## 4 Evolution of the Shale Characteristics of AE
### Characteristics of AE b-Values  
The \(b\)-value that is associated with AE originates from seismology studies. [PERSON] and [PERSON] (2008) first proposed the well-known G-R relationship between earthquake frequency \(N\) (AE event count was identified by the system according to specific acoustic emission characteristics) and magnitude \(M\), with the similarity between the distribution characteristics of AE events during rock compression and the mechanisms of earthquake evolution resulting in the following equation:  
\[\begin{array}{c}\lg N\ =\ a\ -bM\\ M\ =\ A_{dB}/20\end{array}\Bigg{\}}, \tag{5}\]  
where \(a\) and \(b\) are constants and \(A_{\rm dB}\) is the maximum amplitude of the AE event. The AE \(b\)-value is not only a statistically analyzed parameter but is also closely related to the damage pattern of a rock. For shale, the further a primary crack penetrates, the less fracture energy is released per unit crack area, resulting in a larger AE \(b\)-value; however, if shear dominates, more energy is dissipated by shear friction, resulting in lower fracture energy release and a larger AE \(b\)-value. Similarly, the more secondary cracks, the more energy is dissipated into the shale, resulting in larger AE \(b\)-values.  
To ensure statistical accuracy, an amplitude interval \(\Delta M\) of 5 dB was used, and the \(b\)-value was determined by fitting all AE events during the shale failure process, as shown in Figure 15a. The variation curve for the \(b\)-value with the \(\theta\) is shown in Figure 15b; the results indicate that with the increase in \(\theta\), the \(b\)-value of shale decreases from 0\({}^{\circ}\) to 45\({}^{\circ}\) and increases from 45\({}^{\circ}\) to 90\({}^{\circ}\). According to the failure mode, a \(\theta\) of 0\({}^{\circ}\) results in two main cracks running through the shale, resulting in a larger \(b\)-value, while all other \(\theta\) values result in only one main crack. Although a tensile crack is observed in shale with a \(\theta\) of 90\({}^{\circ}\), the associated \(b\)-value is higher than it is for the other \(\theta\) conditions. This may be due to the fact that considerably less fracture energy is released when failure occurs along rather than through the bedding plane. In addition, the range of \(b\)-values for the shale specimens with different \(\theta\) values is 0.6-0.8, which indicates a higher proportion of small amplitude AE events during the failure of shale samples, which mainly consist of small-scale cracks.  
Figure 14: Variation curve showing Felicity ratio with number of cycles.  
To further analyze the crack evolution characteristics of shale during incremental cyclic loading and unloading, the dynamic \(b\)-value variation curve over time is shown in Figure 12. A window length of 10% and step size of 1% were used to calculate the total number of AE events. The dynamic \(b\)-value of shale corresponds to the ring count response; thus, the \(b\)-value is observed to fluctuate. The significant decrease followed by an increase near the first four unloading points remains stable during unloading and reloading, indicating that the large cracks are mostly generated around the unloading points of each cycle. Finally, a through-crack is generated at the point of destruction, with the corresponding dynamic \(b\)-value decreasing sharply to its lowest value. Compared with the cases in which the \(\theta\) is \(0^{\circ}\) and \(90^{\circ}\), the time window during which the dynamic \(b\)-values remain stable before failure is shorter when \(\theta\) = \(30^{\circ}\), \(45^{\circ}\), or \(60^{\circ}\). These results indicate that the distribution of large- and small-scale microfractures within the sample is more balanced at a \(\theta\) of \(0^{\circ}\) and \(90^{\circ}\).