucate to anaverage absolute error of about 13 percent. This means thatpredicted values of bubblepoint pressure could be in errorby 25 percent or more in some instances.Errors this large will cause unacceptably high errors in theprediction by correlation of the other oil fluid properties ofinterest: oil formation volume factor, oil density, oilviscosity, and oil compressibility.The only options currently available for obtaining accuratevalues of bubblepoint pressure are either regular fieldmeasurement of average reservoir pressures or laboratorymeasurement with a sample representative of the originalreservoir oil.

4 W. D. MCCAIN JR., R. B. SOTO, P. P. VALKO, AND T. A. BLASINGAME SPE 51086

Nomenclature

API =&b =

T=

x=

Xn =Y=

.Zn=

Y23 =

stock-tank gravity, ‘APIsolution gas-oil-ratio at the bubble-point,scf/STBreservoir temperature, ‘Ftemporary variable used in Petrosky2

correlationindependent variables, various unitsdependent variabletransforms of the independent are

dependent variablesgas specific gravity, air = 1

References

1.

2.

3.

Standing, M.B.: “A Pressure-Volume-T&mperatureCorrelation for Mixtures of California Oil and Gases;’Drill. & Prod. Prac., API (1947) 275-87.Petrosky, G.E. and Farshad, F.F.: “Pressure-Volume-Temperature Correlations for Gulf of Mexico Crude Oils,”paper SPE 26644 presented at the 1993 SPE ATCE,Houston, 3-6 October.Velarde, J.J., Blasingame, T.A., and McCain, W.D., Jr.:“Correlation of Black Oil Properties at Pressures BelowBubble Point Pressure - A New Approach,” paper 97-93

4.

5.

6.

7.

8.

9.

104

11.

12<

presented at the 48th ATM of The Petroleum Society,Calgary, 8-11 June 1997.Breiman, L. and Friedman, J.H.: “Estimating OptimalTransformations for multiple Regression and Correlation,”Journal of the American Statistical Association, (September1985), 580-619.Xue, G., Datta-Gupta, A., Valko, P., and Biasingame, T.A.:“Optimal Transformations for Multiple Regression:Application to Permeability Estimation from Well Logs,”SPEFE (June 1997), 85-93.Demuth, H. and Beale, M.” “Neural Network Toolbox forUse With Matlab,” version 3, Mathworks, Inc., Natick, MA(1998).,Vasquez, M.E. and Beggs, H.D.: “Correlations for FluidPhysical Property Prediction,” JPT (June 1980), 968-70.Kartoatmodjo, T. and Schmidt, Z.: “Large Data BankImproves Crude Physical Property Correlations,” Oil andGas Jour. (July 1994), 51-55.Lasater, J.A.: “Bubble-Point Pressure”Correlation,” Trans.,AIME (1958) 213,379-81.Glaso, O.: “Generalized Pressure-Volume-TemperatureCorrelations,” JPT(May 1980), 785-95.A1-Marhoun, M.A.: “PVT Correlations for Middle EastCrude Oils,” JPT (May 1988), 650-666.McCain, W.D., Jr.: The Properties of Reservoir Fluids, 2ndEd., PennWell, Tulsa, OK (199)).

Table 1- Ranges of Data in Data Set Used to 11De{elop p~ Correlations

(728 data sets)

70 < Pb < 6700 psia

74 < T < 327 ‘F

10 < Rsb < 1870 scf/STB

12 < API < 55 “API

0.556 y~< < 1.367 (air= 1)

270

SPE 51086 CORRELATION OF BUBBLEPOINT PRESSURES FOR RESERVOIR OILS--A COMPARATIVE STUDY 5

TABLE 2- Description of the Trained Neural Network

Theneural networkmodelhas the followingform:

p@n~~,li,,d= {w3*.tanh*[w2*tanh_’(w1* zI,.Om,lti,N+bl) + b2] + b3)

And the equationthat used for normalizedthe inputand outputvariablesduringthe trainingprocesswas: .

z@~~li@= 2 * zr- zi,~)/zi Mm- zil~)

Where23representseachof the inputvariables:

Z(1)= reservoirtemperature, Z(2)= API, Z(3)= gas gravity, andZ(4)= GOR.

The maximumand minimumvaluesusedwere:

Maximum

Reservoirtemperature 327API 55Gasgravity 1.367GOR 1870Pob 6700

The valuesfor the bias ( b1,b2 and b3) and for the weights(w1,w2, andw3)are:

ZJl={-3.283148798632070.8024301701303761.08076979327138-2.83265706363712 1.68691436705254}

b2={l.92765824477335 0.37973612690’9036-0.0897286359210449-1.7822122415044 -1.61216953831698}

b3=-0.39379012457655

w1={-O.795250144548125 -1.09209856833608 -0.97316703561262 2.25817004067108

0.0113831825310405 -0.0220112913878158 -0.0755903427477344 0.820670010457579-0.0443089522480293 0.465761383477398 0.897960024862847 0.107674068592107-0.132352081142938 0.0374516997795222 2.82233244250817 -1.050999523703681.61344233199094 -1.56179989187548 -0.595446499153064 0.0780285404462807)

w2={-I.09646815273972 -0.854864648278981 -1.11247171493734 -0.1380206485916960.7597860207174260.077022164481693 -0.671113407900328 0.519448286341125 1.22400502388773= -1.372215707271921.47930665237534 1.13868511497154 1.41822922194687 0.197907870996054 0.790664068001338

-0.62378427951244-1.24915231641691 .95526556348521 -0.4o0971121833293 -0.0963149005694486 _.

-1.12898030283311 -0.795224736513933 -1.09624496663587 o.@3155099188247990.203771.22773846 }

md

~3={-O.25763606379853-0.1027177075256732.50555221016993-0.334182893301272-2.688487244623}

Minimum

7411.60.5561070

TabIe3 -Comparison ofResuIts of the Three Prediction Models WithMeasuredBubblepointPressure fortheDataBase Usedin theDevelopments oftheModels

&he 728 Sample data setAverage Absolute

Average Errorin pbpercent

Non-linear regression-Standing-type 11.4equationNon-parametric regression 3.1 11.8

Neuralnetwork -0.3 6.0

271

6 W. D. MCCAIN JR., 1?.B. SOTO, F’. P. VALKO, AND T. A. BLASINGAME SPE 51086

Table 4- Ranges of Data in the Independent Data SetUsed to Vaiidate the pb Correlations

~

121 < Pb < 5990 psia

88 < T c 320 ‘F

12 < R,~ < 1905 scf/STB

14.4 < API < 56.2 ‘API

0.555 < ‘v.. c 1.245 (air = 1) II

Table 5- Comparison of Results of the Three Prediction Models with Measured~mple data set)

Average AbsoluteAverage Error Error hl pb

percent

Non-linear regression-Standing-type equation 12.8Non-parametric regression 13.0Neural network 25.3

Table 6- Comparison of the Results of Various Bubble-Point Pressure Correlations With Data From

~Standin -6.1 13.2

Vasquez and Beggs7 2.8 13.1

●

6000 1

~f9#; ~b:~::mgw+’e!mw Cormw.n (728 pmt.)

0 1000 2000 3000 4000 Smo 6000

Measured Bubbie-Point Preesure, ph paia

Fig. l.- Comparison of calculated and measured bubblepointpressures for Velarde p~ correlation. Correlation data set.

-1.5~

Stock Tank Oil Gravity, “API

Fig. 2.- Grace optimal transformation of oil gravity ~APl).

272

SPE 51086 CORRELATION OF BUBBLEPOiNTPRESSURES FOR RESERVOIR OILS--A COMPARATIVE STUDY 7

Optimal Transform of ln(T,A

1.0 I,, ,lt, ,1, tml, t.!, mmlt .Vl,tttt,,f,.,uw o,dtmal T,mskmn d l,[Tmd (728 WINS)

0.8(frcm QmM Prcwm)

i=

-’”~4.o 4.2 4.4 4.6 4.8 5.0 5.2 5.4 5.6 5.8 6.0

In(T#, lncF)

Fig. 3.- Grace optimal transformation of reservoir temperature, T.

Optimal Transform of Gas Gravity

20

~ 1.5- t

i~ l.o - .~=

z~ 0.5

a$? 0.0~~~ .0.5

~ .1,0 ~ \ ~%*

z0 ●*

-1.5- %. I.&acdOWMTrmdcm040ssGravity (728 pints) -“ “’

(hum mm+ Plcqrm)

-2.0 l,,,,l,,,~l,,~,ib,,,l-1 ““1~0.5 0.6 0.7 0.8 0.9 1.0 1.1 1,2 1.3 1.4

Gas GMvItY (alr=l .0)

Fig. 4. - Grace optimal transformation of gas specific gravity(air=l .0).

-7~

ln(Ff.b), ln(scUSTB)

Optlmrd Transform of kr(p~

9.0

~

bud Opb’d TmmfmrI cdtnlm (728 MIS)

8.5 mmera- Prcwwn)

S.o

7.5

37.0z~ ~,~

$~ 6.0

5.5

5.0●

4,5~ .“”

40~-6-7-6-5-4-3-2-10 123

Optimal Transform: In@d, Ifl(pda)

Fig. 6.- Grace optimal transformation of bubblepoint pressure, Pb.

0 1000 2000 30”04 ‘f&a 5000 m

Measured Bubble-Point Pressure, pb P&a

Fig. 7.- Comparison of calculated and measured bubblepointpressures for Grace p~ correlation-no model. Correlation data set.

●

i loio Z& 30”20 MO 5000 @M@aaured❑ubble-Point Pressure,pb psla

Fig. 8. - Comparison of calculated and measured bubblepointpressures for Grace p~ correlation-quadratic equation model.Correlation data set.

Fig. 5.- Grace optimal transformation of gas-oil-ratio (Rsb).

273

8 W, D. MCCAIN JR., R. B. SOTO, P. P. VALKO, AND T. A. BLASINGAME SPE 51086

6000 I t,. t I T 1,, s fdi L&. Mud Network Curml.llo. (728 minld 1

o-pQmoo2m 30004000 sooo Oooo

MWIswed Bubb!%-Point Pressure, pb psia

. - Comparison of calculated and measured bubblepoint~r%!lwes for neural network f)b correlation. Correlation data sat.

rilhJ2&3&Q’tOk& &Measured Bubbka-Point Pressure, p~ psia

Fig. 10. - Comparison of calculated and measured bubblepointpressure for Velarde f)b correlation. Independent data set.

w I, 1 1 I I

ol&2ck13&14000 5000~Measurwd Bubble-Point Pressure, ph Psla

Fig. 11. - Comparison of calculated and measured bubblepointpressure for non-parametric Pb correlation. Independent data set.

.’i:iZ?.QOO

-i +aloio2&o&4L”w 5ok EJiLl

Measured Bubble-Point Pressure, ph PSla

Fig. 12. - Comparison of calculated and measured bubblepointpressures fOrneural network Pb COITddiOt’k Independent data Set.

010&2&l L?&340& S&OowoMeasured Bubble-Point Pressure, ph psh

Fig. 13. - Comparison of calculated and measured bubblepointpressure for Standing p&correlation. Independent data set.

rile& 2&03&40& &&A!-aasumd ❑ubble.Point Pressure, ph Psia

Fig. 14. - Comparison of calculated and measured bubblepointpressure for Vasquez and Beggs p~ correlation. Independent dataset.

274

SPE 51086 CORRELATION OF BUBBLEPOINT PRESSURES FOR RESERVOIROILS--ACOMPARATIVESTUDY 9

6000U&am KaliorAlwa’lMchtiu Ctm!nlkm

ControlD.m*M6a(54 paints)

s 5ooo -&$

g 4ooo -

~

n ‘%

L

r., . . . . .

0 looozooo3Wo~- ~Measured Bubble-Point Pressure, ph Psia

Fig. 15. - Comparison of calculated and measured bubblepointpressure for Kartoatmodjo and Schmidt p~ correlation.Independent data St?t.

275