Combination of a gamma radiationbased system and the adaptive networkbased fuzzy inference system (ANFIS) for calculating the volume fraction in stratified regime of a threephase flow
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Abstract
Background
Understanding the volume fraction of wateroilgas threephase flow is of significant importance in oil and gas industry.
Purpose
The current research attempts to indicate the ability of adaptive networkbased fuzzy inference system (ANFIS) to forecast the volume fractions in a wateroilgas threephase flow system.
Method
The current investigation devotes to measure the volume fractions in the stratified threephase flow, on the basis of a dualenergy 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 MCNPX 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 wateroilgas threephase flow system.
Keywords
Stratified regime Threephase flow Volume fraction Accuracy Fuzzybased inference system ForecastIntroduction
Water–oil–gas threephase 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 threephase flow. Understanding the volume fraction of water–oil–gas threephase flow is of significant importance to flow measurement as well as to develop oil–gas–water threephase 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 nuclearbased 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 threephase 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 dualenergy gammaray source including \(^{152}\)Eu and \(^{133}\)Ba with energies of 121 and 356 keV in order to estimate volume fraction in gas–oil–water threephase flows without any previous knowledge about the flow regime [2]. In 2010, Salgado et al. utilized a fan beam geometry, consisting of a dualenergy gammaray 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 gammaray 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 gammaray 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 dualenergy broad beam gammaray attenuation technique was applied by Roshani et al. (two transmission 1inch NaI detectors and a also dualenergy gammaray source including \(^{241}\)Am and \(^{137}\)Cs) to forecast the volume fraction of water, oil and gas in threephase 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 dualenergy 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 radiationbased 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 gammaray 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 threephase flows; therefore, we tried to find an approach to determine the volume fraction in a threephase flow with a known flow regime (stratified regime) using just one detector. For this purpose, a dualenergy 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: 1oil and water, 2water and gas, 3gas 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
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 multichannel 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 dualenergy 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.
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 ifthen rules, rules to a set of output characteristics, output characteristics to output membership functions and the output membership function to a singlevalued output or a decision associated with the output.
 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 \(f_{2} = p_{2}x+q_{2}y+r_{2}\)
The reasoning mechanism for the Sugeno model is described in Fig. 3. Moreover, the corresponding equipollent ANFIS structure is presented in Fig. 4.

Layer 1 Every node involved in this layer is considered as an adaptive node with a node function defined as follows:

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:
 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:
 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)
In this regard, ANFIS employs a hybrid algorithm to recognize the parameters during a twostep process. As the first step, the adaptive parameter sets are supposed to be constant and the consequent parameter sets are computed by leastsquares 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 backpropagation 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 leastsquares 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 backpropagation gradient descent algorithm alone or in combination with a leastsquares 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 backpropagation 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
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
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).
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 
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 
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 
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 
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 twoinput/twooutput 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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