PageRank versatility analysis of multilayer modality-based network for exploring the evolution of oil-water slug flow

Numerous irregular flow structures exist in the complicated multiphase flow and result in lots of disparate spatial dynamical flow behaviors. The vertical oil-water slug flow continually attracts plenty of research interests on account of its significant importance. Based on the spatial transient flow information acquired through our designed double-layer distributed-sector conductance sensor, we construct multilayer modality-based network to encode the intricate spatial flow behavior. Particularly, we calculate the PageRank versatility and multilayer weighted clustering coefficient to quantitatively explore the inferred multilayer modality-based networks. Our analysis allows characterizing the complicated evolution of oil-water slug flow, from the opening formation of oil slugs, to the succedent inter-collision and coalescence among oil slugs, and then to the dispersed oil bubbles. These properties render our developed method particularly powerful for mining the essential flow features from the multilayer sensor measurements.

presents a review of complex network analysis of time series. Specifically, in view of that many real-world systems exhibit multiple characteristics among a same set of components [29][30][31][32][33] , large attentions have been focused on the multilayer network [34][35][36] , where a set of nodes simultaneously exist in multiple network layers with different types of edges. This burgeoning network framework has exhibited its strong capacity in exploring the structure, dynamics and function of diverse complex systems involving epidemic spreading 37 , brain system 38 , transportation system 39 and social system 40 .
In this paper, aiming at characterizing the detailed flow structures and complicated spatial flow behaviors underlying the evolution of oil-water slug flow, we first carry out oil-water flow experiments and use our designed DLDSC Sensor system to acquire the abundant spatial-temporal flow information. And then we construct the multilayer modality-based network (MMBN) from the double-layer multi-channel measurements, where the modality determined in terms of the correlation between short-term measurements has close-knit connection with the flow structure. Moreover, we calculate the PageRank versatility 41 and multilayer weighted clustering coefficient, respectively, for all derived MMBNs. PageRank versatility allows characterizing the multilayer network from the perspective of node versatility. The results validate the compact connection between the local features of MMBNs and the intricate spatial flow behaviors in the oil-water slug flow. These properties render our developed method particularly useful for probing the intrinsic flow behaviors from multilayer sensor measurements.

Experiments
In order to capture the spatial flow information of the oil-water slug flow, we design a DLDSC Sensor and conduct oil-water flow experiments in a vertical 20-mm-diameter pipe. As displayed in Fig. 1, the flow loop facility mainly contains three parts: namely, three tanks for storing tap-water, oil and mixture fluid, respectively; the pipe and valves for connecting and building the flow loop; the designed sensor (denoted as DLDSC Sensor) and a high-speed video camera for capturing the requisite spatial flow image information. The DLDSC Sensor, consisting of the up-structure sectors and down-structure sectors, is specially designed for acquiring spatial oil-water flow information from different pipe positions. In our experiments, the water-cut and total flow velocity are set as two pivotal parameters to adjust and generate flow conditions. Note that, we increasingly study 8 discrete total flow velocities in the range of 0.0184 m/s-0.2579 m/s for each specified water-cut and the water-cut is set as 60%, 70%, 80%, 82% and 84%, respectively. For each flow condition, we first pump the tap-water and oil into the vertical pipe via two peristaltic metering pumps, respectively. Before the mixture flow arrives at the measuring section, namely the sensor position, oil and water adequately mix with each other and reach a stable flow state (flow structure). Then the double-layer multi-channel measurements of oil-water flows are acquired via the DLDSC Sensor system. Meanwhile, the snapshots from high-speed video camera are used for helping classify different experimental flow structures. Finally, the mixture flow is drained into the mixing tank where oil and water would naturally separate due to the action of gravity.
Exploring the evolution of oil-water slug flow via MMBN. Based on the experimental double-layer multi-channel measurements, we infer MMBN for each experimental flow condition, aiming at uncovering the intricate spatial flow structures and dynamical flow behaviors associated with the evolution of oil slugs in oil-water flows. A schematic diagram of our method is presented in Fig. 2. We emphasize here that each modality in our MMBNs exactly corresponds to one specific configuration of oil and water phase in the experimental circular pipe.
From the perspective of node versatility, PageRank versatility 41 successfully serves as a good descriptor of dynamical aspects of complex system. We here use a 4th-order tensor 42 β α M j i to encode a directed, weighted connection between node i from layer α to any other node j in layer β (including β = α). The network has L layers and each layer has N nodes. We indicate with β α D j i the strength tensor whose entries are all zeros, except for i = j and which allows to provide a one-to-one relationship between the intra-and inter-layer links in β α M j i with the transition probabilities in β α T j i . We use β α A j i to denote the appearance of dangling nodes which have no forward links. Particularly, if node i in layer α is a dangling node, we set β α A j i to 1 for all variational j and β, otherwise 0. Note that a dangling node corresponds to a row in β α M j i with all entries equal to 0, while a row in β α A j i with all entries equal to 1. To eliminate the interference of dangling node, we conduct the following operation: This means that the random surfer escapes from the dangling page by jumping to a randomly chosen page. β α S j i is called resulting matrix.
Assuming that the walker jumps to a neighbor with rate r and teleports to any other nodes in the multilayer network β α M j i with rate 1−r, we construct the transition tensor β α R j i (a rank-4 tensor): where β α u j i is a rank-4 tensor with all components equal to 1. We use r = 0.85 as in the classical PageRank algorithm. Specifically, mathematically, the iterative procedure can be replaced via calculating the largest eigenvalue and corresponding eigenvector of the adjacency matrix. So then we calculate the eigentensor Ω iα (related to the largest eigenvalue) of the transition tensor β α R j i , denoting the steady-state probability to find the walker at node i of layer α. The multilayer PageRank versatility is obtained by contracting the layer index of the eigentensor with the 1−vector u α : PR i = Ω iα u α , i.e., summing up over layers.
The PageRank versatility distribution for two different flow conditions, belonging to two typical oil-water flow patterns respectively, are displayed in Fig. 3. The interval of the PageRank versatility is 0.02. The distribution of these two typical flow patterns display similar variation trend, arising from the fact that they all belong to oil-water flows with water as the continuum. For instance, they all present an obvious peak at the start of the distribution. This phenomenon can be considered as the mirror of the common and mutual flow structures in both flow patterns, such as continuous water with tiny oil droplets. However, we also find that different significant fluctuations appear after the peak, attributed to the fact that each flow pattern has its typical and special flow structures and behaviors. In order to further explore these significant fluctuations in the complicated evolution of oil slugs, we calculate the standard deviation of the part PageRank versatility distribution series (i.e., the PageRank versatility distribution series except for the first 9 points in that these 9 points in the peak correspond to the common features of oil-in-water flows) for each flow condition. And the results for these two flow patterns are shown in Fig. 4

via boxplots.
Clustering coefficient, serving as one of the most popular network measures, can effectively quantify the tendency of nodes to form triangles. According to ref. 43 , the C MW of node i is defined as:  We note here that L = 2 in our MMBN. The concept of two-triangle is defined as follows: a triangle which is formed by an edge belonging to one layer and two edges belonging to a second layer 43 . Averaging this quantity over all the nodes in the network, we get the multilayer average weighted clustering coefficient:

MW i MW
Correspondingly, standard deviation of C MW (i) can be presented as: where the number of nodes in each layer M k is N. We note that <C MW > provides an entire evaluation for the connectivity of the considered MMBN, and the σ(C MW ) can exactly reflect the discrepancy among the modalities (i.e., nodes). We display the calculated network measures in Figs 5, 6, 7, 8 and 9 for different fixed water-cuts, respectively. The distributions of network measures provide a quantitative mapping of the evolution of oil slugs in oil-water flows. In concrete terms, when the total flow velocity is low, the mixture flow appears as oil-water slug flow. As can be seen in     become dispersed oil bubbles flowing in a water continuum. Namely, the oil slugs gradually break into stochastic oil droplets. In addition, based on the standard deviation of the part PageRank versatility distribution series for different flow patterns, we calculate the p-value based on the t-test, as shown in Fig. 4. The p-value is obviously smaller than 0.05 (a benchmark for determining the significant difference from the p-value), indicating that these two oil-water flow patterns have obvious difference from the perspective of PageRank versatility. From the perspective of clustering coefficient and PageRank versatility, we effectively uncover and present the complicated evolution of oil slugs, from the opening formation of oil slugs, to the succedent inter-collision and coalescence among oil slugs, and then to dispersed oil bubbles. In short, for a specified water-cut (K w ), the increase of total flow velocity (V m ) from 0.0184 m/s to 0.2579 m/s leads to a series of complicated changes of flow states underlying the evolution of oil-water slug flow. One step closer, we study the top 6 critical modalities, determined in terms of the PageRank versatility, in each flow condition. The statistical and analytical processes refer to Fig. 10. The majority critical modalities are shared in both flow patterns, as shown in Table 2 and Table 4 of Fig. 10, which agrees with our above-mentioned analysis that there exist many common flow structures between these two flow patterns arising from continuous water phase. However, some special critical modalities only appear in specific flow pattern declaring the differences between the oil-water slug flow and bubble flow. Then we break down the critical modalities and figure out some interesting phenomena. We note that a modality is a permutation of 6 different correlation coefficients (see methodology part for details). Particularly, in the slug flow, there are significant differences (e.g., A1 is 9, while D5 is 2) in the occurrence frequency of different correlation coefficients at different locations, while in the bubble flow, the number of each correlation coefficient in each position is relatively uniform and all close to 5. All these features again validate the heterogeneous in the slug flow while the randomness in the bubble flow.

Discussion
How a big oil slug can break into oil droplets with diverse sizes and shapes constitutes a fundamental but challenging problem of great importance in field of oil exploration. Based on the double-layer multivariate measurements acquired from our designed DLDSC Sensor system, we infer the MMBN and correspondingly calculate the PageRank versatility and multilayer weighted clustering coefficient, aiming at quantitatively characterizing the detailed flow structures and complicated spatial flow behaviors underlying the evolution from oil-water slug flow to bubble flow. The results demonstrate that these two multilayer network measures can effectively characterize the complicated evolution of oil-water slug flow. Our method provides a novel way for fusing the spatial multivariate sensor measurements, which allows characterizing the complicated dynamical behaviors of complex systems.

Methods
Multilayer modality-based network. As a development of previously modality transition-based network theory 1 , we infer the multilayer modality-based network (MMBN) from the double-layer multivariate measurements. Spatial complicated transitional behaviors of oil-water flows are efficiently mapped into the intra-and inter-layer edges in MMBN.
In concrete terms, we firstly partition the double-layer measurements via a sliding window (length 20), which slides along time by a step of 1. As shown in Fig. 2, the resulting sub-time series in each window consist of 4 channels for each layer, respectively. And we then determine the modality from the sub-time series in each window as follows: Firstly, we calculate the correlation between any two sub-time series by the following equation where n′ = 20 and all six correlation coefficients including r 12 ,r 13 ,r 14 ,r 23 ,r 24 and r 34 are set as the elements of a modality: We then rank the six correlation coefficients in one layer incrementally to obtain a modality. For instance, if  Fig. 2), corresponding to the double-layer measurements, respectively. Based on these two modality sequences, the directed and weighted intra-and inter-layer edges in MMBN are determined in terms of the direction and times of the transition among modalities. We define each modality as a node and set all the nodes in a fixed and identical order in all layers. Taking the up-layer W U of the MMBN as an example, we construct a directed intra-edge from M i U (e.g., modality α) to + M i U 1 (e.g., modality β) in i-th step and the αβ W U pluses one. So repeated transition between two different modalities (e.g., α and β) in different steps would let αβ W U have a large value. The above procedure allows us to infer a two-layer multilayer network with 720 nodes in each layer. In order to reduce the effect of unavoidable noise arising from the non-characteristic flow in different flow patterns, we first remove the edges with weight 1 and then delete the isolated nodes in the resulting MMBNs. This operation allows maintaining and highlighting the critical modalities for further exploration of spatial flow structures and behaviors.  Table 1 presents the top 6 critical modalities in 19 oil-water slug flow conditions; Table 2 shows the unique modalities from Table  1 (i.e., all the modalities that appear in Table1 are listed only once in Table 2); and we break down the unique modalities in Table 2 and map the results into Table 3, where the element A1 in the A row and the 1st column shows the statistic of the correlation coefficient A in the 1st position in unique modalities. And similarly, Table  4, Table 5, and Table 6 are for 29 oil-water bubble flow conditions.