## 4 Discussion
### Effect of Lateral Confining Pressure on Fracture Conductivity
This transport leads to a reduction in pore throat size within the fracture filling layer and consequently lowers permeability. A value of \(m\) greater than 1 indicates more significant debris transport. Notably, shale formations with elevated clay content, particularly montmorillonite or illite, tend to exhibit higher migration coefficients. In this study, considering that the clay mineral content is not very high, we set the value of \(m\) to 1. Moreover, owing to their inherent physical and chemical stability, proppantsexhibit minimal reactivity when in contact with fracturing fluid. Consequently, the mechanical properties and deformation characteristics of these proppants can be regarded as autonomous variables unaffected by the presence of fracturing fluids. Figure 11 shows the proppant embedment considering the swelling and migration effects, assuming that the embedment depth of the proppant at this moment is denoted as \(h^{\prime}\). Additionally, the swelling thickness generated by debris is represented by \(H\), which progressively increases over time during the interaction between hydraulic fracturing fluid and shale. It should be noted that the mechanical parameters of this specific portion of shale are considered negligible or non-existent in this study. The embedded volume of the proppant is \(V_{1}\), and the swelling volume is \(V_{3}\), and then the fracture conductivity can be inferred:  
\[V_{1}=\pi h^{\prime 2}\bigg{(}R_{1}-\frac{h^{\prime}}{3}\bigg{)} \tag{24}\]  
\[V_{3}=sV_{1} \tag{25}\]  
\[V_{1}+V_{2}=\pi\big{(}H+h^{\prime}\big{)}^{2}\bigg{(}R_{1}-\frac{H+h^{\prime}} {3}\bigg{)} \tag{26}\]  
\[S_{0}=2\sqrt{3}R_{1}^{2} \tag{27}\]  
\[F_{\text{RCD}}=kW \tag{19}\]0  
\[F_{\text{RCD}}=kW \tag{19}\]1  
\[F_{\text{RCD}}=kW \tag{19}\]2  
\[F_{\text{RCD}}=kW \tag{19}\]3  
\[F_{\text{RCD}}=kW \tag{19}\]4  
\[F_{\text{RCD}}=kW \tag{19}\]5  
By combining Equations (18), (20), and (24)-(33), we can obtain the following results:  
\[F_{\text{RCD}}=kW \tag{19}\]6  
\[F_{\text{RCD}}=kW \tag{19}\]7  
where \(\alpha\) is the half width of the fracture. \[F_{\text{RCD}}=kW \tag{19}\]8 Schematic of proppant embedment considering the swelling and migration effects. Figure 12 shows the relationships between closure pressure and fracture conductivity under different proppant sizes, Young's moduli, and swelling thicknesses. The impact of closure pressure and swelling thickness on fracture conductivity varies nonlinearly under the same proppant size. Moreover, the effect of different confining pressures on shale's Young's modulus influences fracture conductivity differently. Firstly, considering that confining pressure enhances shale's Young's modulus, it results in a reduction in proppant embedment depth and subsequently improves fracture conductivity compared to cases that do not consider confining pressure. The maximum increase in fracture conductivity for various proppant sizes is observed at \(p=50\) MPa, corresponding to a confining pressure of 50 MPa. For a proppant radius of \(R_{1}=1.25\) mm, the maximum increases are 3.17%, 3.44%, and 4.23% at \(H=0.1\), 0.25, and 0.5 mm, respectively. Similarly, for \(R_{1}=0.75\) mm, the maximum increases are 5.43%, 6.51%, and 11.24% at \(H=0.1\), 0.25, and 0.5 mm, respectively. In the case of \(R_{1}=0.5\) mm, the maximum increases are 8.55%, 12.15%, and 27.33% at \(H=0.1\), 0.25, and 0.4 mm, respectively. These findings indicate that the impact of confining pressure on fracture conductivity is minimal for larger proppant sizes. However, for smaller proppant sizes, the variation in shale mechanical properties due to confining pressure significantly affects fracture conductivity calculations, especially when larger values of \(H\) are induced by a long-term interaction between hydraulic fracturing fluid and shale. Consequently, such calculations often underestimate fracture conductivity due to the selection of smaller shale elastic modulus values.