Reservoir fluid PVT properties modeling using Adaptive Neuro-Fuzzy Inference Systems

Journal of Natural Gas Science and Engineering 21 (2014) 951e961

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Reservoir fluid PVT properties modeling using Adaptive Neuro-Fuzzy Inference Systems Ehsan Ganji-Azad a, Shahin Rafiee-Taghanaki a, Hojjat Rezaei a, Milad Arabloo b, *, Hossein Ali Zamani a a b

Department of Petroleum Engineering, Petroleum University of Technology, Ahwaz, Iran Young Researchers and Elite Club, North Tehran Branch, Islamic Azad University, Tehran, Iran

a r t i c l e i n f o

a b s t r a c t

Article history: Received 12 June 2014 Received in revised form 8 October 2014 Accepted 9 October 2014 Available online 6 November 2014

Knowledge of prediction of PVT properties of reservoir oil is of primary importance in many petroleum engineering studies such as inflow performance calculations, production engineering studies, numerical reservoir simulations and design of proper improved oil recovery techniques. Ideally these parameters should be determined experimentally in laboratory under the reservoir conditions such as pressure and temperature. But owing to the fact that experimental methods are very expensive and time consuming, numerical models are developed for prediction of PVT properties. In this study several predictive models, based on a large data bank from different geographical regions were developed to predict the reservoir oil bubble point pressure as well as oil formation volume factor (OFVF) at bubble pressure. Developed models were successfully applied to the data set and the predicted values were in a good agreement with experimental values. Also a comparative study has been carried out to compare the result of this study to previously proposed correlations in terms of accuracy. Results show that the proposed models are more accurate than the available approaches. © 2014 Elsevier B.V. All rights reserved.

Keywords: Bubble point pressure OFVF ANFIS Subtractive clustering Hybrid optimization

1. Introduction Knowledge of the physical properties of reservoir fluid at reservoir condition (elevated pressure and temperature) is the key factor for monitoring reservoir performance problems at various stages of reservoir life. The prediction of pressure volume temperature (PVT) properties of crude oil such as bubble pressure and oil formation volume factor (OFVF) is achieved by one of three major approaches including equations of state, empirical correlations; and artificial intelligence networks (Arabloo et al., 2014; Rafiee-Taghanaki et al., 2013). The artificial intelligence networks have found a wide application in different engineering fields (Chamkalani et al., 2013; Kumar, 2009; Roosta et al., 2011; Zendehboudi et al., 2013). High accuracy and flexibility, fast data processing and easy adaption to wide range of data sets has made the artificial intelligent system very popular among researchers especially as a means of developing predictive tools (HemmatiSarapardeh et al., 2014; Nejatian et al., 2014; Rafiee-Taghanaki et al., 2013; Talebi et al., 2014).

* Corresponding author. Tel.: þ98 917 1405706. E-mail address: [email protected] (M. Arabloo). http://dx.doi.org/10.1016/j.jngse.2014.10.009 1875-5100/© 2014 Elsevier B.V. All rights reserved.

Contrary to many proposed prediction models which use local data points or use limited number of data points to develop their models, in this study a large data bank of various crudes from all over the world is used to develop comprehensive computer based models. Thus proposed models can be confidently applied to universal crude samples. In this study two data series were used for prediction of bubble point pressure and OFVF. Four Takagi-Sugeno fuzzy inference systems (TS-FIS) coupled with subtractive clustering method were developed and optimized by hybrid method (a combination of back-propagation and least squares technique) to produce accurate predictive models. For the sake of more accuracy, similar to the works of Vazquez and Beggs (1980) and Kartoatmodjo and Schmidt (1994), the whole data bank (bubble point pressure data and OFVF data) were subdivided in two API category; below 30
API and above 30
API. Each model was trained with five different train-test data ratio with five different train-test data ratio and finally the best model with the optimized ratio was selected. To achieve this mission, the rest of the paper is organized as follows; in the Section 2 an extensive review on the previous works on the subject of the study is presented. After that, in Section 3 the compiled data bank is presented. Then, the background and analysis procedure are discussed in detail in Section 4. After that, results of modeling work are illustrated in a great detail in Section

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5. In this section also a comparative study is presented to figure out the performance of the developed models against well-known literature correlations. Finally, the last section concludes the paper. 2. Literature review Due to importance of empirical correlations for PVT properties of crudes, since 1940's numerous studies have been carried out in this field which resulted in the development of several correlations for prediction of these properties. Correlations developed by Katz (1942), Standing (1947) and Lasater (1958) are among the most known correlations proposed for prediction of these properties and were considered as the only source for prediction of PVT properties in the absence of experimental data for a long time. In 1942, a graphical correlation was published by Katz (1942) which used mid-continent crudes of U.S.A. The correlation uses reservoir pressure and temperature, oil and gas gravity and solution gas oil ratio to predict the oil formation volume factor (OFVF). In 1947, correlations for prediction of both OFVF and bubble pressure were proposed by Standing (1947). For the first time he developed a correlation that predicted bubble point pressure and OFVF as a function of gas oil ratio, oil and gas gravity and reservoir temperature. Later after this work these four parameters were widely used in developing empirical correlations for prediction of PVT properties of crudes. In 1980, Vazquez and Beggs (1980) used a large data bank consisting of 6000 data points from 600 laboratory measurements to develop their correlations for gas oil ratio and OFVF. For the first time they categorized their databank into two categories of crudes with API above and below 30. In order to eliminate the dependency of gas gravity measurements to the separation condition, they used a reference pressure of 100 psi. In 1980, by using 45 oil samples mostly from the North Sea region, Glaso (1980) developed a graphical and regression analysis for OFVF at bubble pressure. Average errors of 1.28% and 20.43% were reported for predicted OFVF and bubble pressure, respectively. In 1988, Al-Marhoun (1988) used characteristics of 69 middle east reservoirs to propose empirical correlations based on nonlinear multiple regression analysis for OFVF, bubble point pressure and total formation volume factor. With the purpose to develop PVT correlations for Middle East reservoirs, he employed 160 data sets for bubble point pressure/OFVF, and 1556 data series for total OFVF. In 1988, Abdul-Majeed et al. (1988) proposed an empirical correlation using the correlation of Al-Marhoun (1988) to predict the OFVF. Their correlation was essentially similar to that of AlMarhoun (1988) with new calculated coefficients. Data set used for their study was 420 data points from unpublished sources. In 1990, Labedi (1990) published a correlation for OFVF, fluid compressibility and oil density. His correlation used a data bank of 129 data points (97 data point from Libya, 28 data points from Nigeria, and 4 points from Angola) to serve as a means for predicting the PVT properties of African crude oils. Labedi (1990) eliminated the use of total gas oil ratio and gas gravity with separation gas oil ratio, pressure and temperature which can be found easily in field reports. In 1992, Dokla and Osman (1992) used regression analysis for 51 bottom hole samples from UAE reservoirs to develop new empirical correlations for estimating bubble pressure and OFVF at bubble pressure. In 1993, Omar and Todd (1993) Modified the coefficients of Standing's bubble point pressure and OFVF correlations based on 93 PVT data from Malaysian offshore oil-fields utilizing both linear and nonlinear regression analyses. They assumed bubble point pressure as a function of OFVF as well as reservoir temperature, gas specific gravity, oil gravity and solution gas oil ratio. In 1993, Petrosky and Farshad (1993) developed a new correlation using 90 data sets from Gulf

of Mexico. They used Standing (1947) correlations of solution gas oil ratio, bubble point pressure and OFVF and changed their coefficient to adapt them as a means for prediction of PVT properties. Also oil compressibility correlation model of Vazquez and Beggs (1980) was used as a basis for oil compressibility correlation. Their model allows each variable to have exponent and multiples for more flexibility and adaption. In 1994, Kartoatmodjo and Schmidt (1994) used the same approach as Petrosky and Farshad (1993) and proposed a new correlation for OFVF and saturation pressure using previous correlations in open literature and adapting them to predict these parameters for another data set by adjusting the coefficients. Their correlation was proposed by using a large data bank with 5392 data sets from 740 different crude oils from all over the world. Standing correlation models and Vazquez and Beggs (1980) oil formation volume factor were taken as the basis for prediction of saturation pressure and OFVF, respectively. Kartoatmodjo and Schmidt (1994) correlation was examined with an extra set of data and the results were compared with other existing correlations. Recently, Arabloo et al. (2014) proposed new empirical correlations to estimate bubble point pressure and OFVF at bubble pressure utilizing more than 750 published worldwide data series gathered from literature. Successive linear programming and generalized reduced gradient algorithm as two constrained multivariable search methods were incorporated for modeling and expediting the process of achieving a good feasible solution. Moreover, branch-and-bound method was applied to overcome the problem of stalling to local optimal points. In the recent years there have been an increasing number of researches regarding the use of intelligent soft computing modeling in reservoir studies (Ghiasi et al., 2014; Arabloo et al., 2013). Many studies has been carried out in this field focuses on prediction of PVT properties for crude oil samples. PVT properties of crudes of different geographical regions with varying compositions may not be accurately predicted with empirical correlations and often there will be huge errors in such predictions (Ikiensikimama and Azubuike, 2012). Thus use of intelligent modeling for prediction of PVT properties of crudes, especially crudes from different geographical regions sounds inevitable. In 1997, one of the earliest utilizations of artificial neural network was published by Gharbi and Elsharkawy (1997) for prediction of bubble point pressure and OFVF. In their study, 498 datasets collected from the literature and unpublished sources were employed for training the models and 22 new datasets from Middle East reservoirs were applied to test the accuracy of the model. In 2002 Al-Marhoun and Osman (2002) presented an ANN model in order to estimate PVT properties for Saudi crude oil. Their 283 unpublished datasets of Saudi reservoirs were used for training, cross validation and testing the back propagation network which was developed for estimation of bubble point pressure and OFVF at bubble pressure. In 2008, Dutta and Gupta (2010) proposed models for bubble point pressure, saturated and under saturated OFVF and some other PVT properties of Indian west coast crude using ANN based on MLP using Bayesian regularization coupled with hybrid genetic algorithm as an optimizer. They used 372 data sets for bubble pressure and 530 and 263 data sets for saturated and under saturated OFVF, respectively. In 2011, Moghadam et al. (2011) developed an artificial neural network model to predict bubble point pressure and bubble point OFVF. In 2013, Moghadasi et al. (2013) developed a feedforward back-propagation ANN model on 157 PVT data sets from southwest Iranian oil fields to determine bubble point pressure by implementing a network with two hidden layers of six and three neurons. In 2013, Numbere et al. (2013) proposed an ANN model for prediction of saturation pressure based on 1248 data sets collected

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from the Niger Delta region of Nigeria. They utilize backpropagation algorithm with Simulated Annealing for the optimization procedure. According to the optimization process one hidden layer with four neurons gave best results. In 2013, Rafiee-Taghanaki et al. (2013) proposed a new soft computing model for prediction of oil formation volume factor and saturation pressure based on 569 PVT data sets originated from several geographical regions. This was the first time that SVM algorithm was applied to predict a PVT property. Least squares support vector machine model optimized with coupled simulated annealing algorithm was implemented as a tool to achieve an improved intelligent model and it was tested successfully with 114 data sets of the database. Neural networks, as well as fuzzy logic approaches, have advantages and deficiencies. However, Adaptive Neuro-Fuzzy inference system (ANFIS) was first introduced by Jang (1993) with combination of artificial neural network (ANN) and fuzzy inference system (FIS) in order to recover the weakness of both of these approaches. With keeping this in mind, the purpose of this study is to develop ANFIS models for improved estimation of PVT properties of reservoir oils. Highlighting the main contribution of present study, our developed ANFIS models cover wider ranges of data compared to the previous models. It is important to note that much higher accuracy in predicting PVT properties is attained by using the proposed ANFIS model compared to the previous predictive models. 3. Data bank collection Similar to previous studies (Arabloo et al., 2014; Standing, 1947; Talebi et al., 2014; Vazquez and Beggs, 1980), bubble point pressure and OFVF at bubble pressure are considered functions of four variables of solution gas oil ratio, oil API gravity, gas gravity and reservoir temperature. In order to develop predictive models, a large data bank from literature as well as unpublished sources was gathered. Literature data were collected from Dokla and Osman (1992), Omar and Todd (1993), Mahmood and Al-Marhoun (1996), Moghadam et al. (2011), Obomanu and Okpobiri (1987), De Ghetto and Villa (1994), Ostermann and Owolabi (1983), Gharbi and Elsharkawy (1997), Bello et al. (2008). In addition to data gathered from literature two series of 157 and 128 datasets from Iranian oil fields were used for modeling bubble point pressure and OFVF, respectively. Fifteen sets of unpublished data are now reported in Table 1. Moreover, Table 2 gives a detailed summary of the data bank used for developing predictive models in this study.

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4. ANFIS background and methodology 4.1. Fundamental of ANFIS The concepts of Fuzzy logic (FL) and fuzzy sets, first proposed by Zadeh (1965). Unlike Aristotle's crisp logic which is a binary behavior such as black and white Fuzzy logic works with partial memberships. FL is a solution to decision-making when imprecise, unclear or absent data (Wei et al., 2007). A FIS is a model that maps input characteristics to input membership functions, input membership function to rules, rules to a set of output characteristics, output characteristics to output membership functions, and the output membership function to a single-valued output or a decision associated with the output. Since among two main types of FIS, Takagi and Sugeno (1985) is more efficient than Mamdani (Mamdani, 1976; Mamdani and Assilian, 1975). In this study the first-order Takagi and Sugeno (1985) fuzzy inference system (TSFIS) was employed for prediction of bubble point pressure and OFVF. Backpropagation (BP) is one of the most common methods for training feed-forward artificial neural networks in which the network calculates the difference between output value and corresponding target value in the training dataset to which was introduced. After that the error is propagated backward through the network and the weights and biases of the network are iteratively tuned to minimize the network objective function. ANFIS (Adaptive Neuro-Fuzzy Inference System) is an adaptive network which permits the usage of neural network topology together with fuzzy (Atmaca et al., 2001). ANFIS has two advantages; first, it includes the features of both methods and second, it eliminates some disadvantages of their only-used case. In order to produce proper output values, this system similar to BPANN uses an input-target data set and generates a FIS whose membership function parameters are adjusted using either of the backpropagation learning method or hybrid learning method which is a combination of back-propagation and least square error methods. In this study the hybrid learning method was implemented which includes two stages: first there is a forward pass in which least square estimation identifies consequent parameters or linear parameters (the parameters of output layer) and in the backward pass, in which the premise parameters or the nonlinear parameters (the parameters of input membership functions) are modified by the gradient descent (Jang and Sun, 1995; Roshani et al., 2013). A typical ANFIS construction is shown in Fig. 1. Parameters to be tuned by ANFIS are located in first layer (premise parameters) and

Table 1 Sample PVT data from Iranian oil fields. Solution gas-oil ratio (SCF/STB)

Gas specific gravity

Oil gravity (ºAPI)

Reservoir temperature (ºF)

Bubble point pressure (psi)

OFVF (bbl/STB)

683 706 571.41 591.7 643.53 650.59 1099.1 1320.6 1314.4 532.75 589.61 599.12 554 597.7 597.7

0.869 0.853 0.877 0.889 0.871 0.904 0.826 0.928 0.920 0.859 0.927 0.917 0.872 0.938 0.937

26.37 31.16 34.47 34.12 35.75 36.63 34.24 30.65 30.93 23.6 22.91 22.17 25.32 24.08 24.08

140 163 150 165 150 165 100 245 245 120 200 200 120 185 185

2551 2766 2110 2181 2158 2209 3306 3945 3943 2516 2878 2880 2393 2745 2747

1.363 1.401 1.342 1.350 1.366 1.385 1.520 1.819 1.843 1.265 1.417 1.417 1.272 1.363 1.363

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Table 2 Summary of the databases utilized in this study. Ref.

Origin of data sets

Ostermann and Owolabi (1983) Obomanu and Okpobiri (1987) Al-Marhoun (1988) Dokla and Osman (1992) Omar and Todd (1993) De Ghetto and Villa (1994) Mahmood and Al-Marhoun (1996) Gharbi and Elsharkawy (1997) Bello et al. (2008) Moghadam et al. (2011) This study

Alaska Nigeria Middle East U.A.E Malaysia All over the world Pakistan Middle East Nigeria Iran Iran

Range of each PVT property Solution gas-oil ratio (SCF/STB) 140e435 7.08e1177.22 26e1602 181e2266 142e1440 8.61e3298.66 92e2496 13e3366 400e2300 125e1485 532.75e1320.6

Surface gas gravity 0.853e1.0943 0.566e0.929 0.75e1.37 0.8e1.29 0.61e1.32 0.624e1.789

25.4e37.1 15.74e43.62 28.3e40.3 28.2e40.3 26.6e53.2 6e56.8

0.825e3.445 0.589e1.367 0.552e0.950 0.815e1.270 0.826e0.938

29e56.5 19.35e50.14 30.144e45.186 21.14e34.58 22.17e36.63

the fourth layer (consequent parameters) while parameters in other layers are constant. Consider a system with two inputs (say x and y) and one output (say f). The Takagi-Sugeno model of such a system is constructed by using five layers and two ifethen rules. Here a detailed description of modeling process in each layer of ANFIS is presented: In the first layer which is an adaptive layer, a membership function is assigned to each node i.



 o1i ¼ mAi x ; For i ¼ 1; 2 or o1i ¼ mBi2 y ; For i ¼ 3; 4

(1)

In this equation x (or y) is the input to node i and Ai (or Bi-2) is a linguistic label associated with this node. The membership function for each node can be any properly parameterized membership function with range in interval of 0e1. For example the Gaussian memberships function: ðxci Þ 2s2 i


 mA x ¼ e

2

(2)

Where, s and c are the parameters of input membership functions. These are the premise parameters of the fuzzy inference system. In the second layer all of the nodes are fixed nodes labeled П. The output is the product of all the incoming signals. Each node output represents the firing strength (weight) of a rule.



 o2i ¼ ui ¼ mAi x mBi y ; For i ¼ 1; 2

(3)

In the third layer, every node is a fixed node labeled N. The i-th node calculates the normalized firing strength or normalized weight which is the ratio of i-th firing strength to the sum of all rules firing strengths:

o3i ¼ ui ¼

ui ; For i ¼ 1; 2 u1 þ u2

API gravity of oil

(4)

OFVF at bubble pressure (bbl/STB)

Bubble point pressure (psia)

122e180 323.71e360.37 74e240 190e275 125e280 90.5e341.6

1.129e1.236 1.03e1.803 1.032e1.997 1.216e2.493 1.085e1.954 1.039e2.887

515e1802 58.02e2514.96 20e3573 590e4640 790e3851 147.94e6613.82

182e296 74e306 130e250 120e250 100e245

1.2e2.916 1.032e2.925 1e2.612 1.1e1.805 1.265e1.843

79e4975 121e7127 2000e5000 348e5155 2110e3945

In the fourth layer all of the nodes are adaptive nodes with membership function of:

  o4i ¼ ui fi ¼ ui pi x þ qi y þ ri ; For i ¼ 1; 2

(5)

where, pi, ri and qi are the consequent parameters. These parameters are also referred to as output membership function parameters. The output f of this layer is either linear or constant which corresponds to zero order and first order TS-FIS, respectively. Finally, the single node in this layer is a fixed node, which computes the overall output by the summation of all incoming signals using the following equation:

o5i ¼

X i

P uf ui fi ¼ Pi i i i ui

(6)

Data clustering is the process of finding similarities between data and putting similar data into groups. This data categorization and organization also is useful for data compression and model construction. All clustering techniques try to find cluster center of each cluster. If system identifies the cluster center, later when an input vector is proposed to the system, after analyzing the vector the most similar cluster to the proposed vector is chosen. The subtractive clustering is an unsupervised method proposed by Chiu (1994). It partitions data points based on density measure function as defined:

Di ¼

n X

exp



  ! x i  x j 2

i¼1

(7)

ðra =2Þ2

Where xi is a data point and ra is a positive constant called neighborhood radius or influential radius or cluster radius. A data point will have a high density value if it has many neighboring data points. The first center of cluster xc1 is the point at which density measure is maximized (Dc1). Next the density measure of other points (xi) near the first cluster center (xc1) will decrease significantly using the following equation:

Di ¼ Di  Dc1 exp

Fig. 1. Schematic of a typical ANFIS structure (Jang and Sun, 1995).

Temperature (
F)



kxi  xc1 k2 ðra =2Þ2

! (8)

After revising the density function, the next cluster center is selected as the point having the greatest density value. It goes on until an adequate number of clusters is reached. The cluster radius is critical for determining the number of clusters. Specifying a

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smaller cluster radius will usually yield more and smaller clusters in the data (resulting in more rules and more calculations). A large cluster radius yields a few large clusters in data (Chiu, 1994). 4.2. Procedure As mentioned before 911 and 940 data series were used respectively for developing predictive models of OFVF and bubble point pressure. These two data series were sorted with respect to API and divided in two groups: API above 30 and API below 30. In 1980, Vazquez and Beggs (1980) was the first who grouped data with respect to API in order to develop an empirical correlation for prediction of OFVF and bubble point pressure. Later, in 1994 Kartoatmodjo and Schmidt (1994) also divided their data set in two groups of API above and below 30. As the result two ANFIS model are developed for each PVT property. Tables 3 and 4 list the domain of variables. Prior to training process of the model, data sets should be normalized in order to make the domain of variables uniform. The data were normalized by dividing the difference between desired value and minimum value of variable by difference between maximum and minimum value of each variable. It makes dimensionless data sets that range from 0 to 1.

x  xmin xn ¼ xmax  xmin

(9)

In order to develop a reliable model, a portion of data should be used as train data and the remaining data will be used as test data to examine the viability and accuracy of the model (Arabloo et al., 2014; Hemmati-Sarapardeh et al., 2014). Trained data was introduced to a Takagi-Sugeno Fuzzy inference system which uses subtractive clustering in order to generate the optimum number of fuzzy clusters and centers of clusters. The generated FIS is trained using hybrid leaning rule for ANFIS architecture. Accuracy of developed models is affected by the ratio of data used for training the model and data with which the model is tested. In order to find the optimum ratio of train-test data in this study data bank is introduced to the model in five different train-test ratios. Radius of cluster is another parameter affecting the accuracy of models produced by subtractive clustering method. Thus, for each ratio of train-test data different radii of clustering from 0.1 to 0.8 in 0.01 steps were introduced to the model. For each ratio parameters of RMSE of the FIS, AAPRE for denormalized outputs of the network are calculated. The following formula was used for denormalization of the data.

x ¼ xn  ðxmax  xmin Þ þ xmin

(10)

Considering the calculated statistical parameters, the best radius of cluster is determined for each train-test ratio. As an example the graph regarding %75e%25 ratio for train test data of Bob for oils with

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API below 30 is illustrated in Fig. 2. Best cluster radius in which we have no overfitting and small error is chosen from this figure. This time RMSE of the FIS and AAPRE for denormalized outputs of the system for each ratio of train-test data are calculated and shown in Table 5 and Table 6. As is clear from Tables 5 and 6, considering all the parameters of RMSE of the FIS and AAPRE, ratio of %75e%25 for train-test ratio is the overall optimum ratio with which the final model is developed. Each time model is exposed to hybrid learning, parameters of membership functions are modified. This process is called epoch. Although increasing the number of epoch decreases the train data error, but it may leave a gap between train error and check error which is responsible for inaccurate response of the system to data which are not similar to that seen in train data. This problem is referred to as overfitting and may be avoided by optimizing the number epoch. To this end each model was trained for 1 to 200 epochs to determine the epoch with least check error. Optimum epoch was determined for all train-test data ratios and different clustering radii. 5. Results and discussion Table 7 lists the characteristics of proposed ANFIS models. Moreover, Appendix A presents a step-by-step procedure for using the developed model. As mentioned before, data sets for development of saturation pressure and OFVF were divided in two subsets of oil samples with API above and below 30. Thus, four different ANFIS models namely Model I, Model II, Model III and Model IV were proposed. Model I and Model II are proposed ANFIS models for calculation of OFVF for crudes with oil API gravities below 30 and above 30, respectively. Similarly, Models III and IV are those developed for calculation of bubble point pressure for oils with oil API gravities below 30 and above 30, respectively. In Tables 8 and 9 various statistical parameters of the proposed models including coefficient of determination (R2), average absolute percent relative error (AAPRE), average percent relative errors (APRE), root mean square error (RMSE), maximum absolute percent relative error (MaxAPRE) and minimum absolute percent relative error (MinAPRE) for prediction of OFVF and bubble point pressure have been reported. The mathematical formula of these statistical criteria can be found elsewhere (Arabloo et al., 2014; RafieeTaghanaki et al., 2013). In the next sections a comparative study is carried out to compare the proposed model in this study with a large number of empirical correlations in order to show the advantages of the proposed models to them. 5.1. OFVF at bubble pressure In order to examine the viability of proposed models for estimation of OFVF, the same data bank introduced to the ANFIS model

Table 3 Statistical description of applied PVT data for developing OFVF predictive models. PVT Parameters

Type of crude oil

Min

Max

Mean

Median

Standard deviation

Solution gas-oil ratio, SCF/STB

API30 API>30 API30 API>30 API30 API>30 API30 API>30 API30 API>30

7.08 1.40 0.5640 0.52 6.00 30.03 80.00 74.00 1.023 1.04

1340.00 3300.00 2.252 3.44 29.85 56.80 360.37 360.93 1.94 2.92

323.7 731.9 1.1 1.1 23.1 37.8 197.6 205.8 1.2 1.5

322.5 627.0 1.1 0.9 24.2 38.1 190.0 210.1 1.2 1.4

253.1 478.7 0.3 0.5 5.7 4.8 66.6 58.6 0.2 0.3

Gas gravity (air ¼ 1) API gravity,
API Reservoir temperature, ºF OFVF at bubble pressure, bbl/STB

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Table 4 Statistical description of applied PVT data for developing predictive models for bubble point pressure. PVT Parameters

Type of crude oil

Min

Max

Mean

Median

Standard deviation

Solution gas-oil ratio, SCF/STB

API30 API>30 API30 API>30 API30 API>30 API30 API>30 API30 API>30

7.08 1.3950 0.5640 0.5210 6.0 30.03 80.0 74.0 58.0152 79.0

1340.0 3.30eþ03 2.2520 3.4445 30.0 56.8 360.4 360.9 4215.0 6.61eþ03

333.5 733.0 1.1 1.1 23.2 37.8 196.3 204.8 1450.0 2190.0

331.0 628.0 1.07 0.92 24.3 38.1 187.6 209.5 1.32eþ03 2100

250.2 477.1 0.3 0.5 5.6 4.8 66.5 59.1 966.0 1360.0

Gas gravity (air ¼ 1.0) Oil API gravity,
API Temperature, ºF Bubble point pressure, psia

was exposed to 13 empirical correlations of OFVF prediction including Standing (1947), Vazquez and Beggs (1980), Al-Marhoun (1988), Petrosky and Farshad (1993), Omar and Todd (1993), Kartoatmodjo and Schmidt (1994), Al-Shammasi (1999), Macary and El-Batanoney (1993), Abdul-Majeed et al. (1988), Labedi (1990), Dindoruk and Christman (2004), Frashad et al. (1996) and Arabloo et al. (2014). The results of this comparison are reported in Table 10.

Table 7 Characteristics of the developed FIS. Model I

Fig. 2. Plot of various cluster radii versus AAPRE for OFVF model with API below 30 (75%e25% train-test ratio).

Table 5 Statistical parameters for the best influential radius and number of epochs for 5 different train-test ratios to find optimum ratio for OFVF model. Statistical parameter Type of Train-test percentage crude oil 67e33 75e25 80e20 RMSE of the FIS

API30 API>30 Best Epoch API30 API>30 AAPRE of Train Data API30 API>30 AAPRE of Test Data API30 API>30 AAPRE of Total Data API30 API>30

85e15

90e10

0.048 0.046 0.053 0.055 0.035 0.026 0.029 0.043 0.055 0.029 200 114 147 84 172 88 167 199 118 200 0.83 0.81 0.75 0.71 0.92 1.45 1.43 1.29 1.81 1.34 1.61 1.39 1.66 1.71 1.17 1.89 2.07 2.54 2.65 1.94 1.09 0.96 0.93 0.86 0.95 1.60 1.59 1.54 1.94 1.40

Table 6 Statistical parameters for the best influential radius and number of epochs for five different train-test ratios to find optimum ratio for bubble pressure model. Statistical parameter

RMSE of the FIS Best Epoch AAPRE of Train Data AAPRE of Test Data AAPRE of Total Data

Type of crude oil

Train-test percentage 67e33

75e25

80e20

85e15

90e10

API30 API>30 API30 API>30 API30 API>30 API30 API>30 API30 API>30

0.052 0.057 77 39 12.78 10.54 16.99 14.72 14.21 11.92

0.056 0.061 4 12 8.99 12.13 12.48 12.39 9.86 12.20

0.053 0.064 200 51 11.51 10.67 13.99 12.27 12.00 10.99

0.059 0.048 200 7 11.34 9.04 14.36 10.53 11.80 9.26

0.059 0.049 33 6 10.84 9.69 14.23 10.53 11.19 9.82

No. of input MFs No. of output MFs No. of rules Input MF type Output MF type No. of nodes No. of linear parameters No. of nonlinear parameters Total no. of parameters Influential radius of cluster Check RMSE for normalized data

Model II

Model III

Model IV

[10 10 10 10] [11 11 11 11] [13 13 13 13] [10 10 10 10] 10 11 13 10 10 11 13 10 Gaussian Gaussian Gaussian Gaussian Linear Linear Linear Linear 107 117 137 107 50 55 65 50 80

88

104

80

130

143

169

130

0.41

0.31

0.34

0.38

0.046

0.029

0.056

0.061

Table 8 Statistical parameters of the Model I and Model II. Statistical Parameter

Train

Test

Total

R2 AAPRE (%) APRE (%) RMSE MaxAPRE (%) MinAPRE (%) No. of data series R2 AAPRE (%) APRE (%) RMSE MaxAPRE (%) MinAPRE (%) No. of data series R2 AAPRE (%) APRE (%) RMSE MaxAPRE (%) MinAPRE (%) No. of data series

Model Model I

Model II

0.9936 0.8 0.016 0.0168 5.04 6.9E04 200 0.8470 1.4 0.399 0.0424 15.86 6.9E03 67 0.9706 1.0 0.089 0.0257 15.86 6.9E04 267

0.9853 1.4 0.048 0.0348 29.52 4.1E03 483 0.9549 2.0 0.028 0.0532 13.81 2.7E02 161 0.9791 1.6 0.043 0.0402 29.52 4.1E03 644

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Table 9 Statistical parameters of the Model III and Model IV. Statistical Parameter

Train

Test

Total

2

R AAPRE (%) APRE (%) RMSE MaxAPRE (%) MinAPRE (%) No. of data series R2 AAPRE (%) APRE (%) RMSE MaxAPRE (%) MinAPRE (%) No. of data series R2 AAPRE (%) APRE (%) RMSE MaxAPRE (%) MinAPRE (%) No. of data series

Model Model III

Model IV

0.96150 9.0 1.3 192.4 54.1 9.9E04 216 0.9333 12.5 2.0 234.6 132.3 3.1E01 72 0.9554 9.9 1.5 203.7 132.3 9.9E04 288

0.9596 12.1 1.6 274.2 143.6 1.4E03 489 0.9212 12.4 4.4 400.9 87.9 3.7E01 163 0.9487 12.2 2.3 310.8 143.6 3.7E01 652

Fig. 3. Comparison between the predictions of Model I and experimental data.

5.2. Bubble point pressure According to this table, the proposed ANFIS models (i.e., Model I and Model II) are much more reliable and precise than empirical correlations. From calculated AAPRE of empirical correlations this fact is obtained that Abdul-Majeed et al. (1988) correlations show worst results while Arabloo et al. (2014) correlation has better performance among all the studied correlations. However, Model I and Model II have the smallest AAPRE and higher values of R2 values. Figs. 3 and 4 presents cross plots of the proposed ANFIS models estimations versus the corresponding experimental data (target values). A tight cloud of points about the 45 line indicates the reliable accuracy of the proposed models.

Similar to the previous section, herein a comparative study is carried out to compare the accuracy of proposed models (i.e., Model III and Model IV) with sixteen well-known empirical correlations available in the literature. These correlations include Standing (1947), Lasater (1958), Vazquez and Beggs (1980), Glaso (1980), Al-Marhoun (1988), Petrosky and Farshad (1993), Kartoatmodjo and Schmidt (1994), Al-Shammasi (1999), Macary and ElBatanoney (1993), McCain (1991), Velarde et al. (1997), Dindoruk and Christman (2004), Ikiensikimama and Ogboja (2009), Frashad et al. (1996), Yi (2000), Arabloo et al. (2014), To be able to make a

Table 10 Statistical quality measures of OFVF predictive correlations. Author

Type of crude oil

APRE (%)

AAPRE (%)

R2

MinAPRE (%)

MaxAPRE (%)

RMSE

Standing (1947)

API30 API>30 API30 API>30 API30 API30 API30 API>30 API30 API>30 API30 API>30 API30 API>30 API30 API>30 API30 API>30 API30 API>30 API30 API>30 API30 API>30 API30 API>30 API30 API>30

2.4 1.4 0.3 4.3 1.7 0.9 29.5 25.5 28.5 41.5 9.4 7.7 1.3 2.0 1.6 2.7 1.2 0.9 0.3 1.3 1.6 0.1 1.9 2.4 0.4 0.3 0.1 0.0

3.3 3.2 4.4 5.8 3.9 2.8 29.6 25.6 28.7 41.6 9.8 8.6 2.6 3.5 4.3 5.1 4.1 2.9 3.4 3.0 3.0 2.3 3.7 5.7 2.4 2.3 1.0 1.6

0.78 0.93 0.66 0.82 0.74 0.93 0.61 0.83 0.78 0.93 0.71 0.86 0.81 0.92 0.72 0.85 0.72 0.92 0.78 0.93 0.79 0.94 0.79 0.80 0.81 0.94 0.97 0.98

5.3E03 2.4E03 1.6E02 2.4E02 2.4E02 1.9E03 2.7Eþ00 6.6E01 7.7Eþ00 3.9Eþ00 2.0E01 2.5E02 3.6E03 1.1E03 5.8E02 1.6E03 4.4E03 1.0E02 6.5E03 3.5E03 1.9E03 5.2E03 3.5E03 1.5E03 4.5E03 1.9E06 6.9E04 4.1E03

63.2 68.9 42.9 63.0 47.7 53.1 70.8 118.6 158.6 181.1 72.0 116.5 59.7 53.1 46.5 77.5 48.0 56.5 48.0 57.6 54.6 58.3 61.4 95.4 53.8 56.2 15.9 29.5

0.082 0.084 0.093 0.138 0.082 0.081 0.360 0.399 0.422 0.756 0.141 0.201 0.075 0.086 0.088 0.123 0.084 0.079 0.076 0.077 0.075 0.071 0.091 0.146 0.070 0.071 0.026 0.040

Vazquez and Beggs (1980) Al-Marhoun (1988) Abdul-Majeed et al. (1988) Labedi (1990) Macary and El-Batanoney (1993) Petrosky and Farshad (1993) Omar and Todd (1993) Kartoatmodjo and Schmidt (1994) Frashad et al. (1996) Al-Shammasi (1999) Dindoruk and Christman (2004) Arabloo et al. (2014) ANFIS(this study)

̊

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Fig. 5. Comparison between the predictions of Model III and experimental data. Fig. 4. Comparison between the predictions of Model II and experimental data.

judgment about reliability and accuracy of developed model in this study and other correlations/models, parameters of average percent relative error (APRE), average absolute percent relative error (AAPRE), coefficient of determination (R2), maximum absolute percent relative error (MaxAPRE) and minimum absolute percent relative error (MinAPRE), and root mean square error (RMSE) are calculated and the results are listed in Table 11. As is clear from the table, most of published correlations have huge error and fails to predict the bubble point pressure accurately. In

between, correlations of Ikiensikimama and Ogboja (2009) and Macary and El-Batanoney (1993) which appeared to have the worst performance and had the largest AAPRE, while proposed models of this study with AAPRE of 9.9% and %12.2% for oils with API below and above 30 oils, respectively. Also proposed models have the smallest root mean square error (RMSE), and have the closer value of coefficient of determination (R2) to 1. High negative values for APRE of correlations of Ikiensikimama and Ogboja (2009) and Macary and El-Batanoney (1993) indicates that these models underestimates bubble points pressure, while high positive values of

Table 11 Statistical quality measures of bubble point pressure predictive correlations. Author

Type of crude oil

APRE (%)

AAPRE (%)

R2

MinAPRE (%)

MaxAPRE (%)

RMSE

Standing (1947)

API30 API>30 API30 API>30 API30 API>30 API30 API>30 API30 API>30 API30 API>30 API30 API>30 API30 API>30 API30 API>30 API30 API>30 API30 API>30 API30 API>30 API30 API>30 API30 API>30 API30 API>30 API30 API>30 API30 API>30

16.4 14.0 7.3 6.9 19.2 25.1 33.0 17.3 3.5 0.0 16.4 14.0 8.3 44.9 73.2 34.0 19.8 29.5 26.4 2.4 13.3 4.6 5.2 1.5 95.7 93.3 13.5 2.2 5.0 4.5 5.0 3.1 1.5 2.3

26.5 25.7 27.1 20.1 27.9 31.1 41.9 25.7 27.3 24.5 26.5 25.7 40.2 48.9 100.2 65.6 30.2 35.6 37.0 19.3 23.2 17.2 30.5 20.4 95.7 93.3 25.0 20.3 26.0 22.7 15.8 17.9 9.9 12.2

0.84 0.83 0.84 0.87 0.86 0.84 0.79 0.87 0.80 0.84 0.84 0.83 0.81 0.84 0.79 0.85 0.85 0.82 0.84 0.86 0.87 0.86 0.72 0.83 0.78 0.78 0.83 0.86 0.82 0.80 0.84 0.86 0.96 0.95

4.0E02 3.2E02 1.2E01 1.5E02 4.7E02 4.5E02 2.8E01 1.1E04 4.7E01 1.1E02 4.0E02 3.2E02 5.2E01 1.6E02 1.5E01 1.3E01 2.1E01 3.0E02 1.6E01 4.3E02 2.6E02 3.2E03 5.5E-2 8.9E03 9.0Eþ01 8.1Eþ01 1.1E03 5.4E03 1.4E01 1.4E01 2.1E02 5.9E05 9.9E04 3.7E01

175.6 372.0 723.7 264.1 152.2 403.0 183.9 247.2 292.1 296.9 175.6 372.0 737.7 614.8 1320.0 1110.0 174.3 487.4 216.2 255.9 177.5 279.3 273.0 276.5 99.5 100.0 254.2 291.7 90.3 224.0 79.2 210.1 132.3 143.6

424.5 648.7 393.1 499.0 480.9 780.8 811.5 661.0 475.5 594.1 424.5 648.7 452.4 636.7 657.3 873.3 596.0 938.0 520.0 520.6 355.7 524.9 518.6 599.2 1670.0 2400.0 402.5 526.9 447.8 640.6 405.2 533.5 203.7 310.8

Lasater (1958) Vazquez and Beggs (1980) Glaso (1980) Al-Marhoun (1988) McCain (1991) Macary and El-Batanoney (1993) Petrosky and Farshad (1993) Kartoatmodjo and Schmidt (1994) Frashad et al. (1996) Al-Shammasi (1999) Velarde et al. (1997) Yi (2000)  and McCain (2003) Valko Dindoruk and Christman (2004) Arabloo et al. (2014) ANFIS (this study)

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properties of the newly developed models. These observations are in agreement with Talebi et al. (2014). 6. Conclusions

Fig. 6. Comparison between the predictions of Model IV and experimental data.

APRE obtained from correlations of Yi (2000) and Petrosky and Farshad (1993) is indicator of overestimation in their prediction. Figs. 5 and 6 are the cross plots of predicted values of saturation pressure by developed ANFIS models for and experimental values. High density of points around the unit slope line is an indication of high accuracy of the model in prediction of bubble point pressure.

5.3. Sensitivity analysis of the new models The influence of the individual independent variables on the new ANFIS models was tested to demonstrate the effect of all input variables, such as temperature, solution gas-oil ratio, gas gravity and oil API gravity, on the bubble pressure as well as OFVF. Fig. 7 shows the results of sensitivity analysis of the new ANFIS models. This figure shows the rank correlation coefficients that were calculated between the output variable (either of saturation pressure or OFVF) and the samples for each of the input parameters. In general, as the correlation coefficient between any input variable and output variable increases, the influence of that input in determining the value of the output increases. From this figure it is obvious that solution gas-oil ratio has the major impact on the PVT

Traditionally PVT properties like OFVF and saturation pressure are predicted either by equations of state or empirical correlations which are known for their limited accuracy. Also many of these correlations are based on local data sets and generate huge errors when they are used for predicting PVT properties of global crudes. To obviate the mentioned deficiencies, new models based on Adaptive Neuro-Fuzzy Inference Systems were developed for accurate prediction of OFVF and bubble point pressure. To serve as a means for predicting the PVT properties of worldwide crude samples, a comprehensive data bank from different geographical regions worldwide was used. The results of predictions were compared to many well-known correlations in literature. Comparison between the performances of proposed models and empirical correlations reveals that developed models of this study are more accurate and reliable than published correlations and in the absence of experimental facilities can serve as fast and reliable predictors. Nomenclature AAPRE average absolute percent relative error, % ANFIS adaptive Neuro-Fuzzy Inference System ANN artificial neural network API oil API gravity APRE average percent relative error, % Bo oil formation volume factor Bob oil formation volume factor at bubble point pressure BP back propagation MaxAPRE maximum absolute percent relative error, % MinAPRE minimum absolute percent relative error, % OFVF oil formation volume factor R2 coefficient of determination Pb bubble point pressure, psia Rs solution gas-oil ration, SCF/STB RMSE root mean square error SCF standard cubic feet STB stock tank barrel T reservoir temperature gg gas specific gravity ra influential radius Di density function value for i-th value Dc1 maximum density function value Appendix A. Instructions for running the proposed models Following example provides instructions for using the proposed models. First of all, the working directory of MATLAB software should be changed to the desired directory (i.e., folder containing the ANFIS models). Example: Calculate bubble point pressure and OFVF at bubble pressure for a sample with the following data:    

Fig. 7. Sensitivity analysis of the proposed models and the dependence of bubble pressure and OFVF on each of the independent variable.

Solution gas-oil ratio ¼ 595.05 SCF/STB Gas gravity ¼ 1.28 Oil API gravity ¼ 27 Temperature ¼ 271
F

Solution: The following commands should be entered in the MATLAB command window:

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Appendix B. Supplementary data Supplementary data related to this article can be found at http:// dx.doi.org/10.1016/j.jngse.2014.10.009.

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