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# Zeitschrift für Naturforschung A

### A Journal of Physical Sciences

Editor-in-Chief: Holthaus, Martin

Editorial Board: Fetecau, Corina / Kiefer, Claus

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# The Slug and Churn Turbulence Characteristics of Oil–Gas–Water Flows in a Vertical Small Pipe

Weixin Liu
• School of Electrical and Information Engineering, Tianjin University, Tianjin 300072, China
• School of Computer Science and Software, Tianjin Polytechnic University, Tianjin 300384, China
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• Other articles by this author:
/ Yunfeng Han
/ Dayang Wang
/ An Zhao
/ Ningde Jin
Published Online: 2017-08-08 | DOI: https://doi.org/10.1515/zna-2017-0119

## Abstract

The intention of the present study was to investigate the slug and churn turbulence characteristics of a vertical upward oil–gas–water three-phase flow. We firstly carried out a vertical upward oil–gas–water three-phase flow experiment in a 20-mm inner diameter (ID) pipe to measure the fluctuating signals of a rotating electric field conductance sensor under different flow patterns. Afterwards, typical flow patterns were identified with the aid of the texture structures in a cross recurrence plot. Recurrence quantitative analysis and multi-scale cross entropy (MSCE) algorithms were applied to investigate the turbulence characteristics of slug and churn flows with the varying flow parameters. The results suggest that with cross nonlinear analysis, the underlying dynamic characteristics in the evolution from slug to churn flow can be well understood. The present study provides a novel perspective for the analysis of the spatial–temporal evolution instability and complexity in oil–gas–water three-phase flow.

## 1 Introduction

Oil–gas–water three-phase flow exists widely in the crude oil production process. The distinct differences in the physical properties of the three oil–gas–water phases result in a sophisticated interaction between the dispersed phases and the continuous phase, leading to the discrepancies in complexity and instability with varying flow parameters especially for slug and churn flows [1], [2], [3]. Therefore, it is very difficult to measure the total flow rate in an oil–gas–water three-phase flow. Understanding the turbulence characteristics of oil–gas–water three-phase flow is of significant importance in designing and optimising the structure of a sensor applied underground, as well as in improving the predicted precision with flow models.

Image observation is one of the basic methods for identifying flow patterns in oil–gas–water three-phase flow. Through observing flow images, Shean [4] found that with the increase of oil volume fraction, the continuous phase changes from water phase to oil phase. Woods et al. [5] utilised visualisation video methods to experimentally identify the flow regimes of oil–gas–water three-phase flow in a small diameter pipe. Based on the previous study of flow structures [6], [7], [8], [9], [10], [11], they defined nine different kinds of flow patterns. Vijayakumar [12] utilised high speed photography to particularly investigate slug and churn flows in oil–gas–water three-phase flow and analysed the discrepancies in the lengths and appearance frequencies of gas slugs (gas blocks) under different flow conditions. Spedding et al. [13] carried out vertical and inclined oil–gas–water three-phase flow experiments to classify different flow patterns, and depicted the flow pattern map of oil–gas–water flows with the variations in oil and water phase velocity, as well as oil fraction.

Electrical sensors have the advantages of high sensitivity and quick response for identifying flow patterns in oil–gas–water three-phase flow [14], [15]. Oddie et al. [1] carried out an experiment of oil–gas–water three-phase flow in an inclined pipe with a large diameter. They used a conductance wire probe to study the flow regimes and analysed the evolution process from transient to steady flow. On the basis of the signals obtained from the ring conductance sensors, our research group investigated the nonlinear characteristics of oil–gas–water three-phase flow [16], [17], [18]. Kee et al. [19] placed the emphasis in their research on the flow regimes and pipeline wettability of oil–gas–water flows with a high-speed camera and conductance sensors, and much attention was paid to the transformation from churn flow to annular flow with the increase in gas volume. Due to the differences in refractive index to light between the water and gas phases, optical probes have been widely utilised in determining local flow parameters of the gas phase [20], [21], [22]. Additionally, tomography technology [23], [24], [25] and a capacitive wire-mesh sensor [26] are also demonstrated to be effective for extracting the phase volume fraction in oil–gas–water three-phase flow.

Generally, three-phase flow meters are installed in concentrating devices with small diameters in the well. An accurate understanding of flow patterns contributes to a high measurement precision in individual phase void fraction and flow rate. So far, the flow characteristics in slug and churn flows have become a research focus [27], [28], [29], [30], [31], [32]. In addition, the slug and churn turbulence characteristics become more complicated with varying flow parameters. Our research group previously reported the dynamic characteristics in the coalescence and breakup phenomenon of the gas phase [33] as well as the flow pattern classification of oil–gas–water flows [34] using conductance sensors. In these studies, a rotating electric field conductance sensor is selected as the applied conductance sensor. Compared to other electrical sensors, a rotating electric field conductance sensor has high sensitivity and is capable of capturing detailed flow structure at the pipe cross section. In addition, for oil–gas–water flows, although both oil and gas phase have non-conductive properties, they have respective and independent conductive electricity paths in the pipe. Therefore, the discrepancy in conductive electricity paths will induce different outputs in sensor signals. With the superiority of high sensitivity, the rotating electric field conductance sensor is of benefit for analysing flow characteristics in oil–gas–water three phase flows using nonlinear time series analysis. Herein, we continue to employ rotating electric field conductance sensor signals to analyse the slug and churn turbulence characteristics in oil–gas–water three-phase flow. Firstly, four-channel dynamic outputs of the sensor under different flow patterns are acquired. Afterwards, we utilise a cross recurrence plot to realise flow pattern identification. Finally, through the analysis of cross recurrence quantification and multi-scale cross entropy (MSCE), the slug and churn turbulence characteristics under low and high gas flow rates are, respectively, elaborated.

## 2 Experimental Facility and Measurement System

Merlio et al. [35] first proposed a rotating electric field sensor to measure the void fraction in two-phase flow. Sinusoidal signals with an interlace phase were generated on the rectangle electrodes to improve the electric field uniformity by forming a rotating electric field in the pipe. By comparing rectangle electrodes with arc-shaped ones, Hu et al. [36] found that arc-shaped electrodes can improve the linearity and standard deviation of void fraction measurement. Rocha and Simões-Moreira [37] proposed an eight-electrode rotating field conductance sensor to measure the void fraction of three typical flow patterns (bubble flow, slug flow and churn flow) in vertical upward gas–liquid flows.

Figure 1 shows the configuration of the rotating electric field conductance sensor. Four pairs of electrodes are stimulated by four sinusoidal signals with different initial phase positions, each of which has a lag of 45 degrees. The compound field intensity is fixed while the field angle changes with time. The measurement circuit of an eight-electrode rotating field conductance sensor includes a stimulus module, a conditioning module, and a collecting module. The stimulus module consists of a direct digital frequency synthesiser chip, AD9959 (Analog Devices, Shanghai, China), to stimulate four sinusoidal signals with an amplitude of 10 V, a frequency of 20 kHz, and a lag of 45 degrees and it is controlled by a single chip microcomputer, STC89LV51 (STC micro, Nantong, Jiangsu, China).

Figure 1:

The configuration of an eight-electrode rotating electric field conductance sensor.

A sketch map of the measurement system is shown in Figure 2. Take channel A for example, by controlling the AD9959 chip, we can obtain a sinusoidal signal with a 20 kHz frequency, which is a 5sin(wt) V on electrode A+. Through the amplification and inversion module, we can impose a -5sin(wt) V sinusoidal signal on electrode A–, and then by measuring the voltage of reference resistor Rref, we can determine the current flowing between A+ and A– which can reflect the conductivity variance. An AD637 chip (Analog Devices, Shanghai, China) is used to measure the effective value of the voltage. The voltages of Channel B, C, and D are acquired using the same principle with only the exciting signals of a lag equal to π/4 in the initial phase position. As for signal sampling, we use a PXI 4472 (National Instruments, Shanghai, China) synchronisation data acquisition card produced by National Instruments with sampling frequency and sampling time set as 2 kHz and 30 s, respectively.

Figure 2:

Measurement system of eight-electrode rotating electric field conductance sensor.

A schematic diagram of the measuring circuit is illustrated in Figure 3. The sinusoidal signal generated by AD9959 becomes a negative signal (A–) through the operational amplifier (OP Amp) U2 and a positive signal (A+) through the OP Amp U1, and the two signals A+ and A– are added to the electrodes A+ and A–, respectively. Then the differential voltage across the resistor R6 is converted to a single ended signal Vout through the OP Amp U3. Finally, through a root mean square demodulation chip, AD637, the true value is output to the PXI data acquisition system.

Figure 3:

A schematic diagram of the measuring circuit.

Due to the presence of other excitation signals and sensor resistances, the differential voltage across the resistor R6 cannot be directly calculated by the circuit. According to Thevenin’s theorem, the electrodes position between A+ and A– is equivalent to a voltage source with a voltage of Vs and an internal resistance of Rs. Because the OP Amp magnification is so large that the positive and negative input ends of U1 and U2 can be considered as an equal voltage:

$Vin=V1$(1)

The OP Amp U2, R1, R2, and R5 form a standard reverse amplifier and V1 is equal to:

$V1=−R5VinR2$(2)

The current in the resistor R6 can be expressed as:

$IR6=(Vo−V1)R6$(3)

The current in the electrodes A+ and A– can be given as:

$Is=(V1−Vs−V2)Rs$(4)

As almost no current flows through the input of the OP Amp, the negative input of U1 and U3 can be considered as an open circuit with:

$IR6=Is$(5)

In practice, the resistance value of R5 is equal to that of R2 and the amplified multiple of amplifier U3 (AD620) is about 2. We can obtain the following formula using simultaneous equations from (1) to (5):

$Vout=R6(2Vin−Vs)Rs$(6)

where Vs is the electromotive force produced by the four pairs of electrodes at the same time, Rs is the internal resistance between the electrode pairs andVin is the output voltage amplitude of the signal source. For the four electrode pairs, Vs is equal toVin and the differences in output voltage between different electrode pairs are caused by the discrepancies in the internal resistance Rs, which is determined by flow pattern distribution between different electrode pairs. It can be inferred that when there exists oil or gas in the fluid, the value of internal resistance Rs becomes bigger, which leads to a lower output voltage amplitude.

The experiment was carried out in a multiphase flow loop facility and sensor system in Tianjin University. The liquid mixture velocity varies from 0.0368 m/s to 1.1776 m/s. The oil-cut is from 2% to 20% and changes with a step of 2%. The range of gas superficial velocity is set from 0.0552 m/s to 0.4784 m/s. During the experimental process, six typical flow patterns, namely slug flow, slug-bubble transition flow, bubble flow, slug-churn transition flow, churn flow, and churn-bubble flow are recorded. Slug flow can be characterised by large gas slugs and liquid slugs alternately flowing upwards, while slug-bubble transition flow is characterised by small gas slugs and a liquid phase where gas bubbles are dispersed. As for bubble flow, the gas phase exists in the liquid phase completely with the formation of gas bubbles. Churn flow can be depicted as quantities of gas blocks drastically oscillating in the liquid phase. Slug-churn flow is characterised by small gas slugs and gas blocks alternately appearing in the liquid phase and churn-bubble flow is defined as gas bubbles and gas blocks take the major form of gas phase in liquid phase. Figure 4 illustrates the sketch maps of experimental flow patterns aforementioned. Detailed descriptions regarding the experimental facility and the measurement system can be found in [34].

Figure 4:

The sketch map of experimental flow patterns in oil–gas–water three-phase flow.

## 3 Fluctuating Signals of the Rotating Electric Field Conductance Sensor

Figure 5 exhibits the fluctuating signals of an eight-electrode rotating electric field conductance sensor under six typical flow patterns. The signal for slug flow is shown in Figure 5a. As can be seen, liquid slugs and gas slugs appear alternately. When liquid slugs flow through the sensor, the output signal has a high value, whilst as the gas slugs pass the sensor, it presents a low output signal. The signals received from the four channels are basically the same, but there is a small difference between the appearance of a gas slug and a water slug which suggests that the concentration distribution of the dispersed phase in liquid slugs is inhomogeneous.

Figure 5:

Signals of typical flow patterns from rotating electric field conductance sensor.

(a) Slug flow: Usg=0.0552 m/s, Usl=0.0368 m/s, fo=0.02 (b) Slug-bubble transition flow: Usg=0.0552 m/s, Usl=0.736 m/s, fo=0.02 (c) Bubble flow: Usg=0.0552 m/s, Usl=1.1776 m/s, fo=0.02 (d) Slug-churn transition flow: Usg=0.4416 m/s, Usl=0.5888 m/s, fo=0.02 (e) Churn flow: Usg=0.4416 m/s, Usl=0.736 m/s, fo=0.02 (f) Churn-bubble transition flow: Usg=0.4416 m/s, Usl=1.0304 m/s, fo=0.02.

The signal for slug-bubble transition flow is given in Figure 5b. As shown, low voltage representing a gas slug shows intermittent characteristics. However, due to the high turbulent energy, gas slugs are broken up into smaller ones, which results in the obvious shortening in the duration of low voltage. Additionally, high voltage representing liquid slugs fluctuates frequently, demonstrating that the number of gas bubbles evidently increases due to the high turbulent energy.

Figure 5c shows the signal of bubble flow, where a more frequent fluctuation in signal can be observed. The output signal is relatively high compared to gas slugs and similar to water slugs in slug flow. The frequent fluctuation indicates that gas bubbles and oil droplets show an obvious stochastic movement which leads to the variance of conductivity. The differences between the four channels also shows the non-uniform distribution characteristics of bubble flow.

The sensor signals regarding slug–churn transition flow are illustrated in Figure 5d. It is obvious that because of the high superficial velocity in both the gas and liquid phase, the sensor signals fluctuate with a higher frequency compared to that in slug flow. Moreover, it is worth noting that low and high voltages lasting relatively longer occasionally appear in the signals, indicating that slug flows with large liquid and gas slugs still exist.

It can be seen from Figure 5e that for churn flow, a huge fluctuation between high voltage and low voltage reflects the oscillation of mixed fluid. In comparison with slug flow, the fluctuation frequency of churn flow is higher with water or gas slugs lasting for a shorter period of time. The differences in the signals between the four channels of churn flow indicate the non-uniform distribution in the dispersed phase at the pipe cross-section as well.

Regarding the signals for churn-bubble transition flow delineated in Figure 5f, the huge fluctuations in voltage with a high frequency reflect the drastic oscillation of liquid slugs and gas blocks in churn flow. Nevertheless, in some period of time, the output voltage fluctuates with an extremely high frequency but small amplitudes around a high voltage value, which is consistent with the phenomenon that quantities of gas bubbles are moving upwards in the continuous liquid phase.

Furthermore, in order to analyse the signal differences between gas–water two-phase flow and oil–gas–water three-phase flow, herein we have conducted a comparison experiment for typical flow patterns, i.e. slug flow and bubble flow as well as churn flow. The comparisons of fluctuating signals using a rotating electric field conductance sensor are shown in Figure 6, where signals coloured black refer to oil–gas–water flows, while signals coloured red denote gas–water flows. Herein, it should be stated that for oil–gas–water flows with a same flow condition, the variation in experimental conditions (for instance, temperature, salinity) will affect the output of the conductance sensor. Therefore, the fluctuating range of signals in Figure 6a–c are not identical with those shown in Figure 5a, c, and e.

Figure 6:

Comparison of conductance sensor signals in gas–water flows and oil–gas–water flows.

(a) Slug flow: Usg=0.0552 m/s, Usl=0.0368 m/s (b) Slug-bubble transition flow: Usg=0.0552 m/s, Usl=0.736 m/s (c) Bubble flow: Usg=0.0552 m/s, Usl=1.1776 m/s (d) Slug-churn transition flow: Usg=0.4416 m/s, Usl=0.5888 m/s (e) Churn flow: Usg=0.4416 m/s, Usl=0.736 m/s (f) Churn-bubble transition flow: Usg=0.4416 m/s, Usl=1.0304 m/s.

For slug flow (Fig. 6a), the injection of oil significantly affects the output signals in both liquid and gas slugs. To be specific, for oil–gas–water three-phase flow, the interaction of gas, oil, and water phases induces more complicated signals for gas slug than gas–water two-phase flow. Besides, with the presence of oil, the output signals of liquid slugs in oil–gas–water three-phase flow are generally smaller than those in gas–water two-phase flow, which can be attributed to the non-conductivity of the oil phase.

For bubble flow (Fig. 6b), the differences in amplitude are also pronounced and this indicates different conductivities in the mixed fluid. The drastic signal fluctuations in bubble flow correspond to random and complex changes in the flow structure of the mixed fluid.

For churn flow (Fig. 6c), liquid slugs are significantly affected by the turbulent trailing vortexes of gas slugs for both oil–gas–water three-phase flow and gas–water two-phase flow. Nevertheless, it can be seen that the signals of liquid slugs and gas blocks between three-phase and two-phase flow are also different.

In summary, detailed flow structures are different between gas–water flows and oil–gas–water flows, and a rotating electric field conductance sensor can effectively uncover the influence of oil on gas–water flows.

## 4 Cross Recurrence Plot Textures for Oil–Gas–Water Three-phase Flow

For a given time series {x1, x2, x3, …, xn}, after reconstructing the phase space, we can define the phase space trajectory of the recurrence plot as follows:

$Xi=[x(t), x(t+τ), x(t+2τ), …, x(t+(m−1)τ)] (i=1, 2, …, N1)$(7)

where m is the embedding dimension and τ represents the time delay, N1=n–(m–1)τ. Based on (7), for another time series {y1, y2, y3, …, yn}, we can introduce a second phase trajectory:

$Yj=[y(t), y(t+τ), y(t+2τ), …, y(t+(m−1)τ)] (j=1, 2, …, N2)$(8)

Comparing each point of Xj and Yj, by calculating its vector distance we can obtain the cross recurrence matrix CR. When the vector distance is larger than a defined threshold value, the corresponding position in CR is 0 and we draw a white point at (i, j) in the cross recurrence plot, while the vector distance being smaller than the threshold value results in the corresponding position in CR being 1 and we draw a black point at (i, j) in the cross recurrence plot. The cross recurrence matrix CR can be calculated as follows:

$CRi,j=Θ(ε−‖Xi−Yi‖)$(9)

where i=1, 2, …, N1, j=1, 2, …, N2, ε refers to the threshold value of distance, ||·|| is Euler norm. Θ(·) is defined as the Heaviside function with the calculation formula given as follows:

$Θ(x)={1 (x>0)0 (x<0)$(10)

A cross recurrence plot denotes the similarity of two trajectories. When two trajectories are close enough, the cross recurrence plot presents a recursive state. According to the textural structure, we can identify the differences between the two trajectories. Unlike the traditional recurrence plot, a cross recurrence plot is not always square and the length of the two trajectories can be different. We utilise the signals of an eight-electrode rotating electric field conductance sensor to calculate the cross recurrence plot of the four channels at the cross section, which represents the distribution characteristics of the mixed fluid in different directions. Due to the orthogonality on the geometry position of channel A and C, conducting cross nonlinear analysis using the signals sampled by channel A and C is more persuasive at uncovering the turbulence characteristics in oil–gas–water three-phase flow.

Define A(t) and C(t) as signals collected by channel A and C, respectively. We can reconstruct a phase space from the time series A(t) and C(t), then the phase space trajectories Xa(i) and Yc(j) can be obtained. Afterwards, a cross recurrence plot algorithm is applied to investigate the recursive features of channel A and C from the eight-electrode rotating electric field conductance sensor. The same method regarding a traditional recurrence plot is used to choose the suitable threshold value for the cross recurrence plot. The formula for determining the threshold value is given as follows:

$εAC=α×std(A)×std(C)$(11)

where std(A) and std(C) denote the standard deviations of A(t) and C(t), α is an empirical coefficient based on the value range of the practical signal. The bigger empirical coefficient leads to a bigger value in the threshold, which causes an increase in the number of points satisfying the criteria that the distances of two vectors are smaller than the threshold value. This leads to an increase in the number of recursive points and more black points in the cross recurrence plot. However, an excessively large threshold value will result in the cross recurrence plot being unclear due to too many recursive points. In order to improve the efficiency of the calculation, we choose 10,000 as the length of the original time series corresponding to the data sampled in 5 s.

For the six typical flow patterns aforementioned, there are significant discrepancies in the textures in the cross recurrence plots. As shown in Figure 7, above the diagonals, j>i denotes the similarity of channel A and C with a certain time lag in channel C. For instance, when ji=1, the data on the lines represents the similarity in the mixed fluid of channels A and C with a lag of one sampling time. Similarly, under the diagonals, j<i illustrates the similarity of channels C and A after a certain time lag in channel A. ij=1 demonstrates the similarity of mixed fluid when channel C is one sampling time ahead of channel A.

Figure 7:

Textural structure of cross recurrence plots under typical flow patterns (fo=0.02).

(a) Slug flow: Usg=0.0552 m/s, Usl=0.0368 m/s (b) Slug-bubble transition flow: Usg=0.0552 m/s, Usl=0.736 m/s (c) Bubble flow: Usg=0.0552 m/s, Usl=1.1776 m/s (d) Slug-churn transition flow: Usg=0.4416 m/s, Usl=0.5888 m/s (e) Churn flow: Usg=0.4416 m/s, Usl=0.736 m/s (f) Churn-bubble transition flow: Usg=0.4416 m/s, Usl=1.0304 m/s.

In this study, the embedding dimension m and delay time τ are set as 3 and 2, while the empirical coefficient α is equal to 0.25. As shown in Figure 7a, the cross recurrence plot of slug flow shows the texture of many intermittent black blocks. In addition, there exist obvious line textures parallel to the diagonal direction in the cross recurrence plot, which means that the phase space trajectories of the two channels are close enough. In the main diagonal, the phase space trajectories of the two channels have no time delay, and the continuous line texture indicates that the fluid structure is symmetrical. However, the blank textures appearing occasionally in the line texture also indicate that the fluid structure is not always symmetrical. In addition, the cross recurrence plot exhibits an intermittent line structure in both the horizontal and vertical directions, which signifies that when one channel is fixed at a certain time, the state of its phase space trajectory at this time intermittently has similar characteristics to the state of the phase space trajectory of the other channel. This can be explained by the alternating movement of the gas slug and the liquid slug. Besides, the intermittent black blocks show that in slug flow the gas slug and the liquid slug appear quasi-periodically.

Figure 7b presents the cross recurrence plot texture structures corresponding to slug–bubble transition flow. It can be seen that due to the increasing turbulent energy, some of the gas slugs are broken up into gas bubbles with obvious random motion. Therefore, scattering points appear in the cross recurrence plot. Notably, in the cross recurrence plot some intermittent rectangular outlines can also be found, which coincides with the characteristics of slug flow.

The cross recurrence plot of bubble flow consists of lots of scattering points (Fig. 7c), indicating that the similarity of the dynamic system states in the two channels is stochastic and will not last for a certain duration of time. This corresponds to the random motion of the gas bubbles which leads to the frequent fluctuation in signals.

The cross recurrence plot of slug–churn transition flow is illustrated in Figure 7d. As seen, the small rectangle blocks appearing intermittently represent the existence of slug flow. However, the rectangle blocks shrink obviously in comparison with the texture of gas slugs shown in Figure 7a. The line textures degenerated from rectangle blocks indicate that small gas blocks oscillate in churn flow.

For churn flow, the cross recurrence plot given in Figure 7e is composed of line textures parallel to the direction of the diagonal but the lines are usually shorter than those in slug flow. This is because churn flow has a characteristic of oscillating dramatically, which leads to a shorter time for gas blocks passing through the sensor and this corresponds to short line structures in the cross recurrence plot. According to the distribution of texture, we discover that the oscillation in churn flow is relatively uniform. In the horizontal or vertical direction, intermittent line textures as shown in slug flow do not exist, which indicates that when one channel is fixed at a certain time, the state of its phase space trajectory is not similar to that of other channels at any time. The foregoing results show that the fluid motion in churn flow is very complex and the flow structure oscillation is violent and not repetitive.

Figure 7f shows the cross recurrence plot texture structures in regard to churn–bubble flow. When both the liquid and gas superficial velocities are high enough, the turbulent energy of the mixed fluid starts to present the capacity of decomposing gas blocks into small gas bubbles. Hence, the moving randomness of gas phase enhances. The line structures further degenerate into scattering points corresponding to the random motion of gas bubbles, whilst the existing line structures indicate the complex flow characteristics in churn flow.

## 5 Recurrence Quantification Analysis for Oil–Gas–Water Flows

Being similar to traditional recurrence plots, recursive rate and determinism can be extracted from the cross recurrence plots as the indexes of quantitative analysis. However, the quantitative indexes of cross recurrence plots are defined at the directions parallel with the main diagonal. They represent the similarity of the signals from channels A and C under the condition with positive or negative delay time aforementioned. For each diagonal CRi, j (ji=k), define the probability distribution of its length as Pk(l)={li; i=1, 2, …, N}, where N is the number of diagonals. Specifically, when k=0, the diagonal corresponds to the main diagonal and is designated as the line of synchronisation (LOS). The definition of cross recursive rate is given as follows [38]:

$RRk=1N−k∑l=1N−klPk(l)$(12)

The cross recursive rate indicates the probability of a similar state in the two channels and k denotes the multiple of sampling time. The bigger cross recursive rate represents a higher degree of similarity in the trajectories of the two channels. The expression for determinism is presented as below:

$DETk=∑l=lminN−klPk(l)∑l=1N−klPk(l)$(13)

Determinism denotes the cross recursive rate ratio of longer diagonals (l>lmin) to that of all the diagonals where lmin is the shortest length which is supposed to form a diagonal construction. The value of lmin is selected with the self-correlation time and, in this paper, it is set as 4. The variation trend of cross recursive ratio and determinism with the increasing liquid phase superficial velocity under Usg=0.0552 m/s and Usg=0.4416 m/s, with a constant oil-cut fo set as 0.02, are shown in Figures 8 and 9 respectively.

Figure 8:

Quantitative indexes of cross recurrence plots under a low gas phase superficial velocity.

Figure 9:

Quantitative indexes of cross recurrence plots under a high gas phase superficial velocity.

According to the results presented in Figures 8 and 9, it can be seen that when k=0, that is no time delay between the fluctuating signals sampled by channel A and C, both cross recursive ratio and determinism reach their peak value for a fixed flow condition, which means that the flow structure of the mixed fluid shows satisfied symmetry. As for low gas and liquid phase superficial velocities (Fig. 8), the flow pattern presents as slug flow, the flow structure of which is quasi-periodic and this corresponds to high cross recursive ratio and determinism. With the increasing liquid phase superficial velocity, the flow pattern gradually evolves to bubble flow. The random motion in gas phase enhances and correspondingly both cross recursive ratio and determinism decrease. Furthermore, at a low gas phase superficial velocity, the fluctuations of cross recursive ratio and determinism in slug flow are weak with the varying k value when |k|>50, indicating an unobvious turbulence characteristic and thereby the flow structure is stable. With gas phase superficial velocity increasing to 0.4416 m/s (Fig. 9), as liquid superficial velocity increases, gas slugs are firstly broken up into gas blocks with irregular shapes, and finally into gas bubbles. Accordingly, the flow pattern evolves from slug flow to churn flow, and eventually to bubble flow. Being similar to the results given in Figure 8, both cross recursive ratio and determinism decrease. However, as for an identical liquid phase superficial velocity, the fluctuations in cross recursive ratio and determinism are drastic with the varying k value when |k|>50 under a high gas phase superficial velocity, especially for slug and churn flows shown in Figure 9. It can be inferred that with a high gas phase superficial velocity, the turbulence characteristics in slug and churn flows are more obvious and correspondingly the flow structure is more unstable.

## 6 Multi-scale Cross Entropy (MSCE) Analysis for Oil–Gas–Water Flows

Entropy, which is regarded as a measure of the complexity in dynamic systems, plays an important role in the investigation of dynamic characteristics. Sample entropy was firstly proposed by Richman and Moorman [39]. Costa et al. [40] developed the theory of MSCE which has been widely applied in the analysis of physiological signals [41] as well as the electromigration noise in metallic material [42]. In order to examine the coupling behaviour between two time series, Yan et al. [43] proposed an MSCE algorithm. In our group, Zhu et al. [44] found that MSCE possesses a satisfactory anti-noise ability, and employed it in the flow pattern analysis of inclined upward oil–water two-phase flow. Herein, with the AC signals sampled by channel A and C in the rotating electric field conductance sensor, we continued to conduct MSCE analysis in oil–gas–water flows to further uncover the turbulence characteristics in slug and churn flows. The detailed procedure on calculating MSCE and the setting of relevant parameters was described by Zhu et al. [44]. The AC signals sampled by channel A and C in the rotating electric field conductance sensor under low and high gas superficial velocities are given in Figures 10 and 11 , respectively.

Figure 10:

AC signals of channel A and C in the sensor under a low gas phase superficial velocity.

Figure 11:

AC signals of channel A and C in the sensor under a high gas phase superficial velocity.

As shown in Figure 10, the flow pattern presents as slug flow with low gas and liquid phase superficial velocities. When gas slugs flow through the conductance sensor, it will cause huge fluctuations in the output voltage. However, due to the low turbulent energy of the mixed fluid, the fluctuating frequency is correspondingly small. With the increasing liquid phase superficial velocity, the turbulent energy enhances and disintegrates gas slugs into gas bubbles, indicating that the flow pattern evolves to bubble flow. The small gas bubbles will induce small fluctuations in the output voltage, while the fluctuating frequency increases as a result of the large number of gas bubbles.

With gas phase superficial velocity increasing to 0.4416 m/s (Fig. 11), because of the low turbulent energy under low liquid phase velocities, the flow pattern corresponds to slug flow as well. As the liquid phase superficial velocity increases, gas slugs are broken up into irregular gas blocks for high gas phase superficial velocities and accordingly the flow pattern evolves to churn flow, in which the drastic oscillation of gas and liquid phase leads to the fluctuations in output voltage with a high frequency. With the further increasing liquid phase superficial velocity, gas blocks are decomposed into gas bubbles and the flow pattern presents as bubble flow. The output voltage fluctuates with the smallest amplitude but the highest fluctuating frequency.

Figures 12 and 13 show the MSCE results under low and high gas superficial velocities, respectively. The flow conditions set in Figure 12 correspond to those in Figure 8, while the flow conditions set in Figure 13 are identical to those in Figure 9. As shown in Figure 12, with a low gas superficial velocity, liquid and gas slugs alternately appear in slug flow. The quasi-periodic motion results in better stability and a weak turbulence characteristic. Correspondingly, the calculated MSCE values are low. As the liquid phase superficial velocity increases, the flow pattern gradually evolves to bubble flow. The high degree of random motion in gas bubbles leads to obvious instability in the mixed fluid. Within the whole scale range, the MSCE values are higher than those in slug flow.

Figure 12:

The variation trend of MSCE with scale under a low gas phase superficial velocity.

Figure 13:

The variation trend of MSCE with scale under a high gas phase superficial velocity.

As gas superficial velocity increases to 0.4416 m/s (Fig. 13), the flow pattern presents as slug flow as well for low liquid phase superficial velocities. The differences compared to the results shown in Figure 12 lie in the phenomenon that the rate of MSCE rises significantly under slug flow, indicating that with a high gas superficial velocity, the flow structure is relatively unstable with an obvious turbulence characteristic. This result is in good agreement with the analysis of cross recurrence quantification given in Figure 9. Moreover, with the increasing liquid phase superficial velocity, the flow pattern evolves to churn flow under a high gas phase superficial velocity. Because gas blocks and liquid slugs dramatically oscillate in churn flow, the turbulence characteristic is more obvious and the flow structure is more complex. Consequently, the MSCE values in churn flow are higher than those in slug flow, which also coincides with the drastic fluctuations of determinism with varying k values in churn flow (Fig. 9). Nevertheless, due to the existence of gas blocks in churn flow, the MSCE values are smaller in contrast to those in bubble flow.

In order to further elucidate the discrepancies in the turbulence characteristics between slug and churn flows, average MSCE and the rate of MSCE in Figures 12 and 13 are extracted, respectively. The rate of MSCE is defined as the slope of the linear fitting line determined by the MSCEs at the scale range from 1 to 4. The average MSCE and rate of MSCE in Figures 12 and 13 are illustrated in Figures 14 and 15 , respectively. It can be seen from Figure 14 that as for low gas and liquid phase superficial velocities, the flow pattern presents as slug flow with small average MSCEs and rate of MSCEs, indicating flow structures with low complexities and weak turbulence characteristics. However, with the increment in liquid phase superficial velocity, the lengths of the gas slugs shorten and the motion instability in the mixed fluid increases. When the liquid phase superficial velocity rises to 0.5888 m/s and 0.736 m/s, the flow pattern evolves to slug-bubble transition flow with obvious increments in both average MSCE and rate of MSCE. As the liquid phase superficial velocity further increases, the flow pattern eventually exhibits bubble flow. As the gas phase exists in the pipe with the formation of bubbles, the motion randomness of which is the strongest. Correspondingly, both average MSCE and rate of MSCE are at the highest values.

Figure 14:

Average MSCE and rate of MSCE under a low gas phase superficial velocity.

Figure 15:

Average MSCE and rate of MSCE under a high gas phase superficial velocity.

The variation trends of average MSCE and rate of MSCE versus liquid phase superficial velocity under a high gas superficial velocity are shown in Figure 15. As can be seen, the flow pattern still presents as slug flow at low liquid phase superficial velocities. Nevertheless, both average MSCE and rate of MSCE are equal to higher values compared with the results in Figure 14. This result shows that the turbulence characteristic in slug flow is more obvious under high gas phase superficial velocity. With the mixture velocity increasing to 0.736 m/s, the flow pattern presents as churn flow with drastic oscillation of gas blocks and liquid slugs, leading to sharp increments in average MSCE and rate of MSCE. The flow pattern finally becomes bubble flow when liquid phase superficial velocity rises to 1.1776 m/s and average MSCE as well as rate of MSCE reach maximum values. In addition, both average MSCE and rate of MSCE in bubble flow are smaller than those with a low gas phase superficial velocity (Fig. 14). This can be attributed to the fact that gas bubbles are large in size under high gas superficial velocity and, correspondingly, the instability in the motion of large gas bubbles is not as strong as that of small ones. Hence, average MSCE and rate of MSCE are comparatively smaller.

## 7 Conclusion

We experimentally investigated the slug and churn turbulence characteristics in oil–gas–water three-phase flow in a vertical upward pipe with a small ID. A cross recurrence plot was applied to allow flow pattern identification. Cross recurrence quantification and MSCE analysis were conducted to explore turbulence characteristics especially in slug and churn flows. Our conclusions can be stated as follows:

1. The motion of gas and liquid slugs present quasi-periodic characteristics in slug flow and the corresponding texture in cross recurrence plot is composed of many black rectangles. The cross recurrence plot texture of bubble flow exhibits scattering points which correspond to the random motion in gas bubbles. Due to the drastic oscillation of gas blocks and liquid slugs in churn flow, the cross recurrence plot texture shows line structures.

2. The turbulence characteristic of slug flow is unobvious under a low gas phase superficial velocity, while it becomes remarkable for a high gas phase superficial velocity. In addition, under a high gas phase superficial velocity, the flow pattern evolves from slug to churn flow with the increasing liquid phase superficial velocity and the turbulence characteristic is more remarkable. Bubble flow presents the most remarkable turbulence characteristic and the strongest instability in flow structure.

3. The research results indicate that a cross recurrence plot is proved to be effective in identifying oil–gas–water three-phase flow patterns, while cross recurrence quantification and MSCE can be regarded as sensitive indexes for indicating the turbulence characteristics under different flow conditions. Moreover, understanding the fluctuations of cross recurrence quantification versus sampling time is conducive to uncovering the spatial–temporal evolution characteristic of flow structures in oil–gas–water three-phase flow.

## Acknowledgements

This study was supported by the National Natural Science Foundation of China (Grant Nos. 51527805, 11572220).

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Accepted: 2017-06-30

Published Online: 2017-08-08

Published in Print: 2017-08-28

Citation Information: Zeitschrift für Naturforschung A, Volume 72, Issue 9, Pages 817–831, ISSN (Online) 1865-7109, ISSN (Print) 0932-0784,

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©2017 Walter de Gruyter GmbH, Berlin/Boston.