CORRELATION OF BUBBLEPOINT COMPARATIVE STUDY

~ society

of Petroleum Engineers

SPE 51086 CORRELATION OF BUBBLEPOINT PRESSURES FOR RESERVOIR OILS--A COMPARATIVE STUDY 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 and Exhibition held in Pittsburgh, PA, 8-11 November 198S. This paper was selected for presentation by an SPE Pregram Comminee followicg review of information contained in an abstract submitted by the author(s). Contente of the paper, as presentad, have not bean reviewad by the Society of Petroleum Engineers and are subject to correction by the author(s). The material, ae presented, does not necessarily reflect any position of the Society of Petroleum Engineers, its officere, or members. Papers presentad at SPE meetings are subject to publication review by Editorial Committees of the Society of Petroleum Enginaers. Electronic reproduction, distrikwtion, or storage of any part of this paper for commercial purposes without the written consent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations may not be copied. The abstract must centain conspicuous acknowledgment of 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

The resuIts, using a variety of techniques (and models), establish a clear bound on the accuracy of bubblepoint pressure coyelations. Thus, we have a validation of error bounds on bubblepoint pressure correlations. Introduction Correlations of the bubblepoint pressures of reservoir oil commonly use reservoir temperature, stock tank oil gravity, separator gas specific gravity, and solution gas/oil ratio at the bubblepoint as independent variables. These are the only data commonly available from field observations. This paper presents a comparative analyses of the correlation of bubble-point pressures for reservoir oils using:

None of the currently proposed correlations for bubblepoint pressure am particularly accurate. Knowledge of bubblepoint pressure is one of the important factors in the primary and subsequent developments of an oil field. Bubblepoint pressure is required for material balance calculations, analysis of well performance, reservoir simulation, and production engineering calculations. In addition, bubblepoint pressure is an ingredient, either directly or indirectly, in every oil properly correlation. Thus an error in bubblepoint pressure will cause errors in estimates of all oil properties. These will propagate additional errors throughout all reservoir and production engineering calculations. Bubblepoint pressure correlations use data which are typically available in the field initial ploducing gas-oil ratio, separator gas specific gravity, stock-tank oil gravity, and The lack of accuracy of current reservoir temperature. bubblepoint pressure correlations seems to be due to an inadequate description of the process – in short, one or more relevant variables are missing in these correlations. We considered three independent means for developing bubblepoint pressure correlations. These are (1) non-linear regression of a model (traditional approach), (2) neural network models, and (3) non-parametric regression (a statistical approach which constructs the functional relationship between dependent and independent variabies, without bias towards a particular model).

1.

Standing-type regression.

models

1-3

fitted

using

non-linear

4,5

2.

Non-parametric regression. This approach yields a “non-parametric” (point-to-point) regression of the data, but can also be adapted to provide functional approximations for the transforms of the individual variables. 3. Neural network modeling. A data set consisting of data from 728 laborato~ reservoir fluid studies (PVT studies) was used to create these three models for prediction of reservoir bubblepoint pressure. The ranges 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 Pb ‘al

k ;;

y~

10x

–

ab

5

r

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

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

267

2

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

“smoothness” for the individual transformations. Although this operation is somewhat hidden in the algorithms (where a “data smoother” code is used), we believe that the ACE algorithm (as well as the slightly different GRACE algorithm) does in fact provide a unique and robust minimization of the expected error-in a non-parametric sense.

This model was fitted to the data set with non-linear regression, the following result was obtained: &161488

~@

SPE 51086

5.354891

. ().74()152 . Pb = 1091.47 [R$O*1465 i .........................................................................................(3)

where,

The procedure for this approach is given by 1. 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.

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

3.

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

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

While the ACE and GRACE algorithms do provide a nonparametric optimization of the dependent and independent variables, this approach does not provide a computation (i.e., predictive) model. Figs. 2 through 5 show the optimal transformations of the individual data for the four independent variables. The optimal transform of the data for the dependent variable, bubblepoint pressure, is given in Fig. 6. This “nonparametric” regression results in a reasonable match of calculated and measured bubblepoint pressures. Fig, 7 shows this comparison of calculated and measured bubblepoint pressures for the non-parametric optimization. The ACE and GRACE algorithms do not provide a predictive model, i.e., they do not result in equations. However simple quadratic polynomials can be fitted to the optimal data tmnsforms, resulting in:

Non-Parametric Regression: (GRACE algorithm4’5) Parametric regression--that is, regression where a prescribed model is fitted to data, is a robust and effective mechanism for representing a data function. However, it provides little insight into the interrelation of the independent variables, nor does it provide a “global” minimum expected error of the dependent and independent variables. The non-parametric regression approach proposed by Breiman and Friedman,4 and refined by Xue, et al,5 provides exactly such a “non-biased” mechanism for the purpose of establishing the minimum error relationship between the dependent - and independent variables. -The method of Alternating Conditional Expectations (ACE)4 is based on the concept of developing an optimal transformation of each variable--both the dependent variable, as well as the independent variable(s). The ACE approach finds individual transformations of the independent variables (xIJ2,.. .,xn) which are of the form:

Z~

=1.2206 – 1.8029 X 10-2 (MY)-3.8682

X 10q (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

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

=fo

(y)

ZO=ZI+Z2+Z3

.....................................................................(6)

The development of these data transforms uses an intuitively straightforward requirement--maximize the correlation of the transformed dependent variabie zo with the transformed independent variables 21,22,...,zn. The most important aspect of the ACE4 (or GRACE5) algorithm is that the data transformations are constructed pointwise, rather than functional--therefore there is no need to associate a specific algebraic form with the individual data transformations. The application of the trivial restrictions of zero mean and unit variance for the individual transformations results in A final requirement is a essentially unique solutions.

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

The inverse transform is defined as:

In addition, the ACE approach also finds a transformation for the dependent variable, y, of the form 20

X

+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 the optimal transforms determined by non-parametric regression does not seriously degrade the quality of the fit to the bubblepoint pressures. Eqs. 7 through 12 reproduce the bubblepoint pressures in the data set to an average error of 3.1 percent and an average absolute error of 11.8 percerit. Fig. 8 gives a comparison of bubblepoint pressures calculated with

268

SPE 51086

CORRELATION OF BUBBLEPOINT PRESSURES FOR RESERVOIR OILS--A COMPARATIVE STUDY

equations 7 through 12 with the measured bubblepoint pressures. Note that it is not necessary to fit the optimal data transforms with quadratic polynomials -- equations of any fictional form may be used. However, in this case these simple functions are adequate. Neural Network Modeling: MatlabTMwas used to build an “intelligent” software packageh for data characterization. The software has three integrated modules: a) a preprocessing module for multivariate statistical analysis; b) a neural network module; in this case the backpropagation algorithm with the Levenberg-Marquardt procedure as an optimization method for convergence, and c) a post-training analysis module for evaluation of the performance of the trained network by calculation of the errors for the training, validation, and testing data sets. The 728-sample data set was divided into three subsets for training, validation and testing. Half the data was used for the training subset and one-quarter each for the validation and testing subsets. The training and validation data subsets are used iteratively to develop the neural network model. The model is “trained” with the training data and the validation data are used to determine when the “training” is optimal. The quality of model is then tested with the testing data subset. The final neural network model has four input nodes and the two hidden layers with 5 and 5 nodes respectively. A complete description of the trained neural network model is given in Table 2. The neural network model showed very good performance The bubblepoint for prediction of bubblepoint pressure. pressures predicted by the trained model agreed with the 728 sample data set to within 0.3 percent average error and 6.0 percent average absolute error. The comparison of calculated and measured bubblepoint pressures is in Fig. 9. After the neural network was “trained,” the weight and bias vectors were incorporated into Fortran-90 and Visual Basic interfaces so that the results could be used in a practical manner. Table 3 shows a comparison of the quality of fit of the three models with the bubblepoint pressures used in their development. Validation of the Predictive Models An independent data set consisting of data from 547 PVT studies was used to test the three models. None of these data were included in the 728 sample data set used in developing the models. Table 4 gives the ranges of data in the independent data set. Notice that the maxima and minima in the several variables are very similar in the two data sets (Table 1, Table 4). The predictive abilities of the three models were tested using the independent variables from the independent data set and comparing the calculated values of bubblepoint pressure with the measured values. Table 5 gives the results. The non-linear regression and non-parametric models worked reasonably well on the independent data set--retaintig their accuracy of

3

approximately 13 percent average absolute error. The average absolute error of the neural network model increased from 6 percent (for the “original” data) to 25 percent for the independent data set. Figs. 10 through 12 show the calculatedmeasured comparison. Especially noteworthy is the comparison between Figs. 9 and 12 which illustrates that a trained neural network will not necessarily provide accurate predictions for data not involved in the “training.” Several other bubblepoint pressure correlations from the literature 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 the measured bubblepoint pressures from the independent data base. Other proposed bubblepoint pressure correlations2’9’1 0’1*were also tested with the independent data set. In each case the average absolute error was larger than those given in Table 6. Unfortunately, the best results for any correlation, whether prepared by non-linear regression, non-parametric regression, or neural network, can only predict bubblepoint pressure (given the four commonly available independent variables) to an average absolute error of about 13 percent. This means that errors of 25 percent or greater are possible for a given situation. Bubblepoint pressure is used, either directly or indirectly, in all oil property correlations. 12’3Thus the errors in estimates of bubblepoint pressure will propagate throughout all estimates of other fluid properties such as oil formation volume factor, oil viscosity, oil density, etc. Correlations for these other oil fluid properties are reasonably accurate given accurate values of bubblepoint pressure. It appears that an accurate bubblepoint pressure correlation is not possible (given the usually available input data). Thus, two options are available: regular measurement of average reservoir pressure (pressure buildup tests, etc.) or obtaining a representative sample of the original reservok fluid and measuring bubblepoint pressure and other properties in the laborato~. Conclusions 1. The best possible correlations of bubblepoint pressure; given the usual input data; solution gas-oil ratio at bubblepoint, stock-tank oil gravity, separator gas specific gravity, and reservoir temperature; are accucate to an average absolute error of about 13 percent. This means that predicted values of bubblepoint pressure could be in error by 25 percent or more in some instances. 2, Errors this large will cause unacceptably high errors in the prediction by correlation of the other oil fluid properties of interest: oil formation volume factor, oil density, oil viscosity, and oil compressibility. 3. The only options currently available for obtaining accurate values of bubblepoint pressure are either regular field measurement of average reservoir pressures or laboratory measurement with a sample representative of the original reservoir oil.

269

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

4

presented at the 48th ATM of The Petroleum Society, Calgary, 8-11 June 1997. 4. Breiman, L. and Friedman, J.H.: “Estimating Optimal Transformations for multiple Regression and Correlation,” Journal of the American Statistical Association, (September 1985), 580-619. 5. 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. 6. Demuth, H. and Beale, M.” “Neural Network Toolbox for Use With Matlab,” version 3, Mathworks, Inc., Natick, MA (1998)., 7. Vasquez, M.E. and Beggs, H.D.: “Correlations for Fluid Physical Property Prediction,” JPT (June 1980), 968-70. 8. Kartoatmodjo, T. and Schmidt, Z.: “Large Data Bank Improves Crude Physical Property Correlations,” Oil and Gas Jour. (July 1994), 51-55. 9. Lasater, J.A.: “Bubble-Point Pressure”Correlation,” Trans., AIME (1958) 213,379-81. 104 Glaso, O.: “Generalized Pressure-Volume-Temperature Correlations,” JPT(May 1980), 785-95. 11. A1-Marhoun, M.A.: “PVT Correlations for Middle East Crude Oils,” JPT (May 1988), 650-666. 12< McCain, W.D., Jr.: The Properties of Reservoir Fluids, 2nd Ed., PennWell, Tulsa, OK (199)).

Nomenclature

API

&b T=

x= Xn

Y= .Zn=

Y23

= stock-tank gravity, ‘API = solution gas-oil-ratio at the bubble-point, scf/STB reservoir temperature, ‘F temporary variable used in Petrosky2 correlation = independent variables, various units dependent variable are of the independent transforms dependent variables gas specific gravity, air = 1 =

References 1.

2.

3.

SPE 51086

“A Pressure-Volume-T&mperature Standing, M.B.: Correlation for Mixtures of California Oil and Gases;’ Drill. & Prod. Prac., API (1947) 275-87. “Pressure-VolumePetrosky, G.E. and Farshad, F.F.: 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 Below Bubble Point Pressure - A New Approach,” paper 97-93

Table 1- Ranges of Data in Data Set Used to De{elop p~ Correlations (728 data sets) 70