1 Introduction
Oil-water two-phase flow is frequently encountered in petroleum, chemical and nuclear reaction industries, and its flow pattern is defined as the spatial distribution behavior of two-phase flow in pipelines. However, because of the slippage effect, turbulent droplets in high water-cut oil-water two-phase flow show remarkable non-equilibrium flow phenomena. An understanding on flow instability is of great significance to the monitoring and control of flow pattern transition in oil-water two-phase flow.
Experimental observation is the traditional method for investigating flow pattern of oil-water flows [1], [2], [3], [4]. With high sensitivity and quick response speed, mini-conductance probe has been widely applied to flow pattern identification [5], [6], [7], [8], [9]. In addition, laser-induced fluorescence [10] and the hot-wire probe method [11] are also used to investigate flow structure of oil-water flows. With onshore oil field exploitation in China entering into the stage of high water-cut, the exploration of high water-cut oil-water flows in oil wells has been a research focus. It has been reported that the oil-water two-phase flow with low flow velocity and high water-cut mainly presents dispersed oil-in-water flow (D O/W) and very fine dispersed oil-in-water flow (VFD O/W) [12]. Specifically, oil droplets in D O/W show a rich diversity and non-uniform distribution, and the diameters of oil droplets in VFD O/W are generally about 1 mm. Because of the limited sensitivity of traditional electrical sensors, it is difficult to capture the detailed information of flow characteristics in high water-cut oil-water flows. So far, the understanding of the interaction mechanism and dynamic instability in oil droplets is still limited.
As a measure of system complexity, entropy plays an important role in the study of dynamic characteristics in system. Traditional entropy algorithms for investigating nonlinear time series include approximate entropy [13], sample entropy [14], complex measure [15] and permutation entropy [16]. Fadlallah et al. [17] proposed the algorithm of weighted permutation entropy, and by calculating the variance of each component of phase space vector, weighted permutation entropy retains most of the amplitude information of the time series as well as reflects the nonlinear system characteristics more comprehensively. In addition, the multi-scale sample entropy (MSE) method proposed by Costa et al. [18] enriches the understanding of the complexity behavior of the system from the microscopic and macroscopic perspectives. In recent years, the multi-scale entropy algorithm has also achieved much progress in uncovering the nonlinear dynamic characteristics of multiphase flow [19], [20], [21], [22], [23]. However, the current works lack comparison of various multi-scale entropy algorithms.
In terms of the complex flow behavior of the water-oil two-phase flow with high water-cut, we first carry out the evaluation of multi-scale entropy algorithms by investigating entropy variability with time scale and anti-noise abilities. Afterwards, in order to reveal the dynamic instability of the high water-cut oil-water two-phase flow, the signals of a high-resolution microwave resonator sensor are acquired in a vertical upward testing pipe. With the analysis of multi-scale weighted permutation entropy (MWPE), the effects of total flow rate and water-cut on the dynamic instability of the oil-water two-phase flow are elucidated.
2 Multi-Scale Entropy Algorithms for One-Dimensional Time Series
In an attempt to illuminate the differences of the variability in nonlinear systems by multi-scale entropy algorithms, approximate entropy [13], sample entropy [14], complexity measure [15] and weighted permutation entropy [17] are first generalised to multi-scale analysis, and the Lorenz system [24], K-noise sequence [15] and fractal Brownian motion (fBm) [25] are used to investigate entropy variability with time scale variation and anti-noise ability. Brief mathematical descriptions of the four entropy algorithms are given below.
2.1 Approximate Entropy
For a one-dimensional time series {x(i)}, approximate entropy can be calculated by the following steps [13]:
- Composing a sequence {x(i)} into a vector X(i) with embedded dimension set as m:
$$X\mathrm{(}i\mathrm{)}=[u\mathrm{(}i\mathrm{)},\text{\hspace{0.17em}}u\mathrm{(}i+1\mathrm{)}\cdots u\mathrm{(}i+m-1\mathrm{)}],\text{\hspace{0.17em}1}\le i\le N-m+1$$ - Define the distance between X(i) and X(j) as:
$$d[X\mathrm{(}i\mathrm{)},X\mathrm{(}j\mathrm{)}]=\underset{0\le k\le m-1}{\mathrm{max}}|u\mathrm{(}i+k\mathrm{)}-u\mathrm{(}j+k\mathrm{)}|$$ - For a threshold r(r>0), counting the ratio of the number of vectors satisfying d[X(i), X(j)]<r to the number of the whole vectors:
$$\begin{array}{l}{C}_{i}^{m}\mathrm{(}r\mathrm{)}=\{\text{The\hspace{0.17em}number\hspace{0.17em}of\hspace{0.17em}vectors\hspace{0.17em}satisfying\hspace{0.17em}}\\ \text{}d[X\mathrm{(}i\mathrm{)},\text{\hspace{0.17em}}X\mathrm{(}j\mathrm{)}]<r\}/\mathrm{(}N-m+1\mathrm{)}\end{array}$$ - Calculate the logarithm value of
and then define its average value as Φ^{m}(r):${C}_{i}^{m}\mathrm{(}r\mathrm{)}$ $${\Phi}^{m}\mathrm{(}r\mathrm{)}=\frac{1}{N-m+1}{\displaystyle \sum _{i=1}^{N-m+1}\mathrm{ln}\text{\hspace{0.17em}}{C}_{i}^{m}\mathrm{(}r\mathrm{)}}$$ - Increase embedded dimension to m+1, repeating (1–4) and calculate Φ^{m+}^{1}(r)
- Approximate entropy can be formulated as:
$$\text{ApEn(}m\text{,\hspace{0.17em}}r\text{,\hspace{0.17em}}N\text{)}={\Phi}^{m}\mathrm{(}r\mathrm{)}-{\Phi}^{m+\text{1}}\mathrm{(}r\mathrm{)}$$
2.2 Sample Entropy
The computational procedure of sample entropy can be depicted as [14]: for embedded dimension set as m and m+1, designate
The formula of sample entropy can be given as follows:
2.3 Complexity Measure
For a one-dimensional time series {x(i)}, a vector denoted as X(i) can be acquired after phase reconstruction:
where m and τ refer to embedded dimension and delay time, respectively. Arrange the vectors in X(i) in an ascending order:
For two vectors with equal values, the permutation is conducted by comparing the value of l. We count the appearing number of the kth permutation n_{k}, 1≤k≤m!, and the occurrence probability of the kth permutation can be calculated as:
Therefore, the permutation entropy S[P] can be expressed as [16]:
We normalise S[P] by dividing it by its maximum ln(m!). The normalised permutation entropy H_{S}(P) is formulated as:
Jensen-Shannon divergence Q_{J}[P, P_{e}] is an effective measure of the difference between two probability distributions and can characterise the distance between probability distribution P and equilibrium probability distribution P_{e}:
where P_{e}={p_{k}=1/m!; k=1, …, M=m!} refers to equilibrium probability distribution, and Q_{0} is a normalisation constant with the value given below:
The expression of complexity measure is formulated as [15]:
2.4 Weighted Permutation Entropy
For a one-dimensional time series {x(i)}, w(t) is defined as the variance of m components after phase space reconstruction using (9):
Denote N_{k} as the occurrence number of kth permutation after the permutation implementation with (10), and the probability of its occurrence is:
The formula of weighted permutation entropy can be expressed as [17]:
2.5 Multi-Scale Algorithm
In order to generalise the foregoing entropy algorithms to multi-scale research field, the coarse-graining procedure is required and can be concluded as follows [18]:
Given a one-dimensional time series {x(i):i=1, 2, …, N}, for a time scale τ, the coarse-graining time series is defined as {y^{τ}(j):j=1, 2, …, N/τ}:
3 Multi-Scale Entropies of Typical Nonlinear Systems
We take the x sequence of the Lorenz system as the investigated time series (Fig. 1). The initial condition is set as x_{0}=−1, y_{0}=0 and z_{0}=1. Afterwards, we add White Gaussian noise signals with signal-to-noise ratios (SNRs) of 10, 20, 30 and 40 dB, respectively.
K-noise is a kind of signal obeying the distribution of f ′^{−}^{k}^{/2}. Figure 2 shows the sequences of K-noise with K=0.5, 1.5, 2.5 and 3, respectively. As seen, with the increasing K values, the fluctuating frequency decreases and the non-uniformity in fluctuating amplitude becomes obvious gradually.
fBm is a non-stationary random process with the characteristics of mean value equaling to zero, obeying Gaussian distribution, self-similarity and possessing stationary increment. Its covariance function can be expressed as:
where s, t∈R, H is the Hurst index, 0<H<1. If H<1/2, then the increment in fBm presents a negative correlation. Conversely, H>1/2 indicates that the correlation of the increment is positive. H=1/2 corresponds to classical Brownian motion. The fluctuating sequences of fBm with H=0.1, 0.3, 0.5, 0.7 and 0.9 are shown in Figure 3. It can be seen that similar to the characteristics of K-noise, high fluctuating frequency and uniform fluctuating amplitude can be observed for low H values, while fBm under high H values can be characterised by low fluctuating frequency and non-uniform fluctuating amplitude.
Figures 4–6 show the results of four multi-scale entropy algorithms in three typical nonlinear systems. It can be seen for the Lorenz system (Fig. 4) that four kinds of multi-scale entropy algorithms all present good anti-noise ability. Compared with multi-scale approximate entropy (MAE) and MSE, multi-scale complexity measure (MCM) and MWPE exhibit higher increasing rates in low scale. Moreover, the value of MCM ranges from 0.1 to 0.5, while the MWPE results are located in the range from 0 to 5.2, which indicates that MWPE is more sensitive to the variation in scale.
Compared with the Lorenz system, K-noise presents higher randomness. As seen from Figure 5, MCM and MSE lose the ability to distinguish the characteristics of the system, and the intersection between entropy curves is obvious. Although MAE can show a certain resolution in a low-scale range, the slopes of curves with k=0.5 and k=1.5 are opposite to those with k=2.5 and k=3.5. In the whole scale range, MWPE values show good resolutions between different k values.
The results in Figure 6 indicate that MCM has poor signal resolution capability for fBms with different values of H. The variation trends of MAE for different H values are different, while MWPE and MSE all present good signal resolution ability. However, compared with the results of MSE located in the range from 0 to 1.2, the results of MWPE are in the range from 0 to 7.5, which also demonstrates the high sensitivity of MWPE to the variation of scale.
4 Experiment of Oil-Water Two-Phase Flow with High Water-Cut
Recently, water holdup measurement using the microwave resonance method has achieved great progress. Guo et al. [26] employed a cylindrical cavity sensor for water content measurement in the mixed fluid of water and alcohol. Through the extraction of a parameter designated as S_{11}, the shift in resonance frequency could be detected. Alahnomi et al. [27], [28] proposed a split ring resonator to investigate the physical characteristics of the medium, and the gain of a parameter designated as S_{21} was determined. The research results indicate that the proposed sensor presented a high measurement resolution. The microwave transmission method was used by Oon et al. [29] to realise water content measurement in horizontal oil-water and gas-liquid flows, and water content was determined by deriving the resonance frequency of S_{21} in the range of 5–5.7 GHz.
We apply a double-parallel-line microwave resonant sensor to investigate the flow characteristics of the high water-cut oil-water two-phase flow. The configuration of the sensor is shown in Figure 7. The structure of the double-parallel-line microwave resonator is a cylindrical waveguide resonator with both ends open. The coupling part of the input and output port of the sensor uses the form of parallel lines, and microwave signal is directed through the mixture in the way of electric coupling, which enhances the sensitivity to the change of water holdup in the mixed fluid. The inner wall material of the sensor is polytetrafluoroethylene (PTFE), and the surface of the parallel conductors immersed in the mixed fluid is coated with Teflon to reduce the influence of water phase conduction.
According to the reports in [26], [27], [28], [29], the resonance frequency of the sensor changes with the varying content of the medium. Besides, the decrease in phase and gain will change in the vicinity of resonance frequency. Therefore, by extracting the decrease in phase and gain near the resonance frequency, the variation in the content of the medium can also be determined. Figure 8 shows the phase and gain characteristics of different flow mediums using a vector network analyser (VNA) (Tianjin Deli Technology Company, Tianjin, China). As seen, significant discrepancies in resonance frequency for different mediums are present. Nevertheless, the measurement circuit of resonance frequency for high frequencies is more complicated. It can be inferred from Figure 8a and b that when resonance frequency is around 1.8 GHz, both the variations of phase and gain versus frequency are stable. Furthermore, phase exhibits a satisfied linear relationship versus frequency. Hence, 1.8 GHz is taken as the resonance frequency. In this manner, the microwave sensor can be stimulated by a fixed frequency, and the decrease in phase can be detected using a simple frequency mixer.
When microwave is coupled to the output by the mixture between the parallel lines, the attenuation is almost linearly correlated with the water-cut in mixed fluid. Because the microwave signal is very sensitive to the small change of the medium, it can be used to detect flow characteristics under high water-cut and can capture the small variation of water holdup in the flow process of very fine oil droplets, which provides high-quality data for identifying the discrepancies between different flow patterns under high water-cuts.
The experiment was carried out in the multiphase flow and sensor system laboratory of Tianjin University. The experimental flow facility is schematically shown in Figure 9. The inner diameter of the pipe is 20 mm, and the total pipe length is 2600 mm. The water and oil phases were transported and metered by two industrial Peristaltic pumps (Baoding Leadfluid Company, Hebei, China) with an accuracy of ±0.2 % and a range of 0.822–2879 mL/min. The experimental fluids were mixed at the bottom of the testing pipe using a “Y-junction” inlet. To ensure full development and stability of the mixed fluid, the double-parallel-line microwave resonant sensor was placed at a height of 2000 mm. The total flow rate of the oil-water two-phase flow was set from 0.347 to 4.861 L/min, and the range of the water-cut was 90–99 %. Our experimental procedure can be depicted as follows: water-cut is first held constant. With the increasing total flow rate, the fluctuating signals of the microwave resonant sensor were collected. After all the mixture velocities were finished, the water-cut was changed to the next value, and in this way, the experiment is completed. The experiment consisted of 80 flow conditions, and the flow patterns observed are D O/W and VFD O/W. As for the measurement circuit, PXI-5661 (National Instruments, Shanghai, China) was used as the microwave signal source. Through the power divider from the mini Cirholdups Company, the microwave signal is divided into two routes. With phase detection circuit demodulation, the outputs were transformed into low-frequency signals. We applied the PXI-4472 data acquisition card (National Instruments, Shanghai, China) to collect the fluctuating signals, with sampling frequency of 2 kHz and sampling time of 30 s, respectively.
Figure 10 illustrates the signals of the microwave resonant sensor under different flow conditions. It can be seen that when the total flow rate is low, the flow pattern shows D O/W, where the number of oil droplets is small and the motion of oil droplets presents a certain quasi-periodicity. Therefore, corresponding sensor signals present intermittent fluctuation characteristics. With the increase of total flow rate, the turbulent energy of the mixed fluid increases gradually, leading to the fact that large oil droplets are broken up into small ones. The number of oil droplets increases significantly, and the fluctuating frequency of the sensor signals increases. With the increase of water-cut, both the number and diameter of oil droplets in the pipeline decrease gradually, and the fluctuating frequency and amplitude of the sensor signals show a declining trend. When total flow rate and water-cut are extremely high, the flow pattern presents VFD O/W, where oil droplets are of very small sizes, and corresponding sensor signal exhibits the lowest fluctuating amplitude. Figure 11a–c present the images of typical flow patterns in the oil-water two-phase with a high-speed camera.
5 Flow Instability Analysis of High Water-Cut Oil-Water Flows
5.1 The Effect of Noise on MWPE Using Signals From a Microwave Sensor
In this section, we examine the effect of noise on MWPE using signals from a microwave sensor, with the results shown in Figure 12. The selected flow condition corresponds to Q_{t}=0.347 L/min, K_{w}=91 %, and white Gaussian noises with SNRs from 40 to 70 dB are added to the signal from the microwave sensor. It can be seen that for SNRs equaling to 40 dB, noise presents a significant influence on MWPE at low scales (1–5). As for high scales, the results of MWPE are approximately identical to those without noise. Besides, with the increasing SNR, the effect of noise on MWPE gradually weakens until it almost disappears for an SNR of 60 dB. Hence, the existence of noise affects the performance of MWPE at low scales, which is more pronounced for a low SNR.
As for the present study, under the circumstance of pure water, the maximum and minimum outputs from the microwave resonant sensor are 1.49544 and 1.49455 V. The noise amplitude can be calculated as 1.49544−1.49455=0.00089 V. Accordingly, experimental SNR can be approximately determined as 20*lg(1.49544/0.00089)=64.5. Additionally, the minimum sensor output for oil-water flows is 0.38074 V, and the corresponding minimum SNR is 20*lg(0.38074/0.00089)=52.62. Regarding the results in Figure 11, it can be seen that when SNR exceeds 50 dB, the added noise only affects the performance of MWPE for scales less than 4, which indicates that MWPE extracted from the signals of the microwave resonant sensor is of high reliability.
5.2 Flow Instability Analysis Using MWPE
The results of MWPE and its entropy rate under different flow conditions are shown in Figure 13. The entropy rate of MWPE is the slope obtained by linearly fitting entropies within the range of scale from 1 to 20. Figure 13a, c and e are the results of MWPE for different water-cuts, while Figure 13b, d and f are entropy rates for identical flow conditions. It can be seen that for a fixed water-cut, when total flow rate is low, flow pattern corresponds to D O/W, where the number of oil droplets is small and oil droplets are of large sizes (Fig. 11a), oil droplets and continuous water phase alternately flow through the sensor measurement area and, thereby, the flow of mixed fluid presents a certain quasi-periodicity. Correspondingly, its flow instability is not obvious. MWPE and its entropy rate are both low in values. With the increase of total flow rate, the size of oil droplets gradually decreases and the number of oil droplets increases. Accordingly, flow pattern evolves to D O/W with small oil droplets or VFD O/W (Fig. 11b and c). The increasing flow frequency in oil droplets enhances their movement randomness. Therefore, MWPE and its entropy rate are all equal to high values, and the flow instability of the oil-water two-phase flow becomes more pronounced.
Figure 14 shows a joint distribution of the mean value and entropy rate in MWPE for all flow conditions. Figure 14a is the result classified by total flow rate, and Figure 14b is the result through flow pattern classification. It can be seen from Figure 14a that with the increase of total flow rate, both the mean value and entropy rate in MWPE increase gradually, indicating the enhancing instability of the oil-water two-phase flow. As shown in Figure 14b, there is a clear dividing line between D O/W and VFD O/W, and the entropy rate for VFD O/W is higher than that for D O/W. The results show that flow instability for VFD O/W is more significant due to the small size and obvious motion randomness in oil droplets. It is also notable that the average MWPE presents a wide distribution range for D O/W, which can be ascribed to the wide range of water-cut for D O/W and the diverse sizes in oil droplets [12].
In addition, the images of the oil-water two-phase flow captured by a high-speed camera indicate that the oil droplet sizes are small under high mixture velocities (Fig. 11c). The traditional conductance or capacitance sensing method has low measurement sensitivity and poor measurement resolution for small oil droplets, while the microwave resonator sensor is very sensitive to the small change of internal flow medium and has a quick response speed, which can well characterise the movement instability of small oil droplets. From Figure 14b, we can see that the joint distribution of the mean value and entropy rate in MWPE with high mixture velocities is more dispersed, which shows that the flow behavior of oil droplets is more complicated.
5.3 Flow Instability Analysis Using AOK-TFR
In attempt to validate the analysis of flow instability in oil-water flows using MWPE, an algorithm designated as adaptive optimal kernel time frequency representation (AOK-TFR) is applied. Our research group has demonstrated that AOK-TFR can satisfactorily uncover the flow characteristics in multiphase flow, and descriptions regarding the calculating process of AOK-TFR can refer to our published literatures [8], [30], [31]. Herein, we take K_{w}=95 % as an example to investigate the effect of total flow rate on flow instability in oil-water flows, and the corresponding results are presented in Figure 15a–c .
It can be seen from Figure 15a that for a low total flow rate, flow pattern exhibits D O/W where large oil droplets can be observed. These large oil droplets will induce large fluctuations in sensor signals, and thereby, high energy represented by red colour in AOK-TFR appears. The high energy also presents an intermittent distribution characteristic in time domain due to the small number in oil droplets. Moreover, as oil droplets are with low flow velocities, high energy only concentrates at a low frequency band (0–5 Hz). Therefore, the low flow frequency in oil droplets indicate that the motion randomness is at a low level. Correspondingly, flow instability in the oil-water two-phase flow is unobvious, and this coincides with the MWPE analysis in Section 5.2.
With the increasing total flow rate, large oil droplets are decomposed into small ones, and thereby, the number of oil droplets significantly increases. As seen from Figure 15b, the intermittent distribution of energy disappears, and instead, energy shows a scattering distribution in time domain. Besides, the increase in total flow rate leads to an obvious extension of frequency band (0–40 Hz). Hence, the flowing frequency in oil droplets locates in a wide range, which substantially enhances the motion randomness, and accordingly, flow instability becomes more pronounced. With total flow rate further increasing to 4.861 L/min, the scattering distribution of energy is more remarkable and the frequency band further extends. Therefore, the flow frequency in oil droplets becomes more abundant. As a result, flow instability in the oil-water flows is at a highest degree, which shows a good agreement with the results from MWPE. From the foregoing analysis, it can be concluded that flow instability analysis using MWPE is consistent with the result from AOK-TFR, and thus, the effectiveness of MWPE in the high water-cut oil-water flows can be validated.
6 Conclusions
In this paper, we applied the MWPE algorithm to analyse the flow instability of the oil-water two-phase flow under high water-cuts. By examining the multi-scale permutation entropy (MPE), the MSE, the multi-scale complex measure (MCM) and the MWPE in typical nonlinear systems, MWPE was found to show the most satisfied anti-noise ability and signal resolution. Afterward, MWPE was used to process the signals of a double-parallel-line microwave resonant sensor sampled in high water-cut oil-water two-phase flow, and the effects of the total flow rate and water-cut on the flow instability of the oil-water flows were analysed. Our conclusions can be stated as follows:
- When total flow rate is low, the number of oil droplets is small and oil droplets are with low velocity. The motion of the mixed fluid is quasi-periodic, and the corresponding flow instability is weak. With the increase of total flow rate, the number of oil droplets increases gradually and the motion randomness enhances, leading to the significant flow instability of mixed fluid.
- The mean value and entropy rate of MWPE for VFD O/W are the highest, and correspondingly, flow instability is the strongest. The entropy rate of MWPE for D O/W is relatively low, indicating weak flow instability. However, the distribution range of mean MWPE is wide for D O/W, and thereby, its flow structure is abundant.
- The fluctuating signals of the double-parallel-line microwave resonant sensor present remarkable advantages on uncovering the variation of oil droplet diameter in the high water-cut oil-water two-phase flow, and by processing the signals with high-resolution MWPE algorithm, the flow instability of the high water-cut oil-water two-phase flow can be well characterised.
This study was supported by the National Natural Science Foundation of China (Funder Id: 10.13039/501100001809, Grant Nos. 51527805 and 11572220).
A^{m}(r) | the number of vectors satisfying d[X(i), X(j)]<r divided by (N−m) for embedded dimension of m+1 |
ApEn | approximate entropy |
B^{m}(r) | the number of vectors satisfying d[X(i), X(j)]<r divided by (N−m) for embedded dimension of m |
the ratio of the number of vectors satisfying d[X(i), X(j)]<r to the number of the whole vectors | |
C_{JS}[P] | complexity measure |
d[X(i), X(j)] | the distance between X(i) and X(j) |
H | Hurst index |
H_{w}(p) | weighted permutation entropy |
k | K-noise index |
K_{w} | water-cut |
m | embedded dimension |
n_{k} | the appearing number of the k^{th} permutation |
P | probability distribution |
P_{e} | equilibrium probability distribution |
p_{k} | the occurrence probability of the k^{th} permutation |
Q_{0} | normalisation constant |
Q_{J}[P, P_{e}] | Jensen-Shannon divergence |
Q_{t} | total flow rate |
r | threshold |
SampEn | sample entropy |
S[P] | permutation entropy |
w(t) | the variance of components after phase space reconstruction |
X(i) | phase space vector |
x(i) | one-dimensional time series |
the average value of the components in phase space vector | |
y(j) | coarse-graining time series |
Φ^{m}(r) | the average of the logarithm value of |
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