## 2 ROP Prediction Based on Mechanical-Specific Energy Theory  
### Drilling Rate Equations Based on Mechanical-Specific Energy Theory  
Mechanical drilling rate prediction based on mechanical-specific energy theory is a method used to estimate the mechanical drilling rate during underground drilling processes. The fundamental idea of this theory is that mechanical drilling rate depends on the mechanical performance of the drill bit and the physical properties of the formation. In the latter half of the 19 th century, [PERSON] [23] proposed the mechanical-specific energy theory through extensive laboratory experiments on rock fragmentation. Its expression is as follows:  
\[M=\frac{4W}{\pi D_{\text{b}}^{2}}+\frac{480\tau T_{b}}{D_{\text{b}}^{2}R} \tag{1}\]  
where \(M\) is the mechanical specific energy, MPa; \(W\) is the drilling pressure, KN; \(D_{b}\) is the diameter of the drill bit, mm; \(r\) is the rotational speed, r/min; \(T_{b}\) is the torque of the drill bit, KN\(\cdot\)m; and \(R\) is the ROP, m/h. The mechanical-specific energy theory assumes that the formation is homogeneous and isotropic. The \(M\) value represents the ease of cutting rock or formation materials under unit cutting energy, and this degree is determined by factors such as the drill bit type and drilling tool parameters. In the process of rock breaking with a drill bit, rock strength plays a crucial role. The unconfined compressive strength (UCS) of rock represents the actual compressive strength of rocks under confinement. However, during drilling calculations, the uniaxial compressive strength is still often used. This is because true triaxial rock mechanic experiments are costly and require strict experimental conditions, making it difficult to obtain a large amount of experimental data. The Mohr-Coulomb criterion is an essential tool for analyzing and predicting the fracture and shear behavior of rocks under stress conditions. It is based on graphical representation and employs the Mohr circle and the angle of friction to describe the shear behavior of materials, as expressed below:  
\[\tau=c+\sigma\cdot\tan(\phi) \tag{2}\]  
where \(\tau\) represents shear strength, MPa; \(c\) is the cohesion of the rock (shear strength typically measured under unconfined conditions), MPa; \(\sigma\) denotes normal stress, MPa; and \(\phi\) is the internal friction angle, \({}^{\circ}\). When subsurface rocks are subjected to confining pressure, their compressive strength increases significantly. Using uniaxial compressive strength to describe subsurface rocks under these conditions would result in significant deviations. Therefore, in practical engineering, it is common to perform an equivalent substitution using the Mohr-Coulomb criterion. This involves replacing the parameters in Equation (2) with stress parameters under lateral confinement to obtain the rock's lateral confinement compressive strength:  
\[\left\{ \begin{array}{l}CCS=UCS+D_{p}+2D_{p}+\frac{\sin\phi}{1-\sin\phi}\\ D_{p}=ECD-P_{p}\end{array} \right. \tag{3}\]  
where \(CCS\) is the confined compressive strength, MPa; \(UCS\) is the uniaxial compressive strength, MPa; \(D_{p}\) is the difference between drilling fluid pressure and formation pressure, MPa; \(\phi\) is the internal friction angle of the rock, \({}^{\circ}\); \(ECD\) is the circulating drilling fluid pressure, MPa; and \(P_{p}\) is the formation pressure, MPa. Based on Equations (1) and (3), that is, based on the mechanical-specific energy theory while considering other drilling parameters, such as drill bit characteristics, the drilling rate equation can be derived as:  
\[R=\frac{480\tau T_{\text{b}}}{MD_{\text{b}}^{2}-\frac{4W}{\pi}}=\frac{4.06\mu \cdot r}{D_{\text{b}}\left(\frac{0.6446 CCS}{EF\cdot W}-\frac{101.6}{\pi D_{ \text{b}}^{2}}\right)} \tag{4}\]  
where \(\mu\) is the specific sliding friction factor of the drill bit, dimensionless; \(EFF\) is the mechanical efficiency, %; and the other variables are the same as those that were previously mentioned.