## 2 Proppant Transport Model  
In this study, a coupled method of CFD and DEM is utilized. A fluid-particle two-phase flow model is established by considering particle-particle, particle-wall, and fluid-particle interaction forces. The following assumptions were made for the model. * The fluid (slippery water) is incompressible, and the fluid rheology and temperature will not change during the simulation. * Ultralow density proppants are regular spherical particles. * The particles were fully mixed with the fluid, and the particles were evenly distributed in the fluid at the fracture entrance. * The particles are rigid bodies, the spheres are not deformed, and the contact between the particles is point contact. When establishing the governing equation of the fluid, the effect of particles on fluid is considered. Therefore, the liquid flow control equation is the Navier-Stokes equation of liquid-solid coupling. Because the simulated fluid inlet velocity is high and the flow field changes drastically, after calculating the Reynolds number, the standard \(\mathrm{k}-\varepsilon\) model is invoked as the turbulent flow model to solve the slurry flow field [25; 26; 27; 28]. The turbulence flow energy equation is as follows:  
\[\frac{\partial(\rho k)}{\partial t}+\frac{\partial(\rho ku_{i})}{\partial x_{ i}}=\frac{\partial}{\partial x_{j}}\bigg{[}\bigg{(}\mu+\frac{\mu_{t}}{\sigma_{k}} \bigg{)}\frac{\partial k}{\partial x_{j}}\bigg{]}+G_{k}-\beta\varepsilon+S_{k} \tag{1}\]  
The turbulence diffusion equation is as follows:  
\[\frac{\partial(\rho\varepsilon)}{\partial t}+\frac{\partial(\rho\varepsilon u _{i})}{\partial x_{i}}=\frac{\partial}{\partial x_{j}}\bigg{[}\bigg{(}\mu+ \frac{\mu_{t}}{\sigma_{\varepsilon}}\bigg{)}\frac{\partial\varepsilon}{ \partial x_{j}}\bigg{]}+\frac{C_{1\varepsilon}}{k}G_{k}-C_{2\varepsilon}\rho \frac{\varepsilon^{2}}{k}+S_{\varepsilon} \tag{2}\]  
where \(k\) is the turbulence kinetic energy of the fluid phase; \(\varepsilon\) is the turbulence dissipation rate; \(G_{k}\) is the turbulence kinetic energy generation term; \(\sigma_{k}\) is the Prandtl number corresponding to the turbulence kinetic energy, which is 1.0; \(\sigma_{\varepsilon}\) is the Prandtl number corresponding to the turbulence dissipation rate, and the value is taken as 1.3; \(S_{K}\) and \(S_{\varepsilon}\) are the turbulence exchange term between the liquid and solid phases, respectively; and \(C_{1\varepsilon}\) and \(C_{2\varepsilon}\) are constants with values of 1.44 and 1.92, respectively. \(\mu_{t}\) is the turbulence viscosity and its expression is as follows:  
\[\mu_{t}=\rho C_{\mu}\frac{k^{2}}{\varepsilon} \tag{3}\]  
The motion of proppant particles follows [PERSON]'s second law. When particles follow the fluid, the movement of each particle is affected by the surrounding fluid and adjacent particles; hence, the particles are mainly subject to the force from the fluid, adjacent particles and fracture walls (Figure 1). The equation of particle motion consists of a translational equation and a rotational equation, following [PERSON]'s second law of motion. The equation of the translational is as follows:  
\[m_{p,i}\frac{\mathrm{d}u_{p,i}}{\mathrm{d}t}=\sum\mathbf{F}_{pc,ij}+\mathbf{F}_{ lp,i}+m_{p,i}g \tag{4}\]  
The rotation equation is as follows:  
\[I_{pc,ij}\frac{\mathrm{d}\omega_{p,i}}{\mathrm{d}t}=\mathbf{T}_{pc,ij} \tag{5}\]  
where \(m_{i}\) is the mass of particle \(i\), \(u_{p,i}\) is the linear velocity of the particle, \(\mathbf{F}_{pc,ij}\) is the contact force produced by particle \(i\) in contact with other particles, \(\mathbf{F}_{lp,i}\) is the force of the fluid on particle \(i\), \(I_{pc,ij}\) is the moment of inertia of particle \(i\), \(\omega_{p,i}\) is the angular velocity of particle \(i\), and \(\mathbf{T}_{pc,ij}\) is the contact moment produced by particle \(i\) in contact with particle \(j\). Fluid-solid coupling requires the establishment of a fluid-particle interaction model and a particle-particle interaction model.