## Numerical Examples
### The Software and Computational Meshes and Grids for the Test Cases  
In Case I, three fractures were connected to the domain top and bottom surfaces so that the DFN model could generate the mesh and directly assign the boundary conditions on the edge of the fractures. Thus, the 2D triangular meshes were created for fractures in the DFN model and no other mesh or gird for the remaining parts, i.e., the rock matrix. This study used the commercial software, FracMan, to simulate the groundwater flow in the DFN model. FracMan is the commercial software developed by Golder Associates for fracture generation and flow and transport simulations. The flow simulation package in the software is based on the finite element numerical scheme. Many previous investigations employed the software to simulate flow and transport in fractured rock systems (e.g., [25; 26]). The ECPM model uses the hexahedron as the computational grid. This study used DarcyTools to simulate the groundwater flow for the ECPM model. The DarcyTools is an integrated computational package that links several computational modules for the hydrogeological analysis of nuclear waste repository in fractured rocks. The integrated package is based on a workflow that allows combining the DFN and CPM models. The workflow first generates a fracture network or reads from a FAB file, a text file for recording the fracture positions and geometry in space. One of the modules then up-scales the fracture network to become the effective hydraulic properties of grid cells. The effective hydraulic properties, including the hydraulic conductivity and diffusivity, can represent the complex features of the fractured rocks. The detailed information for the computational package can be obtained in many previous investigations (e.g., [27; 28; 29]). In this study, the proposed HD model uses the 2D triangular mesh for fractures and tetrahedron mesh for the 3D rock matrix. We developed a program to read the fractured file (the FAB file) from FracMan software and the DarcyTools computational package to ensure the compatibility of generated mesh for the fractures in the test cases. For the rock matrix in the test cases, the hydraulic properties were considered homogeneous and isotropic in the entire simulation domain. In the developed HD model, the 3D tetrahedron mesh was generated associated with the 2D fracture meshes. Therefore, the fractures and rock matrix could share the same nodes on the fractures and the matrix interfaces. The coupled solutions then rely on solving the system of equations developed based on all the nodes and elements in the fractures and matrix. In this study, Case I is relatively simple as compared to Case II. We used the DFN model (i.e., FracMan) to generate the fracture geometry and the hydraulic parameters in Case I. In the DFN model, there was no mesh generated for the matrix. The generated nodes and elements for the DFN model are 6433 and 12,624, respectively. The averaged node distance for the fractures is 0.1 m. In the HD model, we had used the averaged node distance of 0.1 m in the matrix to connect triangular fracture mesh. Figure 5 shows the generated mesh for the HD model. Figure 5a is the exterior view of the domain composed of generated fracture and rock matrix elements. Figure 5b,d present the inner views of the generated mesh for Case I. The mesh for three intersected fractures is shown in Figure 5c. Based on the average node distance for 2D triangular mesh and 3D tetrahedron mesh, the number of nodes is 49,747, while the numbers of 2D triangular and 3D tetrahedron elements are 9147 and 290,324, respectively. In Case I, a similar cell size of 0.1 m was used in the ECPM model for the up-scaled hydraulic properties and flow and transport simulations. The total cell number of the ECPM model is 131,072, based on the cell numbers of 32, 64, and 64 in x-, y-, and z-directions. Figure 6 shows the generated mesh for Case II. Figure 6a shows the inner view of the test Case II.