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Investigation on sulfur solubility in sour gas at elevated temperatures and pressures with an artificial neural network algorithm
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Investigation on sulfur solubility in sour gas at elevated temperatures and pressures with an artificial neural network algorithm ⁎
Liang Fua, Jinghong Hua, , Yuan Zhanga, Qian Lib a b
Beijing Key Laboratory of Unconventional Natural Gas Geology Evaluation and Development Engineering, China University of Geosciences, Beijing 100083, China Research Institute of Petroleum Exploration and Development, PetroChina Southwest Oil & Gas Field Company, Chengdu 610041, China
A R T I C LE I N FO
A B S T R A C T
Keywords: Elemental sulfur solubility Sour gas reservoirs T-S fuzzy neural network Prediction model
Accurate prediction of the solubility of elemental sulfur in sour gas mixtures has been recognized as a key issue in the development of sour gas fields. Many experimental measurements and empirical models have shown the complicated relationships between sulfur solubility and sour gas properties. However, the accurate model that can be used to predict sulfur solubility in sour gas over a wide range of temperature and pressure is rare. Therefore, the objective of this work is to build an efficient model, namely T-S fuzzy neural network (T-S FNN), to investigate the sulfur solubility in sour gas. The model considers the reservoir pressure, temperature, the mole fraction of methane, hydrogen sulfide and carbon dioxide as input parameters and the sulfur solubility as target parameters. Subsequently, multiple experimental sulfur solubility data sets accessible to the literature are employed to train and test the model respectively. Finally, a series of studies are conducted to appraise the accuracy and generalization capability of the model. The result shows that the predicted solubility data had great agreement with experimental sulfur solubility with overall average absolute relative deviation of 5.35%, which proves the model in this work is feasible and effective. Additionally, it provides a new idea to the prediction of sulfur solubility in sour gas and helps operators to develop the sour gas reservoir better.
1. Introduction Fossil fuels play an important role in the world energy structure. Many problems should be solved during the development of oil and gas reservoirs [1,2]. Natural gas with high sulfur content is one of the significant clean energy resources with abundant geological reserves and is widely distributed all over the world [3]. As the reservoir pressure and temperature decrease elemental sulfur precipitates from sour gas in reservoir formation, tubing string and so on, consequently sulfur deposition reduces the permeability of the formation and blocks the production transportation equipment, which makes the inflow performances of gas wells decreased during the sour gas field development [4,5]. Therefore, accurate measurement of the sulfur solubility in sour gas is critical to hinder the sulfur deposition. Models for predicting sulfur solubility are generally classified into four categories: experimental measurement, empirical models, thermodynamic models and intelligent algorithm models. The sulfur solubility in sour gas can be accurately obtained by experiments, and many experimental tests have been conducted by scholars at home and abroad since 1960 [6–15], Kennedy and Wieland [6] firstly performed the experimental test in pure gas (CH4, CO2, H2S)
⁎
and three binary gas mixtures over a range of temperatures (339.15 ~ 394.15 K) and pressures (6.89 ~ 41.4 MPa). After that, Roof [7], Swift et al. [8], Brunner and Woll [9], Brunner et al. [10], Gu et al. [11], Sun and Chen [12], Hu et al. [13], Bian et al. [14], Zhang [15] respectively measured the sulfur solubility in sour gas at different temperature and pressure. A detailed summary of experimental tests is listed in Table 1. Although experimental measurement of sulfur solubility in sour gas is the most accurate, it is highly toxic, time-consuming and costly. Compared with experimental measurement, empirical models are easier to acquire, which can be obtained by correlating with experimental data. Chrastil [16] firstly proposed an association model for the correlation of solid solubility in supercritical fluid. Then Roberts [17] regressed the parameters of Chrastil model by means of two groups of experimental data obtained by Brunner and Woll [9] and Brunner et al. [10]. Though the Roberts model was commonly applied to predict the sulfur solubility in sour gas, such model tends to produce large errors due to the vagueness of application conditions. Similarly, Hu et al. [13] regressed the three coefficients (k, a, b) of Chrastil model for different sour gas at different pressure and temperature and functions well. Considering the influence of multiple factors such as the gas density and
Corresponding author. E-mail address: [email protected] (J. Hu).
https://doi.org/10.1016/j.fuel.2019.116541 Received 28 August 2019; Received in revised form 17 October 2019; Accepted 29 October 2019 0016-2361/ © 2019 Elsevier Ltd. All rights reserved.
Please cite this article as: Liang Fu, et al., Fuel, https://doi.org/10.1016/j.fuel.2019.116541
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Table 1 Summary of experiments in measurement of sulfur solubility in sour gas. List
Authors
Temperature (℃)
Pressure (MPa)
Gas Component
Refs.
1 2 3 4 5 6 7 8 9
Kennedy and Wieland (1960) Roof (1971) Swift et al.(1976) Brunner and Woll (1980) Brunner et al. (1988) Gu MX (1993) Sun CY (2003) Yang XF (2009) Bian XQ (2010)
100 ~ 160 125 ~ 212 93.3 ~ 121.1 43.3 ~ 110 121 ~ 204 90 ~ 110 30 ~ 90 100 63 ~ 123
10 ~ 60 6.7 ~ 155 7 ~ 41.4 7 ~ 31 34.5 ~ 138 10 ~ 50 20 ~ 45 10 ~ 36 10 ~ 55.2
Pure H2S, Pure H2S Pure H2S H2S, CH4, H2S, CH4, H2S, CH4, H2S, CH4, H2S, CH4, H2S, CH4,
[6] [7] [8] [9] [10] [11] [12] [13] [14]
CH4, CO2, N2 and different ratio of the four mixed gas
CO2 CO2 CO2 CO2 CO2 CO2
and N2 at different ratio of mixed gas and N2 at different ratio of mixed gas and N2 at different ratio of mixed gas and N2 at different ratio of mixed gas and N2 at different ratio of mixed gas
[17] and Guo-Wang equation [20]) and an intelligent algorithm model (Bian et al. model [14]) reported in previous literature were used to further verify the accuracy of the model. In summary, this work provides a better intelligent algorithm model for the prediction of sulfur solubility in sour gas.
temperature, Bian et al. [18,19] proposed new empirical models. Inspired by the Hu et al model, Guo and Wang [20] considered the constant coefficient k as a temperature function and introduced a new parameter M to lower the importance of sour gas density, hence the new model produces more accurate result than the Hu et al model and the Roberts model. In short, the empirical models mentioned above are piecewise formulas. It is widely acknowledged that empirical models are user-friendly, however, the majority of them can easily produce large errors compared with experimental data, because it is difficult for these models to reflect the intrinsic relationships between input parameters and sulfur solubility. Similar to the empirical model, thermodynamic model is also a kind of mathematical model, which requires not only complicated calculation but also the knowledge of critical parameters of the solute and solvent [21–23], as a result, reservoir engineers rarely adopt them for the engineering calculation of sulfur solubility in sour gas. The fourth category is intelligent algorithm model such as artificial neural network (ANN). In the last 40 years, ANN has been widely applied in engineering calculation such as chemical engineering, environmental engineering, electronic engineering and so on [24,25]. However, the ANN models that can be used for the investigation on sulfur solubility is insufficient, which has only experienced about 10 years of development. Mohammadi and Richon [26] firstly introduced the feed-forward neural network (FNN) to calculate sulfur solubility in gaseous solvents at elevated temperatures (316–433 K) pressure (60 MPa). However, the model can only estimate sulfur solubility in pure H2S. Then Mehrpooya et al. [27], ZareNezhad and Aminian [28] extended the FNN to estimate sulfur solubility in sour gas at different temperatures and pressures with acceptable results. Recently, Bian et al. [14] built an intelligent algorithm model namely grey wolf optimizer-based support vector machine (GWO-SVM) with 170 data sets, the prediction results proves that the model is reliable for the use of predicting sulfur solubility in sour gas. However the H2S content is preferably between 1 and 20 when using the model, since the model greatly acquire the relationship between sulfur solubility and input variables where the H2S is between 1 and 20 in the training sets. Intelligent algorithm models have strong nonlinear correlation capacity and relatively simple formulas hence can be widely used in prediction of sulfur solubility. Additionally, a comparison for the explanation of differences of these models mentioned above was listed in Table 2. In this work, a T-S fuzzy neural network (T-S FNN) was built to calculation of the sulfur solubility in sour gas. The calculation flow chart was shown in Fig. 1. Firstly, the methodology and detailed procedure were explained. The reservoir temperatures, pressures and the mole fractions of CH4, H2S and CO2 were considered as input parameters of the model with the method of gray correlation coefficient. Then 222 data sets accessible to the literature were employed to develop, validate and evaluate the new model. Afterwards, several evaluation parameters were adopted to evaluate the predicted results. The outlier diagnostics was employed by means of the leverage approach to detect the doubtful sulfur solubility data. Additionally, three widely used empirical correlations (Hu et al. equation [13], Roberts equation
2. Methodology 2.1. T-S fuzzy neural network Fuzzy neural network is the combination of fuzzy control theory and artificial neural network. It has the advantages of both neural network and fuzzy theory consequently can be used for learning, association, recognition and information processing [29,30]. The Takagi-Sugeno (TS) type fuzzy logic system is employed for the construction of the fuzzy neural network, shown in Fig. 2, whose structure consists of two parts: antecedent network and consequent network [29–31]. Many studies have been proved that T-S fuzzy neural network can be appeared as a universal approximate and can approximate the nonlinear systems with arbitrary precision [32–34]. Therefore, T-S fuzzy neural network can be employed for the prediction of sulfur solubility in sour gas. 2.1.1. The structures of T-S FNN There are 4 layers in the antecedent network, known as input layer, membership function layer, product inference rule layer and weighted average defuzzifier layer [30,31,35,36]. Each neuron in layer 1 represents input variable xi(i = 1,2,…,n), determined by the number of input variables n. Layer 2 is the membership function layer, in which each neuron stands for one membership function. The Gaussian function is applied as the membership function, and the number of neurons is calculated as follows.
μi j = exp { −(x i − cij )2 / bij2}, i = 1, 2, ...,n; j = 1, 2, ...,n
(1)
h=n×m
(2)
j
where μi is the jth membership function in the ith neuron, h is the number of neurons in layer 2, cij and bij are the center and width of the Gaussian function respectively. m is the fuzzy segmentation number of each input variable xi Layer 3 is the fuzzy rule layer. Each neuron is employed for the calculation of the firing strength by an algebraic operation.
wj = μ1j μ2j . ..μnj
(3)
where wj (j = 1,2,…,m) is the firing strength as well as the adaptability of each rule, m is the number of fuzzy rules. Layer 4 is the normalized calculation layer, which can be calculated as follows.
w¯j = wj ∑m wj, j = 1, 2, ...,m . j=1
(4)
where w¯ j (j = 1,2,…,m) is the normalized firing strength as well as the adaptability of each rule. 2
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Table 2 Comparison between empirical equations and intelligent algorithm models. Classification
Models
Differences
Empirical equations
Roberts model (1997)
A non-piecewise equation, the input variables are gas density and temperature, the model is acquired from only two groups of experimental data. A piecewise equation, model parameters vary with temperature, pressure and H2S content in sour gas. A piecewise equation, similar to Hu et al model, a new parameter M is introduced to lower the importance of sour gas density in the model. The input parameters are temperature and pressure, the model can only predict sulfur solubility in pure H2S gas. Based on Mohammadi et al model, gravity of sour gas and H2S content are added to the input variables of the model. Namely SVM-GWO model, the model considers 5 main factors as input variables and can be given priority where the H2S content is between 1 and 20.
Hu et al model (2014) Guo-Wang model (2016) Intelligent algorithm models
Mohammadi et al model (2008) Mehrpooya et al model (2010) Bian et al model (2018)
membership of the inputnode and the rule, yj is the jth consequent part of fuzzy rule, and pijis system parameters. Layer 3, known as output layer, function as the calculation of the output of the whole system, described as follows. m
y=
∑ w¯ j y j (6)
j=0
where y is the whole system output, w¯ j is the same as layer 4 in antecedent network, yj is equal to layer 2 in consequent network. 2.1.2. The learning algorithm of T-S FNN In order to ensure the T-S FNN model produce a more accurate result, one learning algorithm namely gradient descent is employed, which can adjust the center and width of membership function (cij, bij) in the antecedent network and the connecting coefficient (pij) in the consequent network in the learning process [30,31,35–37]. The detailed algorithm principle of T-S FNN was shown in Fig. 3. The cost function is specified as performance function, in fact, the goal of the learning process is to minimize the performance function, setted as follows.
Fig. 1. Flow chart for the calculation procedure in this work.
e=
1 d (y − y )2 2
(7) d
where e is the cost function, y is the expected output of the network, y is the actual output. Parameter pi j ,cij, bij, is updated using the gradient descent method, expressions are as follows.
Rj : IF x1 is
THEN y j =
...,xn is A nj, + ···+pnj x n
(8)
pi j (k + 1) = pi j (k ) − α
∂e ∂pi j
(9)
cij (k + 1) = cij (k ) − α
∂e ∂cij
(10)
bij (k + 1) = bij (k ) − α
∂e ∂bij
(11)
j
pi (k ), cij (k ), bij (k ) are the network parameters at the kth where iteration step, and their starting values are usually generated randomly; α , known as the learning rate, is a constant at 0–1.
There are 3 layers in the consequent network [30,31,35,36]. Layer 1 is the input layer who has the same function as the corresponding layer in antecedent network. Layer 2 is the fuzzy rule layer that is used to match the consequent part of fuzzy rule.
x2 is A2j, p0j + p1j x1
∂e ∂p0j
p0j (k ),
Fig. 2. Structure of T-S fuzzy neural network.
A1j,
p0j (k + 1) = p0j (k ) − α
3. Data analysis and model training 3.1. Experimental data 222 data sets of experimental sulfur solubility, listed in Table 3, were collected from previous literature [9,12–15] to develop and evaluate the model, among them, about 75 percent for training and the rest for evaluation. The collected data sets covers wide ranges of
(5)
Where Rjis the jth fuzzy ruleAij (i = 1,2,…,n; j = 1,2,…,m) is Gaussian 3
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Fig. 3. The algorithm principle chart of T-S FNN.
approximately positive linear correlation with sulfur solubility. Additionally, the content of CO2 should also be considered while calculating the sulfur solubility, although having smaller impact than that of H2S. To investigate the impact of the main factors on sulfur solubility in sour gas, the gray correlation coefficient is employed. The greater value of gray correlation coefficient of input variable, the greater effect on determining the value of output variable [14,38,39]. Through gray correlation analysis, the most sensitive factors on sulfur solubility, listed in Fig. 4, are H2S content, followed by CO2 content, reservoir pressure, temperature and CH4 content, While N2 content and C2H6 content show least impact on sulfur solubility consequently will not be considered as input parameters. Therefore, it is clear that the new T-S FNN model was designed to acquire the most desirable regression between sulfur solubility and the H2S mole fraction, CO2 mole fraction, pressure, temperature and CH4 mole fraction.
temperatures (30–160 °C), pressures (10–66.52 MPa), H2S content (1–26.62%) and sulfur content (0.012–6.413 g/m3). The collected data sets were split to training sets with 167 data points, testing sets with 51 data points and checking sets with 4 data points. The training sets were used for updating the parameters ( pi j ,cij, bij,) with a goal to minimize the cost function. The testing sets, not appeared in training sets, were used for evaluating the generalization capability of the trained model. And the checking sets, brand-new sets for the model, were applied to appraise the applicable capability of the model in gas field. And in order to get more accurate results, all of the data sets were normalized to between −1 to 1, the expression is as follows.
yi =
2(xi − xmin) −1 (xmax − xmin)
(12)
where yi is normalized data, xi is the collected experimental data, xmax and xmin is the maximum and minimum value of the data sets respectively.
3.3. Simulation on sulfur solubility prediction During the training process, the structure and parameters of T-S FNN model is determined by trial-and-error method with a goal to minimize the cost function. Finally, the structure and parameters, listed in Table 4, were set as follows: the number of membership function is 2 for each input variable, the number of fuzzy rules is 32, membership type is Trim type function, max iterations is 50, and other parameters remain the defaults.
3.2. Influential factors of sulfur solubility in sour gas According to previous researches [9,10,13,14], the chief factors affecting sulfur solubility in sour gas are temperature, pressure and gas composition including H2S, CO2, etc. As the reservoir temperature and pressure descend, the sulfur solubility in sour gases decreases. And the H2S content which has the greatest influence on sulfur solubility has an Table 3 Summary of the database of sulfur solubility used in this work. Author
Temperature (°C)
Pressure (MPa)
Element sulfur solubility (g/m3)
Training data sets
Testing data sets
Brunner and Woll (1980) Sun CY (2003) Yang XF (2009) Bian XQ (2010) Zhang GD (2014)
100–160 30–90 100 63–123 100–152.5
10–60 20–45 24–36 10–55.2 20–66.52
0.043–4.29 0.012–1.455 0.042–0.2682 0.0083–2.067 0.034–6.413
59 56 0 34 18
11 0 4 16 24
4
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Fig. 5. The relationship between sulfur solubility and temperature and pressure.
Fig. 4. Gray correlation coefficients for sulfur solubility in supercritical sour gas. Table 4 Parameters of the trained T-S FNN model. Parameters
Sour gas mixture
Input variables Input data form Max iterations Membership type Fuzzy rules Cost function: e
5 [-1, +1] 50 Trimf 32 0.0069
4. Results and discussion 4.1. Outputs analysis Fig. 6. The relationship between sulfur solubility and H2S mole fraction.
In order to evaluate the model in this work quantitatively, four evaluation parameters, which appears as an effective mathematical tool when comprehensively evaluating the accuracy of a model, were employed as a comparison of experimental and predicted data, including the average absolute relative deviation (AARD), the root mean squared error (RMSE), the standard deviation (SD), and determination coefficient (R2) [14]. The formulas of these evaluation parameters are shown as follows.
AARD =
RSME =
100 N
1 N
N
∑
Data sets
AARD(%)
SD
RSME
R2
Training sets Testing sets Checking sets
3.68 5.35 7.04
0.06 0.08 0.09
0.02 0.06 0.01
0.9996 0.9984 0.8855
yiexp − yical yiexp
i=1
(13)
N
∑ (yiexp − yical )2 i=1 exp
SD =
Table 5 Comparison of data sets calculated by the model in this paper.
(14)
cal 2
1 ⎛ yi − yi ⎞ ⎟ N − 1 ⎜⎝ yiexp ⎠
(15)
N
R2 = 1 −
∑i = 1 (yiexp − yical )2 N
exp − yical )2 ∑i = 1 (yave
(16)
where N is the number of all experimental sulfur solubility data points; exp yiexp , yical and yave are experimental and experimental sulfur solubility and average value of all experimental sulfur solubility respectively. The relationships, obtained by the proposed T-S FNN model, between sulfur solubility and temperature, pressure, H2S content were shown in Figs. 5 and 6, respectively. As can be seen, the input parameters have approximately positive correlations with the output sulfur solubility, which is similar to the previous conclusion: (reached by Brunner et al. [9,10], Hu et al. [13] and Bian et al. [14]) sulfur solubility in sour gas increases, as the system temperature, pressure or H2S mole fraction increase.
Fig. 7. Comparison of the calculated and experimental sulfur solubility.
For the training data sets, the comparison between predicted results and experimental data was shown in Table 5, Figs. 7 and 8. As can be seen, the calculated data points have an excellent agreement with experimental data points with AARD = 3.7% and R2 = 0.999, which reveals the model has strongly high fitting capability. 5
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H = X (X t X )−1X t
(17)
where X is a two-dimensional matrix consisting of n data (rows) and k input variables (columns) and t stands for the transpose matrix. The Williams plot (plot of standardized versus Hat indices), who denotes the correlation between the standardized cross-validated residuals and Hat indices, was depicted, through which the outlier data can be identified. In the Williams plot there is a squared area within ±3 standard residuals and a leverage threshold H* (H* generally equals to 3n/(k + 1)), inside which distribute the valid data. Generally, the majority of data points distribute in the squared area (0⩽H⩽H* and −3⩽SR⩽3) which not only indicates the model has wide applicability domain, but also proves the validity of the model in statistics field. The Williams plot for the outputs of the model in this paper was presented in Fig. 14, from which it can be seen that the great majority of sulfur solubility utilized in this paper locate in the valid domain of [−3,3] and [0,H*], except for 5 points fell outside of the valid domain, considering as outliers, and the Williams plot also demonstrates the validity and reliability of the model in statistics field.
Fig. 8. Absolute relative deviation of the calculated sulfur solubility in training data sets.
4.3. Comparison of estimation accuracy
For the testing data sets, the comparison between predicted results and experimental data was shown in Table 6–8, Figs. 7 and 9, as revealed, the predicted data points show a great agreement with experiment data points with AARD = 5.35% and R2 = 0.998. And another comparison between the model predicted results and the experimental data of Brunner [9], Bian et al [14] and Zhang [15] at isothermal conditions was illustrated in Figs. 10–12, confirming the model in this work enjoys great prediction accuracy and generalization capability. Additionally, in order to appraise the applicable capability of the model in gas field, a group of data sets of Yang [13], not appeared in the training data sets, was employed. The prediction results were shown in Table 9 and Fig. 13, revealing the prediction results have a satisfactory agreement with experimental data points with AARD = 7.04% and R2 = 0.8855. Therefore it proves that the proposed T-S FNN model can function well in the prediction of sulfur solubility in sour gas in gas fields.
In order to further verify the accuracy and reliability of the model in this work, a comparison between the T-S FNN model and three widely used empirical equations (Hu et al equation [13], Roberts equation [17] and Guo-Wang equation [20]) and an intelligent algorithm model (Bian et al. model [14]) was conducted with the testing data sets in this paper. The comparison results were shown in Table 10 and Fig. 15, as can be seen, the data points predicted by T-S FNN model have the best agreement with experimental data points than that of another four models. Both Hu et al model and Guo-Wang model provide closer match to the experimental data compared to Roberts model. While Bian et al model is much better than above three empirical equations, indicating GWO-SVM model is reliable and applicable. The proposed T-S FNN model has advantages over the other four models. It can accurately predict the sulfur solubility in sour gas without adjusting the parameters of the model, while the three empirical equations cannot achieve the same goal, since they are all piecewise formulas. When using the empirical models, the parameters of the equations require to be adjusted as the values of input variables change, which increases the risk of errors. Both the T-S FNN model and GWO-SVM model are intelligent algorithm models based on the theory of statistics, and both can automatically acquire the desirable regression between input variables and output variables. Once fully trained, both models can enjoy high accuracy, which is evidenced in this paper. However, the T-S FNN model
4.2. Outlier diagnosis Outlier diagnosis [14,40] is a significant statistical approach that can diagnose individual outlier datum from the sample set, for which the leverage statistics is an accurate and efficient method. The standardized residual (SR), the Hat indices (H) and the critical Leverage limit (H*) are used in this method. The Hat indices can be defined as follows.
Table 6 Comparison between predicted results and experimental sulfur solubility: Bian et al. [14]. Component
Temperature (K)
Pressure (MPa)
Sulfur solubility (in this work) g/m3
Sulfur solubility (experiment) g/m3
Absolute relative deviation (%)
13.79% H2S 9.01% CO2 0.52% N2 76.64% CH4 0.04% C2H6
351.15
25
0.104
0.113
8.11
35 45 55.5 25 35 45 55.2 25 35 45 55.2 25 35 45 55.2
0.225 0.393 0.610 0.149 0.333 0.600 0.955 0.225 0.501 0.900 1.432 0.339 0.741 1.314 2.070
0.2419 0.389 0.5965 0.1589 0.3359 0.5887 0.9246 0.2216 0.4856 0.899 1.4044 0.3357 0.748 1.3196 2.0672
6.84 0.97 2.18 6.38 0.88 1.88 3.27 1.51 3.12 0.16 2.00 0.96 0.88 0.39 0.14
366.15
381.15
396.15
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Table 7 Comparison between predicted results and experimental sulfur solubility: Brunner [9]. Component
Temperature (K)
Pressure (MPa)
Sulfur solubility (in this work) g/ m3
Sulfur solubility (experiment) g/ m3
Absolute relative deviation (%)
7% H2S 20% CO2 8% N2 65% CH4
373.15
6% H2S 9% CO2 4% N2 81% CH4
393.15 413.15 433.15 373.15
30 48 45 45 30 30 50 40 25 45 25
0.235 0.532 0.925 1.394 1.133 0.115 0.468 0.701 0.500 1.351 0.943
0.202 0.621 1.006 1.43 1.04 0.136 0.476 0.635 0.562 1.42 0.752
16.45 14.26 8.04 2.53 8.98 15.37 1.62 10.42 11.04 4.86 25.41
393.15 413.15 433.15
Table 8 Comparison between predicted results and experimental sulfur solubility: Zhang [15]. Component
Temperature (K)
Pressure (MPa)
Sulfur solubility (in this work) g/m3
Sulfur solubility (experiment) g/m3
Absolute relative deviation (%)
2.93% H2S 4.37% CO2 0.34% N2 92.32% CH4
373.15
30
0.109
0.095
15.09
50 64.88 66.52 20 40 60 66.52 30 50 64.88 66.52 20 40 60 66.52 30 50 64.88 66.52 20 40 60 66.52
0.342 0.531 0.490 0.106 0.510 1.274 1.482 0.489 1.409 2.379 2.411 0.412 1.441 3.162 3.687 1.388 3.510 5.519 5.600 0.676 2.664 5.444 6.231
0.338 0.578 0.615 0.112 0.549 1.278 1.545 0.472 1.436 2.326 2.444 0.374 1.453 3.167 3.778 1.359 3.543 5.558 5.767 0.779 2.655 5.428 6.413
1.13 8.09 20.27 5.13 7.12 0.33 4.08 3.67 1.87 2.30 1.36 10.18 0.80 0.17 2.40 2.14 0.92 0.71 2.89 13.28 0.34 0.29 2.84
393.15
403.15
413.15
423.15
425.65
Fig. 10. Predicted sulfur solubility and experimental data in sour gas: Brunner [9].
Fig. 9. Absolute relative deviation of the predicted sulfur solubility in testing data sets.
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Fig. 13. Predicted sulfur solubility and experimental data in sour gas: Yang et al [13].
Fig. 11. Predicted sulfur solubility and experimental data in sour gas: Bian et al [14].
Fig. 14. Detection of the doubtful calculated sulfur solubility in sour gas. Fig. 12. Predicted sulfur solubility and experimental data in sour gas: Zhang [15].
Table 10 Comparison of T-S FNN model with four widely used models considered in this work.
has some advantages that GWO-SVM does not have. As mentioned above, T-S FNN has the advantages of both neural network and fuzzy theory, and it can use fuzzy reasoning to take advantage of existing information for machine learning, which has positive effect on the improvement of prediction accuracy. Additionally, the H2S content in the training sets in this work is between 1 and 26.62% which is wider than that of Bian et al model, suggesting the model has a lager scope of application. In summary, two different kind of intelligent algorithm models (T-S FNN model and GWO-SVM model) obtain the similar results in predicting sulfur solubility in sour gas, therefore they can refer to each other.
Models
AARD(%)
SD
RSME
R2
Roberts model Hu et al model Guo-Wang model Bian et al model This work
65.36 17.3 12.84 6.68 5.35
0.86 0.22 0.15 0.08 0.08
0.67 0.21 0.17 0.09 0.06
0.6792 0.9731 0.9833 0.9961 0.9983
gas in gas field usually varies from 0 to 100%, whereas the H2S content in the data sets used in this work is only between 1 and 26.62%, thus 167 data sets are still insufficient to train the proposed T-S FNN model for the accurate prediction of sulfur solubility in sour gas regardless of the content of H2S. When the H2S content is greater than 26.62%, the prediction accuracy of the proposed model will deteriorate, in other words, it is the most desirable occasion for the model to predict sulfur solubility, where the H2S content is between 1 and 26.62%.
4.4. Model limitation The T-S FNN model enjoys high accuracy and reliability with the data sets shown in this work hence has advantages over other models in predicting sulfur solubility in sour gas. However, H2S content in sour
Table 9 Comparison between predicted results and experimental sulfur solubility: Yang et al. [13]. Component
Temperature (K)
Pressure (MPa)
Sulfur solubility (in this work) g/m3
Sulfur solubility (experiment) g/m3
Absolute relative deviation (%)
6.86% H2S 2.76% CO2 0.5% N2 89.63% CH4 0.21% C2H6
373.15
24 28 32 36
0.150 0.180 0.218 0.264
0.169 0.194 0.201 0.268
10.95 7.01 8.49 1.68
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Study on features of phase behaviors and seepage mechanism in high sour gas reservoir-take samples from changxing gas reservoir in yuanba as study objects. Chengdu University of Technology; 2014. p. 22–33. [16] Chrastil J. Solubility of solids and liquids in supercritical gases. J Phys Chem 1982;86(15):3016–21. https://doi.org/10.1021/j100212a041. [17] Roberts BE. The effect of sulfur deposition on gas well inflow performance. SPE J Reserv Eng 1997;12(02):118–23. https://doi.org/10.2118/36707-PA. [18] Bian XQ, Du ZM. An association model for the correlation of the solubility of elemental sulfur in sour gases. The First International Acid Gas Injection Symposium, Calgary, Canada 2009; 10: 112–119. [19] Bian XQ, Zhang Q, Du ZM, et al. A five-parameter empirical model for correlating the solubility of solid compounds in supercritical carbon dioxide. Fluid Phase Equilib 2016;411:74–80. https://doi.org/10.1016/j.fluid. 2015.12.017. [20] Guo X, Wang Q. A new prediction model of elemental sulfur solubility in sour gas mixtures. J Nat Gas Sci Eng 2016;31(04):98–107. https://doi.org/10.1016/j. jngse. 2016.02.059. [21] Heidemann RA, Phoenix AV, Karan K, et al. A chemical equilibrium equation of state model for elemental sulfur and sulfur-containing fluids. Ind Eng Chem Res 2001;40(09):2160–7. https://doi.org/10.1021/ie000828u. [22] Karan K, Heidemann RA, Behie L. Sulfur solubility in sour gas: predictions with an equation of state model. Ind Eng Chem Res 1988;37(05):1679–84. https://doi.org/ 10.1021/ie970650k. [23] Cézac P, Serin J, Mercadier J, et al. Modeling solubility of solid sulphur in natural gas. Chem Eng J 2007;133(01):283–91. https://doi.org/10.1016/j.cej.2007.0 2. 014. [24] Jain LC, Fanelli AM. Recent advances in artificial neural networks. CRC; 2000. doi: 10. 9774/GLEAF. 978-1-909493-38-4_2. [25] Almasi AD, Wozniak S, Cristea V, et al. Review of advances in neural networks: Neural design technology stack. 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Fig. 15. Comparison of the predicted and experimental sulfur solubility by the model in this work and four widely used models.
5. Conclusions (1) A T-S fuzzy neural network (T-S FNN) model was established to predict sulfur solubility in sour gas. The prediction results demonstrated the accuracy and reliability of the T-S FNN model in the prediction of sulfur solubility in sour gas. (2) 222 data sets collected from open published literature were used to train and test the model, among them, 167 for training set and 55 for testing set. And with the method of gray correlation analysis, it proved that reservoir temperature, pressure and the mole fraction of H2S, CO2 and CH4 are the main influential factors on sulfur solubility in sour gas. (3) A series of evaluation parameters were considered to evaluate the accuracy and validity of the model, and a comparison between T-S FNN model and another four widely used models was conducted. The results indicated that the T-S FNN model has the closest agreement with experimental data with AARD of 5.35% and R2 of 0.998. (4) In addition, as a statistical tool, outlier diagnostics were also employed. The result confirmed the validity of the proposed T-S FNN model in statistics field and strong capability of regressing the intrinsic relationships between input parameters and the output result. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper Acknowledgments The authors would like to acknowledge the support provided by the National Natural Science Foundation of China (51804282 and 51304174), National Science and Technology Major Project (2017ZX05009-005), Fundamental Research Funds for the Central Universities, China (2652018209). References [1] Li Y, Li HT, Chen SN, et al. The second critical capillary number for chemical flooding in low permeability reservoirs: experimental and numerical investigation. Chem Eng Sci 2019;196:202–13. https://doi.org/10.1016/j.ces.2018.11.004. [2] Li Y, Li HT, Chen SN, et al. Investigation of the dynamic capillary pressure during displacement process in fractured tight rocks. AIChE J 2019. https://doi.org/10. 1002/aic.16783. In Press. [3] Su J, Zhang SC, Zhu GY, et al. Geological reserves of sulfur in China's sour gas fields and the strategy of sulfur markets. Petrol Explor Dev 2010;37(3):369–77. https:// doi.org/10.1016/S1876-3804(10)60039-0.
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decision-making problems. Comput Ind Eng 2007;55(01):80–93. https://doi.org/ 10.1016/j.cie. 2007.12.002. [39] Bian XQ, Han B, Du ZM, et al. Integrating support vector regression with genetic algorithm for CO2-oil minimum miscibility pressure (MMP) in pure and impure CO2. Fuel 2016;182:227–557. https://doi.org/10.1016/j.fuel.2016.05.124. [40] Eslamimanesh A, Gharagheizi F, Mohammadi AH, et al. Assessment test of sulfur content of gases. Fuel Process Technol 2013;110:133–40. 10.10 16/j.fuproc.2012.12.005.
076825. [36] Li PF, Ning YW, Jing JF. Research on the detection of fabric color difference based on T-S fuzzy neural network. Color Res Appl 2017;42(5):609–18. https://doi.org/ 10.1002/col.22113. [37] Qiao JF, Li W, Han HG. Soft computing of biochemical oxygen demand using an improved T-S fuzzy neural network. Chin J Chem Eng 2014;22(11):1254–9. https:// doi.org/10.1002/col.22113. [38] Kuo YT, Huang GW. The use of grey relational analysis in solving multiple attribute
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Full Length Article
Investigation on sulfur solubility in sour gas at elevated temperatures and pressures with an artificial neural network algorithm ⁎
Liang Fua, Jinghong Hua, , Yuan Zhanga, Qian Lib a b
Beijing Key Laboratory of Unconventional Natural Gas Geology Evaluation and Development Engineering, China University of Geosciences, Beijing 100083, China Research Institute of Petroleum Exploration and Development, PetroChina Southwest Oil & Gas Field Company, Chengdu 610041, China
A R T I C LE I N FO
A B S T R A C T
Keywords: Elemental sulfur solubility Sour gas reservoirs T-S fuzzy neural network Prediction model
Accurate prediction of the solubility of elemental sulfur in sour gas mixtures has been recognized as a key issue in the development of sour gas fields. Many experimental measurements and empirical models have shown the complicated relationships between sulfur solubility and sour gas properties. However, the accurate model that can be used to predict sulfur solubility in sour gas over a wide range of temperature and pressure is rare. Therefore, the objective of this work is to build an efficient model, namely T-S fuzzy neural network (T-S FNN), to investigate the sulfur solubility in sour gas. The model considers the reservoir pressure, temperature, the mole fraction of methane, hydrogen sulfide and carbon dioxide as input parameters and the sulfur solubility as target parameters. Subsequently, multiple experimental sulfur solubility data sets accessible to the literature are employed to train and test the model respectively. Finally, a series of studies are conducted to appraise the accuracy and generalization capability of the model. The result shows that the predicted solubility data had great agreement with experimental sulfur solubility with overall average absolute relative deviation of 5.35%, which proves the model in this work is feasible and effective. Additionally, it provides a new idea to the prediction of sulfur solubility in sour gas and helps operators to develop the sour gas reservoir better.
1. Introduction Fossil fuels play an important role in the world energy structure. Many problems should be solved during the development of oil and gas reservoirs [1,2]. Natural gas with high sulfur content is one of the significant clean energy resources with abundant geological reserves and is widely distributed all over the world [3]. As the reservoir pressure and temperature decrease elemental sulfur precipitates from sour gas in reservoir formation, tubing string and so on, consequently sulfur deposition reduces the permeability of the formation and blocks the production transportation equipment, which makes the inflow performances of gas wells decreased during the sour gas field development [4,5]. Therefore, accurate measurement of the sulfur solubility in sour gas is critical to hinder the sulfur deposition. Models for predicting sulfur solubility are generally classified into four categories: experimental measurement, empirical models, thermodynamic models and intelligent algorithm models. The sulfur solubility in sour gas can be accurately obtained by experiments, and many experimental tests have been conducted by scholars at home and abroad since 1960 [6–15], Kennedy and Wieland [6] firstly performed the experimental test in pure gas (CH4, CO2, H2S)
⁎
and three binary gas mixtures over a range of temperatures (339.15 ~ 394.15 K) and pressures (6.89 ~ 41.4 MPa). After that, Roof [7], Swift et al. [8], Brunner and Woll [9], Brunner et al. [10], Gu et al. [11], Sun and Chen [12], Hu et al. [13], Bian et al. [14], Zhang [15] respectively measured the sulfur solubility in sour gas at different temperature and pressure. A detailed summary of experimental tests is listed in Table 1. Although experimental measurement of sulfur solubility in sour gas is the most accurate, it is highly toxic, time-consuming and costly. Compared with experimental measurement, empirical models are easier to acquire, which can be obtained by correlating with experimental data. Chrastil [16] firstly proposed an association model for the correlation of solid solubility in supercritical fluid. Then Roberts [17] regressed the parameters of Chrastil model by means of two groups of experimental data obtained by Brunner and Woll [9] and Brunner et al. [10]. Though the Roberts model was commonly applied to predict the sulfur solubility in sour gas, such model tends to produce large errors due to the vagueness of application conditions. Similarly, Hu et al. [13] regressed the three coefficients (k, a, b) of Chrastil model for different sour gas at different pressure and temperature and functions well. Considering the influence of multiple factors such as the gas density and
Corresponding author. E-mail address: [email protected] (J. Hu).
https://doi.org/10.1016/j.fuel.2019.116541 Received 28 August 2019; Received in revised form 17 October 2019; Accepted 29 October 2019 0016-2361/ © 2019 Elsevier Ltd. All rights reserved.
Please cite this article as: Liang Fu, et al., Fuel, https://doi.org/10.1016/j.fuel.2019.116541
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Table 1 Summary of experiments in measurement of sulfur solubility in sour gas. List
Authors
Temperature (℃)
Pressure (MPa)
Gas Component
Refs.
1 2 3 4 5 6 7 8 9
Kennedy and Wieland (1960) Roof (1971) Swift et al.(1976) Brunner and Woll (1980) Brunner et al. (1988) Gu MX (1993) Sun CY (2003) Yang XF (2009) Bian XQ (2010)
100 ~ 160 125 ~ 212 93.3 ~ 121.1 43.3 ~ 110 121 ~ 204 90 ~ 110 30 ~ 90 100 63 ~ 123
10 ~ 60 6.7 ~ 155 7 ~ 41.4 7 ~ 31 34.5 ~ 138 10 ~ 50 20 ~ 45 10 ~ 36 10 ~ 55.2
Pure H2S, Pure H2S Pure H2S H2S, CH4, H2S, CH4, H2S, CH4, H2S, CH4, H2S, CH4, H2S, CH4,
[6] [7] [8] [9] [10] [11] [12] [13] [14]
CH4, CO2, N2 and different ratio of the four mixed gas
CO2 CO2 CO2 CO2 CO2 CO2
and N2 at different ratio of mixed gas and N2 at different ratio of mixed gas and N2 at different ratio of mixed gas and N2 at different ratio of mixed gas and N2 at different ratio of mixed gas
[17] and Guo-Wang equation [20]) and an intelligent algorithm model (Bian et al. model [14]) reported in previous literature were used to further verify the accuracy of the model. In summary, this work provides a better intelligent algorithm model for the prediction of sulfur solubility in sour gas.
temperature, Bian et al. [18,19] proposed new empirical models. Inspired by the Hu et al model, Guo and Wang [20] considered the constant coefficient k as a temperature function and introduced a new parameter M to lower the importance of sour gas density, hence the new model produces more accurate result than the Hu et al model and the Roberts model. In short, the empirical models mentioned above are piecewise formulas. It is widely acknowledged that empirical models are user-friendly, however, the majority of them can easily produce large errors compared with experimental data, because it is difficult for these models to reflect the intrinsic relationships between input parameters and sulfur solubility. Similar to the empirical model, thermodynamic model is also a kind of mathematical model, which requires not only complicated calculation but also the knowledge of critical parameters of the solute and solvent [21–23], as a result, reservoir engineers rarely adopt them for the engineering calculation of sulfur solubility in sour gas. The fourth category is intelligent algorithm model such as artificial neural network (ANN). In the last 40 years, ANN has been widely applied in engineering calculation such as chemical engineering, environmental engineering, electronic engineering and so on [24,25]. However, the ANN models that can be used for the investigation on sulfur solubility is insufficient, which has only experienced about 10 years of development. Mohammadi and Richon [26] firstly introduced the feed-forward neural network (FNN) to calculate sulfur solubility in gaseous solvents at elevated temperatures (316–433 K) pressure (60 MPa). However, the model can only estimate sulfur solubility in pure H2S. Then Mehrpooya et al. [27], ZareNezhad and Aminian [28] extended the FNN to estimate sulfur solubility in sour gas at different temperatures and pressures with acceptable results. Recently, Bian et al. [14] built an intelligent algorithm model namely grey wolf optimizer-based support vector machine (GWO-SVM) with 170 data sets, the prediction results proves that the model is reliable for the use of predicting sulfur solubility in sour gas. However the H2S content is preferably between 1 and 20 when using the model, since the model greatly acquire the relationship between sulfur solubility and input variables where the H2S is between 1 and 20 in the training sets. Intelligent algorithm models have strong nonlinear correlation capacity and relatively simple formulas hence can be widely used in prediction of sulfur solubility. Additionally, a comparison for the explanation of differences of these models mentioned above was listed in Table 2. In this work, a T-S fuzzy neural network (T-S FNN) was built to calculation of the sulfur solubility in sour gas. The calculation flow chart was shown in Fig. 1. Firstly, the methodology and detailed procedure were explained. The reservoir temperatures, pressures and the mole fractions of CH4, H2S and CO2 were considered as input parameters of the model with the method of gray correlation coefficient. Then 222 data sets accessible to the literature were employed to develop, validate and evaluate the new model. Afterwards, several evaluation parameters were adopted to evaluate the predicted results. The outlier diagnostics was employed by means of the leverage approach to detect the doubtful sulfur solubility data. Additionally, three widely used empirical correlations (Hu et al. equation [13], Roberts equation
2. Methodology 2.1. T-S fuzzy neural network Fuzzy neural network is the combination of fuzzy control theory and artificial neural network. It has the advantages of both neural network and fuzzy theory consequently can be used for learning, association, recognition and information processing [29,30]. The Takagi-Sugeno (TS) type fuzzy logic system is employed for the construction of the fuzzy neural network, shown in Fig. 2, whose structure consists of two parts: antecedent network and consequent network [29–31]. Many studies have been proved that T-S fuzzy neural network can be appeared as a universal approximate and can approximate the nonlinear systems with arbitrary precision [32–34]. Therefore, T-S fuzzy neural network can be employed for the prediction of sulfur solubility in sour gas. 2.1.1. The structures of T-S FNN There are 4 layers in the antecedent network, known as input layer, membership function layer, product inference rule layer and weighted average defuzzifier layer [30,31,35,36]. Each neuron in layer 1 represents input variable xi(i = 1,2,…,n), determined by the number of input variables n. Layer 2 is the membership function layer, in which each neuron stands for one membership function. The Gaussian function is applied as the membership function, and the number of neurons is calculated as follows.
μi j = exp { −(x i − cij )2 / bij2}, i = 1, 2, ...,n; j = 1, 2, ...,n
(1)
h=n×m
(2)
j
where μi is the jth membership function in the ith neuron, h is the number of neurons in layer 2, cij and bij are the center and width of the Gaussian function respectively. m is the fuzzy segmentation number of each input variable xi Layer 3 is the fuzzy rule layer. Each neuron is employed for the calculation of the firing strength by an algebraic operation.
wj = μ1j μ2j . ..μnj
(3)
where wj (j = 1,2,…,m) is the firing strength as well as the adaptability of each rule, m is the number of fuzzy rules. Layer 4 is the normalized calculation layer, which can be calculated as follows.
w¯j = wj ∑m wj, j = 1, 2, ...,m . j=1
(4)
where w¯ j (j = 1,2,…,m) is the normalized firing strength as well as the adaptability of each rule. 2
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Table 2 Comparison between empirical equations and intelligent algorithm models. Classification
Models
Differences
Empirical equations
Roberts model (1997)
A non-piecewise equation, the input variables are gas density and temperature, the model is acquired from only two groups of experimental data. A piecewise equation, model parameters vary with temperature, pressure and H2S content in sour gas. A piecewise equation, similar to Hu et al model, a new parameter M is introduced to lower the importance of sour gas density in the model. The input parameters are temperature and pressure, the model can only predict sulfur solubility in pure H2S gas. Based on Mohammadi et al model, gravity of sour gas and H2S content are added to the input variables of the model. Namely SVM-GWO model, the model considers 5 main factors as input variables and can be given priority where the H2S content is between 1 and 20.
Hu et al model (2014) Guo-Wang model (2016) Intelligent algorithm models
Mohammadi et al model (2008) Mehrpooya et al model (2010) Bian et al model (2018)
membership of the inputnode and the rule, yj is the jth consequent part of fuzzy rule, and pijis system parameters. Layer 3, known as output layer, function as the calculation of the output of the whole system, described as follows. m
y=
∑ w¯ j y j (6)
j=0
where y is the whole system output, w¯ j is the same as layer 4 in antecedent network, yj is equal to layer 2 in consequent network. 2.1.2. The learning algorithm of T-S FNN In order to ensure the T-S FNN model produce a more accurate result, one learning algorithm namely gradient descent is employed, which can adjust the center and width of membership function (cij, bij) in the antecedent network and the connecting coefficient (pij) in the consequent network in the learning process [30,31,35–37]. The detailed algorithm principle of T-S FNN was shown in Fig. 3. The cost function is specified as performance function, in fact, the goal of the learning process is to minimize the performance function, setted as follows.
Fig. 1. Flow chart for the calculation procedure in this work.
e=
1 d (y − y )2 2
(7) d
where e is the cost function, y is the expected output of the network, y is the actual output. Parameter pi j ,cij, bij, is updated using the gradient descent method, expressions are as follows.
Rj : IF x1 is
THEN y j =
...,xn is A nj, + ···+pnj x n
(8)
pi j (k + 1) = pi j (k ) − α
∂e ∂pi j
(9)
cij (k + 1) = cij (k ) − α
∂e ∂cij
(10)
bij (k + 1) = bij (k ) − α
∂e ∂bij
(11)
j
pi (k ), cij (k ), bij (k ) are the network parameters at the kth where iteration step, and their starting values are usually generated randomly; α , known as the learning rate, is a constant at 0–1.
There are 3 layers in the consequent network [30,31,35,36]. Layer 1 is the input layer who has the same function as the corresponding layer in antecedent network. Layer 2 is the fuzzy rule layer that is used to match the consequent part of fuzzy rule.
x2 is A2j, p0j + p1j x1
∂e ∂p0j
p0j (k ),
Fig. 2. Structure of T-S fuzzy neural network.
A1j,
p0j (k + 1) = p0j (k ) − α
3. Data analysis and model training 3.1. Experimental data 222 data sets of experimental sulfur solubility, listed in Table 3, were collected from previous literature [9,12–15] to develop and evaluate the model, among them, about 75 percent for training and the rest for evaluation. The collected data sets covers wide ranges of
(5)
Where Rjis the jth fuzzy ruleAij (i = 1,2,…,n; j = 1,2,…,m) is Gaussian 3
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Fig. 3. The algorithm principle chart of T-S FNN.
approximately positive linear correlation with sulfur solubility. Additionally, the content of CO2 should also be considered while calculating the sulfur solubility, although having smaller impact than that of H2S. To investigate the impact of the main factors on sulfur solubility in sour gas, the gray correlation coefficient is employed. The greater value of gray correlation coefficient of input variable, the greater effect on determining the value of output variable [14,38,39]. Through gray correlation analysis, the most sensitive factors on sulfur solubility, listed in Fig. 4, are H2S content, followed by CO2 content, reservoir pressure, temperature and CH4 content, While N2 content and C2H6 content show least impact on sulfur solubility consequently will not be considered as input parameters. Therefore, it is clear that the new T-S FNN model was designed to acquire the most desirable regression between sulfur solubility and the H2S mole fraction, CO2 mole fraction, pressure, temperature and CH4 mole fraction.
temperatures (30–160 °C), pressures (10–66.52 MPa), H2S content (1–26.62%) and sulfur content (0.012–6.413 g/m3). The collected data sets were split to training sets with 167 data points, testing sets with 51 data points and checking sets with 4 data points. The training sets were used for updating the parameters ( pi j ,cij, bij,) with a goal to minimize the cost function. The testing sets, not appeared in training sets, were used for evaluating the generalization capability of the trained model. And the checking sets, brand-new sets for the model, were applied to appraise the applicable capability of the model in gas field. And in order to get more accurate results, all of the data sets were normalized to between −1 to 1, the expression is as follows.
yi =
2(xi − xmin) −1 (xmax − xmin)
(12)
where yi is normalized data, xi is the collected experimental data, xmax and xmin is the maximum and minimum value of the data sets respectively.
3.3. Simulation on sulfur solubility prediction During the training process, the structure and parameters of T-S FNN model is determined by trial-and-error method with a goal to minimize the cost function. Finally, the structure and parameters, listed in Table 4, were set as follows: the number of membership function is 2 for each input variable, the number of fuzzy rules is 32, membership type is Trim type function, max iterations is 50, and other parameters remain the defaults.
3.2. Influential factors of sulfur solubility in sour gas According to previous researches [9,10,13,14], the chief factors affecting sulfur solubility in sour gas are temperature, pressure and gas composition including H2S, CO2, etc. As the reservoir temperature and pressure descend, the sulfur solubility in sour gases decreases. And the H2S content which has the greatest influence on sulfur solubility has an Table 3 Summary of the database of sulfur solubility used in this work. Author
Temperature (°C)
Pressure (MPa)
Element sulfur solubility (g/m3)
Training data sets
Testing data sets
Brunner and Woll (1980) Sun CY (2003) Yang XF (2009) Bian XQ (2010) Zhang GD (2014)
100–160 30–90 100 63–123 100–152.5
10–60 20–45 24–36 10–55.2 20–66.52
0.043–4.29 0.012–1.455 0.042–0.2682 0.0083–2.067 0.034–6.413
59 56 0 34 18
11 0 4 16 24
4
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Fig. 5. The relationship between sulfur solubility and temperature and pressure.
Fig. 4. Gray correlation coefficients for sulfur solubility in supercritical sour gas. Table 4 Parameters of the trained T-S FNN model. Parameters
Sour gas mixture
Input variables Input data form Max iterations Membership type Fuzzy rules Cost function: e
5 [-1, +1] 50 Trimf 32 0.0069
4. Results and discussion 4.1. Outputs analysis Fig. 6. The relationship between sulfur solubility and H2S mole fraction.
In order to evaluate the model in this work quantitatively, four evaluation parameters, which appears as an effective mathematical tool when comprehensively evaluating the accuracy of a model, were employed as a comparison of experimental and predicted data, including the average absolute relative deviation (AARD), the root mean squared error (RMSE), the standard deviation (SD), and determination coefficient (R2) [14]. The formulas of these evaluation parameters are shown as follows.
AARD =
RSME =
100 N
1 N
N
∑
Data sets
AARD(%)
SD
RSME
R2
Training sets Testing sets Checking sets
3.68 5.35 7.04
0.06 0.08 0.09
0.02 0.06 0.01
0.9996 0.9984 0.8855
yiexp − yical yiexp
i=1
(13)
N
∑ (yiexp − yical )2 i=1 exp
SD =
Table 5 Comparison of data sets calculated by the model in this paper.
(14)
cal 2
1 ⎛ yi − yi ⎞ ⎟ N − 1 ⎜⎝ yiexp ⎠
(15)
N
R2 = 1 −
∑i = 1 (yiexp − yical )2 N
exp − yical )2 ∑i = 1 (yave
(16)
where N is the number of all experimental sulfur solubility data points; exp yiexp , yical and yave are experimental and experimental sulfur solubility and average value of all experimental sulfur solubility respectively. The relationships, obtained by the proposed T-S FNN model, between sulfur solubility and temperature, pressure, H2S content were shown in Figs. 5 and 6, respectively. As can be seen, the input parameters have approximately positive correlations with the output sulfur solubility, which is similar to the previous conclusion: (reached by Brunner et al. [9,10], Hu et al. [13] and Bian et al. [14]) sulfur solubility in sour gas increases, as the system temperature, pressure or H2S mole fraction increase.
Fig. 7. Comparison of the calculated and experimental sulfur solubility.
For the training data sets, the comparison between predicted results and experimental data was shown in Table 5, Figs. 7 and 8. As can be seen, the calculated data points have an excellent agreement with experimental data points with AARD = 3.7% and R2 = 0.999, which reveals the model has strongly high fitting capability. 5
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H = X (X t X )−1X t
(17)
where X is a two-dimensional matrix consisting of n data (rows) and k input variables (columns) and t stands for the transpose matrix. The Williams plot (plot of standardized versus Hat indices), who denotes the correlation between the standardized cross-validated residuals and Hat indices, was depicted, through which the outlier data can be identified. In the Williams plot there is a squared area within ±3 standard residuals and a leverage threshold H* (H* generally equals to 3n/(k + 1)), inside which distribute the valid data. Generally, the majority of data points distribute in the squared area (0⩽H⩽H* and −3⩽SR⩽3) which not only indicates the model has wide applicability domain, but also proves the validity of the model in statistics field. The Williams plot for the outputs of the model in this paper was presented in Fig. 14, from which it can be seen that the great majority of sulfur solubility utilized in this paper locate in the valid domain of [−3,3] and [0,H*], except for 5 points fell outside of the valid domain, considering as outliers, and the Williams plot also demonstrates the validity and reliability of the model in statistics field.
Fig. 8. Absolute relative deviation of the calculated sulfur solubility in training data sets.
4.3. Comparison of estimation accuracy
For the testing data sets, the comparison between predicted results and experimental data was shown in Table 6–8, Figs. 7 and 9, as revealed, the predicted data points show a great agreement with experiment data points with AARD = 5.35% and R2 = 0.998. And another comparison between the model predicted results and the experimental data of Brunner [9], Bian et al [14] and Zhang [15] at isothermal conditions was illustrated in Figs. 10–12, confirming the model in this work enjoys great prediction accuracy and generalization capability. Additionally, in order to appraise the applicable capability of the model in gas field, a group of data sets of Yang [13], not appeared in the training data sets, was employed. The prediction results were shown in Table 9 and Fig. 13, revealing the prediction results have a satisfactory agreement with experimental data points with AARD = 7.04% and R2 = 0.8855. Therefore it proves that the proposed T-S FNN model can function well in the prediction of sulfur solubility in sour gas in gas fields.
In order to further verify the accuracy and reliability of the model in this work, a comparison between the T-S FNN model and three widely used empirical equations (Hu et al equation [13], Roberts equation [17] and Guo-Wang equation [20]) and an intelligent algorithm model (Bian et al. model [14]) was conducted with the testing data sets in this paper. The comparison results were shown in Table 10 and Fig. 15, as can be seen, the data points predicted by T-S FNN model have the best agreement with experimental data points than that of another four models. Both Hu et al model and Guo-Wang model provide closer match to the experimental data compared to Roberts model. While Bian et al model is much better than above three empirical equations, indicating GWO-SVM model is reliable and applicable. The proposed T-S FNN model has advantages over the other four models. It can accurately predict the sulfur solubility in sour gas without adjusting the parameters of the model, while the three empirical equations cannot achieve the same goal, since they are all piecewise formulas. When using the empirical models, the parameters of the equations require to be adjusted as the values of input variables change, which increases the risk of errors. Both the T-S FNN model and GWO-SVM model are intelligent algorithm models based on the theory of statistics, and both can automatically acquire the desirable regression between input variables and output variables. Once fully trained, both models can enjoy high accuracy, which is evidenced in this paper. However, the T-S FNN model
4.2. Outlier diagnosis Outlier diagnosis [14,40] is a significant statistical approach that can diagnose individual outlier datum from the sample set, for which the leverage statistics is an accurate and efficient method. The standardized residual (SR), the Hat indices (H) and the critical Leverage limit (H*) are used in this method. The Hat indices can be defined as follows.
Table 6 Comparison between predicted results and experimental sulfur solubility: Bian et al. [14]. Component
Temperature (K)
Pressure (MPa)
Sulfur solubility (in this work) g/m3
Sulfur solubility (experiment) g/m3
Absolute relative deviation (%)
13.79% H2S 9.01% CO2 0.52% N2 76.64% CH4 0.04% C2H6
351.15
25
0.104
0.113
8.11
35 45 55.5 25 35 45 55.2 25 35 45 55.2 25 35 45 55.2
0.225 0.393 0.610 0.149 0.333 0.600 0.955 0.225 0.501 0.900 1.432 0.339 0.741 1.314 2.070
0.2419 0.389 0.5965 0.1589 0.3359 0.5887 0.9246 0.2216 0.4856 0.899 1.4044 0.3357 0.748 1.3196 2.0672
6.84 0.97 2.18 6.38 0.88 1.88 3.27 1.51 3.12 0.16 2.00 0.96 0.88 0.39 0.14
366.15
381.15
396.15
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Table 7 Comparison between predicted results and experimental sulfur solubility: Brunner [9]. Component
Temperature (K)
Pressure (MPa)
Sulfur solubility (in this work) g/ m3
Sulfur solubility (experiment) g/ m3
Absolute relative deviation (%)
7% H2S 20% CO2 8% N2 65% CH4
373.15
6% H2S 9% CO2 4% N2 81% CH4
393.15 413.15 433.15 373.15
30 48 45 45 30 30 50 40 25 45 25
0.235 0.532 0.925 1.394 1.133 0.115 0.468 0.701 0.500 1.351 0.943
0.202 0.621 1.006 1.43 1.04 0.136 0.476 0.635 0.562 1.42 0.752
16.45 14.26 8.04 2.53 8.98 15.37 1.62 10.42 11.04 4.86 25.41
393.15 413.15 433.15
Table 8 Comparison between predicted results and experimental sulfur solubility: Zhang [15]. Component
Temperature (K)
Pressure (MPa)
Sulfur solubility (in this work) g/m3
Sulfur solubility (experiment) g/m3
Absolute relative deviation (%)
2.93% H2S 4.37% CO2 0.34% N2 92.32% CH4
373.15
30
0.109
0.095
15.09
50 64.88 66.52 20 40 60 66.52 30 50 64.88 66.52 20 40 60 66.52 30 50 64.88 66.52 20 40 60 66.52
0.342 0.531 0.490 0.106 0.510 1.274 1.482 0.489 1.409 2.379 2.411 0.412 1.441 3.162 3.687 1.388 3.510 5.519 5.600 0.676 2.664 5.444 6.231
0.338 0.578 0.615 0.112 0.549 1.278 1.545 0.472 1.436 2.326 2.444 0.374 1.453 3.167 3.778 1.359 3.543 5.558 5.767 0.779 2.655 5.428 6.413
1.13 8.09 20.27 5.13 7.12 0.33 4.08 3.67 1.87 2.30 1.36 10.18 0.80 0.17 2.40 2.14 0.92 0.71 2.89 13.28 0.34 0.29 2.84
393.15
403.15
413.15
423.15
425.65
Fig. 10. Predicted sulfur solubility and experimental data in sour gas: Brunner [9].
Fig. 9. Absolute relative deviation of the predicted sulfur solubility in testing data sets.
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Fig. 13. Predicted sulfur solubility and experimental data in sour gas: Yang et al [13].
Fig. 11. Predicted sulfur solubility and experimental data in sour gas: Bian et al [14].
Fig. 14. Detection of the doubtful calculated sulfur solubility in sour gas. Fig. 12. Predicted sulfur solubility and experimental data in sour gas: Zhang [15].
Table 10 Comparison of T-S FNN model with four widely used models considered in this work.
has some advantages that GWO-SVM does not have. As mentioned above, T-S FNN has the advantages of both neural network and fuzzy theory, and it can use fuzzy reasoning to take advantage of existing information for machine learning, which has positive effect on the improvement of prediction accuracy. Additionally, the H2S content in the training sets in this work is between 1 and 26.62% which is wider than that of Bian et al model, suggesting the model has a lager scope of application. In summary, two different kind of intelligent algorithm models (T-S FNN model and GWO-SVM model) obtain the similar results in predicting sulfur solubility in sour gas, therefore they can refer to each other.
Models
AARD(%)
SD
RSME
R2
Roberts model Hu et al model Guo-Wang model Bian et al model This work
65.36 17.3 12.84 6.68 5.35
0.86 0.22 0.15 0.08 0.08
0.67 0.21 0.17 0.09 0.06
0.6792 0.9731 0.9833 0.9961 0.9983
gas in gas field usually varies from 0 to 100%, whereas the H2S content in the data sets used in this work is only between 1 and 26.62%, thus 167 data sets are still insufficient to train the proposed T-S FNN model for the accurate prediction of sulfur solubility in sour gas regardless of the content of H2S. When the H2S content is greater than 26.62%, the prediction accuracy of the proposed model will deteriorate, in other words, it is the most desirable occasion for the model to predict sulfur solubility, where the H2S content is between 1 and 26.62%.
4.4. Model limitation The T-S FNN model enjoys high accuracy and reliability with the data sets shown in this work hence has advantages over other models in predicting sulfur solubility in sour gas. However, H2S content in sour
Table 9 Comparison between predicted results and experimental sulfur solubility: Yang et al. [13]. Component
Temperature (K)
Pressure (MPa)
Sulfur solubility (in this work) g/m3
Sulfur solubility (experiment) g/m3
Absolute relative deviation (%)
6.86% H2S 2.76% CO2 0.5% N2 89.63% CH4 0.21% C2H6
373.15
24 28 32 36
0.150 0.180 0.218 0.264
0.169 0.194 0.201 0.268
10.95 7.01 8.49 1.68
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Fig. 15. Comparison of the predicted and experimental sulfur solubility by the model in this work and four widely used models.
5. Conclusions (1) A T-S fuzzy neural network (T-S FNN) model was established to predict sulfur solubility in sour gas. The prediction results demonstrated the accuracy and reliability of the T-S FNN model in the prediction of sulfur solubility in sour gas. (2) 222 data sets collected from open published literature were used to train and test the model, among them, 167 for training set and 55 for testing set. And with the method of gray correlation analysis, it proved that reservoir temperature, pressure and the mole fraction of H2S, CO2 and CH4 are the main influential factors on sulfur solubility in sour gas. (3) A series of evaluation parameters were considered to evaluate the accuracy and validity of the model, and a comparison between T-S FNN model and another four widely used models was conducted. The results indicated that the T-S FNN model has the closest agreement with experimental data with AARD of 5.35% and R2 of 0.998. (4) In addition, as a statistical tool, outlier diagnostics were also employed. The result confirmed the validity of the proposed T-S FNN model in statistics field and strong capability of regressing the intrinsic relationships between input parameters and the output result. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper Acknowledgments The authors would like to acknowledge the support provided by the National Natural Science Foundation of China (51804282 and 51304174), National Science and Technology Major Project (2017ZX05009-005), Fundamental Research Funds for the Central Universities, China (2652018209). References [1] Li Y, Li HT, Chen SN, et al. The second critical capillary number for chemical flooding in low permeability reservoirs: experimental and numerical investigation. Chem Eng Sci 2019;196:202–13. https://doi.org/10.1016/j.ces.2018.11.004. [2] Li Y, Li HT, Chen SN, et al. Investigation of the dynamic capillary pressure during displacement process in fractured tight rocks. AIChE J 2019. https://doi.org/10. 1002/aic.16783. In Press. [3] Su J, Zhang SC, Zhu GY, et al. Geological reserves of sulfur in China's sour gas fields and the strategy of sulfur markets. Petrol Explor Dev 2010;37(3):369–77. https:// doi.org/10.1016/S1876-3804(10)60039-0.
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