Nonlinear multi-scale dynamic stability of oil–gas–water three-phase flow in vertical upward pipe

Chemical Engineering Journal 302 (2016) 595–608

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Chemical Engineering Journal journal homepage: www.elsevier.com/locate/cej

Nonlinear multi-scale dynamic stability of oil–gas–water three-phase flow in vertical upward pipe Lian-Xin Zhuang, Ning-De Jin ⇑, An Zhao, Zhong-Ke Gao ⇑, Lu-Sheng Zhai, Yi Tang School of Electrical Engineering and Automation, Tianjin University, Tianjin 300072, China

h i g h l i g h t s
We design a rotating electric field conductance sensor to measure three-phase flow.
We use recurrence plot and AOK TFR to recognize three-phase flow patterns.
We propose a MS-WCECP to study the stability and nonlinearity of three-phase flow.
Our analysis yields novel insights into fluid dynamics from the disequilibrium view.

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 5 February 2016 Received in revised form 16 April 2016 Accepted 18 May 2016 Available online 20 May 2016

Characterizing stability and nonlinearity underlying oil–water–gas three-phase flow is a challenging problem of significant importance. We carry out experiments and measure the fluctuation signals from a rotating electric field conductance sensor with eight electrodes. We use recurrence plot and adaptive optimal kernel time–frequency representation to recognize different oil–water–gas three-phase flow patterns from experimental measurements. Then we employ multi-scale weighted complexity entropy causality plane (MS-WCECP) to explore the nonlinear characteristics for five typical oil–water–gas three-phase flow structures. The results suggest that our method enables to indicate flow pattern transitions. In particular, with the increase of scales, more information will be lost. Slug flow ends up in chaotic region, representing high complexity; Churn flow falls down from the chaotic to the random noise area, indicating the decreasing stability; while the drop degree of bubble flow is the biggest, suggesting that bubble flow has the most randomness. These findings demonstrate that multi-scale weighted complexity entropy causality plane can effectively depict the transitions of three-phase flow structures and serve as a useful tool for probing the nonlinear dynamics of the three-phase flows. Ó 2016 Elsevier B.V. All rights reserved.

Keywords: Oil–gas–water three-phase flow Flow regime Stability Multi-scale Weighted complexity entropy causality plane

1. Introduction The exploitation of onshore oil and gas field has come into its middle-late stage, there is going to be a big change in oil and gas reservoirs as well as the distribution of oil, gas and water. On one hand, most of the water flooded oil fields have already been in their high water-cut period, due to pumping, the pressure of flow in the reservoir is low which degases the crude oil; on the other hand, in oil deposits, the top structures and the places near fault also have gas production, this undoubtedly will lead to oilgas–water three-phase flow in the reservoir. The complex flow structure of three-phase flow give rise to many difficulties for oil well production as well as oil gas transportation. Furthermore, ⇑ Corresponding authors. E-mail addresses: (Z.-K. Gao).

[email protected]

(N.-D.

http://dx.doi.org/10.1016/j.cej.2016.05.081 1385-8947/Ó 2016 Elsevier B.V. All rights reserved.

Jin),

[email protected]

pressure gradient, hold-up and the pipe parameters have a strong correlation with the flow pattern [1–4]. Any variances of flow structures will have a great influence on the industrial production, such as, artificial lift, pipe installation and many other situations. Therefore, the study of the stability and nonlinearity of flow patterns is of significant importance to both industrial production and practical application. The conventional approaches in identifying the flow patterns include direct observation [5–8], high speed photography [9–10] and local probing [11–13]. Through estimating the existing state of different flow structures we can distinguish various flow patterns. With the development of time series analysis methods [14–16], many works dedicated to characterizing flow patterns from experimental measurements [17–18]. Wu et al. [19] adopted wavelet theory to de-noise differential pressure signals and then obtained the characteristics vectors of various flow regimes in terms of fractal theory. Wang et al. [20] used vertical

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Nomenclature Notation pi pe Q Q0 J½P; P e  Cw m, s XðtÞ pk S½P Hw wðtÞ

probability distribution uniform distribution disequilibrium normalized constant expansion of Jenson–Shannon Divergence statistical complexity measure embedding dimension, time delay mean value of XðtÞ weighted probability weighted permutation entropy normalized weighted permutation entropy weight value

multi-electrode array conductance sensor to obtain signals from oil–gas–water three-phase flow and then employed chaotic attractor morphological description and complexity measures to study the dynamical characteristics of four water dominated flow patterns. Gao et al. [21–22] developed complex network-based analytical frameworks to uncover the complicated flow behaviors underlying the transitions of two-phase/three-phase flow patterns. Mukherjee et al. [23] utilized the probability density function (PDF) to identify the range of existence of different patterns from the probe signals. Dispersed flow and slug flow can be identified from the PDF analysis, and this method is particularly useful at high flow velocity where visualization techniques fail to work. There still exist significant challenges in the characterization of transitional flow behaviors underlying three-phase flow which has attracted wide attention [24–26]. A study has been made to analyze the influence of gas injection on the phase inversion between oil and water in a vertical upward pipe by Descamps et al. [27]. The results show that gas injection does not significantly change the critical concentration, but the influence on the pressure drop is considerable. Zhao et al. [28] applied multi-scale long-range magnitude and sign correlations to analyze the signals, and the results suggest that the magnitude series relates to nonlinear properties of the original time series, whereas the sign series relates to the linear properties. Despite the huge breakthrough in the researches of oil–gas–water three-phase flow, it is still a challenge to investigate the stability and determinacy of three-phase flow due to its complex structure. In this regard, a powerful tool integrating complexity and entropy remains to be developed. Entropy is a physical quantity for describing the degree of randomness. Pincus [29] came up with the approximate entropy, which has been widely used in climate prediction, medical science as well as mechanical equipment. Balil et al. [30] developed the weighted permutation entropy by improving the permutation entropy introduced by Bandt and Pompe [31]. In the previous study of multi-phase flow, some indexes such as permutation entropy and Kolmogorov entropy have been extracted to identify flow patterns and characterize flow stability. Moreover, multiscale sample entropy has been demonstrated to be an efficient indicator for the transitions of two-phase flow [32]. Nevertheless, neither sample entropy nor permutation entropy can describe the system from the perspective of both disequilibrium and entropy. To solve this problem, Rosso et al. [33–34] proposed the complexity entropy causality plane (CECP). Zunino et al. [35] used the complexity entropy causality plane to quantify the stock market inefficiency. Dou et al. [36] further developed the CECP by taking the advantage of multi-scale technique to map a time series into a multi-scale complexity-entropy causality plane which has

h H ys ðjÞ Hð
Þ k
k

angle of electrode height of electrode coarse-grained time series Heaviside function Euclidean norm e threshold value of distance Aðt; s; v Þ time-localized short-time ambiguity function UAOK ðs; v Þ Gaussion function Greek letters water superficial velocity U sw U sg gas superficial velocity oil superficial velocity U so

enriched the knowledge of the stability and nonlinear dynamics of two-phase flow. As a development of previous methods, we proposed a multiscale weighted complexity entropy causality plane (MS-WCECP). Our method allows acquiring another important information about peculiarities of a probability distribution, a task that entropy-based methods fail to work and improves the performance of the CECP by combining multi-scale technique, most importantly, MS-WCECP introduce the permutation entropy into the weighted permutation entropy which performs much better in anti-noise ability as well as distinguishing different signals. The MS-WCECP has demonstrated to be a useful tool for exploring the stability and complexity of gas– liquid two phase flow [37]. In this paper we utilize multi-scale weighted complexity entropy causality plane (MS-WCECP) to analyze the signals measured from vertical upward oil–gas–water three-phase flow. Based on our analysis of experimental data, we can draw the conclusion that our method improves the signal classification ability and shows a good algorithm robustness compared to the previous work. The results indicate that the multi-scale weighted complexity entropy causality plane enables to effectively characterize the transitions of three-phase flow structures and provides an efficient analytical framework for investigating the nonlinear dynamics of the three-phase flow.

2. Multi-scale complexity entropy causality plane 2.1. Statistical complexity measure Jaynes has established the relevance of information theory for theoretical physics [33]. Two essential ingredients of this content are: (1) Shannon’s logarithmic information measure regarded as the general measure of the uncertainty associated to probabilistic physical processes. (2) Maximum entropy principle. Later, from another point of view, Kolmogoro and Sinai converted information theory into a powerful tool for the study of dynamical systems. Ascertaining the degree of uncertainty and randomness of a system is not equal to adequately grasp the correlative structures that maybe present. Randomness and structural correlations are not totally independent aspects of this physics. It is obvious that there is no structural information of perfect order and maximal randomness. There exists a wide range of possible degree of physical structure. The characteristics of system are reflected under the probability distribution. When the probability distribution is given, the entropy of a system is zero. We can gain the most information from it. The other way round, for a uniform distribution, very little information can

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be obtained from it, the system is under the most stochastic condition at this time. Under the analytical framework of complexity entropy causality plane, even though period and randomness are in opposite position, the values of complexities of two systems are both zero. Between these two conditions there exist wide states, therefore, complexity measure can be used to describe these behaviors.

wðtÞnk Pk ¼ Pn=sðm1Þs t¼1

wðtÞ

597

ð12Þ

Therefore, the weighted permutation entropy S½P can be defined as follows

S½P ¼ 

M X

Pk lnðPk Þ; P ¼ fPk ; k ¼ 1; 2 . . . M ¼ m!g

ð13Þ

k¼1

2.2. Weighted complexity entropy causality plane The definition of complexity measure is given in Ref. [33]

C JS ½P ¼ Q J ½P; Pe HS ½P

ð1Þ

where Q represents the disequilibrium, which refers to the distance in the probability space. It measures the distance between fpi g and equilibrium fpe g in the phase space. pe ¼ f1=m!; . . . ; 1=m!g, which obeys the uniform distribution, m is the embedding dimension.

Q J ½P; Pe  ¼ Q 0 J½P; Pe 

ð2Þ

where Q 0 is the normalized constant, which can be expressed as follow


1 m! þ 1 lnðm! þ 1Þ  2 lnð2m!Þ þ lnðm!Þ Q 0 ¼ 2 m!

ð3Þ

the definition of J½P; Pe  is the expansion of Jenson–Shannon Divergence, In probability theory and statistics, Jenson–Shannon divergence is defined as the distance between two different probability distributions to describe the disequilibrium of a system.

J½P; Pe  ¼ fS½ðP þ Pe =2  S½P=2  S½P e =2g

ð4Þ

HS ½P represents the normalized permutation entropy.

HS ½P ¼ S½P=Smax

ð5Þ

where Smax ¼ S½P e  ¼ ln m!: We extend the normalized permutation entropy [31] to normalized weighted permutation entropy [30] and attain a new complexity measure

C w ½P ¼ Q ½P; Pe Hw ½P

ð6Þ

For a given one dimensional discrete time series, we reconstruct phase space and attain a vector

XðiÞ ¼ ½xðiÞ; xði þ sÞ;    ; x½i þ ðm  1Þs

ð7Þ

where m denotes the embedding dimension and s represents the time delay. Then the variance can be calculated as follows [30]

wðtÞ ¼

m h i2 1X xðt þ ðj  1ÞsÞ  XðtÞ m j¼1

ð8Þ

when pk ¼ 1=m!; k ¼ 1; . . . M, S½P attains the maximum value ln (m!). We can normalize S½P by ln(m!), and denote the normalized weighted permutation entropy as follow

Hw ½P ¼ S½P= lnðm!Þ

ð14Þ

Bandt and Pompe [31] recommended m = 3, 4,. . .,7 and s ¼ 1. In this paper, we select m = 6 on the basis of false nearest neighbors. The choice of length of time series should ensure the calculation accuracy of entropy. Based on the C JS  HS plane proposed by Rosso et al. [33], we put forward the weighted complexity entropy causality plane C w  Hw with its horizontal and vertical axis being the normalized weighted permutation entropy Hw and the new complexity measure C w , respectively. Weighted permutation entropy Hw can depict the complexity and quantify the degree of randomness of a system, which has a prominent ability to extract complexity information of nonlinear system, and significantly improve the robustness and stability compared with the previous entropy methods. Statistical complexity measure C w provides a quantitative description about the degree of structural correlations and the randomness behind the system. Moreover, it provides additional information on probability distribution, describing the difference between the characteristic of system and the state of maximal randomness. In order to give a better understanding of the WCECP, we analyze three types of signals: (a) periodic process (b) chaotic system (c) stochastic process. (1) The square symbols represent the WCECP results from sin signal x ¼ sin 100pt and the cosine signal x ¼ cos 50pt. (2) The triangle symbols represent the WCECP results from the logistic map and Henon map. The logistic map: x ¼ rxn ð1  xn Þ, where r = 3.7, x(1) = 0.55 Henon map: nþ1 xnþ1 ¼ 1  axn þ yn , where a = 1.4, b = 0.3, and the initial ynþ1 ¼ bxn value ðx0 ; y0 Þ ¼ ð0:1; 0:1Þ. (3) The circle symbols represent the WCECP results from Colored noises and Gaussion white noise. Colored noises eðkÞ ¼ xðkÞ þ 0:5xðk  1Þ; where xðkÞ is the white noise, the length N = 1000. Gaussion white noise is also located on the right lower part of MS-WCECP.

where XðtÞ can be denoted as the arithmetic mean value

XðtÞ ¼

m 1X xðt þ ðj  1ÞsÞ m j¼1

ð9Þ

Then, the components of vector XðtÞ can be arranged in an increasing order

x½t þ ðj1  1Þs 6 x½t þ ðj2  1Þs 6    6 x½t þ ðjm  1Þs

ð10Þ

when encountering the state of equality, we consider the quantities according to the j value. Hence, any vector XðtÞ has a permutation

AðgÞ ¼ ½j1 ; j2 ; . . . ; jm 

ð11Þ

where g ¼ 1; 2; . . . ; k; k 6 m!. There are m! permutations for m distinct symbols. The appearing number of ever permutation nk has been calculated ðk 6 m!Þ; the weighted relative frequency for each permutation can be denoted as follows

As can be seen from Fig. 1, the result of sine signal and cosine signal are located on the lower left side of the plane. The low values of Hw and C w are consistent with the low complexity and regular structure of sine and cosine signals. In this way, we regard this part of area the regular region. For the chaotic system logistic map, it shows that both the values of Hw and C w are between 0.4 and 0.5, near to the maximum value which is caused by the intricate structures immersed in the chaotic system. The higher the C w ; the more complex the system will be. Therefore, we name this region as the chaotic region. For the colored noise and Gaussion white noise, the results are located in the right lower part of the picture which represents higher entropy but lower complexity. If the lengthen of N is long enough, it will almost reach the maximum Hw related to its irregular and unpredictable characteristic and the minim C w for it dose not contain any non-trivial structures. In this regard, we name this the noise region. Note that, under this analyt-

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ical framework, even though regular and randomness are in opposite position, the values of complexities of two systems exhibit very small values. Therefore, the MS-WCECP allows measuring the complexity induced by the correlational structures and distinguishing different types of system such as regular systems, chaotic systems and noise system.

A sin(ω t + 135 )

2.3. Multi-scale complexity-entropy causality plane In order to better describe the details of system, the multi-scale analysis method is introduced to this paper. The time series coarsegrained method [38] is described as follows:

Asin(ω t+ 90 )

(1) For a given time series fxðiÞ; i ¼ 1; 2; . . . ; Ng; (2) We can reconstruct it with the scale s and obtain the coarsegrained time series fys ðjÞ; j ¼ 1; 2; . . . ; N=sg which is generated by

ys ðjÞ ¼

js 1 X xðiÞ; s i¼ðj1Þsþ1

1 6 j 6 N=s

A sin(ω t + 45 )

A sin( ω t)

Fig. 2. Electric rotating field schematic diagram.

ð15Þ

ating a rotating field. This overcomes the non-uniform distribution of single sensor and reduce the measurement error. Fig. 5 demonstrates the configuration of sensor. There are mainly three structural parameters of the conductance sensor, including angles, height and thickness of an electrode. In order to guarantee a good measurement with the rotating field conductance sensor, we applied finite element analysis method and single factor alternate method to optimize the structural parameters of the sensor. According to the analysis result, we finally determine the optimal structural parameters as an angle of 22.5°, a height of 0.004 m and a thickness of 0.001 m.

(3) We calculate complexity measure and weighted permutation entropy from the coarse-grained time series at each scale and then plot the weighted complexity causality plane for all scales, which is the multi-scale weighted complexity causality plane (MS-WCECP). 3. Experiment equipment and data acquisition 3.1. Sensor configuration Rotating electric field conductance sensor with eight electrodes has been employed to measure the voltage signal in the experiment. The inner diameter of the sensor is 20 mm. Four pairs of electrodes are stimulated by four sinusoidal signals with different phase positions, each of them has a lag of 45°. The compound field intensities are the same but its angle changes with time which can be described by / ¼ wt. Figs. 2 and 3 demonstrate the schematic diagram of rotating field, and provides the incentive method of rotating field. Fig. 4 presents the compound electric field intensity of four pairs of sensor at four different moments. Each electrode is at a different initial phase, the variance of electric potential is in accordance with sinusoidal signal. We can see from Fig. 4 that the electric field intensity at each position varies with time, gener-

3.2. Experiment equipment and data acquisition Oil–gas–water three-phase flow experiment was carried out in a 20 mm inner diameter vertical upward transparent Plexiglas pipe in the multi-phase flow loop of Tianjin University. The experimental mediums are white oil and tap water, the gas medium is air. Peristaltic pumps and air compressor were applied to transport the three-phase respectively. The length of testing pipe is 1.8 m, making sure that the three-phase flow was fully mixed. At the entrance of the pipe, a gas distributor and a check valve were installed to ensure that the gas phase was flowing to the pipe homogeneously. The equipment figure is given in Fig. 6.

0.5 cos 50πt sin100πt logistic(x)

chaotic region

0.4

Henon Colored noise Gaussian white noise

Cw

0.3

0.2

noise region 0.1

regular region 0.0 0.0

0.2

0.4

0.6

0.8

Hw Fig. 1. WCECP for several typical dynamic systems.

1.0

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A+

B+

C+

D+

Asinwt

Asin( wt + 45°)

Asin( wt + 90°)

Asin( wt + 135°)

A-

B-

C-

D-

- Asinwt

− Asin( wt + 45°) - Asin( wt + 90°) - Asin( wt + 135°)

Fig. 3. Electrodes structure and incentive method of rotating electric field conductivity sensor.

Fig. 4. The distribution of electric field at different moment.

Fig. 5. The configuration parameters of the conductivity sensor.

Firstly, we fix the water superficial velocity and gas superficial velocity, and gradually increase the oil superficial velocity, then we obtain the conductance fluctuating signals from different flow patterns. Next we change the gas superficial velocity and fix the water superficial velocity and oil superficial velocity to conduct experiment; finally, we change the water superficial velocity and fix the other two phase to acquire flow information. Through the above experimental procedure, we can obtain the conductance fluctuating signals under different flow conditions from the rotating electric field conductance sensor with eight electrodes. In our experiment, four-channel data can be acquired from the sensor, the values of gas superficial velocity ranges from 0.055 m/s to 0.442 m/s (1.5 m3/day–12 m3/day), the water superficial velocity are 0.037 m/s–0.737 m/s (1 m3/day–20m3/day), and the oil superficial velocity changes from 0.001 m/s to 0.147 m/s (0.2 m3/day–4 m3/day). There are totally 450 flow conditions, covering five flow patterns: slug flow, slug-bubble transitional flow, bubble flow, churn flow and slug-churn transitional flow. The experimental measurements are shown in Fig. 7.

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Fig. 6. Oil–gas–water three-phase flow experiment equipment.

probe A

0.4

probe B

probe C

probe D

Usw=0.737 m/s Uso=0.014 m/s Usg=0.055 m/s Bubble

0.2 0.0 -0.2 -0.4 1.0

Usw=0.037 m/s Uso=0.004 m/s Usg=0.055 m/s Slug

0.5 0.0

Signals (V)

-0.5 -1.0 1.0

Usw=0.737 m/s Uso=0.147 m/s Usg=0.331 m/s Churn

0.5 0.0 -0.5 -1.0 0.6 0.3

Usw=0.442 m/s Uso=0.071 m/s Usg=0.055 m/s Slug-Bubble Transition

0.0 -0.3 -0.6 1.0

Usw=0.221 m/s Uso=0.013 m/s Usg=0.055 m/s Slug-Churn Transition

0.5 0.0 -0.5 -1.0 5

6

7

8

9

10

11

12

13

14

15

Time(s) Fig. 7. Measurement signal of rotating electric field sensor.

We can see from the Fig. 7(a), for the flow condition Usw = 0.737 m/s, Uso = 0.015 m/s, Usg = 0.055 m/s, the flow pattern is bubble flow. There exist very little fluctuations in the Fig. 7(a), normally between 0.2 V and 0.2 V, suggesting the stochastic behavior. This reflects that the bubble flow consists of random minute bubbles and this pattern always occurs at a high water flow velocity and a low gas flow velocity. As can be seen from Fig. 7(b) for slug flow (Usw = 0.037 m/s, Uso = 0.004 m/s, Usg = 0.055 m/s), its

fluctuation is relatively high compared with bubble flow, in the range of 0.5V  0.5V with its peak reaching 1 V. The peak value indicates a large gas slug flowing through the sensor. The fluctuation varies very little between the two peaks, reflecting a water slug is flowing through the sensor. The signals for churn flow are shown in Fig. 7(c), for the flow condition Usw = 0.737 m/s, Uso = 0.015 m/s, Usg = 0.332 m/s. Compared to slug flow, the fluctuation signal of churn flow normally

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is in the range of 0.5 V and 0.5 V, and the peak is lower, reflecting the gas slug flowing through the sensor become smaller than that in slug pattern, correspondingly, the voltage is lower. Meanwhile, instead of fluctuating between two peaks, it swings back and forth. With the values of total flow rate increase it has caused turbulence which attributes to the constant oscillation. Fig. 7(d) presents the signals of slug-bubble transitional flow for the flow condition of Usw = 0.442 m/s, Uso = 0.071 m/s, Usg = 0.055 m/s. During a period of time, the gas slug is passing through the sensor together with some small water slug occasionally and after that water slug flow is passing for a long time. It passes alternately which reflects the characteristics of transitional flow. Fig. 7 (e) shows the signals of slug-churn transitional flow for flow condition Usw = 0.221 m/s, Uso = 0.013 m/s, Usg = 0.055 m/s. We can see that for a long duration of time, the signal swings between a relatively high voltage and during that time there exists some minute fluctuations which typically reflect the slug flow pattern, then it comes to the churn flow, it swings back and forth between the two peaks in a short time. Hence, it is slug-churn transitional flow.

1

In order to clarify how the flow patterns change with the threephase flow velocity varies, here we give the voltage signals of conductivity sensor with the increase of water flow velocity. Fig. 8(a) delineates fluctuating signals measured from conductivity sensor with the increase of water phase velocity for a fixed gas superficial velocity 0.553 m/s and a fixed oil-in-liquid phase volume fraction. As illustrated in Fig. 8(a), the relatively low water phase flow velocity leads to the low turbulent kinetic energy, small gas slug coalescing into big ones at this time, the flow pattern is slug flow. As the water phase velocity increases, the turbulent kinetic energy is enhanced. Gas slugs are disintegrated into small pieces, thereby transforming to bubble flow. When it becomes bubble flow completely, the magnitude of voltage signal is the lowest but with relatively high frequency, which means the signal is quite stochastic. Under low gas phase velocity, slug flow, slug-bubble transitional flow and bubble flow are dominated. Fig. 8(b) represents the signals when the gas flow velocity and the oil phase volume fraction is fixed at 0.332 m/s and 0.02, respectively. Slug flow appears when the water phase flow velocity is

Usg=0.055m/s

(a) Usw=0.737m/s bubble flow

signal(v)

0 1 0 1 0 -1 1 0 -1 1 0 -1 1 0 1 0 -1 1 0 -1 1 0 -1 10

Usw=0.589m/s bubble flow Usw=0.443m/s slug-bubble transitional flow Usw=0.368m/s slug-bubble transitional flow Usw=0.295m/s slug-bubble transitional flow Usw=0.221m/s slug flow Usw=0.147m/s slug flow Usw=0.074m/s slug flow Usw=0.037m/s slug flow 12

14

16

18

20

time(s)

signal(v)

Usg=0.331m/s

1 0 -1 1 0 -1 1 0 -1 1 0 -1 1 0 -1 1 0 1 0 -1 1 0 -1 1 0 -1 10

(b) Usw=0.737m/s churn flow Usw=0.589m/s churn flow Usw=0.442m/s slug-churn transitional flow Usw=0.368m/s slug-churn transitional flow Usw=0.295m/s slug-churn transitional flow Usw=0.221m/s slug-churn transitional flow Usw=0.147m/s slug flow Usw=0.074m/s slug flow Usw=0.037m/s slug flow

12

14

16

18

20

time(s) Fig. 8. Fluctuating signals measured from conductivity sensor which Usg is fixed at: (a) Usg = 0.055 m/s; (b) Usg = 0.331 m/s.

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(a) bubble flow

(b) slug flow

(Usw = 0.737m/sˈUso = 0.103m/sˈUsg = 0.055m/s)

( Usw = 0.074m/sˈUso = 0.006m/sˈUsg = 0.332m/s)

(c) churn flow

(d) slug-bubble transitionalflow

(Usw = 0.589m/sˈUso = 0.047m/sˈUsg = 0.442m/s)

(Usw= 0.442m/s, Uso = 0.071m/s, Usg = 0.055m/s)

(f) slug-churn transitional flow (Usw = 0.221 m/sˈUso = 0.018m/sˈUsg = 0.055m/s) Fig. 9. Recurrence plot of five typical flow patterns.

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low. The flow structure of slug flow is relatively stable. With the increase of water flow velocity, the turbulence energy increases at the same time, the large gas slugs are broken into small pieces, then the slug flow evolves into churn flow, i.e., slug-churn transitional flow. The gas block drives the water phase to rise, while due to the influence of gravity, the water phase fall down periodically. This attributes to higher indeterminacy of churn flow than that of slug flow. Compared with bubble flow, it is more stable because there still exist some blocks moving periodically. 4. Identification of oil–gas–water three-phase flow patterns

the interface between liquid phase and gas phase becomes irregular, which can be reflected by the decrease of black block as well as the increase of lines along the diagonal in the recurrence plot of Fig. 11(e). We can draw the following conclusions from the above analysis: slug flow presents the features of intermittent black block while the texture of churn flow shows numbers of linellae along with the diagonal, and bubble flow exhibits the features of a large number of isolated points. The recurrence plot of transitional flow is normally the combination of three typical flow patterns. The recurrence plot provides us with a convenient and obvious way for identifying different flow patterns.

4.1. Recurrence plot analysis 4.2. Time–frequency joint distribution analysis Recurrence plot is a nonlinear time series analysis method, which employs phase space reconstruction to investigate the recursive features of dynamic system [39]. Generally this method project high dimensional space to two dimensions which means the two dimensional recurrence plot can reflect the trajectory of m dimensions. Our research group firstly employed this method to identify different two-phase flow patterns and have achieved good results [40–41]. We in this paper utilize the recurrence plot to analyze the voltage signals of oil–gas–water three-phase flow. The outcome elucidates that the texture structure of recurrence plot is rather clear and obvious. Recurrence plot provides a useful tool to identify the oil–gas–water three-phase flow. Based on Takens embedded theory [42], we can reconstruct a phase space from a time series xðtÞ; t ¼ 1; . . . ; n as follows:

XðtÞ ¼ ½xðtÞ; xðt þ sÞ; . . . xðt þ ðm  1ÞsÞ;

t ¼ 1; . . . N

i; j ¼ 1    N

Z Z PAOK ðt; f Þ ¼

ð16Þ

where m is embedded dimensions, s is time delay, N is the number of phase space vector. The representation of recurrence plots is give below

Ri;j ¼ Hðe  jjXðiÞ  XðjÞjjÞ;

Time–frequency joint distribution method can describe the flow state from two aspects: the energy and frequency of the movement. Adaptive optimal kernel (AOK) is proposed by Jones and Baraniuk [43], they used AOK to analyze the time–frequency characteristics of voltage signal. Zhao et al. [28] utilized AOK TFR to analyze the voltage signal of three-phase flow and found out this method can effectively identify different flow patterns. The approach can also automatically match the parameters of core algorithm with the signals, restraining the interference of cross term and maintaining the high frequency at this time. The original formula of AOK can be expressed as follows

ð17Þ

Hð
Þ is Heaviside function, while e is the threshold; k
k is the Euclidean norm; The matrix Ri;j consists of the values 1 and 0 only. The graphical representation for recurrence plots is N  N grid of points, which are presented as black for 1 and white for 0. We use recurrence plot to analyze our experimental signals measured from eight electrodes rotating field sensor. The results are as follows: Bubble flow (Usw = 0.737 m/s, Uso = 0.103 m/s, Usg = 0.055 m/s): As shown in Fig. 11(a), due to abundant minute blister, randomness is the most obvious characteristic of bubble flow, thus the recurrence plot represents some scattered points. Slug flow (Usw = 0.074 m/s, Uso = 0.006 m/s, Usg = 0.332 m/s): In Fig. 11(b), there are many black blocks in the recurrence plot, which can be explained by the alternating movement of gas slug and water slug leading to the quasi periodicity of voltage signal. Churn flow (Usw = 0.590 m/s, Uso = 0.047 m/s, Usg = 0.442 m/s): When the gas volume and the total flow velocities are high, the recurrence plot develops the texture along side the diagonal, i.e., the features of churn flow as presented in Fig. 11(c). Churn flow has some inner certainty which leads to the texture lines. Slug-bubble transitional flow (Usw = 0.442 m/s, Uso = 0.071 m/s, Usg = 0.055 m/s): Based on slug flow condition, the transitional flow occurs with the increase of water flow velocity. Black blocks still exit, which just corresponds to the gas slug flow pasting through the sensor. At the same time, there are also some scattered points, indicating the transition to bubble flow, as shown in Fig. 11(d). Slug-churn transitional flow (Usw = 0.221 m/s, Uso = 0.018 m/s, Usg = 0.055 m/s): when the gas flow velocity is high, with the total flow velocity increasing, slug flow evolves into churn flow. During the flow transition, large gas slugs are broken into small ones and

Aðt; s; mÞUAOK ðs; v Þej2pðtmþsf Þ dudv

ð18Þ

where UAOK ðs; mÞ ¼ er =2r ðwÞ is the function that control the extension of Gaussian Kernel at w direction, where r 2 ¼ s2 þ v 2 ; s represent the time-delay, and t stands for frequency shift, t means time and f is frequency. Aðt; s; mÞ is time-localized short-time ambiguity function. Usually the adaptive optimal kernel is solved under polar coordinates, the adaptive optimal kernel is determined by the following conditions. 2

Z Z max U

2

jAðt; r; sÞUðt; r; WÞ2 jr drdW

ð19Þ

The AOK TFR has the advantages of contributing to high concentration in the time–frequency representation and suppressing the effect induced by the cross term. We perform AOK time–frequency analysis to the data (the same data as in the recurrence plot analysis). Slug flow: We can see from the time–frequency plane in Fig. 9 (a), its frequency mainly focuses on the range of 0–15 Hz, with one or more peaks. The energy reveals the periodical intervals of slug flow, e.g., high and low energy appear alternatively, reflecting the movement characteristics of water slug and gas slug, and a great change of energy can be seen from the Fig. 9(a). Bubble flow: Fig. 9(b) shows that the frequency band is quite wide and the frequency distribution of different signals is relatively average. The energy intensity changes slowly with the variance of frequency. This indicates the disorder movement of bubble flow. As can be seen from the AOK energy distribution, there is no interval in time domain, indicating the continuity of bubble flow. Meanwhile the average energy of bubble flow is relatively low. Churn flow: Fig. 9(c) shows the time–frequency joint distribution of churn flow. There exists obvious frequency peaks associated with the periodic oscillation. The energy distribution in time domain is not only continues and uniform, but also present a high value, corresponding to the flow behavior that gas slugs passing through the sensor. The periodicity of the movement is reflected by the oscillation of high and low voltage.

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Churn-slug transitional flow: As illustrated in Fig. 9(d), the frequency of this flow mainly gathered around 0–15 Hz, however there is a peak between 0 and 5 Hz and the frequency distribution is rare before the peak, normally there is a mutation when the signal comes to peak. After the peak, the frequency distribution decreased rapidly. Furthermore, energy distribution presents a tendency of continuous distribution but with occasional low energy. This explains the fact that the flow pattern is not stable in both churn flow and slug flow, passing the sensor with liquid slug as well as gas slug, occasionally appears alternatively, so this can be treated as the churn-slug transitional flow. Bubble-slug transitional flow: As shown in Fig. 9(e), high energy and low energy appear alternatively in the time–

frequency joint distribution. For a fixed gas flow velocity, with the increase of water flow velocity, the period of low energy elongates which declares that the water slug becomes longer, and high energy region has been shortened indicating the gas slug becomes shorter at this time. The results illustrate that the transition from slug flow to bubble flow. Whereas, when the water flow velocity is decreasing, flow pattern evolves into bubble flow. In this flow pattern, the volume of gas slug reduces, leading to the decrease of voltage signal measured from the experiment. Besides, because of the movement of minute bubbles, the signal frequency of bubble-slug transitional flow mainly clustered in the areas between 0 and 15 Hz, without any obvious peak.

Fig. 10. Time frequency joint distribution of five typical flow patterns .(a) slug flow (Usw = 0.037 m/s, Uso = 0.007 m/s, Usg=0.055 m/s); (b) bubble flow (Usw = 0.589 m/s, Uso = 0.012 m/s, Usg = 0.055 m/s); (c) churn flow(Usw = 0.368 m/s, Uso = 0.036 m/s, Usg = 0.442 m/s); (d) churn-slug transitional flow(Usw = 0.073 m/s, Uso = 0.014 m/s, Usg = 0.221 m/s); (e) bubble-slug transitional flow(Usw = 0442 m/s, Uso = 0.071 m/s, Usg = 0.055 m/s).

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5. Three-phase flow pattern characterization by using MSWCECP We now employ multi-scale weighted complexity entropy casualty plane (MS-WCECP) to analyze the signals measured from the rotating electric field sensor. The embedding dimension m is 6, and the time delay is 1. The length of time series is set 100,000 to ensure the calculation accuracy of entropy. In the experiments, the range of gas superficial velocity U sg is from 0.055 m/s to 0.442 m/s, the water phase superficial velocity changes from 0.037 m/s to 0.074 m/s, and the range of oil superficial velocity U so varies from 0.001 m/s to 0.147 m/s. We carried out 450 flow conditions in total, and five typical flow patterns are obtained, including slug flow, slug-churn transitional flow, churn flow, slug-bubble transitional flow, bubble flow. Fig. 10 demonstrate the results of MS-WCECP for five different flow conditions. Fig. 11(a) shows the MS-WCECP of five typical flow patterns. As shown in Fig. 11(a), the distributions of MS-WCECP for five typical flow patterns are different, indicating that the MS-WCECP can efficiently distinguish the differences among these flow patterns. This method dose not only provide a new way for identifying flow structures but also help to know more about the nonlinearity and stability of each flow pattern. This yields new insights into the nonlinear dynamics underlying bubble flow, slug flow, churn flow as well as transitional flow. The MS-WCECP of slug flow is

illustrated in Fig. 11(b). With the increase of scale, the curve of MS-WCECP extends from regular region to chaotic region, and the value almost reaches the highest complexity C w , indicating the structure of slug flow presents a rather high complexity and high determinacy. The reason can be attributed to the alternately movement of gas slug and water slug, and the motion with a feature of quasi period leading to its high certainty and stability. Fig. 11(c) presents the MS-WCECP of slug-churn transitional flow. As can be seen, the higher the scale is, the lower it will drop, and the more information it will be lost. Compared with slug flow, it drops deeper suggesting the transitional flow presents more random but it still locates in the chaotic region. Fig. 11(d) exhibits the MS-WCECP of slug-bubble transitional flow. With the increase of scale, the curve of MS-WCECP ends up in chaotic region and presents little difference with the slugchurn transitional flow. The MS-WCECP of churn flow is shown in Fig. 11(e). As can be seen, with the increase of scale, the curve of MS-WCECP rises from regular region to chaotic region and finally drops to noisy region, which implies that the signal is rather random. This is because the fact that with the increase of water velocity, gas slug is broken into small ones, oscillating up and down which formulate into churn flow. Besides, churn flow is similar to bubble flow but not as stochastic as bubble flow. Therefore, the dropping degree of churn flow is less than bubble flow in MS-WCECP.

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Fig. 11(f) shows the MS-WCECP of bubble flow. The dropping degree of bubble flow is the highest among all the flow patterns. The normalized weighted permutation entropy Hw entropy of bubble flow almost reaches the maximum. The larger Hw is, the more uncertainty it becomes, and the more information it will be lost. The stochastic behavior of bubble flow results from the irregularly movements of amounts of small bubbles. Therefore, from the MS-WCECP analysis, we can identify different three-phase flow patterns, and meanwhile characterize the dynamical characteristics of complex flow behavior. For example, slug flow is located in the chaotic region which denotes high certainty and complexity; Churn flow rises from chaotic region to noise region, which becomes more stochastic compared with slug flow, due to the fact that large gas slugs are broken into smaller ones and the small pieces oscillate up and down; Bubble flow is the most stochastic among all these flow patterns as demonstrated Fig. 11(f), it is situated in the noise region and the dropping degree in MS-WCECP is the deepest. The state of the transitional lies in between slug flow and bubble flow. In order to give a better understanding of the flow patterns under various flow conditions, we exhibit two more MS-WCECP for each flow pattern in Fig. 12, and the results suggest that the MS-WCECP from different flow conditions corresponding to the same flow pattern are pretty the same, indicating the robustness of our method. Multi-scale complexity entropy causality plane consists of two parameters which are entropy and complexity. We thus present the error estimation from two perspectives, respectively. We utilize the error bar to analyze the error estimations of the same flow patterns under various flow conditions. We pick the last scale to calculate the error in the sense that the last scale shows the biggest difference among various flow conditions compared to the previous scales. The results of each parameter are shown in Fig. 13. We can see from Fig. 13 that the error estimations of bubble flow are smallest for both C w and Hw which indicates that the inner structure of bubble flow is quite simple. The error estimations of churn flow is comparatively large, reflecting its relatively complex dynamic characteristics.

uncertainty, and improve the signal classification ability as well as anti-noise ability. We can also gain the details of continuous loss of dynamic structures underlying different flow patterns, which reflect the stability of system. These three approaches reveal different information from the flow patterns and provide us a better understanding of the system. To sum up, the conclusions are stated as follows: (1) Slug flow extends from periodic region to chaotic region and ends up in chaotic region, indicating that the structure of slug flow has a high determinacy, meanwhile the dynamic state is complex. (2) As the scale increases, churn flow rises up to chaotic region and drop to the noise area but not as deep as bubble flow, therefore the flow dynamic characteristic has a high degree of determinacy compared to the bubble flow. (3) Bubble flow extends from periodic to chaotic region and ends up in noise region. Compared with churn flow, the degree of bubble flow extending to noise region is the highest, indicating the obviously stochastic structure of this flow pattern, but its inner structure is not complicated. (4) Transitional flow includes slug-bubble transitional flow and slug-churn transitional flow. Compared with slug flow, these two transitional flow patterns are more stochastic, but are more stable than churn flow and bubble flow reflected by a higher certainty. The result declares that there is not a significant difference between these two transitional flow patterns. The MS-WCECP analysis of the oil–gas–water three-phase flow has enriched the knowledge of three-phase bubble flow, slug flow as well as churn flow, and contributes to the understanding of structure uncertainty. These yield deep insights into the complex dynamic flow behavior of three-phase flow from the disequilibrium view, and also establish a novel approach for identifying and characterizing the flow patterns. Acknowledgments

6. Conclusions We have employed recurrence plot and AOK time–frequency analysis to characterize oil–water–gas three-phase flow patterns. Recurrence plot allows identifying the different flow patterns from the perspective of nonlinear characterization, while AOK TFR enables to recognizing different flow behaviors from the perspective of energy and frequency. Furthermore, we propose multiscale complexity entropy casualty plane (MS-WCECP) to analyze the experimental measurements. MS-WCECP enables to precisely describe the dynamical characteristics such as complexity and

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