## 2 Proppant Transport Model
The buoyancy and drag forces of the fluid on the particles are mainly considered in the fluid-particle force model. By calculating the particle Reynolds number, the drag force is obtained using the Gidspow model [29]. The calculation formula is as follows:  
\[F_{\mathrm{Is}}=\left\{ \begin{array}{ll}\frac{3}{4}C_{D}\frac{\rho_{l}(1- \varepsilon_{s})\varepsilon_{s}|\overrightarrow{u_{s}}\overrightarrow{u_{l}} |}{d_{s}}(1-\varepsilon_{s})^{-2.65}&\varepsilon_{l}>0.8\\ \frac{150\varepsilon_{s}\overrightarrow{u_{l}}}{(1-\varepsilon_{s})d_{s}^{2}} +\frac{1.75\rho_{l}\varepsilon_{s}|\overrightarrow{u_{s}}\overrightarrow{u_ {l}}|}{d_{s}}&\varepsilon_{l}\leq 0.8\end{array} \right. \tag{6}\]  
\[C_{D}=\frac{24}{\varepsilon_{l}\mathrm{Re}_{s}}\Big{(}1+0.15(\varepsilon_{l} \mathrm{Re}_{s})^{0.687}\Big{)} \tag{7}\]  
\[\mathrm{Re}_{s}=\frac{\varepsilon_{l}\rho_{l}|u_{s}-u_{l}|d_{s}}{\mu_{l}} \tag{8}\]  
where \(\rho_{l}\) is the density of the liquid, \(\varepsilon_{s}\) is the volume fraction of the particles, \(\overrightarrow{u_{s}}\) is the moving speed of the particles, \(\overrightarrow{u_{l}}\) is the moving speed of the liquid phase, \(d_{s}\) is the particle size of the particle phase, and \(\mu_{l}\) is the fluid viscosity. When particle \(i\) migrates in the fracture, it receives contact force from particle \(j\) and noncontact force from particle \(k\). As the particle size of this article is small, only the contact force between particles and particles is considered. When fracturing fluid flows in a fracture, its boundary conditions include inlet conditions and wall conditions. The inlet conditions were set as a constant speed, and the outlet conditions used a pressure of 0 MPa. The fracture wall surface was set to have no slip boundary; that is, the normal velocity and tangential velocity of the fluid at the wall surface were both zero. In the  
Figure 1: Schematic of collision forces between particles \(i\) and \(j\). fracturing process, prefluid is first injected into the formation to open the fracture, and then uniformly mixed slurry is injected. In order to accurately simulate this process, the stable flow is first calculated to converge in CFD, and stable flow field is used as the initial condition. Based on the abovementioned two-phase coupling model, a fracture physical model (4000 mm \(\times\) 600 mm \(\times\) 6 mm) was established. As the fracture model has a simple shape and is a rectangular fracture, according to the previous research experience, the mesh refinement was performed at the inlet and exit of the fracture model, a hexahedron was used to mesh the fracture model. Two hundred, 30 and 4 grids were divided in sequence along the length, height and width of the fracture, with a total grid number of 24,000. Based on the Rayleigh time step calculation results, the proppant particle simulation time step was set to 0.2 times the Rayleigh time step, which is \(2\times 10^{-5}\) s, and the fluid simulation time step was set to 20 times the Rayleigh time step, which is \(2\times 10^{-3}\) s. The specific steps in the CFD-DEM coupling solution are as follows:  
* CFD iteratively calculates the fluid flow field distribution and the fluid-particle interaction force according to the initial conditions, and transmits information such as drag and buoyancy to the DEM solver. * The DEM solver calculates the contact force of each particle, including the particle-particle and particle-wall forces, and updates the particle position and velocity according to the combined force. The volume fraction and the force of the particles on the fluid were calculated and passed to the CFD solver. * Based on the updated particle volume fraction and the force between the two phases, the CFD solver starts the iterative solution of the next time step, repeating the processes of (1) and (2) until convergence or reaching a preset number of simulation steps.