## 4 Discussion
### Effect of Lateral Confining Pressure on Fracture Conductivity  
The efficiency of hydraulic fracturing in shale gas extraction is critically assessed based on the fracture conductivity of the formation. To determine this influential factor, the  
Figure 10: (**a**) Relations between the closure pressures and proppant embedment across various Young’s moduli; (**b**) relationship among confining pressures; Young’s moduli and reduction in proppant embedment under each closure pressure. Carman-Kozeny equation is commonly employed as a dependable method for calculating fracture conductivity. The equation is represented as follows [44]:  
\[F_{\text{RCD}}=kW \tag{19}\]  
\[k=\frac{\varphi r^{2}}{8\tau} \tag{20}\]  
where \(F_{\text{RCD}}\) represents the fracture conductivity, \(k\) denotes permeability, \(W\) signifies the fracture aperture when subjected to closure pressure, \(\varphi\) refers to porosity, \(r\) indicates the equivalent pore throat radius, and \(\tau\) represents pore tortuosity. The porosity, pore throat, and tortuosity of the fracture are influenced by different closure pressures (\(p\)). Taking the arrangement of proppants in a single layer as an example and disregarding the variations in interlayer porosity, when \(p=0\) and \(K=1\), the values for fracture porosity, equivalent pore throat radius, and tortuosity can be obtained:  
\[\varphi_{0}=\frac{4\sqrt{3}R_{1}^{3}-\frac{4}{3}\pi R_{1}^{3}}{4\sqrt{3}R_{1}^ {3}}=1-\frac{\pi}{3\sqrt{3}} \tag{21}\]  
\[r_{0}(K=1)=\sqrt{\frac{2K}{\pi}-\frac{1}{2}}R_{1}=0.37R_{1} \tag{22}\]  
\[\tau_{0}=\frac{4}{3} \tag{23}\]  
where \(\varphi_{0}\), \(r_{0}\), and \(\tau_{0}\) represent the fracture porosity, equivalent pore throat radius, and tortuosity when the closure pressure is zero. \(R_{1}\) corresponds to the radius of the proppant. During hydraulic fracturing, the application of closure pressure induces proppant embedment and fracture face damage, leading to the generation of debris particles. These particles exhibit different behaviors: some adhere to the fracture or proppant surfaces, while others disperse and migrate with the fluid flow. In the studied shale, which contains water-sensitive clay minerals (specifically illite and chlorite), these debris particles undergo volumetric expansion upon contact with fracturing fluid. This expansion subsequently reduces the fracture porosity. Moreover, under closure pressure, the transported debris particles accumulate and undergo diagenesis, resulting in reconsolidation and uneven distribution of proppant filling pore throat sizes. As a consequence, certain locations experience a significant decrease in pore throat sizes, leading to a substantial decline in fracture permeability. To assess the extent of expansion caused by swelling debris on fracture surfaces, this research introduces the shale swelling coefficient (\(s\)) as a parameter. This coefficient quantifies the ratio between the volume of swelling and the embedded volume, assuming that the swelling layer is impermeable. Higher values of \(s\) indicate more pronounced swelling. With an increased clay content in the shale, there is an enhancement in its hydration swelling capacity, ultimately resulting in a larger swelling coefficient. Consequently, this augmentation has a greater impact on the porosity and pore throat radius of the fillings. Regarding the composition of the shale sample investigated in this study, it consists of 10.6 wt% illite and 2.6 wt% chlorite. Notably, this specific sample does not contain expandable layers such as smectite or vermiculite, thus rendering interlayer swelling absent. However, osmotic swelling still occurs, causing the partial separation of clay particles when subjected to external forces [45; 46]. Therefore, the determined value of \(s\) for this sample is 1. In addition, the migration coefficient (\(m\)) is used to quantify the level of the dispersed transport of fragmented debris in the presence of fluid.