# Acoustic Emissions in Rock Deformation and Failure: New Insights from Q-Statistical Analysis

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## Abstract

**:**

## 1. Introduction and Motivations

## 2. Experimental Setup and Data Analysis

_{B}at which the breakdown occurs, the total time t

_{TOT}of the experiment, the number N* of AE events before breakdown and the total number N of AE events present in the corresponding time series. In the following, we will investigate if q-statistics can help in revealing different structures in these data sets for increasing levels of confinement.

#### 2.1. AE Amplitudes

- -
- From the beginning to 30% t
_{B}; - -
- From 30% t
_{B}to 70% t_{B}; - -
- From 70% t
_{B}to breakdown; - -
- After breakdown.

- (i)
- Events in the first time interval (within 30% t
_{B}) are always too few to give consistent distributions, regardless of the material; - (ii)
- Distributions before and after failure are quite similar, again regardless of the material, with an initial sudden increase, a peak and a slow decrease for high amplitudes;
- (iii)
- Distributions after failure are more peaked for both AG and DDS.

#### 2.2. AE Inter-Event Times

_{1}or by q

_{2}.

#### 2.3. AE Positions and Inter-Event Distances

^{7}and B = 132) are independent of the type of materials and of the confinement, thus revealing again some kind of universal behavior. On the other hand, it is well visible a single narrow peak around zero for some AG samples, in particular those with intermediate levels of confinement, while this peak is completely absent for DDS samples. This different behavior might be explained by the single fracture nucleation mechanisms observed for AG, which implies high spatial clustering. For DDS samples, instead, multiple fracture nucleation mechanisms due to multiple fracturing regions have been observed [16] and these imply a diffused seismicity and a low clustering.

## 3. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

AE | Acoustic Emissions |

DDS | Darley Dale Sandstone |

AG | Alzo Granite |

3D | Three dimensional |

Probability density function | |

Lin-Lin | Linear-Linear |

Log-Lin | Logarithmic-Linear |

Log-Log | Logarithmic-Logarithmic |

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**Figure 1.**AE Amplitude and inter-event time as functions of time (s) for both AG 40 MPa, panels (

**a**) and (

**b**), and DDS 20 MPa, panels (

**c**) and (

**d**).

**Figure 2.**Amplitude probability distributions for both AG (

**a**) and DDS (

**b**) and for the four increasing levels of confinement. Each data set has been divided into four time intervals, three before breakdown (0–30%, 30–70%, 70–100%) and the last one after failure. In the legend, for each interval and each level of confinement, the number of AE events is reported in parentheses. Only distributions extracted from more than 50 events are shown in the panels.

**Figure 3.**Amplitude probability distributions for the entire time series of AG (

**a**–

**c**) and DDS (

**b**–

**d**) at the four levels of confinement are reported, both in Lin-Lin (

**a**,

**b**) and in Log-Lin (

**c**,

**d**) scale. The Log-Lin curves have also been fitted with the non-linear function reported in Equation (1). Correlation coefficients of 0.68 for panel (

**c**) and 0.74 for panel (

**d**) confirm the good quality of the fits.

**Figure 4.**AG samples: double q-exp fits for the decumulative PDFs of inter-event times. All the correlation coefficients are above 0.8, thus confirming the good quality of the fits.

**Figure 5.**DDS samples: double q-exp fits for the decumulative PDFs of inter-event times. All the correlation coefficients are above 0.8, thus confirming the good quality of the fits.

**Figure 6.**Scatter plots of the time-ordered positions before rupture (blue scale of decreasing intensity), projected on the three coordinate planes X–Y, X–Z and Y–Z, for AE events in the samples AG–5 MPa (

**a**), AG–10 MPa (

**b**), AG–20 MPa (

**c**), AG–40 MPa (

**d**).

**Figure 7.**Scatter plots of the time-ordered positions before rupture (blue scale of decreasing intensity), projected on the three coordinate planes X–Y, X–Z and Y–Z, for AE events in the samples DDS–5 MPa (

**a**), DDS–10 MPa (

**b**), DDS–20 MPa (

**c**), DDS–40 MPa (

**d**).

**Figure 8.**PDFs of the AE inter-event distance for AG (top panel) and DDS (bottom panel) at the various levels of confinement. Curves in both panels have been fitted with the function in Equation (4). Correlation coefficients of, respectively, 0.81 (top panel) and 0.87 (bottom panel), confirm the good quality of the fits.

**Figure 11.**The entropic index ${q}_{\tau}$ is reported as a function of ${q}_{d}$ for the eight considered samples (colored circles) and it is compared with the line ${q}_{\tau}=2-{q}_{d}$. The $\pm $10% area around the line is colored in brown. Notice that the points corresponding to AG-10 MPa and DDS-10 MPa coincide.

Sample | t_{B} | t_{TOT} | N* Events before Breakdown | Tot. N Events |
---|---|---|---|---|

AG-5 MPa | 2398 s | 3395 s | 2367 | 2751 |

AG-10 MPa | 1767 s | 2445 s | 1577 | 1956 |

AG-20 MPa | 2160 s | 2973 s | 1874 | 2367 |

AG-40 MPa | 2857 s | 4566 s | 4419 | 5533 |

DDS-5 MPa | 2400 s | 2540 s | 1067 | 1100 |

DDS-10 MPa | 5808 s | 6248 s | 4802 | 5334 |

DDS-20 MPa | 2409 s | 3348 s | 5760 | 6714 |

DDS-40 MPa | 9710 s | 11,736 s | 10,659 | 11,696 |

Sample | $\mathit{A}$ | ${\mathit{\beta}}_{1}$ | ${\mathit{q}}_{1}$ | ${\mathit{\beta}}_{2}$ | ${\mathit{q}}_{2}$ | $\mathit{\lambda}$ |
---|---|---|---|---|---|---|

AG-5 MPa | 0.975 | 3.5 | 1.48 | 0.01 | 1.7 | 0.09 |

AG-10 MPa | 0.98 | 2.0 | 1.32 | 0.02 | 2.5 | 0.19 |

AG-20 MPa | 0.98 | 1.3 | 1.2 | 0.04 | 2.0 | 0.13 |

AG-40 MPa | 0.983 | 2.6 | 1.16 | 0.017 | 1.4 | 0.12 |

DDS-5 MPa | 0.975 | 1.5 | 1.4 | 0.0001 | 1.7 | 0.055 |

DDS-10 MPa | 0.98 | 4.0 | 1.32 | 0.02 | 1.7 | 0.19 |

DDS-20 MPa | 0.98 | 4.1 | 1.24 | 0.017 | 1.5 | 0.20 |

DDS-40 MPa | 0.983 | 4.6 | 1.16 | 0.015 | 2.4 | 0.19 |

**Table 3.**The sum of the entropic indices ${q}_{\tau}$ and ${q}_{d}$ for the 8 considered samples oscillates around 2.

Sample | ${\mathit{q}}_{\mathit{\tau}}$ | ${\mathit{q}}_{\mathit{d}}$ | ${\mathit{q}}_{\mathit{\tau}}+{\mathit{q}}_{\mathit{d}}$ |
---|---|---|---|

AG-5 MPa | 1.48 | 0.76 | 2.24 |

AG-10 MPa | 1.32 | 0.75 | 2.07 |

AG-20 MPa | 1.2 | 0.69 | 1.89 |

AG-40 MPa | 1.16 | 0.75 | 1.91 |

DDS-5 MPa | 1.4 | 0.78 | 2.18 |

DDS-10 MPa | 1.32 | 0.75 | 2.07 |

DDS-20 MPa | 1.24 | 0.69 | 1.93 |

DDS-40 MPa | 1.16 | 0.79 | 1.95 |

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**MDPI and ACS Style**

Vinciguerra, S.C.; Greco, A.; Pluchino, A.; Rapisarda, A.; Tsallis, C.
Acoustic Emissions in Rock Deformation and Failure: New Insights from Q-Statistical Analysis. *Entropy* **2023**, *25*, 701.
https://doi.org/10.3390/e25040701

**AMA Style**

Vinciguerra SC, Greco A, Pluchino A, Rapisarda A, Tsallis C.
Acoustic Emissions in Rock Deformation and Failure: New Insights from Q-Statistical Analysis. *Entropy*. 2023; 25(4):701.
https://doi.org/10.3390/e25040701

**Chicago/Turabian Style**

Vinciguerra, Sergio C., Annalisa Greco, Alessandro Pluchino, Andrea Rapisarda, and Constantino Tsallis.
2023. "Acoustic Emissions in Rock Deformation and Failure: New Insights from Q-Statistical Analysis" *Entropy* 25, no. 4: 701.
https://doi.org/10.3390/e25040701