Modeling viscosity of methane,nitrogen,and hydrocarbon gas mixtures at ultra-high pressures and temperatures using group method of data handling and gene expression programming techniques

Farzaneh Rezaei ,Saeed Jafari ,Abdolhossein Hemmati-Sarapardeh,2,3,,Amir H.Mohammadi

1 Department of Petroleum Engineering,Shahid Bahonar University of Kerman,Kerman,Iran

2 Institute of Research and Development,Duy Tan University,Da Nang 550000,Vietnam

3 Faculty of Environment and Chemical Engineering,Duy Tan University,Da Nang 550000,Vietnam

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

5 Discipline of Chemical Engineering,School of Engineering,University of KwaZulu-Natal,Howard College Campus,King George V Avenue,Durban 4041,South Africa

Keywords:Gas Viscosity High pressure high temperature Group method of data handling Gene expression programming

ABSTRACT Accurate gas viscosity determination is an important issue in the oil and gas industries.Experimental approaches for gas viscosity measurement are time-consuming,expensive and hardly possible at high pressures and high temperatures(HPHT).In this study,a number of correlations were developed to estimate gas viscosity by the use of group method of data handling (GMDH)-type neural network and gene expression programming(GEP)techniques using a large data set containing more than 3000 experimental data points for methane,nitrogen,and hydrocarbon gas mixtures.It is worth mentioning that unlike many of viscosity correlations,the proposed ones in this study could compute gas viscosity at pressures ranging between 34 and 172 MPa and temperatures between 310 and 1300 K.Also,a comparison was performed between the results of these established models and the results of ten well-known models reported in the literature.Average absolute relative errors of GMDH models were obtained 4.23%,0.64%,and 0.61% for hydrocarbon gas mixtures,methane,and nitrogen,respectively.In addition,graphical analyses indicate that the GMDH can predict gas viscosity with higher accuracy than GEP at HPHT conditions.Also,using leverage technique,valid,suspected and outlier data points were determined.Finally,trends of gas viscosity models at different conditions were evaluated.

1.Introduction

Nowadays,natural gases play an important role in our daily life as well as petroleum industry because of their various sources,easy accessibility,and also light burning.Hence,the use of natural gases has increased considerably during the past decades and it has led to exploration of new deeper reservoirs at high pressures and temperatures [1].Exact estimation of thermo-physical properties of natural gases (e.g.,density,solubility and viscosity) is very essential for chemical and petroleum engineers.Gas viscosity plays a major role in description of phase behavior,sketching of surface facilities,pipeline design and also optimized fuel consumption[2–6].Various approaches have been suggested for gas viscosity determination such as experimental tests,mathematical correlations,soft computing methods and equations of state(EOSs)based models [7,8].Experimental measurements have been conducted with different apparatuses including vibrating tube viscometer,capillary tube viscometer,and falling body[9].In general,experimental measurements present the most trustworthy and accurate results.However,this approach may not be economical and it is timeconsuming especially at HPHT conditions.

Due to the disadvantages mentioned earlier,a number of researchers have developed gas viscosity correlations.Carr et al.(CKB) [10] developed a model utilizing specific gravity,pressure and temperature while temperature and specific gravity range between 311.15 K and 422.15 K and 0.55 and 1.55,respectively.Also,this correlation was developed using only 30 data points.Hence,this model suffers from insufficient number of data points.The model consists of three graphical steps in which the use of the numerous charts are not convenient.After that,Lohrenz-Bray Clark(LBC) [11] developed a mathematical model to estimate gas mix-ture viscosity based on the coefficients as well as the correlation proposed by Jossi et al.[12].Also,they utilized density and three other common parameters to calculate gas viscosity.Therefore,accurate measurement of density in laboratory is highly vital.Another model developed for gas mixture viscosity estimation was proposed by Dean and Stiel [13].They worked on non-polar gas mixture and developed this model at moderate to high pressures.They used molecular weights and critical constants to develop their model.One of the most accurate models for gas viscosity estimation was proposed by Lee et al.utilizing temperature,molecular weight,and gas density [14].The results obtained from the aforementioned model for high values of specific gravity,are not accurate.Also,in order to obtain relatively accurate results,pressure and temperature should range between 0.68 MPa and 55 MPa and 311.15 K and 444.15 K,respectively.Standing [15]proposed some correlations for estimating gas viscosity.He also used the charts proposed by Carr et al.[10]and Dempsey equation[16].Lucas[17]established a relationship to compute gas viscosity.Some parameters including dipole moment and critical compressibility factor are required in this model.Also,the average deviation of dense gases was reported more than 8.0%.A model was developed by Chen [18] utilizing molecular weight,pressure,and temperature.Then,Gurbanov and Dadash-zade [19] developed a model for associate and condensate gases using Chen correlation[18]for model optimization.Sutton[20]correlation was developed based on molecular weight,gas density,temperature and pressure.After that,an intelligent/smart model was proposed by Shokir &Dmour [6] based on Genetic programming to calculate gas viscosity.This model is only applicable within the range of 0.02 to 26.6 for pseudo reduced pressure(Ppr)and 0.6 to 3.4 for pseudo reduced temperature(Tpr).Afterwards,a model was proposed by Heidaryan et al.[21] by using reduced pressure and temperature for estimating the viscosity of lean gases such as methane.Heidaryan et al.[22] presented another mathematical relation to calculate the viscosities of natural gases by the use of density,molecular weight and temperature.Sanjari et al.[4] introduced a numerical model utilizing reduced pressure and temperature.A robust smart model was developed by Hajirezaie et al.[23] utilizing gene expression programming (GEP).This model is reliable while pseudo reduced pressure and pseudo reduced temperature range between 0.020 and 29.298 and 0.5408 and 2.6815,respectively.Recently,Dargahi et al.[24] developed a correlation to estimate viscosities of gases.This model calculates viscosity using pseudo-reduced pressure,pseudo-reduced temperature,gas density,and molecular weight.Rostami et al.[25] proposed some intelligent models by the use of pseudo-reduced pressure and (pseudo reduced) temperature and molecular weight.An average error for the aforementioned model was 1.46%.A smart model was developed by Chico et al.[26] for gas viscosity utilizing two evolutionary algorithms such as genetic algorithm (GA) and particle swarm optimization (PSO).

In this study,two sets of reliable,robust and accurate correlations for estimating gas viscosity were developed using a huge data set (more than 3000 experimental data points) including viscosity data for methane,nitrogen,and hydrocarbon gas mixtures (containing methane,propane,butane,and pentane).The models were developed using two correlative methods including group method of data handling(GMDH)and gene expression programming(GEP)based on different properties namely reduced pressure,reduced temperature,and molecular weight of components.The proposed models do not use gas density as an input parameter as its measurement is costly especially at HPHT conditions.It is worth to mention that the proposed models were developed at high pressures (34 MPa to 172 MPa) and high temperatures (310 K to 1300 K).The reason for selecting these systems was that only these hydrocarbon gases at high pressure and temperature conditions are available in literature.Also,most of the models presented to date have been developed based on data at pressures lower than 34 MPa and few models are available for higher pressures.Lastly,the Leverage approach distinguishes valid,suspected and outlier data points and proves the validity of the proposed models.

2.Model Development

2.1.Data assembly

In this work,a comprehensive data set was prepared from various sources for the viscosities of methane,nitrogen and hydrocarbon gas mixtures with various compositions.The data bank includes 3017 experimental data points and different compositions for hydrocarbon gas mixtures and also varying pseudo reduced temperature and pseudo reduced pressure for all of components which are considered as input parameters[1,27,28].The proposed models for methane and nitrogen use reduced temperature (Tr)and reduced pressure (Pr) as inputs,while for hydrocarbon gas mixtures pseudo reduced temperature (Tpr),pseudo reduced pressure (Ppr) and average molecular weight of gas are considered as inputs.Table 1 shows the statistical explanation for data points.As it is clear from this table,the data bank covers a wide range of pressures and temperatures and gas compositions and is mostly for HPHT conditions.

Table 1 Statistical explanation of the data set used in this study

2.2.Group method of data handling

The artificial neural networks are applicable in different fields.Smart algorithms including group method of data handling(GMDH) and genetic algorithm (GA) can transform complicated problems into easy ones [29].Ivakhnenko [30] was the first researcher who proposed the first article about these algorithms.For exploratory self-organized and complex systems,polynomial theory determines which of the proposed theoretical approaches are able to solve simple problems in engineering cybernetics[30].GMDH algorithm is known as a self-organized system.Shankar conducted a research on this algorithm in 1972[31].Other kind of this algorithm was developed by Japanese and Polish scientists[32].They found that GMDH is the most accurate method to identify AI issue as well as random procedures predicting long and short-term future and realize the pattern in multiplex systems.Also,regression analysis is known as a special type of GMDH using statistical GMDH theory.GMDH plays a significant role in conducting researches on economic systems,analysis and forecasting events related to ecological systems,medical diagnostics,meteorology,and demographic systems [33–36].It is note-worthy that GMDH algorithm is rarely used in petroleum engineering.

GMDH is noted as a polynomial neural network(PNN)with layered structure and each layer contains separate neurons.Each pair of independent neurons is mixed utilizing a quadratic polynomial expression.Thereafter,in the adjoining layers,other neurons are made and finally,through the joint independent parameters,a new structure is produced [36].The following relationship between input and output is proposed utilizing Volterra-Kolmogorov Gabor [37] series:

The output is shown by Yi.Also,xixj﹒﹒﹒xkare representatives of input parameters and a,bi,cij,di,j,﹒﹒﹒,kshow the coefficients of polynomial.The number of independent parameters is shown by M.

At first,this matrix is generated using GMDH algorithm.The output parameters are demonstrated on the left side of matrix and the input parameters are shown on the right side of the matrix.After that,using a quadratic polynomial expression as well as a combination of a number of independent variables,the new variables are created and placed instead of current variables.(Z1,Z2,﹒﹒﹒,Zn).

The latest matrix can be defined as:vZ=(Z1,Z2,﹒﹒﹒Zn)

Utilizing the least square method(LSM),Eq.(2)coefficients are computed.Predicted values should be close to the experimental data,thus,minimum square of deviation from the experimental data must be obtained [38].

The number of data points and independent variables are shown by Ntand M,respectively.

In the fowling,the common equation of matrix is written utilizing a quadratic polynomial:

The quadratic polynomial vector is defined as A={a,b,c,d,e,f};T is the transform of matrix;Y={Y1,Y2,﹒﹒﹒,YN},and n represents the input numbers.

Using the least square technique,Eq.(4) is solved:

By dividing the data set into two categories,development and validation sets are made.Using development data points,the coefficients of Eq.(2) are determined and validation data points helps to validate the obtained variables.

An accurate model should estimate gas viscosity with low deviation,as following equation implies:

The new variable will be replaced if the condition is met.Otherwise,it should be removed.At the end of iteration,the algorithm captures the difference between the estimated value and the experimental one and once the minimum difference is obtained,the algorithm ends.

2.3.Hybrid GMDH-type neural network

In this study,a hybrid GMDH was utilized to model the complicated systems.Hybrid GMDH can combine more than two independent variables simultaneously and also nodes from various layers are joined.

For general GMDH algorithm,there are continuous nodal interconnections.Hybrid GMDH algorithm with more connections among independent variables leads to the more practical model comparing with the original model [38–41].The last correlation for GMDH algorithm is presented as follows:

The schematics of the proposed GMDH models for hydrocarbon gas mixtures,methane,and nitrogen are demonstrated in Fig.1(a,b,and c),respectively.Also,in Table 2(a,b,and c)the correlations generated by the GMDH model are reported to estimate hydrocarbon gas mixtures,methane,and nitrogen viscosities.

2.4.Gene expression programming (GEP) technique

Finding the best prediction using the actual data without any need to a special pre-determined function is considered as the most important property of GEP technique.The GEP strategy determines the most effective mathematical form using evolutionary algorithm (EA) [42].Therefore,in this study,GEP algorithm was used to propose a correlation to estimate gas viscosity.The EA consists of four implementation methods including:Evolutionary Programming (EP) [43],Genetic Algorithm (GA) [44],Genetic Programming (GP) [45],and Evolutionary Strategies (ES) [46].In these algorithms,there are different ways to represent chromosomes and in each algorithm,chromosomes have specific meanings.

Eq.(8) shows a mathematical example of a GEP model:

Two genes generate this chromosome(model)and a plus function (+) connects these genes together.Obviously,this chromosome (model) is created from some functions such as/,Q,+,a,and b.The function part of this chromosome is defined by F=(/,+,Q)that includes division,addition,and square root.T=(a,b)represents the terminal function.Also,‘‘a”and‘‘b”are forecaster variables.In addition,these genes and chromosomes can be described by a tree diagram named Expression Trees(ET).Fig.2 indicates the ET for the aforementioned example.

Fig.1.A schematic of the proposed GMDH models to estimate (a) hydrocarbon gas mixtures,(b) methane,and (c) nitrogen viscosities.

The stage-by-stage performance of GEP technique is described as follows [47,48]:

(1) Initialize the population randomly using generating chromosomes.

(2) Consider the chromosomes as ET and use an autonomous fitness assessment.

(3) Select the best elements based on their capability for duplication reform.

(4) Establish the best solution using genetic operators.

(5) This procedure is replicated till the algorithm finds the best answer based on the predefined criteria.

GEP was primarily invented by Ferreira[47].In addition,regeneration procedures were conducted based on crossover and modification operators.This procedure restricts the algorithm processes.

The mutation ratio changes the chromosome components.Mutation percentage differs in various systems.As an example,the mutation ratio of eukaryotes is lower than bacterias.More detailed information about GEP technique can be found in the literature[49,50].Table 3 shows the correlations established by GEP for estimating hydrocarbon gas mixtures,methane,and nitrogen viscosities.

Table 2 Correlations of the GMDH model for hydrocarbon gas mixtures,methane,and nitrogen viscosities

Table 3 Correlations of the GEP model for hydrocarbon gas mixtures,methane,and nitrogen viscosities

2.5.Outliers and inaccurate experimental data points

Outlier data may occupy a considerable portion of a dataset.Definitely,some of the data points can impact the accuracy and validity of models.Hence,identification of outlier can cause a significant effect on the developed models[51–56].Despite the number of existing methods for detection and separation of outlier data,the Leverage approach was used in this study.The technique[52,53,55,56] includes computation of model deviations from the experimental data.

The deviations are named‘‘standardized cross-validated residuals”and also a matrix(the Hat matrix)is computed based on input data points.Comprehensive information about this method could be found in the literature [51–56].

3.Model Assessment

3.1.Statistical error examination

Fig.2.A normal expression tree indicating the algebraic expression.

Statistical error calculations were conducted for determination of average percent relative error (APRE),average absolute percent relative error (AAPRE),standard deviation (SD),and root mean square error (RMSE) in order to determine the accuracy of the established models.Statistical parameters include:

3.2.Graphical error evaluation

In order to evaluate the applicability of the developed models visually,graphical analysis methods were used:

(1) Cross plot:Experimental data are plotted versus predicted/estimated data and the accuracy of the model is determined using the unit slop line.

(2) Error distribution:Accuracy and coherence of the models are obtained by measuring the distributions of points around the zero-error line.

4.Results and Discussion

In this work,by using 3017 data points for methane,nitrogen,and hydrocarbon gas mixtures,six models were developed for estimating gas viscosity at HPHT conditions by applying GEP and GMDH techniques.For each model,80% and 20% of data points were selected randomly as training and testing sets,respectively.In contrary to most of literature models,the models proposed in this study do not use gas density to estimate gas viscosity,as mentioned earlier,whereas the measurement of gas density in laboratory is costly and time-consuming.Also,it should be mentioned that gas density models suffer from high errors in some conditions.In addition,most of previous models for gas density were developed at low to moderate pressures and temperatures.

Table 4 indicates the results of a comparison between the proposed models in this study and ten well-known existing models for hydrocarbon gas mixture,methane,and nitrogen viscosities estimations.For hydrocarbon gas mixture as it is clear from Table 4 and based on AAPRE and R2,the GMDH model is more accurate than the other models.AAPRE indicates that the model developed by Sanjari et al.[4] has the highest error.Average percent relative error data shows that Lee modified by Londono [3] model as well as Sanjari et al.[4] model overestimate hydrocarbon gas mixture viscosity.GEP model was ranked as the second accurate model in estimating viscosities of hydrocarbon gas mixtures.

Also,for methane gas according to Table 4,the most and the least accurate models to estimate methane viscosity are GMDH model (AAPRE=0.6436%)and Sanjari et al.model(AAPRE=2692.72%).APRE values indicate that,Lee modified by Londono [3],Sanjari et al.[4],and Lucas [17] models overestimate the value of methane viscosity.However,methane viscosity is considerably underestimated by using Chen [18] correlation.

As can be obviously noted in Table 4,based on AAPRE,APRE,RMSE,SD,and R2,the GMDH model is the most efficient model to estimate nitrogen viscosity.The results obtained for the hydrocarbon gas mixture and methane show that Lee modified by Londono[3] and Sanjari et al.[4] models highly overestimate nitrogen viscosity;whereas Heidaryan A [21] and Chen [18] models considerably underestimate it.The main advantage of the established GMDH model is the reduced number of input parameters as well as presenting the highest accuracy.

Fig.3.Average absolute relative error of the proposed models as well as the ten mentioned models for hydrocarbon gas mixture,Methane,and Nitrogen.

Fig.4.Cross plots of the proposed models in this study as well as the three other well-known models for hydrocarbon gas mixture viscosity estimation.

Fig.3 shows the bar plot of AAPRE of different models as another technique for evaluation of the efficiency of the presented models.Obviously,AAPRE of the models proposed in the study are lower than the ones of pre-existing models.However,Lucas [17],Hajirezaie et al.[23],and Heidaryan B[22]models are much more accurate than Lee modified by Londono[3],and Sanjari et al.[4]models.The correlations developed by Sanjari et al.[4],Lee modified by Londono[3],Standing[15],Sutton[20],Heidaryan A[21],and Chen[18]present high error values and are not recommended.The aforementioned models were established in a limited condition and used a small group of data points.Based on the results mentioned earlier,Lee et al.[14],Heidaryan B [22],Lucas [17],and Hajirezaie et al.[23]models,which are determined as the most accurate literature models,were compared visually with the developed models in this study.

Fig.5.Cross plots of the proposed models in this study as well as the three other well-known models for methane viscosity estimation.

Fig.6.Cross plots of the proposed models in this study as well as the three other well-known models for nitrogen viscosity estimation.

To efficiently analyze the performance of the developed models in this study and the other models for hydrocarbon gas mixture viscosity estimation,the estimated values using GDMH and GEP models as well as the four aforesaid models were plotted versus the experimental data in Fig.4.It can be recognized that the estimated values by GMDH and GEP models better match with experimental data points compared to results of other models.However,Lee et al.[14]and Heidaryan B[22]models underestimate hydrocarbon gas mixture viscosity;whereas Lucas[17]model overestimates it.

The superiority of the proposed models in estimating methane and nitrogen viscosities is clearly observed in Figs.5 and 6.Lee et al.[14],Heidaryan B [22],Lucas [17],and Hajirezaie et al.[23]correlations estimate gas viscosity with higher errors (Figs.5 and 6).It should be noted that Hajirezaie et al.[23] correlations could not predict a huge number of data points of methane and nitrogen viscosities.Also,Lee et al.[14] and Heidaryan B [22] correlations underestimate methane viscosity,while Lucas [17] correlation overestimates it,as shown in Fig.5.

Fig.7.Error distribution of best proposed models in this study as well as the three other well-known models for hydrocarbon gas mixture viscosity estimation as a function of pseudo reduced temperature.

In addition,Fig.6 indicates that all models underestimate nitrogen viscosity except the GMDH and GEP and the results obtained from these two models are closely accumulated around the unit slop line.Thus,GMDH and GEP models estimate gas viscosity with the highest accuracy.

Figs.7 to 9 evaluate the distribution of data points around the zero-error line for the proposed models as well as the aforementioned models at various temperatures.As these figures indicate,there is a compaction of data points around the zero-error line for GMDH and GEP models,which again emphasis the excellent performance of these models.

It should be noted that there are two main reasons why the models presented by GMDH and GEP are better than the other models reported in the literature for gas viscosity:

(1) The databank used in this study is much broader and covers a wide range of temperatures,pressures and compositions which has a huge impact on the accuracy of the model.

Fig.8.Error distribution of the proposed models in this study as well as the three other well-known models for methane viscosity estimation as a function of reduced temperature.

(2) The structure of GMDH and GEP is such that it automatically detects a mathematical relationship among the inputs and the output (gas viscosity).While in most of the literature correlations,the formats of the mathematical equations were proposed by a trial and error procedure and only their coefficients were optimized,in this work,both format of the correlations and their coefficients were developed by advanced computing algorithms.

Fig.9.Error distribution of the proposed models in this study as well as the three other well-known models for nitrogen viscosity estimation as a function of reduced temperature.

It is important to statistically check the validity of the proposed models and recognize outliers.To this end,the Leverage approach was used.The Leverage approach is a standard statistical method for outlier detection.In this method,it is statistically suggested that the data whose standardized residuals (SR) are between -3 to 3 and their Hats are between 0 and h*are in the valid region and the data outside this range are experimentally suspected data or out of model application domain.The leverage approach uses several formulas listed below:

Fig.10.William’s plot for detecting probable outliers and the applicability domain of the developed GMDH models to estimate(a)gas mixture,(b)methane,and(c)nitrogen viscosities,respectively.

Fig.11.Experimental data as well as the estimated viscosities using GMDH model for methane versus Pr and Tr respectively.

where X is a two-dimensional matrix consisting of m rows and n columns.m here is the total number of data used and n is the input parameters of model.

First,using Eq.(14),the H matrix was calculated for hydrocarbon gas mixture,methane,and nitrogen.Then according to the number of data for each gas,which is equal to 1265,702,and 1049 respectively,and also according to the input parameters,which is 3 for hydrocarbon gas mixture and 2 for pure gases,the values of h*and standardized residual were calculated and finally the valid and suspected data were determined.

In Fig.10a,the feasibility domain of the GMDH model as the most accurate model for hydrocarbon gas mixture is indicated.Existence of most of the data in the range of 0 ≤hat ≤hat*(0.0094)and-3 ≤SR ≤3 means that the proposed model is statistically valid and the experimental data have good quality.This figure shows that GMDH model has considerable reliability and precision.A few numbers of data points (0.31%) are considered out of Leverage for the GMDH model.Apart from that,the data points located in the range of R<-3 or R>3(without considering the value of hat compared to hat*) are called ‘‘Bad High Leverage”and they are outliers or experimentally suspected data.The suspected data are shown by red color(diamond).In addition,the data points placed in area that hat ≥hat*and -3 ≤R ≤3,are recognized as ‘‘Good High Leverage”.A good high leverage data points do not have negative impacts on the correlations [50–53].

As it is clear in Fig.10b,there is no suspected or out of leverage data for methane viscosity and all data are located in valid area.In Fig.10c,approximately 3% of data points of nitrogen viscosity are out of Leverage.In summary,the developed models by GMDH are statistically valid and only few data points are located out of applicability domain of the proposed models.

In order to estimate hydrocarbon gas viscosity variation with changing pressure and temperature,Fig.11(a and b) is plotted.Since the dataset presents viscosity data at high pressures and high temperatures,therefore,monitoring gas viscosity variation at low pressures was impossible.As Fig.11(a) shows,gas viscosity increases as pressure increases and the GMDH also follows the same trend.In Fig.11(b),it is observed that gas viscosity decreases with increasing temperature and the GMDH follows the same trend,which is physically valid.It should be noted that at high pressure conditions,gases behave as liquids and by increasing temperature,their viscosities decrease.

5.Conclusions

In this study,two robust algorithms namely GMDH and GEP were developed to estimate gas viscosity at HPHT conditions using a comprehensive experimental data set collected from different literature sources.The proposed models for methane and nitrogen use reduced temperature and reduced pressure as inputs,while for hydrocarbon gas mixture pseudo reduced temperature,pseudo reduced pressure and average molecular weight of gas are considered as inputs.One of the main merits of these models is their few numbers of input variables and also,they do not need density as input,the measurement of which is costly and time-consuming.To validate the developed models,statistical and graphical analyses were used.Also,the developed models were compared with ten well-known empirical correlations.The results of GMDH are more accurate than the ones obtained by GEP.Average absolute percent relative error of the GMDH models proposed for hydrocarbon gas mixture,methane and nitrogen were obtained 4.23%,0.64%,and 0.61%,respectively,which illustrate the excellent performance of GMDH algorithm.The models proposed in the study are able to estimate gas viscosity more accurately than the literature models.The results indicate Lucas [17] is the second most accurate model for estimating viscosities of hydrocarbon gas mixtures and methane.Also,graphical analyses demonstrate that the GMDH and GEP models do not show notable error trends at different pressures and temperatures,while pre-existing correlations underestimate or overestimate gas viscosity at HPHT conditions.In addition,using the Leverage approach technique the validity of the proposed GMDH models was confirmed and also the experimental data show good quality.Finally,the trend of gas viscosity with respect to pressure and temperature was physically validated.