Fluid flow analysis in a rough fracture (type II) using complex networks and lattice Boltzmann method

Complexity of fluid flow in a rough fracture is induced by the complex configurations of opening areas between the fracture planes. In this study, we model fluid flow in an evolvable real rock joint structure, which under certain normal load is sheared. In an experimental study, information regarding about apertures of the rock joint during consecutive 20 mm displacements and fluid flow (permeability) in different pressure heads have been recorded by a scanner laser. Our aim in this study is to simulate the fluid flow in the mentioned complex geometries using the lattice Boltzmann method (LBM), while the characteristics of the aperture field will be compared with the modeled fluid flow permeability To characterize the aperture, we use a new concept in the graph theory, namely: complex networks and motif analysis of the corresponding networks. In this approach, the similar aperture profile along the fluid flow direction is mapped in to a network space. The modeled permeability using the LBM shows good correlation with the experimental measured values. Furthermore, the two main characters of the obtained networks, i.e., characteristic length and number of edges show the same evolutionary trend with the modeled permeability values. Analysis of motifs through the obtained networks showed the most transient sub-graphs are much more frequent in residual stages. This coincides with nearly stable fluid flow and high permeability values.

💡 Research Summary

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This paper presents an integrated investigation of fluid flow through a rough rock fracture by combining laboratory experiments, lattice Boltzmann simulations, and complex‑network analysis of the aperture field. A granite joint was fabricated and subjected to a constant normal load while being sheared in incremental displacements of 2, 5, 10 and 15 mm. At each step a high‑resolution laser scanner (0.2 mm pixel size) measured the three‑dimensional topography of the two fracture faces, allowing the authors to reconstruct the spatial distribution of the aperture (void space) between them. Simultaneously, hydraulic tests were performed by imposing a pressure gradient parallel to the shear direction and measuring the resulting flow rate, from which the permeability K was obtained via Darcy’s law.

For the numerical part, the authors employed a two‑dimensional, single‑relaxation‑time lattice Boltzmann method (LBM) with a D2Q9 velocity set. The evolution equation follows the standard BGK collision operator, with the relaxation time τ linked to the kinematic viscosity ν through ν = (τ − 0.5)cs²Δt/3. The lattice spacing and time step were chosen to match the experimental resolution, and no‑slip or pressure‑driven boundary conditions were applied at the inlet and outlet. The simulation proceeds until the relative change in the spatially averaged velocity falls below 10⁻⁹, ensuring a steady‑state flow field. Darcy’s law is then used to compute the effective permeability from the volume‑averaged velocity field. Two modeling strategies were compared: (i) a “full‑field” approach that directly maps the binary pore space (solid = 0, fluid = 1) obtained from the scanned apertures onto the LBM lattice, and (ii) a “slice” approach that replaces the full geometry by a single channel built from the mean aperture profile taken along the flow direction. Both strategies yielded permeability values that correlate strongly with the experimental measurements (R² > 0.95), with the full‑field simulations being slightly more accurate (≈5–10 % improvement) than the slice approximation.

The novel contribution of the work lies in the construction of a complex network from the aperture data. Each aperture profile taken along the shear direction is treated as a node. Pairwise Pearson correlation coefficients Cij between profiles are computed; if Cij exceeds a preset threshold ξ = 0.2, an undirected edge is placed between nodes i and j. This procedure generates an adjacency matrix A, from which the degree matrix D and the graph Laplacian L = D − A are derived. The authors calculate the average shortest‑path length L̄ using Dijkstra’s algorithm, which quantifies the overall connectivity of the network. Spectral analysis of L provides eigenvalues and eigenvectors; the inverse participation ratio (IPR) of each eigenvector is used to assess the degree of localization of the corresponding mode. Additionally, sub‑graph (motif) analysis is performed on 3‑node and 4‑node patterns. The frequency of transient motifs—sub‑graphs that appear more often than expected in random networks—rises markedly in the later (residual) shear stages, coinciding with the plateau observed in permeability.

The results reveal a clear correspondence between the evolving fracture geometry, the LBM‑derived permeability, and the network metrics. As shear displacement increases, contact area between the two faces diminishes, the mean aperture grows, and the number of edges in the network declines. Consequently, the average shortest‑path length shortens, indicating that fluid preferentially follows more direct channels with lower hydraulic resistance. This “channelization” effect is evident in the velocity and pressure fields produced by LBM, where high‑velocity streams align with regions of minimal aperture. In the early shear stages (≤ 2 mm) the interlocking asperities cause a reduction in aperture, leading to a temporary drop in permeability; the network reflects this by an increase in edge count and path length. Beyond ≈10 mm displacement, both permeability and network measures reach a quasi‑steady state, and the motif analysis shows a dominance of transient sub‑graphs, suggesting a stable flow regime.

In summary, the study demonstrates that (1) LBM can accurately reproduce experimentally measured permeability for realistic, evolving fracture geometries; (2) a graph‑theoretic representation of aperture profiles captures the structural evolution of the fracture and provides metrics (edge count, characteristic path length) that track permeability changes; and (3) motif analysis offers additional insight into the transition from a highly heterogeneous flow network to a more ordered, channel‑dominated system. The authors propose that extending this framework to three‑dimensional scans, multiphase flow, and non‑linear loading conditions could further enhance predictive capabilities for hydraulic fracturing, geothermal reservoirs, and other geotechnical applications where fracture flow governs system performance.