## 4 Discussion: Analysis of Factors Affecting Drilling Forces
### Analysis of Drill Bit Size on Drilling Force  
During the drilling process, the variation in drill bit dimensions can have different effects on drill bit forces. In this study, we selected these drill bit sizes because they are relatively common on the market and can meet most practical drilling conditions. These drill bit models are denoted as 59#, 75#, and 110#. Their corresponding outer diameters are 59 mm, 75 mm, and 110 mm, while their inner diameters are 43 mm, 59 mm, and 91 mm, respectively. Additionally, the number of water outlets for these drill bits is 6, 7, and 10, respectively. Finite element mesh models for these three drill bit models interacting with the rock are depicted in Figure 10.  
Figure 9: Relationship between torque and thrust force during drilling. By setting drilling speed and rotational speed, we control the drilling and rock-breaking process. We will separately discuss the variations in thrust force and torque with drilling speed for different-sized drill bits when the rotational speed is set at 200 rpm. Additionally, we will examine the changes in thrust force and torque with rotational speed for different-sized drill bits when the drilling speed is set at 2 mm/s. When the drilling speed is 2 mm/s, the rotation speed is 200 rpm, and the drilling time is 2.5 s. The large principal stress cloud diagrams of drilling with three sizes of drill bits are shown in Figure 11. The diameter of the drill bit in Figure 11a is 59, the diameter of the drill bit in Figure 11b is 75, and the diameter of the drill bit in Figure 11c is 110. There is a small difference in the stress values of the three sizes of drill bits. The maximum stress values appear at the contact point between the drill bit and the rock, mainly distributed at the bottom of the well and around the wellbore. Notably, the stress distribution region for the larger-sized drill bit extends over a wider range along the wellbore compared to the smaller-sized drill bits. Figure 12 illustrates the relationship curves between thrust force, torque, and drilling speed for the three drill bit sizes, 59#, 75#, and 110#, all at the same rotational speed of 200 rpm. It is evident that both thrust force and torque exhibit a linear increase with the augmentation of drilling speed. Under the same drilling speed conditions, larger drill bit sizes result in higher thrust force and torque. However, the slope of the relationship curve between thrust force, torque, and drilling speed remains largely unaffected by drill bit size. The increase in drill bit size enlarges the effective contact area between the drill bit and the rock during drilling, consequently leading to increased thrust force and torque. Therefore, it is possible to calculate the required thrust force and torque for a given drill bit size by considering the effective contact area between the drill bit and the rock. For various drill bit sizes, the following relationship expressions between thrust force, torque, and drilling speed, taking into account the drill bit size effect, can be established:  
\[\left\{ \begin{array}{l}F=\alpha(\xi_{1}v+F_{0})=\alpha\xi_{1}v+\alpha F\\ T=\alpha(\xi_{2}v+T_{0})=\alpha\xi_{2}v+\alpha T\end{array} \right. \tag{15}\]  
Figure 10: Finite element mesh model with different drill sizes. (**a**) 59# Drill bit (**b**) 75# Drill bit (**c**) 110# Drill bit. \[\alpha=\frac{A_{n}}{A_{0}} \tag{16}\]  
where \(F\) is thrust force, \(T\) is torque, \(v\) is drilling speed, \(\xi_{1}\) is the slope of the relationship curve between thrust force and drilling speed, \(\xi_{2}\) is the slope of the relationship curve between torque and drilling speed, \(F_{0}\) is the thrust force constant, \(T_{0}\) is the torque constant, \(\alpha\) represents the drill bit size influence factor, \(A_{0}\) represents the effective contact area between the current drill bit and the rock, and \(A_{n}\) represents the effective contact area between the required drill bit size and the rock.