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Natural gas hydrate has been widely of concern due to its great potential in application to address problems including gas storage, transmission, separation techniques, and also as energy resource. Accurate prediction of hydrate formation phase equilibrium conditions is essential for the optimized design during natural gas production, processing, and transportation. In this study, a novel graphical alternating conditional expectation (ACE) algorithm was proposed to predict hydrate formation phase equilibrium conditions for sweet and sour natural gases. The accuracy and performance of the presented ACE model were evaluated using 1055 data points (688, 249, and 118 data points for sweet natural gas, CO_{2}-CH_{4}, and H_{2}S-CO_{2}-CH_{4} systems, respectively) collected from literature. Meanwhile, a comparative study was conducted between the ACE model and commonly used correlations, including thirteen models for sweet natural gases, three models for CO_{2}-CH_{4} binary system, and seven thermodynamic models for H_{2}S-CO_{2}-CH_{4} ternary system. The obtained results indicated that the proposed ACE model produces the best results in prediction of hydrate phase equilibrium temperature for sweet natural gases and pressure for CO_{2}-CH_{4} system with average absolute relative deviation (AARD) of 0.134% and 2.75%, respectively. The proposed quick and explicit ACE model also provides a better performance in prediction of hydrate phase equilibrium pressure for H_{2}S-CO_{2}-CH_{4} ternary systems with AARD=5.20% compared with seven thermodynamic methods considered in this work, except for CPA/Electrolyte/Chen–Guo combined model (AARD=4.45%).

Gas hydrates are regarded as the representatives of a class of compounds widely known as clathrates or inclusion compounds, which occur in appearance of a crystalline solid like water ice or dry ice. Generally, hydrates concomitantly consist of some guest molecules trapped in a cage structure built by host molecules. The most leading host molecule is water and the most common guest molecules include light hydrocarbons (CH_{4}), hydrogen sulfide (H_{2}S), carbon dioxide (CO_{2}), hydrogen (H_{2}), and nitrogen (N_{2}). Accurate calculation of natural gas hydrate formation phase equilibrium conditions not only is essential for an optimized design scheme of natural gas production, processing, and transportation (flow assurance), but also is of great significance for preventing [

For the natural gas hydrate formation phase equilibrium conditions, the primary problem is how to rapidly and precisely predict the hydrate phase equilibrium pressure or temperature as gas hydrates form. Up to now, various approaches have been presented to determine hydrate phase equilibrium pressure or temperature. These approaches can be mainly divided into five types: laboratory tests [

Recently, with the increase of global energy demand and the decrease of conventional gas, more attentions have been paid to explore and exploit new sour natural gas fields. Natural gas containing H_{2}S and/or CO_{2} is defined as sour gas, and also H_{2}S or/and CO_{2} hydrates are more easily formed under lower pressure and higher temperature conditions in the presence of water. Therefore, it is more critical to calculate the hydrate phase equilibrium conditions of sour natural gas mixtures to avoid the subsea or land oil/gas pipeline hydrate plugging problems. Thus, several methods (K-factor [_{2} but without H_{2}S. In addition, the CO_{2} mole concentration should be limited to no more than 5%. Note that when the temperature is below 55°F(12.8°C), all methods mentioned above fails to obtain error range of ±3°F(1.7°C) in the process of hydrate phase equilibrium temperature prediction. In 1996, a classical hydrate thermodynamic model was presented by Chen and Guo [_{2} and/or H_{2}S dissolution and reactions with water indicated that the water activity coefficient cannot be assumed to be unity. Afterward, according to Chen–Guo model, Sun and Chen [

Recently, ZareNezhad and Aminian utilized an adaptive neurofuzzy inference system (ANFIS) [_{2}S mole fraction is higher than 10%. Similarly, an improved artificial neural network (ANN) was also provided by Soroush et al. [

However, there is still few reports of an empirical correlation to predict hydrate phase equilibrium conditions especially for sour natural gases. The purpose of this study is to develop a novel correlation of hydrate phase equilibrium conditions for sweet and sour natural gases using a graphical alternating conditional expectation (ACE) algorithm. The ACE algorithm and data acquisition are introduced in Section

The ACE algorithm is a multivariate nonparametric regression method originally proposed by Breiman and Friedman [

Through converting the original linear function problem of an

Then a series of single functions is employed to minimize the error variance (

With respect to equations mentioned above, the alternating conditional expectation algorithm developed by (

In the process of ACE transformation, the minimum regression error,

Assuming that the transformation function

In consequence, the following criteria may indicate most of the advantages of the newly proposed ACE algorithm over the conventional multivariate parametric regression methods and artificial intelligence algorithms (e.g., ANN and SVM):

No requirement for a prior assumption of a function form and the optimal transformations are merely relying on the data set.

Normalization and stabilization of error variance distribution.

Applicable for the dual cases of bivariate and multivariate regression.

Different from ANN and SVM, such transformations can result in an explicit formula instead of “black box”.

It can effectively avoid probability for convergence to the local optimum, and it leads to fewer adjustable parameters.

A graphical interface program has been efficiently developed to compute ACE simulation results (GRACE) [

Acceptable generalization performance.

Adequate data on sweet and sour natural gases hydrate phase equilibrium conditions are crucial for developing an empirical correlation since the comprehensiveness, diversity, and validity of the collected database have a dominating influence on the accuracy and reliability of the correlation. Meanwhile, due to the toxicity of H_{2}S and the corrosion of H_{2}S/CO_{2} to the device, the experimental data on the hydrate phase equilibrium conditions of sour gases with H_{2}S/CO_{2} content are scarce in the open literature, which has been also identified as an important and challenging task. In this work, 1055 experimental data sets of hydrate formation conditions for sweet (688 datasets) and sour (367 datasets) natural gases were collected from published literature [

The ranges of all the experimental gas hydrate formation conditions.

Constituent | Minimum | Maximum | Average | Standard deviation |
---|---|---|---|---|

C_{1} (mole percent) | 0 | 100 | 71.38 | 26.01 |

C_{2} (mole percent) | 0 | 25 | 10.56 | 7.49 |

C_{3} (mole percent) | 0 | 13.3 | 5.63 | 4.83 |

C_{4} (mole percent) | 0 | 8.5 | 2.86 | 2.83 |

C_{5+} (mole percent) | 0 | 6.04 | 2.00 | 2.02 |

H_{2}S (mole percent) | 0 | 26.62 | 10.91 | 6.38 |

CO_{2} (mole percent) | 0 | 100 | 37.02 | 32.46 |

| 0.54 | 1.03 | 0.74 | 0.15 |

| 367.65 | 289900 | 21647.01 | 63.96 |

| 272.66 | 299.7 | 284.27 | 22.73 |

The superiority of ACE algorithm to other conventional multiple regressions is that it can flexibly employ actual and logarithmic space of the variables (e.g., ln(CH_{4}), ln(

The flowchart of the newly proposed ACE model.

For sweet natural gases hydrate formation condition, temperature is set as dependent output variable. Among the 688 data points, 563 points were selected to establish the ACE model and the remaining 125 points were used to validate the model. Meanwhile, all possible combinations of the output and input variables in either logarithmic or actual space were tested. After a series of tests, the best relationship is acquired as follows:

Figure

Fitted coefficients of each independent variable for sweet natural gases.

Variables | | | |
---|---|---|---|

Equation order | 5 | 3 | 5 |

a_{0} | 0.0328 | -1.6722 | -344.7917 |

a_{1} | -9.2945E-06 | -0.9369 | 2157.1707 |

a_{2} | 3.8474E-10 | 0.1715 | -5381.1832 |

a_{3} | 1.7341E-14 | -0.0041 | 6685.0702 |

a_{4} | -1.9660E-18 | 0 | -4132.0454 |

a_{5} | 4.2860E-23 | 0 | 1016.1832 |

R^{2} | 0.99960 | 0.99995 | 0.99503 |

Optimal transformation of variables as determined by ACE for (a) pressure, (b) pressure in the logarithmic space, (c) gas gravity, and (d) temperature.

Optimal transformation of

CO_{2} hydrate has attracted increasing researchers’ attention due to its great application potentials in many fields including CO_{2} capture and sequestration from flue gases, steam reforming processes, hydrogen (H_{2}) storage, water desalination, etc. These promising applications are held back for the problems related to the CO_{2} hydrate formation condition. In this section, hydrate formation condition data for CH_{4}-CO_{2} binary system was sorted and collected from different literature [_{2}-CH_{4} system, hydrate phase equilibrium pressure is set as an output variable. Similarly, working with CO_{2}-CH_{4} streams, 189 out of the 249 data points were chosen to construct the ACE model and the remaining 60 points were employed to validate the model. The best relationship is given as follows:

Figure _{2}),_{4}) for CH_{4}-CO_{2} binary system. Figure _{2}), and_{4}).

Optimal transformation of variables as determined by ACE for (a) CH_{4}, (b) CO_{2}, (c) temperature, and (d) pressure.

Optimal transformation of_{2}),_{4}) for CO_{2}-CH_{4} system.

According to the same principles above, the prediction model of CO_{2}-CH_{4} system hydrate formation pressure can be obtained from the inversion of the optimal transformation of CO_{2}-CH_{4} hydrate formation pressure:

Fitted coefficients of each independent variable for CO_{2}-CH_{4} systems.

Variables | CO_{2} | CH_{4} | |
---|---|---|---|

b_{0} | 3.02180508E-01 | -1.40918983E-01 | 2.28446201E+05 |

b_{1} | -5.27602603E-03 | -4.0110216E-03 | -3.37125293E+03 |

b_{2} | 9.84237658E-05 | 4.82131738E-04 | 1.86483689E+01 |

b_{3} | 3.19929491E-06 | -2.09059521E-05 | -4.58298269E-02 |

b_{4} | -3.53114022E-08 | 4.11983407E-07 | 4.22234349E-05 |

b_{5} | 1.30009467E-10 | -3.49154166E-09 | 0 |

b_{6} | 0 | 1.07521804E-11 | 0 |

R^{2} | 0.99996 | 0.99996 | 0.99993 |

In recent years, due to the decrease of conventional natural gas, sour natural gases have been becoming subsequent candidates for economical and clean energy supply. Kashagan field, Shah field, Limestone field [_{2}S content. The precise knowledge of hydrate phase equilibrium conditions for sour natural gas with high H_{2}S content is an essential prerequisite for any sour gas processing and pipeline engineering activity. To the best of our knowledge, there are few reports of an empirical correlation for predicting sour natural gases hydrate equilibrium conditions. Thus, the ACE algorithm was proposed for predicting the sour gas hydrate phase equilibrium conditions. For H_{2}S-CO_{2}-CH_{4} ternary system, hydrate formation pressure is set as an output variable. In view of the data for H_{2}S-CO_{2}-CH_{4} ternary system, the 118 data points were used for training data sets and 59 points were used for validating the feasibility and effectiveness of the model. The best expression form is shown as follows:

Figure _{4}),_{2}S),_{2}),_{4}-CO_{2} ternary system. Figure _{2}S),_{2}), and_{4})). It can be observed from Figure _{2}S-CO_{2}-CH_{4} ternary system can be obtained using the inverse of the optimal transformation of H_{2}S-CO_{2}-CH_{4} system hydrate formation pressure by means of the relationship between

Fitted coefficients of each independent variable for H_{2}S-CO_{2}-CH_{4} systems.

Variables | H_{2}S | CO_{2} | | |
---|---|---|---|---|

Equation order | 2 | 2 | 2 | 2 |

c_{0} | 4.91981951 | 3.99716313 | 40.13531836 | 95.41442257 |

c_{1} | -4.52315342E-01 | -3.92997270E-01 | 8.83934459 | -0.84492771 |

c_{2} | 1.15488583E-04 | 9.20810840E-04 | -4.13616715 | 0.00178585 |

R^{2} | 0.99991 | 0.99965 | 0.99950 | 0.99995 |

Optimal transformation of variables as determined by ACE for (a) _{2} mole percent fraction, (c) H_{2}S mole percent fraction, (d) temperature, and (e) pressure.

Optimal transformation of_{2}S-CO_{2}-CH_{4} system.

To evaluate the performance of the presented ACE model and other commonly used models considered in this work, statistical analyses such as average absolute relative deviation (AARD), root mean square error (RMSE), standard deviation (SD), and coefficient of determination (R^{2}) were performed between the experimental and predicted values as follows.

where

The scatter diagram (cross-plot) of a comparison between the experimental results and calculated temperature using ACE model is presented in Figure

Comparison of experimental data with ACE testing outputs.

Moreover, the predicted hydrate formation temperature of sweet natural gases for testing data sets from the proposed ACE model was compared to 13 commonly used correlations listed in Figure

Global AARD of different correlations: (1) this work; (2) Ghayyem et al.; (3) Ghiasi; (4) Amin et al. (2016); (5) Amin et al. (2015, first); (6) Amin et al. (2015, second); (7) Bahadori and Vuthaluru; (8) Zahedi et al.; (9) Towler and Mokhatab; (10) Kobayashi et al.; (11) Motiee; (12) Berge; (13) Hammerschmidt; and (14) Makogon correlation.

It can be concluded from Figure

The cross-plot (Figure _{2}-CH_{4} system. The excellent agreement between ACE model prediction and the experimental values validate our ACE model for the CO_{2}-CH_{4} system.

Comparison of the experimental_{2} -CH_{4} system.

Moreover, a comparison between predictions proposed by Adisasmito et al. [

Comparison of ACE model with Adisasmito et al.’s equation to predict hydrate formation pressure for CO_{2}-CH_{4} binary system.

Train set | AADR | SD | RMSE | R^{2} |
---|---|---|---|---|

ACE model | 2.20 | 0.033 | 134.00 | 0.9954 |

Adisasmito et al. | 2.45 | 0.038 | 140.00 | 0.9952 |

ACE model | 3.43 | 0.065 | 483.87 | 0.9724 |

Adisasmito et al. | 4.53 | 0.063 | 411.40 | 0.9711 |

ACE model | 2.75 | 0.043 | 251.53 | 0.9873 |

Adisasmito et al. | 2.95 | 0.045 | 235.93 | 0.9862 |

In Table

In addition, a comparison between the proposed ACE model and two commonly software-based thermodynamic models, including CSMGem [_{2}-CH_{4} binary system.

Comparison between experimental _{2}-CH_{4} system.

| ACE model(kPa) | ARD% | CSMGem model(kPa) | ARD% | HWHYD model(kPa) | ARD% |
---|---|---|---|---|---|---|

4030 | 3953 | 1.90 | 3980 | 1.24 | 4180 | 3.72 |

5480 | 5248 | 4.24 | 5670 | 3.47 | 5860 | 6.39 |

8270 | 8083 | 2.26 | 8710 | 5.32 | 8840 | 6.89 |

2720 | 2716 | 0.15 | 2780 | 2.21 | 2970 | 9.19 |

3610 | 3373 | 6.57 | 3680 | 1.94 | 3940 | 9.14 |

6090 | 6737 | 10.62 | 7340 | 20.53 | 7620 | 25.12 |

2720 | 2757 | 1.37 | 2730 | 0.37 | 2850 | 4.78 |

3210 | 3160 | 1.56 | 3260 | 1.56 | 3420 | 6.54 |

4700 | 4834 | 2.86 | 5120 | 8.94 | 5390 | 14.68 |

AARD | 3.50 | 5.06 | 9.67 |

In Figure _{2}S-CO_{2}-CH_{4} system. The overall AARD, RMSE, SD, and R^{2} are 7.20%, 453.29, 0.10, and 0.97, respectively. The predicted results show a good agreement with the experimental values. Careful examination of Figure _{2}S-CO_{2}-CH_{4} ternary systems.

Comparison of the experimental _{2}S-CO_{2}-CH_{4} system.

In order to evaluate the accuracy of ACE model, the prediction results are also compared with values calculated using other thermodynamic models from ZareNezhad and Ziaee [

Comparison of AARD of the predicted hydrate formation pressures by different models for H_{2}S-CO_{2}-CH_{4} systems.

Gas mixture | AARRD% | |||||||
---|---|---|---|---|---|---|---|---|

ACE | M2 | M3 | M4 | M5 | M6 | M7 | M8 | |

87.65%CH_{4}+7.40%CO_{2}+4.95%H_{2}S | 5.98 | 4.13 | 4.39 | 8.24 | 9.27 | — | — | 7.31 |

82.45%CH_{4}+10.77%CO_{2}+6.78%H_{2}S | 8.86 | 2.84 | 2.89 | 7.67 | 9.36 | 4.97 | 5.81 | 8.11 |

82.91%CH_{4}+7.16%CO_{2}+ 9.93% H_{2}S | 3.07 | 3.91 | 4.02 | 7.59 | 6.98 | 2.48 | 4.48 | 8.43 |

77.71%CH_{4}+7.31%CO_{2}+14.98%H_{2}S | 2.05 | 2.77 | 4.31 | 10.0 | 11.0 | 5.55 | 6.81 | 10.0 |

75.48%CH_{4}+6.81%CO_{2}+17.71%H_{2}S | 4.78 | 4.35 | 7.65 | 11.1 | 12.5 | 7.62 | 9.96 | 12.2 |

66.38%CH_{4}+7.00%CO_{2}+26.62%H_{2}S | 6.44 | 8.67 | 14.6 | 16.6 | 19.0 | 15.3 | 17.6 | 17.6 |

Mean AARD% | 5.2 | 4.45 | 6.31 | 10.2 | 11.35 | 7.18 | 8.93 | 10.6 |

(M2: CPA/Electrolyte/Chen–Guo model; M3: SRK/Electrolyte/Chen–Guo model; M4: SRK/Electrolyte/vdW-P model; M5: SRK/vdW-P model; M6: Patel-Teja/Electrolyte/Chen–Guo model; M7: Patel-Teja/Chen–Guo model; M8: CSMGem software)

It can be observed in Table

In addition, compared to the thermodynamic models, the ACE model has one enormous advantage: it provides quick and explicit calculations of the hydrate phase equilibrium pressure for H_{2}S-CO_{2}-CH_{4} ternary system over a wide range of phase equilibrium temperature and H_{2}S content, which avoids complex calculations by the thermodynamic models. Careful examination of Table _{2}S content range from 9.93% to 26.62%, which covers the mostly of natural gas fields with high H_{2}S content. Note that the result of (_{2}-CH_{4} hydrate formation pressure by setting the H_{2}S content as zero, indicating that (_{2}S component, while H_{2}S content has an important influence on the results of hydrate phase equilibrium conditions for H_{2}S-CO_{2}-CH_{4} ternary system.

In this work, the ACE model was proposed to predict hydrate formation phase equilibrium conditions for both sweet and sour nature gases. Among the 1055 experimental data from literature, three data groups (688, 249, and 118 data points) were used to train, test, and evaluate the performances of the ACE model for sweet and sour natural gases. The presented ACE model gives the best matching result among 13 commonly used correlations with mean AARD of 0.134% for sweet natural gases. The ACE model produces better results than Adisasmito et al., CSMGem and HWHYD models with AARD less than 3.5% for CO_{2}-CH_{4} binary system. The proposed ACE model has better performance in prediction of hydrate formation pressure for H_{2}S-CO_{2}-CH_{4} ternary systems with mean AARD=5.20% with respect to seven thermodynamic methods considered in this work, except for CPA/Electrolyte/Chen–Guo combined model (AARD=4.45%). Moreover, the most advantage of the ACE model is that it provides quick and explicit calculations of the hydrate formation conditions.

Here are the empirical correlations for hydrate formation condition representation.

Hammerschmidt (1934)

Makogon (1981)

Berge (1986)

Kobayashi et al. (1987)

Riazi (2014)

The data used to support the findings of this study are included within the supplementary information file.

The authors declare that they have no conflicts of interest.

This work was supported by National Science and Technology Major Project (2016ZX05048) and National Key Basic Research Development Plan (973 program) “Well completion and test optimization methods for deepwater oil and gas wells” (2015CB251205).

Supplementary Materials folder is the supplementary information, unzip the folder, it contains the data files for all the figures (Figure 2-Figure 11) in the manuscript. The name of file in the folder corresponds to the figure title in the manuscript.

_{2}, N

_{2}and tetrahydrofuran

_{4}, C

_{2}H

_{6}, C

_{3}H

_{8}, N

_{2}, CO

_{2}and their mixtures using the PRSV2 equation of state and obtaining the Kihara potential parameters for these components

_{2}S and CO

_{2}containing sour gas hydrates formation conditions considering hydrolytic and hydrogen bonding association effects