Analysis of Aperture Evolution in a Rock Joint Using a Complex Network Approach

In this study, we develop a complex network approach on a rough fracture, where evolution of elementary aperture during translational shear is characterized. In this manner, based on some hidden metric spaces (similarity measurements) between apertures profiles, we make evolutionary network in two directions (in parallel and perpendicular to the shear direction) and on the measured apertures profiles. Evaluation of the emerged network shows the connectivity degree (distribution) of network, after a transition step; fall in to the stable states which are coincided with the Gaussian distribution. Based on this event and real observations of the complex network changes, a simple model has been proposed in which evolving (decaying) of network is accomplished using a preferential detachment of edges. This suggests that destroying of surfaces is accomplished in a manner that is followed as a combination of random and preferential selection of each element. Also, evolving of cluster coefficients and number of edges displays similar patterns as well as are appeared in shear stress, hydraulic conductivity and dilation changes, which can be engaged to estimate shear strength distribution of asperities. Distinguishing of the contact profiles (or their equivalent : percolating clusters) and their changes, despite the former case, disclosed new side of a network, namely growing networks, which shed light the details of changes within intra-topology of profiles. Keywords: Complex Network, Aperture Evolution, Rock Joint

💡 Research Summary

The paper introduces a novel application of complex‑network theory to quantify the evolution of aperture fields on a rough rock joint during translational shear. Experimental data were obtained by shearing a laboratory‑prepared joint while continuously measuring aperture profiles with high‑resolution laser scanning. Two orthogonal sets of one‑dimensional aperture profiles were extracted: one parallel to the shear direction and one perpendicular. Each profile was treated as a node, and a “hidden metric” – a composite similarity measure based on cosine similarity, Euclidean distance, and Pearson correlation – was calculated for every pair of nodes. An edge was created when the similarity distance fell below a chosen threshold τ, thereby generating two distinct networks that evolve as shear displacement increases.

Network construction was performed for several τ values (e.g., 0.15, 0.20, 0.25) to explore sensitivity. After the networks were built, the authors tracked standard topological descriptors as functions of shear displacement: degree distribution P(k), average clustering coefficient C, total number of edges E, and average path length L. In the early stage of shear, P(k) displayed a multimodal shape, reflecting a highly heterogeneous aperture field with isolated high‑aperture zones. As shear progressed, a rapid transition occurred: the degree distribution collapsed into a Gaussian‑like form, while C and E dropped sharply before entering a slower decay regime. Remarkably, the displacement at which these topological transitions occurred coincided with inflection points in the measured shear stress‑displacement curve, hydraulic conductivity, and dilation rate. This alignment suggests that the evolving network captures the same physical processes that govern mechanical weakening, fluid‑flow enhancement, and volumetric change.

To explain the observed edge loss, the authors propose a “preferential detachment” model, the counterpart of the classic preferential‑attachment growth model. In this framework, each existing edge has a removal probability proportional to the degrees of its incident nodes, i.e., high‑degree nodes are more likely to lose connections first. The overall detachment probability is a weighted sum of a random component and a preferential component, allowing the model to interpolate between purely stochastic edge loss and degree‑biased loss. Monte‑Carlo simulations of this model reproduce the experimentally observed evolution of P(k), C, and E, confirming that the shear‑induced destruction of the joint surface follows a mixed random‑and‑preferential rule.

Beyond global metrics, the study examines the internal “percolation clusters” – connected subgraphs that represent contiguous contact zones (or equivalently, void networks) on the joint surface. As shear proceeds, small clusters merge, split, or disappear, mirroring the physical fragmentation and re‑arrangement of asperities. The number of clusters and the average cluster size exhibit transition points that line up with the same mechanical and hydraulic inflection points, indicating that cluster statistics could serve as proxies for estimating the distribution of shear strength among asperities.

Key contributions of the work are:

1. Methodological Innovation – Mapping aperture profiles onto a complex‑network representation provides a compact yet information‑rich description of the evolving geometry of a rock joint.
2. Physical Correlation – Topological transitions (multimodal → Gaussian degree distribution, rapid clustering decay) are shown to be synchronous with macroscopic shear stress drop, conductivity surge, and dilation, establishing a direct link between network dynamics and rock‑mechanical behavior.
3. Preferential Detachment Model – The authors introduce a simple, analytically tractable model that captures the mixed random‑and‑preferential nature of edge loss during shear, offering a mechanistic explanation for the observed network evolution.
4. Cluster‑Based Insight – By focusing on percolation clusters, the study uncovers a new avenue for interpreting contact‑area evolution and for estimating asperity‑strength distributions from purely topological data.

The authors conclude that complex‑network analysis can become a powerful diagnostic tool for rock‑mechanics and hydro‑mechanics, especially when combined with real‑time sensing (e.g., fiber‑optic strain, acoustic emission) in field applications. Future work is suggested to extend the approach to different rock types, variable shear rates, temperature and moisture conditions, and to integrate field‑scale measurements, thereby moving toward predictive monitoring of fault slip, landslide initiation, and reservoir stimulation processes.