# A New Model for Cleaning Small Cuttings in Extended-Reach Drilling Based on Dimensional Analysis
## 2 Model Building and Solution Methods
### Cleaning Model for a High-Inclination Well Section of the Wellbore  
From the analysis in the previous section, it is clear that the van der Waals forces of particle interactions need to be considered for small cuttings relative to large cuttings. In addition, [PERSON] et al. conducted extensive experiments on the motion of small cuttings  
Figure 2: Arrangement angle diagram of small-sized cuttings. in the borehole annulus using an experimental setup similar to field scale ([PERSON] et al., 2008) [14]. Ultimately, it was found that an error of up to 80% could be reached between the experimental results and the predicted values from the commonly used model built by performing a macroscopic forces analysis of the cutting bed. Therefore, it is necessary to develop a wellbore cleaning model that is easy to apply in the field for high-inclination sections of extended-reach drilling with small cutting particles. The physical quantities involved in the wellbore cleaning problem in high-inclination sections of extended-reach drilling are wellbore size, well inclination angle, drill pipe eccentricity, drilling fluid flow rate, drilling fluid density, drilling fluid rheology, mechanical drilling speed, drill pipe speed, cutting size and cutting density ([PERSON] et al., 2013) [15]. Using the method of dimensional analysis, each of these parameters is summarized and analyzed to derive a wellbore cleaning model, and then the model can be used more easily in the field with the help of regression analysis. In this paper, it is assumed that the thickness of the cutting bed is uniform at rest, the cutting particles are spheres of equal diameter, the rheological properties of the drilling fluid are consistent with a steady-state incompressible non-Newtonian fluid, i.e., a power-law model drilling fluid, and the cutting particles do not affect the flow rate distribution of the drilling fluid. The dimensionless cutting bed thickness was used as a discriminant to evaluate the drilling fluid's efficiency in carrying small cutting. The various parameters involved in the process are summarized, and the relationships between all parameters can be written as:  
\[f\left( \begin{array}{c}H,D_{w},D_{p},\theta,e,Q,v_{d},\rho_{d},\mu_{e},\\ ROP,rpm,C_{c},\rho_{c},d_{c},g,\Delta P,A_{bit}\end{array} \right)=0 \tag{9}\]  
Of which:  
\[Q=\frac{\pi}{4}\left(D_{w}^{2}-D_{p}^{2}\right)v_{d} \tag{10}\]  
\[ROP=\frac{C_{c}R_{t}Q}{A_{bit}} \tag{11}\]  
\[A_{bit}=\frac{\pi{d_{w}}^{2}}{4} \tag{12}\]  
\[D_{hyd}=D_{w}-D_{p} \tag{13}\]  
For power-law fluids:  
\[\mu_{e}=K\left(\frac{2n+1}{3n}\frac{D_{w}-D_{p}}{12v_{d}}\right)^{1-n} \tag{14}\]  
where \(H\) is the thickness of the cutting bed, \(D_{w}\) is the diameter of the borehole and \(D_{p}\) is the outer diameter of the drill pipe. \(\theta\) is the well inclination angle, \(e\) is the drill pipe eccentricity, \(Q\) is the drilling fluid discharge, \(v_{d}\) is the drilling fluid upward return velocity, \(\mu_{e}\) is the apparent drilling fluid viscosity, \(ROP\) is the drilling speed, \(rpm\) is the drilling pipe speed, \(C_{c}\) is the small cutting volume concentration, \(\Delta P\) is the pressure difference, \(A_{bit}\) is the drill bit area, \(R_{t}\) is the quotient of cutting velocity and drilling fluid velocity, \(D_{hyd}\) is the annulus hydraulic diameter, \(n\) is the fluidity coefficient of the drilling fluid and \(K\) is the drilling fluid consistency coefficient. Since drilling fluid discharge is a function of drilling fluid return rate and wellbore size; differential pressure is a function of wellbore size and shear stress, which in turn is a function of drilling fluid viscosity, drilling fluid return rate, and wellbore size; mechanical drilling speed is a function of small cutting volume concentration, bit area and drilling fluid discharge, which in turn is a function of borehole diameter; for power-law fluids,drilling fluid viscosity is a function of the drilling fluid rheological parameters.