## 3 Development of a Novel Model Cutting Transport Ratio
Equation (6) is used to determine the \(V_{ann}\)[40]:  
\[V_{ann}=\frac{24.5 GPM}{OH^{2}-OD_{pipe}}^{2}=V_{cr}+V_{sc} \tag{6}\]  
where \(V_{cr}\) is the is the cutting rise velocity (ft/min), which can be calculated by Equation (7), and \(V_{sc}\) is the cutting slip velocity with the effect of rate of penetration (ft/min), which can be obtained from Equations (6) and (7) \(V_{sc}=V_{ann}-V_{cr}\)[41; 42]. \[V_{cr}=\frac{60}{\left(1-\left(\frac{OD_{pipe}}{OH}\right)^{2}\right)*\left( 0.64+\frac{18.16}{ROP}\right)} \tag{7}\]  
\[V_{sc}=\frac{24.5 GPM}{OH^{2}-OD_{pipe}}^{2}-\frac{60}{\left(1-\left(\frac{OD_ {pipe}}{OH}\right)^{2}\right)\left(0.64+\frac{18.16}{ROP}\right)} \tag{8}\]  
In Equation (7), \(V_{cr}\) represents the rate at which cuttings are produced and elevated with the mud as a result of the \(ROP\). Equation (9) is used to determine the apparent viscosity of the drilling fluid [42]:  
\[M_{App}=\frac{2.4V_{ann}}{OH-OD}\left(\frac{2n+1}{3n}\right)^{n}\left(\frac{200 K(OH-OD)}{V_{a}}\right) \tag{9}\]where the drilling fluid's \(K\) is a consistency factor (cP) and the flow behavior index is \(n\), which can be obtained by the Equations (10) and (11), respectively [40,43]. \[\mathrm{CTR}=1-\frac{V_{s}}{V_{a}} \tag{2}\]0  
\[\mathrm{CTR}=1-\frac{V_{s}}{V_{a}} \tag{2}\]1  
The \(k_{m}\) and \(n_{m}\) factor can be replaced and used in Equation (9). The drilling fluid's effective viscosity can be obtained from Equation (12) (cP) [42]:  
\[\mathrm{CTR}=1-\frac{V_{s}}{V_{a}} \tag{2}\]2  
where \(d_{c}\) is the cutting diameter (inch) and can be calculated by Equation (13) [40]:  
\[\mathrm{CTR}=1-\frac{V_{s}}{V_{a}} \tag{2}\]3  
where \(RPM\) is the rotary speed of bit (rpm). In Equation (13), based on [42,44], the maximum cutting diameter is 0.3-0.5 inch. Due to the use of directional drilling tools such as mud motors or the combination of a rotary steerable system (RSS) and a mud motor under a general name, which is referred to in the field as motorized RSS, due to the importance of considering the downhole \(RPM\) while drilling using the mud motor or the motorized RSS, Equation (13) was modified to Equation (14) to predict the modified average cutting diameter by including the specifications of the mud motor, such as the revolution per gallon ratio (\(x\)) and the \(GPM\) of the mud pump flow rate. Thus, the modified cutting diameter can be obtained as follows:  
\[\mathrm{CTR}=1-\frac{V_{s}}{V_{a}} \tag{2}\]4  
As was explained above, the CTR is calculated by dividing the transport velocity by the drilling fluid velocity. However, this method falls short because it overlooks a number of factors that affect the CTR, such as the cuttings density (\(Wc\)) and the low shear yield point (\(LSYP\)). Thus, from Equation (5), the \(ECD\) can be modified as follows based on \(PV\) and \(YP\). Moreover, the Fann viscometer is a tool commonly utilized to measure shear stress readings at various rotational speeds, including 600 RPM, 300 RPM, 200 RPM, 100 RPM, 6 RPM, and 3 RPM. The \(PV\) and \(YP\) values can be calculated using Equations (15) and (16), with the shear stress readings obtained at 300 RPM and 600 RPM [7,42]:  
\[\mathrm{CTR}=1-\frac{V_{s}}{V_{a}} \tag{2}\]5  
\[\mathrm{CTR}=1-\frac{V_{s}}{V_{a}} \tag{2}\]6  
where \(R600\) and \(R300\) are the viscometer reading at 600 RPM and 300 RPM, respectively. From Equations (5) and (12), the \(PV_{m}\) and \(YP_{m}\) can be replaced, and the following Equation (17) of \(ECD_{m}\) can be obtained:  
\[ \begin{array}{ll}ECD_{m}=MW_{eff}&+((\frac{0.085}{OH-OD_{pipe}})\\ &\cdot((2R300-LSYP)-(R600-LSYP)\\ &+\frac{(R600-LSYP)-(R300-LSYP)Vann.m)}{300(OH-OD_{pipe}})).7.481\end{array}  \tag{17}\]  
In Equation (6), which represents the original annular mud velocity applied in the vertical hole section alone, \(V_{amn}\) is given as a function of \(GPM\), \(OH\), and \(OD\).