## 3 Results
### Fracture Toughness  
Fracture toughness is a critical parameter in the design of hydraulic fracturing operations. This study utilizes an energy-based approach to determine the fracture toughness of shale. The total energy (\(U_{\text{t}}\)) during the indentation process combines elastic (\(U_{\text{e}}\)), plastic (\(U_{\text{pp}}\)), and fracture (\(U_{\text{crack}}\)) energies. Analyzing the areas under the loading and unloading curves enables the quantification of total (\(U_{\text{t}}\)) and elastic (\(U_{\text{e}}\)) energies, respectively. Additionally, the fitting of the \(F\)-\(h\) curve is accomplished using power law functions, while Equation (1) is employed for fitting the unloading curve, and fitting the loading curve can be achieved by the following [35; 36; 37]:  
\[F=kh^{n}(\text{loading}) \tag{10}\]  
Consequently, it is possible to calculate \(U_{\text{e}}\) and \(U_{\text{t}}\) using the following equations:  
\[U_{\text{e}}=\int_{h_{\text{f}}}^{h_{\text{max}}}\eta(h-h_{\text{f}})^{m}_{d}dh \tag{11}\]  
\[U_{\text{t}}=U_{\text{loading}}+U_{\text{unloading}}=\int_{0}^{h_{\text{h}}} kh^{n}dh+P_{\text{max}}(h_{\text{max}}-h_{\text{h}}) \tag{12}\]  
where \(h_{\text{h}}\) represents the displacement at the onset of the holding stage; \(\eta\), \(k\), \(m\), \(n\) are the parameters derived through data fitting.  
Within this approach, the irreversible energy (\(U_{\text{ir}}\)) is defined as the combined total of plastic (\(U_{\text{pp}}\)) and fracture (\(U_{\text{crack}}\)) energies [38]. Subsequently, the fracture energy (\(U_{\text{crack}}\)) can be ascertained through the following equation:  
\[U_{\text{crack}}=U_{\text{ir}}-U_{\text{pp}} \tag{13}\]  
Figure 8: Typical microindentation results showing the variations in Young’s moduli and Brinell hardness over confining pressures: (**a**) Young’s moduli; (**b**) Brinell hardness.  
The pure plastic energy (\(U_{\rm{PP}}\)) is evaluated through the utilization of the following equation:  
\[\frac{U_{\rm{PP}}}{U_{\rm{I}}}=1-\frac{U_{\rm{e}}}{U_{\rm{I}}} \tag{14}\]  
The determination of the critical energy release rate, denoted as \(G_{\rm{c}}\), is achieved by the following:  
\[G_{\rm{c}}=\frac{\partial U_{\rm{crack}}}{\partial A}=\frac{U_{\rm{crack}}}{A_{ \rm{max}}} \tag{15}\]  
where \(A_{\rm{max}}\) is the maximum crack area. In the case of a spherical indenter, this area can be described by the following:  
\[A_{\rm{max}}=\pi\times\left(2h_{\rm{max}}R-h_{\rm{max}}^{2}\right) \tag{16}\]  
Ultimately, the fracture stress intensity, represented by \(K_{\rm{c}}\), is determined by the following:  
\[K_{\rm{c}}=\sqrt{G_{\rm{c}}E_{\rm{r}}} \tag{17}\]  
Fracture toughness quantifies a rock's resistance to brittle fractures when cracks are present. Figure 9 depicts the relationship between confining pressure and fracture toughness based on microindentation curves obtained from the tested sample. The findings indicate that fracture toughness generally exhibits a linear increase with increasing confining pressure. Prior studies have indicated that Young's modulus directly correlates with fracture strength, with a higher Young's modulus indicating increased fracture resistance and, therefore, higher fracture toughness [27; 35]. In this study, both Young's modulus and fracture toughness exhibit a positive linear association with confining pressure, thus revealing a positive correlation between Young's modulus and fracture toughness, and these observations are consistent with previous studies. Overall, the shale subjected to confining pressure displays elevated Young's moduli and fracture toughness. As a result, embedding proppants in the shale becomes more challenging, and crack nucleation and propagation occur at a slower pace.