## 2 Methodology  
### Governing Equations  
The Eulerian-Eulerian Two-Fluid Model (TFM) is used for the numerical simulation. The TFM method has been fully proved in the literature and is widely recognized in engineering [17; 18; 19]. For simplicity, it is assumed that the solid phase consists of spherical particles with the same particle size. The mass conservation equation for drilling fluid and drill cuttings are as follows:  
\[\frac{\partial}{\partial t}(\alpha_{\mathrm{I}}\rho_{\mathrm{I}})+\
abla\cdot (\alpha_{\mathrm{I}}\rho_{\mathrm{I}}\overset{\rightarrow}{u_{\mathrm{I}}})=0 \tag{1}\]  
\[\frac{\partial}{\partial t}(\alpha_{\mathrm{s}}\rho_{\mathrm{s}})+\
abla\cdot (\alpha_{\mathrm{s}}\rho_{\mathrm{s}}\overset{\rightarrow}{u_{\mathrm{s}}})=0 \tag{2}\]  
where \(\alpha\) is the volume fraction; \(\rho\) is the density; \(\overset{\rightarrow}{u}\) is the velocity; and subscripts \(\mathrm{I}\) and \(\mathrm{s}\) represent drilling fluid and cuttings, respectively. The momentum conservation equation for drilling fluid and drill cuttings are as follows:  
\[\frac{\partial}{\partial t}(\alpha_{\mathrm{I}}\rho_{\mathrm{I}}\overset{ \rightarrow}{u_{\mathrm{I}}})+\
abla\cdot(\alpha_{\mathrm{I}}\rho_{\mathrm{I}} \overset{\rightarrow}{u_{\mathrm{I}}}\overset{\rightarrow}{u_{\mathrm{I}}})=- \alpha_{\mathrm{I}}\
abla p+\alpha_{\mathrm{I}}\
abla\cdot\overline{\overline{ \alpha}}_{\mathrm{I}}+\alpha_{\mathrm{I}}\rho_{\mathrm{I}}\overset{ \rightarrow}{g}-\beta(\overset{\rightarrow}{u_{\mathrm{I}}}-\overset{ \rightarrow}{u_{\mathrm{s}}}) \tag{3}\]  
\[\frac{\partial}{\partial t}(\alpha_{\mathrm{s}}\rho_{\mathrm{s}}\overset{ \rightarrow}{u_{\mathrm{s}}})+\
abla\cdot(\alpha_{\mathrm{s}}\rho_{\mathrm{s}} \overset{\rightarrow}{u_{\mathrm{s}}}\overset{\rightarrow}{u_{\mathrm{s}}})=- \alpha_{\mathrm{s}}\
abla p-\
abla p_{\mathrm{s}}+\alpha_{\mathrm{s}}\
abla \cdot\overline{\overline{\alpha}}_{\mathrm{s}}+\alpha_{\mathrm{s}}\rho_{ \mathrm{s}}\overset{\rightarrow}{g}+\beta(\overset{\rightarrow}{u_{\mathrm{ I}}}-\overset{\rightarrow}{u_{\mathrm{s}}}) \tag{4}\]  
where \(g\) is the acceleration of gravity; \(p\) is the pressure; \(\tau\) is the stress tensor; and \(\beta\) is the inter-phase momentum exchange coefficient, predicted by the Huilin-Gidaspow model [20]. \[\beta=\varphi\beta_{\mathrm{Ergun}}+(1-\varphi)\beta_{\mathrm{Wenk}\mathrm{Yu}} \tag{5}\]  
\[\varphi=\frac{\arctan[262.5(\alpha_{\mathrm{s}}-0.2)]}{\pi}+0.5 \tag{6}\]  
for \(\alpha_{\mathrm{I}}\leq 0.8\),  
\[\beta_{\mathrm{[PERSON]}}=150\frac{\alpha_{\mathrm{s}}^{2}\mu_{\mathrm{I}}}{ \alpha_{\mathrm{I}}d_{\mathrm{s}}^{2}}+1.75\frac{\rho_{\mathrm{I}}\alpha_{ \mathrm{s}}\big{|}\overset{\rightarrow}{u_{\mathrm{I}}}-\overset{ \rightarrow}{u_{\mathrm{s}}}\big{|}}{d_{\mathrm{s}}} \tag{7}\]  
for \(\alpha_{\mathrm{I}}>0.8\),  
\[\beta_{\mathrm{[PERSON]}\mathrm{[PERSON]}}=\frac{3}{4}C_{\mathrm{D}}\frac{\alpha_{ \mathrm{s}}\alpha_{\mathrm{I}}\rho_{\mathrm{I}}\big{|}\overset{\rightarrow}{u _{\mathrm{I}}}-\overset{\rightarrow}{u_{\mathrm{s}}}\big{|}}{d_{\mathrm{s}}} \alpha_{\mathrm{I}}^{-2.65} \tag{8}\]  
where \(d_{\mathrm{s}}\) is the cuttings diameter and \(C_{\mathrm{D}}\) is the drag coefficient:  
\[C_{\mathrm{D}}=\left\{ \begin{array}{ll}\frac{24}{R\varepsilon_{\mathrm{s}}} \big{(}1+0.15R\varepsilon_{\mathrm{s}}^{0.687}\big{)}&Re_{s}\leq 1000\\ 0.44&Re_{s}>1000\end{array} \right.