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Estimating the Isothermal Compressibility Coefficient of Undersaturated Middle East Crudes Using Neural Networks Ridha Gharbi Department of Petroleum Engineering, College of Engineering and Petroleum, Kuwait University, P.O. Box 5969, Safat 13060, Kuwait Received August 5, 1996X
The isothermal compressibility coefficients are required in several reservoir engineering applications such as transient fluid flow problems and also in the determination of physical properties of crude oils. Over the years, several correlations to estimate PVT properties have been reported in the literature for different types of hydrocarbon systems. All of these correlations were developed using conventional regression or graphical techniques that may not lead to the highest accuracy. On the other hand, the use of neural networks to develop such correlations can be excellent and reliable for the prediction of these properties. This paper presents an artificial neural network model to predict the isothermal compressibility coefficient of undersaturated crude oils of the Middle East region. The back-propagation algorithm with momentum for error minimization was used in this study. The data set, on which the network was trained, contain 520 experimentally obtained PVT data sets, representing 102 different crudes from the region of the Middle East. It is the largest data set ever collected in the Middle East to be used in developing a model to estimate the isothermal compressibility coefficients. An additional set of 35 PVT data points was used to test the effectiveness of the neural network to accurately predict outputs for data not used during the training process. The neural network model is able to predict the isothermal compressibility coefficient as a function of the solution gas/oil ratio, the gas specific gravity, the oil specific gravity, the reservoir temperature, and the reservoir pressure. A detailed comparison between the results predicted by this model and those predicted by others are presented for these Middle East crude oil samples.
1. Introduction The development of correlations for the prediction of fluid properties has been subjected to extensive research.1-12 Several graphical and mathematical correlations have been proposed. The isothermal compressibility coefficient (C0) is defined as the rate of change in volume with pressure per unit volume of liquid at constant temperature. The isothermal comAbstract published in Advance ACS Abstracts, January 15, 1997. (1) Standing, M. B. Volumetric and Phase Behavior of Oil Field Hydrocarbon Systems; Millet Print: Dallas, TX, 1977; pp 70-95. (2) Standing, M. B. A Pressure-Volume-Temperature Correlation for Mixtures of California Oils and Gases. Drill., Prod. Prac. API 1947, 275-287. (3) Standing, M. B. Oil-System Correlations. Petroleum Production Handbook; Frick, T. C., Ed.; SPE: Richardson, TX, 1962; pp 1-42. (4) Al-Marhoun, M. A. PVT Correlations for Middle East Crude Oils. J. Pet. Technol. 1988, May, 650-666. (5) Al-Marhoun, M. A. New Correlation for Formation Volume Factors of Oil and Gas Mixtures. J. Can. Pet. Technol. 1992, 31, 2226. (6) Dokla, M. E.; Osman, M. E. Correlation of PVT Properties for UAE Crudes. SPE Form. Eval. 1992, March, 41-46 . (7) Lasater, J. A. Bubble Point Pressure Correlation. Trans. AIME 1958, 213, 379-381. (8) Labedi, R. Use of Production Data to Estimate Volume Factor Density and Compressibility of Reservoir Fluids. J. Pet. Sci. Eng. 1990, 4, 357-390. (9) Glasø, O. Generalized Pressure-Volume Temperature Correlations. J. Pet. Technol. 1980, May, 785-95. (10) Vasquez, M. E.; Beggs, H. D. Correlations for Fluid Physical Property Prediction. J. Pet. Technol. 1980, June, 968-970. (11) Trube, A. S. Compressibility of Undersaturated Hydrocarbon Reservoir Fluids. Trans. AIME 1957, 210, 341-344. (12) Ahmed, T. Hydrocarbon Phase Behavior; Gulf Publishing: Houston, TX, 1989. X
S0887-0624(96)00123-5 CCC: $14.00
pressibility is mathematically given by the following equation:
1 ∂V C0 ) V ∂p T
( )
(1)
Generally, the isothermal compressibility coefficient for an undersaturated crude oil is determined experimentally. A sample of the crude oil is placed in a PVT cell at the reservoir temperature and a pressure greater than the bubble point pressure of the crude oil. The pressure is decreased, and the volume is allowed to increase. The volume and pressure are recorded. A plot of pressure versus volume allows the calculation of C0 using eq 1. Very often, reliable experimental data are not available, and correlations must be used to predict properties. The importance of accurate predictions is very crucial in material balance calculations and in production operation and design. Correlations to estimate the isothermal compressibility coefficients are essentially based on the assumption that C0 is a function of the solution gas/oil ratio (Rs), the gas specific gravity (γg), the oil specific gravity (γo), the reservoir temperature (T), and the reservoir pressure (p), or
C0 ) f(Rs, γg, γo, T, p)
(2)
In 1957, Trube11 presented a correlation of the form
C0 ) Cr/ppc
(3)
where ppc is the pseudocritical pressure and Cr is the © 1997 American Chemical Society
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isothermal pseudoreduced compressibility of undersaturated oils. Trube presented a graphical correlation of this property with the pseudoreduced pressure and temperature, ppr and Tpr, respectively. In his paper, Trube did not specify how many data points were used to develop his correlation. However, an average absolute error of 7.9% between the calculated and the measured values was reported. In 1980, Vasquez and Beggs10 used 4036 experimental data points, which are collected from all over the world, to correlate the isothermal compressibility coefficients with Rs, γg, γo, T, and p. Vasquez and Beggs used linear regression technique to develop a model with the form
C0 )
-1433 + 5Rs + 17.2T - 1180γgs + 12.61 °API 105p
(4)
where γgs is the gas gravity at a reference separator pressure. If the average field separator pressure is 100 psig, then γgs is equal to γg. Vasquez and Beggs’s correlation is the most widely used correlation for the prediction of the isothermal compressibility coefficient of undersaturated crude oils. In 1985, Ahmed12 used nonlinear regression analysis to develop a model for the prediction of the isothermal compressibility coefficients. Ahmed used 245 experimentally obtained data points. The correlation has the form
C0 )
[
]
a1 + a2[Rs(γg/γo)0.5 + 1.25T]1.175 exp(a3p) a4γo + a5Rsγg
(5)
where
works comes from the fact that neural networks have large degrees of freedom, which allows them to capture the nonlinearity of the system being studied better than regression techniques. Artificial neural networks also have the ability to learn and adapt themselves to new situations in which if additional data become available, the neural networks can be further trained and refined to include these new data. Furthermore, neural networks can map multiple-input multiple-output behavior systems. These are some of the qualities that makes neural networks superior to conventional regression techniques. Recently, we have developed a neural-network-based model23 that was able to successfully predict the breakthrough oil recovery of immiscible displacement of oil by water in a two-dimensional cross section as a function of the system’s dimensionless scaling groups. This model is able to reproduce the results of fine mesh numerical simulations without actually performing the simulation runs. We have also developed neural network models13 for the estimation of the bubble-point pressure and the formation-volume factor of crude oils from the Middle East region as functions of the solution gas/oil ratio, Rs, the reservoir temperature, T, the gas specific gravity, γg, and the oil specific gravity, γo. These models were based on the largest PVT data set (498 data points) ever collected for Middle East crudes and were shown to be more accurate than existing PVT models. The objective of this paper is to further extend our previous work and use artificial neural networks to develop a model to estimate the isothermal compressibility coefficient of undersaturated Middle East crude oils as a function of Rs, γg, γo, T, and p.
a1 ) 1.026638
2. Artificial Neural Networks
a2 ) 0.0001553
The theory behind artificial neural networks has been the subject of numerous studies. In an artificial neural network, a training set of examples of input and output are entered and neural network algorithms attempt to map the process by which the input becomes output. It is desired that the difference between the predicted and the observed (actual) outputs be as small as possible. Artificial neural networks are composed of many simple elements called neurons. These neurons may be arranged in multiple layers as shown in Figure 1. The
a3 ) -0.0001847272 a4 ) 62400 a5 )13.6 The author reported an average absolute error of 3.9% between the measured and the predicted values. All of these correlations mentioned above have been developed using conventional regression techniques, which may not give accurate results. However, artificial neural networks were shown to be excellent predictive tools in various petroleum engineering applications. Such applications include the prediction of fluid properties,13-16 well logging,17-19 well testing,20,21 drilling,22 and prediction of reservoir performance.23 The excellent predictive capability of artificial neural net(13) Gharbi, R.; Elsharkawy, A. M. Neural-Network Model for Estimating the PVT Properties of Middle East Crude Oils. In Situ 1996 20, 367-394. (14) Briones, M. F.; Rojas, G. A.; Moreno, J. A.; Hidaigo, O. Thermodynamic Characterization of Volatile Hydrocarbon Reservoirs by Neural Networks. Proceedings of the SPE Latin Am and Caribbean Petroleum Engineering Conference, Buenos Aires, Argentina; SPE: Richardson, TX, 1994; pp 235-243. (15) Habiballah, W. A.; Startzman, R. A.; Barrufet, M. A. Use of Neural Networks for the Prediction of Vapor-Liquid Equilibrium K-Values. In Situ 1993, 17, 227-242. (16) Mohaghegh, S.; Arefi, R.; Ameri, S. Design and Development of an Artificial Neural Network for the Prediction of Formation Permeability. Proceedings of the SPE Petroleum Computer Conference, Dallas, TX; SPE: Richardson, TX, 1994.
(17) Baldwin, J. L. Using a Simulated Bi-Directional Associative Neural Network Memory with Incomplete Prototype Memories to Identify Facies from Intermittent Logging Data Acquired in a Siliciclastic Depositional Sequence. A case Study. Proceedings of the SPE Annual Technical Conference Exhibition; SPE: Richardson, TX, 1991; pp 273-286. (18) Garcia, G.; Whitman, W. W. Inversion of a Lateral Log Using Neural Networks. Proceeding of the SPE Petroleum Computer Conference; SPE: Richardson, TX, 1992; pp 295-304. (19) Smith, M.; Carmichael, N.; Reid, I.; Bruce, C. Lithofacies Determination from Wire-Line Log Data Using a Distributed Neural Network. Proceedings of the IEEE Workshop Neural Networks Signal Processing, Princeton, NJ; IEEE: Piscataway, NJ, 1991; pp 482-292. (20) Juniardi, I. R.; Ershaghi, I. Complexities of Using Neural Network in Well Test Analysis of Faulted Reservoirs. Proceedings of the SPE Annual Western Regional Meeting, Anchorage, AK; SPE: Richardson, TX, 1993; pp 711-721. (21) Kumoluyi, A. O.; Daltaban, T. S. Higher-Order Neural Networks in Petroleum Engineering. Proceedings of the SPE Annual Western Regional Meeting, Long Beach, CA; SPE: Richardson, TX, 1994; pp 555-570. (22) Waller, M. D.; Rowsell, P. J. Intelligent Well Control. Trans. Am. Inst. Min. Metall. 1994, 103, a47-a51. (23) Gharbi, R.; Karkoub, M.; Elkamel, A. An Artificial Neural Network for the Prediction of Immiscible Flood Performance. Energy Fuels 1995, 9, 894-900.
374 Energy & Fuels, Vol. 11, No. 2, 1997
Gharbi Table 1. Range of Data Used To Train the Neural Network reservoir pressure, psia isothermal compressibility coefficient, microsip solution gas/oil ratio, scf/stb gas specific gravity (air ) 1) oil specific gravity, (water ) 1) reservoir temp, °F
Figure 1. One-hidden-layer neural network model.
network, shown in Figure 1, has an input layer, an output layer, and one hidden layer. Artificial neural networks are trained by adjusting the input weights (connection weights) by some algorithm so that the calculated outputs approximate the desired outputs. The output from a given neuron is calculated by applying a transfer function to a weighted summation of its input to give an output, which can serve as input to other neurons. The output, ajk, from neuron j in layer k may be obtained by Nk-1
ajk ) Fk(
wijkai(k-1) + bjk) ∑ i)1
(6)
where bjk is the bias weight for neuron j in layer k and is associated with the back-propagation algorithm. The coefficients wijk in the summations are the connection weights of the neural network model. These connection weights are the model fitting parameters. The transfer function, Fk, is a nonlinear function, usually the sigmoid function, which is the s-shaped curve of logistic growth equations. This function gives values that range from 0 to 1. One commonly used algorithm used for the system learning “training”, which is used in this study, is the back-propagation algorithm with momemtum.24 Backpropagation is used so that input connection weights become modified on the basis of the error signal arising from the output layer. The momentum procedure is used to avoid the search process getting stuck in a local minimum. Back-propagation is not the only method used in the literature, but it is by far the most common one. A detailed description of this algorithm can be found in a paper by Gharbi et al.23 3. Input Data To develop a neural network model to accurately predict C0 for Middle East crudes, it is necessary that the neural network model is exposed to a large data set during its training phase. For this reason, a total of 520 PVT data points were collected, representing 102 different crudes from the region of the Middle East. Each PVT data point contains the isothermal compress(24) Rumelhart, D. E.; Hinton, G. E.; Williams, R. J. Learning Internal Representation by Error Propagation. Parallel Data Processing; The M.I.T. Press: Cambridge, MA, 1986; Vol. 1, Chapter 8, pp 318-362.
1025-9260 3.62-29.18 367-1568 0.807-1.234 0.826-0.907 130-243
ibility coefficient C0, the solution gas/oil ratio Rs, the gas specific gravity γg, the oil specific gravity γo, the reservoir temperature T, and the reservoir pressure p. The range of the data is listed in Table 1. The general relationship of C0 given by eq 2 was considered. Therefore, we are attempting to map the relations between the isothermal compressibility coefficient C0, which is the output variable, as function of Rs, γg, γo, T, and p, which are the input variables. The data of the isothermal compressibility coefficients were given in units of microsip or 10-6 psi-1. Since the domain of variation for C0 (output) in units of microsip is high, then the neural network tends to be unstable during the training phase.13 To eliminate this problem, a normalized compressibility coefficient is used to help the neural network be stable.25 This normalized compressibility coefficient is calculated using the equation
C0new )
C0old - min(C0old) max(C0old) - min(C0old)
(7)
where C0old is the original data in microsip and C0new is the normalized output. This equation yields values in dimensionless form between 0 and 1. 4. Developing the Neural Network Model Several neural network architectures were investigated to obtain the most accurate model for predicting C0 as a function of the other five variables. Only onehidden layer and two-hidden layer architectures were considered in this study. The number of neurons in each hidden layer was systematically varied to obtain a good estimate of the data being trained. Initially, an attempt was made to train the neural network model using all 520 data points. However, we were not able to obtain an accurate model using all of the collected data in the training phase. The reason is that the data contain too much noise, and consequently the neural network was not able to satisfactorally map the system’s behavior. To have a successfully trained neural network, the noise in the data must be reduced. Principal component analysis,26 which is used extensively in pattern recognition problems, was used in this study to reduce the noise in the data. Several different names have been given to this method in the literature. Such names include Karhunen-Loe´ve (K-L) decomposition,27 used in dynamic flow problems, or the Hotelling transform,28 used in image processing problems. Regardless of the name used, the method is essentially the same. The description and the mathematical details of (25) Hertz, J.; Krogh, A.; Palmer, R. G. Introduction to the Theory of Neural Computation; Addison-Wesley Publishing: New York, 1991. (26) Jolliffe, I. T. Principal Component Analysis; Springer-Verlag: New York, 1986. (27) Aubry, N.; Holmes, P.; Lumley, J. L.; Stone, E. The Dynamics of Coherent Structures in the Wall Region of a Turbulent Boundary Layer. J. Fluid Mech. 1988, 192, 115-173. (28) Gonzalez, R. C.; Wintz, P. Digital Image Processing, 2nd ed.; Addison-Wesley: Reading, MA, 1987; pp 122-130.
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Table 2. Results of the Training Process for Various Neural Network Architectures no. of neurons no. of layers
first layer
1 1 1 1 1 1 1 2 2 2 2 2 2 2
5 8 10 15 18 20 25 5 6 6 7 7 8 8
second layer
sum-squared error
5 5 6 6 7 7 8
1.0634 0.9103 0.7906 0.2431 0.1511 0.0887 0.0342 0.1687 0.1433 0.0832 0.0655 0.0414 0.0033 0.0022
the principal component analysis are well-known and are given in several publications.26-28 The primary objective of the principal component analysis is to determine a set of optimal eigenfunctions of the data set. To each of these eigenfunctions is calculated an energy percentage from each eigenvalue that is associated with each eigenfunction. The higher the energy percentage for an eigenvalue, the more energetic is the eigenfunction associated with it. Using the most energetic eigenfunctions, it is possible then to reconstruct another set of data that is smooth compared to the original data without changing the characteristics of the original data. After this method was carried out on the original data, only 426 data points were left. This new data set was then used to train the neural network. Using the new data set, the training time taken by the neural network was substantially reduced. Table 2 shows the various neural network designs that were investigated in this study. Also listed in Table 2 are the sum-squared errors (sse) obtained from each neural network design. Some of the neural network architectures did not yield acceptable fit, whereas more than one architecture yielded good fit for estimating the isothermal compressibility coefficient. An acceptable fit is the one that has a sum-squared error of
