SPE-68233-MS

Copyright 2001, Society of Petroleum Engineers Inc. This paper was prepared for presentation at the 2001 SPE Middle East Oil Show held in Bahrain, 17–20 March 2001. This paper was selected for presentation by an SPE Program Committee following review of information contained in an abstract submitted by the author(s). Contents of the paper, as presented, have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material, as presented, does not necessarily reflect any position of the Society of Petroleum Engineers, its officers, or members. Papers presented at SPE meetings are subject to publication review by Editorial Committees of the Society of Petroleum Engineers. Electronic reproduction, distribution, 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 contain conspicuous acknowledgment of where and by whom the paper was presented. Write Librarian, SPE, P.O. Box 833836, Richardson, TX 75083-3836, U.S.A., fax 01-972-952-9435.

Abstract

Reservoir fluid properties are very important in reservoir engineering computations such as material balance calculations, well test analysis, reserve estimates, and numerical reservoir simulations. Ideally, these properties should be obtained from actual measurements. Quite often, however, these measurements are either not available, or very costly to obtain. In such cases, empirically derived correlations are used to predict the needed properties. All computations, therefore, will depend on the accuracy of the correlations used for predicting the fluid properties.

This study presents Artificial Neural Networks (ANN) model for predicting the formation volume factor at the bubble point pressure. The model is developed using 803 published data from the Middle East, Malaysia, Colombia, and Gulf of Mexico fields. One-half of the data was used to train the ANN models, one quarter to cross-validate the relationships established during the training process and the remaining one quarter to test the models to evaluate their accuracy and trend stability. The results show that the developed model provides better predictions and higher accuracy than the published empirical correlations. The present model provides predictions of the formation volume factor at the bubble point pressure with an absolute average percent error of 1.789%, a standard deviation of 2.2053% and correlation coefficient of 0.988. Trend tests were performed to check the behavior of the predicted values of Bob for any change in reservoir temperature, Gas Oil Ratio (GOR), gas gravity and oil gravity. The trends were found to obey the physical laws.

Introduction Reservoir fluid properties are very important in petroleum engineering computations, such as material balance calculations, well test analysis, reserve estimates, inflow performance calculations and numerical reservoir simulations. Ideally, these properties are determined from laboratory studies on samples collected from the bottom of the wellbore or at the surface. Such experimental data are, however, not always available or very costly to obtain. Then, the solution is to use the empirically derived correlations to predict PVT properties. There are many empirical correlations for predicting PVT properties, most of them were developed using linear or non-linear multiple regression or graphical techniques. Each correlation was developed for a certain range of reservoir fluid characteristics and geographical area with similar fluid compositions and API gravity. Thus, the accuracy of such correlations is critical and it is not often known in advance.

Among those PVT properties is the bubble point Oil Formation Volume Factor (Bob), which is defined as the volume of reservoir oil that would be occupied by one stock tank barrel oil plus any dissolved gas at the bubble point pressure and reservoir temperature. Precise prediction of Bob is very important in reservoir and production computations. The objective of this study is to develop a new predictive model for Bob based on Artificial Neural Networks (ANN) using worldwide experimental PVT data.

A new algorithm for training feed forward neural networks was used. That algorithm was found to be faster and more stable than other schemes reported in the literature. Database of 803 published data from the Middle East, Malaysia, and Gulf of Mexico fields was used to develop the present model. Of the 803 data points, 402 were used to train the ANN models, 201 to cross-validate the relationships established during the training process and the remaining 200 to test the model to evaluate its accuracy and trend stability. Using the same 200 data points, several empirical correlations were used to predict Bob. The results show that the present model outperforms all the existing models in terms of absolute average percent error, standard deviation, and correlation coefficient.

SPE 68233

Prediction of Oil PVT Properties Using Neural Networks E.A. Osman, SPE, O.A. Abdel-Wahhab, and M.A. Al-Marhoun, SPE, King Fahd University of Petroleum and Minerals, Saudi Arabia

2 EL-SAYED A. OSMAN, OSAMA A. ABDEL-WAHHAB AND MOHAMMED AL-MARHOUN SPE 68233

Empirical Models and Evaluation Studies For the last 60 years engineers realized the importance of developing and using empirical correlations for PVT properties. Studies carried out in this field resulted in the development of new correlations. Standing 1,3 presented correlations for bubble point pressure and for oil formation volume factor. Standing’s correlations were based on laboratory experiments carried out on 105 samples from 22 different crude oils in California. Katz2 presented five methods for predicting the reservoir oil shrinkage. Vazquez and Beggs4 presented correlations for oil formation volume factor. They divided oil mixtures into two groups, above and below thirty degrees API gravity. More than 6000 data points from 600 laboratory measurements were used in developing the correlations. Glaso5 developed correlation for formation volume factor using 45 oil samples from North Sea hydrocarbon mixtures. Al-Marhoun6 published correlations for estimating bubble point pressure and oil formation volume factor for the Middle East oils. He used 160 data sets from 69 Middle Eastern reservoirs to develop the correlation. Abdul-Majeed and Salman7 published an oil formation volume factor correlation based on 420 data sets. Their model is similar to that of Al-Marhoun6 oil formation volume factor correlation with new calculated coefficients.

Labedi8 presented correlations for oil formation volume factor for African crude oils. He used 97 data sets from Libya, 28 sets from Nigeria, and 4 sets from Angola to develop his correlations. Dokla and Osman9 published set of correlations for estimating bubble point pressure and oil formation volume factor for UAE crudes. They used 51 data sets to calculate new coefficients for Al-Marhoun6 Middle East models. Al-Yousef and Al-Marhoun10 pointed out that the Dokla and Osman9,11 bubble point pressure correlation was found to contradict the physical laws. In 1992, Al-Marhoun12 published a second correlation for oil formation volume factor. The correlation was developed with 11,728 experimentally obtained formation volume factors at, above, and below bubble point pressure. The data set represents samples from more than 700 reservoirs from all over the world, mostly from Middle East and North America.

Macary and El-Batanoney13 presented correlations for bubble point pressure and oil formation volume factor. They used 90 data sets from 30 independent reservoirs in the Gulf of Suez to develop the correlations. The new correlations were tested against other Egyptian data of Saleh et al.14, and showed improvement over published correlations. Omar and Todd15 presented oil formation volume factor correlation, based on Standing1 model. Their correlation was based on 93 data sets from Malaysian oil reservoirs. In 1993, Petrosky and Farshad16 developed new correlations for Gulf of Mexico crude oils. Standing1 correlations for bubble point pressure, solution gas oil ratio, and oil formation volume factor were taken as a basis for developing their new correlation coefficients. Ninety data sets from Gulf of Mexico were used in developing these correlations.

Kartoatmodjo and Schmidt17 used a global data bank to develop new correlations for all PVT properties. Data from 740 different crude oil samples gathered from all over the world provided 5392 data sets for the correlation development. Al-Mehaideb18 published a new set of correlations for UAE crudes using 62 data sets from UAE reservoirs. These correlations were developed for bubble point pressure and oil formation volume factor. The bubble point pressure correlation like Omar and Todd15 uses the oil formation volume factor as input in addition to oil gravity, gas gravity, solution gas oil ratio, and reservoir temperature.

Saleh et al.14 evaluated the empirical correlations for Egyptian oils. They reported that Standing1 correlation was the best for oil formation volume factor. Sutton and Farshad19, 20 published an evaluation for Gulf of Mexico crude oils. They used 285 data sets for gas-saturated oil and 134 data sets for undersaturated oil representing 31 different crude oils and natural gas systems. The results show that Glaso5 correlation for oil formation volume factor perform the best for most of the data of the study. Later, Petrosky and Farshad16 published a new correlation based on Gulf of Mexico crudes. They reported that the best performing published correlation for oil formation volume is Al-Marhoun6 correlation. McCain21 published an evaluation of all reservoir properties correlations based on a large global database. He recommended Standing1 correlations for formation volume factor at and below bubble point pressure.

Ghetto et al.22 performed a comprehensive study on PVT properties correlation based on 195 global data sets collected from the Mediterranean Basin, Africa, Middle East, and the North Sea reservoirs. They recommended Vazquez and Beggs4 correlation for the oil formation volume factor. Elsharkawy et al.23 evaluated PVT correlations for Kuwaiti crude oils using 44 samples. Standing1 correlation gave the best results for bubble point pressure while Al-Marhoun6 oil formation volume factor correlation performed satisfactory. Mahmood and Al-Marhoun24 presented an evaluation of PVT correlations for Pakistani crude oils. They used 166 data sets from 22 different crude samples for the evaluation. Al-Marhoun12 oil formation volume factor correlation gave the best results. The bubble point pressure errors reported in this study, for all correlations, are among the highest reported in the literature. Hanafy et al.25 published a study to evaluate the most accurate correlation to apply to Egyptian crude oils. For formation volume factor Macary and El-Batanoney13 correlation showed an average absolute error of 4.9% while Dokla and Osman9 showed 3.9%. The study strongly supports the approach of developing a local correlation versus a global correlation.

Al-Fattah and Al-Marhoun26 published an evaluation of all available oil formation volume factor correlations. They used 674 data sets from published literature. They found that Al-Marhoun12 correlation has the least error for global data set. Also, they performed trend tests to evaluate the models’ physical behavior. Finally, Al-Shammasi27 evaluated the published correlations and neural network models for bubble point pressure and oil formation volume factor for accuracy

SPE 68233 PREDICTION OF OIL PVT PROPERTIES USING NEURAL NETWORKS 3

and flexibility to represent hydrocarbon mixtures from different geographical locations worldwide. He presented a new correlation for bubble point pressure based on global data of 1661 published and 48 unpublished data sets. Also, he presented neural network models and compared their performance to numerical correlations. He concluded that statistical and trend performance analysis showed that some of the correlations violate the physical behavior of hydrocarbon fluid properties. Published neural network models miss major model parameters to be reproduced.

Neural Network Models Artificial neural networks are parallel-distributed information processing models that can recognize highly complex patterns within available data. In recent years, neural network have gained popularity in petroleum applications. Many authors discussed the applications of neural network in petroleum engineering28-32. Few studies were carried out to model PVT properties using neural networks. In 1996, Gharbi and Elsharkawy33 published neural network models for estimating bubble point pressure and oil formation volume factor for Middle East crude oils. They used two hidden layers neural networks to model each property separately. The bubble point pressure model had eight neurons in the first layer and four neurons in the second. The formation volume factor model had six neurons in both layers. Both models were trained using 498 data sets collected from the literature and unpublished sources. The models were tested by other 22 data points from the Middle East. The results showed improvement over the conventional correlation methods with reduction in the average error for the bubble point pressure oil formation volume factor.

Gharbi and Elsharkawy34 presented another neural network model for estimating bubble point pressure and oil formation volume factor for universal use. They used three-layer neural network model to predict the two properties. They developed the model using 5200 data sets collected from all over the world representing 350 different crude oils. Another set of data consisting of 234 data sets was used for verifying the results of the model. The reported results for the universal model showed less improvement than the Middle East neural model over the conventional correlations. The bubble point pressure average error was lower than that of the conventional correlations for both training and test data. The oil formation volume factor on the other hand was better than conventional correlations in terms of correlation coefficient. The average error for the neural network model is similar to conventional correlations for training data and higher for test data than the best performing conventional correlation.

Elsharkawy35 presented a new technique to model the behavior of crude oil and natural gas systems using a radial basis function neural network model (RBFNM). The model can predict oil formation volume factor, solution gas-oil ratio, oil viscosity, saturated oil density, undersaturated oil compressibility, and evolved gas gravity. He used differential PVT data of ninety samples for training and another ten novel

samples for testing the model. Input data to the RBFNM were reservoir pressure, temperature, stock tank oil gravity, and separator gas gravity. Accuracy of the model in predicting the solution gas oil ratio, oil formation volume factor, oil viscosity, oil density, undersaturated oil compressibility and evolved gas gravity was compared for training and testing samples to all published correlations. The comparison shows that the proposed model is much more accurate than these correlations in predicting the properties of the oils. The behavior of the model in capturing the physical trend of the PVT data was also checked against experimentally measured PVT properties of the test samples. He concluded that although, the model was developed for specific crude oil and gas system, the idea of using neural network to model behavior of reservoir fluid can be extended to other crude oil and gas systems as a substitute to PVT correlations that were developed by conventional regression techniques.

Finally, Varotsis et al.36 presented a novel approach for predicting the complete PVT behavior of reservoir oils and gas condensates using Artificial Neural Network (ANN). The method uses key measurements that can be performed rapidly either in the lab or at the well site as input to an ANN. The ANN was trained by a PVT studies database of over 650 reservoir fluids originating from all parts of the world. Tests of the trained ANN architecture utilizing a validation set of PVT studies indicate that, for all fluid types, most PVT property estimates can be obtained with a very low mean relative error of 0.5-2.5%, with no data set having a relative error in excess of 5%. This level of error is considered better than that provided by tuned Equation of State (EOS) models, which are currently in common use for the estimation of reservoir fluid properties. In addition to improved accuracy, the proposed ANN architecture avoids the ambiguity and numerical difficulties inherent to EOS models and provides for continuous improvements by the enrichment of the ANN training database with additional data.

Data Acquisition and Analysis Data used for this work are collected from published sources. After dropping the repeated data sets, we ended up with 803 data sets as follows: Katz2 (53), Vazquez and Beggs4 (254), Glaso5 (41), Ghetto et al.22 (173), Omar and Todd15 (93), Gharbi and Elsharkawy33 (22), and Farshad et al.37 (146). Each data set contains reservoir temperature, oil gravity, total solution gas oil ratio, and average gas gravity, bubble point pressure and oil formation volume factor at the bubble point pressure. The repeated data sets were reported by other investigators26,27. Of the 803 data points, 403 were used to train the ANN models, the remaining 200 to cross-validate the relationships established during the training process and 200 to test the model to evaluate its accuracy and trend stability. A statistical description of training and test data are given in Table 1 and Table 2, respectively.

4 EL-SAYED A. OSMAN, OSAMA A. ABDEL-WAHHAB AND MOHAMMED AL-MARHOUN SPE 68233

Neural Networks An artificial neural network is a computer model that attempts to mimic simple biological learning processes and simulate specific functions of human nervous system. It is an adaptive, parallel information processing system, which is able to develop associations, transformations or mappings between objects or data. It is also the most popular intelligent technique for pattern recognition to date. The basic elements of a neural network are the neurons and their connection strengths (weights). Given a topology of the network structure expressing how the neurons (the processing elements) are connected, a learning algorithm takes an initial model with some “prior” connection weights (usually random numbers) and produces a final model by numerical iterations. Hence “learning” implies the derivation of the “posterior” connection weights when a performance criterion is matched (e.g. the mean square error is below a certain tolerance value). Learning can be performed by “supervised” or “unsupervised” algorithm. The former requires a set of known input-output data patterns (or training patterns), while the latter requires only the input patterns. This is commonly known as the feed forward model, in which no lateral or backward connections are used38. Advantages of Artificial Neural Networks Several advantages can be attributed to ANNs rendering them suitable to applications such as considered here. Firstly, an ANN learns the behavior of a database population by self-tuning its parameters in such a way that the trained ANN matches the employed data accurately. Secondly, if the data used are sufficiently descriptive39, the ANN provides a rapid and confident prediction as soon as a new case, which has not been “seen” by the model during the training phase, is applied. Possibly, the most important aspect of ANNs is their ability to discover patterns in data that are so obscure as to be imperceptible to normal observation and standard statistical methods. This is particularly the case for data exhibiting significantly unpredictable nonlinearities40. Traditional correlations are based on simple models which often have to be stretched by adding terms and constants in order for them to become flexible enough to fit experimental data, whereas neural networks are marvelously self-adaptable. Using a sufficiently large database for training, ANNs allow property values to be accurately predicted over a very wide range of input data36.

An ANN model can accept substantially more information as input to the model, thereby, improving significantly the accuracy of the predictions and reducing the ambiguity of the requested relationship. Moreover, ANNs are fast-responding systems. Once the model has been “educated” predictions about unknown fluids are obtained with direct and rapid calculations without the need for tuning or iterative computations. Furthermore, an outstanding attribute of the ANNs is their capability of becoming increasingly “expert” by retraining them using larger databases. Continuous enrichment of the ANN “knowledge” eventually leads to a predictive model exhibiting accuracy comparable to the PVT data itself36.

Neural Network Architecture In this study, a backpropagation network (BPN) is used. A backpropagation network is multi-layered and information flows from the input to the output through at least one hidden/middle layer. Each layer contains neurons that are connected to all neurons in the neighboring layers. The connections have numerical values (weights) associated with them. During the training phase, the weights are adjusted according to the generalized delta rule. Training is completed when the network is able to predict the given output. A new algorithm was used to train the three-layer network41. The first layer consists of four neurons representing the input values of reservoir temperature, solution gas oil ratio, gas specific gravity and API oil gravity. The second (hidden) layer consists of 5 neurons, and the third layer contains one neuron representing the output values of the bubble-point formation volume factor Bob. A simplified schematic of the used neural network is illustrated in Fig. 1. Details of the learning algorithm are given in Appendix A.

The data were divided into two groups: training group (603 data sets) and testing group (200 data sets). The training group is split into two groups: the first (402 data sets) was used to train the network; the second set was used to test the error during the training, this was called cross validation. It gives the ability to monitor the generalization performance of the network and prevent the network to over fit the training data38. In a BPN, the input activity is transmitted forward while the error is propagated backwards. The neurons in the BPN use a transfer function that is sigmoid or S shaped. A key feature of the sigmoid function is that it has a minimum value of 0 and a maximum value of 1 and is differentiable everywhere with a positive slope. The derivative of the transfer function is required to calculate the error that is backpropagated and the derivative of the sigmoid function is easy to calculate.

In designing the neural network, many important parameters that will control its overall performance such as the leaning constant and the number of middle layer neurons. The learning constant must be kept sufficiently low enough to ensure good training with minimum oscillations without compromising on the speed of the training procedure. The exponential decline method of varying the learning constant is an efficient process to obtain a well-trained network. The exercises involving the middle layer neurons have shown that for case studies involving little or no noise, using the least possible number (usually 1 or 2) of middle layer neurons result in a well trained network.

Over-training a network must be avoided and it is important to frequently monitor the error as training progresses. It has been shown that over training a network causes the network to memorize results rather than generalize. Then, the resulted model can perfectly predict the data similar to training data, but it will perform badly if new cases submitted to the network. The cross-validation method used in this study utilized as a checking mechanism in the training algorithm to prevent over-training.

SPE 68233 PREDICTION OF OIL PVT PROPERTIES USING NEURAL NETWORKS 5

Statistical Error Analysis: To compare the performance and accuracy of the new model to other empirical correlations, statistical error analysis is performed. The statistical parameters used for comparison are: average percent relative error, average absolute percent relative error, minimum and maximum absolute percent error, root mean square and the correlation coefficient. Equations for those parameters are given below: 1. Average Percent Relative Error: It is the measure of the relative deviation from the experimental data, defined by:

∑=

=N

iir E

nE

1

1 (1)

Where iE is the relative deviation of an estimated value from an experimental value

( ) ( )( ) 100

exp

exp ×

−=

ob

estobobi B

BBE i = 1, 2, ………n (2)

2. Average Absolute Percent Relative Error: It measures the relative absolute deviation from the experimental values, defined by:

∑=

=n

iia E

nE

1

1 (3)

3. Minimum and Maximum Absolute Percent Relative Error: To define the range of error for each correlation, the calculated absolute percent relative error values are scanned to determine the minimum and maximum values. They are defined by:

i

n

iEE min

1min

== (4)

i

n

iEE max

1max

== (5)

4. Root Mean Square Error: Measures the data dispersion around zero deviation, defined by:

21

1

21

= ∑=

n

iiE

nRMS (6)

5. The Correlation coefficient: It represents the degree of success in reducing the standard deviation by regression analysis, defined by:

( ) ( )[ ] ( )∑ ∑= =

−−−=n

i

n

iobobiestobob BBBBr

1 1

_

exp2

exp /1

(7)

Where

( )[ ]i

n

iobob B

nB ∑

==

1exp

_ 1 (8)

Results and Discussion After training the neural networks, the model becomes ready for testing and evaluation. To perform this, the last data group (200 data sets), which was not seen by the neural network during training, was used. To compare the performance and accuracy of the new model to other empirical correlations, five correlations were selected. Those are: Standing (1947)1, Vazquez and Beggs (1980)4, Glaso (1980)5, Al-Marhoun (1988)6, Al-Marhoun (1992)12. Equations describing those models are found in Appendix B. The statistical results of the comparison are given in Table 3. The Artificial Neural Network Model outperforms all the empirical correlations. The proposed model showed high accuracy in predicting the Bob values, and achieved the lowest absolute percent relative error, lowest minimum error, lowest maximum error, lowest root mean square error, and the highest correlation coefficient among other correlations.

The absolute percent relative error is an important indicator of the accuracy of the models. Figure 2 shows a comparison of the absolute percent relative error for all correlations. Its value for ANN was 1.7886%, while other correlations indicates higher error values of 2.2053% for Al-Marhoun (1992), 2.334 for Al-Marhoun (1988), 2.7238% for Standing1, 2.9755% for Vazquez and Beggs4, and 3.3743 for Glaso5 Correlation. Figure 3 shows a comparison of the correlation coefficient for all correlations. ANN achieves the highest correlation coefficient of 98.78%, while other values range between 97.15% for Glaso5, and 98.42% for Vazquez and Beggs4 correlation. It should be noted that the ANN model outperforms other empirical correlations despite the fact that the ANN model did not see the testing data during training. On the other hand, some of these data sets were already used in developing the other empirical correlations. The higher accuracy of the predicted results indicates that the neural network was successfully trained. Also, these results demonstrated the efficiency of the training algorithm.

Figures 4-9 illustrate scatter diagrams of the predicted versus experimental Bob values. These cross plots indicates the degree of agreement between the experimental and the predicted values. If the agreement is perfect, then all points should lie on the 45º degrees line on the plot. Compared to other cross plots, Fig. 9 shows the most tight cloud of points around the 45º degrees line indicating the excellent agreement between the experimental and the calculated data values. The most scattered points were found in Fig. 4, representing Standing1 Correlation, and Fig. 6, which represent Glaso5

6 EL-SAYED A. OSMAN, OSAMA A. ABDEL-WAHHAB AND MOHAMMED AL-MARHOUN SPE 68233

correlation, indicating their poor performance for this set of data. Again, this indicates the superior performance of the ANN model compared to other empirical correlations.

Trend Analysis The above discussion shows that the developed model outperforms other empirical correlations in predicting Bob values. Now, the model must be tested to see whether it is physically correct or not, and to make sure that the model is stable. In order to perform these tests, trend tests must be conducted. The model was tested using hypothetical intermediate data points, and the dependence of Bob on solution gas oil ratio (Rs), reservoir temperature (T), oil gravity (γo) and gas relative density (γg) was studied. To study the effect of solution gas oil ratio Rs on Bob, Rs was varied between 20 and 4000 SCF/STB, while fixed values of other parameters were used ( T = 200ºF, γg = 0.9, γo = 0.85 (API=34.95) ). Fig. 10 illustrates the performance of the ANN model and other empirical correlations. The graph demonstrates that Bob is an increasing function of Rs, and the results of the ANN model are stable. To study the effect of reservoir temperature T on Bob, temperature was varied between 80 and 400ºF, while fixed values of other parameters were used (Rs = 500 SCF/STB, γg = 0.9, γo = 0.85 (API=34.95)). Fig. 11 illustrates the performance of the ANN model and other empirical correlations. The graph demonstrates that Bob is an increasing function of T, and that the results of the ANN model are stable.

To study the effect of oil gravity γo on Bob, γo was varied between 20 and 56 API, while fixed values of other parameters were used (T = 200ºF, γg = 0.9, Rs = 500 SCF/STB). Fig. 12 illustrates the performance of the ANN model and other empirical correlations. Again, the plot indicates that Bob is an increasing function of API oil gravity (decreasing function of oil relative density), and the results of the ANN model are stable. The break in Vazquez and Beggs4 correlation curve occurs at API = 30, which is the criterion defined by the authors to apply either equation to predict Bob

4. Finally, the effect of gas relative density (γg) on Bob was studied. γg was varied between 0.5 and 1.5, while fixed values of other parameters were used (Rs = 500 SCF/STB, T = 200ºF, γo = 0.85 (API=34.95) ). Fig. 13 illustrates the performance of the ANN model and other empirical correlations. The plot demonstrates that except for Al-Marhoun 19926 correlation, Bob is an increasing function of γg. Also, the results of the ANN model are stable but the function increased slower than the correlations of Standing1, Vazquez and Beggs4, and Al-Marhoun 19886. Conclusions

1. A new model was developed to predict the oil formation volume factor at the bubble-point pressure. The model was based on artificial neural networks, and developed using 803 published data sets from the Middle East, Malaysia, Colombia, and Gulf of Mexico fields.

2. Of the 803 data sets, 403 were used to train the ANN model, 200 to cross-validate the relationships established during the training process and the remaining 200 to test the model to evaluate its accuracy.

3. A new algorithm was used to train a feedforward three-layer network. The first layer consists of four neurons representing the input values of reservoir temperature, solution gas oil ratio, gas specific gravity and API oil gravity. The second (hidden) layer consists of 5 neurons, and the third layer contains one neuron representing the output values of the bubble-point formation volume factor Bob.

4. The results show that the developed model provides better predictions and higher accuracy than the published empirical correlations. The present model provides predictions of the formation volume factor at the bubble point pressure with an absolute average percent error of 1.789%, and correlation coefficient of 0.988.

5. Trend analysis was performed to check the behavior of the predicted values of Bob for any change in reservoir temperature, solution gas oil ratio (Rs), gas gravity and oil gravity. The model was found to be physically correct. The stability of the model indicated that the neural network model does not over fit the data, which implies that it was successfully trained.

6. Incorporating additional data sets during training and cross-validation stages can further refine the new model to cover a wider range of input variables. Nomenclature Bob = OFVF at the bubble- point pressure, RB/STB Rs = on solution gas oil ratio, SCF/STB T = reservoir temperature, degrees Fahrenheit γo = oil relative density (water=1.0) γg = gas relative density (air=1.0) Er = average percent relative error Ei = percent relative error Ea = average absolute percent relative error Emax = Maximum absolute percent relative error Emin = Minimum absolute percent relative error RMS = root mean square error r = correlation coefficient Acknowledgement The authors wish to thank King Fahd University of Petroleum and Minerals for the facilities utilized to perform the present work and for their support.

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SPE 68233 PREDICTION OF OIL PVT PROPERTIES USING NEURAL NETWORKS 7

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the 6th Abu Dhabi International Petroleum Exhibition & Conference, Oct. 16-19, 1994.

24. Mahmood, M. M. and Al-Marhoun, M. A.: “Evaluation of empirically derived PVT properties for Pakistani crude oils,” J. Pet. Sci. & Engg. 16 (1996) 275.

25. Hanafy, H.H., Macary, S.A., Elnady, Y. M., Bayomi, A.A. and El-Batanoney, M.H.: “ Empirical PVT Correlation Applied to Egyptian Crude Oils Exemplify Significance of Using Regional Correlations,” paper SPE 37295 presented at the SPE Oilfield Chemistry International Symposium, Houston, Feb. 18–21, 1997.

26. Al-Fattah, S. M. and Al-Marhoun, M. A.: “Evaluation of empirical correlation for bubble point oil formation volume factor,” J. Pet. Sci. & Engg. 11(1994) 341.

27. Al-Shammasi, A.A.,: “Bubble Point Pressure and Oil Formation Volume Factor Correlations,” paper SPE 53185 presented at the 1997 SPE Middle East Oil Show and Conference, Bahrain, March 15–18.

28. Kumoluyi, A.O. and Daltaban, T.S.: “High-Order Neural Networks in Petroleum Engineering,” paper SPE 27905 presented at the 1994 SPE Western Regional Meeting, Longbeach, California, USA, March 23-25.

29. Ali, J. K.: “Neural Networks: A New Tool for the Petroleum Industry,” paper SPE 27561 presented at the 1994 European Petroleum Computer Conference, Aberdeen, U.K., March 15-17.

30. Mohaghegh, S. and Ameri, S.,:" A Artificial Neural Network As A Valuable Tool For Petroleum Engineers," SPE 29220, unsolicited paper for Society of Petroleum Engineers, 1994.

31. Mohaghegh, S.:" Neural Networks: What it Can do for Petroleum Engineers," JPT, (Jan. 1995) 42.

32. Mohaghegh, S.:" Virtual Intelligence Applications in Petroleum Engineering: Part 1 - Artificial Neural Networks,” JPT (September 2000).

33. Gharbi, R.B. and Elsharkawy, A.M.: “Neural-Network Model for Estimating the PVT Properties of Middle East Crude Oils,” paper SPE 37695 presented at the 1997 SPE Middle East Oil Show and Conference, Bahrain, March 15–18.

34. Gharbi, R.B. and Elsharkawy, A.M.: “Universal Neural-Network Model for Estimating the PVT Properties of Crude Oils,” paper SPE 38099 presented at the 1997 SPE Asia Pacific Oil & Gas Conference, Kuala Lumpur, Malaysia, April 14-16.

35. Elsharkawy, A.M.: “Modeling the Properties of Crude Oil and Gas Systems Using RBF Network,” paper SPE 49961 presented at the 1998 SPE Asia Pacific Oil & Gas Conference, Perth, Australia, October 12-14.

36. Varotsis N., Gaganis V., Nighswander J., and Guieze P.,: “A Novel Non-Iterative Method for the Prediction of the PVT Behavior of Reservoir Fluids,” paper SPE 56745 presented at the 1999 SPE Annual Technical Conference and Exhibition, Houston, Texas, October 3–6.

37. Farshad, F.F, Leblance, J.L, Garber, J.D. and Osorio, J.G.: “Empirical Correlation for Colombian Crude Oils,” paper SEP 24538, Unsolicited (1992), Available from SPE book Order Dep., Richardson, TX.

38. Bishop, C.: Neural Networks for Pattern Recognition, Oxford University Press, NY (1995).

39. Fauset, L.: Fundamentals of Neural Networks, Prentice Hall, NJ, USA, 1996.

8 EL-SAYED A. OSMAN, OSAMA A. ABDEL-WAHHAB AND MOHAMMED AL-MARHOUN SPE 68233

40. Hornik, K., “Multilayer Feedforward Networks are Universal Approximators”, Neural Networks, (1989), Vol. 2, 359.

41. Abdel-Wahhab, O. and Sid-Ahmed, M.A.: “A New Scheme for Training Feedforward Neural Networks,” Pattern Recognition, (Feb. 1998) 15.

Appendix A: Training Algorithm Used in the Present Study: In Abdel-Wahhab and Sid-Ahmed41, a new algorithm for training feed-forward neural networks was developed. This algorithm is shown to be faster and more stable than other schemes presented in the literature. The algorithm is summarized below. Figure 10 in the original paper is used for notations. 1. Initialization:

• Randomize all weights in the network.

• For layers j = 1 through L, initialize the mj×mj matrices Sj

by small non-zero random number. Where mj is the length

of the vectors xj-1.

• Equate the node offsets xj-1,0 of every node to some non-

zero constant for layers j = 1 through L.

2. Choose a training pattern:

• Randomly select an input/output pair (x0, o) to present to

the network.

• For each layer j, and for every node k, calculate the

summation output

y x wjk j i jkii

N

= −=∑ ( . ),1

0

(A-1)

the function output

[ ][ ]f y

a y

a yjk

jk

jk

( )( . )

( . )=

− −

+ −

1

1

exp

exp (A-2)

and the derivative of f( yjk)

f ya a y

a yjk

jk

jk

' ( ). ( . )

( . )=

−

+ −

2

12

exp

exp. (A-3)

Here, N is the number of inputs to a node, and the constant a is

the sigmoid slope.

3. Invoke the unweighted gain equations:

• For each layer j from 1 through L, the predicted residual

variance inverse is given by

( )σjj j

Tj j jb

=( ) ( ) +

1

x S x S (A-4)

• and the unweighted Kalman gain is given by

kj = Sj (xj Sj) (A-5)

4. Update the inverse matrix:

• Calculate:

SS k x S

jj j j j

T

jb+j =

- ( ) ( ) 1

γ (A-6)

where γσ

σjj

j

=+1

(A-7)

5. Back-propagate error signals:

• Calculate error signals, for every node k.

= ( ) ( - )e f y o xLk Lk k Lk′ for the output layer, L. (A-8)

e f y e wjk jk j i j i ki

= + +∑'

, , ,( ) ( )1 1 for the hidden

layers from L-1 to 1 (A-9)

6. Find the desired summation output:

• For every node, k, in the output layer, calculate the

desired summation output by using the inverse function

da

ook

k

k

= +−

1 11

ln (A-10)

6. Update the weights::

For every node, k,

• The weight vectors in the output layer L are updated by

wLk = wLk + kL (dk - yLk) (A-11)

• For each hidden layer j = 1 through L - 1, weight vectors

are updated by

wjk = wjk + kj ejk µj. (A-12)

8. Stopping criteria:

• Use the mean-square error of the network output as a

convergence test, or

• Run the algorithm for a fixed number of iterations, or

• Split the data into 2 sets: one set to train the network and

the other set to test the error. This method is called cross

SPE 68233 PREDICTION OF OIL PVT PROPERTIES USING NEURAL NETWORKS 9

validation. It gives the ability to monitor the generalization

performance of the network and prevent the network to over

fit the training data.

Appendix B: Empirical PVT Correlations Used for Comparison:

1. Standing (1947):

( )[ ] 53421 /

aaogsob TaRaaB ++= γγ (B-1)

Where 975.01 =a 5

2 1012 −×=a 5.03 =a

25.14 =a 2.15 =a

2. Vazquez and Beggs (1980):

( )( )( )( )gs

gsob

APITRcAPITcRcB

γγ

/60/601

3

21

−

+−++= (B-2)

Where for API ≤ 30 4

1 10677.4 −×=c 52 10751.2 −×=c

83 10811.1 −×−=c

and for API >30 4

1 10670.4 −×=c 52 101.1 −×=c

93 10337.1 −×=c

3. Glaso (1980):

( ) ( )[ ]2321 logloglog1 GaGaaantiBob +++=

(B-3) Where

( ) TaRG aogs 5

4/ += γγ

5811.61 −=a 91329.22 =a

27683.03 −=a 526.04 =a

968.05 =a

4. Al-Marhoun (1988):

( ) 24321 60 MbMbTbbBob ++−+= (B-4)

Where

765 bo

bg

bsRM γγ=

945810.01 =b 32 10862963.0 −×=b

23 10182594.0 −×=b 5

4 10318099.0 −×=b

74239.05 =b 323294.06 =b

20204.17 −=b

5. Al-Marhoun (1992):

( ) ( )( )( )60

160/1

4

321

−+

−−+++=

TaTRaRaRaB osogssob γγγ

(B-5) Where

31 10177342.0 −×=a 3

2 10220163.0 −×=a

64 10292580.4 −×=a 3

4 10528707.0 −×=a

TABLE 1: STATISTICAL DESCRIPTION OF THE INPUT DATA USED FOR TRAINING (603 POINTS)

Property Min Max Average St. Dev Skewness Kurtosis Oil FVF At Pb 1.028 3.562 1.342 0.284 2.116 8.001 Bubblepoint Pressure 107.33 7127 2015.958 1284.455 0.996 1.052 Temperature, ºF 58 341.6 183.333 52.228 0.222 -0.523 Gas-Oil Ratio 8.61 3617.27 549.192 495.508 1.969 6.251 Gas Relative Density 0.511 1.789 0.89 0.184 1.019 1.16 Oil Relative Density 0.725 0.993 0.854 0.046 0.544 -0.045 Bp Oil Relative Density 0.432 0.992 0.732 0.098 -0.191 -0.528 API Oil Gravity 11 63.7 34.593 8.765 -0.276 -0.211

10 EL-SAYED A. OSMAN, OSAMA A. ABDEL-WAHHAB AND MOHAMMED AL-MARHOUN SPE 68233

TABLE 2: STATISTICAL DESCRIPTION OF THE INPUT DATA USED FOR TESTING (200 POINTS)

Property Min Max Average St. Dev Skewness Kurtosis

Oil FVF At Pb 1.038 2.478 1.338 0.262 1.425 2.302 Bubblepoint Pressure 148 7127 1964.05 1312.422 1.128 1.528 Temperature, ºF 60 341.6 184.302 55.815 0.322 -0.46 Gas-Oil Ratio 12 2191.33 533.75 447.483 1.233 1.366 Gas Relative Density 0.525 1.789 0.9 0.198 1.042 1.286 Oil Relative Density 0.741 0.99 0.854 0.047 0.475 -0.284 Bp Oil Relative Density 0.48 0.991 0.732 0.101 -0.129 -0.572 API Oil Gravity 11.4 59.5 34.757 8.996 -0.225 -0.386

TABLE 3: STATISTICAL ANALYSIS OF THE RESULTS FOR DIFFERENT EMPIRICAL CORRELATIONS

Correlation E r E A E min E max E rms r

Standing (1947) -0.1696 2.7238 0.0081 20.1795 4.2025 0.9742 Vazquez & Beggs (1980) 2.3083 2.9755 0.0136 15.5368 4.0417 0.9842 Glaso (1980) 1.8186 3.3743 0.0030 17.7763 4.5663 0.9715 Al-Marhoun (1988) 0.3395 2.3334 0.0112 13.2590 3.3810 0.9811 Al-Marhoun (1992) -0.1152 2.2053 0.0033 13.1794 3.4162 0.9806 ANN 0.3024 1.7886 0.0076 11.7751 2.7193 0.9878

Fig.1- Schematic of an artificial neural network with one-hidden layer.

INPUT

OUTPUT

HIDDEN LAYER

SPE 68233 PREDICTION OF OIL PVT PROPERTIES USING NEURAL NETWORKS 11

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Standing (1947) Vazquez andBeggs (1980)

Glaso 1980 Al-Marhoun(1988)

Al-Marhoun(1992)

Artificial NeuralNetworks

AA

PRE

Fig. 2- Comparison of average absolute percent error for different correlations.

0.950

0.955

0.960

0.965

0.970

0.975

0.980

0.985

0.990

Standing 1947 Vazquez andBeggs 1980

Glaso 1980 Al-Marhoun1988

Al-Marhoun1992

ArtificialNeural

Networks

Cor

rela

tion

coef

ficie

nt

Fig. 3- Comparison of correlation coefficient for different correlations.

12 EL-SAYED A. OSMAN, OSAMA A. ABDEL-WAHHAB AND MOHAMMED AL-MARHOUN SPE 68233

0.00

0.50

1.00

1.50

2.00

2.50

3.00

0.00 0.50 1.00 1.50 2.00 2.50 3.00

Measured Bob

Pred

icte

d B

ob

0.00

0.50

1.00

1.50

2.00

2.50

3.00

0.00 0.50 1.00 1.50 2.00 2.50 3.00

Measured Bob

Pred

icte

d B

ob

Fig. 4- CrossPlot of Standing Correlation. Fig. 5-Cross Plot of Vazquez & Beggs Correlation.

0.00

0.50

1.00

1.50

2.00

2.50

3.00

0.00 0.50 1.00 1.50 2.00 2.50 3.00

Measured Bob

Pred

icte

d B

ob

0.00

0.50

1.00

1.50

2.00

2.50

3.00

0.00 0.50 1.00 1.50 2.00 2.50 3.00

Measured Bob

Pred

icte

d B

ob

Fig. 6- Cross Plot of Glaso Correlation. Fig. 7- Cross Plot of Al-Marhoun (1988) Correlation.

0.00

0.50

1.00

1.50

2.00

2.50

3.00

0.00 0.50 1.00 1.50 2.00 2.50 3.00

Measured Bob

Pred

icte

d B

ob

0.00

0.50

1.00

1.50

2.00

2.50

3.00

0.00 0.50 1.00 1.50 2.00 2.50 3.00

Measured Bob

Pred

icte

d B

ob

Fig 8- Cross Plot of Al-Marhoun (1992) Correlation. Fig. 9- Cross Plot of Artificial Neural Networks Model.

SPE 68233 PREDICTION OF OIL PVT PROPERTIES USING NEURAL NETWORKS 13

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0 1000 2000 3000 4000 5000

Solution Gas Oil Ratio, SCF/STB

Pred

icte

d Bo

b

StandingVazquez&BeggsGlasoAl-Marhoun 88Al-Marhoun 92ANN

Fig. 10-Trend Analysis - Effect of Solution Gas Oil Ratio on Bob.

1.20

1.25

1.30

1.35

1.40

1.45

1.50

0 100 200 300 400 500

Reservoir Temperature, Degrees F

Pred

icte

d B

ob

Standing

Vazquez & Beggs

Glaso

Al-Marhoun 88

Al-Marhoun 92

ANN

Fig. 11- Trend Analysis -Effect of Reservoir Temperature on Bob.

14 EL-SAYED A. OSMAN, OSAMA A. ABDEL-WAHHAB AND MOHAMMED AL-MARHOUN SPE 68233

1.25

1.30

1.35

1.40

10 20 30 40 50 60

Oil API Gravity

Pred

icte

d Bo

bStandingVazquez & Beggs

GlasoAl-Marhoum 88Al-Marhoun 92

ANN

Fig. 12- Trend Analysis, Effect of Oil Gravity on Bob.

1.20

1.24

1.28

1.32

1.36

1.40

1.44

0.4 0.6 0.8 1.0 1.2 1.4 1.6

Gas Relative Density

Pred

icte

d B

ob

StandingVazquez&BeggsGlasoAl-Marhoun 88Al-Marhoun 92ANN

Fig. 13- Trend Analysis, Effect of Gas Relative Density on Bob.