## 4 Results and Discussion
### Simulations of Advective Transport
The result indicates that most particles travel longer distances in the HD model because the fractures dominate the transport when the particles reach the fractures in the simulation domain. In this study, the STD of the trace length for the HD model is higher than that of the ECPM model. In general, the results show that the ECPM model underestimates the particle trace lengths and travel times in the case. The particle movement statistics in Table 2 also show that the mean trace length and STD of the HD model are 0.98 and 0.59 times less than those of the DFN model. The minimal trace lengths for HD and DFN models are identical. However, the maximal trace length of the HD model is less than that of the DFN model. We found that the relatively significant local flow circulations in the DFN model lead to a longer trace length because of the particles that move in the horizontal fracture F3 [47]. Table 2 also shows the statistics of the traveling time for the released particles. Similar to the results of trace length, the HD model obtained lower mean and STD values of the particle travel time than those obtained from the ECPM model. Specifically, the minimum travel time of the HD model is two orders of magnitude less than that of the ECPM model, while the observed maximum travel time of the HD model is approximately half of the time obtained from the ECPM  
Figure 9: The particle traces and lengths obtained from the ECPM, DFN, and HD models for Case I: (**a,b**) The particle traces for the ECPM model; (**c,d**) The particle traces for the HD model; (**e,f**) The particle traces for the DFN model; and (**g,h**) The particle trace along the fractures for the HD model. model. The results indicate that most particles in the HD model have fast traveling paths dominated by the fracture network. The short travel time is critical in conducting the safety assessment at a specific site. In general, the DFN model obtains relatively short travel times. The mean, minimum, and maximum travel times in the DFN model are generally one to two orders of magnitude smaller than those in the ECPM. However, the travel time variation (i.e., STD and CV) for the DFN model is relatively small compared with the results obtained from the HD and the ECPM models. This result is reasonable because the DFN model has constrained the particles to transport in the fracture network. Table 2 shows that the travel time statistics for the HD model vary between the results obtained from the ECPM and DFN models. The results in Table 2 also show that the mean velocity of the HD model is 7.62 times higher than that of the ECPM model. In addition, the STD of the HD model is 9.62 times higher than that of the ECPM model. Note that the velocity calculation is based on the trace length and the travel time for a specific particle. The minimum velocity for the ECPM and HD models are in the same order. However, the maximum velocity of the HD model is considerably higher than that of the ECPM model because the particles in the HD model might move in the fractures. For the results obtained from the fracture-based DFN model, the HD model could reproduce well the overall behavior of the advective transport in fractures. The HD model enables the particles to move in fractures and the rock matrix with the feature of interactions between fractures and the rock matrix. We had discussed the computational issues for different approaches. The ECPM model requires the cell sizes to be small to resolve the detailed flow dynamics near fractures. The DFN ignores the influence of the rock matrix, which might partially decrease the velocity of advective transport. In this study, the developed HD model considers the virtual fractures embedded on the element face could be a feasible approach for problems with practical scales and complexity. In Case II, the objective is to evaluate the influence of the DH added in the simulation domain. Figure 10 shows the released location and the particle traces for the ECPM and HD model.