## 5 Cutter Spacing Optimization and Bit Rock Breaking Simulation
### Full-Size Bit Rock Breaking Simulation  
To fully understand the rock-breaking effect of the obtained optimal solution, the cutting structure of the PDC bit embedded by a triangular prism cutter, axe cutter, and mixed triangular prism and axe cutter with a diameter of 215.9 mm was established, respectively, and the related finite element model was constructed in ABAQUS software as shown in Figure 19, with a formation diameter of 600 mm and a height of 300 mm. The layout of the cutter on the bit follows the principle of equal cutting volume: the grid size of the cutting structure of the bit is 4 mm, the size of the contact with the rock is 4 mm, and the grid size of the rest of the rock is 12 mm. The wear of the cutting structure of the bit is ignored in the simulation process, and it is considered a rigid body. The cutter is arranged with a back rake angle of 15\({}^{\circ}\) and a side rotation angle of 0\({}^{\circ}\). Its rotation and vertical motion in the well are considered. The WOB is 80 kN, and the RPM is 210 r/min. The rock is fixed except for the upper surface, and the simulation time is 15 s.  
The contact between the simulated bit and the rock is a highly nonlinear process that includes: (1) Material nonlinearity: the rock begins to deform plastically when it is subjected to more than the maximum stress it can withstand. (2) Geometric nonlinearity: The position of the bit is constantly changing during the interaction between rock and bit contact. (3) Contact nonlinearity: The contact surface between the bit and the rock is constantly changing as the rock disappears after breaking during the simulation. The above nonlinear process can cause the simulation to be difficult to converge. Based on D'Alembert's principle, the mechanical model of the contact process between the drill bit and the rock is established as follows:  
\[ \begin{pmatrix}T_{Q}&0\\ 0&T_{Y}\end{pmatrix} \left\{ \begin{array}{cc}A^{t}_{Q}\\ A^{t}_{Y}\end{array} \right\}+ \begin{pmatrix}C_{Q}&0\\ 0&C_{Y}\end{pmatrix} \left\{ \begin{array}{cc}V^{t}_{Q}\\ V^{t}_{Y}\end{array} \right\}+ \begin{pmatrix}K_{Q}&0\\ 0&K_{Y}\end{pmatrix} \left\{ \begin{array}{cc}U^{t}_{Q}\\ U^{t}_{Y}\end{array} \right\}=\left\{ \begin{array}{cc}S^{t}_{Q}-F^{t}_{Q}\\ S^{t}_{Y}-F^{t}_{Y}\end{array} \right\} \tag{11}\]  
where \(T\) is the overall mass matrix of the finite element model; \(C\) is the overall damping matrix of the finite element model; \(K\) is the overall stiffness matrix; \(A\) is acceleration; \(V\) is velocity; \(U\) is displacement; \(S^{t}\) is the WOB vector; \(F^{t}\) is the contact force vector; the subscript \(Q\) represents the PDC cutter; and \(Y\) represents the rock. Figure 18: Rock-breaking comparison of drill bits before and after using an optimized cutter arrangement: (**a**) Finite element model; (**b**) Bit footage comparison. The dependability of the full-scale drill bit simulation outcomes was authenticated by applying a PDC bit equipped with an axe cutter with a WOB of 60 kN, an RPM of 90 r/min, and a simulation duration of 1 s. Subsequently, the mechanical ROP curve was acquired, as shown in Figure 20a, and the mean ROP was computed at 11.615 mm/s. Following this, a constant mechanical ROP of 11.615 mm/s was administered to the bit, with the resultant reaction force illustrated in Figure 20b. Under constant WOB, the average reaction force exerted on the bit was calculated to be 47.65 kN, while the same under constant ROP was 51.92 kN. Observing a discrepancy of 8.9% between the two values provided partial validation of the numerical simulation's precision. Figure 21 delineates the rock stress contour during and subsequent to the completion of rock fragmentation for the three bits. It can be inferred from the contour that the bit implemented with mixed cutters exerts the maximum stress on rock action, thus indicating that the PDC bit with mixed cutters can more effortlessly attain the yield stress of the rock during drilling.