Triggering Processes in Rock Fracture

We study triggering processes in triaxial compression experiments under a constant displacement rate on sandstone and granite samples using spatially located acoustic emission events and their focal mechanisms. We present strong evidence that event-event triggering plays an important role in the presence of large-scale or macrocopic imperfections, while such triggering is basically absent if no significant imperfections are present. In the former case, we recover all established empirical relations of aftershock seismicity including the Gutenberg-Richter relation, a modified version of the Omori-Utsu relation and the productivity relation—despite the fact that the activity is dominated by compaction-type events and triggering cascades have a swarmlike topology. For the Gutenberg-Richter relations, we find that the $b$ value is smaller for triggered events compared to background events. Moreover, we show that triggered acoustic emission events have a focal mechanism much more similar to their associated trigger than expected by chance.

Figure 1

PDF of the interevent time ratio $R$ for different conditions on the magnitudes of the events evaluated for the different experiments. In all panels, the thick solid and dotted lines correspond to the 95% confidence intervals of a uniform distribution (based on Poissonian errors) for the largest $\mathrm{\Delta }m$ threshold ($\mathrm{\Delta }m=m_{i+1}-m_i$) and the smallest $m$ threshold (or the unconditioned case as indicated by color), respectively. These typically correspond to the largest and smallest uncertainties of all data sets shown in a panel.

Figure 2

Density plots of the set $\{n_j^{*}={\tau }_i{l}_i\}$ for experiments with and without large scale imperfections (using $D_f=2.3$ and the respective $b$ values [see, e.g., Fig. 3)], represented in $\log \tau –\log l$ space as defined in Eq. (1). Data are denser in yellow and red regions and tend to zero density in blue regions. Time is measured in seconds, distances in millimeters. The straight lines (obtained from shuffled catalogs, Fig. S1 [37]) separate the two different populations of triggered events (below the line) and background events (above the line).

Figure 3

Sample Be7a: (a) distribution of distances of triggered events from a large-scale imperfection (notch) for different vertex depths $d$ in the triggering cascades. (b) Scatter plot of the average leaf depth $⟨d{⟩}_f$ as a function of the number of events in the triggering cascade $N_{\mathrm{TC}}$. (c) Distribution of rotation angles $\delta$ between the trigger and triggered event and for randomly selected pairs (background events only and all events, respectively). (d) Frequency-magnitude distribution for different AE events. $M$ is the magnitude of the trigger (main shock). The estimated $b$ values are $1.08\pm 0.08$ (background events) and $0.75\pm 0.1$ (triggered events).

Figure 4

Sample Be7a: triggering rate as a function of time averaged over different magnitude ranges of the trigger (main shock). Solid symbols correspond to a shorter subcatalog containing only about the first third of events. Lower inset: rates rescaled according to the magnitude of the trigger. An exponential (dashed line) is shown for comparison. Upper inset: productivity relation giving the average number of events triggered as a function of the trigger magnitude.