Predicting mining collapse: Superjerks and the appearance of record-breaking events in coal as collapse precursors

The quest for predictive indicators for the collapse of coal mines has led to a robust criterion from scale-model tests in the laboratory. Mechanical collapse under uniaxial stress forms avalanches with a power-law probability distribution function of radiated energy $P\sim {E^{-}}^{\varepsilon }$, with exponent $\varepsilon =1.5$. Impending major collapse is preceded by a reduction of the energy exponent to the mean-field value $\varepsilon =1.32$. Concurrently, the crackling noise increases in intensity and the waiting time between avalanches is reduced when the major collapse is approaching. These latter criteria were so-far deemed too unreliable for safety assessments in coal mines. We report a reassessment of previously collected extensive collapse data sets using “record-breaking analysis,” based on the statistical appearance of “superjerks” within a smaller spectrum of collapse events. Superjerks are defined as avalanche signals with energies that surpass those of all previous events. The final major collapse is one such superjerk but other “near collapse” events equally qualify. In this way a very large data set of events is reduced to a sparse sequence of superjerks (21 in our coal sample). The main collapse can be anticipated from the sequence of energies and waiting times of superjerks, ignoring all weaker events. Superjerks are excellent indicators for the temporal evolution, and reveal clear nonstationarity of the crackling noise at constant loading rate, as well as self-similarity in the energy distribution of superjerks as a function of the number of events so far in the sequence $E_{sj}\sim n^{\delta }$ with $\delta =1.79$. They are less robust in identifying the precise time of the final collapse, however, than the shift of the energy exponents in the whole data set which occurs only over a short time interval just before the major event. Nevertheless, they provide additional diagnostics that may increase the reliability of such forecasts.

Figure 1

Time sequence of jerk events in coal under uniaxial stress. The spectrum contains 18 968 jerks (blue) and 21superjerks as record-breaking events. Superjerks are more energetic than any of the previous jerks. (a) shows the full data set; (b), (c) are enlarged images for initial and last stages.

Figure 2

Average number of superjerks $\langle N_n\rangle$ that occur before the $n\mathrm{th}$ jerk as a function of $n$ (as double logarithmic plot). The line represents a power law with an exponent $\alpha =0.29$. The relative increase of $\langle N_n\rangle$ at the highest value of $n$ may be correlated with the occurrence of the superjerk with $k=17$ which is indicated by the broken line.

Figure 3

Energy of superjerks $E_{sj}$ after the $n\mathrm{th}$ jerk as a function of $n$ on a double logarithmic scale. A power law $E_{sj}\sim n^{\delta }$ with $\delta =1.79$ is found over a large energy scale of eight decades.

Figure 4

Average value of the energy and the energy increment $\delta E^k$ [see Eq. (1)] of superjerks as a function of the rank $k$. A weak break of slope occurs near $k=17$.

Figure 5

Waiting time distribution functions for all events (black circles) and for jerks between superjerks. The data have been collapsed for large waiting times by a scaling factor $\lambda$. The probability distribution has been normalized by the total number of waiting times $N\left(m\right)$ which has been measured in each interval [bin size for each data set is twice the interquartile range divided by $N{\left(m\right)}^{1/3}$]. The (negative) exponents are near 2 for large waiting times and 1 for short waiting times, as indicated by the slopes of the triangles. Data for late events (18–21) in the dotted area show higher relative probabilities for short waiting times than all other curves.

Figure 6

Probability distribution functions $P\left(E\right)$ of AE energies for sequences of events between consecutive superjerks. Curves were fitted with simple power laws. The slopes of the curves correspond to the energy exponents. The number of bins is 100 with bin width 100 aJ for each subset.

Figure 7

Energy exponent as a function of the lower energy cutoff recorded determined by ML. (a) Energy exponents determined by the ML method for $k$ intervals 13–14 and 20–21 with error bars and estimate values; (b) energy exponents for all intervals between $k=13$ and $k=21$. Sparse data sets lead to less well defined plateaus. We used data near ${10}^3\mathrm{aJ}$ to determine the energy exponents.

Figure 8

Evolution of the waiting time renormalization factor $\lambda$ and the energy exponent $\varepsilon$ with increasing rank $k$ of superjerks. Two plateaus can be distinguished for large and small $\lambda$ and exponents near $\varepsilon =1.5$ and $\varepsilon =1.32$.