## 3 Materials and Methods
### Mechanism of Free Settling of Proppants under Multi-Fluid Conditions  
The distribution characteristics of proppants in tight sandstone fracturing are directly reflected in the fracturing quality and have an impact on the conductivity of the fractures [29]. It can be inferred that optimizing the distribution characteristics of proppants in fractures is of great value in improving fracturing quality. The physical properties of the support are closely related to changes in fluid conditions. When the external liquid is relatively stationary, the support agent mainly settles. If there is relative movement of the proppant, it is also accompanied by migration. Here, taking Newtonian fluids and power-law fluids as examples, we analyze their motion mechanisms [30]. Free settlement of proppant particles in Newtonian fluids:  
Considering the acceptance of a single proppant in a relatively stationary Newtonian fluid, it can be concluded that in the vertical direction, the Earth's gravity (downward), liquid buoyancy (upward), and fluid resistance (upward) should be considered. Assuming the particle is a sphere, the vertical downward direction is defined as the reference direction, and the resultant force can be described as follows:  
\[F_{1}=\frac{\pi}{6}d^{3}(\rho_{s}-\rho_{l})g-\frac{\pi}{4}d^{2}C_{d}\frac{v^{ 2}}{2}\rho_{l} \tag{4}\]\(v\)--the settling rate of a single particle. When the resistance, buoyancy, and self-weight of the proppant particles reach equilibrium, the velocity of the particles is shown in Formula (5):  
\[v_{\rm s}=\left[\frac{4g(\rho_{s}-\rho_{l})d}{3C_{d}\rho_{l}}\right]^{0.5} \tag{5}\]  
The formula for calculating the drag coefficient is as follows:  
\[C_{d}=\frac{4 gd_{s}}{3v_{p}^{2}}\frac{\rho_{s}-\rho_{l}}{\rho_{l}} \tag{6}\]  
According to the Reynolds number, settlement is divided into three flow regions (Stokes region, intermediate Reynolds number region, inertia region). In different fluid regions, there are different relationships between Cd and Re (Table 1). Sedimentation of proppant particles in non-Newtonian liquids:  
Fracturing sand-carrying fluid is generally a power-law fluid. If the viscosity coefficient and rheological index of a power-law fluid are K and n, respectively, then the Reynolds number's expression is as follows:  
\[N_{\rm Re}=\frac{\rho_{l}d_{s}v_{s}}{\mu_{a}} \tag{7}\]  
Different shear rates result in differences in the velocity settlement equation under laminar flow conditions. For a Daneshy shear rate \(D=\frac{3v_{s}}{d_{s}}\), \(\mu_{a}=K(\frac{3u}{2n+1})^{n}D^{n-1}\), the settlement velocity equation is as follows:  
\[v_{s}=\frac{2n+1}{9n}\bigg{[}\frac{(\rho_{s}-\rho_{l})gd_{s}^{n+1}}{6K}\bigg{]} ^{\frac{1}{n}} \tag{8}\]  
Figure 1 shows the relevant information on the force analysis of a single particle proppant. The settlement of proppants is also affected by viscosity, sand ratio, and wall efficiency. The viscous force will increase with an increase in fracturing fluid viscosity, hindering the settling movement of the proppant and weakening its speed [31]. When reaching a certain critical value, the proppant may be in a static suspension state. Therefore, low-density or small-particle-size proppants should be selected in actual construction to ensure that they can stay in the rock crevices for a long time. After fracturing tight sandstone, the proppant settles and migrates in the form of a group, and the mathematical relationship between particle concentration and settling speed can be described as follows:  
\[v_{\phi}=v_{\rm s}(1+6.88\phi)^{-1} \tag{9}\] \begin{table}
\begin{tabular}{c c} \hline
**Cd** & **Re** \\ \hline \(\frac{24}{K}\) & \(\leq\)1 \\ \hline \(\frac{18.5}{K^{6}}\) & 1-500 \\ \hline
0.44 & 500–200,000 \\ \hline \end{tabular}
\end{table} 
Table 1: Relationship table between Cd and Re. \(\varnothing\) represents concentration, indicating that the sedimentation rate decreases with increasing concentration.