## 2 Materials and Methods
Based on the grid shown in Figure 1, the model domain was also analyzed considering cross sections drawn along the entire spring catchment area to highlight the groundwater flow in depth. To analyze the flow in depth, and go over the _Dupuit-Forchheimer conditions_ used to build the plan model, a 2D analysis was carried out along a vertical plane. In this case, the boundary conditions fixed to build the general model were taken from the result obtained by the horizontal model. In particular, the hydraulic head values along the cross-sections of Figure 1 (AS, BS and CS) were imposed into the 2D vertical model to find the geometry of the flow net in depth. Figure 2 shows the results of the modelling along the vertical cross sections AS, BS and CS, where each different flow pipe is distinguished by different colors, for a total of n.7 flow pipes in this example. The flow net in the vertical plane was obtained using MODPATH package. Of course, the geometry of the flow net and the number of flow pipes depend on the thickness of the aquifer's saturated zone (TSZ), which plays a crucial role in flow paths, as the more thickly saturated zone implies the higher the travel time. The model was also run for different values of the effective porosity \(\mathrm{n_{eff}}\) considering the range 0.01-0.10 [30]; even if the effective porosity ruled the flow net geometry, in particular the groundwater travel time, no significant differences were found for the considered \(\mathrm{n_{eff}}\)  
Figure 1: 2D plan-view (horizontal) groundwater model of a homogenous square-shape aquifer, n.22550 cells, with a single drain (spring outlet). (1) spring; (2) active cell (flow zone); (3) inactive cell (no flow zone); (4) flow net with equipotential and flow lines (blue and red color respectively); (5) section trace. range, representative of these kinds of aquifers; as described in the following sections, the hydraulic conductivity, K, has an important role in controlling the flow net shape. Even if this procedure could be an approximation of a more complex 3D model, it could provide a useful simplification of the flow net geometry both horizontally and vertically, allowing for the further considerations that are described below. Considering the common flow pipe of different cross-sections (Figure 2), in a plan-view it is possible to extrapolate the origin of flow pipes by interpolation, considering the beginning of the flow line boundaries of each flow pipe. Figure 3 provides a plan-view of the starting path of each flow pipe, which corresponds to zones 1 to 7; this representation has been obtained through a combined use of the GIS-MODFLOW tools, interpolating the flow pipe boundaries on the horizontal plane. Figure 2: 2D vertical groundwater model along the cross-section traces AS, BS, and CS of Figure 1. Flow pipes have been drawn with different colors and marked with the relative number. The geometry and size of the cross-section are defined by the depth (h) and length (l) of the grid. In Figure 3, the different zones (from 1 to 7) contribute differently to aquifer discharge: the most distant ones from the spring are also wider, and they are characterized by a deeper flux, whereas the closest to the hydraulic outlet are smaller and are affected by a shallower flux. The simple zonation of the aquifer shown in Figure 3, considering an isotropic and homogenous aquifer, allowing one to investigate the contribution of different hydraulic sectors to the spring, in terms of the percentage of total discharge and the time that the flow needs to reach the final outlet. In particular, the catchment area zoning in different flow pipes allows one to estimate the different recharge volumes and different contributions to spring discharge (Equation (3)). The contribution to the discharge \(Q_{i}\) of each flow pipe can expressed by:  
\[Q_{i}=FPA_{i}\times R \tag{3}\]  
where \(R\) is the recharge amount, expressed as a length/time, [L/t], imposed as constant on the entire catchment (_Neumann condition_), and \(FPA_{i}\) is a generic flow pipe area.