# 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
Therefore, Equation (9) can be simplified to:  
\[f\Big{(}H,D_{hyd},\theta,e,v_{d},\rho_{d},n,K,rpm,C_{c},\rho_{c},d_{c},g\Big{)}=0 \tag{15}\]  
There are 13 physical quantities involved during the transport of small cuttings, and the magnitudes and corresponding symbols of each variable are described in Table 1.  
In the wellbore cleaning problem of extended-reach drillings with high-inclination sections, \(D_{hyd}\), \(D_{hyd}\) and \(D_{hyd}\) are chosen as the basic physical quantities, and all other quantities can be obtained by these three basic quantities; the three basic measures chosen are the mass basic measure \([M]\), the time basic measure \([T]\), and the distance basic measure \([L]\) (Sun, 2013) [16]. We can know that \(n=13\) and \(m=3\) according to the \(\Pi\) theorem, so there exist \(n-m=10\) dimensionless physical quantities. To solve for the dimensionless term \(\Pi_{5}\) of the drilling pipe speed, we have the following equation according to \(\Pi\) theorem:  
\[\Pi_{5}=D_{hyd}^{a}v_{d}^{b}\rho_{d}^{c}(rpm) \tag{16}\]  
where \(a\), \(b\) and \(c\) are factors to be determined. Find the magnitudes of the variables from Table 1 and bring in:  
\[\Pi_{5}=(L)^{a}\Big{(}LT^{-1}\Big{)}^{b}\Big{(}ML^{-3}\Big{)}^{c}T^{-1}=L^{a+b- 3c}T^{-b-1}M^{c} \tag{17}\]  
By making the exponents of the three basic measures zero, we have:  
\[\left\{ \begin{array}{l}a+b-3c=0\\ -b-1=0\\ c=0\end{array} \right. \tag{18}\]  
The solution gives: \(a=1\), \(b=-1\), \(c=0\). The dimensionless term of drill pipe speed is:  
\[\Pi_{5}=D_{hyd}v_{d}^{-1}(rpm)=\frac{D_{hyd}(rpm)}{v_{d}} \tag{19}\]  
The other dimensionless terms can be found in the same way and are expressed as follows:  
\[\Pi_{1}=\theta \tag{20}\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Symbols of Variables** & **Corresponding Scale** \\ \hline \(D_{hyd}\) & L \\ \(\theta\) & 1 \\ \(e\) & 1 \\ \(v_{d}\) & LT\({}^{-1}\) \\ \(\rho_{d}\) & ML\({}^{-3}\) \\ \(n\) & 1 \\ \(K\) & ML\({}^{-1}\)Tn\({}^{-2}\) \\ \(rpm\) & T\({}^{-1}\) \\ \(C_{c}\) & 1 \\ \(\rho_{c}\) & ML\({}^{-3}\) \\ \(d_{c}\) & L \\ \(g\) & LT\({}^{-2}\) \\ \(H\) & L \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The symbols and dimensions of the variables. \[\Pi_{2}=e \tag{21}\] \[\Pi_{3}=n\] (22) \[\Pi_{4}=\frac{K}{D_{hyd}{}^{n}\rho_{d}{}\
u_{d}{}^{2-n}}\] (23) \[\Pi_{6}=C_{cs}\] (24) \[\Pi_{7}=\frac{\rho_{c}}{\rho_{d}}\] (25) \[\Pi_{8}=\frac{d_{c}}{D_{hyd}}\] (26) \[\Pi_{9}=\frac{D_{hyd}g}{v_{d}^{2}}\] (27) \[\Pi_{10}=\frac{H}{D_{hyd}} \tag{28}\]  
The relationship between each of the above dimensionless terms can be expressed as:  
\[f(\Pi_{1},\Pi_{2},\Pi_{3},\Pi_{4},\Pi_{5},\Pi_{6},\Pi_{7},\Pi_{8},\Pi_{9},\Pi _{10})=0 \tag{29}\]  
Separate the cutting bed thickness dimensionless term:  
\[\Pi_{2}=e \tag{21}\]0  
The physical quantities are nonlinearly related to the thickness of the dimensionless cutting bed, so that Equation (24) can be expressed as:  
\[\Pi_{2}=e \tag{21}\]1  
where \(a_{0}\), \(a_{1}\), \(a_{2}\), \(a_{3}\), \(a_{4}\), \(a_{5}\), \(a_{6}\), \(a_{7}\), \(a_{8}\) and \(a_{9}\) are factors to be determined. Regression calculations were performed using the experimental data from the article by [PERSON] et al. (2008) [PERSON] et al. (2008). The data relevant to the calculation are shown in \[\Pi_{2}=e \tag{21}\]2 \begin{table}
\begin{tabular}{c c} \hline \hline
**Symbols of Variables** & **Corresponding Scale** \\ \hline Parameters & Values \\ Drilling fluid density & 1180 kg/m\({}^{3}\) \\ Drilling fluid flowability index & 0.55 \\ Drilling fluid consistency factor & 0.64 kg\(\cdot\)s\({}^{n-2}\)/m \\ Drill pipe speed & 10.47 s\({}^{-1}\) \\ Displacement & 0.03 m\({}^{3}\)/s \\ Drill pipe OD & 0.127 m \\ Borelode diameter & 0.2159 m \\ Cutting diameter & 0.0003 m \\ Parameters & Values \\ Drilling fluid density & 1180 kg/m\({}^{3}\) \\ Drilling fluid flowability index & 0.55 \\ Drilling fluid consistency factor & 0.64 kg\(\cdot\)s\({}^{n-2}\)/m \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The basic data used in the example well.