Self-affine roughness and spatial-temporal correlation of shear-induced micro-fractures in rough DEM rock joints

Abstract

We investigate the spatial and temporal correlations of micro-fractures generated during shear simulations of rough rock joints using the Discrete Element Method (DEM). Eight numerical joints were meticulously designed, characterized by distinct self-affine roughness parameters: the Hurst exponent $H$, self-affine correlation length $L_c$, and variance of heights ${\sigma }^2$. Shearing was conducted under quasi-static conditions, maintaining constant normal load (CNL) and constant shear speed. The DEM model underwent calibration to faithfully replicate the mechanical behavior observed in experimental compression, tension and shear tests on synthetic rock. Remarkably, our results unveil a power-law behavior in both spatial and temporal dimensions. The probability distribution $P\left(\Delta d\right)$ describing the separation distance $\Delta d$ between two fractures follows a power-law relationship: $P\left(\Delta d\right){\Delta d}^{-q}$, where the exponent $q$ ranges between 0.4 and 0.8, displaying a marked sensitivity to joint roughness. Additionally, the probability $p\left(\Delta t\right)$ of two fractures occurring within a time interval $\Delta t$ during shearing also conforms to a power-law distribution: $P\left(\Delta t\right){\Delta t}^{-p}$. For mechanically interacting micro-fractures, $n$ ranges from 0.716 to 0.869, while for micro-fractures generated across the entire joint area (not necessarily interacting fractures), $n$ takes values between 1.458 and 1.895. Notably, these findings underscore a remarkable analogy with the Omori law in seismic activity.