## 2 Materials and Methods
### Microindentation Testing  
The microindentation technique enables the characterization of mechanical properties for diverse materials. In this study, a single microindentation test involves the controlled loading and unloading of a highly rigid tip into the surface of the shale specimen, and to simulate the interaction between proppants and shale, a spherical tip was chosen as the indenter. Figure 3 depicts a representative loading-unloading cycle of a single microindentation. The mechanical characteristics of the shale samples were evaluated using the Oliver-Pharr method, with a specific focus on analyzing the \(F\)-\(l\) curve [29]. For fitting the upper segment of the unloading curve, a power law equation can be employed, as shown below:  
\[F=\eta(h-h_{\mathrm{f}})^{m} \tag{1}\] \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Samples**} & \multicolumn{8}{c}{**Mineral (wt\%)**} \\ \cline{2-10}  & **Quartz** & **Orthoclase** & **Albite** & **Calcite** & **Dolomite** & **Pyrite** & **Chlorite** & **Illite** \\ \hline Shale & 12.9 & 0.2 & 2.4 & 15.5 & 55.5 & 0.4 & 2.6 & 10.6 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Mineralogical composition of the studied sample. Figure 2: Unconfined compression test by RMT–150C servo test system. where \(F\) represents the applied load, \(h\) denotes the indentation depth, \(\eta\) and \(m\) refer to the fitting parameters, and \(h_{\text{f}}\) represents the final indentation depth. The contact depth, designated as \(h_{\text{c}}\), is expressed as follows [30]:  
\[\varepsilon(m)=\frac{0.08158}{\sqrt{m-0.94}}-\frac{0.061679}{(m-0.94)^{0.02}}+ \frac{1.26386}{(m-0.94)^{0.001}} \tag{2}\]  
\[h_{\text{c}}=h_{\text{max}}-\varepsilon(m)\times\frac{F_{\text{max}}}{S} \tag{3}\]  
where \(\varepsilon(m)\) is a factor that accounts for the interplay between the indenter's geometry and the plastic characteristics of the material being tested; \(h_{\text{max}}\) represents the maximum indentation depth; \(F_{\text{max}}\) denotes the maximum applied load; and \(S\) stands for contact stiffness. The contact stiffness, labeled as \(S\), is determined by the slope of the upper segment of the unloading curve:  
\[S=\frac{\text{d}F}{\text{d}h}\mid h=h_{\text{max}} \tag{4}\]  
The projected area of the contact surfaces, \(A_{\text{c}}\), can be determined by calculating the area of the contact circle with radius, \(a\), given by the following:  
\[A_{\text{c}}=\pi\times a^{2} \tag{5}\]  
The radius of the circle of contact, \(a\), is computed as follows [31]:  
\[a=\sqrt{2h_{\text{c}}R-h_{\text{c}}^{2}} \tag{6}\]  
where \(R\) is the radius of the spherical indenter. The Brinell hardness, \(HB\), is performed by the following [32]:  
\[HB=\frac{F_{\text{max}}}{9.80665A_{\text{c}}} \tag{7}\]  
The reduced modulus, \(E_{\text{r}}\), is subsequently computed using the following equation:  
\[E_{\text{r}}=\frac{\sqrt{\pi}}{2\sqrt{A_{\text{c}}}}S \tag{8}\]  
To calculate [PERSON]'s modulus of the sample, the following equation is employed [33]:  
\[\frac{1}{E_{\text{r}}}=\frac{1-\
u^{2}}{E}+\frac{1-\
u_{\text{i}}^{2}}{E_{ \text{i}}} \tag{9}\]  
Figure 2:0 Schematic of microindentation testing: (**a**) typical load–displacement curve of an individual indentation; (**b**) indent geometry by the spherical indenter. where \(E\) and \(E_{\rm i}\) represent [PERSON]'s modulus of the tested sample and the indenter, respectively, whereas \(\
u\) and \(\
u_{\rm i}\) denote [PERSON]'s ratio of the tested sample and indenter, with defined values of \(E_{\rm i}\) = 700 GPa and \(\
u_{\rm i}\) = 0.3. Furthermore, [PERSON]'s ratio for shale samples can be derived from the results obtained through unconfined compression testing. The microindentation testing system comprised two principal components: a bidirectional loading apparatus and a spherical penetration apparatus (Figure 4). The spherical penetration apparatus utilized in this study comprised a specialized spherical microindenter (Wance Technology Ltd., Shenzhen, China).