Complex network analysis of phase dynamics underlying oil-water two-phase flows

Characterizing the complicated flow behaviors arising from high water cut and low velocity oil-water flows is an important problem of significant challenge. We design a high-speed cycle motivation conductance sensor and carry out experiments for measuring the local flow information from different oil-in-water flow patterns. We first use multivariate time-frequency analysis to probe the typical features of three flow patterns from the perspective of energy and frequency. Then we infer complex networks from multi-channel measurements in terms of phase lag index, aiming to uncovering the phase dynamics governing the transition and evolution of different oil-in-water flow patterns. In particular, we employ spectral radius and weighted clustering coefficient entropy to characterize the derived unweighted and weighted networks and the results indicate that our approach yields quantitative insights into the phase dynamics underlying the high water cut and low velocity oil-water flows.

obtain local flow information (multi-channel signals) from different flow conditions. We first use multivariate pseudo Wigner distribution to probe the transient flow behaviors of different flow patterns from the perspective of energy and frequency. Then, we infer unweighted and weighted complex networks from multivariate measurements in terms of the phase lag index. We employ spectral radius and weighted clustering coefficient entropy to access to the networks derived from different flow conditions. The results demonstrate that our analysis allows characterizing the coupled phase dynamic behaviors associated with the evolution and transition of different oil-water flow patterns. These properties render our designed sensor and analytical framework particularly useful for uncovering the phase dynamics of high water cut and low velocity oil-water two-phase flow.

Results
Experimental design and data acquisition. We carry out the oil-water two-phase flow experiment in a vertical upward 20 mm-inner-diameter plexiglass pipe at Tianjin University. The experiential media are tap-water and No. 3 white oil. Figure 1 shows our experimental flow loop facility which mainly consists of the following parts: a water tank, an oil tank, two peristaltic metering pumps, a mixing tank, a vertical testing pipe, and our designed HCMC sensor (High-speed Cycle Motivation Conductance sensor). The inlet mixture flow velocity and water cut can be accurately measured by the high precision peristaltic metering pumps. Experiments are conducted by fixing a water cut and then adjusting the mixture flow velocity increasingly. For each experimental run, oil and water are drawn out from their own tanks, respectively. Thereafter the two phases mix themselves at the mixer section before they are delivered together to the vertical testing pipe. At the testing part, for each flow condition, 48-channel signals are acquired from different spatial positions and then are stored by our designed data acquisition devices. The high-speed camera accompanying with the visual observation is employed to help identify flow patterns. Three oil-in-water flow patterns have been observed, i.e., oil-in-water slug flow, oil-in-water bubble flow, oil-in-water VFD flow (very fine dispersed oil-in-water bubble flow). Specially, the installation positions of our designed sensor and high-speed camera are chosen elaborately, aiming to record fully developed flow structures. Finally, the mixture flow is discharged to the mixing tank where oil and water will separate from each other due to gravity difference.
Multivariate time-frequency representation of flow patterns. Based on the experimental multi-channel signals measured from high-speed cycle motivation 8-electrodes sensor, we using multivariate pseudo Wigner distribution 24 calculate the multivariate time-frequency representation for three typical oil-in-water flow patterns. Multivariate time-frequency representation allows us to access to the coupled transient features of multi-channel signals. The calculated results are shown in Figs 2-4. As can be seen, the distributions of energy and frequency for different flow patterns exhibit distinct features. Vertical oil-in-water slug flow occurs at low oil-water mixture flow velocity and high oil cut, where the oil phase exists in the form of slugs whose diameters nearly equal to the pipe diameter. Its dominant characteristics lie in slowly quasi-periodic movements and obvious intermittent oscillations. Consequently, as shown in Fig. 2, the multivariate time-frequency representation of oil-in-water slug flow exhibits the features of low frequency band, i.e., 0-2 Hz, and high energy values with an intermittent distribution (in the range of 0-0.6). With an increase in mixture flow velocity, the oil slug gradually becomes unstable resulting from the increased turbulent energy, and then oil slugs are dispersed into small oil droplets, i.e., oil-in-water bubble flow occurs. Figure 3 shows the multivariate time-frequency representation of oil-in-water bubble flow. Notably, the energy distribution of oil-in-water bubble flow presents a relative dispersive feature with the upper energy value decreasing to 0.12, and meanwhile the frequency band increases to 0-4 Hz, suggesting the intermittent oscillation becomes unobvious and lots of oil droplets flow in a faster and stochastic way. With the further increase of mixture flow velocity, the oil droplets are eventually broken into much smaller oil droplets as the flow evolves into oil-in-water VFD flow. The energy of this flow pattern further decreases (with the upper energy value being 0.001), as shown in Fig. 4, and meanwhile the dispersive feature in the energy distribution becomes more obvious and the frequency band increases to 0-8 Hz. These features attribute to the fact that the local transient flow behavior becomes more stochastic induced by the increased turbulent energy. These Complex network analysis of phase dynamics underlying different flow patterns. Our method allows inferring phase-based two-phase flow networks from experimental multi-channel measurements. The derived complex network can be unweighted or weighted. We in this paper construct both unweighted and weighted networks to characterize the phase dynamics underlying different oil-water flow patterns. In order to quantitatively characterize the inherent phase dynamics, we employ spectral radius 25 to analyze the derived unweighted networks and develop a unique weighted clustering coefficient entropy to assess the inferred weighted networks. The spectral radius ρ (A) is defined as: are the eigenvalues of the unweighted matrix A. We recently proposed a clustering coefficient entropy 15 , and here we develop it to analyze weighted network, named as weighted clustering coefficient entropy: where N is the number of nodes and C W (v) is the weighted clustering coefficient 26 of a node v: where w v,j is the weight between node v and j, i.e., the element of weighted matrix W. We using the above method construct unweighted and weighted networks from our multi-channel measurements for different flow conditions, and then calculate the spectral radius and weighted clustering coefficient entropy. The results are shown in Figs 5-8, in which K w denotes the fixed water cut and V m represents mixture flow velocity. We can see that, the distributions of spectral radius and weighted clustering coefficient entropy present distinct features for three typical oil-in-water flow patterns. Vertical oil-in-water slug flow occurs at low water cut and oil-water mixture flow velocity, where small numbers of oil droplets simultaneously follow big cap shaped oil slugs. Its local flow behavior presents the features of intermittent oscillations and non-homogenous distribution. As can be seen from Fig. 5, the spectral radius and weighted clustering coefficient entropy gradually increase as the mixture flow velocity increasing from 0.0184 m/s to 0.1105 m/s, and then they fall down when the flow velocity reaches to 0.1474 m/s. Actually, as the flow velocity increases from a very low value, numbers of small slugs and oil droplets interact with each other and more coalescences occur to form long slugs, and flow behavior exhibits a more obvious quasi-periodic feature. Consequently, the increase of network measures at low flow velocities reflects that the intermittent oscillations and coupled phase dynamic behaviors become more obvious. When the flow velocity reaches to a critical value, large slugs are broken into small oil slugs due to the large turbulence energy and the quasi-periodic movements of oil slugs become weak. Therefore, the two network measures will fall down as the flow velocity reaching to 0.1474 m/s. With the continuously increase of mixture flow velocity, oil-in-water bubble flow occurs, where oil phase that exists in the form of discrete droplets randomly flows in a water continuum. As can be seen from Figs 5 and 6, the spectral radius and weighted clustering coefficient entropy gradually decrease in the transition from oil-in-water slug flow to oil-in-water bubble flow, indicating the intermittent oscillations of oil slugs gradually disappear and the stochastic movements of oil droplets become dominant, and meanwhile the coupled phase dynamic behavior has been weakened. With a further increase in mixture flow velocity, the oil droplets are broken into even smaller oil droplets as the flow evolves from oil-in-water bubble flow to VFD flow. In this flow pattern, lots of smaller oil droplets uniformly dispersed in the water continuous phase and randomly flow from the bottom up. Consequently, as shown in the Figs 7 and 8, the spectral radius and weighted clustering coefficient entropy decrease to the lowest values as the flow pattern evolves to oil-in-water VFD flow, suggesting the local flow behavior of VFD flow becomes more stochastic and correspondingly the coupled phase dynamic behavior gradually dies away. In addition, we interestingly find that for the oil-in-water bubble type flows at a high water cut, the spectral radius is sensitive to the change of water cut. For example, the spectral radius will decrease as the water cut increase from 93% to 96%. These interesting findings demonstrate that our phase-based network analysis allows characterizing the evolution and transition of flow patterns arising from high water cut and low velocity oil-water flows.

Discussions
In summary, we design a high-speed cycle motivation conductance sensor to measure very local flow information from different oil-water flow conditions. Based on the multi-channel measurements, we investigate the Then we introduce a methodology for inferring phase-based two-phase flow complex networks from multi-channel signals. In particular, we derive unweighted and weighted networks from different flow conditions and use spectral radius and weighted clustering coefficient entropy to characterize the inferred networks. Our results suggest that the clustering coefficient entropy and spectral radius allow faithfully representing and characterizing the change of phase dynamic states in the transitions of three oil-in-water flow patterns, which yields novel insights into the complicated coupled local flow behavior associated with flow velocity and water cut, especially for high water cut and low velocity vertical oil-water flows.

Methods
Multivariate pseudo Wigner distribution. For a multi-channel signal + t H j and its inverse can be described by: H j is the Hermitian transpose of a vector + t x ( ). The central frequency of the Wigner distribution for a multi-channel signals + t x ( ) at time t is as follows: We using the inverse Wigner distribution rewrite the formula (8) in the form of where the symbol "I" represents the imaginary part of a complex signal and φ ′ t ( ) n is the instantaneous frequency for each channel signal. In a similar way, the instantaneous bandwidth follows from Finally, we use a multivariate extension of the pseudo Wigner distribution 28 , where a window function is employed to evaluate formula (6), and therefore the multivariate pseudo Wigner distribution can be realized. , containing N sub-signals of equal length L. We first use the phase lag index (PLI) 16,17 to measure the phase correlation between the two sub-signals i and j as follows: [ sin( ( ))] (14) φ ∆ is the phase difference at time-point k between two sub-signals i and j, sign stands for signum function, < > denotes the mean value and || represents the absolute value. Instantaneous phases can be determined by Hilbert transformation. PLI is in the range of 0~1. If there does not exist any coupling between two signals, the value of the PLI equals to 0; In contrast, PLI will be 1 if there exists a completely consistent phase-lag coupling between two signals. We can obtain a phase correlation measure matrix P in terms of the rule P ij = PLI (i, j). We then construct a phase-based complex network by regarding each sub-signal as a node and determining the edges in terms of the strength of phase correlation between each pairs of signals. The derived phase-based network can be in the form of unweighted or weighted. By choosing a threshold, we can obtain an unweighted network matrix A or a weighted network matrix W from the phase correlation measure matrix P, and the rules of which read In particular, we select the threshold in the following rule 29 , σ = + r M n (17) where M is the mean of all PLI values and σ is the corresponding standard deviation and n = 0.15. An unweighted or weighted edge between node i and j exists if ≥ P r ij ; on the contrary, node i and j are disconnected if < P r ij . The above procedures allow us to infer phase-based two-phase flow networks from multi-channel signals.