## 5 Conclusions  
This study has presented the concept of the HD approach for the simulation of advective transport in fractured rocks. Specifically, the developed HD model uses the 2D triangular mesh for fractures and tetrahedral mesh for the 3D rock matrix in a simulation domain and allows the system of equations to be solved simultaneously. The study employed two synthetic cases to assess the developed HD model. Solutions of flow and advective transport were compared with those obtained from the DFN model and ECPM model. In this study, the advective transport was conducted based on the particle tracking algorithm. The simulation results show that the HD model is flexible in considering the concepts of DFN, ECPM, or both. Based on the test cases, the flow fields obtained from the HD model agree well with those obtained from the ECPM and DFN models. However, the local flow dynamics led to significant variations in the advective transport. Fractures in the simulation domain play an important role in controlling the local flow patterns. In addition, the released particle locations could also influence the particle traces and the travel times. The ECPM model might need fine grids to resolve the interaction dynamics between the fractures and rock matrix. The DFN model focuses on the fractures only and has the limited feature to account for the interaction between fractures and rock matrix. The local flow circulations are the special characteristic obtained from the DFN and the developed HD model. The transport statistics relied on collecting the particle traces and travel times obtained from different models. The results in Case I indicated that the HD and DFN models obtained similar particle traces and travel times because the particles migrations have been \begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{1}{c}{**Parameters**} & \multicolumn{2}{c}{**ECPM Model**} & \multicolumn{2}{c}{**HD Model**} \\ \hline \multirow{4}{*}{Trace length} & Mean (m) & 10.69 & 9.07 \\  & STD (m) & 3.16 & 2.74 \\  & CV & 0.296 & 0.302 \\  & Min. (m) & 7.25 & 5.85 \\  & Max. (m) & 15.70 & 15.10 \\ \hline \multirow{4}{*}{Travel time} & Mean (s) & 9.70 \(\times\) 10\({}^{9}\) & 4.25 \(\times\) 10\({}^{9}\) \\  & STD (s) & 2.40 \(\times\) 10\({}^{9}\) & 1.05 \(\times\) 10\({}^{9}\) \\  & CV & 0.247 & 0.247 \\  & Min. (s) & 6.50 \(\times\) 10\({}^{9}\) & 2.69 \(\times\) 10\({}^{9}\) \\  & Max. (s) & 1.55 \(\times\) 10\({}^{10}\) & 7.10 \(\times\) 10\({}^{9}\) \\ \hline \multirow{4}{*}{Velocity} & Mean (m/s) & 1.09 \(\times\) 10\({}^{-9}\) & 2.18 \(\times\) 10\({}^{-9}\) \\  & STD (m/s) & 1.56 \(\times\) 10\({}^{-10}\) & 1.10 \(\times\) 10\({}^{-9}\) \\ \cline{1-1}  & CV & 0.143 & 0.505 \\ \cline{1-1}  & Min. (s) & 8.31 \(\times\) 10\({}^{-10}\) & 1.15 \(\times\) 10\({}^{-9}\) \\ \cline{1-1}  & Max. (s) & 1.37 \(\times\) 10\({}^{-9}\) & 4.33 \(\times\) 10\({}^{-9}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The statistics of the particle trackings for Cases II. restricted in the fractures. The particle released in the matrix and fractures might obtain different results from the ECPM and HD models. For the case with the particle release in the matrix (i.e., Case I), the HD model considered the detailed fracture flow and obtained a longer mean trace length than that obtained from the ECPM model. In Case II, the particles were released on the intersections between the DH and fractures. The HD model obtained a relatively small mean trace length and travel time. The identified highest velocity in the HD model is significantly greater than that in the ECPM model. The results have shown the feasibility of the HD model for accurate simulations of flow and advective transport in complex fractured rock systems. Conceptualization, Y.-C.Y., C.-F.N. and I.-H.L. ; methodology, Y.-C.Y., I.-H.L. and C.-F.N. ; software, Y.-C.Y., I.-H.L., Y.-H.S. and C.-Z.T. ; validation, Y.-C.Y. ; formal analysis, Y.-C.Y. ; data curation, Y.-C.Y., I.-H.L. and C.-F.N. ; writing--original draft preparation, Y.-C.Y. ; writing--review and editing, C.-F.N.