A Methodology for Analysis and Prediction of Volume Fraction of Two-Phase Flow Using Particle Swarm Optimization and Group Method of Data Handling Neural Network

Abstract

Determining the volume percentages of flows passing through the oil transmission lines is one of the most essential problems in the oil, gas, and petrochemical industries. This article proposes a detecting system made of a Pyrex-glass pipe between an X-ray tube and a NaI detector to record the photons. This geometry was modeled using the MCNP version X algorithm. Three liquid-gas two-phase flow regimes named annular, homogeneous, and stratified were simulated in percentages ranging from 5 to 95%. Five time characteristics, three frequency characteristics, and five wavelet characteristics were extracted from the signals obtained from the simulation. X-ray radiation-based two-phase flowmeters’ accuracy has been improved by PSO to choose the best case among thirteen characteristics. The proposed feature selection method introduced seven features as the best combination. The void fraction inside the pipe could be predicted using the GMDH neural network, with the given characteristics as inputs to the network. The novel aspect of the current study is the application of a PSO-based feature selection method to calculate volume percentages, which yields outcomes such as the following: (1) presenting seven suitable time, frequency, and wavelet characteristics for calculating volume percentages; (2) the presented method accurately predicted the volume fraction of the two-phase flow components with RMSE and MSE of less than 0.30 and 0.09, respectively; (3) dramatically reducing the amount of calculations applied to the detection system. This research shows that the simultaneous use of time, frequency, and wavelet characteristics, as well as the use of the PSO method as a feature selection system, can significantly help to improve the accuracy of the detection system.

1. Introduction

Determining volume percentages of flows traveling through oil transmission lines is one of the most important problems in the oil, gas, and petrochemical industries. Therefore, there has been much study in this area. Photon attenuation methods, capacitive tomography, resistance tomography, X-rays, and so on are only some of the numerous non-invasive techniques available for determining this parameter. The photon attenuation approach has been studied extensively for the detection of these factors in recent years [1,2,3,4,5,6,7,8,9,10,11,12,13]. In [1], researchers used the MCNP algorithm to examine three different flow regimes—stratified, homogeneous, and annular—across a range of volume fractions. This investigation utilized two radioisotope sources and three NaI detectors to calculate volume fractions and recognize various flow regimes. To identify different flow regimes, a neural network was developed. Three different networks were developed to calculate the volume fractions of oil–water–gas three-phase flow. Similarly, another study presented a different methodology for calculating volume fractions in a stratified regime. An energy spectrum of the backscattered gamma radiation was recorded by placing a NaI detector near a cesium-137 source. MLP neural network was also utilized to accurately estimate volume fractions [2]. The extraction of features from detector signals has garnered much interest from scientists in recent years. To introduce the most suitable time-domain features, Sattari et al., for instance, conducted a study [3]. The percentages of volume and the types of flow regimes were determined using a measurement equipment consisting of a cesium-137 source, two NaI detectors, and a Pyrex glass. After collecting the data, several characteristics in the time domain were extracted for better interpretation. Then, the MLP neural network was used to accurately identify the flow regime types and provide accurate predictions of the volume fractions. Time-domain features and the GMDH neural network were utilized to calculate volume fractions and flow regime types in a different study [4]. Roshani et al., conducted their research on three-phase flows using the dual energy approach. A pair of 241 Am and 137 Cs sources and two NaI detectors made up their detecting system. They trained the neural network using the features of recorded counts of 241 Am and 137 Cs in two detectors, and it was able to make predictions about volume percentages with an MRE% of less than 5.68% [5]. Several characteristics in the time [6] and frequency [7] domains were presented by Hanus et al. to identify the kind of flow regimes in two-phase flows, and the most effective features were then determined. Using these features and other kinds of neural networks, they attempted to classify the different flow regimes in their subsequent study [8]. Time-domain features were obtained from frequency domain data using the FFT by Hosseini et al. With these features, they accurately predicted volume percentages in two-phase flows and detected all flow regimes [9].

Many researchers have shifted in recent years to using an X-ray source instead of gamma in their structures to avoid issues such as requiring personnel to wear protective clothing when working with this device (due to its inability to be turned off). X-ray tubes were used to distinguish multiphase flow characteristics [10,13]. In [10], for instance, the aforementioned complementary source was used. With the use of an X-ray tube and a sodium iodide detector, the researchers were able to determine the regime type and volume fraction of two-phase flows. Temporal features extracted from the signals received by the detector were also used to train two MLP neural networks in their proposed structure. The study [11] looked at three-phase flows and simulated them in three different regimes (annular, stratified, and homogeneous) at varying volumes. Three RBF neural networks were developed in this study and trained using the frequency characteristics of the received signals, yielding satisfactory results. X-ray tubes were used to model a control system in [12]. The MCNP code was used to model the interactions between four petroleum products when they were combined in pairs at different volumes. Three MLP neural networks were fed the recorded signals to make predictions about the volume distribution ratio of the three products. Researchers claimed that knowing the volume ratios of the first three products would allow them to compute the volume ratio of the fourth product. While the introduced method could make predictions regarding the product types and amounts, the lack of feature extraction techniques limited the accuracy of those predictions. In order to build upon earlier studies [12], wavelet transform was used for feature extraction by Balubaid et al. in [13], which effectively decreased computational load and improved accuracy. The key flaw in the majority of the aforementioned studies is the absence of feature extraction and feature selection methods that could be employed to artificial neural networks. This study makes an effort to propose a technique for extracting time-domain, wavelet, and frequency-domain features and choosing the most useful characteristic using the PSO algorithm. Figure 1 illustrates the recommended technique. The main contributions of current study are summarized here.

• Extraction of time, wavelet, and frequency features for investigating the two-phase fluid;
• Using a feature-selection system based on the PSO algorithm to introduce useful characteristics;
• A significant boost in volume percentage calculations’ accuracy;
• Selecting the optimal characteristics to utilize as the neural network’s inputs will minimize the volume of calculations that must be done on the system.

2. Proposed Methodology

2.1. Detection System

In this study, we use the MCNP algorithm to simulate the detection system, which consists of an X-ray tube and a NaI detector on opposite ends of a Pyrex-glass pipe. Instruments that use ionizing radiation as a measuring standard have been modeled using this code [14,15,16,17,18]. The basic geometry of the planned setup is shown in Figure 2. When analyzing two-phase flow, photons from an X-ray tube are sent to a pipe and, after being somewhat attenuated by the pipe walls and contents, are picked up by a detector. The amount of gas and liquid within the pipe accounts for the attenuation of the radiation beam. This study simulates three different flow patterns (shown in Figure 3) and nineteen various volume fractions, ranging from a 5% void fraction to a 95% volume fraction with a 5% void fraction step (57 simulations were used in total). It is worth noting that prior research has corroborated the results of this investigation’s simulations [19]. In this study, several laboratory structures were constructed and compared with the data acquired from the MCNP code. To facilitate a direct comparison between actual and simulated findings, the Tally output from the MCNP algorithm was converted to units per source particle. The difference between the results of the simulation and the experimental setup represented the biggest relative error, at 2.2%.

This study established a more efficient architecture that involves a photon source positioned within a metal shield because accurate modeling of an industrial X-ray tube using the MCNPX code, which includes a cathode (electron source) and an anode (tungsten target) contained in a cylindrical shield, is time demanding. That is to say, instead of simulating the cathode–anode development, a photon source placed in a metal shield was considered in the current study since photon tracking in the MCNPX algorithm is significantly quicker than electron tracking. TASMIC, a free piece of the software shown by Hernandez et al. [20] was used to acquire the X-ray energy spectrum that was then used to characterize the photon source. The used X-ray spectrum is shown in Figure 4, with the X-ray peaks that are characteristic of the tungsten anode. Radiation shields for X-ray tubes, essentially hollow cylinders, are often composed of steel or lead. This output window is a hole in the shield’s surface that allows for the release of X-ray photons that have been generated. In this investigation, the simulated X-output ray’s window has a 5 cm radius. An aluminum filter of thickness 2.5 mm was placed in front of the output window to filter the low-energy photons with the purpose of decreasing scattering.

2.2. Feature Extraction

The term “feature extraction” describes converting unstructured data into a collection of quantifiable characteristics that may be further processed without losing any of the original data’s context. It produces superior outcomes compared to using machine learning on the raw data directly. To extract features manually, one must first determine which characteristics are essential for a specific situation, and then devise a strategy to extract them. Knowing the context or domain is often helpful when determining which features to implement. Optimization methods can also be used to select the best features. In this research, inspired by the previous research [3,4,10,11,13], the received signals in three domains of time, frequency, and wavelet transformation were investigated.

2.2.1. Time-Domain Feature Extraction

Figure 5 shows the signals received from the NaI detector for three flow regimes: annular, homogeneous, and stratified in volume percentages from 5 to 95%. From these signals, five time characteristics have been extracted with the following formulas:

• skewness:

$$g_1=\frac{m_3}{{\sigma }^3} , m_3=\frac{1}{N}{{\displaystyle \sum }}_{n=1}^N{\left[x_n-m\right]}^{3}m=\frac{1}{N}{{\displaystyle \sum }}_{n=1}^{N}x\left(n\right) \sigma =\sqrt{\frac{1}{N}{{\displaystyle \sum }}_{n=1}^N{\left(x_n-m\right)}^2}$$

(1)

• kurtosis:

$$g_2=\frac{m_4}{{\sigma }^4} m_4=\frac{1}{N}{\displaystyle \sum }_{n=1}^N{\left[x\left(n\right)-m\right]}^4$$

(2)

• WL:

$$WL={{\displaystyle \sum }}_{n=0}^{N-1}\left|x_{n+1}-x_n\right|$$

(3)

• ASS:

$$ASS=\left|{{\displaystyle \sum }}_{n=1}^N{\left(x_n\right)}^{0.5}\right|$$

(4)

• MSR:

$$MSR=\frac{1}{N}{{\displaystyle \sum }}_{n=1}^N{\left(x_n\right)}^{0.5}$$

(5)

2.2.2. Frequency-Domain Feature Extraction

To extract frequency features, the FTT (Equation (1) [21]) was employed to transform the received signal into the frequency domain. The transmitted signals in the frequency domain for three regimes, annular, homogeneous, and stratified, are shown in Figure 6. AFDF, ASDF, and ATDF were extracted from frequency-domain signals.

$$Y\left(k\right)={\displaystyle \sum }_{J=1}^{n}x\left(J\right)w_n^{\left(y-1\right)\left(k-1\right)}$$

(6)

where Y(k) = FFT(X), and $w_n={\mathrm{e}}^{\left(-2\pi \mathrm{i}\right)∕n}$ is one of n roots of unity.

The frequency domain signals for the three flow regimes are shown in Figure 6.

2.2.3. Wavelet Transform

DWT is helpful for studying data with characteristics that change at various sizes. Signal characteristics might be fluctuating frequencies, brief fluctuations, or long-term tendencies. The primary motivation for developing wavelet transforms was to overcome limitations in the Fourier transform. Signals are broken down into scaled and shifted copies of wavelets in wavelet analysis instead of sine waves in Fourier analysis. In contrast to a sine wave, a wavelet is an oscillation that decomposes quickly. The ability to represent data at different scales makes wavelets so valuable. Depending on the application, different wavelets might be utilized. The DWT is computed by first applying a low-pass filter with impulse response g to the signal, leading to the convolution of the two as shown in [22,23]:

$$y\left[n\right]=\left(x\ast g\right)\left[n\right]={{\displaystyle \sum }}_{k=-\infty }^{\infty }x\left[k\right]g\left[n-k\right]$$

(7)

Signal is also decomposed by a second high-pass filter (h) operating concurrently. The breakdown of a signal is seen in great detail in Figure 7. Both exact coefficients (the result of a high-pass filter) and approximations (the result of a low-pass filter) are produced using this method. There is a downsampler with two steps at the filter output. Approximation (a) is provided by the low-pass filter’s downsampled output, while detail (d) is provided by the high-pass filter’s low-pass samplers. The approximate component may be split into smaller and smaller pieces at each stage. The analysis in this study has reached the fourth step at this point. The wavelet operation determines, for a given mother wavelet ψ(t), a distinct group of child wavelet coefficients. The discrete wavelet transform involves scaling and shifting the mother wavelet by powers of two [22,23].

$${\psi }_{j,k}=\frac{1}{\sqrt{2^j}}\psi \left(\frac{t-k{2}^j}{2^j}\right)$$

(8)

Scale is indicated by j, and shift by k. The wavelet coefficients obtained from x(t) may be regarded as the projection of x(t) onto a wavelet, given that x(t) represents a 2 N-length signal. The following conditions must be met for a member of the aforementioned discrete wavelet family to be considered a child wavelet [22,23]:

$${\gamma }_{jk}={{\displaystyle \int }}_{-\infty }^{\infty }x\left(t\right)\frac{1}{\sqrt{2^j}}\psi \left(\frac{t-k{2}^j}{2^j}\right)dt$$

(9)

Then, j is a constant, and the ${\gamma }_{jk}$ is derived only in terms of a function of k. Sampled at points 1, 2j, 2j,..., 2N, ${\gamma }_{jk}$ is calculated by convolving x(t) with the mother wavelet signal, $h\left(t\right)=\frac{1}{\sqrt{2^j}}\psi \left(\frac{-t}{2^j}\right)$.

Discrete wavelet coefficients at the j^th level are specified here. Thus, for a given mother wavelet ψ(t), a perfect match between the detail coefficients of the filter bank and a wavelet coefficient of a discrete collection of child wavelets is guaranteed by selecting h[n] and g[n] appropriately. Average a₄ and average d₁–d₄ characteristics were derived from the investigated signals and employed in the following procedures.

2.3. PSO-Based Feature Extraction

The PSO algorithm is among the essential tools for researchers into swarm intelligence [24]. Because of its effectiveness and simplicity, the PSO algorithm has rapidly become a go-to solution for feature-selection problems. This strategy is inspired by the cooperative lifestyles of animals in the wild, such as fish and birds. In order to solve the problem, the algorithm needs the contribution of the whole population. Each member of the population is denoted by the term “particle”, and the whole group is spread out evenly throughout the search space of the function being optimized. The position of each particle is measured against the goal function. Next, a path forward is decided upon by integrating information on the present location, the optimal past location, and the most promising available particles. When all particles have updated their positions, the program will proceed. The desired outcome may be achieved by repeating these steps numerous times. Like a flock of birds looking for food, a collection of particles seeking the optimal value of a function is like a swarm of particles. This algorithm’s central notion may be stated as follows: at each given time, particles move to the best available position in the search space based on their previous observations and the locations of their neighbors. Like other evolutionary algorithms, PSO first generates a fully random initial population. The initial population has N particles picked at random. Particle positions and velocities are both expressed as vectors. After the objective function’s value is calculated, these particles start moving throughout the problem space in search of a more suitable location. The ability to search requires each particle to have a dual-memory system. One memory stores the optimal location for all particles and the best past positions of each particle. The particles use this information to plan their next course of action. Each particle adjusts its speed and location at each iteration in order to optimize the biggest absolute and local solutions. Each particle adjusts its speed and location at each iteration to optimize the biggest absolute and local solutions [25]. In PSO, each particle represents a possible solution. Two vectors at each iteration support the search for the i^th particle: the location vector $X_i^t$ = [$X_{i1}^t$,$X_{i2}^t$,..., $X_{iD}^t$] and the velocity vector $V_i^t$ = [$V_{i1}^t$, $V_{i2}^t$,..., $V_{iD}^t$]. During motion, each particle’s position and velocity are updated depending on the best position (or solution) of the particle itself (denoted by pbesti = [pbesti1, pbesti2, …, pbestiD]) and the best position (or solution) of the population as a whole (denoted by gbest = [gbest1, gbest2, …, gbestD]). At the next iteration, $t$+1, the $i$^th particle’s velocity, and position are modified according to pbest and gbest, using the following formulas:

$$V_{id}^{t+1}=\omega \ast V_{id}^t+c_1\ast r_1\ast \left(pbes{t}_{id}^t-X_{id}^t\right)+c_2\ast r_2\ast \left(gbes{t}_d^t-X_{id}^t\right)$$

(10)

$$X_{id}^{t+1}=X_{id}^t+V_{id}^{t+1}$$

(11)

where $t$ and $\omega$ represent the iteration number and the iteration weight, respectively, c₁ and c₂ are acceleration constants (the cognitive and social parameters, respectively); the r₁ and r₂ are uniformly distributed and fall between [0, 1]. Defining the cost function is a crucial step in establishing optimization systems. The cost function of the PSO system is established in this study as the MSE of an MLP neural network with one hidden layer and ten neurons in the hidden layer. First, the network is fed a random sample of features collected from the data. Then, the inputs are gradually improved toward ideal values using the established optimization method to minimize the cost function. The PSO system employs an iterative process in which it first attempts to forecast the goal using a characteristic, then raises the number of inputs accordingly, and finally implements the system for all modes using various inputs.

2.4. GMDH Neural Network

M.G. Ivakhnenko, a mathematician from Ukraine, developed a mathematical strategy for handling prediction and classification problems; he dubbed it the GMDH [26]. Ivakhnenko proposes a model that, because of its inherent capacity for self-organization, may have its many parameters—such as network architecture, adequate inputs, number of hidden layers, and number of neurons in hidden layers—automatically chosen. Its input–output relationship is described by the Kolmogorov–Gabor polynomial, which is given below:

$$y=a_0+{\displaystyle \sum }_{i=1}^m{a}_i{x}_i+{\displaystyle \sum }_{i=1}^m{\displaystyle \sum }_{j=1}^m{a}_{ij}{x}_i{x}_j+{\displaystyle \sum }_{i=1}^m{\displaystyle \sum }_{j=1}^m{\displaystyle \sum }_{k=1}^m{a}_{ijk}{x}_i{x}_j{x}_k+\cdots$$

(12)

The output y is the network’s calculated value based on the input vectors X (x₁, x₂,..., x_m) and the weights a (a₁, a₂,..., a_m) of the vector. There are five stages to implementing the GMDH neural network.

• For each admixture ($\begin{array}{c}m\\ 2\end{array}$), two inputs (extracted characteristics) are fitted using Equation (13). This process is responsible for extracting the C coefficients from the least squares approach. The solutions to the quadratic polynomials could be used as estimates for the desired answer. The neurons will compute these polynomials in the neural network.

  $$Z=c_1+c_2{x}_i+c_3{x}_j+c_4{x}_i^2+c_5{x}_j^2+c_6{x}_i{x}_j$$

  (13)
• The neurons with the highest erroneous predictions of future output are removed.
• Like the first, the selected neurons are regarded as inputs characterized by quadratic polynomials. In this method, a polynomial of higher order is created by combining polynomials of smaller orders.
• The second step is repeated, and the most defective neurons are eliminated. The generation of polynomials from polynomials is repeated until the desired inaccuracy is achieved.
• Validating the network’s performance using a test dataset. The neural network is trained using around 70% of the data and then tested using the remaining 30%. If the created neural network can show accuracy on these data sets, it is guaranteed to provide acceptable performance under operational conditions. Numerous aspects of chemical and petrochemical engineering [27,28,29,30,31,32], electrical engineering [33,34,35,36,37,38], civil engineering [39,40], instrumentation and control engineering [41,42,43], and nanoelectronic [44,45,46,47] problems have recently been addressed by computational and numerical calculations, as well as Digital Signal Processing (DSP) and particularly Artificial Neural Networks (ANN), a very potent mathematical tool.

3. Results

As explained in the previous sections, 13 characteristics were extracted from the recorded signals in terms of time, frequency, and wavelet transform. The PSO method was used to determine the suitable mode. First, a characteristic was applied to the PSO system, and the cost function was calculated. Then, the number of characteristics increased to 13, and the PSO algorithm found the suitable mode for each of the characteristics and calculated the cost function. Figure 8 shows the value of the cost function according to the number of input characteristics. Carefully, in this figure, it can be seen that the value of the cost function in the selection of seven characteristics has the lowest value. Therefore, seven characteristics have been chosen as the suitable mode. The lowest specified cost function is obtained for selecting the following characteristics: Mean{a₄}, Mean{d₃}, Mean{d₂}, ASDF, WL, kurtosis, and skewness. These characteristics were introduced as the input of the GMDH neural network to predict the void fraction inside the pipe. This study trained many neural networks to estimate percentages of volume. In order to determine the void percentage, a GMDH neural network was trained. The GMDH network’s self-organization capacity is used to automatically create the trained network with a topology consisting of an input layer with seven neurons, an output layer, and four hidden layers, which have eight neurons, six neurons, four neurons, and two neurons, respectively. The structure of this network can be seen in Figure 9.

A pair of regression and error histogram diagrams depicting the network’s performance throughout training and testing datasets are shown in Figure 10. The red circles in the regression diagram represent the network’s output values, while the blue line represents the expected output. Matching both of them with each other indicates high accuracy in neural network prediction. The distribution of the error value is graphically represented in the error histogram diagram. The MSE and RMSE were computed for both the training and testing data. The maximum MSE and RMSE values are, respectively, 0.09 and 0.30. The diagnosis system presented in this study was compared with previous studies in

Table 1. A comparison of the accuracy of the proposed detection system and previous studies.

Ref	Extracted Features	Feature Selection Method	Type of Neural Network	Maximum MSE	Maximum RMSE
[2]	No feature extraction	Lack of feature selection	MLP	2.56	1.6
[3]	Time features	Lack of feature selection	MLP	0.21	0.46
[4]	Time features	Lack of feature selection	GMDH	1.24	1.11
[9]	Frequency features	Lack of feature selection	MLP	0.67	0.82
[15]	Lack of feature extraction	Lack of feature selection	GMDH	7.34	2.71
[48]	Full energy peak (transmission count), photon counts of Compton edge in transmission detector and total count in the scattering detector	Lack of feature selection	MLP	1.08	1.04
[49]	Compton continuum and counts under full energy peaks of 1173 and 1333 keV	Lack of feature selection	RBF	37.45	6.12
[proposed method]	Time, wavelet, and frequency features	PSO-based feature selection	GMDH	0.09	0.30

.

4. Conclusions

System optimization and improved oil industry performance could be achieved by determining the volume percent of each condensate phase that goes through the oil pipe. As a result, developing and implementing a system to calculate volume percentages could be helpful in solving the problems in the petroleum industry. In this research, the volume percentage-detection structure was simulated using MCNP code, which consisted of an X-ray tube, a Pyrex pipe, and a sodium-iodide detector. Three flow regimes named annular, homogeneous, and stratified were simulated at void fraction percentages of 5 to 95%, and the signals of each simulation were collected and labeled. A total of thirteen time, frequency, and wavelet characteristics were extracted from the collected signals. A PSO-based feature selection system was implemented to identify the best features. Among the extracted features, the simultaneous use of seven features was selected as the suitable mode. The seven selected features were defined as the inputs of the GMDH neural network to predict the percentage of void fraction inside the pipe. The methodology presented in this research was able to predict volume percentages with MSE of less than 0.09. High accuracy in determining volume percentages, the use of only one detector in the structure of the detection system, and the use of the PSO-based feature selection method are among the advantages and innovations of the current research.