# A Novel Efficient Borehole Cleaning Model for Optimizing Drilling Performance in Real Time
## 3 Mathematical Development of the Model for the TI\({}_{\text{m}}\)
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{2}{c}{\(AF\) by [PERSON] [38]} & \multicolumn{2}{c}{\(AF\) by [PERSON] [21]} \\ \hline
**Borehole Angle** & **Angle Factor** & **Borehole Angle** & **Angle Factor** \\ \hline
0 & - & 0 & 2.03 \\
25 & 1.51 & 25 & 1.51 \\
30 & 1.39 & 30 & 1.39 \\
35 & 1.31 & 35 & 1.31 \\
40 & 1.24 & 40 & 1.24 \\
45 & 1.18 & 45 & 1.18 \\
50 & 1.14 & 50 & 1.14 \\
55 & 1.1 & 55 & 1.1 \\
60 & 1.07 & 60 & 1.07 \\
65 & 1.05 & 65 & 1.05 \\
70–80 & 1.02 & 70 & 1.02 \\
80–90 & 1 & 80 & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Angle factor values at different borehole angles. Figure 5: Graphical chart for the \(RF\) in borehole size 17–12′′. Chart is based on [PERSON]’s chart [38]. of the drilling process in removing cuttings and maintaining a clean wellbore. By using this model, drilling operators can quickly assess the state of borehole cleaning and take corrective measures and actions if necessary [21; 37; 38]. Accordingly, the novel modified model transport index (\(TI_{m}\)) was developed based on the effective mud weight (\(MW_{eff}\)) calculated using Equation (1) [8]. \[MW_{eff}=MW\cdot CCA+MW \tag{3}\]  
where the variable \(CCA\) represents the cuttings concentration in an annulus, and it is defined by Equation (4) [8; 40]. \[CCA\;or\;CCA_{API}=\frac{ROP\cdot OH^{2}}{1471\cdot GPM\cdot TR} \tag{4}\]  
where \(ROP\) represents the rate of penetration (ft/h), \(OH\) represents the diameter of the borehole size (inch), and 1471 is a conversion factor. \(GPM\) represents the pump flow rate in gal/min, and \(TR\) represents the transport ratio. The transport ratio can be substituted with a value of 0.55 according to [8]. From Equation (3), in real time, the \(ECD\) can be found based on \(MW_{eff}\) (pcf). Specifically, Equation (4) can be used to determine the \(ECD\)[39]. \[ECD=MW+\left(\left(  
In particular, [PERSON] developed a more accurate _CCA_ model for steady-state lifting of materials in a vertical tube by using Equation (4), which may be found in Equation (10) [8; 35]. [PERSON] provided evidence that an annular concentration model may be derived by taking into account both the circulation that takes place before a connection but after drilling has been halted and the circulation that takes place after connections but before drilling resumes. The time known as the preconnection circulation period is what is known as the later circulation. The annulus is where his Equation tells us to go and obtain the average cutting volume percent that was computed for us (see Equation (11)) [8].  
\[CCA_{1}=-\frac{1}{2}\bigg{(}\frac{Vann_{m}}{Vs}-1\bigg{)}+\bigg{(}\frac{1}{4} \bigg{(}\frac{Vann_{m}}{Vs}-1\bigg{)}^{2}+\frac{Vann_{m}}{Vs}\frac{Vc}{\frac{ CPM}{\mathcal{J}8}}\bigg{)}^{0.5} \tag{10}\]  
\[CCA_{2}=\frac{1}{1+\Big{(}1-\frac{OD}{OH}\Big{)}\Big{(}\frac{Vann_{m}-Vs}{30} \Big{)}\Big{(}\frac{1800}{1+ROP}+\frac{Vs}{Vs_{an.dc}-Vs_{an}T_{PC}}\Big{)}} \tag{11}\]  
where \(Vann_{m}\) represents the annular modified velocity of the drilling fluid (ft/min) (Equation (12)), \(V_{ann.dc}\) is the annular velocity across the drill collar (ft/min), \(T_{PC}\) is the preconnection circulation time, which specifies the amount of time required to circulate the cuttings to a height that will prevent them from sinking to the bottom of the borehole during the process of making that connection (min), \(V_{sa}\) is the average velocity of cutting slip (ft/min), and \(V_{c}\) is the volumetric rate of cuttings entering the annulus (ft/min). \(Vann_{m}\), \(V_{ann.dp}\), \(V_{mem.dc}\), \(T_{PC}\), \(Vc\), and \(V_{sa}\) can be obtained from Equations (12)-(17), respectively.