Rupture Cascades in a Discrete Element Model of a Porous Sedimentary Rock

We investigate the scaling properties of the sources of crackling noise in a fully dynamic numerical model of sedimentary rocks subject to uniaxial compression. The model is initiated by filling a cylindrical container with randomly sized spherical particles that are then connected by breakable beams. Loading at a constant strain rate the cohesive elements fail, and the resulting stress transfer produces sudden bursts of correlated failures, directly analogous to the sources of acoustic emissions in real experiments. The source size, energy, and duration can all be quantified for an individual event, and the population can be analyzed for its scaling properties, including the distribution of waiting times between consecutive events. Despite the nonstationary loading, the results are all characterized by power-law distributions over a broad range of scales in agreement with experiments. As failure is approached, temporal correlation of events emerges accompanied by spatial clustering.

Figure 1

(a) Preparation of the sample by sedimenting spherical particles with randomly distributed radius $R$. The color code corresponds to the radius of the particles: the smallest particles are dark blue whereas the biggest ones have light red color. (b) Comparison of the radius distribution $p(R)$ obtained numerically (symbols) to the desired log-normal functional form (continuous line). (c) Histogram $p(n_c)$ of the number of contacts $n_c$ in a sample of $N=20000$ particles. Exponential function (green line) was fitted for $n_c>3$.

Figure 2

Constitutive behavior $\sigma (ϵ)$ of a single sample together with the series of burst size $\mathrm{\Delta }$ (red bars) as a function of the strain of their appearance. The moving average of $\mathrm{\Delta }$ (blue line) was calculated over 50 consecutive events. The inset illustrates the loading condition.

Figure 3

Probability distributions of the characteristic quantities of bursts occurring before and after the peak of $\sigma (ϵ)$: distribution of burst (a) size $p(\mathrm{\Delta })$, (b) energy $p(E)$, (c) duration $p(T)$, and (d) waiting time $p(t_w)$. The red lines represent fits with the functional form of Eq. (2).

Figure 4

(a) The average energy and duration of bursts as function of their size. (b) Average waiting times separately calculated before $⟨t_w^b⟩$ and after $⟨t_w^a⟩$ events as a function of $\mathrm{\Delta }$. The straight lines in (a) and (b) are fits with Eqs. (3) and (4). (c) Average distance between consecutive events and the average size of bursts as function of strain. The vertical line indicates the peak position of $\sigma (ϵ)$. (d) The correlation integral $C(r)$ for windows of 200 consecutive events.