A soft computing approach for the determination of crude oil viscosity: Light and intermediate crude oil systems

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Journal of the Taiwan Institute of Chemical Engineers 000 (2015) 1–10

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A soft computing approach for the determination of crude oil viscosity: Light and intermediate crude oil systems Abdolhossein Hemmati-Sarapardeh a,∗, Babak Aminshahidy a, Amin Pajouhandeh b, Seyed Hamidreza Yousefi a, Seyed Arman Hosseini-Kaldozakh a a b

Department of Petroleum Engineering, Amirkabir University of Technology, Tehran, Iran Department of Petroleum Engineering, Shahid Bahonar University of Kerman, Kerman, Iran

a r t i c l e

i n f o

Article history: Received 8 May 2015 Revised 8 July 2015 Accepted 10 July 2015 Available online xxx Keywords: Crude oil Viscosity Least square support vector machine Temperature, Oil API gravity

a b s t r a c t Crude oil viscosity is a key property needed for petroleum engineering analysis such as evaluation of fluid flow in porous media, reservoir performance, reservoir simulation, etc. This property is traditionally measured through expensive and time consuming laboratory measurements. In this communication, about 1500 dead oil viscosity data points of light and intermediate crude oil systems from various geological locations have been collected. Afterward, a soft computing approach, namely least square support vector machine (LSSVM), has been utilized to develop two distinct viscosity models for temperatures below and above 313.15 K. The parameters of these models have been optimized using coupled simulated annealing (CSA) optimization tool. The results of this study indicated that the developed models can predict dead oil viscosity at all temperatures and oil API gravities with enough accuracy. In addition, statistical and graphical error analyses illustrated that the proposed CSA-LSSCM models outperform all of pre-existing models. Besides, the relevancy factor showed that oil API gravity has the greatest effect on dead oil viscosity. Finally, the Leverage approach demonstrated that the proposed models are statistically valid and acceptable, and only 2% of the data points may be regarded as the probable outliers. © 2015 Taiwan Institute of Chemical Engineers. Published by Elsevier B.V. All rights reserved.

1. Introduction Crude oil consumption is projected to increase year after year. Problems of crude oil consumption in the next decades will revolve around two main factors consisting of population and the increasing use in developing countries. For this reason, accurate and reliable oil properties are required. One of these properties is viscosity which is defined as the internal friction of the fluid to flow [1–4]. Reservoir oil viscosity is a key property needed for petroleum engineering analysis such as evaluation of fluid flow in porous media, reservoir performance, reservoir simulation, well testing, and design of production facilities and transport equipment [5–14]. Crude oil viscosity is a function of several thermodynamic and physical properties such as type and the nature of its chemical composition, pressure, bubble point pressure, temperature, oil specific gravity, gas gravity, and gas solubility [1,2,15,16]. Traditionally, this property is measured experimentally on subsurface or surface (recombined) samples at the reservoir temperature and pressure. Laboratory measurements of this property are always money and time consuming [2,10,11,15,17–20]. ∗

Corresponding author. Tel.: +989132437785. E-mail address: [email protected], [email protected], [email protected] (A. Hemmati-Sarapardeh).

Moreover, viscosity data at other temperatures for production equipment and pipelines design, and for planning thermal enhanced oil recovery methods is required. Due to these problems, developing predictive models for estimation of crude oil viscosity have been much attended. Several correlations have been proposed to predict crude oil viscosity. These correlations are generally categorized into two classes, depending on the input variables. The first type refers to black oil type correlations which use oilfield data, such as oil API gravity, reservoir temperature, solution gas–oil ratio, saturation pressure, and pressure to predict oil viscosity. The second type is empirical and/or semiempirical correlations which use reservoir fluid composition, normal boiling point, pour point temperature, critical temperature, molar mass, and acentric factor of components [1,13,14,19,21,22]. In the past decades, several empirical and semi-empirical correlations have been derived mostly from corresponding state equations to predict crude oil viscosity. Most of presented correlations were developed for a given region. Application of these correlations to crude oils of different sources leads to large errors. This difference is attributed to the difference in the nature of crude oil chemical composition. Above the bubble point pressure, increasing pressure causes an increase in crude oil viscosity. However, below the bubble point, increasing pressure causes a decrease in oil viscosity. Crude oil has its

http://dx.doi.org/10.1016/j.jtice.2015.07.017 1876-1070/© 2015 Taiwan Institute of Chemical Engineers. Published by Elsevier B.V. All rights reserved.

Please cite this article as: A. Hemmati-Sarapardeh et al., A soft computing approach for the determination of crude oil viscosity: Light and intermediate crude oil systems, Journal of the Taiwan Institute of Chemical Engineers (2015), http://dx.doi.org/10.1016/j.jtice.2015.07.017

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A. Hemmati-Sarapardeh et al. / Journal of the Taiwan Institute of Chemical Engineers 000 (2015) 1–10 Table 1 The origin and PVT data ranges used in dead oil viscosity models. Author

Source of data

T, K

API

μod , cP

Beggs and Robinson (1975) [27] Glasø (1980) [28] Labedi (1982) [31] Kaye (1985) [29] Al-Khafaji et al. (1987) [30] Egbogah and Ng (1990) [9] Labedi (1992) [32] Kartoatmodjo and Schmidt (1994) [33] Petrosky (1995) [34] Bennison (1998) [35] Elsharkawy and Alikhan (1999) [36] Hossain et al. (2005) [38] Naseri et al. (2005) [13] Hemmati-Sarapardeh et al. (2013) [5] This study

– North Sea Nigeria and Angola Offshore California – – Libya Worldwide Gulf of Mexico North Sea Middle East Worldwide Iran Iran Worldwide

294.2–419.2 283.1–422.0 313.1–378.1 334.8–412.0 288.7–422.0 288.1–353.1 310.9–425.4 299.8–433.1 318.7–415.3 277.0–422.0 310.9–422.0 273.1–374.8 313.7–420.9 283.1–416.5 273.1–533.1

16–58 20–48 25–45.5 7–41 15–51 5–58 32–48 14–59 25–46 11–20 20–48 7–22 17–44 17–44 20–50

– 0.60–39 0.72–21.15 – – – 0.66–4.79 0.50–586 0.72–10.25 6.40–8396 0.60–33.7 12–451 0.75–54 0.39–70 0.34–1393

highest viscosity value at atmospheric pressure. Thus, based on pressure, these correlations can be classified into three categories: dead oil (stock tank), saturated (below and at bubble point) and undersaturated (above bubble point) [1,21,23]. The evaluation of dead oil viscosity is an important step in the design of various operations. Previous studies have demonstrated that correlations developed for dead oil viscosity cannot predict dead oil viscosity with enough accuracy. Moreover, dead oil viscosity is the input parameter for saturated oil viscosity correlations; therefore, inaccurate estimation of dead oil viscosity leads to inaccurate prediction of saturated oil viscosity as well as under-saturated oil viscosity. Hence, seeking for a more accurate dead oil viscosity model is important. Literature search and review have shown that most of dead oil viscosity correlations use oil API gravity and temperature for viscosity prediction, while some of authors correlated viscosity to molar mass, normal boiling point, critical temperature, and acentric factor [24,25]. The most popular empirical models which are used in petroleum engineering calculations for predicting dead oil viscosity are those ones developed by Beal [26], Beggs and Robinson [27], Glasø [28], Kaye [29], Al-Khafaji et al. [30], Egbogah and Ng [9], Labedi [31,32], Kartoatmodjo and Schmidt [33], Petrosky [34], Bennison [35], Elsharkawy and Alikhan [36], Bergman [37], Hossain et al. [38], Naseri et al. [13] and Hemmati-Sarapardeh et al. [5]. The ranges and origin of data used by these authors to develop their correlations are listed in Table 1. These models can be categorized into two sections: 1—models developed for heavy and extra heavy crude oil systems (API < 20) 2—models developed for intermediate and light crude oil systems (API ≥ 20). Bennison [35] and Hossain et al. [38] correlations are identified for the first group. In 1998, Bennison used viscosity data to derive a new dead oil viscosity correlation [35]. In 2005, Hossain proposed an empirical correlation for dead oil, which is applicable for the heavy oils with API gravity ranging from 10 to 22.3 [38]. Beggs and Robinson [27], Glasø [28], Labedi [31,32], Petrosky [34], Elsharkawy and Alikhan [36], Naseri et al. [13], and HemmatiSarapardeh et al. [5] correlations have been proposed to predict the viscosity of intermediate and light crude oils. In 1975, Beggs and Robinson published a correlation for prediction of dead oil viscosity. The correlation developed based on analyzing 460 dead oil viscosity measurements [27]. In 1980, Glasø presented another correlation for predicting dead oil viscosity. Glasø used the data from six North Sea oil samples [28]. In 1982, Labedi derived a correlation for the viscosity of Nigeria and Angola [31]. In 1992, Labedi published a new correlation for dead oil viscosity of Libya [32]. In 1999, Elsharkawy and Alikhan developed empirical correlations for estimating dead oil viscosity of Middle East crude oils [36]. In 2005, Naseri et al. proposed a correlation for prediction of Iranian dead oil viscosity. Naseri’s correlation is considered to be a function of oil API gravity and

reservoir temperature [13]. Very recently, Hemmati-Sarapardeh et al. presented a simple correlation for dead oil viscosity using 120 dead oil viscosity data points [5]. Afterward, Hemmati-Sarapardeh et al. used the same data bank and developed an intelligent model for prediction of crude oil viscosity [7]. Some of the models including Kaye [29], Al-Khafaji et al. [30], Egbogah and Ng [9], Kartoatmodjo and Schmidt [33] can be used to estimate all API range of crude oil (heavy to light crude oils). In 1985, Kaye derived a correlation for the viscosity of California dead oil [29]. In 1987, Al-Khafaji et al. presented a viscosity correlation for dead oil by modifying Beal correlation [30]. In 1988, Egbogah and Ng developed two different correlations to compute dead oil viscosity. In 1994, Kartoatmodjo and Schmidt modified Glasø correlation [33]. Most of previously published models have been developed based on a specific region and cannot be used globally for prediction of dead oil viscosity, because characteristics of fluids are different in each region. In addition, most of these correlations have been derived based on limited data points and limited range of parameters. The main purpose of this study is to propose a universal model for accurate prediction of dead oil viscosity of light and intermediate crude oil systems (API ≥ 20) as a function of temperature and oil API gravity. It should be pointed out that heavy and extra-heavy crude oil systems (API < 20) cannot be modeled using only temperature and API, as these crude oils contain a large amount of heavy components such as asphaltenes and resins, which control their rheological behaviors. In this study, a novel soft computing modeling approach, namely least square support vector machine (LSSVM), is employed to model dead oil viscosity of light and intermediate crude oil systems. The parameters of the model are optimized using a global optimizer, namely coupled simulated annealing (CSA). In summary, the following are the main goals of this study: 1. Providing a large data bank of dead oil viscosity from various geological locations, covering a wide range of temperature and oil API gravity. 2. Developing a robust and accurate model for prediction of dead oil viscosity as a function of temperature and API, using CSA-LSSVM modeling procedure. 3. Comparing the results obtained by the proposed CSA-LSSVM model with those obtained by pre-existing correlations through statistical and graphical error analyses. 4. Checking the validity of the proposed model to see whether the model can capture the physically expected trends with variation of input parameters. 5. Investigating the relative effect of each input parameter on dead oil viscosity by means of relevancy factor. 6. Identifying the probable suspected data and applicability domain of the applied CSA-LSSVM model.

Please cite this article as: A. Hemmati-Sarapardeh et al., A soft computing approach for the determination of crude oil viscosity: Light and intermediate crude oil systems, Journal of the Taiwan Institute of Chemical Engineers (2015), http://dx.doi.org/10.1016/j.jtice.2015.07.017

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2. Database

set of equations [55,56]. The cost function of the least square support vector machine (LSSVM) is determined as:

Dead oil viscosity is assumed to be a function of two variables: reservoir oil gravity (API) and reservoir temperature (T).

μod = f (API, T )

Cost function =

(1)

In this study, almost all experimental dead oil viscosity data, which are available in open literature sources, have been used [5,6,13,14,21,23,28,38–45]. As can be seen, these data are from various geological locations and cover a wide range of temperature and API. These data points include oil API gravity, reservoir temperature, and dead oil viscosity. The PVT data ranges used in this study are summarized in Table 1.

1 1 T w w+ 2 2

ϕ(x) + b

(2)

where w and ϕ(x) show the transposed output layer vector and the kernel function, respectively, and b is the bias. x is the input of the model which has a dimension of N × n, where N and n represent the number of data points and number of input variables, respectively. Vapnik used minimization of the following cost function to calculate w and b [54]:

 1 T ( ξk − ξk∗ ) w +c 2 N

(3)

k=1

Subjected to the below constraints:

⎧ T ⎨yk − w ϕ( xk ) − b ≤ ε + ξk , ∗ T w ϕ( xk ) + b − yk ≤ ε + ξk , ⎩ ∗ ξk , ξk ≥ 0,

k = 1, 2, 3, . . . , N k = 1, 2, 3, . . . , N

(4)

k = 1, 2, 3, . . . , N

ϕ( xk ) + b + ek

N N 1 1  2  L(w, b, e, a) = wT w + γ ek − ak (wT ϕ(xk ) + b + ek − yk ) 2 2

ε

(

ak − a∗k

k=1 N 

)+

N 

(

yk ak − a∗k

)

(5)

(5a)

k=1 T

ϕ ( xl ) , k = 1, 2, . . . , N

(5b)

αk∗

where αk and are Lagrangian multipliers. Consequently, the final form of the SVM is obtained as follows:

f (x) =

N 

( ak − a∗k )K ( x − xk ) + b

⎧ N  ∂L ⎪ ⎪ = 0 ⇒ w = a ϕ( x ) ⎪ ⎪ ∂w k k ⎪ ⎪ k=1 ⎪ ⎪ N ⎪  ⎪ L ∂ ⎨ =0⇒ ak = 0 ∂b k=1 ⎪ ⎪ ∂L ⎪ ⎪ = 0 ⇒ a = γ ek , k = 1, 2, . . . , N ⎪ ⎪ ∂ ek k ⎪ ⎪ ⎪ ⎪ ∂L ⎩ = 0 ⇒ wT ϕ(xk ) + b + ek − yk = 0 k = 1, 2, . . . , N ∂ ak

(6)

(10)

where γ is a tuning parameter of LSSVM. As can be found in Eq. (10), there are 2N + 2 equations and 2N + 2 unknown parameters ( ak , ek , w and b). Therefore, the parameters of LSSVM are achieved by solving the system of equations depicted in Eq. (10). In this work, the radial basis function (RBF) Kernel was employed, which is presented as below: (11)

where σ 2 is the other tuning parameter. Hence, in LSSVM with RBF kernel function two tuning parameters exist, which can be obtained by minimizing deviation of the experimental data from the predicted values. Normally, minimization of mean square error is considered as the objective function to find the tuning parameters of LSSVM model. This error is defined as follows:

n

k=1

( ak − a∗k ) = 0, ak , a∗k ∈ [0, c]

K ( x k − xl ) = ϕ ( x k )

k=1

where ak is Lagrangian multipliers. In order to solve the problem, the derivatives of Eq. (9) should be equated to zero. Thus, the following equations are obtained:

MSE =

k,l=1

−

(8)

where γ and ek are tuning parameter in LSSVM method and the variable error, respectively. The Lagrangian for this problem is as follows:

2

N 1  ( ak − a∗k )( al − a∗l )K ( xk − xl ) 2 N 

(7)

k=1

K ( x, xk ) = exp (−|| xk − x || / σ 2 )

where xk , yk and ε are the kth data input, kth data output, and fixed precision of the function approximation, respectively. ξk and ξk∗ stand for slack variables, which should be used to determine the allowed margin of error. c is considered as the tuning parameter of the SVM. In order to minimize the cost function, the Lagrangian of this problem should be used as follows [54]: ∗ L ( a, a ) = −

e2k

(9)

T

Cost function =

N 

k=1

Support vector machine (SVM) methodology is a tool for both classification and regression analysis [46,47] and recently has been applied in several fields [48–53]. According to SVM primary formulations, any function f(x) can be written as follows [54]:

f (x) = w

γ

Subjected to the following constraint:

yk = wT

3. Model development

T

3

i=1

( Orep./ pred.i − Oexp .i )2 n

(12)

where O is the output, subscripts rep./pred. and exp. denote the represented/predicted and experimental values, respectively, and n expresses the number of data points. In this study, coupled simulated annealing has been used to optimize the parameters of the developed LSSVM model. More details about this optimization technique can be found elsewhere [57]. A schematic flowchart of the applied CSA-LSSVM strategy is demonstrated in Fig. 1. 4. Performance evaluation Statistical and graphical error analyses can be used to evaluate the performance of the proposed models. 4.1. Statistical error analysis

k,l=1

SVM are usually solved by finding solutions to quadratic programming issues with linear inequality constraints. For this reason, in 1999, Suykens and Vandewalle [55,56] proposed a least square modification to the original SVM (LSSVM) in order to improve the SVM method. In LSSVM method, solution is obtained by solving a linear

These statistical parameters are normally used to quantitatively evaluate the performance of a model: mean percentage error (MPE), mean absolute percentage error (MAPE), root mean square error (RMSE), and standard deviation (SD). Formulation of these parameters is given below.

Please cite this article as: A. Hemmati-Sarapardeh et al., A soft computing approach for the determination of crude oil viscosity: Light and intermediate crude oil systems, Journal of the Taiwan Institute of Chemical Engineers (2015), http://dx.doi.org/10.1016/j.jtice.2015.07.017

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Input data Random division of data into training and testing

Implement Coupled Simulated Annealing (CSA)

Training data

Select model features (σ2,γ)

Testing data

Employ feature subset

(σ 2,γ)

Construct dead oil viscosity model

No

Model evaluation using both training and testing data

Meet stopping criterion?

Yes Retrain the LSSVM model using the optimum features

Optimum model features (σ 2,γ) obtained

Final LSSVM model Fig. 1. A typical flowchart for the applied CSA-LSSVM algorithm in this study.

1. Mean percentage error (MPE).

4.2. Graphical analysis

n 1 Ei n

Er =

(13)

i=1

where n and Ei are the total number of experimental data and the relative deviation of an estimated value from the corresponding experimental value (Ei is expressed as percent relative error), respectively:



Ei =

(μ)exp . − (μ)est. × 100 ⇒ i = 1, 2, 3, . . . , n (μ)exp .

(14)

2. Mean absolute percentage error (MAPE).

Ea =

n 1 |Ei | n

(15)

Graphical tools can be used to check the accuracy and validity of a model. In this study, error distribution curve and cumulative frequency plot are used. 1. Error distribution curve: it is a tool to quantify error distribution around the zero error line to designate if the model has an error trend or not. 2. Cumulative frequency plot: in this technique, the proportion of the data points having absolute percent relative errors below different ascending values is plotted versus the absolute percent relative error in a cumulative manner for each model on the same figure to facilitate the visual comparison. 5. Results and discussion

i=1

3. Root mean square error (RMSE):



RMSE =

n 2 1  μiexp . − μiest. n

(16)

i=1

4. Standard deviation (SD).



SD =

n 1  n−1 i=1

μiexp . − μiest. μiexp .

2 (17)

In this study, viscosity of dead oil is predicted by a robust soft computing approach. Selecting appropriate input parameters for model development is a key factor. Similar to most of previously published correlations, temperature and oil API gravity were selected as the input parameters of the proposed model. Many attempts were undertaken to develop a single model for the prediction of all viscosity data; however, the proposed models were not able to satisfactorily predict dead oil viscosity of all data set. Afterward, the data bank were divided into two subsections; viscosity data at temperatures below 313.15 K (40 °C) and viscosity data at temperatures above 313.15 K (40 °C). Therefore, two distinct models were developed for

Please cite this article as: A. Hemmati-Sarapardeh et al., A soft computing approach for the determination of crude oil viscosity: Light and intermediate crude oil systems, Journal of the Taiwan Institute of Chemical Engineers (2015), http://dx.doi.org/10.1016/j.jtice.2015.07.017

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5

Table 2 Statistical parameters of the proposed models for the determination of viscosity.

90

Statistical parameter Below 313.15 K

Above 313.15 K

All data

70 Training set MAPE% MPE% RMSE, cP SD Number of data Test set MAPE% MPE% RMSE, cP SD Number of data Total MAPE% MPE% RMSE, cP SD Number of data

17.92 –3.57 29.26 0.24 746

16.56 1.83 8.80 0.21 451

16.61 –1.27 35.17 0.23 187

16.56 –2.08 8.22 0.22 113

17.66 –3.11 30.54 0.24 933

16.38 –1.30 8.35 0.22 564

50 30 10

17.17 –2.43 24.65 0.23 1497

Mean Absolute Percentage Error, %

110

Fig. 2. Mean absolute percentage error of the models.

temperatures below 313.15 K (Model 1) and above 313.15 K (Model 2). To achieve the most efficient and accurate models, the data points in each region were randomly distributed into two subsets as training and testing sets. In each region, the training set was used to construct the model, to achieve the optimal model structure, and to educate existing physical rules in the system. The testing set was used to evaluate the performance of the developed model in terms of accuracy and reliability. 80% of the data points were used for model development and the remaining 20% were used as testing set. In distribution of data points into training and testing sets, several distributions were done to escape from local accumulation of the data points in the feasible region of the problem. It should be noted here that, 933 of the viscosity data points have a temperature below 313.15 K, while 565 of the data points are for crude oil systems with temperatures above 313.15 K. The values of σ 2 and γ for Model 1 (temperature below 313.15 K) were obtained 0.6807 and 4469.1858, respectively. As pointed out earlier, coupled simulated annealing was used to find the optimal model parameters. For Model 2 (temperature above 313.15 K), the values of σ 2 and γ were obtained 22.8155 and 950.1749, respectively. It should be noted that it is not necessary to develop a distinct model for higher temperatures (for example, temperatures higher than 500 K) as viscosity is not strongly affected by temperature at very high temperatures. Model 1 predicts dead oil viscosity data with an MAPE of 17.66%, while Models 2 can predict viscosity data of dead oils with an MAPE of 16.38 %, which shows that Model 2 is slightly more accurate than Model 1. On the other hand, RMSE (cP) obtained by Model 1 (30.54 cP) is much higher than that of Model 2 (8.35 cP). For a given crude oil system, viscosity data at lower temperatures have very large values compared to those of high temperatures. As it is known, RMSE (Eq. 16) is strongly affected by the values of experimental data and predicted values; therefore, as expected, the value of RMSE (cP) for Model 1 is higher than that of Model 2. The other statistical quality measures of the proposed models are summarized in Table 2.

50

RMSE, cP

70

30

10

Fig. 3. Root mean square error of the models. Table 3 Statistical parameters of the proposed model and previously published correlations for the determination of viscosity. Author

MPE (%)

MAPE (%)

RMSE, cP

SD

Beggs and Robinson Glasø Labedi—Libya Labedi—Nigeria and Angola Egbogah and Ng Kaye Al-Khafaji Petrosky Kartoatmodjo and Schmidt Bennison Elsharkawy Hossain Naseri Hemmati-Sarapardeh This study

–1643.47 99.97 16.08 –17.96 17.64 21.08 5.18 25.60 15.12 84.80 –6.93 85.65 29.13 31.09 –2.43

1651.05 99.97 36.86 41.49 26.42 32.78 21.55 30.38 22.81 88.84 23.08 87.68 32.24 32.95 17.17

595573.92 69.29 61.52 57.82 42.58 543.62 54.96 59.93 48.69 40.32 321.41 39.73 55.68 56.68 24.65

444.13 1.00 0.61 0.53 0.36 0.55 0.29 0.37 0.29 0.91 0.40 0.90 0.38 0.38 0.23

5.1. Validation of the developed models compared to pre-existing models To pursue the objective of this study, performance of the developed models were compared to pre-existing correlations by means of statistical and graphical error analyses. Generally speaking, MAPE and RMSE are the most relevant statistical quality measures. MAPE of different models have been sketched in Fig. 2, demonstrating that the proposed models outperform all of pre-existing models. Besides,

it is found that Glasø [28], Hossain [38], and Bennison [35] correlations predict dead oil viscosity with an MAPE more than 80%. Also, Beggs and Robinson [27] correlation predicts dead oil viscosity with an MAPE of 1651%, which was not shown in this figure. Fig. 3 illustrates RMSE (cP) of different models in predicting viscosity of dead oils. As can be seen, the CSA-LSSVM models predict dead oil viscosity with the smallest RMSE (cP), showing their excellent performance.

Please cite this article as: A. Hemmati-Sarapardeh et al., A soft computing approach for the determination of crude oil viscosity: Light and intermediate crude oil systems, Journal of the Taiwan Institute of Chemical Engineers (2015), http://dx.doi.org/10.1016/j.jtice.2015.07.017

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300

300

200

200

Relative Error, %

Relative Error, %

6

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100 0

-100

0

-100

-200

-200

Al-Khafaji

-300

Kartoatmodjo and Schmidt

-300 20

30

API

40

50

20

300

300

200

200

Relative Error, %

Relative Error, %

100

100 0

-100

30

API

40

50

100 0

-100

-200

-200

Elsharkawy

-300

This study

-300 20

30

API

40

50

20

30

API

40

50

Fig. 4. Error distribution curves of the models.

1 0.9

Cumulative Frequency

0.8 0.7 0.6 0.5 0.4 Hemmati-Sarapardeh Naseri Elsharkawy Kartoatmodjo and Schmidt Al-Khafaji This study

0.3 0.2 0.1 0 0

5

10

15

20

25

30

35

40

45

50

Absolute Relative Error, % Fig. 5. Cumulative frequency of data points versus absolute relative error for different models.

The other result obtained in this figure is that Glasø [28], Labedi— Libya [32], Labedi—Nigeria and Angola [31], Kaye [29], Al-Khafaji [30], Petrosky [34], Elsharkawy [36], Naseri [13], Hemmati-Sarapardeh [5] correlations cannot satisfactorily predict dead oil viscosity and provides RMSEs (cP) more than 50. Also, Beggs and Robinson [27] correlation estimates dead oil viscosity data with an RMSE (cP) more than 595,000, which was not shown in this figure. From Figs. 2 and 3, it is evident that the proposed models in this study are more robust, reliable, and accurate than pre-existing correlations, in terms of MAPE (%) and RMSE (cP). The other statistical parameters obtained by the proposed models in this study and pre-existing correlations are reported in Table 3. As illustrated, the proposed models provide the smallest MPE, MAPE, RMSE, and SD, indicating the better efficiency of the proposed models over the existing correlations. Care should be taken that the proposed models in this study are only for interme-

diate and light crude oil systems (i.e., API ≥ 20). This study does not aim to develop a model for heavy crude oils (API < 20) as viscosity of heavy and extra heavy crude oils (API < 20) are strongly affected by heavy components such as resins and asphaltenes, and cannot be modeled by using only API and temperature. By comparison of MAPE and MPE, it can be concluded that Beggs and Robinson [27], Labedi— Nigeria and Angola [31] correlations overestimate dead oil viscosity, while Glasø [28], Labedi—Libya [32], Egbogah and Ng [9], Kaye [29], Petrosky [34], Kartoatmodjo and Schmidt [33], Bennison [35], Hossain [38], Naseri [13], Hemmati-Sarapardeh [5] correlations underestimate this property. To graphically compare the performance of CSA-LSSVM models against the existing correlations, two graphical techniques were employed. The proposed models in this study were compared to three of the most accurate correlations in terms of mean absolute percentage

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7

1000 Experimental data at T=294.94 K Predicted values at T=294.94 K Experimental data at T=310.46 K Predicted values at T=310.46 K

Viscosity, cP

100

10

1 20.00

25.00

30.00

35.00

40.00

45.00

50.00

Oil API gravity

Fig. 6. Experimental viscosity data and the predicted values by the proposed model (Model 1) at two temperatures below 313.15 K.

100 Experimental data at T=333.15 K Predicted values at T=333.15 K Experimental data at T=422.03 K Predicted values at T=422.03 K

Viscosity, cP

10

1

0.1 20.00

25.00

30.00

35.00

40.00

45.00

50.00

Oil API gravity

Fig. 7. Experimental viscosity data and the predicted values by the proposed model (Model 2) at two temperatures above 313.15 K.

error, namely Al-Khafaji [30] correlation, Kartoatmodjo and Schmidt [33] correlation, and Elsharkawy [36] correlation. Error distribution curves of these models have been plotted in Fig. 4. As can be observed, the proposed CSA-LSSVM models provide the least scattering around the zero error line, which reveals the better performance of the proposed models against the existing correlations. Furthermore, five of the most accurate correlations as well as CSA-LSSVM models were selected to compare their cumulative frequency plots. Fig. 5 shows that the proposed models in this study, as the most accurate models, predict 70% of data points with absolute relative error less than 22%. Elsharkawy [36] correlation predicts 60% of data points with absolute relative error less than 22%. The proposed models in this study predict only 6% of data points with absolute relative error more than

50%. Care should be taken that the high errors of the developed models in predicting viscosity of some crude oils are due to high dependency of oil viscosity on oil nature and source; as already mentioned, the data points in this study have been collected from variety of geological and geographical locations. Therefore, it is not possible to develop very accurate models for these data points as the nature of crude oils are different. 5.2. Checking the physically expected trends of viscosity for the developed models To check the validity of the proposed models against the input parameters of the developed models (temperature and API),

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0.1 0

Temperature

API

-0.2

-0.13 -0.3

-0.19

-0.4 Above 313.15 K

-0.5

Below 313.15 K

-0.6

-0.51

Relevancy Factor

-0.1

The relevancy factor was firstly used by Chen et al. [58] to understand the relative effect of their model inputs on the minimum miscibility pressure. Moreover, this factor was also used by other researchers to find the relative impact of the input parameters on their models’ outputs [59–62]. The higher the absolute value of r between any input and output, the greater the effect of that input on the output. It is interesting to note that, r value with directionality provides clearer understanding about the overall impact; therefore, it was used in this study. Using the following formula, the r values are obtained:

r(Inpk , μ) =



n

i=1

n

-0.7 -0.64

Fig. 8. Relative effect of each input parameter on viscosity at temperatures below and above 313.15 K.

experimental viscosity data and predicted values by the proposed models were sketched in Figs. 6 and 7. Fig. 6 shows the experimental data and predicted values by Model 1 at two temperatures of 294.94 and 310.46 K. It is clear that, crude oils with lower API (higher density) have higher viscosity. However, as oil API gravity increases, oil viscosity decreases exponentially. Moreover, the higher the temperature, the lower the viscosity. These physical rules exist in crude oil systems and the proposed model can obey these trends. It is worth noting that the vertical axes in Figs. 6 and 7 are in logarithmic scales; therefore, in these figures the viscosity decreases linearly as a function of API. Fig. 7 illustrates the experimental data and predicted values by Model 2 at temperatures of 333.15 and 422.04 K. As can be seen, the proposed model can capture the physically expected trends with variation of temperature and oil API gravity. Besides, the predicted values by Model 2 are in good agreement with the experimental data. 5.3. Sensitivity analysis To quantitatively assess the relative impact of temperature and oil API gravity on dead oil viscosity by means of the developed models, a sensitivity analysis was carried out. To this end, the relevancy factor (r) was employed to evaluate the influence degree of temperature and API on both of the proposed models (Model 1 and Model 2) [57,58].

(Inpk,i − Inpk )(μi − μ) 2 n i=1

i=1 (Inpk,i − Inpk )

(18)

(μi − μ)2

where μi and μ are the ith value of the predicted dead oil viscosity and the average value of the predicted dead oil viscosities, respectively. Inpk , i and Inpk denote the ith value and the average value of the kth input variable, respectively (k = temperature and API). In this study, the relevancy factor was calculated for both of the models (Model 1 and Model 2). The relative effect of each parameter (temperature and API) on the dead oil viscosity is presented in Fig. 8. As can be seen, the effect of both temperature and API are negative, meaning that an increase in temperature/API leads to a decrease in dead oil viscosity. Furthermore, it was found that, for both of the models, API has larger effect on dead oil viscosity. In addition, API in Model 2 has a greater effect on the dead oil viscosity than that in Model 1. Generally speaking, API plays a key role in viscosity behavior of crude oil systems. 5.4. Applicability domain of the models and outlier detection In the last part of this study, outlier detection, which is of a vital importance in developing mathematical models/correlations, has been conducted. Some methods have been proposed for outlier detection and identifying the applicability domain of a model/correlation of which Leverage is known as one of the most popular ones [63–65]. In this method, residual is defined as the difference between the experimental data points and their corresponding values predicted by the model. Afterward, to calculate Hat or Leverage indices, the following H matrix should be calculated [63,64]:

H = X (X t X )−1 X t

(19)

where X is an N × k matrix in which N is the number of data points (rows) and k stands for the number of model parameters (columns),

9 Valid Data Suspected Data Out of Leverage Upper Suspected Limit Upper Suspected Limit Leverage Limit

Standardized Residuals

6

3

0

-3

-6

-9 0

0.002

0.004

0.006

0.008

0.01

hat Fig. 9. Recognition of the probable outlier data and applicability domain of the developed model for crude oil systems with temperatures below 313.15 K.

Please cite this article as: A. Hemmati-Sarapardeh et al., A soft computing approach for the determination of crude oil viscosity: Light and intermediate crude oil systems, Journal of the Taiwan Institute of Chemical Engineers (2015), http://dx.doi.org/10.1016/j.jtice.2015.07.017

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6 Valid Data Suspected Data Out of Leverage Upper Suspected Limit Lower Suspected Limit Leverage Limit

Standardized Residuals

4

2

0

-2

-4

-6 0

0.005

0.01

0.015

0.02

0.025

hat Fig. 10. Recognition of the probable outlier data and applicability domain of the developed model for crude oil systems with temperatures above 313.15 K.

and t represents the transpose of matrix X. The diagonal elements of the H matrix are called hat in the feasible region of the problem. Williams plot is sketched for graphical presentation of the applicability domain of the used model and suspected data. In fact, Williams plot illustrates the correlation of Hat indices and standardized residuals (R). A warning Leverage (H∗ ) is fixed at a value equal to 3(k + 1)/N, in which N (Model 1: N = 933; Model 2: N = 564) and k (k = 2 for both models) are the number of data points and variables, respectively. Thus, H∗ of Model 1 and Model 2 are set to be 0.0096 and 0.0158, respectively. Existence of most of the data points in the range of 0 ≤ H ≤ H∗ and –3 ≤ R ≤ 3 reveals that both the data used for the model development and the data predicted by the applied model are in the applicability domain; and consequently, the model is statistically valid. Williams plots for Model 1 and Model 2 have been sketched in Figs. 9 and 10, respectively. For Model 1, only 11 data points have R > 3 or R < –3, and 4 data points have hat > H∗ . These data points (1.6% of data points) may be regarded as the probable outlier ones, and are located out of the applicability domain of Model 1. For Model 2, only 9 data points have R > 3 or R < –3, and 8 data points have hat > H∗ . Therefore, 3% of data points are located out of the applicability domain of Model 2, and may be regarded as the probable outliers. In conclusion, both of the models are statistically acceptable and valid. It is worthwhile to note that data points with lower values of R and hat are recognized as more reliable ones. 6. Conclusions In this study, a supervised learning algorithm, namely least square support vector machine, has been used to develop reliable and accurate models for estimation of dead oil viscosity of light and intermediate crude oil systems. About 1500 data points from various geological locations were used to develop these models as a function of temperature and oil API gravity. Two distinct models for crude oil systems with temperatures lower than 313.15 K and higher than 313.15 K have been developed. According to the results obtained in this study, the following conclusions can be drawn: 1. The proposed CSA-LSSVM models estimate crude oil viscosity of light and intermediate systems with enough accuracy. 2. The statistical and graphical error analyses reveal that the developed CSA-LSSVM models are more accurate, reliable, and superior to all of the existing correlations. 3. The CSA-LSSVM models can follow the physically expected trends with variation of temperature and oil API gravity.

4. The results demonstrate that both of API and temperature have negative effects on dead oil viscosity, which means that an increase in API/temperature causes a decrease in viscosity. 5. The relevancy factor indicates that API has the most important effect on dead oil viscosity. 6. The Leverage approach shows that both of Model 1 and Model 2 are statistically valid and acceptable. In addition, less than 2% of data points may be regarded as the probable outliers.

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Please cite this article as: A. Hemmati-Sarapardeh et al., A soft computing approach for the determination of crude oil viscosity: Light and intermediate crude oil systems, Journal of the Taiwan Institute of Chemical Engineers (2015), http://dx.doi.org/10.1016/j.jtice.2015.07.017