## Numerical Examples
### The Conceptual Models of the Test Cases  
Figure 4 shows the conceptual models for the test cases. In test Case I, the simple benchmark case includes three orthogonal fractures in the fractured rock (see Figure 4a). The domain size is a cuboid with 3 m \(\times\) 4 m \(\times\) 5 m in x-, y- and z-directions. There are  
Figure 3: The main steps involved in the workflow to model flow and advective transport in fractured rocks. three fracture plates in the simulation domain, i.e., F1 to F3 (F1: x = 1.5 m; F2: y = 2 m; F3: z = 2.5 m). All remaining zones of the domain represent the rock matrix, and the hydraulic properties are assumed to be homogenous and isotropic. A constant head condition was assigned to a rectangular area on the top plane, and a similar constant head boundary condition was specified on a rectangular area on the bottom of the simulation domain. The specified hydraulic heads of H\({}_{\text{in}}\) = 4 m and H\({}_{\text{out}}\) = 1 m are on the inlet and outlet areas, respectively (see Figure 4a). All remaining areas on the boundaries were assigned no-flow boundary conditions. In this study, assigning the in- and out-stresses on the matrix zones aims to evaluate the behavior of the flow that moves in and out of the interfaces between the fractures and matrix. Note that the DFN model focuses on the fractures only. However, the ECPM model considers the lumped behavior of the fractures and rock matrix. The assigned boundary conditions are either the fracture edges (i.e., DFN) or the cell faces (i.e., ECPM). Table 1 lists the hydraulic properties for the fractures and the rock matrix. Note that the apertures of the fractures were assumed to be constant for two test cases. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters** & **Case I** & **Case II** \\ \hline Fracture transmissivity (m\({}^{2}\)/s) & 5.0 \(\times\) 10\({}^{-10}\) & 5.0 \(\times\) 10\({}^{-7}\) \\ Matrix hydraulic conductivity (m/s) & 1.0 \(\times\) 10\({}^{-10}\) & 1.0 \(\times\) 10\({}^{-10}\) \\ Deposition hole hydraulic conductivity (m/s) & - & 1.0 \(\times\) 10\({}^{-10}\) \\ Fracture aperture (m) & 1.0 \(\times\) 10\({}^{-4}\) & 1.0 \(\times\) 10\({}^{-1}\) \\ Fracture porosity (-) & 4.0 \(\times\) 10\({}^{-1}\) & 4.0 \(\times\) 10\({}^{-1}\) \\ Rock matrix porosity (-) & 5.4 \(\times\) 10\({}^{-3}\) & 5.4 \(\times\) 10\({}^{-3}\) \\ Convergence criteria (m) & 1.0 \(\times\) 10\({}^{-8}\) & 1.0 \(\times\) 10\({}^{-8}\) \\ Particle numbers (-) & 1000 & 1;48 ** \\ \hline \hline \end{tabular}  
* There is a subcase with 48 particles for Case II.  
\end{table} 
Table 1: The physical and simulation parameters for the test cases in this study. Figure 4: The conceptual model of the benchmark cases in this study: **(a)** Case I includes three orthogonal fractures (F1 to F3) in the simulation domain; **(b)** Case II includes three fractures (F4 to F6) and a deposition hole (DH) in the fractured rock. Note that the contours show the elevation of the fractures. With the available flow field in Case I, there are 1000 particles randomly released in the inlet area. Note that the starting locations of particles and the particle movements are different in different models. In the DFN model, which only simulates the fractures, the particles are released on the fracture edges connecting to the inlet boundary area (see the marked solid pink lines on the top plane in Figure 4a). For the ECPM model, the particles are randomly released on the cell faces on the inlet boundary area (see Figure 4a). These two types of particle release scenarios are conducted in the HD model for comparison purposes. Figure 4b shows the conceptual model of the second benchmark case. In Case II, the domain size is relatively larger than that in Case I. A cylinder object represents a deposition hole (DH) of a disposal facility in the simulation domain, and two intersected fractures cut the DH.