## 5 Anisotropy of Shale Energy Evolution
### Elastic Energy Density Evolution Model  
The energy accumulation effect of the rock is triggered only when the energy absorbed by the rock reaches the minimum activation energy \(U_{\text{d0}}\), with \(U_{\text{d0}}\) and the total energy \(U_{0}\) satisfying the following criteria [31]:  
\[U_{0}-U_{\text{d0}}>0. \tag{8}\]  
The more the external work applied, the more energy is absorbed, which is conducive to energy storage. Therefore, the cumulative rate of change in the elastic energy density \(dU_{\text{e}}/U_{\text{e}}\) is related to the total energy \(U_{0}\), and the elastic energy density \(U_{\text{e}}\) is the axial strain increment \(\mathrm{d}\varepsilon_{1}\), i.e.,  
\[\frac{1}{U_{e}}\frac{dU_{\text{e}}}{d\varepsilon_{1}}\propto(U_{0}-U_{\text{ d0}}). \tag{9}\]  
Accumulated elastic energy has an inhibitory effect on the subsequent energy storage. This inhibitory effect increases sharply as the amount of accumulated elastic energy increases [32] in a manner that is non-linear and can be described using the following:  
\[\frac{1}{U_{\text{e}}}\frac{dU_{\text{e}}}{d\varepsilon_{1}}\propto-U_{\text{ e}}. \tag{10}\]  
Finally, the following equation is used to establish the evolution of shale energy density:  
\[\frac{1}{U_{\text{e}}}\frac{dU_{\text{e}}}{d\varepsilon_{1}}\ =\ r(U_{0}-U_{\text{d0}})-sU_{\text{e}}, \tag{11}\]  
where \(r\) and \(s\) are the coefficients representing the degree of acceleration and limitation of the internal energy conversion process, respectively. Different values were obtained for different rock samples and energy conversion processes.  
Integrating Equation (11) yields  
\[U_{\text{e}}\ =\ \frac{k}{1+e^{-m\varepsilon_{1}-c}}. \tag{12}\]  
Equation (12) is the energy evolution equation for shale failure. The elastic strain energy exhibits an exponential relationship with axial strain, where \(c\) is a constant and \(m\ =\ r(U_{0}-U_{\text{d0}})\), \(k\ =\ \frac{r(U_{0}-U_{\text{d0}})}{s}\ =\ \frac{m}{s}\).  
Figure 19: Ratio of shale damage factor to \(0^{\circ}\) bedding inclination angle.  
Regression analysis of the energy evolution curve of the shale loading process was performed to obtain the fitting curve between elastic energy density and the axial strain during the three-axis incremental cyclic loading and unloading processes, as shown in Figure 20. The ranges of the fitted parameters are \(0.18\leq m\leq 0.49\), \(758.8\leq k\leq 1211.7\), and \(-6.71\leq c\leq-5.93\). The elastic energy density calculation results for the shale-loading process with three confining pressures and five \(\theta\) values are in good agreement with the experimental data, with all correlation coefficients being above 0.97, indicating that the theoretical model can accurately describe the variation in shale elastic energy with axial strain under different confining pressures and \(\theta\) values.  
In the shale elastic energy density evolution model, \(m\) represents the rate at which elastic energy accumulates, with a larger m indicating faster rates of accumulation. The variation curve of m with \(\theta\) under three confining pressures is shown in Figure 21. When the confining pressure is the same, m shows a U-shaped variation as \(\theta\) increases, with the minimum value observed at 60\({}^{\circ}\), indicating that the energy accumulation rate has clear dependence on \(\theta\). When the \(\theta\) is the same, the value of m increases with the confining pressure, which is because the pressure compacts the initial pores, suppresses the circumferential deformation, reduces crack development, and improves the accumulation rate of elastic energy density.  
Figure 20: Fitting curve for elastic energy density and axial strain.