# A Novel Efficient Borehole Cleaning Model for Optimizing Drilling Performance in Real Time
## 3 Mathematical Development of the Model for the TI\({}_{\text{m}}\)  
The transport index is a model used in drilling operations to evaluate the effectiveness of the drilling fluid in removing cuttings and maintaining a clean wellbore. It is a measure of the ability of the fluid to transport cuttings out of the wellbore and to the surface and was developed by [PERSON] in 1992 [37; 38]. The first model is described by the following Equation [37]:  
\[\text{TI}=\frac{GPM\cdot RF\cdot MW}{100} \tag{1}\]  
where \(GPM\) is the flow rate of the mud pump (gal/min), \(RF\) is the rheology factor, and \(MW\) is the static drilling fluid density (lb/cf). RF was added to Equation (1) to describe the combined effect of the \(PV\) and \(YP\) of drilling fluids on fluid rheology. Moreover, [PERSON] proved that the penetration rate depends on the mud weight, flow rate, and the rheology factor. As seen in Figure 4, from the chart, the TI and borehole angle can be applied to determine the maximum ROP while drilling based on the results obtained from Equation (1) [37]. More importantly, in 1994, [PERSON] showed a straightforward graphical approach that could be used on the rig to evaluate borehole cleaning for different sized boreholes. The model employs a series of charts to assess the efficacy of the borehole cleaning procedure [38]. The full model was obtained by combining the effects of the mud weight (\(MW\)), angle factor (\(AF\)), and rheology factor (\(RF\)), as shown in Equation (2) [38]. \[\text{TI}=RF\cdot AF\cdot MW \tag{2}\]  
Based on [37; 38], using Equations (1) and (2), a series of charts were developed for each of the borehole diameters to calculate the value of the \(RF\) based on the mud \(PV\) and \(YP\). A higher \(RF\) value indicates a mud that is more viscous and thicker, which translates to better suspension of cuttings and more efficient transport to the surface [10]. As an example, Figure 5 shows the \(RF\) for the borehole size 17-12\({}^{\prime\prime}\). Figure 4: Read of maximum ROP based on the TI values and the borehole angle. Chart is based on [PERSON]’s chart [37]. From Equations (1) and (2), the \(AF\) is a term used in directional drilling to describe the effect of the borehole angle on the amount of cuttings that can be transported by the drilling fluid. A higher wellbore angle can result in a lower \(AF\) value, indicating that borehole cleaning may become more difficult. The \(AF\) can be obtained based on the borehole angle, as shown in Table 2[38]. More importantly, Unegbu developed a model based on the flow rate in 2010 to determine the equivalent RF based on the borehole angle for vertical and deviated wells, as shown in Table 2[21]. Despite the fact that the model was able to evaluate the borehole cleaning conditions in real time, [PERSON]'s model considers only the mud weight in static conditions, flow rate, and rheology factor. Moreover, the \(RF\) can only be obtained from the chart, which also depends on the borehole size [37; 38]. The \(AF\) can only be obtained from Table 2[21; 37; 38]. Therefore, the main novelty of the novel modified model transport index (\(TI_{m}\)) is to consider the \(MW\) in static and dynamic conditions (\(ECD\)), develop a new model by interpolation to calculate the \(AF\) at any borehole angle, and develop a novel model to calculate the \(RF\). More importantly, the novel \(TI_{m}\) model considers a range of additional factors, including hydraulic velocities, the rheological characteristics of the drilling fluids with regard to the low shear yield point (\(LSP\)), the flow regime, the properties of the cuttings, and \(ECD\)[39]. Moreover, \(TI_{m}\) is an automated indicator based on two criteria to evaluate real-time borehole cleaning during drilling operations. If the value of \(TI_{m}\) is above 1, borehole cleaning is satisfactory. On the other hand, if \(TI_{m}\) falls below 1, borehole cleaning is inadequate. In essence, the \(TI_{m}\) model provides an easy-to-use indicator of the efficiency