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Combination of a gamma radiation-based system and the adaptive network-based fuzzy inference system (ANFIS) for calculating the volume fraction in stratified regime of a three-phase flow

  • G. H. Roshani
  • A. Karami
  • E. Nazemi
Original Paper
  • 167 Downloads

Abstract

Background

Understanding the volume fraction of water-oil-gas three-phase flow is of significant importance in oil and gas industry.

Purpose

The current research attempts to indicate the ability of adaptive network-based fuzzy inference system (ANFIS) to forecast the volume fractions in a water-oil-gas three-phase flow system.

Method

The current investigation devotes to measure the volume fractions in the stratified three-phase flow, on the basis of a dual-energy metering system consisting of the 152Eu and 137Cs and one NaI detector using ANFIS. The summation of volume fractions is equal to 100% and is also a constant, and this is enough for the ANFIS just to forecast two volume fractions. In the paper, three ANFIS models are employed. The first network is applied to forecast the oil and water volume fractions. The next to forecast the water and gas volume fractions, and the last to forecast the gas and oil volume fractions. For the next step, ANFIS networks are trained based on numerical simulation data from MCNP-X code.

Results

The accuracy of the nets is evaluated through the calculation of average testing error. The average errors are then compared. The model in which predictions has the most consistency with the numerical simulation results is selected as the most accurate predictor model. Based on the results, the best ANFIS net forecasts the water and gas volume fractions with the mean error of less than 0.8%.

Conclusion

The proposed methodology indicates that ANFIS can precisely forecast the volume fractions in a water-oil-gas three-phase flow system.

Keywords

Stratified regime Three-phase flow Volume fraction Accuracy Fuzzy-based inference system Forecast 

Introduction

Water–oil–gas three-phase flows are extensively applied in the oil well production and gas–oil transportation. Because of the significant differences of physical properties and mutual interactions among three phases, the flow structures are more complicated compared with gas–liquid or oil–water two phase flow, which brings great difficulties to measure flow parameters in oil–gas–water three-phase flow. Understanding the volume fraction of water–oil–gas three-phase flow is of significant importance to flow measurement as well as to develop oil–gas–water three-phase flow models. In recent years, different methods such as volumetric, optical, electrical, ultrasonic and radiation techniques have been introduced to determine volume fraction in multiphase flows, but there has been an increasing interest in using gamma radiation attenuation technique. Because utilizing gamma radiation technique has some advantages compared to other techniques such as being non intrusive, relatively inexpensive and portable [1]. The nuclear-based meters are known as the gold standard of metering in flow measurement devices. Gamma meters utilize the concept of gamma attenuation in matter where the magnitude of attenuation is directly related to the density (in special range) and atomic number (in special range) of the material through which the gamma ray passes, and to the intensity of the ray itself.

In early studies of oil–gas–water volume fraction measuring in three-phase flow using gamma radiation technique, Salgado et al. used a detection system comprised of three detectors (one of them for registering transmitted photons and other two detectors for registering scattered photons) and one dual-energy gamma-ray source including \(^{152}\)Eu and \(^{133}\)Ba with energies of 121 and 356 keV in order to estimate volume fraction in gas–oil–water three-phase flows without any previous knowledge about the flow regime [2]. In 2010, Salgado et al. utilized a fan beam geometry, consisting of a dual-energy gamma-ray source and also two NaI (Tl) detectors adequately located to measure scattered and transmitted photons [3]. They tried to identify three basic flow regime of annular, stratified and homogenous and also predict volume fraction in water–oil–gas multiphase systems. They used four artificial neural networks (ANNs), the first one for identifying the flow regime and the other three ANNs were applied for forecasting the volume fraction. They implemented total spectrum of gamma-ray registered in detectors as the inputs of all four ANNs. Using this methodology, all three different regimes were correctly identified with the satisfactory prediction of the volume fraction. In 2014, Salgado et al. studied the response of gamma-ray attenuation and scattering in volume fraction estimation of annular and stratified flow regimes of water–gas–oil multiphase flows considering variations in salinity of water [4]. In that study, they used a detection geometry same as their previous work (fan beam geometry and two detectors). Using this approach, they could predict volume fractions in stratified and annular flow regimes under salinity of water component variations up to 16.5% with error of less than 10%. In 2016, a dual-energy broad beam gamma-ray attenuation technique was applied by Roshani et al. (two transmission 1-inch NaI detectors and a also dual-energy gamma-ray source including \(^{241}\)Am and \(^{137}\)Cs) to forecast the volume fraction of water, oil and gas in three-phase flows independent of the flow regime [5]. A multilayer perceptron (MLP) neural network was applied to develop the ANN model in MATLAB 8.1.0.604 software. They used registered count under full energy peaks of \(^{241}\)Am and \(^{137}\)Cs in both detectors as the inputs and volume fraction of gas and oil as the output of ANN. The volume fractions were obtained precisely independent of flow regime with mean absolute error (MAE) of less than 2.24%. In 2017, Roshani et al. utilized one dual-energy source and two transmitted detectors, in which, the associated orientations had been optimized [6]. Four ANNs were employed in the study, the first one for identifying the flow regime and the other three ANNs were for forecasting the volume fraction. For first ANN, they used just two inputs (count under full energy peak of \(^{137}\)Cs in both detectors), but for other three ANNs they used four inputs (count under full energy peaks of \(^{241}\)Am and \(^{137}\)Cs of both detectors). Applying this method, all flow regimes were recognized with 100% accuracy and also the volume fraction percentages were estimated with MAE of less than 2.59. More investigations on the gamma radiation-based multiphase flow meters and also some applications of ANN related to nuclear engineering problems can be found elsewhere [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31].

The most part of industrial gamma radiation gauge’s price is related to gamma-ray detectors, and consequently, by reducing the number of gamma detectors, the economical expenses will be also decreased. By reviewing the previous studies (some of them mentioned above), we understood that in all of them more than two detectors have been at least implemented for determining the volume fraction of three-phase flows; therefore, we tried to find an approach to determine the volume fraction in a three-phase flow with a known flow regime (stratified regime) using just one detector. For this purpose, a dual-energy metering system including the \(^{152}\)Eu and \(^{137}\)Cs and one NaI detector combined with ANFIS were utilized. In the research, because the summation of volume fractions is constant (equal to 100%), a modeling problem of constrained type exists. Therefore, ANFIS just forecast two volume fractions and the other one is simply obtained from the constraint relation. Since in this way three different networks can be considered (the network output is: 1-oil and water, 2-water and gas, 3-gas and oil volume fractions), at first all of possible networks should be trained and compared to each other. Finally, the network in which predictions have the most agreement with the numerical simulation results, is characterized as the best forecast model.

Methodology

Generating required data

In order to train and test the network, Monte Carlo N-Particle eXtended (MCNPX) code was utilized in this research [32]. The simulation geometry is comprised of one dual-energy source including \(^{152}\)Eu and \(^{137}\)Cs with energies of 121 and 662 keV, one 7.62 cm (3 inch) sodium iodide (NaI) scintillator detector and one polyethylene pipe with outer diameter of 10.16 cm (4 inch). The polyethylene pipe was placed between source and detector. The simulation geometry is shown in Fig. 1. It should be noted that, the simulation geometry in this paper was validated in one of our previous studies in which some experiments were done and compared with the simulation results and there was a good agreement between them [7].
Fig. 1

a The schematic view of simulated geometry, b interaction of gamma ray with material

Hydrocarbon (with molecular formula of \(\hbox {C}_{5}\hbox {H}_{10})\), usual water and air with densities of 0.896, 1 and 0.00125 g/cm\(^{3}\) were considered and defined as the oil, water and gas phases, respectively. Various volume fractions were acquired by changing the thickness’s value of each material (oil, water and air) in the input file of MCNPX code. Totally 33 various combinations of volume fractions were simulated. In each simulation, counts under full energy peaks of \(^{152}\)Eu and \(^{137}\)Cs were extracted using Pulse Height Tally F8. The output tally was divided to 35 sections; consequently, 35 channels were considered as multi-channel analyzer. The results of the numerical simulation are shown in Table 1.

Registered counts in the transmission detector were calculated as per one source particle in the MCNPX code using Pulse Height Tally F8. The STOP card was used to terminate calculations when a desired tally precision was reached. In all simulations, a maximum 0.005 relative error was set using the STOP card; therefore, all of the Monte Carlo results meet this standard of precision. A precise description of the benchmark simulation is provided in [7]. The maximum difference between experimental and simulated results was 3.8% [7]. In the present study, the previously validated simulation was repeated in multiphase with a dual-energy source. To account for both photoelectric and Compton interactions, 152Eu (121 KeV) and 137Cs (662 KeV) sources were used. Photoelectric interactions, with a strong dependency on the atomic number, are dominant at low energies, while Compton interactions, which are proportional to the density, dominate at higher energies. Based on the validated simulated dual modality densitometry setup for detector output, other volume fractions with different densities could easily be simulated by the MCNPX code, thus avoiding difficulties related to experimental conditions. A typical output signal with two extracted features is shown in Fig. 2.

As mentioned previously, MCNP code results are presented per one source particle and this is the reason of low values of Table 1. The MCNP code is a static code, and there is no definition for measurement time in simulations. The full energy peaks of \(^{152}\)Eu and \(^{137}\)Cs mention to counts of gamma rays which deposited all of their energy in the crystal of detector.
Table 1

Validated simulation results for different volume fractions of oil, water gas using MCNPX code

No.

Gas percentage

Oil percentage

Water percentage

Full energy peak of \(^{152}\)Eu

Full energy peak of \(^{137}\)Cs

1,TR

0

0

100

5.97E\(-\) 02

8.18E\(-\) 02

2,TR

20

0

80

9.78E\(-\) 02

1.08E\(-\) 01

3,TR

40

0

60

1.38E\(-\) 01

1.32E\(-\) 01

4,TR

60

0

40

1.88E\(-\) 01

1.59E\(-\) 01

5,TR

80

0

20

2.60E\(-\) 01

1.92E\(-\) 01

6,TR

100

0

0

4.10E\(-\) 01

2.56E\(-\) 01

7,TS

10

10

80

8.21E\(-\) 02

9.74E\(-\) 02

8,TS

30

10

60

1.19E\(-\) 01

1.20E\(-\) 01

9,CH

50

10

40

1.63E\(-\) 01

1.45E\(-\) 01

10,TS

70

10

20

2.23E\(-\) 01

1.75E\(-\) 01

11,TR

90

10

0

3.19E\(-\) 01

2.18E\(-\) 01

12,TS

10

20

70

8.32E\(-\) 02

9.81E\(-\) 02

13,CH

30

20

50

1.20E\(-\) 01

1.21E\(-\) 01

14,CH

50

20

30

1.65E\(-\) 01

1.47E\(-\) 01

15,TS

70

20

10

2.26E\(-\) 01

1.77E\(-\) 01

16,TR

0

30

70

6.27E\(-\) 02

8.37E\(-\) 02

17,CH

20

30

50

1.02E\(-\) 01

1.10E\(-\) 01

18,TR

40

30

30

1.42E\(-\) 01

1.35E\(-\) 01

19,TR

60

30

10

1.95E\(-\) 01

1.62E\(-\) 01

20,TR

0

40

60

6.33E\(-\) 02

8.39E\(-\) 02

21,TR

20

40

40

1.03E\(-\) 01

1.11E\(-\) 01

22,CH

40

40

20

1.44E\(-\) 01

1.35E\(-\) 01

23,TR

60

40

0

1.99E\(-\) 01

1.64E\(-\) 01

24,TS

10

50

40

8.58E\(-\) 02

1.00E\(-\) 01

25,CH

30

50

20

1.25E\(-\) 01

1.24E\(-\) 01

26,TR

50

50

0

1.72E\(-\) 01

1.50E\(-\) 01

27,TS

10

60

30

8.68E\(-\) 02

1.01E\(-\) 01

28,TS

30

60

10

1.25E\(-\) 01

1.25E\(-\) 01

29,TR

0

70

30

6.60E\(-\) 02

8.63E\(-\) 02

30,TS

20

70

10

1.07E\(-\) 01

1.13E\(-\) 01

31,TR

0

80

20

6.59E\(-\) 02

8.61E\(-\) 02

32,TR

20

80

0

1.09E\(-\) 01

1.14E\(-\) 01

33,TR

10

90

0

9.10E\(-\) 02

1.02E\(-\) 01

Modeling approach: ANFIS network

ANFIS is regarded as a fuzzy inference system implemented in the mold of neural networks [33, 34, 35, 36, 37]. In fact, ANFIS is a hybrid model which provides the capabilities of neural network topology along with fuzzy logic. The ANFIS involves all the components of a simple fuzzy system except that the computations at each step are performed by a layer including hidden neurons and also the neural network’s training capacity is provided as an special advantage to enhance the system knowledge. The ANFIS attempts to find a model which can model the inputs with the outputs accurately. The ANFIS is utilized to map out, input characteristics to input membership functions, input membership function to a set of fuzzy if-then rules, rules to a set of output characteristics, output characteristics to output membership functions and the output membership function to a single-valued output or a decision associated with the output.

For simplicity, suppose that the fuzzy inference system has two inputs of (xy) and only one output (f) [38, 39, 40, 41, 42]. For the first-order Sugeno model, a single fuzzy if-then rule assumes the form
Rule1:

if x is \(A_{1}\)and y is \(B_{1}\), then \(f_{1} = p_{1}x+q_{1}y+r_{1}\)

Rule2:

if x is \(A_{2}\)and y is \(B_{2}\), then \(f_{2} = p_{2}x+q_{2}y+r_{2}\)

where \({A}_{i}, {B}_{i}\) and \({f}_{i}\) indicate fuzzy sets and system’s output, respectively. Also \({p}_{i}, {q}_{i }\)and \({r}_{i}\) represent the designing parameters which are obtained during the training process for i, \(j=1,2\).

The reasoning mechanism for the Sugeno model is described in Fig. 3. Moreover, the corresponding equipollent ANFIS structure is presented in Fig. 4.

The layers involved in Fig. 4 are expressed by the following relations:
  • Layer 1 Every node involved in this layer is considered as an adaptive node with a node function defined as follows:

$$\begin{aligned} {O_{1,i}}= & {} {\mu _A}_{i} (x),\quad i=1,2, \end{aligned}$$
(1)
$$\begin{aligned} {O_{1,i}}= & {} {\mu _B}_{i-2} (y) , \quad i=3,4 \end{aligned}$$
(2)
where i represents the membership grade of a fuzzy set (\({A}_{1}, {A}_{2}, {B}_{1}, {B}_{2})\) and also \(O_{1,i}\) indicates the output of the node i in the layer 1.
  • Layer 2 Every node in this layer is regarded as a fixed node labeled as\(\Pi \), the nodes multiply all incoming signals and the outputs are obtains as follows:

$$\begin{aligned} O_{2,i} =w_i ={\mu _A}_{i} (x){\mu _B}_{i} (y) ,\quad i=1,2 \end{aligned}$$
(3)
Each node in this layer indicates the firing strength of a rule.
  • Layer 3 Every node in this layer is a fixed node labeled as N which gives the ratio of the ith rule’s firing strength to the sum of all rule’s firing strengths. The output of each node in this layer is called as normalized firing strength and is obtained as follows:
    $$\begin{aligned} O_{3,i} =\overline{w_i } =\frac{w_i }{w_1 +w_2 },\quad i=1,2 \end{aligned}$$
    (4)
  • Layer 4 In this layer, all nodes are adaptive nodes with a node function defined as follows:

$$\begin{aligned} O_{4,i} =\overline{w_i } f_i =\overline{w_i }({p_i} x+{q_i} y+{r_i} ),\quad i=1,2 \end{aligned}$$
(5)
where \(\overline{w_i } \) represents a normalized firing strength from layer 3 and the set of \(\left\{ {p_i ,q_i ,r_i } \right\} \) stand for the modifiable parameter set, called as consequent parameters.
Fig. 2

A typical output signal with two extracted features

Fig. 3

The inference system based on the Sugeno model

  • Layer 5 This layer is a fixed node labeled \(\sum , \)calculating the overall output as the summation of all incoming signals:
    $$\begin{aligned} O_{5,i} =\sum \limits _{i} {\overline{w_i } } f_i =\frac{\sum \nolimits _{i} {w_i f_i } }{\sum \nolimits _{i} {w_i } },\quad i=1,2 \end{aligned}$$
    (6)
Similar to ANNs, an ANFIS is trained on the basis of the supervised learning to reach from a specified input to a specific target output. In the training process, at first, the parameters need to be learned. In the ANFIS structure, the parameters sets are categorized into two groups, one called as adaptive parameters and the other as consequent ones. If the modeling is performed properly, using both sets of parameters, then the difference between the forecasted outputs and available ones should give the lowest possible level of error.
Fig. 4

The ANFIS structure on the basis of the Takagi-Sugeno Model

Fig. 5

A representation of the suggested ANFIS model to forecast the oil and water volume fractions

Fig. 6

A representation of the suggested ANFIS model to forecast the water and gas volume fractions

In this regard, ANFIS employs a hybrid algorithm to recognize the parameters during a two-step process. As the first step, the adaptive parameter sets are supposed to be constant and the consequent parameter sets are computed by least-squares technique. The aforementioned process is called as forward pass. As a matter of fact, the process is used to optimize the consequent parameters with the premise nonlinear parameters fixed. As the optimal consequent parameters are computed, the second process which is called as backward pass begins. The process is used to optimally adjust the premise parameters through a back-propagation gradient descent method, corresponding to the fuzzy sets in the input domain. In this step, the consequent parameters are supposed to be constant and the adaptive ones set are computed by gradient descends method (GDM). In fact, the hybrid algorithm combines the gradient descent method and the least-squares method to train parameters. Whenever the parameters sets of the model are acquired, the values of the model output are computed for each regular pair of training data and compared with the values that have been predicted by the model. It has already been demonstrated that the suggested hybrid algorithm is highly efficient in training the ANFIS. Briefly, it can be expressed that, in the ANFIS structure, the optimal distribution of membership functions is established using either a back-propagation gradient descent algorithm alone or in combination with a least-squares method, via the training process. In the last step, the output of the ANFIS is calculated, based on the obtained consequent parameters in the forward pass. The output error is utilized to adapt the premise parameters via a standard back-propagation algorithm. In better understanding, ANFIS is regarded as a flexible mathematical structure that is the best choice to be applied for modeling and simulation in various engineering fields [43, 44, 45, 46, 47, 48].

Volume fraction determination

In order to determine the volume fractions, three different models are presented. The inputs of the models are full energy peak of \(^{152}\)Eu and full energy peak of \(^{137}\)Cs. In the current study, since the summation of volume fractions is constant (equal to 100%), a modeling problem of constrained type is considered, meaning that ANFIS just forecast two volume fractions and the other one is simply obtained from the constraint relation. The first model is utilized to forecast the oil and water volume fractions. The other to forecast the water and gas volume fractions, and the last one to forecast the gas and oil volume fractions. Indeed, three two-input/two-output models are employed in the investigation. The schematic overview of the proposed ANFIS models is shown in Figs. 56 and 7.
Fig. 7

A representation of the suggested ANFIS model to forecast the gas and oil volume fractions

In order to build up the suggested ANFIS models for the stratified regime, 33 data were utilized. Whole data are categorized into three sets: training, testing and checking. About 55% of the whole data (18 data) were considered for training, 27% for testing (9 data) and the rest 18% (6 data) were selected as checking data for the evaluation of the proposed ANFIS models. The training and testing sets are chosen stochastically. The best architecture of suggested ANFIS models for forecasting the volume fractions is described in Table 2.
Table 2

The best structure and characteristics of the suggested ANFIS model for forecasting the volume fractions

No

Fuzzy inference system (FIS) type

Sugeno

1

Inputs/output

Oil and water

2/2

Water and gas

2/2

Gas and oil

2/2

2

Input membership function types

Oil and water

Gaussian

Water and gas

Gaussian

Gas and oil

Gaussian

3

Number of input membership functions

Oil and water

6/6

Water and gas

6/6

Gas and oil

6/6

4

Number of output membership functions

Oil and water

36

Water and gas

36

Gas and oil

36

5

Number of fuzzy rules

Oil and water

36

Water and gas

36

Gas and oil

36

6

Number of linear parameters

Oil and water

40

Water and gas

41

Gas and oil

38

7

Number of nonlinear parameters

Oil and water

22

Water and gas

21

Gas and oil

20

8

Number of epochs (iterations)

Oil and water

290

Water and gas

300

Gas and oil

295

Results and discussion

A comparison between the forecasted values using the ANFIS model and validated simulation results obtained from MCNPX code for training, testing and checking data in the stratified regime is shown in Fig. 8. As can be observed, the forecasted values using the suggested ANFIS model are in good consistency with the numerically obtained data with most accuracy. Similar comparisons can be found in Figs. 9 and 10 for two other ANFIS models.
Fig. 8

Comparisons between the numerical simulation results and the ANFIS forecasted values for the a training set of oil and water outputs, b testing set of oil and water outputs and c checking set of oil and water outputs

Fig. 9

Comparisons between the numerical simulation results and the ANFIS forecasted values for the a training set of water and gas outputs, b testing set of water and gas outputs and c checking set of water and gas outputs

Fig. 10

Comparisons between the numerical simulation results and the ANFIS forecasted values for the a training set of gas and oil outputs, b testing set of gas and oil outputs and c checking set of gas and oil outputs

Generally, the forecast performances of the suggested ANFIS models are appraised through the mean relative error (MRE), sum-squared errors (SSE), the correlation coefficients (\(R^{2}\)%) root-mean-square error (RMSE), and standard deviation (STD) values, which are obtained as follows:
$$\begin{aligned} \mathrm{SSE}= & {} {{\sum \limits _i}^{n}} {(Y_{\mathrm{Num},i} -Y_{\mathrm{Pred},i})^{2}} \end{aligned}$$
(7)
$$\begin{aligned} R^{2}= & {} 1-\frac{\hbox {SSE}}{\sum \nolimits _{i=1}^N {Y_{\mathrm{Num},i}^{2}}} \end{aligned}$$
(8)
$$\begin{aligned} \mathrm{MRE}\%= & {} \frac{1}{N}\sum \limits _{i=1}^N {\left| {\frac{Y_{\mathrm{Num},i} -Y_{\mathrm{Pred},i} }{Y_{\mathrm{Num},i} }} \right| } \times 100 \end{aligned}$$
(9)
$$\begin{aligned} \mathrm{RMSE}= & {} \sqrt{\frac{\sum \nolimits _i^N {(Y_{\mathrm{Num},i} -Y_{\mathrm{Pred},i} )^{2}} }{N}} \end{aligned}$$
(10)
$$\begin{aligned} \mathrm{STD}= & {} \sqrt{\frac{\sum \nolimits _i^N {(Y_{\mathrm{Num},i} -Y_{\mathrm{Pred},i} )^{2}} }{N-1}} \end{aligned}$$
(11)
where N stands for the number of data and ‘\(Y_{\mathrm{Num},i} \)’ and ‘\(Y_{\mathrm{Forec},i} \)’ are numerical results and forecasted values, respectively.

The error information of the proposed ANFIS models in forecasting the volume fractions is given in Tables 3, 4 and 5.

According to Table 3, in the stratified regime and training set, MRE% and SSE for forecasting the oil output are 0.4073% and 0.3375, respectively, and \(0.1675\%\) and 0.0425 for the water output. Also, for the testing set, these errors are 0.9338% and 1.2125, respectively, for oil output, and 0.9330% and 2.4525 for the water output. For the checking set, these errors are \(1.6527\%\) and 1.3325, respectively, for oil output, and 0.775% and 1.1536 for the water output.

Moreover, based on the results presented in Table 4, in the stratified regime and training set, MRE% and SSE for forecasting the water output are 0.1675% and 0.0425, respectively, and 0.1965% and 0.27 for the gas output. Also, for the testing set, these errors are 0.9330% and 2.4525, respectively, for water output, and 0.6455% and 0.56 for the gas output. For the checking set, these errors are 0.775% and 1.1536, respectively, for water output, and 1.28% and 1.27 for the gas output.

Furthermore, according to Table 5, in the training set, MRE% and SSE for forecasting the gas output are 0.1965% and 0.27, respectively, and 0.4073% and 0.3375 for the oil output. Also, for the testing set, these errors are 0.6455% and 0.56, respectively, for gas output, and 0.9338% and 1.2125 for the oil output. For the checking set, these errors are 1.6527% and 1.3325, respectively, for gas output, and 1.28% and 1.27 for the oil output.

For each model, the average MRE% is obtained as the arithmetic mean error value of the model in forecasting the two corresponding volume fractions associated with each data set (testing or checking).

According to the results presented in Table 6, the average MRE% of the model which forecasts the oil and water outputs is less as compared to two other networks which forecast the corresponding volume fractions. Therefore, the model which forecasts the water and gas outputs is characterized as the best model in the investigation.
Table 3

The accuracy of the suggested ANFIS model in comparison with the numerical simulation results, for the set of oil and water outputs

No

Output

Error bounds

MRE%

SSE

\(R^{2}\)%

RMSE

STD

1

Oil

Mean

Training

0.4073

0.3375

99.9989

0.1448

0.1490

Testing

0.9338

1.2125

99.9922

0.3670

0.3893

Checking

1.6527

1.3325

99.9774

0.4712

0.5162

2

Water

Mean

Training

0.1675

0.0425

99.9998

0.0486

0.0500

Testing

0.9330

2.4525

99.9933

0.5220

0.5537

Checking

0.775

1.1536

99.9839

0.4384

0.4803

Table 4

The accuracy of the suggested ANFIS model in comparison with the numerical simulation results, for the set of water and gas outputs

No

Output

Error bounds

MRE%

SSE

\(R^{2}\)%

RMSE

STD

1

Water

Mean

Training

0.1675

0.0425

99.9998

0.0486

0.0500

Testing

0.9330

2.4525

99.9933

0.5220

0.5537

Checking

0.775

1.1536

99.9839

0.4384

0.4803

2

Gas

Mean

Training

0.1965

0.27

99.9993

0.1224

0.1260

Testing

0.6455

0.56

99.9954

0.2494

0.2645

Checking

1.2800

1.27

99.9855

0.4600

0.5039

Table 5

The accuracy of the suggested ANFIS model in comparison with the numerical simulation results, for the set of gas and oil outputs

No

Output

Error bounds

MRE%

SSE

\(R^{2}\)%

RMSE

STD

1

Gas

Mean

Training

0.1965

0.27

99.9993

0.1224

0.1260

Testing

0.6455

0.56

99.9954

0.2494

0.2645

Checking

1.2800

1.27

99.9855

0.4600

0.5039

2

Oil

Mean

Training

0.4073

0.3375

99.9989

0.1448

0.1490

Testing

0.9338

1.2125

99.9922

0.3670

0.3893

Checking

1.6527

1.3325

99.9774

0.4712

0.5162

Table 6

The accuracy of the suggested ANFIS model in comparison with the numerical simulation results

No

Data type

Model

Avergae MRE%

1

Training

ANFIS model with the oil and water outputs

0.2874

ANFIS model with the water and gas outputs

0.182

ANFIS model with the gas and oil outputs

0.3019

2

Testing

ANFIS model with the oil and water outputs

0.9334

ANFIS model with the water and gas oil outputs

0.7892

ANFIS model with the gas and oil outputs

0.7896

3

Checking

ANFIS model with the oil and water outputs

1.2138

ANFIS model with the water and gas oil outputs

1.0275

ANFIS model with the gas and oil outputs

1.4663

Moreover, the aforementioned ANFIS models have been trained and tested by triangular membership functions (Trimfs) and generalized bell membership functions (Gbellmfs) along with Gaussian membership functions. Figures 8, 9 and 10 show the results. As can be observed, for testing data, utilizing Gaussian membership functions (Gaussmfs), leads to better results due to lower error values. For example, utilizing Gaussian membership functions, decreases 11% of the MRE in comparison with similar one by triangular memberships and 22% decrease by generalized bell memberships, for forecasting the set of gas and water outputs, based on the testing data. Generally, using Gaussian membership functions, helps ANFIS models to forecast the gas and oil outputs with more accuracy (Fig. 11). The characteristics of the compared ANFIS models including triangular memberships and generalized bell membership functions are applied, and are presented in Table 7.
Fig. 11

The MRE% of the ANFIS model versus the membership functions type the set of water and gas outputs

Table 7

The best structure and characteristics of the compared ANFIS models for forecasting the oil and water outputs

No

Fuzzy inference system (FIS) type

Sugeno

1

Inputs/output

First compared model

2/2

Second compared model

2/2

2

Input membership function types

First compared model

Triangular

Second compared model

Generalized bell

3

Number of input membership functions

First compared model

4/4

Second compared model

4/4

4

Number of output membership functions

First compared model

16

Second compared model

16

5

Number of fuzzy rules

First compared model

16

Second compared model

16

6

Number of linear parameters

First compared model

37

Second compared model

39

7

Number of nonlinear parameters

First compared model

19

Second compared model

22

8

Number of epochs (iterations)

First compared model

280

Second compared model

300

Concluding remarks

In this study, the ANFIS was employed in order to model the volume fractions in a water–oil–gas multiphase system versus the input parameters including the full energy peak of \(^{152}\)Eu and full energy peak of \(^{137}\)Cs. The main idea of this paper was that three accurate and precise ANFIS models to be extended for modeling the output parameters as a function of input parameters. The first model was employed to forecast the oil and water outputs. The next to forecast the water and gas outputs, and the last one to forecast the oil and water outputs. Indeed, three two-input/two-output models were utilized in the investigation. The aforementioned networks were developed on the basis of the validated numerical simulation data from MCNPX code. It was concluded that, the ANFIS model which forecasts the water and gas outputs is the best model due to its low error in as compared to the other two models. Furthermore, the results revealed that, the best model in which the Gaussian membership functions are applied, provides better results than those in which other two types of membership functions are employed.

The main advantage of this work prior to our previous published works [5, 6] is using just one detector for determining the volume fraction instead of two detectors or more. In addition, the obtained error in this study which is 0.8%, is less than obtained errors in references [5, 6] that are 2.24 and 1.63%, respectively. Therefore, it can be said that the proposed algorithm in this study is more efficient and robust in relation to methods used in our previous published works.

Notes

Compliance with ethical standards

Conflict of interest

The authors have no conflict of interest.

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Copyright information

© Institute of High Energy Physics, Chinese Academy of Sciences; Nuclear Electronics and Nuclear Detection Society and Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  1. 1.Electrical Engineering DepartmentKermanshah University of TechnologyKermanshahIran
  2. 2.Mechanical Engineering DepartmentRazi UniversityKermanshahIran
  3. 3.Nuclear Science and Technology Research Institute (NSTRI)TehranIran

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