## 3 Development of a Novel Model Cutting Transport Ratio
The modified annular velocity (\(V_{amn,m}\)), as defined in Equation (18), depends on the weight and flow rate of the drilling fluid, the size of the drilled hole, the outer diameter of the drill pipe, the rate of penetration, the rotation of the drill string, the plastic viscosity, the yield point, the Fann viscometer readings at 3 and 6 rpm, the wellbore inclination, and the azimuthal directions [7; 42]. \[ \begin{array}{c}V_{ann,m}\\ =(\frac{24.5\cdot GPM}{OH^{2}-OD_{pipe}}cos(\alpha)\\ +\Bigg{(}\frac{60}{\big{(}1-\big{(}OD_{pipe}\big{)}^{2}\big{)}*(0.64+\frac{18 \alpha}{\mathrm{R}\mathrm{D}D}\big{)}}+\frac{ROP\big{(}OH^{2}\big{)}}{60\big{(} OH^{2}-OD_{pipe}\big{)}}\Bigg{)}\sin(\beta))\\ +\frac{175(4_{cm})\Big{(}22-\frac{MW_{eff}}{7.481}\Big{)}^{2m}}{(MW_{eff}/7.481 )^{m}\cdot(\frac{24.5\cdot Var_{cm}}{OH^{2}-OD_{pipe}}(\frac{2m+1}{34m})^{m} \Big{(}\frac{200K_{m}\big{(}OH-OD_{pipe}\big{)}}{V_{ann}}\Big{)}\Big{)}^{m}} \end{array}  \tag{18}\]  
where \(\alpha\) is the open hole angle (degrees) and \(\beta\) is the open hole azimuth (degrees). Based on worldwide experience, while drilling, there is a recommended pump flow which can be applied while drilling [45]. Table 1 shows the recommended pump flow (_GPM_) based on the hole size diameters. In Equation (8), \(V_{scm}\) represents the modified slip velocity of the cuttings (ft/min), which takes into account the influence of the mud weight and other factors. The slip velocity of the cuttings, along with the effect of the mud weight, is determined using Equation (19). This equation employs \(V_{s1}\), \(V_{s2}\), and \(V_{st}\), which are cuttings velocities calculated based on the effective viscosity, the apparent viscosity, the rate of penetration, and the cutting in turbulent flow [40]. Moreover, the \(V_{sc}\) can be added to the equation to take into account the influence of the cutting rise and the average mud annular velocity to consider the hydraulic velocity influence. To calculate \(V_{s1}\), \(V_{s2}\), \(V_{st}\), and \(V_{sc}\) Equation (19) can be utilized with the modified \(d_{cm}\)[46; 47; 48]. \[V_{scm}=\frac{V_{s1}+V_{s2}+V_{st}+V_{sc}}{4} \tag{19}\]  
where  
\[V_{s1}=0.45\Bigg{(}\frac{Meff}{\Big{(}MW_{eff}dcm\Big{)}}\Bigg{)}\left(\Bigg{(} \frac{36800 MW_{eff}dcm^{3}\Big{(}Wc-MW_{eff}\Big{)}}{Meff^{2}}\Bigg{)}+1 \right)^{0.5}-1 \tag{20}\]  
\[V_{s2}=\Bigg{(}\frac{175(dcm)\Big{(}Wc-MW_{eff}\Big{)}^{0.667}}{MW_{eff}{}^{0. 333}Mapp^{0.333}}\Bigg{)} \tag{21}\]  
\[V_{st}=113.4\cdot\Bigg{(}\frac{d_{cm}\Big{(}pp-MW_{eff}\Big{)}}{CD\cdot MW_{ eff}}\Bigg{)} \tag{22}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Hole Size, inches** & **Recommended GPM, gal/min** & \(V_{ann}\)**, ft/hr** \\ \hline
22 & 1000–1200 & 54–65 \\
16 & 1000 & 108 \\
12.5 & 950–1000 & 185–195 \\
8.5 & 500–600 & 292–350 \\
6.125 & 250–300 & 285–342 \\
5 7/8 & 250–300 & 330–397 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The recommended GPM based on the hole size diameters. where \(pp\) is the particle density (lb/cf), \(CD\) is the drag coefficient, which can be calculated using Equation (23), and \(Wc\) is the weight cutting (lb/cf), which can be calculated using Equation (24) as follows [36; 49]:  
\[\text{CD}=\frac{24}{Re}+(1+Re)^{0.4}+e^{(1-Re^{0.39})} \tag{23}\]  
\[Wc=\left(MW_{eff}CCA+MW_{eff}\right)+(1-CCA)MW_{eff} \tag{24}\]  
where \(Re\) is the Reynolds number, which can be calculated using Equation (25) as follows [36; 49]:  
\[Re=\frac{2.4\cdot MW_{eff}\cdot V_{ann.m}(OH-OD)}{\mu_{eff}} \tag{25}\]  
where \(\mu_{eff}\) is the effective viscosity (cp). Equations (19)-(22) demonstrate that the average slip velocity is determined by averaging the values of the cuttings slip velocities under different flow regime types, each calculated using a respective equation. These equations also consider real-time drilling parameters, including the density of the cuttings, the drilling fluid weight, and the rheology of the drilling fluid.