## 3 CFD Model of the Large Diameter Modular Drill Bit
### Governing Equations and Boundary Conditions for the CFD Simulation  
Model parameters and boundary conditions play critical roles in CFD model calculation and analysis, which directly affect the accuracy of the final results [26; 27]. There are three kinds of solutions for the turbulent \(k\)-\(\varepsilon\) model in the COMSOL software, which are the standard \(k\)-\(\varepsilon\) turbulence solver, the realizable \(k\)-\(\varepsilon\) turbulence solver and the low Reynolds number \(k\)-\(\varepsilon\) turbulence solver. When the average strain rate is large, which may lead to negative normal stress [28; 29], for which it is almost impossible to achieve a precise solution, accurate dissipation scales will be barely predicted using the solution for standard \(k\)-\(\varepsilon\) turbulence. The realizable k-\(\varepsilon\) turbulence solver has a higher accuracy and better turbulence consistency in fluid models and is more accurate in simulating flows with large velocity gradients and strong curvatures [30]. The third \(k\)-\(\varepsilon\) turbulence solver can solve models in all regions up to the wall and is often used to obtain higher precision solutions, but finer mesh and more accurate boundary conditions are needed to improve convergence. When the Reynolds number is high, the boundary layer mesh needs to be very precise, and high computer performance will be required compared with the first two methods. Considering that the gradients of fluid shear rate at the nozzles of the external bit and the nozzles of the internal non-coring bit are large, the realizable k-\(\varepsilon\) turbulent solver is selected to improve the accuracy of the bit model in this paper. Since drilling fluids flowing in a borehole follow conservation laws, including mass conservation, energy conservation, and momentum conservation, the fluids can be regarded as a continuous, incompressible turbulent flow of a single medium [31; 32]. Then, the governing equations can be derived. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Mesh Tape** & **Number of Elements** & \
\begin{tabular}{c} **Degrees of Freedom** \\ **for Solving** \\ \end{tabular} & **Computation Time** &
\begin{tabular}{c} **Outlet Return** \\ **Velocity (m/s)** \\ \end{tabular} & **Inlet Pressure (Pa)** \\ \hline Coarse mesh & 1,211,801 & 2,477,407 & 5 min, 10 s & 0.72 & 19,868 \\ Medium mesh & 243,296 & 521,245 & 11 min, 24 s & 0.70 & 19,657 \\ Fine mesh & 133,327 & 294,889 & 1 h, 19 min, 8 s & 0.71 & 18,760 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Mesh independence study. Figure 2: Fluid domain model meshing of the modular drill bit: (**a**) coring bit; (**b**) non-coring bit. Firstly, the turbulent kinetic energy, \(k\), in the transport equation can be expressed as:  
\[\rho(\mathbf{u}\cdot\
abla)k=\
abla\cdot\left[\left(\mu+\frac{\mu_{T}}{\sigma_{k}} \right)\
abla k\right]+P_{k}-\rho\varepsilon. \tag{1}\]  
The transport equation of \(\varepsilon\) is:  
\[\rho(\mathbf{u}\cdot\
abla)\varepsilon=\
abla\cdot\left[\left(\mu+\frac{\mu_{T}} {\sigma_{k}}\right)\
abla\varepsilon\right]+C_{1}\rho S\varepsilon-C_{\varepsilon 2 }\frac{\rho\varepsilon^{2}}{k+\sqrt{\
u\varepsilon}}, \tag{2}\]  
in which \(\rho\) is liquid density in kg/m\({}^{3}\); \(\mathbf{u}\) represents drilling fluid velocities in m/s; \(\mu\) denotes liquid viscosity in Pa-s; \(k\) stands for turbulent kinetic energy in m\({}^{2}\)/s\({}^{2}\); \(\mu_{T}\) is turbulent dynamic viscosity, whose unit is Pa-s; \(\varepsilon\) represents the turbulent kinetic energy dissipation rate in m\({}^{2}\)/s\({}^{3}\); \(P_{k}\) represents turbulent kinetic energy in kg/(s\({}^{3}\cdot\)m); \(S\) denotes the average strain rate tensor; \(C_{\varepsilon 2}\) is an empirical constant; \(\sigma_{k}\) and \(\sigma_{\varepsilon}\) represent Prandtl numbers for the \(k\) equation and \(\varepsilon\) equation, respectively; and \(\
u\) denotes turbulent eddy viscosity in N\(\cdot\)S/m\({}^{2}\), which can be expressed as \(\
u=\frac{\mu_{T}}{\rho}\).