# Simulation of the Infiltration of Fractured Rock in the Unsaturated Zone
## 3 Discrete Fracture and Matrix Models (DFM)
### Three-Dimensional Model of Fractured Volcanic Tuff (Model TUFF3D)  
To study the effect of fracture aperture variations on the saturation front advancing in the fracture plane, a three-dimensional model was constructed (Figure 9a, Table 3). The dimensions of this model are 0.5 \(\times\) 0.5 \(\times\) 2 m. Similar to the previous cases, a cover layer of soil stays at the top model, and a fracture goes through a lower permeable rock layer. The finite element mesh divides the matrix into 249,750 tetra elements and the fracture surface into 17,258 triangle elements (Figure 9b).  
\begin{table}
\begin{tabular}{l l} \hline \hline
**Properties** & **Value** \\ \hline Soil & \\ \hline Porosity (-) & 0.28 \\ Permeability (m\({}^{2}\)) & 0.4\(\cdot\)10\({}^{-11}\) \\ van Genuchten parameter (-) & 0.8 \\ Entry pressure (Pa) & 100.0 \\ Residual water saturation (-) & (a) 0.08, (b) 0.15 \\ Residual air saturation (-) & 0.00 \\ \hline Matrix & \\ \hline Porosity (-) & 0.25 \\ Permeability (m\({}^{2}\)) & 0.5\(\cdot\)10\({}^{-15}\) \\ van Genuchten parameter (-) & 0.8 \\ Entry pressure (Pa) & 24,400 \\ Residual water saturation (-) & (a) 0.08, (b) 0.45 \\ Residual air saturation (-) & 0.00 \\ \hline Fracture 1 & \\ \hline Porosity (-) & 1.0 \\ Permeability (m\({}^{2}\)) & 2.1\(\cdot\)10\({}^{-8}\) \\ Aperture (mm) & 0.5 \\ van Genuchten parameter (-) & 0.2 \\ Entry pressure (Pa) & 1000 \\ \hline Fracture 2 & \\ \hline Porosity (-) & 1.0 \\ Permeability (m\({}^{2}\)) & 1.00\(\cdot\)10\({}^{-7}\) \\ Aperture (mm) & 1.1 \\ van Genuchten parameter (-) & 0.2 \\ Entry pressure (Pa) & 330 \\ \hline Water & \\ \hline Density (kg m\({}^{-3}\)) & 997 \\ Viscosity (Pa s) & 8.9\(\cdot\)10\({}^{-4}\) \\ \hline Air & \\ \hline Density (kg m\({}^{-3}\)) & 1.17 \\ Viscosity (Pa s) & 1.87\(\cdot\)10\({}^{-5}\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Parameterisation and constitutive relationships used in the model TUFF2D. Water properties calculated at 25 \({}^{\circ}\)C, and atmospheric pressure.  
We employ a Perlin noise generator to create a spatially correlated random variable field used as a possible realisation of fracture aperture. Then other features of the fracture are assessed by the local thickness, stored as the cell's attributes. As a result, fracture properties spread smoothly from cell to cell. The average aperture generated is 0.5 mm, and the ratio between the maximum and minimum aperture is ten folds (Figure 10). Fluid properties are the same as in the previous test case, and the media properties are shown in Table 2. Unlike the previous models, we use mixed relative permeability models for this 3D model. For the matrix and the soil layer, the van Genuchten model describes the relative permeability of the aqueous phase and capillary pressure, while the Brooks-Corey model is used to describe the relative permeability of the aerial phase. And the [PERSON]'s model describes both the relative permeability and the capillary pressure for the fracture (Figure 11).  
Figure 8: Capillary pressure curves of matrix and fracture of model TUFF2D.  
Figure 9: Model TUFF3D, a three-dimensional single-fracture version of fractured tuff model. (**a**) boundary conditions and (**b**) tetrahedral finite element mesh with split boundaries.  
To setup the model, a 125 mm/(m\({}^{2}\cdot\) day) water volume source is applied at the top of the model, and the lateral boundaries are treated as impermeable. The bottom boundary is assigned a constant pressure.  
A second constant aperture version of this model (a = 0.5 mm) was created to facilitate a comparison with the variable aperture one.  
Figure 11: Relative permeability curves assigned to rock matrix and fracture in model TUFF3D.  
Figure 10: Fracture plane in model TUFF3D-VA with synthetically generated aperture variations between 0.09 and 0.91 mm plotted onto it.