## 2 Materials and Methods
### Distributed Hydrological-Hydraulic Modelling  
The two-dimensional shallow-water equations, better known as the Saint-Venant equations, were adopted to describe the motion of surface waters. To this end, we configured the hydrological module of IBER+ [26]. It solves numerically the depth-averaged inviscid conservation laws of mass and momentum for clear water in a two-dimensional, Cartesian system of coordinates, given in compact form by  
\[\frac{\partial h}{\partial t}+\
abla\cdot\left(h\,\mathbf{u}\right)=P-I\,, \tag{1}\]  
\[\frac{\partial h\,\mathbf{u}}{\partial t}+\
abla\cdot\left(h\,\mathbf{u}\,\mathbf{u} \right)+\
abla\left(\frac{g\,h^{2}}{2}\right)=-g\,h\,\
abla z-\frac{\mathbf{\tau}_ {b}}{\rho}\,, \tag{2}\]  
where \(t\) is the time, \(h\) is the depth of the water measured along the vertical coordinate, \(\mathbf{u}\) is the depth-averaged velocity vector, \(z\) is the bed altitude, and \(g\) is the acceleration due to gravity. The source terms in the continuity Equation (1) represent the rainfall intensity, \(P\), and the soil infiltration rate, \(I\). In the momentum balance Equation (2), the source terms are the bed slope \(\
abla z\) and the bottom shear stress \(\mathbf{\tau}_{b}/\rho\), in which \(\rho\) is the water density. [PERSON]'s friction law was used for evaluating the hydraulic resistance in (2) by setting  
\[\frac{\mathbf{\tau}_{b}}{\rho}=\frac{g\,n^{2}}{h^{1/3}}\mathbf{u}|\mathbf{u}|\,, \tag{3}\]  
where \(n\) represents [PERSON]'s roughness coefficient.  
The assumption of small bottom slopes is the basic hypothesis of the two-dimensional shallow-water Equations (1) and (2), which imply that the pressure distribution on the vertical is hydrostatic. Although such a constraint is not formally satisfied in all the situations, as in mountain areas, the most sophisticated GPU numerical codes for distributed hydrological simulations solve standard Equations (1) and (2) [26; 27; 28; 29]. Nowadays, there is no consensus on the theoretical formulations for steep slopes, and several theories have been proposed (see the review by [PERSON] and [PERSON] [56]). So, on the numerical side, the existing numerical codes are in-house and still based on CPU instead of GPU [57; 58], preventing their use for the current purpose.  
Figure 2: (**a**) DEM, (**b**) orthophoto, and (**c**) DEM slope in a small area surrounding the Arquillos town. We built the DEM from scratch using filtered LiDAR data and in-situ measurements with Leica Zeno 20 Global Positioning System (GPS). (**d**) Zoom of the computational mesh for the artificial channel and the floodplains. Note the hybrid topology of the cells with rectangular elements in the channel and triangles otherwise.  
The two-dimensional shallow water model (1)-(3), together with homogeneous initial conditions for \(h\) and \(u\), were discretised using a finite volume method based on the DHD solver [59] implemented in the free-software IBER+ [26]. The Reynolds stresses were neglected as suggested by [PERSON] and [PERSON] [59] for overland flow applications. We ran all the simulations using single-precision calculations in a Quadro RTX5000 GPU. In so doing, we achieved a reasonable computational time of three days per simulation. We set the Courant-Friedrich-Lewy number of 0.45 to satisfy the stability constraint of the DHD scheme.  
In the following subsections, we describe the configuration of the physical parameters of the hydrological model. We give details on the computational grid, and the DEM in Section 2.2.1. Then, in Section 2.2.2, we used historical data of extreme hydrological events to characterise the design storm parametrised with \(P\) in (1). Lastly, we configured the Manning roughness \(n\) (2) and the infiltration rate \(I\) (1) in Section 2.2.3 and Section 2.2.4, respectively, borrowing existing results from well-established studies. The analysis of the possible values of \(I\) and \(n\) for the present-day situation and the future with NbSs served to vary them systematically and configure a set of fifteen numerical simulations.