# 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
Since the experimental eccentricity of the drill pipe, the concentration of cutting injection, the density of cutting and the acceleration of gravity are constant values, as is the well inclination angle, the range of the well inclination angle obtained in the experiment has little impact on the thickness of the cutting bed, based on previous experience; thus, the dimensionless terms \(\Pi_{1}\), \(\Pi_{2}\), \(\Pi_{6}\), \(\Pi_{7}\) and \(\Pi_{9}\) are not considered. Then, Equation (31) can be simplified as:  
\[\frac{H}{D_{hyd}}=a_{0}[\Pi_{3}{}^{a_{3}}\Pi_{4}{}^{a_{4}}\Pi_{5}{}^{a_{5}}\Pi_ {8}{}^{a_{8}}] \tag{32}\]  
When using water as the drilling fluid, using the least squares method yields fitting with the experimental data:  
\[\frac{H}{D_{hyd}}=1.2082\left[ \begin{array}{c}n^{-0.1068}\left(\frac{k}{D_{ hyd}^{n}\rho_{d}v_{d}^{2-n}}\right)^{0.0686}\\ \left(\frac{D_{hyd}rpm}{v_{d}}\right)^{-0.0034}\left(\frac{d_{c}}{D_{hyd}} \right)^{0.0059}\end{array} \right] \tag{33}\]  
The thickness of the dimensionless cutting bed is expressed as:  
\[\frac{H}{D_{w}}=1.2082\left[ \begin{array}{c}n^{-0.1068}\left(\frac{k}{D_{ hyd}^{n}\rho_{d}v_{d}^{2-n}}\right)^{0.0686}\\ \left(\frac{D_{hyd}rpm}{v_{d}}\right)^{-0.0034}\left(\frac{d_{c}}{D_{ hyd}}\right)^{0.0059}\end{array} \right]\frac{D_{hyd}}{D_{w}} \tag{34}\]  
Figure 3 shows the comparison between the predicted values of the model and the experimental data when water is used as the drilling fluid. A total of 42 sets of data were used for data fitting. After the calculation and comparison, the mean square error for the three data fit lines was 7.438%, 5.247% and 6.731%, respectively, and it can be found that the error of most of the data is less than 8%, which meets the requirements of the model error at the drilling construction site. Figure 3: Comparing the experimental and predicted values of the dimensionless cuttings bed thickness. When using Polyanionic Cellulose (PAC) as the drilling fluid, the same least-squares fit with the help of experimental data yields:  
\[\frac{H}{D_{hyd}}=3.4331\left[ \begin{array}{c}n^{-0.3553}\left(\frac{k}{D_{ hyd}^{n}\rho_{d}\upsilon_{d}^{2-n}}\right)^{0.2228}\\ \left(\frac{D_{hyd}rpm}{v_{d}}\right)^{-0.0154}\left(\frac{d_{c}}{D_{hyd}} \right)^{0.0875}\end{array} \right] \tag{35}\]  
The thickness of the dimensionless cutting bed is expressed as:  
\[\frac{H}{D_{w}}=3.4331\left[ \begin{array}{c}n^{-0.3553}\left(\frac{k}{D_{ hyd}^{n}\rho_{d}\upsilon_{d}^{2-n}}\right)^{0.2228}\\ \left(\frac{D_{hyd}rpm}{v_{d}}\right)^{-0.0154}\left(\frac{d_{c}}{D_{hyd}} \right)^{0.0875}\end{array} \right]\frac{D_{hyd}}{D_{w}} \tag{36}\]  
Figure 4 shows the comparison between the predicted values of the model and the experimental data when PAC is used as the drilling fluid; 52 sets of data were used for data fitting. After the calculation and comparison, the mean square error for the three data fit lines was 5.949%, 4.354% and 6.003%, respectively, and it can be found that the error of most of the data is less than 8%, which meets the requirements of the model error at the drilling construction site.