# A Novel Efficient Borehole Cleaning Model for Optimizing Drilling Performance in Real Time
## 3 Mathematical Development of the Model for the TI\({}_{\text{m}}\)
Thus, the average modified RF\({}_{am}\), incorporating the effects of temperatures and \(CCA_{am}\), is defined by Equation (33)  
\[CCA\;or\;CCA_{API}=\frac{ROP\cdot OH^{2}}{1471\cdot GPM\cdot TR} \tag{4}\]0  
where \(T_{2}\) is the borehole temperature, and \(T_{1}\) is the flow line temperature at the surface. \[CCA\;or\;CCA_{API}=\frac{ROP\cdot OH^{2}}{1471\cdot GPM\cdot TR} \tag{4}\]1  
Moreover, from Equation (2) and based on [PERSON]'s model [21], \(AF\) was found in accordance with Table 2 based on the changes in the borehole angle [21; 37; 38]. Accordingly, \(AF\) was modified, and a new model was developed by interpolation to calculate \(AF\) at any borehole angle (see Figure 6). As shown in Equation (34), the \(AF\) regime is characterized by two important parameters: the \(\alpha\) borehole angle and \(CCA_{am}\). The \(\alpha\) borehole angle represents the angle between the wellbore axis and the horizontal plane and \(CCA_{am}\) is a critical parameter that impacts the ability of the drilling fluid to effectively transport cuttings out of the wellbore. Hence, the \(AF\) regime and the parameters associated with it must be carefully monitored and evaluated to ensure optimal borehole cleaning efficiency [21; 37; 38]. Figure 6 shows the interpolation of the AF based on the borehole angle. \[ \begin{array}{c}AF_{m}=-0.0208\alpha+2.03\\ Or\\ (0.0001(\alpha)^{2}-0.03(\alpha)+2.1)(1-CCA_{am})\end{array}  \tag{34}\]  
More importantly, it is important to take into consideration real-time drilling factors such as the density of the cuttings, the weight of the drilling fluid, and the rheology of the drilling fluid [53]. Furthermore, \(ECD_{m}\) can be obtained from Equation (35) using  
Figure 6: The interpolation of AF based on the borehole angle. parameters defined by Equations (5)-(9). Moreover, \(MW\) defined by Equation (3) can be obtained using the original lifting capacity (\(LC\)) as shown in Equation (36) [53] and be modified (\(LC_{m}\)). Furthermore, \(LC_{m}\) can be included in \(RF_{av}\) to incorporate the influence of bouncy on rheology in accordance with [53]. \[ \begin{array}{rcl}ECD_{m}=MW_{eff}&+&\left(\left(\frac{0.085}{OH-OD_{pipe}} \right)\right.\\ &\cdot&\left(\left(2R300-LSYP\right)-\left(R600-LSYP\right)\right.\\ &+&\left.\frac{\left(\left(R600-LSYP\right)-\left(R300-LSYP\right)Vam.m}{300 \left(OH-OD_{pipe}\right)}\right)\right).7.481\\ \\ \end{array}  \tag{35}\]  
\[LC=\frac{MW}{Wc}\to LC_{m}=\frac{MW_{eff}}{Wc} \tag{36}\]  
As a direct result, Equation (37) may be applied to compute and finally determine the updated \(\Pi_{m}\):  
\[ \begin{array}{rcl}TI_{m}=MW\cdot(RF_{av}&+&LC_{m})\cdot AF_{m}\\ &=&\left(\frac{MW_{eff}}{ECD_{m}}\right)\left(\frac{RF_{1}+RF_{2}}{2}+\left( \frac{MW_{eff}}{Wc}\right)\right)\cdot(-0.0208\alpha+2.03)\end{array}  \tag{37}\]  
where \(MW\) is the specific gravity of the used drilling fluids, which can be computed as \(SG=MW/62.4\), (pcf). The development of a novel \(TI_{m}\) model represents a significant advancement in drilling operations. The model contains different parameters, such as hydraulic velocities, rheological characteristics of drilling fluids (including the low shear yield point, \(k\) and \(n\) factor based on \(CCA_{m}\)), flow regime, cuttings properties such as diameter and weight, \(ECD\), lifting capacity, and angle factor, to create a unique \(TI_{m}\) model. As depicted in Figure 7, initial parameters such as \(CCA\) and \(MW_{eff}\) can be used to determine \(TI_{m}\). The \(TI_{m}\) model provides a reliable indication of the state of borehole cleaning, and two different standards are applied to judge the performance of the model for evaluating the conditions of borehole cleaning. A \(TI_{m}\) value of more than 1 indicates that the borehole cleaning process was carried out correctly, while a \(TI_{m}\) value of less than 1 indicates unsatisfactory borehole cleaning in accordance with [21; 37; 38].