# Simulation of the Infiltration of Fractured Rock in the Unsaturated Zone
## 2 Mathematical Model
### Conceptual Fracture Modelling  
Back of the envelope calculations provide some first insights on how fast water might infiltrate an impervious rock mass intersected by vertical continuous fractures of a given aperture. The fluid is assumed to be Newtonian, and capillary (pinning) and inertia effects are ignored. It is further assumed that the fractures are fully saturated, i.e., that fluid supply does not constrain flow rate. For film flow that might be common in centimetre-aperture fractures [PERSON] and [PERSON] [68] show that flow velocities might be up to 1000 times faster than for saturated flow. This phenomenon will not be considered here. The very much simplified scenario is intended for a preliminary verification of the more sophisticated numeric model. For isolated opening-mode fractures (penny-shaped cracks), aperture is correlated with fracture length ([PERSON] and [PERSON] [69]), following linear or sublinear trends established by many field studies (e.g., [PERSON] [70]). Given the power-law length-frequency statistics of many natural fracture patterns (e.g., [PERSON] et al., 2001), short fractures often  
Figure 3: The flux over the fracture stays between two material patches R\({}_{1}\) and R\({}_{2}\). Velocity field \(\mathbf{v}\) are constant within elements; Nodal saturation values of matrix and fracture are generally not the same. are orders of magnitude more common than long ones, with implications for aperture statistics. An example are aperture measurements in the cross-drifts of the exploratory research facility at Yucca Mountain (Figure 4). While the average fracture aperture is only 0.1 mm, [PERSON] (2017) observed sixty five 1 mm aperture fractures for every 1-centimetre aperture fracture, and mapped tens of 2 cm aperture fractures and even larger dilatant jogs. A wide-spread field observation that also applies at Yucca Mountain is that longer fractures tend to be better interconnected than shorter ones (e.g., [PERSON] et al. (2017)). As a consequence only a small fraction of the overall fracture population plays an active role for fluid flow ([PERSON] et al. (2017)). These systematics indicate that the flow properties of fractured rocks in the vadose zone are highly heterogeneous, promoting the development of highly localised flow paths. From the fracture aperture, a, an upper bound on the discharge through the fracture conduit, \(q(m^{3}/s)\), can be calculated using the parallel plate law (e.g., [PERSON], 1965),  
\[q=\frac{a^{3}}{12\mu}\rho\mathbf{g}\mathbf{I} \tag{13}\]  
see [PERSON] and [PERSON] (1996). Here, \(\rho\) and \(\mu\) are the fluid density (kg/m\({}^{3}\)) and dynamic viscosity (Pa \(\cdot\) s), and \(\mathbf{g}\) is the acceleration of gravity (m/s\({}^{2}\)) acting in the vertical direction I. We used this relationship to calculate the gravity-driven advance of the wetting front in vertical fractures as a function of fracture aperture, for verification of our DFM simulator for the case of an impermeable solid rock matrix. Regarding fracture-matrix transfer in the case of a permeable porous rock matrix, the cubic increase of discharge with aperture suggests that there should be a threshold aperture above which, the flow rate in the fracture should no longer be influenced by the transfer. This important threshold cannot be determined analytically because it depends on the nonlinear saturation-capillary pressure relationships of the matrix and its initial water saturation. We will evaluate this further by numeric simulation assuming that co- and counter-current imbibition can be approximated by Van Genuchten saturation functions. Apart from computing frontal advance in individual fractures, we investigate how multiple fractures with the distribution of apertures given in Figure 4, drain a shared fluid reservoir at their top. This reservoir might be a water saturated coarse sand or gravel overlying the fractured bedrock. The objective of the multi-fracture calculation is to get an idea of the maximum infiltration rate that the fractured bedrock can sustain.