Prediction of the sulfur solubility in pure H2S and sour gas by intelligent models

Journal Pre-proof Prediction of the sulfur solubility in pure H2S and sour gas by intelligent models

Xiao-Qiang Bian, Yi-Lun Song, Martin Kelvin Mwamukonda, Yu Fu PII:

S0167-7322(19)34293-X

DOI:

https://doi.org/10.1016/j.molliq.2019.112242

Reference:

MOLLIQ 112242

To appear in:

Journal of Molecular Liquids

Received date:

31 July 2019

Revised date:

15 October 2019

Accepted date:

27 November 2019

Please cite this article as: X.-Q. Bian, Y.-L. Song, M.K. Mwamukonda, et al., Prediction of the sulfur solubility in pure H2S and sour gas by intelligent models, Journal of Molecular Liquids(2018), https://doi.org/10.1016/j.molliq.2019.112242

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© 2018 Published by Elsevier.

Journal Pre-proof

Prediction of the sulfur solubility in pure H2S and sour gas by intelligent models Xiao-Qiang Bian1, Yi-Lun Song, Martin Kelvin Mwamukonda, Yu Fu (Petroleum Engineering School, Southwest Petroleum University, Chengdu 610500, China)

Highlights

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Four meta-heuristic models were proposed for sulfur solubility in H 2S and sour gas.

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Pearson correlation analyzed the influential factors of sulfur solubility.

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The Williams’ plot was used to search the outlier data for reliability analysis. The GWO-LSSVM gives the best results among all models considered in this

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Abstract

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work.

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In order to reduce time and enhance accuracy, four intelligent models named

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Grey Wolf Optimizer based Least Squares Support Vector Machine (GWO-LSSVM), Grey Wolf Optimizer based Radial Basis Function (GWO-RBF), Genetic Algorithm based Adaptive Network Fuzzy Inference System (GA-ANFIS), and Particle Swarm Optimization based Adaptive Network Fuzzy Inference System (PSO-ANFIS) were applied to predict sulfur solubility in pure H2S and sour gas. According to Pearson correlation analysis, the content of H2S, critical temperature, temperature, gas density, and pressure were selected as input variables and sulfur solubility was selected as an output variable in sour gas. The contradistinction among the four models reveals

1

Corresponding author at Petroleum Engineering School, Southwest Petroleum University, Chengdu 610500,

China (X.-Q. Bian). E-mail addresses: [email protected] (X.-Q. Bian)

Journal Pre-proof GWO-LSSVM behaves the best performance with the minimum average absolute relative deviation (AARD=3.5029%), and the maximum determination coefficient (R2=0.9976) in all 239 data. But according to the minimum root mean squared error (RMSE), PSO-ANFIS performs best in pure H2S and sour gas among the four models. The leverage method was used to search outlier data for sulfur solubility, indicating that there are only 5 anomalous data points of all 239 data for the best GWO-LSSVM model. Keywords: Sulfur solubility; Intelligent algorithm; GWO-LSSVM; GWO-RBF;

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GA-ANFIS; PSO-ANFIS

eTraining set

300

Testing set

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250

45° Line

200 150 100

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Estimated sulfur solubility /g·m-3

350

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Graphical abstract

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50

GWO-LSSVM

0

0

50 100 150 200 250 300 350

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Experimental sulfur solubility /g·m-3

1. Introduction Gas reservoirs with high sulfur content are one of the major resources throughout the world[1]. With the increasing overall demand for energy, it is imperative to explore the new natural gas fields with high sour gas contents [2]. During the development of high sulfur gas reservoirs, sulfur deposition will occur with the decrease of temperature and pressure. The sulfur deposition will cause porosity and permeability of the formation to decrease, and affect the productivity of gas wells[3]. The sulfur solubility in high sulfur gas is the precondition and basis for the study of sulfur deposition prediction and treatment technique[4,5].

Journal Pre-proof Therefore, a good knowledgement of sulfur solubility in pure hydrogen sulfide (H2S) and sour gas is significant to resist sulfur deposition. The main methods used to determine the sulfur solubility in pure H2S and sour gas were divided into four aspects: experimental methods, thermodynamic methods, empirical methods, and intelligent arithmetic. The experimental method is especially trustworthy one among four methods. In 1960, Kennedy and Wieland[6] gauged sulfur solubility in pure CO2, CH4, H2S gas, and covered wide ranges of pressure from 7 to 41 MPa, temperature from 339 to

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394K. The results indicated that the sulfur content would become higher along with the addition of pressure and temperature. In 1971, Roof[7] gauged the sulfur solubility

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in H2S indicated that along an isotherm the sulfur solubility to increase with pressure. In 1976, Swift[8] gauged sulfur solubility in pure H2S gas with wide ranges of pressure

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from 35 to 140 MPa, temperature from 390 to 450K. In 1980, Brunner and Woll[9]

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gauged sulfur solubility in supercritical CH4-H2S-CO2-N2 gas mixtures within wide ranges of pressure from 20 to 60 MPa, temperature from 373 to 433K. In addition,

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Roof pointed out that the experimental approach used by Kennedy and Wieland is

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questionable. Until now, many scholars performed experiments to test the sulfur solubility in pure H2S[10,11] and sour gases

[12-16]

. The literature[17,18] described an

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exhaustive process to estimate the sulfur solubility. Although the experimental test of sulfur solubility is usually accurate in pure H2S and sour gas, it is expensive, cumbersome and time-consuming[19]. Thermodynamic models not only demand a good knowledge of mathematical principles but also contain complicated calculations. Among the thermodynamic models, cubic equations of state[20-24] were usually applied to estimate the sulfur solubility. In 1993, Gu et al.[25] proposed a novel two-constant equation of state to estimate sulfur solubility in sour gas, which displayed superior to the SRK equation in many cases. In 1998, Karan et al.[26] exploited the PR equation to depict the sulfur phase performance and the sulfur solubility in gas mixtures, but they used S8 to replace the sulfur in all phases. In 2001, Heidemann et al. [27] regarded the sulfur as an admixture of eight components and premeditated the reaction effects among H 2S and

Journal Pre-proof eight components by making use of the PR equation of state. In 2003, Sun and Chen[14] proposed the PR EOS to estimate sulfur solubility in gas mixtures. The calculated results were in corking conformity with experimental data and the average absolute deviation is only 6.5%. In 2007, Cézac et al.[28] connected a flash reaction mold with the PR equation of state to depict the phase performance, which was appropriate to estimate the solubility of sulfur in gas mixtures. Empirical models, without having a good knowledge of sulfur properties, were proverbially applied to sulfur solubility precipitation. In 1982, Chrastil[29] developed a

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connection mold to match the solid sulfur solubility in sour gas. In 1997, Roberts [30,31] regressed the critical reservoir parameters that affect the degree of damage by sulfur

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deposition were determined by the derivation of an analytical expression for the rate of sulfur set up presuming ideal flow status. In 2011, Eslamimanesh et al., [32] contrast

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the results estimated by frequently applied correlations (Adachi and Lu [33], Valle and

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Aguilera[34], Mѐndez-Santiago and Teja equations[35]) with advanced performance. In 2014, Hu et al.,[30] applied a section mathematical means to optimize the mentioned

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models’ parameters in sour gas at different temperatures and pressure and introduced

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the corking performance. In 2016, Guo and Wang[36], thinking over the constant modulus based on the Chrastil model as a temperature function and leading into a

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novel coefficient to lighten the significance of gas density, developed a novel mold to measure sulfur solubility in sour gas, the total average relative error of the entire data is 5.63%.

Intelligent algorithm, especially artificial neural networks (ANNs) [37] has been developed dramatically and executed to connect the sulfur solubility with other parameters. In the temperature range of 303.2 K to 433.15 K, pressure range of 10 to 60 MPa, the R2 value of 0.99791 indicates a superior performance between the experimental sulfur solubility and predicted values. Recently, The feedforward neural network（FNN）was also applied to estimate sulfur solubility in sour gas[38] with excellent performance. GWO as one of superior Intelligent arithmetics is becoming especially appealing for scholars and has gradually applied to settle sundry optimization problems, such as non-convex economic load dispatch[39]. Nevertheless,

Journal Pre-proof it’s not very ordinary that scholars take advantage of an intelligent algorithm to work out the sulfur solubility in articles. Besides the application of PSO to regress the modulus of ANFIS, in the article, a strange framework is proposed to regress coefficients by making use of a novel intelligent algorithm (GWO) that was proved to be more effective compared with the previous intelligent algorithms. The goal in the article is to develop and check the reliability of four meta-heuristic models utilizing experimental sulfur solubility 239 data from published literature. In this work, four models were developed to predict the sulfur solubility.

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Pressure and temperature as an input variable and the sulfur solubility as an output variable in pure H2S. According to Pearson correlation analysis, the content of H2S,

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critical temperature, temperature, gas density, and pressure as an input variable and

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sulfur solubility as an output variable in sour gas. This article was arranged as follows:

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Part 2, the concise characterization of GWO-LSSVM, GWO-RBF, GA-ANFIS, PSO-ANFIS models.

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Part 3, the data source and the influential factors of sulfur solubility in pure H 2S

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and sour gas were described.

Part 4, Compasting the consequence of developed models to experimental dates,

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the capacity of proposed meta-heuristic models was evaluated. The outlier diagnosis was applied to identify the dubious sulfur solubility data. Part 5, The corresponding conclusions were given.

2 Methodology 2.1 GWO 2.1.1 Grey Wolf Optimizer Mirijalili[40] proposed an original GWO majorization model, which employed novel intelligent arithmetic stimulated by gray wolves in the light of imitating the

Journal Pre-proof hierarchic connection and chasing performance of gray wolves spontaneously. With the profit of other Intelligent means, for example, the Whale Optimization Algorithm(WOA) and Invasive Weed Optimization(IWO), the evolution procedure of those arithmetic populations relates stochastic conditions, while GWO stands for superior performance compared with those classic means. GWO arithmetic mostly contains four aspects: community stratum, encircling, tracing and assaulting the pert procedure. In community stratum, GWO imitates the wolf levels involving four kinds

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of gray wolves: α、β、δ and ω, as described in Figure 1.

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Fig. 1. Grey wolf hierarchy (dominance decreases from the top down)

ω.

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The most appropriate results are considered as, in descending order, α, β, δ, and With the development of arithmetic, α, β, and δ wolves guide the attacking, and

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the ω wolves come behind.

On the moment of hunting quarry, the performance can be described as follows:     D  C  X p (t )  X (t )

    X (t  1)  X p (t ) - A  D

(1) (2)

X stand for the current location vector of gray wolves, Xp is the current quarry location, and C and A are the modulus vectors and modeled as follows:     A  2a  r1 - a   C  2  r2

(3) (4)

ascents from 0 to 2 linearly with the subjoin of r1 and r2. r1 and r2 are the stochastic vector from 1 to 2. wherefore A is a stochastic result from -0.9a to 1.8a. To

Journal Pre-proof simulate the attacking quarry, A ranged from -1 to 1. In the case of A<1, the wolves will assault the quarry and replace the current location of the quarry. α wolves guide gray wolves to find and assault quarry, we can get the α, β, and δ wolves in the all calculation process and execute other seek mediums to renew their locations based on present first-three hierarchy locations. the procedure is described as follows： (5)

    X j  X i  Aj  Di (i   ,  ,  ; j  1, 2, 3)

(6)

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    Di  C j  X j  X (i   ,  ,  ; j  1, 2, 3)

(7)

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    X1  X 2  X 3 X (t  1)  3

To sum up, the GWO arithmetic arbitrarily produces a group of gray wolves.

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Based on the consideration of relevant adaptation value and the probable location of

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the quarry with the superior schemes, the δ, β, and α wolves could be acquired. Every alternative scheme renews its location with eq 7. While attacking a quarry, the hunting

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performance, and underlined search are influenced by vector A and vector C. At last,

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the GWO arithmetic can be accomplished when the final norm is pleased.

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2.1.2 Least Square Support Vector Machine The Least Square Support Vector Machine (LSSVM) classifier is applied for experimental researches in this study. LSSVM classifiers are one special specimen of Support Vector Machine (SVM)[41]. Table 1 Parameters of the GWO-LSSVM model to predict the solubility of sulfur in pure H2S and sour gas

Parameter

In pure H2S

In sour gas

Input data form

[-1,1]

[-1,1]

Max iteration

200

200

Range of ε, γ

[2-10, 210]

[2-10, 210]

No. of search Agents

30

30

Best ζ2

7.9190

7.8687

Journal Pre-proof Best γ

0.0907

0.7815

The LSSVM classifier is employed to seek out a hyperplane, which separates numerous classes. The LSSVM classifier acquires this superior hyperplane by applying maximum Euclidean removing to the nearest dot. The LSSVM classifier draws the input vectors into a high dimensional characteristic vacuum for nonseparable data. Soon after, the LSSVM classifier looks for a superior separating hyperplane in this higher spatial vacuum. xi is the Ith input variable and yi∈{1, 1} is the Ith output variable. An SVM

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classifier is shown as follows: L

y ( x)  sign[  i yi ( x, xi )  b]

(8)

pr

i 1

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Where y(x) is an output vector. xi are support vectors, which belongs to the training series. The training series is applied to train the classifier. αi are LaGrange

Pr

multipliers, and b is a real constant. A series of linear equations are applied for the LSSVM classifier. Where ei are shown as follows: L

(9)

i 1

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ei  [ yi  (  i yi ( x, xi )  b)]

The LSSVM classifiers consist of two parameters, which can be optimized.

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These parameters are the breadth of the Gaussian kernels (ζ) and the regularization parameter C. In these experimental researches, the results of the parameter (ζ) is ranged between 0.2 and 24 and the result of C parameter is ranged between 1 and 10000 for majorization of these parameters.

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Journal Pre-proof

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Fig. 2. Flowchart of the newly proposed GWO-LSSVM model

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2.1.3 Radial Basis Function net

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RBF net is a type of forward feedback NNs. On account of the uncomplicated topologic construction and the capability to show how to learn profits in a definite mode, the RBF net has been diffusely applied as the general performance approach to work out non-linear questions. Also, because of its virtues over accustomed multilayer perceptrons, it has been a fast expand on the amount of adoption of RBF nets to prediction. The knowledge scheme of RBF nets is to determine the structure and to seek out the mold coefficients. Determining the structure is to identify the variables of closet nerve cells. Seeking out the mold variables proposes to look for the connection scales among the closet and output levels [42].

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Journal Pre-proof

Fig. 3. Graphic illustration of the proposed RBF-based model.

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The structure of an RBF net contains three levels. The closet level changes data from the input vacuum into the closet vacuum using a non-linear function. The

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non-linear function of closet nerve cells is balanced in the input area and the output of

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every closet nerve cell relies on the radial range among the core of the closet nerve cell and the input vector. The output of the jth closet nerve cells are shown as follows: (10)

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h j ( x)   ( x  c j )

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where ||.|| represents the Euclidean area, x represents the p-spatial input vector, cj represents the core of the jth closet nerve cell, andΦ(.) represents the energizing

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performance. The energizing function is a non-linear performance that is symmetrical in the input area. The Gaussian function, which is a widely applied energizing function, is adapted hereof. The formula is shown as follows:

 ( x  c j )  exp( 

1 2

2

2

x  cj )

(11)

ζ is the breadth of the closet nerve cell that indicates the breadth of the acceptable area of the Gaussian energizing performance. A core and a breadth are related to every closet nerve cell in the RBF net. Breadth ζ is shown as follows:

d 2 max  2M

(12)

where dmax is the maximum area among two cores of closet nerve cells of RBF net. M is the amount of closet nerve cells. The answer to every closet nerve cell is

Journal Pre-proof measured by connection scale and is considered to cause the whole net output. The kth output of net is shown as follows: M

yˆ k  wo   w jk h j ( x)

(13)

j 1

where wjk is the connection scale among the jth closet unit and the kth output neuron. w0 is the deviation. The w0 wjk and wjk are predicted using the least square error (LSE) means. Table 2 Parameters of the GWO-RBF model to predict the solubility of sulfur in pure H2S and sour gas

In pure H2S

Input data form

[-1,1]

Max iteration

200

Range of goal, spread, MN

[2-10, 210]

goal

0.0018

spread

0.4354

0.6126

MN

178

121

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Pr

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pr

[-1,1]

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In sour gas

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Parameter

200

[2-10, 210] 0.0163

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Pr

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pr

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Journal Pre-proof

Fig. 4. Flowchart of the newly proposed GWO-RBF model

2.2 ANFIS 2.2.1 Adaptive Network-based Fuzzy Inference System According to uniting the Fuzzy Inference System (FIS) and Artificial Neural Network (ANN), Jang[43] modeled ANFIS to handle with the disadvantages of FIS and ANN in 1993. ANFIS is a logic vague mold operating under specific means and

Journal Pre-proof which was proposed by certain regulations while developing the frame. ANFIS consists of a five-level economy that is interconnected. Under these circumstances, level 0 represents the input variable and level 5 implies the output variable. Among the output and input levels, the covert levels consist of multiple flexible and stable contacts. These contacts function with the regulations. ANFIS regards x1 and x2 as two input variables and f is regarded as the output variable. The connection of output and input contacts is determined using “if and then” vague regulations. levels are ascertained as follows:

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Level 1: Fuzzy Level Q1,i   Ai ( x)

(14)

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Where Q1, i is the output of each contact;μAi is a membership function and

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ascertained as follows:

 A ( x)  exp[((

Pr

i

x  ci 2 bi ) ) ] ai

is

(15)

membership function.

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Level 2: Rule Level

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a, b and c are hypothetical coefficients, which determine the form of the

Every contact in rule level is a changeless contact, moreover, the output contact

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is the result of the reaching signals in level 2 : Q2,i  Wi   Ai ( x). Bi ( x)

(16)

Where Wi, and Q2, i are the output variables of rule level 2. Level 3: Normalization Level Wi is renormalized by the ith contact in level 3. Moreover, the rate of shooting intensity of ith regulation to the total of shooting intensity of entire regulations should be decided with E.g. 17:

Q3,i  W1 

W1 W2

(17)

Q3, i and Wi are the output variables if level 3 and renormalized shooting intensity separately.

Journal Pre-proof Level 4: De-fuzzification level The node in level 4 is an adaptive contact linked with a contact performance, which stands for the affection of ith regulation with the ordinary output. The output of level 4 should be ascertained as follows:

Q4,i  W1 fi  W1 ( pi .x  qi . y  r1 )

(18)

Level 5: Output level In general, for counting the output, we should apply the E.g. 19. in level 5:

Wf W i

i i

(19)

i

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i

f

Q5,i   i W1 f i 

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al

Pr

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applied with blended studying arithmetic.

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To restrict the membership function coefficients, ANFIS arithmetic should be

Fig. 5. An ANFIS architecture of two inputs, four rules, and first-order Sugeno model

2.2.2 Particle Swarm Optimization Particle Swarm Optimization[44-46] is capable of optimizing an impartial performance employing the undertaking an investigating on the base of the population. The people consist of the hidden scheme, which is an analogy of birds in groups. These corpuscles are arbitrarily initialized and pass through the multi-dimension

Journal Pre-proof investigation area with liberty. When flying, every corpuscle renews their personal speed and location according to its superior performance and the entire people. The multiple procedures by PSO arithmetic are described as follows: Procedure 1: The speed and location of whole corpuscles are arbitrarily setting with pre-defined scopes. Procedure 2: renewing speed–after every iterance, the speed of each corpuscle is replaced by: (20)

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v i  v i  c1  R1  ( pi ,best  pi )  c2  R2  ( g i ,best  pi )

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pi and vi are the location and speed of corpuscle i; pi, best and gi, best is the location with the best result found until now by corpuscle i and whole people separately. w is a

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coefficient managing the behaviors of flight; R1 and R2 range arbitrarily from 0 to 1;

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c1 and c2 are elements dominating the association scale of homologous terms. The stochastic variables are favorable to the random searching of Particle Swarm

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Optimization.

Procedure 3: Renewing location– Locations of whole particles are replaced with pi  pi  v i

al

(21)

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pi should be verified in the limited scope when renewing,

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Procedure 4: Renewing storage–gi, best and pi, best will be replaced when pleased, p i ,best  pi if f ( pi )  f ( pi ,best )

(22)

g i ,best  g i , if f ( g i )  f ( g i ,best )

(23)

Procedure 5: Halting–The arithmetic replicates from step 2 to 4 till precise halting circumstances are pleased, for example, a pre-defined count of iterations. While halting, the arithmetic gains the results as their values. Table 3 Parameters of the PSO-ANFIS model to predict the solubility of sulfur in pure H2S and sour gas

Parameter

In pure H2S

In sour gas

Maximum number of particles

1000

1000

The maximum number of iterations

800

800

Initial inertia weight ω

0.5

0.6

Journal Pre-proof Acceleration C

C1=0.5,C2=1

C1=0.6,C2=0.9

Vector r1 and r2

random

random

Number of fuzzy rules

10

10

PSO[47] utilizes numerous searching data reaching the total vintage data. Indigenous locations of gbest and pbest are distinct. Nevertheless, employing the distinct directions of pbest and gbest, entire agents progressively reach the overall optimal value.

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2.2.3 Genetic Algorithm

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In the beginning, we can express that development should be delimited as majorization mentation. Therefore, as the same as other intelligent models, the root of

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development could be noticed in nature. The evolution of counting data was applied to calculate commands of arithmetic and evolution procedure [48]. Darwin’s theory [49]

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and Mendel’s genetics are widely acceptable evolution principles as a known set of parameters[50]. The significant features of the genetic algorithm (GA) are able to seek

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out another area in a particular region[51]. The primary and major behaviors of GA are

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a crossover, variation, and reversion[52]. The equilibrium of the development of

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shaping procedure by looking for discovering novel areas is applied by the primary manipulators. To manipulate GA, the major population is established with the attentiveness of a question. Soon after, the target function thinks over the singularity of each answer procedure for questions. Henceforth, the primary population is kept in order pertinent to every individual’s fitness. Thinking of crossover and variation operators on the proper individual continues until the implementation of the final standard, the next generation can be set up. Table 4 Parameters of the GA-ANFIS model to predict the solubility of sulfur in pure H2S and sour gas

Parameter

In pure H2S

In sour gas

Population size

200

200

Genetic post-algebra

1000

1000

Selection probability

0.9

0.85

Journal Pre-proof Cross probability

0.65

0.75

Mutation probability

0.1

0.08

Number of fuzzy rules

10

10

3. Data analysis

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3.1 Experimental data To exploit and validate the proposed models, 239 experimental data of sulfur

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solubility gathered from the literature 5, 7, 10-13 were applied in the article. The applied experimental data of sulfur solubility included 55 data in pure H2S and 184

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data in sour gas. Generally speaking, the gathered experimental data sets covered

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extensive ranges of pressure from 7 to 60 MPa, and temperature from 303 to 433K. The data sets were randomly divided into training data with 70% of entire data and

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testing data with 30% of entire data as two parts. The training data were applied to regulate the regression parameters with a purpose to decrease the average absolute

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relative deviation.

Table 5 summary of the database of sulfur solubility used in this article

Author

H2S content

Temperature(K)

Pressure(MPa)

N

Ref

Roof (1971)

100

316.48–383.15

7–31.15

21

7

Brunner and Woll (1980)

100，6-20

373.15–433.15

10–60

28+72

11

Gu et al. (1993)

100

363.2–383.2

10–50

6

13

Sun and Chen (2003)

4.95-26.62

303.2–363.2

20–45

57

14

Yang et al. (2009)

6.86

373.15

16–36

5

16

Bian et al. (2010)

13.79

336.2–396.6

10–55.3

50

19

Journal Pre-proof

3.2 influential factors As far as the pure H2S was concerned, the main influential factors of sulfur solubility were pressure and temperature. In terms of sour gas, the factors influencing the solubility of elemental were temperature, pressure, gas density, gas gravity, critical temperature, and gas composition, etc. We make use of the Pearson correlation analysis to determine the key influential factors on sour gas. The larger the result of the Pearson correlation

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coefficient among the output variable and input variable, the more major effect that the input variable in deciding the result of measurements. The result of Pearson

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correlation analysis was shown in Fig 6, from which we can know the influential factors on the sour gas were, in ascending order, gas gravity, the content of CO 2, the

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content of CH4, the content of H2S, critical temperature, temperature, gas density, and

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pressure. Hence, the developed four models were designed to obtain superior performance among the content of H2S, critical temperature, temperature, gas density,

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and pressure as an input variable and the sulfur solubility as an output variable in sour

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gas. 0.8

Pearson correlations coefficient

0.6

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0.653

0.57

0.433

0.4

0.336

0.329

0.2

0.005 0 T

P

ρ

CH₄

H₂S

CO₂ -0.051

-0.2 -0.246 -0.4

Fig. 6. Pearson correlations for sulfur solubility

g

Tc

Journal Pre-proof

4. Result and discussion To verify the regression of the proposed models, a contrast between the GWO-LSSVM, GWO-RBF, PSO-ANFIS, and GA-ANFIS were made by the entire 239 data. As illustrated in Fig.5-8a, the data calculated by the developed GWO-LSSVM were arranged more closely around the 45° line than the other three models, illustrating data forecasted by GWO-LSSVM had a superior performance with experimental sulfur solubility.

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Many assessment parameters such as the average absolute relative deviation (AARD), the standard deviation (SD), the root mean squared error (RMSE),

pr

determination coefficient (R2), and error percentage (EP) were used to assess

100 N yiexp  yical  y exp N i 1 i

Pr

AARD 

e-

proposed models. The formula should be described as follows:

1 N exp ( yi  yical ) 2  N i 1

(25)

1 N yiexp  yical 2  ( y exp ) N  1 i 1 i

(26)

rn

al

RMSE 

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SD 

(24)

N

R2  1

(y i 1 N

exp i

(y i 1

EP  100 

 yical ) 2 (27)

exp ave

y )

cal 2 i

yiexp  yical yiexp

(28)

Where N describes the number of all experimental sulfur solubility data; yical describes calculated sulfur solubility, yiexp is experimental sulfur solubility, and yavecal means the average result of entire experimental sulfur solubility. The comparative performance of sulfur solubility between the developed GWO-LSSVM, GWO-RBF, GA-ANFIS, and PSO-ANFIS models were described in

Journal Pre-proof Table 6-8.

Total

PSO-ANFIS

AARD/%

3.2058

10.0122

5.9948

3.2808

SD

0.0416

0.1845

0.1005

0.0435

RMSE

3.6463

6.3380

3.6509

3.3062

R2

0.9978

0.9933

0.9978

0.9982

EPmax/%

8.7613

66.2160

42.6378

9.9179

EPmin/%

0.0035

0.0743

0.1641

0.0219

AARD/%

4.1968

7.4341

8.1454

5.3573

SD

0.0721

0.0958

0.1238

0.0816

RMSE

6.8134

9.3715

9.7099

7.2426

R2

0.9936

0.9876

0.9863

0.9927

EPmax/%

17.0531

19.0286

26.9670

17.8287

EPmin/%

0.0278

0.7373

0.0047

0.2374

AARD/%

3.5121

9.2153

6.6595

3.9226

SD

0.1571

0.1077

0.0553

4.6252

7.2756

5.5237

4.5230

0.9965

0.9915

0.9943

0.9965

EPmax/%

17.0531

66.2160

42.6378

17.8287

EPmin/%

0.0035

0.0743

0.0047

0.0219

RMSE

oo

pr e-

0.0510

Jo u

R2

f

GA-ANFIS

rn

Testing set

GWO-RBF

Pr

Evaluation Training set

GWO-LSSVM

al

Table 6 Comparison of four models to predict sulfur solubility in pure H 2S(55 data)

Table 7 Comparison of four models to predict sulfur solubility in sour gas(184 data) Evaluation Training set

GWO-LSSVM

GWO-RBF

GA-ANFIS

PSO-ANFIS

AARD/%

3.0593

8.9021

4.4522

5.6018

SD

0.0789

0.1722

0.0779

0.0965

RMSE

0.0100

0.0248

0.0156

0.0368

R2

0.9998

0.9991

0.9996

0.9981

EPmax/%

68.2103

122.1659

37.2381

40.2438

Journal Pre-proof 0.0087

0.0813

AARD/%

4.5617

7.7997

8.4798

22.7788

SD

0.0759

0.1203

0.1265

0.6012

RMSE

0.0589

0.0530

0.0928

0.0464

R2

0.9932

0.9949

0.9855

0.9949

EPmax/%

27.5177

41.6817

48.5051

335.0072

EPmin/%

0.0014

0.2446

0.0470

0.1578

AARD/%

3.5002

8.5786

5.6342

11.2964

SD

0.0780

0.1570

0.0922

0.2638

RMSE

0.0244

0.9979

0.0383

0.0340

R2

0.9979

0.9915

0.9955

0.9970

EPmax/%

68.2103

122.1659

48.5051

335.0072

EPmin/%

0.0014

0.0087

0.0813

pr

oo

f

0.0307

0.0307

Pr

Total

0.0301

e-

Testing set

EPmin/%

al

Table 8 Comparison of four models to predict sulfur solubility with all 239 data GWO-RBF

AARD/%

3.0924

9.1532

4.8011

5.0540

SD

0.0705

0.1750

0.0830

0.0840

RMSE

0.8325

1.4528

0.8379

0.8085

R2

0.9994

0.9978

0.9992

0.9981

EPmax/%

68.2103

122.1659

42.6378

40.2438

EPmin/%

0.0035

0.0307

0.0087

0.0219

AARD/%

4.4743

7.7122

8.3997

18.9818

SD

0.0750

0.1144

0.1259

0.4880

RMSE

1.6762

2.2842

2.3955

1.6148

R2

0.9933

0.9932

0.9857

0.9944

EPmax/%

68.2103

122.1659

37.2381

40.2438

EPmin/%

0.0014

0.2446

0.0047

0.1578

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Training set

GWO-LSSVM

rn

Evaluation

Testing set

GA-ANFIS

PSO-ANFIS

Journal Pre-proof AARD/%

3.5029

8.7251

5.8701

9.5995

SD

0.0718

0.1570

0.0958

0.2158

RMSE

1.0832

1.6998

1.3006

1. 0716

R2

0.9976

0.9964

0.9952

0.9969

EPmax/%

68.2103

122.1659

48.5051

335.0072

EPmin/%

0.0014

0.0307

0.0047

0.0219

oo

f

Total

pr

Testing set 45° Line

300 250

200 150 100

GWO-LSSVM 50

100

GWO-RBF 50 0

rn

0 50 100 150 200 250 300 350 Experimental sulfur solubility /g·m-3

Training set Testing set

45° Line

200 150 100

GA-ANFIS 50 0

(b) 350 Estimated sulfur solubility /g·m-3

Estimated sulfur solubility /g·m-3

Jo u

(a)

250

45° Line

150

0 50 100 150 200 250 300 350 Experimental sulfur solubility /g·m-3

300

Testing set

200

al

0

350

Training set

e-

250

Estimated sulfur solubility /g·m-3

300

350 Training set

Pr

Estimated sulfur solubility /g·m-3

350

Training set

300

Testing set

250

45° Line

200 150 100

PSO-ANFIS 50 0

0 50 100 150 200 250 300 350 Experimental sulfur solubility /g·m-3

0 50 100 150 200 250 300 350 Experimental sulfur solubility /g·m-3

(c)

(d)

Fig. 7 (a-d). Estimated sulfur solubility versus experimental sulfur solubility by the proposed models in pure H2S(55 data)

5

5

4.5

4.5

4 3.5

Estimated sulfur solubility /g·m-3

Estimated sulfur solubility /g·m-3

Journal Pre-proof

Training set Testing set 45° Line

3 2.5 2 1.5

GWO-LSSVM

1 0.5

Training set

4 Testing set

3.5 45° Line

3 2.5 2 1.5

GWO-RBF

1 0.5 0

0

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Experimental sulfur solubility /g·m-3

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Experimental sulfur solubility /g·m-3

(b)

45° Line

2.5

3.5 3

Training set Testing set

45° Line

GA-ANFIS

0.5

2.5

2

e-

2

1.5 1

4

pr

3

Testing set

Estimated sulfur solubility /g·m-3

3.5

4.5

1.5

1

PSO-ANFIS

0.5

Pr

Estimated sulfur solubility /g·m-3

Training set

4

oo

5

5 4.5

f

(a)

0

0

(c)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Experimental sulfur solubility /g·m-3

al

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Experimental sulfur solubility /g·m-3

(d)

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Fig. 8. (a-d) Estimated sulfur solubility versus experimental sulfur solubility by the proposed

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models in sour gas(184 data)

As illustrated in Table 8 that the developed GWO-LSSVM model has the outstanding results in proposed models and provides predictions in particularly superior conformity with entire experimental sulfur solubility with AARD of 3.5029% and R2 of 0.9976 comparing with other models (GWO-RBF, GA-ANFIS and PSO-ANFIS ) with AARD of 8.7251%, 5.8701%, 9.5995% and R2 of 0.9964, 0.9952, 0.9969 respectively. Therefore, we can know that GWO-LSSVM reveals the best agreement in all 239 datasets (the result of R2 is bigger and AARD is smaller than the other three models). To further assess the proposed models, a contrast of estimated result of the models to the experimental sulfur solubility data in pure H2S (55 data)and sour gas(184 data) is respectively shown in Fig. 7-8 and Table 6-7. Fig. 7-8 shows that GWO-LSSVM has a great agreement between the predicted and experimental sulfur

Journal Pre-proof solubility (arranged more closely around the 45° line) in different systems. And Table 6-7 indicates specific results about the performance of developed models. The AARD of GWO-LSSVM in pure H2S is 3.5121 lower than other models’ AARD, AARD for GWO-RBF, GA-ANFIS, PSO-ANFIS is 9.2153, 6.6595, 3.9226 respectively from Table 6; The R2 of GWO-LSSVM in pure H2S is 0.9965 bigger than other models’ R2, R2 for GWO-RBF, GA-ANFIS, PSO-ANFIS is 0.9915, 0.9943, 0.9965 respectively from Table 6. The AARD of GWO-LSSVM in sour gas is 3.5002 smaller than the other three models’ AARD, where AARD for GWO-RBF, GA-ANFIS, PSO-ANFIS is

oo

f

8.5786, 5.6342, 11.2964 respectively from Table 7; The R2 of GWO-LSSVM in sour gas is 0.9979 bigger than other models’ R2, R2 for GWO-RBF, GA-ANFIS,

pr

PSO-ANFIS is 0.9915, 0.9955, 0.9970 respectively from Table 7. Therefore, we can know that GWO-LSSVM has the best performance no matter in pure H 2S or sour gas.

e-

The leverage method[53,54] was used to search the outlier data in the dataset for

Pr

reliability analysis. To seek out suspicious data, the Williams’ plot was drafted by gauging the Hat values, as shown in Figure 9-10. 6

6

Leverage limit

rn

This work

2

Jo u

0

-2

-4

Leverage limit

4

Standardized Residuals

4 Standardized Residuals

Suapected limit

al

Suapected limit

This work

2

0

-2

-4

GWO-LSSVM

-6 0

0.04

0.08

0.12 H

GWO-RBF -6 0.16

0.2

0

0.04

0.08

0.12

0.16

H

(a)

(b)

0.2

Journal Pre-proof 6

6 Suapected limit

4

Leverage limit

4

This work Standardized Residuals

Standardized Residuals

Suapected limit

Leverage limit

2

0

-2

This work

2

0

-2

-4

-4

GA-ANFIS

PSO-ANFIS

-6

-6 0

0.04

0.08

0.12

0.16

0.2

0

0.04

0.08

H

0.12

0.16

0.2

H

(d)

f

(c)

6

oo

Fig.9 Detection of doubtful data in four models using Hat values in pure H 2S(55 data) 6 Suapected limit

Leverage limit

This work

This work

2

e-

2

pr

4

Standardized Residuals

Standardized Residuals

Suapected limit

Leverage limit

4

0

-2

-4

Pr

-2

0

-4

GWO-LSSVM -6

GWO-RBF

-6 0.02

0.04

0.06 H

0.1

0.12

0

6

0.02

0.04

0.06

Jo u

Leverage limit

4 Standardized Residuals

Standardized Residuals

Suapected limit

This work

0

0.12

6

Leverage limit

2

0.1

(b)

Suapected limit

4

0.08

H

rn

（a）

0.08

al

0

-2

This work

2

0

-2

-4

-4

GA-ANFIS

PSO-ANFIS

-6

-6 0

0.02

0.04

0.06 H

(c)

0.08

0.1

0.12

0

0.02

0.04

0.06

0.08

0.1

0.12

H

(d)

Fig.10 Detection of doubtful data in four models using Hat values in sour gas(184 data)

As is described in Fig 9-10, The greater part of sulfur solubility data in pure H2S and sour gas is valid, although there are several suspicious data in these four models. William’s plot confirm stability, robustness, reliability of four models and suspicious

Journal Pre-proof values. We can know that the developed models not only efficacious in the mathematics area but also have superior robustness in mirroring the internal connections among influential factors and sulfur solubility.

5. Conclusion In the article, four intelligent models (GWO-LSSVM, GWO-RBF, PSO-ANFIS, GA-ANFIS) were developed to estimate the result of sulfur solubility. Pressure and

oo

f

temperature are the influential factors with sulfur solubility in pure H 2S. According to Pearson correlation analysis, the content of H2S, critical temperature, temperature, gas

pr

density, and pressure are the influential factors with sulfur solubility in sour gas. All dates collected from the literature were divided into training sets with 70% of total

e-

data and testing sets with 30% of total data as two parts. Furthermore, the developed

Pr

GWO-LSSVM model comparing with other intelligent models (GWO-RBF, PSO-ANFIS, GA-ANFIS). The results show that no matter what under any conditions,

al

in pure H2S or sour gas, the GWO-LSSVM model gives the excellent performance, and the data calculated by the developed GWO-LSSVM were arranged more closely

rn

around the 45° line than other three models, illustrating data forecasted by

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GWO-LSSVM had a superior performance with experimental sulfur solubility. But according to the minimum root mean squared error (RMSE), PSO-ANFIS performs best in pure H2S and sour gas among the four models. Also, GWO-LSSVM has the characters of superior stability and robustness according to Williams’ plot used to confirm the reliability and robustness. Moreover, GWO-LSSVM has a better performance in sour gas than in pure H2S. As a result, the developed methods provide a functional mode in petroleum engineering and have an advanced effect on the development of gas reservoirs.

Acknowledgments This work was supported by National Natural Science Foundation of China (No.

Journal Pre-proof 51404205), and the Program for Innovative Research Team of the Education Departm ent of Sichuan Province, China−The Greenhouse Gas Carbon Dioxide Storage and Resource Utilization (No. 16TD0010).

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Conflict of interest statement We declare that we have no financial and personal relationships with other people or organizations that can inappropriately influence our work, there is no professional or other personal interest of any nature or kind in any product, service and/or company that could be construed as influencing the position presented in, or

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the review of, the manuscript entitled“Prediction of the sulfur solubility in pure H2S

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rn

al

Pr

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and sour gas by intelligent models”.