## 4 Discussion
### Effect of Lateral Confining Pressure on Propnant Embedment  
[PERSON] et al. (2015) introduced a novel mathematical model based on elastic contact theory to compute proppant embedment [43]. The model assumes that the proppant embedding process entails the solely elastic deformation of the rock, with no rock fracture occurring and disregarding any fractures in the proppant itself. Additionally, it is assumed that the upper and lower layers in contact with the proppant deform according to Hooke's law under closure pressure. As the thickness of the rock layer greatly exceeds the size of the proppant particles, the impact of changes in fracture width resulting from rock layer deformation is limited solely to the fracture surface, aligning with the scale of the proppant. Then, the value of proppant embedment can be estimated as follows:  
\[h^{\prime}=1.04D\left(K^{2}p\right)^{\frac{2}{3}}\left[\left(\frac{1-\
u_{1}^{ 2}}{E_{1}}+\frac{1-\
u^{2}}{E}\right)^{\frac{2}{3}}-\left(\frac{1-\
u_{1}^{2} }{E_{1}}\right)^{\frac{2}{3}}\right]+D\frac{p}{E} \tag{18}\]  
where \(h^{\prime}\) is the value of proppant embedment, in mm; \(p\) is the closure pressure, in MPa; \(K\) is the distance coefficient; \(E_{1}\) and \(E\) are [PERSON]'s modulus of the proppant and shale, respectively, in MPa; \(\
u_{1}\) and \(\
u\) are [PERSON]'s ratio of the proppant and shale, respectively; and \(D\) is the diameter of the proppant, in mm. From the above equation, it can be observed that factors influencing the elastic embedment depth of proppants include proppant particle size, proppant distance coefficient, closure pressure, and Young's modulus of both the proppant and shale. This section focuses on examining the effect of lateral confining pressure on proppant embedment depth. To accomplish this, a simplified arrangement of proppants is employed, assuming that they are uniformly distributed in a single layer within the reservoir. Furthermore, the weight of the proppants is disregarded. The diameter of the proppant (\(D\)) is chosen as 2.5 mm, which matches the diameter of the microindentation probe. The proppant spacing factor (\(K\)) is set to 1, indicating that the proppants are tightly packed with no gaps in between. The closure pressure (\(p\)) ranges from 0 to 50 MPa. Young's modulus (\(E\)) of the shale is determined based on microindentation experiments under different confining pressures, following a linear relationship between confining pressure and elastic modulus. The values of Young's modulus are selected within the range of 39.50-55.38 GPa, corresponding to confining pressures from 0 to 50 MPa. Furthermore, due to the negligible impact of variations in shale's Poisson's ratio on proppant embedment depth, it can be disregarded. Therefore, it is hypothesized that Poisson's ratio (\(\
u\)) of the shale remains consistent across varying confining pressures, maintaining a constant value of 0.25. Young's modulus (\(E_{1}\)) and Poisson's ratio (\(\
u_{1}\)) of the proppant are established as 100 GPa and 0.2, respectively. Figure 10a illustrates the relationship between closure pressure and proppant embedment depth, considering changes in Young's modulus of the shale under different confining pressures. The results indicate a nonlinear increase in embedment depth as closure pressure rises. Notably, the maximum embedment depth occurs when disregarding the influence of confining pressure (\(E=39.50\) GPa), while the minimum embedment depth is observed at a confining pressure of 50 MPa (\(E=55.38\) GPa). Specifically, at a closure pressure of 20 MPa, the embedment depths without considering confining pressure and with a confining pressure of 50 MPa (i.e., \(E=39.50\) and 55.38 GPa) are measured as 0.0125 mm and 0.00933 mm, respectively, resulting in a reduction of 25.36%. It should be noted that during microindentation testing, the simulated proppant embedment depths (\(h_{\text{max-1}}\)) account for the combined effects of various confining pressures, including a closure pressure of 20 MPa (i.e., \(F_{\text{max}}=100\) N).