A reservoir bubble point pressure prediction model using the Adaptive Neuro-Fuzzy Inference System (ANFIS) technique with trend analysis.

The bubble point pressure (Pb) could be obtained from pressure-volume-temperature (PVT) measurements; nonetheless, these measurements have drawbacks such as time, cost, and difficulties associated with conducting experiments at high-pressure-high-temperature conditions. Therefore, numerous attempts have been made using several approaches (such as regressions and machine learning) to accurately develop models for predicting the Pb. However, some previous models did not study the trend analysis to prove the correct relationships between inputs and outputs to show the proper physical behavior. Thus, this study aims to build a robust and more accurate model to predict the Pb using the adaptive neuro-fuzzy inference system (ANFIS) and trend analysis approaches for the first time. More than 700 global datasets have been used to develop and validate the model to robustly and accurately predict the Pb. The proposed ANFIS model is compared with 21 existing models using statistical error analysis such as correlation coefficient (R), standard deviation (SD), average absolute percentage relative error (AAPRE), average percentage relative error (APRE), and root mean square error (RMSE). The ANFIS model shows the proper relationships between independent and dependent parameters that indicate the correct physical behavior. The ANFIS model outperformed all 21 models with the highest R of 0.994 and the lowest AAPRE, APRE, SD, and RMSE of 6.38%, -0.99%, 0.074 psi, and 9.73 psi, respectively, as the first rank model. The second rank model has the R, AAPRE, APRE, SD, and RMSE of 0.9724, 9%, -1.58%, 0.095 psi, and 13.04 psi, respectively. It is concluded that the proposed ANFIS model is validated to follow the correct physical behavior with higher accuracy than all studied models.

1. Introduction

Determination or measurement of an accurate reservoir bubble point pressure (Pb) is essential for achieving accurate reservoir and petroleum production calculations [1-4]. As a result, obtaining the Pb with high accuracy is necessary.

Numerous researchers studied the Pb for different crude oils. In North America, Standing [5], Lasater [6], Glaso [7], Petrosky and Farshad [8], De Ghetto et al. [9], Velarde et al. [4], and Dindoruk and Christman [10] showed correlations applied to determine the Pb based on Rs, γg, API, and Tf. Standing [5] and Lasater [6] utilized 105 and 158 datasets from the USA and Canada to develop their models. Glaso [7] applied some regressions methods to create a correlation for Pb with a standard deviation (SD) of 6.98. Petrosky and Farshad [8] used 90 Gulf Mexico datasets to develop their Pb model by applying regression methods (involving Statistical Analysis System (SAS) software). De Ghetto et al. [9] and Velarde et al. [4] used regressions techniques to create their equations to determine the Pb, and they mentioned that their correlations have AAE of 12.8% and 11.7%. Dindoruk and Christman [10] showed a correlation employed to determine the Pb using 100 datasets and MS-Excel software.

Al-Marhoun [11], Dokla and Osman [12], Almehaideb [13], Mehran et al. [14], Bolondarzadeh et al. [15], Hemmati and Kharrat [16], Mazandarani and Asghari [17], Khamehchi et al. [18], and Gomaa [19] developed their Pb correlations depended on the Middle East crude oils. Al-Marhoun [11] utilized Rs, γg, API, and Tf as independent parameters to create a correlation to determine the Pb by applying the non-linear multiple regression method using 160 data points. Dokla and Osman [12] and Almehaideb [13] displayed Pb correlations using 51 and 62 data points from the United Arab Emirates, and their equations have AAE of 7.61% and 4.997%, respectively. Mehran et al. [14], Bolondarzadeh et al. [15], Hemmati and Kharrat [16], Mazandarani and Asghari [17], Khamehchi et al. [18] operated regression methods to create their Pb equations using datasets from Iranian fields. Gomaa [19] developed the correlation based on Rs, γg, API, and Tf and disclosed that their equation has the AAE and the SD of 8.12% and 10.69.

In Africa, Macary and EL-Batanoney [20] showed an equation used to predict the Pb with AAE of 7.04% using Rs, γg, API, and Tf as independent variables and 90 datasets from Egypt. Hanafy et al. [21] used only the Rs as input parameter, the regression methods, and 324 datasets from Egyptian fields to determine the Pb. Sharrad and Abd-Alrahman [22] found a Pb equation using more than thirty Libyan datasets and EViews software and displayed their correlation with the AAE of 8.7%.

Frashad et al. [23] showed the Pb correlation with SD of 37.02 using regression methods and 43 datasets from Colombia. Omar and Todd [24] applied non-linear regression analysis and more than ninety Malaysian datasets to display their Pb correlation and indicated that the correlation has AAE and SD of 7.17% and 9.54.

Vasquez and Beggs [25], Kartoatmodjo and Schmidt [26], Al-Shammasi [27], and Arabloo et al. [28] proposed equations for predicting the Pb based on Rs, γg, API, and Tf and utilizing data points from different places. Kartoatmodjo and Schmidt [26] employed more than 5000 datasets from different regions in North America and used a regression approach to build the Pb correlation with 20.17% (AAE). Al-Shammasi [27] utilized a regression approach, 1661 datasets from different places to develop a Pb correlation, and stated that the correlation could predict the Pb with 17.849% AAE and 17.16 SD. Arabloo et al. [28] represented a Pb correlation with an AAE of 18.9, operating LINGO software and more than 700 global datasets. Fig 1 illustrates the previously published models based on used data locations.

Fig 1

Previous models based on used data locations [recreated from copyright free open source [29]].

Nowadays, machine learning and deep learning methods are used to develop the Pb model. Alakbari et al. [30] used artificial neural networks and fuzzy logic approaches for predicting the Pb based on Rs, γg, API, and Tf. Yang et al. [31] represented a correlation that can be used to predict the Pb using some artificial intelligent algorithms, namely neural networks. Alakbari et al. [32] created their model based on the Rs, γg, API, and Tf as inputs and more than 700 datasets, and they showed that their model has the absolute average percent relative error and the (R) were 8.422% and 0.990. Nonetheless, the previous models are required to improve their accuracy in obtaining the Pb.

Numerous researchers successfully applied the adaptive neuro-fuzzy inference system (ANFIS) method in engineering calculations. A noise assessment of wind turbine was predicted using the ANFIS [33]. The ionic and electronic conductivity of materials was estimated utilizing the ANFIS [34]. Ayoub et al. [35] developed a model to obtain the drilling rate of penetration using the ANFIS technique. The wind power density was determined by applying the ANFIS [36]. Sambo et al. [37] used ANFIS to determine water saturation from seismic attributes. Hamdi and Chenxi [38] proposed an ANFIS model to predict CO2 minimum miscibility pressure (MMP) with higher accuracy. A recent study has applied ANFIS to model the isothermal oil compressibility below the Pb Ayoub et al. [39].

This research aims to build a robust and higher accurate model that can be used to determine the Pb using the ANFIS method with the trend analysis (TrA). The only attempt to apply ANFIS for developing Pb correlations is the one proposed by Shojaei et al. [40], who used 750 data points to build the Pb model. However, they have not studied the TrA to prove the proper physical behavior for their model. Therefore, in this study, a robust and highly accurate ANFIS model was developed to predict the Pb through TrA. More than 700 global datasets and the ANFIS method were applied with the trend analysis that is used to find the relationships between the independent variables (Rs, γg, API, and Tf) and dependent variable (Pb) to indicate the correct physical behavior to build our ANFIS model with the trends analysis that is used for the first time to a robustly and accurately determine the Pb. Moreover, statistical error analyses such as R were utilized to compare the ANFIS and all existing models’ accuracy.

2. Methodology

2.1 Data collection and pre-processing

More than seven hundred data sets were gathered from existing sources [11, 24, 28] to build the proposed ANFIS model. The Rs, γg, API, and Tf are utilized as independent parameters in this study because most of the studies in the literature consider these parameters as inputs; however, Hanafy et al. [21] used only the Rs as the input to predict the Pb, Table 1. Furthermore, the (R) for independent parameters (Rs, γ, API, and T) to the dependent parameter (Pb) was found to evaluate the importance of the independent and dependent parameters as shown in Fig 2. From this figure, we can see the (R) of 0.876 for the Rs, and the Pb means that the Pb can be a strong function of the Rs. As displayed in Fig 2, the (R) of -0.513 for the γ and the Pb indicates that the Pb can be a moderate function of the γ and the (R) of 0.383 and 0.315 for the API and T proves that the Pb can be a weak function of the API and T.

Table 1

Comparison of input parameters used in the published correlations and the proposed ANFIS model.

No	Model	Input parameters
		Bubble point oil volume factor (Bob) (bbl/STB)	Gas to oil ratio (Rs) (scf/STB)	Gas-specific gravity (γg)	Oil-specific gravity (API) (oAPI)	Reservoir temperature (Tf) (° F)
1	Standing (1947) [5]		√	√	√	√
2	Lasater (1958) [6]		√	√	√	√
3	Glaso (1980) [7]		√	√	√	√
4	Vazquez and Beggs (1980) [25]		√	√	√	√
5	Al-Marhoun (1988) [11]		√	√	√	√
6	Kartoatmodjo and Schmit (1991) [26]		√	√	√	√
7	Dokla and Osman (1992) [12]		√	√	√	√
8	Petrosky and Farshed (1993) [8]		√	√	√	√
9	Macary and El-Batanoney (1993) [20]		√	√	√	√
10	Omar and Todd (1993) [24]	√	√	√	√	√
11	De Ghetto et al. (1994) [9]		√	√	√	√
12	Frashad et al. (1996) [23]		√	√	√	√
13	Almehaideb (1997) [13]	√	√	√	√	√
14	Hanafy et al. (1997) [21]		√
15	Velarde et al. (1997) [4]		√	√	√	√
16	Al-Shammasi (1999) [27]		√	√	√	√
17	Dindoruk and Christman (2001) [10]		√	√	√	√
18	Mehran et al. (2006) [14]		√	√	√	√
19	Bolondarzadeh et al. (2006) [15]		√	√	√	√
20	Hemati and Kharrat (2007) [16]	√	√	√	√	√
21	Mazandarani and Asghari (2007) [17]		√	√	√	√
22	Khamechchi et al. (2009) [18]		√	√	√	√
23	Arabloo et al. (2014) [28]		√	√	√	√
24	Gomaa (2016) [19]		√	√	√	√
25	Sharrad and Abd-Alrahman (2019) [22]		√	√	√	√
26	Proposed ANFIS		√	√	√	√

Fig 2

Relative importance of inputs with Pb output.

Before the ANFIS model was applied, the collected data were split into two parts 70% for training the model and 30% for testing the proposed ANFIS model. The statistical description of the training and testing datasets is shown in Table 2. As in the table, the training and testing datasets are at the same ranges to build and evaluate the ANFIS model with the same data ranges. It is essential to avoid the over-fitting and under-fitting issues; data randomization was used to overcome these issues. In addition, all parameters for the training and testing datasets were normalized between -1 and 1 to scale them in this range based on the following equation:

Table 2

Statistical description of the data.

Parameters	Training data			Testing data
	Minimum	Maximum	SD	Minimum	Maximum	SD
Bubble point pressure (P b ) psi	126	7127	1151.55	130	4432	1135.4
Gas to oil ratio (Rs) SCF/STB	9	2637	423.50	26	1850	424.93
Gas-specific gravity (γ g )	0.5890	1.367	0.1593	0.5890	1.367	0.1622
Oil-specific gravity (API) oAPI	15.30	59.50	7.32	19.40	51.70	6.38
Reservoir temperature (Tf)°F	74	294	49.46	74	271	45.36

Where:

Y: the normalized parameter.

Y: the maximum normalized value (1).

Y: the minimum normalized value (-1).

X: the input variable.

X: the minimum of the variable.

X: the maximum of the variable.

2.2 Proposed ANFIS model strategy

ANFIS is a combination of artificial neural networks (ANN) and fuzzy logic (FL), and it is one of the neural networks that use the Takagi-Sugeno fuzzy inference system. The Takagi-Sugeno fuzzy model applies two fuzzy rules [41]:

rule 1: if (x1 is A1) and (x2 is B1), then Eq (2) is used. rule 2: if (x1 is A2) and (x2 is B2), then Eq (3) is applied. where:

x1 and x2: inputs.

A1, A2, B1, and B2: membership values.

p1, q1, r1, p2, q2, and r2: parameters of the output functions f1 and f2, respectively.

As displayed in Fig 3, the ANFIS structure is constructed of five layers. These layers are the fuzzification layer, rule layer, normalization layer, defuzzification layer, and output layer. ANFIS is a multilayer feedforward neural network with supervised learning capability (a hybrid learning rule) [42, 43]. For the Sugeno fuzzy reasoning, the default defuzzification technique was applied. It can be a weighted average of all rule outputs. The fuzzified input values can be an algebraic sum of consequent fuzzy sets for the used aggregate technique. Firstly, input characteristics transfer to input membership functions. Then, they move to rules. After that, they shift to a set of output characteristics. Next, they go to output membership functions. Finally, the output membership functions provide output [44].

Fig 3

The workflow of MATLAB ANFIS structure.

The ANFIS technique has advantages of showing better results than other methods. The ANFIS shows a better learning ability. It can perform a highly non-linear mapping. It has fewer adjustable parameters than those needed in other machine learning. Its structure can allow for parallel computation. Its networks show a well-structured knowledge representation and can also allow better integration with other control design methods [45]. ANFIS can combine ANN and Fl in a single tool to make the technique superb in reaching a quicker decision about the mapped relationship between the feature and target parameters [46]. The ANFIS has the benefit of decreased training time not only because of its smaller dimensions but also because the network is initialized with parameters in relation to the problem domain [47].

The proposed ANFIS model in this work was built using MATLAB R2019b. Fig 4 demonstrates the ANFIS output generated from MATLAB 2019b. The type of membership function applied in this proposed ANFIS model is Gaussian curve membership. The optimal hyperparameters of ANFIS were selected by using the manual method. In the manual method, each parameter changed in its different types or values. Then, the model accuracy and the correct trend analysis were checked. Finally, the optimal hyperparameters were selected with the proper trend analysis for the highest accuracy, as shown in Table 3.

Fig 4

ANFIS system results with four input parameters, three rules, and one output, (generated from MATLAB R2019b).

Table 3

Descriptions of the optimal ANFIS model hyperparameters.

Parameter	Description/value
Fuzzy structure	Sugeno-type
Initial FIS for training	genfis2
Membership function type	Dsigmf
Output membership function	Linear
Cluster centre’s range of influence	0.459
Number of inputs	4
Number of outputs	1
Optimization method	Hybrid
Number of fuzzy rules	10
Training epoch number	24
Initial step size	0.3555
Step size decrease rate	0.2
Step size increase rate	2

3. Results and discussion

The ANFIS model was evaluated by conducting two tests. The proposed ANFIS model was first investigated by conducting TrA to ensure that all inputs follow the proper physical behavior. After that, the ANFIS model and studied correlations were compared. Statistical error analysis, namely, (R), standard deviation (SD), average percent relative error (APRE), average absolute percentage relative error (AAPRE), and root mean square error (RMSE), were performed to show the performance of the ANFIS and studied models.

3.1 Trend analysis (TrA)

The trend analysis (TrA) can be used to study the reliability of models. TrA can be applied by changing the studied input between the minimum and maximum values while keeping the other inputs at their constant mean values. The studied input, such as Rs, is plotted as the x-axis and the output Pb as the y-axis [27, 48–50]. The TrA is an essential part of this work, as some researchers used ANFIS, but they have not applied the trend analysis [40]. Without considering the trend analysis, it was clear that the ANFIS model may show fake high accuracy. As a result, the models developed without considering the trend analysis should not be considered as a reliable tool.

The trend analysis was conducted for the ANFIS, and 21 studied models to study the relationships between the inputs (Rs, γ, API, T) and output Pb to show the physical behavior.

In the TrA study, the four independent variables (Rs, γ, API, T) were selected because most previous models used these variables; nevertheless, the oil formation volume factor was not considered in our model because it is only utilized by [13, 16, 24]. The TrA was performed to represent the proper relationships between the Rs, γ, API, T and the Pb to show the actual physical behavior for the studied parameters and validated the ANFIS model.

Fig 5 presents the Rs TrA for the ANFIS and all existing models. As shown in Fig 5, the ANFIS and all the previous models show the proper relationships between the Rs and the Pb. Increasing the Rs increases the Pb. However, Farshad’s [23] and Almehaideb’s [13] correlations indicate that the Pb was -812.6 and -207.5 psi at Rs 26 SCF/STB (as shown in Fig 5) because they built their correlation based on Rs ranges from 217 to 1406 and from 128 to 3871 SCF/STB, respectively. Fig 6 indicates that the developed ANFIS model follows the proper relationships between the Rs and the Pb to correct physical behavior. Li et al. [51] showed that increasing the Rs increased the Pb.

Fig 5

Rs TrA of the ANFIS and existing models.

Fig 6

Rs TrA of proposed ANFIS model.

The TrA of γ for the ANFIS and all current models is demonstrated in Fig 7. The ANFIS and most existing models revealed that the γ is inversely proportional to the Pb, which proves the proper relationships between the γ and the Pb; nevertheless, Hanafy et al.’s [21] correlation displayed that changing the γ does not change the Pb as indicated by the constant trend. This indicates an incorrect relationship between the γ and the Pb because γ was not considered as input in their model. Goma’s [19] correlation showed that the Pb was slightly increased by increasing the γ and the correlation indicate improper TrA for γ. Omar and Todd’s [24] correlation represented that the Pb decreases and then increases by increasing the γ, which is also improper relationships between the γ and the Pb. Therefore, Omar and Todd’s [24], Hanafy et al.’s [21], and Goma’s [19] models represent incorrect relationships between the γ and the Pb, and hence, improper physical behavior for γ trend. Fig 8 illustrated the correct trend γ for the ANFIS model. Al-Shammasi [27] proved that growing the γ declines the Pb.

Fig 7

γ TrA of the ANFIS and existing models.

Fig 8

γ TrA of proposed ANFIS model.

Fig 9 shows the TrA of API for the ANFIS and all current models. The ANFIS and most models display the proper relationships between the API and the Pb. The higher the API, the lower the Pb is (Fig 9); however, Dokla and Osman [12], Hanafy et al. [21], and Gomaa [19] models do not show the correct relationships between the API and the Pb, indicating incorrect physical behavior. Dokla and Osman’s [12] correlation showed that the Pb was slightly decreased by rising the API, (Fig 9). Gomaa’s [19] correlation demonstrated that increasing the API also drops the Pb slightly (Fig 9). Hanafy et al.’s [21] equation displayed that the Pb is constant with changing the API (Fig 9). Petrosky and Farshad’s [8] correlation shows that the Pb is -37.37 psi and -145.91 psi at 48.11 and 51.7°API, Fig 9 because they developed the equation in (16.3–45°API) range. The ANFIS model presents the correct relationships between the API and the Pb, indicating proper physical behavior, as shown in Fig 10. Al-Shammasi [27] also revealed that increasing the API drops the Pb.

Fig 9

API TrA of the ANFIS and existing models.

Fig 10

API TrA of the ANFIS model.

The TrA of the T for the ANFIS and all current models is illustrated in Fig 11. As shown in Fig 11, the ANFIS and most current models follow the proper relationships between the T and the Pb, increasing the T increases the Pb; nonetheless, Dokla and Osman’s [12] equation indicates that the Pb declines by increasing T indicating incorrect relationships between the T and the Pb. Hanafy et al.’s [21] correlation also displays a constant Pb with increasing the T to indicate the improper relationships between the T and the Pb. Dindoruk and Christman’s [10] and Arabloo et al.’s [28] correlations represent that the Pb is slightly changed by growing the T to show incorrect physical behavior for the T trend. The correct T trend for the proposed ANFIS model is clearly represented in Fig 12. The temperature can drop the gas density; therefore, the temperature can increase the Pb.

Fig 11

T TrA of the ANFIS and existing models.

Fig 12

T TrA of the ANFIS model.

From the TrA study, we can conclude that all independent parameters (Rs, γ, API, T) of the ANFIS model represent the proper relationships with the Pb to indicate the correct physical behavior; however, Dokla and Osman’s [12], Omar and Todd’s [24], Hanafy et al.’s [21], and Goma’s [19] correlation show the improper relationships between the independent parameters and the Pb to indicate the incorrect physical behavior. Petrosky and Farshad’s [8] and Almehaideb’s [13] correlations display some negative Pb because the Rs and API as inputs for these negative values do not include in their study ranges.

3.2 Comparison of the ANFIS model against other models

3.2.1 Cross-plot

Fig 13 shows the cross-plot for the training datasets of the ANFIS model. Most training data are closer to the 45° line to indicate that the ANFIS is a higher accurate model for the training datasets. The (R2) for the training datasets of the ANFIS model is 0.9725. Fig 14 presents the cross plot for the testing datasets of the ANFIS model, and most of the testing data are also closer to the 45° line to show that the ANFIS model can accurately predict the Pb for the testing datasets with the (R2) of 0.9878. Fig 15 displays the cross-plot for the ANFIS and all current models studied in this paper. As shown in Fig 15, the ANFIS model is the highest accurate model with (R2) of 0.9878 compared to all studied models.

Fig 13

Cross-plot of training ANFIS model.

Fig 14

Cross-plot of testing ANFIS model.

Fig 15

Cross-plot of the ANFIS and existing models.

3.2.2 Statistical error analysis

Some statistical analysis has been used along with trend analysis and cross-plotting analysis to validate and describe the efficiencies of the proposed ANFIS model. In addition, the ANFIS was compared against the 22 studied models that follow the correct physical behavior. The statistical error analysis applying in this study are (R), RMSE, SD, APRE, AAPRE, maximum and minimum absolute percent relative error (E) and (E). The statistical criterion explanations are presented in the appendix (S1 Appendix). The AAPRE and R were used in this research as the leading indicators to compare the ANFIS model’s accuracy with the current models.

The ANFIS and existing models were compared by plotting the AAPRE and R (Fig 16). As display in Fig 16, the ANFIS model is the first rank model and has the lowest AAPRE of 6.378% and APRE of -0.99%, and the highest (R) of 0.994. The second rank model is Velarde et al.’s [4] model with the AAPRE of 9%, the APRE of -1.58%, and R of 0.9724. The third rank model is Mehran et al.’s [14] correlation with the AAPRE of 9.75%, the APRE of -3.91%, and R of 0.9699. The last rank model is Petrosky and Farshad’s [8] model with the AAPRE of 76.59%, the APRE of 57.39%, and R of 0.9703.

Fig 16

Comparing the ANFIS and existing models using (R) and AARE (%).

The ANFIS and all existing models are compared using statistical error analyses AAPRE, APRE, RMSE, SD, E, and E., Table 4. The ANFIS model and all studied models are ranked based on the leading indicators AAPRE and R. The ANFIS model is the first rank model and has the lowest AAPRE of 6.38%, APRE of -0.99, RMSE of 9.73, SD of 0.074, Eof 0.021%, and E. of 50.19% and the highest R of 0.9939. The results indicate that the ANFIS model outperformed all existing models (22 models). The second rank model is Velarde et al.’s [4] correlation that has the AAPRE of 9%, APRE of -1.58, RMSE of 13.04, SD of 0.094, E of 0.039, E. of 62.47, and R of 0.9724. The third rank model is Mehran et al.’s [14] correlation and has the AAPRE of 9.75%, APRE of -3.91%, RMSE of 13.60, SD of 0.095, E of 0.035%, E. of 63.86%, and R of 0.9699. The last rank model is Petrosky and Farshed’s [8] correlation that has the AAPRE of 76.59%, APRE of 57.39%, RMSE of 159.87, SD of 1.406, E of 0.295%, E. of 784.59%, and R of 0.9703. Comparing the ANFIS and existing models conducts an important means of evaluating all the models’ performance.

Table 4

Statistical error analysis of the ANFIS and existing models.

Rank	Model	APRE (%)	AAPRE (%)	Emax. (%)	Emin. (%)	RMSE (psi)	SD (psi)	R
1	Proposed ANFIS	-0.99	6.38	50.19	0.021	9.73	0.074	0.9939
2	Velarde et al. (1997) [4]	-1.58	9.00	62.47	0.039	13.04	0.095	0.9724
3	Mehran et al. (2006) [14]	-3.91	9.75	63.86	0.035	13.60	0.095	0.9699
4	Lasater (1958) [6]	-1.83	11.07	66.08	0.016	15.31	0.106	0.9742
5	Standing (1947) [5]	-3.95	12.35	69.28	0.032	16.26	0.106	0.9753
6	Arabloo et al. (2014) [28]	1.51	12.66	72.98	0.000	17.12	0.116	0.9589
7	Hemati and Kharrat (2007) [16]	6.35	13.76	85.01	0.026	22.13	0.174	0.9741
8	Vazquez and Beggs (1980) [25]	-13.07	16.88	74.79	0.493	21.65	0.136	0.9767
9	Kartoatmodjo and Schmit (1991) [26]	-9.33	16.94	78.37	0.085	22.74	0.152	0.9722
10	Al-Shammasi (1999) [27]	-11.20	17.33	62.95	0.205	22.60	0.145	0.9663
11	Frashad et al. (1996) [23]	-8.03	18.23	74.23	0.042	24.30	0.161	0.9621
12	De Ghetto et al. (1994) [9]	-14.18	18.37	73.97	0.007	24.83	0.167	0.9720
13	Dindoruk and Christman (2001) [10]	-3.72	20.89	77.83	0.432	25.81	0.152	0.9369
14	Glaso (1980) [7]	-14.33	23.02	79.52	0.281	27.70	0.154	0.9701
15	Mazandarani and Asghari (2007) [17]	-19.19	23.91	120.93	0.127	34.19	0.245	0.9462
16	Almehaideb (1997) [13]	22.89	26.15	234.92	0.037	44.18	0.357	0.9482
17	Macary and El-Batanoney (1993) [20]	-25.03	31.20	149.75	0.111	42.62	0.291	0.9499
18	Khamechchi et al. (2009) [18]	-29.55	31.24	97.52	0.059	37.27	0.204	0.9652
19	Bolodarzadeh et al. (2006) [15]	28.31	40.42	434.20	0.175	84.69	0.746	0.9694
20	Sharrad and Abd-Alrahman (2019) [22]	45.92	45.93	72.46	0.346	47.96	0.139	0.8929
21	Al-marhoun (1988) [11]	54.06	54.06	79.22	27.176	54.40	0.148	0.9538
22	Petrosky and Farshed (1993) [8]	57.39	76.59	784.59	0.295	159.87	1.406	0.9703

4. Conclusions

With 760 global datasets used, the ANFIS model was developed with the trend analysis to robustly and accurately predict the Pb. In addition, the ANFIS mode’s accuracy was compared with 21 existing models utilizing statistical error analysis. In this research, we can conclude the following:

The trend analysis results of the ANFIS model indicate that the ANFIS model can describe the correct relationships between the independent parameters (Rs, γ, API, T) and dependent parameter Pb to show the proper physical behavior.

Some previous correlations fail to represent the proper relationships between the independent parameters and the Pb to indicate incorrect physical behavior.

The proposed ANFIS model outperformed all 21 existing models and has the lowest AAPRE of 6.38%, APRE of -0.99, RMSE of 9.73, SD of 0.074, E of 0.021%, and E. of 50.19% and the highest R of 0.9939 compared to 21 studied correlations that follow the correct physical behavior. The ANFIS model shows better results than other models because of its combination of the FL and ANN performances and better learning ability. The ANFIS can perform a highly non-linear mapping.

The data randomization was conducted to prevent the model from overfitting or underfitting to obtain the robust and accurate ANFIS model to predict the Pb.

(PDF)

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1 May 2022

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Reviewer #1: Yes

Reviewer #2: Partly

**********

2. Has the statistical analysis been performed appropriately and rigorously?

Reviewer #1: Yes

Reviewer #2: Yes

**********

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Reviewer #1: Yes

Reviewer #2: Yes

**********

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Reviewer #1: No

Reviewer #2: Yes

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5. Review Comments to the Author

Please use the space provided to explain your answers to the questions above. You may also include additional comments for the author, including concerns about dual publication, research ethics, or publication ethics. (Please upload your review as an attachment if it exceeds 20,000 characters)

Reviewer #1: The abstract needs a little bit more numerical result. Please add them.

Please explain why did you chose ANFIS among other models? Why did not you use ANN, SVM, etc?

Page 3, line 59: please use a uniform mode when reporting the numbers. Please correct “one hundred and 158 datasets”.

Considering the fact that your research only uses ANFIS, I suggest you to explain more about ANFIS including its steps and equations. Use following papers for that.

Azad, A., Farzin, S., Sanikhani, H., Karami, H., Kisi, O., & Singh, V. P. (2021). Approaches for optimizing the performance of adaptive neuro-fuzzy inference system and least-squares support vector machine in precipitation modeling. Journal of Hydrologic Engineering, 26(4), 04021010.

Azad, A., Manoochehri, M., Kashi, H., Farzin, S., Karami, H., Nourani, V., & Shiri, J. (2019). Comparative evaluation of intelligent algorithms to improve adaptive neuro-fuzzy inference system performance in precipitation modelling. Journal of hydrology, 571, 214-224.

Please change Table 3 to a casual table in terms of colour.

How did you select the setting parameters of ANFIS. For instance, what was the reason training epoch is 10?

I think figure 4 has been copied from another article. Please cite them and ask for their permission.

The novelty of this work is not clear enough yet. I mean, it should be cleared that how much ANFIS improved the modeling accuracy in comparison with other simpler models? Why we need to use ANFIS?

Results and discussion is satisfying.

We do not usually cite other papers in the Conclusion section. Please remove them and modify the conclusion accordingly.

Reviewer #2: Please give explanation why the developed ANFIS gave low or high values of reservoir bubble point pressure

when compared against the 22 studied models that follow the correct physical behavior.

**********

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Reviewer #2: No

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6 Jul 2022

Reviewer #1:

The abstract needs a little bit more numerical result. Please add them.

Done, the numerical results were added to the abstract accordingly.

Please explain why did you choose ANFIS among other models? Why did not you use ANN, SVM, etc?

The ANFIS technique has the advantage of showing better results than other methods, which is why it is chosen in this study. The ANFIS offers a better learning ability. It can perform a highly nonlinear mapping. It has fewer adjustable parameters than those needed in other machine learning. Its structure can allow for parallel computation. Its networks show a well-structured knowledge representation and can also allow better integration with other control design methods [1]. ANFIS can combine ANN and Fl in a single tool to make the technique superb in reaching a quicker decision about the mapped relationship between the feature and target parameters [2]. The ANFIS benefits from decreased training time because of its smaller dimensions and because the network is initialized with parameters about the problem domain [3].

Page 3, line 59: please use a uniform mode when reporting the numbers. Please correct “one hundred and 158 datasets”.

Done, the numbers were corrected accordingly.

Considering the fact that your research only uses ANFIS, I suggest you to explain more about ANFIS, including its steps and equations. Use the following papers for that.

Azad, A., Farzin, S., Sanikhani, H., Karami, H., Kisi, O., & Singh, V. P. (2021). Approaches for optimizing the performance of adaptive neuro-fuzzy inference system and least-squares support vector machine in precipitation modeling. Journal of Hydrologic Engineering, 26(4), 04021010.

Azad, A., Manoochehri, M., Kashi, H., Farzin, S., Karami, H., Nourani, V., & Shiri, J. (2019). Comparative evaluation of intelligent algorithms to improve adaptive neuro-fuzzy inference system performance in precipitation modelling. Journal of Hydrology, 571, 214-224.

Done, more explanation about ANFIS, including its steps and equations, is added accordingly using the recommended papers. Please, you can see highlighted yellow color in subsection (2.2 Proposed ANFIS model strategy).

Please change Table 3 to a casual table in terms of colour.

Done, the Table was changed to a casual table in terms of color accordingly.

How did you select the setting parameters of ANFIS? For instance, what was the reason the training epoch is 10?

The optimal hyper-parameters of ANFIS were selected by using the manual method. In the manual way, each parameter changed in its different types or values. Then, the model accuracy and the correct trend analysis were checked. Finally, the optimal hyper-parameters were selected with the appropriate trend analysis for the highest accuracy, as shown in Table 3. The statement is added in subsection 2.2 Proposed ANFIS model strategy accordingly.

I think figure 4 has been copied from another article. Please cite them and ask for their permission.

Figure 4 was generated from MATLAB R2019b; it is mentioned accordingly.

The novelty of this work is not clear enough yet. It should be clear how much ANFIS improved the modelling accuracy compared to cleared and how much ANFIS improved the modeling accuracy compared to the other simpler models?

The novelty of this study is to apply the ANFIS model with the trend analysis. The trend analysis study shows that the model follows the correct relationships between the inputs and output to prove the proper physical behavior. In addition, the ANFIS model is higher accuracy than all studied models. Please, see the added statement in the abstract accordingly. The second rank model that is the best in the previous models has R, AAPRE, APRE, SD, and RMSE of 0.9724, 9%, -1.58%, 0.095 psi, and 13.04 psi, respectively. However, the proposed ANFIS model has the highest R of 0.994 and the lowest AAPRE, APRE, SD, and RMSE of 6.38%, -0.99%, 0.074 psi, and 9.73 psi, respectively, as the first rank model. These values are added in the abstract accordingly.

Why we need to use ANFIS?

It is the first time to use the ANFIS model with the trend analysis to determine the Pb because the ANFIS shows better results as discussed in the advantages of the ANFIS in the subsection (2.2 Proposed ANFIS model strategy). In addition, the trend analysis is used to validate the ANFIS model to follow the correct relationships between the inputs and output to prove the proper physical behavior or to show the effects of the inputs on the output.

Results and discussion are satisfying.

Thank you a lot.

We do not usually cite other papers in the Conclusion section. Please remove them and modify the conclusion accordingly.

Done, the conclusion was modified accordingly.

References

[1] R. Isanta Navarro, “Study of a neural network-based system for stability augmentation of an airplane,” 2013.

[2] P. Tahmasebi and A. Hezarkhani, “A hybrid neural networks-fuzzy logic-genetic algorithm for grade estimation,” Comput. Geosci., vol. 42, pp. 18–27, 2012.

[3] L. P. Maguire, B. Roche, T. M. McGinnity, and L. J. McDaid, “Predicting a chaotic time series using a fuzzy neural network,” Inf. Sci. (NY)., vol. 112, no. 1–4, pp. 125–136, 1998.

Reviewer #2:

Please give explanation why the developed ANFIS gave low or high values of reservoir bubble point pressure when compared against the 22 studied models that follow the correct physical behavior.

The ANFIS model shows better results than other models because of combining the FL and ANN performances and better learning ability. The ANFIS can perform a highly non-linear mapping. These statements are added to the conclusion accordingly. The other advantages of the ANFIS are added in the subsection (2.2 Proposed ANFIS model strategy).

Submitted filename: Response to Reviewers.docx

Click here for additional data file.

27 Jul 2022

A reservoir bubble point pressure prediction model using the Adaptive Neuro-Fuzzy Inference System (ANFIS) technique with trend analysis

PONE-D-22-03484R1

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Ardashir Mohammadzadeh, Phd

Academic Editor

PLOS ONE

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Reviewer #1: All comments have been addressed

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The manuscript must describe a technically sound piece of scientific research with data that supports the conclusions. Experiments must have been conducted rigorously, with appropriate controls, replication, and sample sizes. The conclusions must be drawn appropriately based on the data presented.

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**********

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Reviewer #1: Yes

**********

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The PLOS Data policy requires authors to make all data underlying the findings described in their manuscript fully available without restriction, with rare exception (please refer to the Data Availability Statement in the manuscript PDF file). The data should be provided as part of the manuscript or its supporting information, or deposited to a public repository. For example, in addition to summary statistics, the data points behind means, medians and variance measures should be available. If there are restrictions on publicly sharing data—e.g. participant privacy or use of data from a third party—those must be specified.

Reviewer #1: Yes

**********

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PLOS ONE does not copyedit accepted manuscripts, so the language in submitted articles must be clear, correct, and unambiguous. Any typographical or grammatical errors should be corrected at revision, so please note any specific errors here.

Reviewer #1: Yes

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Reviewer #1: I think the authors addressed my concerns carefully and the manuscript is ready to be published now.

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Reviewer #1: No

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1 Aug 2022

PONE-D-22-03484R1

A reservoir bubble point pressure prediction model using the Adaptive Neuro-Fuzzy Inference System (ANFIS) technique with trend analysis

Dear Dr. Alakbari:

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