Eigenvector centrality for geometric and topological characterization of porous media

Solving flow and transport through complex geometries such as porous media is computationally difficult. Such calculations usually involve the solution of a system of discretized differential equations, which could lead to extreme computational cost depending on the size of the domain and the accuracy of the model. Geometric simplifications like pore networks, where the pores are represented by nodes and the pore throats by edges connecting pores, have been proposed. These models, despite their ability to preserve the connectivity of the medium, have difficulties capturing preferential paths (high velocity) and stagnation zones (low velocity), as they do not consider the specific relations between nodes. Nonetheless, network theory approaches, where a complex network is a graph, can help to simplify and better understand fluid dynamics and transport in porous media. Here we present an alternative method to address these issues based on eigenvector centrality, which has been corrected to overcome the centralization problem and modified to introduce a bias in the centrality distribution along a particular direction to address the flow and transport anisotropy in porous media. We compare the model predictions with millifluidic transport experiments, which shows that, albeit simple, this technique is computationally efficient and has potential for predicting preferential paths and stagnation zones for flow and transport in porous media. We propose to use the eigenvector centrality probability distribution to compute the entropy as an indicator of the “mixing capacity” of the system.

Figure 1

(a) Binary image of the 2D porous medium, with pores or voids in black and grains in white. (b) Distances to nearest objects (DNO function) inside the pore space. (c) Pore network composed of edges connecting nodes (centroid of the pore) determined by the local maxima of the DNO function. (d) Corresponding graph for the defined pore network. The minimum distance normal to the edge (i.e., pore-throat size) provides the weight. The termination of a path, both in the pore network and in the graph, is called a dead-end.

Figure 2

Left: Millifluidic transport experiments under variably saturated conditions [26]. Area of the wetting phase reached by the tracer (flowing from left to right; entering across the entire $y/\varepsilon$ width at $x/\varepsilon =0)$ over the duration of the experiment (red areas), for different water saturation $S_{\mathrm{w}}$ values: (a) 1.00, (b) 0.83, (c) 0.77, and (d) 0.71. Areas without tracer present are shown in blue. Grains are depicted in gray, and air clusters in black. Right: Two-dimensional maps showing the $x$-biased SPEC function $\left[c\right(\mathbf{r}\left)\right]$ for each water saturation value. Warm colors (red and yellow) represent nodes and regions of the pore network with high eigenvector centrality values, i.e., high relative importance for flow and transport. Cold colors (green and blue) represent nodes and regions of the wetting phase with low eigenvector centrality values, i.e., low relative importance in the network. The pore network, including edges and nodes, defined for the wetting phase is plotted with black lines.

Figure 3

Eigenvector centrality $\left(c_i\right)$ probability density functions for different water saturation values $S_{\text{w}}$.

Figure 4

(a) Transport experiment and SPEC function $[c\left(\mathbf{r}\right)\ge 5.2\times {10}^{-3}]$ superimposed on the same image (for $S_{\text{w}}=0.83)$. The brown area has been invaded by the tracer and falls above the $c\left(\mathbf{r}\right)$ threshold considered; the purple area has not been invaded by the tracer and falls below the $c\left(\mathbf{r}\right)$ threshold; the red area has been invaded in the experiment but falls below the $c\left(\mathbf{r}\right)$ threshold; and the blue area has not been invaded in the experiment but falls above the $c\left(\mathbf{r}\right)$ threshold. (b) Variation of $\xi$ with $S_{\text{w}}$. Comparison between the fraction of the wetting phase with a low eigenvector centrality value (gray region defined from centrality thresholds, $c\left(\mathbf{r}\right)\ge 4\times {10}^{-3}$ and $c\left(\mathbf{r}\right)\ge 5.2\times {10}^{-3})$ and the fraction of the wetting phase not involved in the solute transport process for the duration of the experiments (red circles) [26].

Figure 5

Different degrees of connectivity. Connectivity is limited to (a) first, (b) second, (c) third, and (d) fourth nearest neighbors. Weights $w_{ij}$ are chosen at random between 0.5 and 1.0.

Figure 6

${\lambda }_N$ as a function of the lattice connectivity (number of neighbors).

Figure 7

Comparison between (a) a pristine first-neighbor-connected lattice and (b) a regular graph with randomized weights where $0.5\le w_{ij}\le 1.0$.

Figure 8

(a) Original compared to (b) corrected weights for a network where $0.5\le w_{ij}\le 1.0$ and with $\delta ={10}^{-4}$.

Figure 9

Correction of the centrality where $0.5\le w_{ij}\le 1.0$ and with $\delta ={10}^{-4}$.

Figure 10

Histogram of edge weights $w_{ij}$ computed for the pore network with $S_{\text{w}}=0.83$ (see the text). A medium value of $w\simeq 0.9$ is shown.

Figure 11

Lorentzian fit of a normalized histogram of the deviation of node position. The Lorentzian (red curve) with $\mathrm{\Gamma }=10.8$ adjusts very well to the normalized histogram. This distribution can be seen as the probability of finding node $i$ at position ${\mathbf{r}}_i$.

Figure 12

(a) Unbiased and (b) $x$-biased spatially projected eigenvector centrality (SPEC) function for a random weighted graph where $0.5\le w_{ij}\le 1.0$ and with $\delta ={10}^{-4}$.

Figure 13

(a) Centrality measure of each node of the pore network and (b) spatially projected eigenvector centrality (SPEC) function for the saturation degree $S_{\text{w}}=1.00$. Note that the scales are different because the SPEC function smooths out very high and low values.