Robust Model for the Determination of Wax Deposition in Oil Systems

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A Robust Model for Determination of Wax Deposition in Oil Systems Arash Kamari, Abbas Khaksar-Manshad, Farhad Gharagheizi, Amir H. Mohammadi, and Siavash Ashoori Ind. Eng. Chem. Res., Just Accepted Manuscript • Publication Date (Web): 27 Sep 2013 Downloaded from http://pubs.acs.org on September 28, 2013

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Industrial & Engineering Chemistry Research

A Robust Model for Determination of Wax Deposition in Oil Systems Arash Kamari,a Abbas Khaksar-Manshad,b Mohammadi,,a,d Siavash Ashoori, e

Farhad

Gharagheizi,a,c

Amir

H.

a

Thermodynamics Research Unit, School of Engineering, University of KwaZulu-Natal, ‎Howard College Campus, King George V Avenue, Durban 4041, South Africa b

Department of Petroleum Engineering, Abadan Faculty of Petroleum Engineering, Petroleum University of Technology, Abadan, Iran c

Department of Chemical Engineering, Buinzahra Branch, Islamic Azad University, Buinzahra, Iran

d

Institut de Recherche en Génie Chimique et Pétrolier (IRGCP), Paris Cedex, France

e

Department of Petroleum Engineering, Petroleum University of Technology, Ahwaz, Iran

Abstract –Wax

deposition is a serious problem during oil production in the petroleum industry. Therefore, accurate predicting this solid deposition problem can result in increasing the efficiency of oil/gas production. In this communication, a novel approach is proposed to develop a predictive model for estimation of wax deposition. An intelligent reliable model is proposed using a robust soft computing approach namely least square support vector machine (LSSVM) modeling optimized with coupled simulated annealing (CSA) optimization approach. The results of prediction operation demonstrate that there is good agreement between the estimation of CSA-LSSVM and the experimental data of wax deposition. Furthermore, the performance of the newly developed model is compared with the performance of the neural network and multi-solid models for wax deposition prediction. Results of this comparison indicate that the proposed method is superior, both in accuracy and generality, over the multisolid and neural network models. Finally, to check whether the newly developed CSA-LSSVM model is statistically correct and valid, Leverage approach, in which the statistical Hat matrix, Williams plot, and the residuals of the model results lead to identification of the probable outliers, is applied. It is found that all of the wax deposition experimental data used in the present study seem to be reliable and only one point of them is out of applicability domain of the developed models for wax deposition.

Keywords: Wax deposition; Multi-solid Modeling; LSSVM; Coupled Simulated Annealing.



Corresponding authors Email: [email protected]

1

ACS Paragon Plus Environment


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1.

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Introduction Wax deposition is a serious problem in the petroleum industry because it may result in

decreasing the efficiency of oil production, the plugging of well bores, production facilities and transportation pipelines during production and even reservoir rock

1-3

. Without an appropriate,

accurate and detailed wax management program which it can be achieved by correctly predicting this solid deposition problem, the production of oil with high wax content can result in reduced amounts of productions, shutdown, and severe economic losses due to production loss and costly remediation. This justifies the necessity for a reliable model that accurately estimates the amount of precipitated wax deposition. To design production processes efficiently, it is of great importance to predict the amount of precipitated wax and wax appearance/disappearance temperatures (WAT/WDT) using a wax model 2. Therefore, a reliable model is required to estimate the wax deposition, as already mentioned. Burger et al. 4 presented one of the commonly used thermodynamic models in which the crude oil is dissolved in a solvent mixture (ether/acetone), cooled at 253 K and filtered afterwards at this temperature. This method is widely accepted to represent the total amount of wax able to precipitate in a crude oil. Among thermodynamics methods, the multi-solid phase one is frequently used in the literature 3. In multi-solid (MS) wax model developed by LiraGaleana et al. 5, each solid phase is considered as a pure component which does not mix with other solid phases. Valinejad and Solaimany Nazar

6

conducted an experimental work to

determine the wax deposition potential of three waxy crude oils during laminar flow in a pipeline system. The Taguchi experimental design approach is applied to evaluate the impact of important operating factors such as inlet crude oil temperature, temperature difference between the oil and the pipe wall, the ﬂow rate of crude oil, wax content and time on wax precipitation phenomena. 2


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The obtained results indicated that a waxy crude oil with higher wax content could lead the more precipitated solid wax in transportation lines. Akbarzadeh and Zougari

7

developed a novel

strategy for modeling wax deposition in fluid flows. They considered several mechanisms as possible mechanisms in the wax deposition process. Among these mechanisms, particle diffusion/deposition played the most significant role in formation of the deposit at the realistic transport conditions. Bai and Zhang

8

applied the vane method to determine the yield stress of

waxy oil gels formed under quiescent or shear conditions, in which an implemented shear stress maintained during the process of cooling and isothermal holding. The obtained results demonstrated that the yield stresses dramatically decrease with increase of average carbon number of wax regardless of the quiescent or shear conditions. Kelechukwu et al. 9 proposed an empirical model for estimating wax precipitation of hydrocarbon production systems. The model exhibited good estimation ability in comparison with the laboratory measurements. However, thermodynamic models estimate wax formation conditions which are not in excellent agreement with experimental data, and they normally over or under-estimate the amount of precipitated wax and WAT/WDT 1. Three types of experiments were undertaken

10

:

(1) wax crystallization experiments on live oil to define the conditions under which wax would deposit, (2) diffusion precipitation experiments on dead oil to determine the contribution of wax diffusion to precipitation rates under various conditions, and (3) shear precipitation experiments, also on dead oil, to identify the rate of transport of deposited wax particles to

the wall.

Consequently, experimental measurements require special equipment along with expensive, difficult and time-consuming procedures. Therefore, introducing a rapid and accurate method than the experimental measurements and the thermodynamic models which can solve the aforementioned problems is necessary.

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Intelligent methods such as the Artificial Neural Network (ANN), Support Vector Machine (SVM), Genetic Algorithms (GA), Fuzzy Logic (FL) and etc for data analysis and interpretation are increasingly powerful and reliable tools that can be used in order to estimate regression and classification problems

11-16

. Least Square Support Vector Machine (LSSVM)

17

is a modification of the original SVM mathematical approach. This method uses a set of linear equations using support vectors (SVs) instead of quadratic programming problems in order to facilitate the solution of the original SVM. So far, LSSVM methodology has been used for several estimation targets in petroleum engineering18-22. However, this intelligent mathematical approach has not yet been applied for performance prediction of wax deposition in crude oil systems. This study presents a new model for the estimation of wax deposition in oil systems based on LSSVM modeling approach using available data set collected from previously published literature23-25.

Moreover, a novel feature selection mechanism base on Coupled

Simulated Annealing (CSA) optimization for tuning the optimal parameters is proposed. The CSA-LSSVM model is an adequate nominee for characterizing the nonlinear behavior and prediction of physical properties such as wax deposition. To evaluate the performance and accuracy of the newly proposed model as well as previously published ones, both statistical and graphical error analyses are applied simultaneously. Next, wax deposition predicted and in order to observe the superiority or failure of our model than traditional methods, the results are compared with the performance of multi-solid model for wax deposition and the experimental data as well as ANN model. Finally, Leverage approach, in which the statistical Hat matrix, Williams Plot, and the residuals of the model results lead to identification of the probable outliers, is implemented.

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2.

Experimental Data Acquisition In general, the applicability, reliability and accuracy of the models for estimation of

physical properties and phase behaviors of fluids depends on the comprehensiveness of the employed data set for their development 12, 26-29. Therefore, the most important parameters which affect deposition of wax must be selected. The change of oil composition, pressure and temperature may cause deposition of wax 2. As previously mentioned, wax deposition is one of the most important flow assurance problems30. The data used in this study

23-25

include nine parameters that also indicate

thermodynamic properties of the oil systems. These input parameters include compositions of C1-C3, C4-C7, C8-C15, C16-C22, C23-C29, C30+, specific gravity, system pressure, and system temperature and output parameter is wax deposition. The parameter ranges are shown in Table 1. The weight percent of precipitated wax is defined as a function of temperature (T), pressure (P), composition, and specific gravity of oil.

3.

Model Development

3.1

Support Vector Machine Strategy Although the neural network models have been generally examined to provide high

accuracy

31, 32

, they may have the disadvantages of non-reproducibility of results, partly as a

result of random initialization of the networks and variation of the stopping criteria during optimization

20

. The SVM mathematical strategy has been recognized as a consistent and

effective method proposed from the machine-learning community 17, 33. There are several criteria 5



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which may demonstrate the superiority of SVM-based models than the ANN-based methods including more probability for convergence to the global optimum, no need to identify the network topology in advance, which can be automatically determined as the training process ends, over-fitting complications are less probable in SVM strategies, no need to choosing the number of hidden neurons, acceptable generalization performance, and fewer adjustable parameters (the developed model in this study has only two adjustable parameters

17, 34, 35

).

Summary of ANN-based models performance is explained in Appendix A. A SVM is a tool for a set of related supervised learning techniques which analyze data and recognize patterns and are also utilized for regression analysis. In other words, the SVM strategy has been preliminary proposed for classification problems using the hyper-planes to define decision boundaries between the experimental data points of different classes 17. On the basis of SVM primary formulations any function f(x) can be regressed as follow36: f ( x)  wT (x)  b

(1)

where wT is transposed output layer vector,  (x) represents the Kernel function, and b stands for the bias. The input of the model, x, is of a dimension N×n in which N and n express the number of data points and number of input parameters, respectively (In case of training set, N may be regarded as the number of training set data points). Vapnik proposed minimization of the following cost function in order to calculate w and b 36:

Cost function 

(2)

N 1 T w  c ( k   k* ) 2 k 1

To satisfy constraints:

6


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 y k  wT  ( x k )  b     k , k  1,2,...., N  T *  w  ( x k )  b  y k     k , k  1,2,...., N   ,  *  0, k  1,2,...., N  k k

(3)

where xk and yk stand for kth data point input, and kth data point output, respectively. The ε denotes the fixed precision of the function approximation. The  k and  k* are slack variables. It should be considered that if we choose a small ε to propose a very accurate model, some data points may be outside of the ε precision. Consequently, this issue may result in infeasible solution. Accordingly, one should utilize slack parameters to determine the allowed margin of error. The c  0 in Eq. (2) is considered as the tuning parameter of the SVM which determines the amount of the deviation from the desired ε. In other words, one of the tuning parameters of the SVM is c. To minimize the cost function illustrated in Eq. (2) along with its constraints defined in Eq. (3), one should use the Lagrangian for this problem as follows 36:





L a, a*  













N N 1 N * * *   a  a a  a K x , x   a  a  yk ak  ak*  k k l l k l   k k 2 k ,l 1 k 1 k 1



(4) (4a)

 a N

k 1

k



 ak*  0 , ak , ak*  0, c (4b)

K xk , xl    xk   xl , k  1,2,..., N T

where ak and ak* denote Lagrangin multipliers. Eventually, the final form of the SVM is obtained as follows:

f ( x) 

N

 (a

k , l 1

k

(5)

 ak* )K ( x, xk )  b

7



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To solve the problem and find ak , ak* , and b , one should solve a quadratic programming problem which is immensely difficult. Later, Suykens and Vandewalle

17, 37

developed the least

square modification of the SVM (LSSVM) to facilitate the original SVM method. In the developed LSSVM approach, Suykens and Vandewalle 17, 37 reformulated the SVM as follows 36:

Cost function 

1 T 1 N w w    ek2 2 2 k 1

(6)

Subjected to the following constraint: yk  wT xk   b  ek

(7)

where γ is tuning parameter in LSSVM method and ek represents the error variable. The Lagrangian for this problem is as follows: Lw, b, e, a  



N 1 T 1 N w w    ek2   ak wT  xk   b  ek  yk 2 2 k 1 k 1



(8)

where ak are Lagrangian multipliers. The derivatives of Eq. (8) should be equated to zero in order to solve the problem. Thus, the following equations are obtained: N  L  0  w  ak ( xk )   w k 1  N  L  0  a  0  k  b k 1  L   0  ak  ek , k  1,2,..., N  ek  L  0  wT  xk   b  ek  yk  0 k  1,2,..., N   a  k

8


(9)

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Eq. (9) indicates that there are 2N+2 equations and 2N+2 unknown parameters (ak , ek , w, and b) . Thus, the parameters of LSSVM can be obtained by solving the system of

equations defined in Eq. (9) 36. As stated earlier, the LSSVM has a tuning parameter  . Since, either of the LSSVM and SVM are kernel-based technique, we should consider the parameters of the Kernel functions as other tuning parameters. In case of RBF kernel function: K ( x, xk )  exp(  || xk  x ||2 /  2 )

(10)

The other tuning parameter is  2 . Therefore, in LSSVM algorithm with RBF Kernel function, there are two tuning parameters which should be achieved by minimization of the deviation of the LSSVM model from experimental values

36

. The mean square error (MSE) of

the results of the LSSVM algorithm has been measured using the following equation: n

MSE 

(X i 1

rep. / predi

 X expi ) 2

(11)

n

where X is the percent of precipitated wax, subscripts rep./pred. and exp. stand for the represented/predicted, and experimental wax deposition, respectively, and n stands for the number of samples from the initial population. In this study, the LSSVM algorithm developed by Suykens and Vandewalle 17 has been used. It is worthwhile to note that the main benefit of the LSSVM over the original SVM is the idea of modifying the inequality constraints of Eq. (3) to the equality constraints of Eq. (7). The parameters of the LSSVM are easily obtained by solving the system of equations presented in Eq. (9) instead of solving a nonlinear quadratic programming 36.

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3.2

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Data Normalization During training phase of the LSSVM, higher valued input variables may be likely to

suppress the impact of the smaller ones. To subdue such an obstacle and in order to make LSSVM perform sufficiently, data must be well processed and sufficiently scaled prior to input to the LSSVM. All of the inputs and their corresponding outputs are normalized as follows.

 x xn    1.5  xmax

   0.8  0.1 

(12)

where, x denotes actual data, xmax expresses the maximum value of the data and xn stands for the normalized data

38

. Normalization procedure which is generally applied in optimization process

has been applied to obtain the parameters of LSSVM mathematical strategy, and it has no effect on the model results 39, 40. At the end, these values were returned to their original values.

3.3

Coupled Simulated Annealing Simulated annealing (SA) 41-43 as the earliest mathematical strategy (algorithm) extending

local search techniques has an explicit algorithm in order to escape from local optima. The primary idea is to allow moves which lead in solutions of worse quality than the present solution to facilitate escaping from local optima. The possibility of doing such a move is decreasing during the search process. Coupled Simulated Annealing (CSA), as a modification of SA, is designed to be able to straightforwardly escape from local optima, and accordingly, improves the accuracy of solutions without slowing down too much the speed of convergence. Suykens et al. 44

presented original principles of this method and demonstrated that coupling among local

optimization processes can be employed to improve gradient optimization approach to escape 10


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from local optima in non-convex problems. Moreover, in order to increase the quality of the ﬁnal solution, Wavier et al.

45

indicated the utilize of coupling in a global optimization technique.

Besides, by reducing a coupling strategy with minimal communication, these coupled strategies can be applied very efficiently in parallel computer architectures, to make them very interesting to the multi-core trend in the new generation of computer architectures 46.

3.4

Computational Procedure To develop our intelligent model nine input parameters have been selected including

compositions of C1-C3, C4-C7, C8-C15, C16-C22, C23-C29, C30+, specific gravity, system pressure, and system temperature, as already mentioned. In this work, first, the database is randomly divided into three sub-data sets involving the “Training” set, “Validating” set and the “Test” set. Normally, the “Training” set is applied to generate the model structure and “Validating” set as well as the “Test (prediction)” set are employed to investigate its prediction validity and capability. To develop CSA-LSSVM model about 80% of the main data set randomly selected for the “Training” set and the 10% and 10% have been applied for validating and testing phases, respectively. In distribution of the existing data into these sub-data sets, several distributions have been implemented to avoid the local minima and accumulations of the data in the feasible region of the problem. As a result, the sufficient distribution is the one with homogeneous accumulations of the data on the domain of the three sub-data sets 33.

11



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4.

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Leverage Approach In developing a model, outlier detection plays a key role to determine individual datum

(or groups of data) which may differ from the bulk of the data present in a dataset

47

. For this

end, the proposed methods, in general, consist of both numerical and graphical algorithms The Leverage approach

47-49

47-52

.

is a well-known method for outlier detection. This method relates

with the values of the residuals (i.e., the deviations of a model results from the experimental data) and a matrix known as Hat matrix consist of the experimental data and the represented/predicted values obtained from a model. The main application criterion of this method is to use a model, which is capable of adequate calculation/ estimation of the data of interest. The Leverage or Hat indices are calculated based on Hat matrix (H) with the following equation 47-52:

H  X ( X t X ) 1 X t

(13)

where X is a (n  k) matrix, in which n stands for the number of data (rows) and k denotes parameters of the model (columns), and t expresses the transpose matrix. The Hat values of the data in the feasible region of the problem are the diagonal arrays of the H value. Afterward, the Williams plot is sketched for graphical identification of the suspended data or outliers based on calculated H value through Eq. (13). This plot demonstrates the correlation of Hat indices and standardized cross-validated residuals (R), which are defined as the difference between the represented/predicted values and the applied data. Normally, a warning Leverage (H*) is fixed at the value equal to 3p/n, in which n stands for the number of training points and p is the number of model parameters plus one. The Leverage equal to three is a ‘‘cut-off’’ value to accept the points within  3 range standard deviations from the mean (to cover 99% normally distributed

12


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data). If majority of data points locate in the ranges of 0  H  H*and -3  R  3, it displays that both model development and its predictions are done in applicability domain and this consequently leads to a statistically valid model. It is worthwhile to note that ‘‘Good High Leverage’’ points are located in the domain of H *  H and -3  R  3. The Good High Leverage can be recognized as the points which are outside of applicability domain of the implemented model. The points which are situated in the range of R  -3 or 3  R (whether they are greater or smaller than the H* value) are identified as outliers of the model or ‘‘Bad High Leverage’’ points. These erroneous representations/predictions may be identified to the doubtful data 52.

5.

Results and Discussion The LSSVM parameters consist of γ and σ2 must be evaluated and optimized in order to

achieve an accurate and reliable value for prediction of wax deposition. Therefore, these parameters have been optimized using CSA mathematical optimization tool. It should be mentioned that the numbers of reported digits of the two aforementioned parameters (σ2 and γ ), in general, are obtained through sensitivity analysis of the overall error of the optimization procedure 34. Finally, the optimized values of σ2 and γ for the newly developed model have been obtained 0.72381 and 615.8804, respectively To evaluate the accuracy of the developed CSA-LSSVM model, statistical error analysis, in which squared correlation coefficients (R2), average absolute relative deviations (AARDs), standard deviation errors STD, and root mean square errors (RMSEs), and graphical error analysis, in which crossplot and error distribution is sketched, have been utilized. Definitions and equations of the aforementioned parameters are presented in Appendix B. Table 2 lists the 13



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Page 14 of 36

statistical parameters of the developed model for prediction of wax deposition. The R 2 and average absolute relative deviation of CSA-LSSVM model in testing phase are reported 0.989 and 36.3, respectively. A comparison between the represented/predicted wax deposition values and the experimental values are illustrated in Figs. 1 and 2. Fig. 1 displays the scatter diagram that compares experimental wax deposition versus CSA-LSSVM model outputs. A tight cloud of points about 45˚ line for training phase, validation and testing data sets illustrate the robustness of the proposed CSA-LSSVM model. The obtained results demonstrate that excellent agreement exists between the prediction of CSA-LSSVM and the experimental data of wax deposition. Moreover, Fig. 2 represents the error distribution of the developed CSA-LSSVM model for prediction of wax deposition. This figure confirms that the developed CSA-LSSVM model has the low scatter around the zero error and the small error range to estimate the wax deposition. These results display that the major advantage of CSA-LSSVM method is appropriate capability for predict and modeling the physical properties. Lira‐Galeana et al.

5

obtained the wax deposition values by using a multi-solid

thermodynamic model just for oils 12 and 15 among the all. Consequently, to represent the trend plot of wax deposition versus temperature and also a comparison between experimental data, the obtained values by CSA-LSSVM model, the ANN model

3

, and the multi-solid wax

thermodynamic model 5, selection of oil 12 and oil 15 from the aforementioned experimental database

23-25

seems adequate. The step-by-step solution of this thermodynamic model is

presented in Appendix C. The solution of set of Eqs. (A8-13) in Appendix C, it is possible to estimate the amount of wax deposition. However, the estimated amounts of deposited wax from the previous equations are not in good agreement with experimental data. Figs. 3 and 4 indicate the trend plot

14


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of wax deposition versus temperature for oil 12 and 15, respectively. As can be seen in Figs. 3 and 4, by increasing in temperature the wax deposition decreases at both oil systems (oil 12 and 15). Moreover, these Figures indicate that there is more agreement between the experimental data and the predicted values by CSA-LSSVM than the predicted values by ANN model

3

and

multi-solid wax thermodynamic model 5. Here, it should be noted that the CSA-LSSVM has been developed by using “two adjustable parameters” while ANN and multi-solid wax thermodynamic models require more adjustable parameters. As previously mentioned, in developing a model, outlier detection plays a key role to determine individual datum (or groups of data) which may differ from the bulk of the data present in a dataset

47, 51, 53

. Therefore, to check whether the CSA-LSSVM model is statistically

correct and valid; the Williams plot has been sketched for the obtained results for wax deposition using the CSA-LSSVM model (Fig. 5). Existence of majority of data points in the ranges 0  H 0.3103 and -3R demonstrate that the applied model for prediction of wax deposition is statistically acceptable and valid. Good high leverage points are located in the domain of 0.3103< H for developed CSA-LSSVM model. These points may be known to be outlier of the applicability domain of the implemented model. The results of the wax deposition predictive model show that only one data point is located in the aforementioned domain (Fig. 5). It may be possible to eliminate these probable outliers from the developed model results and propose more accurate ones; however, our aim, herein, has been to study the ability of all of the investigated models to estimate the whole wax deposition values from a data set in the literature. In final, it should be mentioned that although the CSA-LSSVM model was developed for the dead oil systems, it can be applied for determination of wax deposition in live oil systems. Therefore, at first, it is assumed that wax deposition does not affect the vapor-liquid equilibrium

15



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Page 16 of 36

and using an appropriate model, a vapor-liquid equilibrium calculation is first performed on live oil system, and then, the LSSVM model is applied on remaining liquid phase.

6.

Conclusion The implement of a LSSVM approach optimized by co-implementation of a CSA

algorithm was used to estimation of wax deposition as a serious problem in oil production. Field and experimental data from the literature were utilized to build the new predictive model for wax deposition at various compositions and temperature. The model was successfully applied to prediction of wax deposition against experimental data. Moreover, to evaluate the capability of the newly developed model the predicted values were compared with a multi-solid wax thermodynamic model 5. The results indicate that the predicted values outperform than the multi-solid wax thermodynamic model 5. Finally, the Leverage approach was applied to evaluate the performance of wax deposition data. The results show that only one of the data points for data points was found to be outliers (doubtful measured data) while all of the investigated data were interpreted to be within the applicability of the developed model. Therefore, the developed model is reliable for prediction of wax deposition in their domain. The results of this study reveal that SVM-based technique with the CSA based parameters tuning approach, described in this research, can result very good generalization and can advantageously be employed for estimation of wax deposition.

Supporting information The predicted values as well as the status of each data point (training, validation, test set). This material is available free of charge via the Internet at http://pubs.acs.org.

16


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Appendix A. Artificial Neural Network Strategy ANNs have large numbers of computational units called neurons, connected in a massively parallel structure and do not need an explicit formulation of the physical or mathematical relationships of the handled problem

54

.

Several types of neural network strategies

have been developed in the literature. One of the most-widely used is called ‘‘Three-Layer Feedforward ANNs. This type is applied to generate a non-linear correlation between output and input parameters. In Feed-forward ANNs, the design is based on one input layer, one output layer and hidden layers

55

. The number of neurons in the input and output layers equals to the

number of inputs and outputs physical quantities, respectively 55. But the number of the neurons in the hidden layer may vary from one to the optimum one. The data from the input neurons are propagated through the network via weighted interconnections

55

.Every i neuron in a k layer is

connected to every neuron in adjacent layers. The activation function of exponential sigmoid function which has generally and traditionally been utilized to develop ANNs 56 as following:

f ( x) 

1 1  ex

(A.1)

where x stands for parameter of activation function. A bias term, b, is associated with every interconnection in order to introduce a supplementary degree of freedom. The expression of the weighted sum, S, to the ith neuron in the kth layer (k ≥ 2) is 55 N k 1



S k ,i   ( wk 1, j ,i I k 1, j )  bk ,i



(A.2)

j 1

where w is the weight parameter between each neuron-neuron interconnection. Using this feedforward network with activation function, the output, O, of the i neuron within the hidden k layer is 55 17



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

1

Ok ,i 

N k 1

1 e



(  ( wk 1, j ,i I k 1, j ) bk ,i j 1





Page 18 of 36

1 S 1  e k ,i

(A.3)

Appendix B. Statistical Error Parameters In this study, to identify the accuracy of the newly CSA-LSSVM model a number of statistical parameters have been applied including squared correlation coefficients (R2), average absolute relative deviations (AARDs), standard deviation errors STD, and root mean square errors (RMSEs). Definitions and equations of aforementioned parameters are as follows: 1. Squared correlation coefficients:

 X N

R2  1

i 1

(A.4)

 X (i) rep./pred 

2

( i ) exp

 X N

(i) rep./pred

X



2

i

2. Average absolute relative deviations:

% AARD 

100 N  N i

X

( i ) rep / pred

X

(A.5)

 X (i ) exp

( i ) exp

3. Standard deviation errors:

STD 

1 N  N i

(A.6)

(( X (i)rep / pred )  average( X (i)rep / pred ))

2

4. Root mean square errors:

(A.7)

18


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RMSE 



1 N  X (i ) exp  X (i ) rep./pred N i



2

Appendix C. Multi-Solid Thermodynamic Model5 In this method, the criterion of vapor-liquid-solid equilibrium is that the fugacities of component i in vapor, liquid, and solid phases are identical and the stability criteria exist: N vapor-liquid isofugacity equations: f i v ( P, T , y1 , y2 ,..., y N 1  f i1 ( P, T , x1 , x2 ,..., x N 1 )  0, i  1, N

(A.8)

Ns liquid-solid isofugacity equations: s f i l ( P, T , x1 , x2 ,..., x N 1 )  f pure ,i ( P, T )  0, i  ( N  N s )  1, N

(A.9)

N-1 material-balance equations: (a) for the non-precipitating components:

 Ns S j V  V zi  xil 1      K ivl xil f i l  0, i  1, ( N  N s ) F F j F 

(A.10)

(b) for precipitating components, where all solid phases are pure:

 Ns S j V  V zi  xil 1      K ivl xil f i l  0, i  ( N  N s )  1, N  1( N s  1) F F j F  where

19


(A.11)


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Page 20 of 36

 il (T , P, x l ) K  v  i (T , P, y )

(A.12)

s f i l (T , P, z j )  f pure ,i ( P, T )  0, i  1, N

(A.13)

vl i

where fli is the liquid fugacity of component i, fvi denotes the vapor fugacity of component i, T and P are temperature and pressure, respectively. X, y, z and s express mole percent of liquid, vapor, feed and solid phases, respectively.

Nomenclature ANN

artificial neural network

GA

genetic algorithm

FL

fuzzy logic

SVM

support vector machine

LSSVM

least square support vector machine

SA

simulated annealing

CSA

coupled simulated annealing

RBF

radial basis function

WAT

wax appearance temperatures

MSE

mean square error

R2

correlation coefficient

AARD

average absolute relative deviations, %

RMSE

root mean square errors

SDE

standard deviation errors

γ

relative weight of the summation of the regression errors

20


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σ2

k

squared bandwidth slack variable

 k*

slack variable

ak

Lagrangin multiplier

ak*

Lagrangin multiplier

ek

error variable

H

Hat matrix

Ski

expression of the weighted sum

Oik

output of a neural network

fl i

liquid fugacity of component i

fvi

vapor fugacity of component i

T

temperature

P

pressure

x

mole percent of liquid

y

mole percent of vapor

z

mole percent of feed

s

mole percent of solid

21



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Table Captions Table 1. Descriptive statistics of data set for prediction wax deposition; data from 23-25. Table 2. Statistical Parameters of the developed CSA-LSSVM model to determine the wax deposition.

29



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Page 30 of 36

Figures Captions Fig. 1. Comparison between the results of the developed model and the experimental values 23-25 of wax deposition. Fig. 2. Relative deviations of the wax deposition values obtained by the proposed model from the database values 23-25. Fig. 3. Trend plot of wax deposition versus temperature and a comparison between the results of the developed model and the experimental values of oil 12 from the database 23-25 and multi-solid model 5. Fig. 4. Trend plot of wax deposition versus temperature and a comparison between the results of the developed model and the experimental values of oil 15 from the database 23-25 and multisolid model 5. Fig. 5. Detection of the probable doubtful data of wax deposition and the applicability domain of the developed CSA-LSSVM model

30


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Table 1 Parameter

Unit

Type

Min.

Avg.

Max.

Weight percent of wax deposition

%

Output

0

3.1419

13

System temperature

K

Input

230

272.657

314.150

System pressure

bar

Input

1

1

1

Input

0.872

0.918

0.963

Specific gravity Composition of C1-C3

%

Input

0.218

1.315

2.127

Composition of C4-C7

%

Input

3.057

18.476

30.952


%

Input

33.468

44.495

49.791


%

Input

16.029

29.005

57.335


%

Input

0

2.811

10

Composition of C30+

%

Input

0

3.538

13.230

31



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Table 2 Statistical Parameter

Value Training Set

R2

0.993

Average absolute relative deviation

16.0

Standard deviation error

0.24

Root mean square error

0.24

Number of used data points

71 Validation Set

R2

0.990


43.7


0.68


0.63


8 Test Set

R2

0.989


36.3


0.47


0.44


8 Total

R2

0.989


20.4


0.32


0.32


87

32


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Fig. 1

33



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Fig. 2

34


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Fig. 3

Fig. 4

35



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Fig. 5

36


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Robust Model for the Determination of Wax Deposition in Oil Systems

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