## 2 Different ROP Models
### Modified Warren Model  
The central idea for this model comes from the scientific fact that under steady-state drilling circumstances, the removal rate is equal to the rate at which chips are formed. Therefore, ROP is regulated by the formation cutter process, the cutting removal process, and a combination of these factors mentioned above. This approach correlates ROP to rotary speed, weight on bit (WOB), bit size, and rock strength from generalized response curves and dimensional analysis, which is formulated as follows [19; 12]:  
\[\text{ROP}=\text{W}_{\text{f}}\left(\text{f}_{\text{c}}(\text{P}_{\text{e}}) \left(\frac{\text{a}\text{S}^{2}\text{D}_{\text{bit}}{}^{3}}{\text{RPM}\cdot \text{WOB}^{2}}+\frac{\text{b}}{\text{RPM}\cdot\text{D}_{\text{bit}}}\right) +\frac{\text{c}\rho\upmu\text{D}_{\text{bit}}}{\text{F}_{\text{jm}}}\right)^ {-1} \tag{1}\]  
where ROP stands for the drilling rate in terms of (ft/h), \(\text{D}_{\text{bit}}\) denotes the bit diameter in terms of (in), S represents the confined rock compressive strength in terms of (psi), WOB represents the weight on the bit in terms of (lb\({}_{\text{f}}\)), RPM stands for the rotary speed in terms of (rev/min), \(\upmu\) stands for the plastic viscosity in terms of (cp), \(\rho\) denotes the mud density in terms of (ppg), \(\text{F}_{\text{jm}}\) represents the modified impact force in terms of (lb\({}_{\text{f}}\)), \(\text{fc}(\text{P}_{\text{e}})\) denotes the chip hold-down function (dimensionless), \(\text{W}_{\text{f}}\) stands for the bit wear dimensionless factor, and a, b, and c represent the bit dimensional parameters. The first part of the equation defines the rock breaking-up rate. The second term accounts for the distribution of WOB,as WOB is increased with the number of teeth and the teeth penetrating further into the rock. As noted earlier, \(\mathrm{F_{jm}}\) stands for the modified impact force, which is formulated as follows:  
\[\mathrm{F_{jm}}=\left(\mathbf{1}-\mathbf{A_{v}}^{-\mathbf{0.122}}\right)\times \mathrm{F_{j}} \tag{2}\]  
where \(\mathrm{A_{v}}\) is the jet-to-fluid returning velocity ratio, and then \(\mathrm{A_{v}}\) (for three jets) can be obtained using Equation (3):  
\[\mathbf{A_{v}}=\frac{\mathbf{V_{n}}}{\mathbf{V_{f}}}=\frac{\mathbf{0.15d_{b}} ^{2}}{3d_{n}^{2}} \tag{3}\]  
where \(\mathrm{d_{n}}\) is the nozzle diameter (in.) and \(\mathrm{d_{b}}\) is the bit diameter (in.). For a fixed value of impact force and a fixed bit size from Equation (4), the measured impact force must be independent of nozzle size as follows:  
\[\mathrm{F_{j}}=\mathbf{0.000516\rho qv_{n}} \tag{4}\]  
where q represents the pump flow rate in terms of (gpm) and \(\mathrm{V_{n}}\) stands for the nozzle velocity in terms of (ft/s). To consider the resultant force on a formation cutting caused by the bit, which is called the chip hold-down function, Equation (5) can be used:  
\[\mathrm{f_{c}(P_{e})}=\mathrm{c_{c}}+\mathrm{a_{c}(P_{e}-120)^{bc}} \tag{5}\]  
where \(\mathrm{P_{e}}\) stands for the effective differential pressure. \(\mathrm{a_{c}}\), \(\mathrm{b_{c}}\), and \(\mathrm{c_{c}}\) are lithology and permeability-dependent parameters, which are demonstrated in Table 2[19]. Circulation pressure at the bottom hole is defined as the summation of the annulus pressure drop and static mud column pressure. This variable can be determined through the following equation:  
\[\mathrm{P_{ECD}}=0.052\rho\mathrm{TVD}+\Delta\mathrm{P_{ann}} \tag{6}\]  
where TVD is true vertical depth in ft and \(\mathrm{P_{ECD}}\) is in psi. During the drilling operation, the teeth of a bit start to wear out. Teeth area increment is due to bit wear that reduces stress on each cutter. [PERSON] developed the below equation to determine the bit wear:  
\[\mathrm{W_{f}}=\mathbf{1}-\frac{\Delta\mathbf{B}\mathbf{G}}{\mathbf{8}} \tag{7}\]  
\[\Delta\mathbf{B}\mathbf{G}=\mathrm{W_{c}}\mathbf{\sum WOB_{i}\cdot RPM_{i} \cdot A_{abtr_{i}}S_{i}} \tag{8}\]  
\(\mathrm{A_{abri}}\) stands for the relative rock abrasiveness and \(\mathrm{W_{c}}\) represents the wear coefficient.