# Simulation of the Infiltration of Fractured Rock in the Unsaturated Zone
## 1 Introduction
Although capillary transfer can be regarded as a nonlinear diffusion process that is more important on the small scale, our domain of interest, the vadose zone, can be hundreds of meters thick. Considering the frequency of major rainfall events in semi-arid regions, the time scales of interest are hours to tens of years. These constraints imply that physical experiments are very difficult to perform, and numeric simulation is the preferred tool to study this system. However, realistic simulation of fracture infiltration remains challenging (e.g., [PERSON] et al. [30]). Dual continua methods, including dual-porosity ([PERSON] and [PERSON] [31]), dual-permeability ([PERSON] et al. [32]), and their successors ([PERSON] and [PERSON] [33], [PERSON] [34], [PERSON] [35], [PERSON] and [PERSON] [36]) have been applied traditionally to model fractured porous media. However, they resolve saturation fronts only in a volume averaged sense. To evaluate fracture-matrix transfer, these methods employ transfer functions, which are nontrivial to construct ([PERSON] et al. [37], [PERSON] et al. [38], [PERSON] and [PERSON] [39]). Spontaneous imbibition is a dynamic process not captured by the pseudo steady-state assumptions that underpin many transfer functions ([PERSON] [7], [PERSON] et al. [40]). Furthermore, transfer functions are zero-dimensional, not discerning whether the transfer process leads to co-current or counter-current imbibition although these modes are characterised by different mobilities ([PERSON] et al. [41]). In this study, we apply the alternative discrete-fracture and matrix (DFM) simulation approach ([PERSON] and [PERSON] [42], [PERSON] et al. [43]). Lower-dimensional and higher-dimensional finite elements are used to represent individual fractures and matrix blocks, respectively. Therefore, fractures in 2D models are represented by line elements and by surface elements in 3D models. In this mixed-dimensional approach described in detail by [PERSON] et al. [44], fracture properties are surface attributes, permitting, for instance, to change fracture aperture without modifying the mesh. This offers flexibility and greatly reduces model complexity. Such DFM models have been applied successfully to investigate single phase flow in fractured rock ([PERSON] [13], [PERSON] and [PERSON] [45], and [PERSON] et al. [44]). [PERSON] and [PERSON] [45] first applied this method to two-phase flow and [PERSON] et al. [46], [PERSON] and [PERSON] [47] extended it to complex field-data based fracture models. [PERSON] et al. [48] introduced support for saturation discontinuities at material interfaces. However, their model did not yet support the decoupling of phase pressure across material interfaces because, in this hybrid finite element - finite volume method, pressure is discretised on the finite-element nodes and these are shared by elements joined at material interfaces. This missing degree of freedom of this so-called FECFVM method can be overcome through the application of an extended version of the embedded discontinuity method of [PERSON] and [PERSON] [49; 50] with additional pressure degrees of freedom along interfaces and a non-linear iterative method to compute the interface fluxes, see [PERSON] et al. [51]. [PERSON]'s embedded discontinuity method is the point of departure of this study, and here we present a further extension of it, supporting unique phase-pressure values for the lower-dimensional fracture and the rock matrix on either side. Capillary pressure is solved implicitly across the different subdomains of the model separated by interfaces. This embedded discontinuity scheme differs from the many formulations that compute local capillary pressure derivatives only (e.g., [PERSON] and Firoozabadi [52]) and often assume that the saturation on either side of the interface reflects capillary equilibrium (e.g., [PERSON] and [PERSON] [53], [PERSON] et al. [54]). That the latter is not necessarily the case during fracture flow will be illustrated by answering the leading questions of this study.