# A New Model for Cleaning Small Cuttings in Extended-Reach Drilling Based on Dimensional Analysis
## 2 Model Building and Solution Methods  
### Force Analysis of Small Cuttings in a High-Inclination Well Section  
Figure 1 illustrates the various forces acting on small cuttings in extended-reach drilling and the direction of the forces. The forces acting on small-size cuttings include mainly net gravity, drag force, lift force, a tensile force due to uneven pressure distribution along the wellbore axis, and van der Waals force due to the interaction of adjacent small cuttings ([PERSON] et al., 2014) [12]. The expressions for the individual forces on the small cuttings are as follows. The relative gravity of small-sized cutting particles in the annulus is the difference between gravity and buoyancy, which can be expressed as:  
\[W=F_{g}-F_{f}=\frac{\pi d_{c}^{2}(\rho_{c}-\rho_{d})g}{6} \tag{1}\]  
where the \(W\) is the relative gravity of small cuttings, \(F_{g}\) and \(F_{f}\) are the gravity and buoyancy of small cuttings, \(d_{c}\) is the diameter of the small cuttings, \(\rho_{c}\) and \(\rho_{d}\) are the density of the small cuttings and the density of the drilling fluid. When small cutting particles move with the drilling fluid in the annular space, they are subjected to the same drag force as the axial direction of the borehole, which is expressed as:  
\[F_{D}=\frac{1}{2}C_{D}A_{p}^{*}\rho_{d}v_{rp}\big{|}v_{rp}\big{|} \tag{2}\]  
where \(F_{D}\) is the drag force on small cutting particles, \(C_{D}\) is the drag coefficient, \(A_{p}^{*}\) is the characteristic area and \(v_{rp}\) is the relative velocity of mud and small cuttings. Due to the viscous effect of the drilling fluid, the velocities at different diameters at the same annular cross-section are different, and the velocity gradient around the small cuttings will produce a lifting force on it, which can be expressed as:  
\[F_{L}=\frac{1}{2}\rho_{d}C_{L}v_{rp}^{2}\bigg{(}\frac{\pi d_{c}^{2}}{4}\bigg{)} \tag{3}\]  
Figure 1: Diagram of forces on small-sized cuttings. where \(F_{L}\) is the lifting force of small cuttings and \(C_{L}\) is the lifting coefficient. When drilling fluid flows axially along the borehole, an axial pressure gradient is generated, and the axial pressure gradient creates a pulling force along the borehole axis for small-sized cutting particles, which can be expressed as:  
\[F_{\Delta P}=\frac{1}{6}\pi d_{p}^{3}G_{p} \tag{4}\]  
where \(G_{p}\) is the pressure gradient. Due to the viscous effect of the drilling fluid, the drilling fluid will make a rotational motion with the rotation of the drill pipe, generating a tangential velocity perpendicular to the axial direction of the wellbore, making the drilling fluid velocity not be the same everywhere any longer, thus subjecting the small cuttings to the rotational lifting force of the drill pipe, expressed as:  
\[F_{R}=\frac{\pi d_{c}^{2}}{8}C_{LR}\rho_{d}v_{PR} \tag{5}\]  
where \(F_{R}\) is the rotational lift force of the drill pipe on small cuttings, \(C_{LR}\) is the rotational lift force coefficient of the drill pipe and \(v_{PR}\) is the flow velocity in the axial direction of the vertical borehole due to the rotation of the drill pipe. The pressure caused by drilling fluid cementation is:  
\[F_{P}=\frac{\pi}{4}d_{c}^{2}v_{y} \tag{6}\]  
where \(F_{P}\) is the pressure on small cuttings and \(\tau_{y}\) is the drilling fluid yield strength. As the cuttings' diameter decreases and the surface area per unit volume increases, the Van der Waals force of molecular interactions between small cuttings has an impact on the cuttings' forces. The Van der Waals force between two small cuttings can be expressed as:  
\[F_{van}=-\frac{A_{H}d_{p}}{24s^{2}} \tag{7}\]  
where \(A_{H}\) is \(4.14\times 10^{-20}\) N \(\cdot\) m. \(s\) is the distance between two small cuttings. Each small cutting is surrounded by an average of six small cuttings, and each one enclosed in the center interacts with each of the surrounding small cuttings in different directions, with the final combined force perpendicular to the surface of the cutting bed ([PERSON], 2014) [13].