# A New Mechanical Specific Energy Model for Composite Impact Drilling
## 2 Materials and Methods
### Mechanical Specific Energy Calculation Model of Composite Impact Drilling  
Based on a large number of experiments, [PERSON] took into account the influence of factors such as bit weight, bit diameter, rotational speed, mechanical penetration rate and torque on the working efficiency of the bit. In 1965, he first proposed the calculation model of _MSE_, which was also the original model of _MSE_ ([PERSON] 1965) [2]:  
\[MSE=\frac{WOB}{A_{b}}+\frac{120\pi\cdot N\cdot T}{A_{b}\cdot ROP} \tag{1}\]  
where _MSE_ is mechanical specific energy, \(N\) is bit rotating speed, \(T\) is torque, \(ROP\) is rate of penetration, \(A_{b}\) is the bit area and \(WOB\) (Weight on bit) is the weight on bits of the surface measurement. According to the rock-breaking mechanism of drilling with the composite impact tool mentioned previously, the energy of rock breaking by composite impact drilling mainly comes from the work done by four forces: weight on bit, torque, axial impact force generated by impact hammer and circumferential torsion force generated by the pendulum. In hard formations, hydraulic energy is mainly used to clean rock cuttings at the bottom of wells, and its contribution to the energy of broken rock is negligible here ([PERSON] 2016) [11]. In the actual drilling process, the mechanical energy provided by the power drilling tool to the bit cannot be used to break rock, so the influence of mechanical efficiency needs to be considered ([PERSON] et al. 2005, [PERSON] 2005, [PERSON] 2012) [5; 6; 7]. Considering the above factors comprehensively, the mechanical-specific energy equation at each impact can be expressed as follows:  
\[MSE=E_{m}\bigg{(}\frac{E_{W}}{A\cdot ROP}+\frac{E_{T}}{A\cdot ROP}+\frac{E_{a} }{A\cdot ROP}+\frac{E_{t}}{A\cdot ROP}\bigg{)} \tag{2}\]  
Figure 1: Working diagram of a composite impactor ([PERSON] 2019) [9]. where \(E_{m}\) is the mechanical efficiency of the bit, which can be measured by core experiments or retrieved from the logging data of adjacent wells. \(E_{W}\) is the axial energy generated by surface \(WOB\) on the bit. [PERSON] pointed out that the \(WOB\) measured by surface \(WOB\) indicator is measured under surface conditions ([PERSON] 2014) [8], which is greatly different from the actual bottomhole \(WOB\), and the relationship between the bottomhole \(WOB\) and surface \(WOB\) is obtained through research:  
\[WOB_{b}=WOB\cdot e^{-\mu T_{b}} \tag{3}\]  
where \(WOB_{b}\) is the bottomhole actual weight on the bits, \(\mu\) is the coefficient of friction of the drill string, usually the value of 0.35 ([PERSON] 2009) [12] and \(\gamma_{b}\) is the inclination of the bottomhole. Therefore,  
\[E_{W}=WOB_{b}\cdot ROP=WOB\cdot e^{-\mu T_{b}}\cdot ROP \tag{4}\]  
\(E_{T}\) is the rotational kinetic energy generated by the surface rotary table on the bit and is a function of torque. [PERSON] took the influence of the sliding friction coefficient into consideration and expressed torque as a function of the bit weight ([PERSON] and [PERSON] 1992) [4]. By substituting Equation (3), we obtain the torque generated by the surface turntable on the bit:  
\[T_{b}=\mu_{b}\frac{D_{b}WOB_{b}}{36}=\mu_{b}\frac{D_{b}WOB\cdot e^{-\mu T_{b} }}{36} \tag{5}\]  
where \(T_{b}\) is the torque on the bit generated by the surface turntable and \(D_{b}\) is the bit diameter. Therefore,  
\[E_{T}=120\pi NT_{b}=3.33\mu_{b}\pi ND_{b}WOB\cdot e^{-\mu T_{b}} \tag{6}\]  
\(E_{a}\) is the impact energy generated by the axial impact hammer in the composite impactor on the bit, and \(E_{t}\) is the torsion energy generated by the circumferential pendulum in the composite impactor on the bit:  
\[E_{a}=F_{a}v_{a} \tag{7}\]  
\[E_{t}=F_{t}v_{t} \tag{8}\]  
where \(F_{a}\) is the axial impact force of an axial hammer on the bit, \(v_{a}\) is the final velocity of the impact motion of the axial impact hammer, \(F_{t}\) is the circumferential torsional force exerted by the circumferential pendulum on the bit and \(v_{t}\) is the final circumferential velocity of a circumferential pendulum.