# Experimental Analysis of Shale Cuttings Migration in Horizontal Wells
## 4 Theoretical Model and Validation  
### Theoretical Model  
The initiation of cuttings movement, or \"start-up\", refers to the point when particles on the surface of the cuttings bed begin to transition from a state of rest to motion. This process is governed by several mechanical forces, including those that promote movement (such as drag and lifting forces from the drilling fluid) and those that resist it (such as gravity and inter-particle adhesion). The geometric characteristics of the cuttings, particularly their shape, significantly influence their movement. The forces acting on the cuttings can be categorized into two groups: forces that facilitate movement and forces that hinder it. As the flow velocity of the drilling fluid increases, the balance between these forces changes, and once a critical threshold is reached, the cuttings begin to move. If the fluid flow is insufficient to mobilize the cuttings adjacent to the wellbore wall, the particles will remain stationary and accumulate, forming a cuttings bed. The migration of these particles can occur in three primary modes: sliding, rolling, or suspension, depending on the flow conditions and particle characteristics (Figure 13). The diagram in Figure 14 illustrates the various forces acting on the surface particles of the sedimentary bed within the annular hole. These forces include gravity, lifting force, drag force, and pressure gradient force. The model assumes that the cuttings particles are cuboid in shape and that the annular fluid is in a steady state. Additionally, it assumes no slippage or mass exchange between the solid and liquid phases, and that both phases are incompressible. Figure 14: Force diagram of cuttings particles. Figure 13: Multilayer cuttings start-up diagram. Given the complexity of the forces acting on the particles within the flow field, the model is simplified to focus on the primary forces influencing particle movement. Minor forces, which have a minimal impact on overall behavior, are disregarded to facilitate analysis and calculation. This approach allows for a more manageable examination of the cuttings migration process while still capturing the key dynamics affecting particle movement. (1) Gravity. \[G=\rho_{\text{s}}\text{g}abc \tag{1}\]  
where \(G\) is the gravity of the cuttings particles, N; \(\rho_{\text{s}}\) is the density of cuttings, kg/m\({}^{3}\); \(a\), \(b\), and \(c\) are the length, width, and height of the cuttings, respectively, m; and g is the acceleration of gravity, 9.81 m/s\({}^{2}\). (2) Buoyancy. The direction is opposite to gravity, and the size is affected by drilling fluid density and particle volume. \[F_{f}=\rho_{l}\text{g}abc \tag{2}\]  
where \(F_{f}\) is the buoyancy force of the cuttings particles, N, and \(\rho_{l}\) is the density of the drilling fluid, kg/m\({}^{3}\). (3) Drag force. The drag force is generated as the drilling fluid flows around the cuttings, causing resistance due to friction and pressure differentials. For spherical particles, this force is well understood, but for non-spherical cuttings like shale, the drag coefficient is adjusted based on their shape. The more irregular the particle, the greater the drag force acting on it. For spherical particles, the following formula is used. \[F_{D}=C_{D}\rho_{l}v^{2}A_{s}/2 \tag{3}\]  
\[Re<0.5\quad C_{\text{D}}=\frac{24}{Re} \tag{4}\]  
\[Re>10,000\quad C_{\text{D}}=0.4 \tag{5}\]  
\[0.5<Re<10,000\quad C_{\text{D}}=\frac{24}{Re}+\frac{3}{\sqrt{Re}}+0.34 \tag{6}\]  
For non-spherical particles, we use the calculation method described in Reference [26].