Development of a least squares support vector machine model for prediction of natural gas hydrate formation temperature

Mohammad Mesbah *,Ebrahim Soroush ,Mashallah Rezakazemi

1 Young Researchers and Elites Club,Science and Research Branch,Islamic Azad University,Tehran,Iran

2 Young Researchers and Elites Club,Ahvaz Branch,Islamic Azad University,Ahvaz,Iran

3 Department of Chemical Engineering,Shahrood University of Technology,Shahrood,Iran

1.Introduction

As water is always present during the production of reservoir fluids,the formation of hydrate is very likely[1,2].The occurrence of hydrates in wellheads,multiphase transfer pipelines,and surface facilities is a common problem.Plugging of pipelines because of hydrate formation can sometimes result in production stoppages even for months[1].

Hydrates are crystalline compounds that in its structure a small gas molecule is trapped in the cage of water molecules joined by hydrogen bonding[3–6].Under low-temperatures and high pressures,hydrate formation is very probable.There are three forms of hydrate structures,sI,sII,andSH[1].

Finding hydrate formation temperature(HFT)for any precise gas composition through experimental methods seems impractical.In this manner,itis crucialto find a methodicalsystemfor exactprognostication of hydrate formation.Many hydrate formation prediction correlations could be found in literature,including:Hammerschmidt[7],Katz gravity and K-value methods[8,9],Baillie and Wichert[10],Mann[11],Makogon[12],Berg[13],Kobayashi[14],Motiee[15],?stergaard[4],Towler and Mokhatab[16],Bahadori and Vuthaluru[17]Safamirzaei[18],and Salufu[19].All the mentioned correlations suffer from,not distinguishing between different types of hydrates.The existence of even a very slight amount ofsIIstructure hydrate former results in predominance ofsIIstructure in the hydrate phase.This is while structural difference,have a very significant effect on HFT.It may be the key reason,in the failure of these correlations in predicting HFT for natural gas mixtures[4].

Although,it is not appropriate for industrial applications,groundwork of thermodynamic based models is van der Waals and Platteeuw's work[3,4,20–22].Later Parrish and Prausnitz suggested a thermodynamic model that suits engineering accuracy[20].A thermodynamic model was presented by Ng and Robinson that could predict HFT when liquid hydrocarbon was present.In addition,Holderet al.,John and Holder,Johnet al.,and Chen and Guo[22–25]suggested different adjustments of van der Waals and Platteeuw.These thermodynamic models assume water activity coefficient of unity.

Sour gas generally denotes naturalgas,which contains H2S.Ifa large amount of carbon dioxide is also contained,it should be referred to as acid gas.The high solubility of CO2and H2S causes reaction with water molecules and thus the activity coefficient could not be unity[3].Sun and Chen coupled their method with Chen and Guo to resolve this problem[25].Nevertheless,these thermodynamic models require numerous parameter adjustments and are tedious.

Recently,due to increasing development in artificial intelligence,predicting HFT has found fascinating alternatives.Particularly,artificial neural networks(ANN)have been in spotlight in predicting hydrate formation[2,26–32].However,because of random initialization and variation of the stopping criteria during the optimization of the model parameters,using ANNs for external prognostications is not a good choice[33].

For having an efficient process and developing a strong model,accurate and reliable data are required.So the quality of experimental data should examine through a general method which be appropriate for different experimental techniques.

In this study,we are intended to propose a novel approach based on the least squares support vector machine(LSSVM)method for prediction of hydrate formation temperature(HFT)of a gas mixture(sweet or sour).This general model will predict the HFT based on the chemical properties and types of hydrate structure that each species,which may be present in natural gas will form.This approach is followed in the hope that the proposed model has the ability of precise prediction of both sour and sweet gases.The statistically based Leverage approach is used for avoiding doubtful data.The validity and sensitivity of the model will be checked through statistical parameters.At the end,the model will be compared with existing correlations and thermodynamic models for HFT.

2.Modeling

2.1.LSSVM model

Support vector machine(SVM)is a non-probabilistic binary linear classifier,which has been developed by the machine learning community[34–36].This robust mathematical tool can be used for solving verity of complex problems from nonlinear function approximation to pattern classification.By projecting the input spectra to a higher/in finite dimension in a nonlinear manner,the algorithm tries to find an optimum hyper-plane with minimum distance from actual data[37].This algorithm assures a quick solution and converging to universal optimum.The topology ofthe network willbe identified during the training procedure automatically,there is less probable chance for over fitting and no need for multiple adjustable parameters[34,35].Nevertheless,the use of SVM algorithm for solving regression problems requires convex optimization with an inequality constraint to find the support vectors.In other words,using SVM method for solving regression problems requires a high computational load to solve the inequality constraint.Consequently,the use of SVM algorithm with large-scale data is not recommended[38].

A modified version of support vector machine,known as least squares support vector machine(LSSVM),is introduced by Suykens and Vandewalle[35].This method has all the bene fits of the conventional SVM but by solving a group of linear equations instead of a quadratic programming problem,or in other words,changing the inequality constraint to equality constraint,the algorithm can handle large data sets with acceptable accuracy[35,38,39].

The regression error,which is the difference between experimental and predicted value,is used as an additional constraint in LSSVM algorithm.In conventional SVM the value of error optimized during calculation steps,but the in LSSVM the error is mathematically de fined[35,39,40].

2.2.Equations

Assuming a dataset with the form of Eq.(1),which shall be approximated by a nonlinear function of Eq.(2):

Here,w∈Rnhis the weight vector in the initial weight space,g(?):Rn→Rnhis the nonlinear mapping function that projects the inputs to a higher dimensional feature space where linear regression is conducted;bindicates the bias term.The dimensionnhof this space is covertly de fined;in other words,it can be in finite dimensional.The optimization problem in LSSVM algorithm is de fined as follows[35]:

Here T denotes the transpose of the weight matrix,μ≥0 signifies regularization constant,and the symbolekindicates error variables.Joining the linear constraint Eq.(4)into the optimization problem in Eq.(3)gives:

Where Lagrangian multipliers βk∈R.The following conditions are required for optimization due to Lagrangian multipliers:

Assuming a liner regression relation between dependent and independent variables of the LSSVM,it can be written:

Which stands only for linear regression problems.The Kernel function should be inserted in Eq.(7)to extend its applicability for nonlinear regression problems:

The Kernel function ofK(xi,x)could be presented as inner product of vectors g(x)and g(xi):

Many Kernel functions could be used in LSSVM algorithm;but RBF function(Eq.(10))and polynomial function(Eq.(11))are generally used in LSSVM models.

Here the symbol σ2represents squared bandwidth that optimize during the optimization step ofthe learning process anddshows the degree of polynomial.

2.3.Computation procedure

The intention of this paper is to propose an intelligent model,with satisfactory performance in predicting HFT of a wide range of natural gas mixtures.A set of 279 experimental data points,from both sweet and sour gases was collected from open literature[9,41–48].Randomly a set of 223 data was chosen for training and a set of 56 for testing.For having a solid model which could support a wide range of natural gas mixtures,input variables are de fined upon the type of hydrate structure they form.The summation of methane and ethane mole percent forms the first input variables.These two light hydrocarbons formsIhydrate structure[3].The second variable is the total mole percent of threesIIformers;propane,butane and nitrogen[3].The third variable isC5+mole percentwhich are non-hydrate formers[4].Mostnaturalgascompounds are hydrophobic,but CO2and H2S are water soluble acid gases,this difference in chemical nature is stipulated by Jeffrey and McMullan[49].So although CO2and H2S both aresIformers[3]butdue to their different chemical nature from other hydrate formers,their composition was each taken as a separate variable.The reason for taking each of their compositions as a separate variable is because of H2S behavior,forming hydrates under rather low pressures and rather high temperatures[25,47].This means thateven a very little amountofH2S presentin a gas mixture,its effect on the hydrate formation behavior of system is significant.The sixth input parameter represents a ratio of acid gases presented in the gas mixture over other components.This variable simply helps the model to have more accurate predictions for sour gases.Gas specific gravity and pressure are the last two variables and HFT is the output of the model.Range and statistical parameters of these variables could be found in Table 1.

Table 1Ranges and corresponding statistical parameters of the input/output data used to construct LSSVM model

Tuning parameters are found by using a combination of Coupled Simulated Annealing(CSA)and a standard simplex method.First,CSA finds good starting values and these are passed to the simplex method in order to fine tune the result[50].LSSMV optimum values of parameters σ2and γ have been evaluated 273.6766 and 110,212.7226,respectively.A typical schematic diagram for the CSA-LSSVM algorithm is shown in Fig.1.

2.4.Leverage approach

A necessity for developing a mathematical model is outlier diagnostics[51,52].Through this process individual data(or groups of data)which may extravagate from the bulk of population in data set will be recognized[51,52].The main causes of outliers are experimental errors.These doubtful data may harm the mathematical model through introduction of some uncertainties and lower the prediction accuracy.Leverage approach is one of the most effectual and authoritative statistical based algorithms for outlier diagnostics[52,53].This model depends on a matrix,known as Hat matrix.Its elements represent the deviation of predicted values,found through a correlation(or a model)from experimental data[51,52].Leverage or Hat indices are de fined as follows[51,52]:

Fig.1.CSA-LSSVM algorithm schematic.

Here X is a two dimensional matrix built fromnrows(number of data)andkcolumns(parameters of the model).In this de finition T denotes transpose matrix.H matrix diagonal elements are the Hat values in the feasible region of the problem.

The correlation of hat indices and standardized cross-validation residuals(R)is shown by Williams plot which is used for graphical identification of dubious data[51,52].With the help of Eq.(12)for evaluatingHvalues the Williams plot could be sketched.Warning Leverage value(H)is generally found through the de finition 3p/nwherenis the number of training points andpis the number of correlation input parameters[51,52].Commonly the acceptable “cut off”value of Leverage is 3[51,52]so the points within the standard deviation range of?3 are accepted.Existence of the majority of the data points within the range of 0＜H＜H*and?3＜R＜3,con firms the model's statistical validity.Good high Leverage points are located in the range ofH≤H*and?3＜R＜3 which are represented as the ones that the model could not predict at all.The points located in the range ofR＜?3 orR＞3 are outliers or bad high Leverage and can be considered as the doubtful data.

3.Result and Discussion

3.1.Accuracy of the model

Statistical parameters of the intelligent model,including squared correlation coefficients(R2),average absolute relative deviations(AARDs)and root mean square errors(RMSE)are shown in Table 2.A comparison between the calculated HFT data and the experimentalvalues of train data set and test data are illustrated in Figs.2 and 3 and the figures are evidence of excellent agreement between predictions of LSSMV and experimental data.

Table 2Statistical parameters of the develop LSSVM model to determine hydrate formation temperature(HFT)

Fig.2.Comparison between the results of the developed model for train data set and the data base values(Train pressure range(MPa)0.5820–62.8500).(a)Scatter plot,(b)relative deviation plot,and(c)results of the developed model for train data set and the data base values versus the number of data.

For evaluating the model's performance,the same data sets were also used to evaluate the accuracy of the most popular empirical correlations that were used for hydrate formation temperature prediction.These correlations are Motiee[15],Towler and Mokhatab[16],Berge[13],Baillie-Wichert[10],Mann[11]Safamirzaei[18],and Salufu[19].The correlations of Mannet al.and Baillie-Wichert which are used in this study are the version implemented in the software package Hydrate Plus from FlowPhase Inc.Fig.4 is demonstrating the degree ofagreement between experimentaldata and predicted values by correlations and the LSSVM model through correlation coefficient(R2).The relative deviation and comparison between the experimental data and results of model and correlations are depicted in Figs.5 and 6,respectively.In addition,statistical parameters of the LSSVM model and correlations could be found in Table 3.These results demonstrate the fact that LSSVM model surpasses all correlations in prediction accuracy.

Fig.3.Comparison between the results of the developed model for test data set and the data base values(Test pressure range(MPa)0.7580–27.3200).(a)Scatter plot,(b)relative deviation plot,and(c)results of the developed model for train data set and the data base values versus the number of data.

Fig.4.Comparison between the results of the developed model and other correlations.(a)LSSVM model,(b)Motiee[15]correlation,(c)Towler and Mokhatab[16]correlation(d)Berge[13]correlation(e)Hydrate Plus Software(Baillie-Wichert Method[10])(f)Hydrate Plus Software(Mann Method[11])(g)Safamirzaei[18]correlation(h)Salufu et al.[19]correlation.

Fig.5.Comparison between relative error deviations for(a)LSSVMmodel,(b)Motiee[15]correlation,(c)Towlerand Mokhatab[16]correlation(d)Berge[13]correlation(e)Hydrate Plus Software(Baillie-Wichert Method[10])(f)Hydrate Plus Software(Mann Method[11])(g)Safamirzaei[18]correlation(h)Salufu et al.[19]correlation.

The applicability of the LSSVM approach for sour gas mixture is examined through a data set of 50[47],covering a wide range of H2S concentrations(i.e.6.78 mol%to 26.62 mol%).Due to H2S characteristics,forming hydrates under rather low pressures and rather high temperatures,a comprehensive study of the model for sour gas data needs a wide range of H2S concentration.Through former correlations and also two thermodynamic base models for sour gas hydrate prediction,Chen-Guo[24],Sun-Chen[25]the performance of the LSSVM model is examined.The results are shown in Table 4.The first point,as it can be seen,is that Motiee,Towler and Mokhatab and Berge correlations have serious deviations from experimental data.The reason is maybe because these correlations are based on gas gravity not composition.The second and more important point is that in low H2S concentrations the accuracy ofthe Mannetal.,Baillie-Wichert,Chen-Guo and Sun-Chen methods is not much different than LSSVM model,but in high H2S concentrations,despite LSSVM excellent accuracy,both correlations and thermodynamic models fail to show acceptable results.

3.2.Outlier detection

From Table 2 it is clear that the deviations of predicted values with LSSVM model from the corresponding experimental data,are almost suited to be used for Leverage approach.Through Eq.(12),theHvalues have been found and warning Leverage(H*)values were found from 3p/nfor the total data set.Finally Williams plots have been sketched in Fig.7.

The cluster of data points between the ranges of 0≤H≤H*and?3≤R≤3 is a sign of statistical validity of LSSVM model for prediction of these experimental values.The figures suggest that,except for two points(one in the sweet gas data set and one in the sour gas data set),in Fig.7,the whole HFT data points are in the applicability domain of proposed LSSVM model.The quality of treated data is different as the data with the lower absolute valuesR(nearR=0 line)and lowerHvalues may be identified as more reliable experimental data.

3.3.Sensitivity analysis

In order to extend our understanding of hydrate formation,a sensitivity analysis was conducted through LSSVM model by Pearson technique.The global effect of each independent parameter on hydrate formation temperature was examined through relevancy factor with directionality.The Pearson correlation is de fined as[54]:

Table 4LSSVM model accuracy in comparison with thermodynamic based models and existing correlations for predicting sour gas HFT

Fig.8.Sensitivity impact analysis of hydrate formation temperature.

For a better visualassessmenton the effectof pressure,using LSSVM model,a data set with 50 data points[47]was used.Fig.9 clearly shows thatthe HFT willincrease by increasing pressure.In addition,in order to gain a more sensible understanding of H2S effect,an imaginary mixture of methane and hydrogen sul fide was presented to the LSSVM model at two constant pressures(3 and 4 MPa).As it could be seen in Fig.10,the model predicted that increasing H2S content of the mixture would increase the hydrate formation temperature.

Fig.10.Graphical illustration of sensitivity impact analysis performed based on H2S content.

4.Conclusions

In this work a mathematical-based method of least square support vector machine was developed for the HFT prediction of natural gas mixtures.The inputparameterswere de fined on the basisofthe hydrate structure type thateach naturalgas componentwas forming.The statisticalparameters revealed thatthe LSSVMalgorithm could prognosticate HFT for both sweetand sour gases.In addition,the proposed modelperformed with an outstanding accuracy in comparison to conventional correlations and thermodynamic models.Especially in the case of sour gases with high H2S concentrations where both correlations and thermodynamic models fail to show any acceptable accuracy while LSSVM model was rigorous.

In an attempt to evaluate the quality of experimental data,outlier detection and finding the applicability range of the LSSVM model,a mathematical algorithm based on Leverage approach was used.This algorithm showed that the applied LSSVM model,for predicting HFT of gas mixtures is statistically valid and correct as well as whole experimentaldata points exceptone were in the applicability domain of the model.

Fig.9.Graphical illustration of sensitivity impact analysis performed based on pressure.

In order to assess the effect of each input variable on the HFT,a sensitivity analysis was conducted.The pressure had the largest effect among the parameters on the HFT.The result showed that increasing pressure could drastically increase the HFT.In addition the results revealed that among hydrate formers,hydrogen sul fide has the biggest effect on HFT.

Nomenclature

AARD average absolute relative deviation,%

blinear regression intercept of the model

ekregression error

H hat matrix

K(x,xk) Kernel function

Pc,icritical pressure of componenti

RBF radial basis function

R2correlation coefficient

T transpose matrix

Tr,ireduced temperature of componenti

wregression weight

xkinput vector

ykoutput vector

αiLagrange multipliers

γrelative weight of the summation of the regression errors σ2squared bandwidth

φfeature map

ωiacentric factor of componenti

Appendix A

Correlation factor(R2):

Average absolute relative deviation(AARD):

Root mean square error(RMSE):

Standard deviation(STD):

Appendix B.Instruction for using the model

A computer program is organized to use the developed model.At if rst,the LSSVM toolbox for MATLAB should be installed,and then,the directory of the toolbox should be inserted as the main directory in the MATLAB environment.After that,the model.mat file is dragged and dropped in the MATLAB workspace.

Example:Calculation of the hydrate formation temperature for following mixture(Table B1).

Table B1The sample set for calculation of hydrate formation temperature

Hydrate formation temperature is calculated simply using the below command line in the command window:

The output result of the program(based on the developed model)will be 295.66 K while the corresponding experimental value is 295.2 K.

[1]E.D.Sloan,Fundamental principles and applications of natural gas hydrates,Nature426(6964)(2003)353–363.

[2]A.Chapoy,A.-H.Mohammadi,D.Richon,Predicting the hydrate stability zones of natural gases using artificial neural networks,Oil Gas Sci.Technol.Rev.l'IFP62(5)(2007)701–706.

[3]E.D.Sloan,C.Koh,Clathrate Hydrates of Natural Gases,Third edition,Taylor&Francis,2007.

[4]J.Carroll,Natural Gas Hydrates:A Guide for Engineers,Elsevier Science,2009.

[5]A.H.Mohammadi,D.Richon,Development of predictive techniques for estimating liquid water-hydrate equilibrium of water-hydrocarbon system,J.Thermodyn.2009(2009)1–12.

[6]A.H.Mohammadi,R.Anderson,B.Tohidi,Carb on monoxide clathrate hydrates:Equilibrium data and thermodynamic modeling,AIChE J.51(10)(2005)2825–2833.

[7]E.Hammerschmidt,Formation of gas hydrates in natural gas transmission lines,Ind.Eng.Chem.26(8)(1934)851–855.

[8]D.L.Katz,Prediction of conditions for hydrate formation in naturalgases,Trans.AIME160(1945)140–149.

[9]W.I.Wilcox,D.Carson,D.Katz,Natural gas hydrates,Ind.Eng.Chem.33(5)(1941)662–665.

[10]C.Baillie,E.Wichert,Chart gives hydrate formation temperature for natural gas,Oil Gas J.85(14)(1987)37–39.

[11]S.L.Mann,Vapor–Solid Equilibrium Ratios for Structure I and II Natural Gas Hydrates,Gas Processors Association,1988.

[12]I.U.r.F.Makogon,et al.,Hydrates of Natural Gas,PennWell Books,Tulsa,Oklahoma,1981.

[13]B.Berge,Hydrate predictions on a microcomputer,In:Petroleum Industry Application of Microcomputers,SPE,Colorado,USA,1986.

[14]R.Kobayashi,K.Y.Song,E.D.Sloan,Phase behavior of water/hydrocarbon systems,Petroleum Engineering Handbook25(1987)e13.

[15]M.Motiee,Estimate possibility of hydrates,Hydrocarb.Process.70(7)(1991)98–99.[16]B.Towler,S.Mokhatab,Quickly estimate hydrate formation conditions in natural gases,Hydrocarb.Process.84(4)(2005)61–62.

[17]A.Bahadori,H.B.Vuthaluru,A novel correlation for estimation of hydrate forming condition of natural gases,J.Nat.Gas Chem.18(4)(2009)453–457.

[18]M.Safamirzaei,Predict gas hydrate formation temperature with a simple correlation,in,2015,http://www.gasprocessingnews.com/features/201508/Predict Gas Hydrate Formation Temperature With a Simple Correlation.aspx.Accessed:18.09.15.

[19]S.O.Salufu,P.Nwakwo,New empirical correlation for predicting hydrate formation conditions,SPE Nigeria Annual International Conference and Exhibition,Society of Petroleum Engineers,2013.

[20]W.R.Parrish,J.M.Prausnitz,Dissociation pressures of gas hydrates formed by gas mixtures,Ind.Eng.Chem.Process.Des.Dev.11(1)(1972)26–35.

[21]H.J.Ng,D.B.Robinson,The measurement and prediction of hydrate formation in liquid hydrocarbon–water systems,Ind.Eng.Chem.Fundam.15(4)(1976)293–298.

[22]V.John,K.Papadopoulos,G.Holder,A generalized model for predicting equilibrium conditions for gas hydrates,AIChE J.31(2)(1985)252–259.

[23]G.J.Chen,T.M.Guo,Thermodynamic modeling of hydrate formation based on new concepts,Fluid Phase Equilib.122(1)(1996)43–65.

[24]G.J.Chen,T.M.Guo,A new approach to gas hydrate modelling,Chem.Eng.J.71(2)(1998)145–151.

[25]C.Y.Sun,G.J.Chen,Modelling the hydrate formation condition for sour gas and mixtures,Chem.Eng.Sci.60(17)(2005)4879–4885.

[26]A.Elgibaly,A.Elkamel,Optimal hydrate inhibition policies with the aid of neural networks,Energy Fuel13(1)(1999)105–113.

[27]A.A.Elgibaly,A.M.Elkamel,A new correlation for predicting hydrate formation conditions for various gas mixtures and inhibitors,Fluid Phase Equilib.152(1)(1998)23–42.

[28]G.Zahedi,Z.Karami,H.Yaghoobi,Prediction of hydrate formation temperature by both statistical models and artificial neural network approaches,Energy Convers.Manag.50(8)(2009)2052–2059.

[29]A.H.Mohammadi,V.Belandria,D.Richon,Use of an artificial neural network algorithm to predict hydrate dissociation conditions for hydrogen+water and hydrogen+tetra-n-butyl ammonium bromide+water systems,Chem.Eng.Sci.65(14)(2010)4302–4305.

[30]M.Ghavipour,M.Chitsazan,S.H.Najibi,S.S.Ghidary,Experimental study of natural gas hydrates and a novel use of neural network to predict hydrate formation conditions,Chemical Engineering Research and Design91(2013)264–273.

[31]M.Moradi,K.Nazari,S.Alavi,M.Mohaddesi,Prediction ofequilibrium conditions for hydrate formation in binary gaseous systems using artificialneural networks,Energy Technol.1(2–3)(2013)171–176.

[32]J.Yang,B.Tohidi,Determination of hydrate inhibitor concentrations by measuring electrical conductivity and acoustic velocity,Energy Fuel27(2)(2013)736–742.

[33]A.Eslamimanesh,F.Gharagheizi,M.Illbeigi,A.H.Mohammadi,A.Fazlali,D.Richon,Phase equilibrium modeling of clathrate hydrates of methane,carbon dioxide,nitrogen,and hydrogen+water soluble organic promoters using support vector machine algorithm,Fluid Phase Equilib.316(2012)34–45.

[34]C.Cortes,V.Vapnik,Support-vector networks,Mach.Learn.20(3)(1995)273–297.

[35]J.A.Suykens,J.Vandewalle,Least squares support vector machine classifiers,Neural.Process.Lett.9(3)(1999)293–300.

[36]M.Curilem,G.Acu?a,F.Cubillos,E.Vyhmeister,Neural networks and supportvector machine models applied to energy consumption optimization in semiautogeneous grinding,Chem.Eng.Trans.25(2011)761–766.

[37]E.Soroush,M.Mesbah,A.Shokrollahi,A.Bahadori,M.H.Ghazanfari,Prediction ofmethane uptake on differentadsorbents in adsorbed naturalgas technology using a rigorous model,Energy Fuel28(10)(2014)6299–6314.

[38]H.Wang,D.Hu,Comparison of SVM and LS-SVM for regression,2005 International Conference on Neural Networks and Brain,IEEE,2005.

[39]J.A.Suykens,J.De Brabanter,L.Lukas,J.Vandewalle,Weighted least squares support vector machines:Robustness and sparse approximation,Neurocomputing48(1)(2002)85–105.

[40]K.Pelckmans,J.A.Suykens,T.Van Gestel,J.De Brabanter,L.Lukas,B.Hamers,B.De Moor,J.Vandewalle,LS-SVMlab:A Matlab/c Toolbox for Least Squares Support Vector Machines.Tutorial,KULeuven-ESAT,Leuven,Belgium,2002.

[41]W.Deaton,E.Frost Jr.,Gas Hydrates and Their Relation to the Operation of Naturalgas Pipe Lines,Helium Research Center,Bureau of Mines,Amarillo,TX(USA),1946.

[42]R.Kobayashi,H.Withrow,G.Williams,D.Katz,Gas hydrate formation with brine and ethanol solutions,Proceeding of the 30th Annual Convention,Natural Gasoline Association of America,1951.

[43]L.J.Noaker,D.L.Katz,M.Aime,Gas hydrates of hydrogen sulphide–methane mixtures,Trans.Am.Inst.Min.Metall.Pet.Eng.201(1954)237–239.

[44]H.McLeod Jr.,J.Campbell,1566-G-natural gas hydrates at pressures to 10,000 psia,J.Pet.Technol.13(6)(1961)590–594.

[45]D.Robinson,J.Hutton,Hydrate formation systems containing methane,hydrogen sulphide and carbon dioxide,J.Can.Pet.Technol.10(1971)33–35.

[46]S.Adisasmito,R.J.Frank III,E.D.Sloan Jr.,Hydrates of carbon dioxide and methane mixtures,J.Chem.Eng.Data36(1)(1991)68–71.

[47]C.Y.Sun,G.J.Chen,W.Lin,T.M.Guo,Ice,1995.

[48]E.Kamari,M.Oyarhossein,Experimental determination of hydrate phase equilibrium curve for an Iranian sour gas condensate sample,J.Nat.Gas Sci.Eng.9(2012)11–15.

[49]G.Jeffrey,R.McMullan,The clathrate hydrates,Prog.Inorg.Chem.8(1967)43–108.

[50]K.De Brabanter,P.Karsmakers,F.Ojeda,C.Alzate,J.De Brabanter,K.Pelckmans,B.De Moor,J.Vandewalle,J.Suykens,LS-SVMlab Toolbox User's Guide,ESAT-SISTA Technical Report,10 2011.

[51]C.R.Goodall,13 computation using the QR decomposition,Handbook of Statistics,9,1993,pp.467–508.

[52]P.J.Rousseeuw,A.M.Leroy,Robust Regression and Outlier Detection,vol.589,Wiley.com,2005.

[53]P.Gramatica,Principles of QSAR models validation:Internal and external,QSAR Comb.Sci.26(5)(2007)694–701.

[54]A.Kamari,M.Arabloo,A.Shokrollahi,F.Gharagheizi,A.H.Mohammadi,Rapid method to estimate the minimum miscibility pressure(MMP)in live reservoir oil systems during CO2flooding,Fuel153(2015)310–319.