# Simulation of the Infiltration of Fractured Rock in the Unsaturated Zone
## 3 Discrete Fracture and Matrix Models (DFM)
### Fractured Volcanic Tuff (Model TUFF2D)  
An idealised two-dimensional DFM model with vertical fractures intersecting a porous volcanic tuff with largely uniform micro-pores and a milli-Darcy permeability (cf., Diner and Bostanci [74]) serves as the second test case for the new DFM embedded-discontinuity solver (Figure 6, Table 2). A 20 cm thick layer of highly permeable and porous sandy soil covers this rock (0.4\({}^{-11}\) m\({}^{2}\), \(\phi\) = 0.28). The model dimensions are 4.0 \(\times\) 3.5 m. The fractured tuff beneath the soil layer accounts for the largest portion of the pore volume of the model. Its permeability is 0.5\(\cdot\)10\({}^{-15}\) m\({}^{2}\) and entry pressure p\({}_{d}\) = 24.4 kPa.  
Figure 5: Model CUBE: (**a**) boundary conditions and (**b**) mixed-dimensional finite element mesh.  
Parallel fractures penetrate the rock layer. Their hydraulic aperture alternates between 0.5 and 1.1 mm. The parallel-plate law determines fracture permeability: \(k_{f}=b^{2}/\) 12, results in permeability of \(k_{1}^{f}=2.083\cdot 10^{-8}\) m\({}^{2}\) and \(k_{2}^{f}=1.008\cdot 10^{-7}\)m\({}^{2}\) respectively, see [PERSON] and [PERSON] [75]. A triangular mesh is used to discretise the domain with 3294 elements and 2002 nodes. The split operator introduces 231 new line elements representing the vertical fractures.  
The van Genuchten relative permeability model and capillary pressure curves are used for the air phase in both fracture and matrix (Figures 7 and 8).  
The model setup consists of a constant pressure bottom boundary condition modelling a water table. The left and right sides of the model are treated as impermeable boundaries.  
Two different precipitation rates are explored to study their effect on water movement in the model. In the first scenario (Case A), a surface source is applied at the top to simulate a small rainfall event at an average precipitation rate of 10.5 mm/h. In the second scenario (Case B), the precipitation is much higher, at 73 mm/h, to simulate a vigorous rainfall event. We assume that all the rainwater imbibes into the flat model top. Thus, runoff is ignored as it will be in the scope of another study.  
In each rainfall scenario, we investigate the effect of residual saturation on the response of the system. For this purpose, we run two simulations corresponding to different residual water saturation for the matrix and the soil layer. The first run uses the same irreducible water saturation of 0.08 for the soil and the rock layers. The values for the soil and the rock layers in the second run are 0.15 and 0.45, respectively.  
\begin{table}
\begin{tabular}{l l} \hline \hline
**Sandstone** & \\ \hline Porosity (-) & 0.24 \\ Permeability (m\({}^{2}\)) & 7.89 \(\cdot 10^{-13}\) \\ Brooks-Corey parameter (-) & 2.0 \\ Entry pressure (Pa) & 1300 \\ \hline Fracture & \\ \hline Porosity (-) & 1.0 \\ Aperture (mm) & 0.1 \\ Permeability (m\({}^{2}\)) & 3.33 \(\cdot 10^{-9}\) \\ Brooks-Corey parameter (-) & 1.0 \\ Entry pressure (Pa) & 100 \\ \hline Fluids * & \\ \hline Water density (kg/cm\({}^{3}\)) & 1.0000 \\ Air density (g/cm\({}^{3}\)) & 0.0012 \\ Water viscosity (cp) & 1.00 \\ Air viscosity (cp) & \(\sim\)0.02 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Properties used in the construction of the model CUBE.  
Figure 6: Model TUFF2D, a two-dimensional model of fractured tuff and its discretisation by a triangular finite-element mesh. Split boundaries were inserted between the soil layer at the top and the fractured tuff below as well as at the fracture–matrix interfaces. Collocated finite volume mesh not shown.  
Figure 7: Relative permeability curves of matrix and fracture of model TUFF2D.