Robust Model for the Determination of Wax Deposition in Oil Systems

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Article

A Robust Model for Determination of Wax Deposition in Oil SystemsArash 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

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1

A Robust Model for Determination of Wax Deposition in Oil

Systems

Arash Kamari,a Abbas Khaksar-Manshad,

b Farhad Gharagheizi,

a,c Amir H.

Mohammadi,,a,d

Siavash Ashoori, e

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 multi-

solid 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]

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1. 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 Lira-

Galeana 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.

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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

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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

:

bwxf T (x))( (1)

where Tw 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

:

N

k

kk

T cw function Cost1

* )(2

1

(2)

To satisfy constraints:

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N1,2,....,k

N1,2,....,k ybxw

N1,2,....,k bxwy

kk

kkk

T

kk

T

k

,0,

,)(

,)(

*

*

(3)

where kk y and x stand for k

th data point input, and k

th 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 0c 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

:

N

k

kkk

N

k

kklkll

N

lk

kk aayaaxxKaaaaaaL1

*

1

**

1,

** ,2

1,

caa ,aa kk

N

k kk ,0,0 *

1

*

N1,2,...,k xxxxK l

T

klk ,,

(4)

(4a)

(4b)

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

as follows:

bxxKaaxf k

N

lk

kk

),()()(1,

*

(5)

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To solve the problem and find b and a a kk ,, * , 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

:

N

k

k

T eww function Cost1

2

2

1

2

1 (6)

Subjected to the following constraint:

kk

T

k ebxwy

(7)

where γ is tuning parameter in LSSVM method and ek represents the error variable. The

Lagrangian for this problem is as follows:

N

k

kkk

T

k

N

k

k

T yebxwaewwaebwL11

2

2

1

2

1,,,

(8)

where ka 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:

N1,2,...,k yebxw a

L

N1,2,...,k ea e

L

a b

L

xa w w

L

kkk

T

k

kk

k

N

1k

k

N

1k

kk

00

,0

00

)(0

(9)

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Eq. (9) indicates that there are 2N+2 equations and 2N+2 unknown parameters

b) and wea kk ,,,( . 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:

)/||||exp(),( 22 xxxxK kk (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

XX

MSEii

n

i

predrep

2

exp

1

/. )(

(11)

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 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.

1.08.05.1 max

x

xxn (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

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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

.

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4. 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 47-52

.

The Leverage approach 47-49

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

:

tt XXXXH 1)( (13)

where X is a (nk) 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

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data). If majority of data points locate in the ranges of 0 HH*and -3R 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 HH *and -3R 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 3R (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

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statistical parameters of the developed model for prediction of wax deposition. The R2 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

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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 -3R 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

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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.

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17

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 Feed-

forward 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:

xexf

1

1)( (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

1

1

,,1,,1, )(kN

j

ikjkijkik bIwS (A.2)

where w is the weight parameter between each neuron-neuron interconnection. Using this feed-

forward network with activation function, the output, O, of the i neuron within the hidden k layer

is 55

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18

ikkN

jikjkijk

SbIw

ike

e

O,1

1,,1,,1 1

1

1

1

)((,

(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: (A.4)

N

i

N

i

i

XX

XX

R2

rep./pred (i)

1

2

rep./pred (i)exp)(2 1

2. Average absolute relative deviations: (A.5)

N

i i

ipredrepi

X

XXN

AARD

exp)(

exp)(/)(100%

3. Standard deviation errors: (A.6)

N

i

XaverageX predrepipredrepiNSTD ))()(( /)(/)(

1 2

4. Root mean square errors: (A.7)

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19

N

i

ii XXN

RMSE2

rep./pred)(exp)(

1

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:

NixxxTPfyyyTPf NiN

v

i ,1,0),...,,,,(,...,,,,( 121

1

121 (A.8)

Ns liquid-solid isofugacity equations:

NNNiTPfxxxTPf s

s

ipureN

l

i ,1)(,0),(),...,,,,( ,121 (A.9)

N-1 material-balance equations:

(a) for the non-precipitating components:

)(,1,01 s

l

i

l

i

vl

i

N

j

jl

ii NNifF

VxK

F

V

F

Sxz

s

(A.10)

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

)1(1,1)(,01

ss

l

i

l

i

vl

i

N

j

jl

ii NNNNifF

VxK

F

V

F

Sxz

s

(A.11)

where

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20

),,(

),,(

yPT

xPTK

v

i

ll

ivl

i

(A.12)

NiTPfzPTf s

ipurej

l

i ,1,0),(),,( , (A.13)

where fli is the liquid fugacity of component i, f

vi 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

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σ2 squared bandwidth

k 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

fli 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

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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.

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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

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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

Specific gravity Input 0.872 0.918 0.963

Composition of C1-C3 % Input 0.218 1.315 2.127




Composition of C23-C29 % Input 0 2.811 10

Composition of C30+ % Input 0 3.538 13.230

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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





Test Set

R2 0.989





Total

R2 0.989





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Fig. 1

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Fig. 2

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Fig. 3

Fig. 4

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Fig. 5

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Robust Model for the Determination of Wax Deposition in Oil Systems

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