## 3 Results
### Flow Vectors  
The flow pipes assessment for the investigated area allowed one to understand the hydraulic behavior of groundwater from the recharge zone toward the spring area. The numerical simulation of the Caposele aquifer also allowed one to estimate the flow vectors; in particular, the groundwater flow net highlighted the different flow directions along the aquifer (Figure 12, Table 2). The analysis focused on the area within 1 km upstream the Caposele spring, where the flow lines converge, following which the flow vectors were geometrically estimated.  
Figure 12 shows the geometric approach followed to estimate the \(x\) and \(y\) vectors, whose values are duly tabulated. A hydraulic approach was proposed to assess the vertical and horizontal contributions of groundwater flow to spring discharge. Assuming that the medium can be considered homogeneous and hydraulically isotropic, and that the fluid is incompressible, the flow equation can be simplified under the _Laplace law_[45, 46]:  
\[\left(\frac{\partial^{2}h}{\partial x^{2}}+\frac{\partial^{2}h}{\partial z^{2 }}\right)=0 \tag{9}\]  
\begin{table}
\begin{tabular}{c c c c c c c} \hline
**Flow pipe** & \(\alpha\) (*) & \(v_{\mathrm{x}}\) (\%) & \(v_{\mathrm{y}}\) (\%) & \(Q_{\mathrm{FP}}\) (l/s) & \(Q_{\mathrm{x}}\) (l/s) & \(Q_{\mathrm{y}}\) (l/s) \\ \hline
1 & 5 & 3.9 & 0.3 & 147 & 135 & 12 \\
2 & 8 & 5.0 & 0.7 & 185 & 163 & 22 \\
3 & 17 & 5.1 & 1.5 & 259 & 199 & 60 \\
4 & 26 & 5.8 & 2.8 & 368 & 246 & 122 \\
5 & 42 & 5.1 & 4.6 & 554 & 288 & 266 \\
6 & 53 & 6.0 & 8.0 & 1142 & 491 & 651 \\
7 & 81 & 2.1 & 3.3 & 1032 & 144 & 888 \\ \hline \end{tabular}
\end{table}
Table 2: The main results obtained by flow vectors analysis; \(\alpha\), slope angle of the flow vector; \(v_{\mathrm{x}}\), horizontal velocity; \(v_{\mathrm{y}}\), vertical velocity; \(Q_{\mathrm{FP}}\), discharge of the flow pipe; \(Q_{\mathrm{x}}\) discharge of the horizontal component of the flow; \(Q_{\mathrm{y}}\) discharge of the vertical component of the flow.  
Figure 11: Thickness of the aquifer’s saturated zone (TSZ) vs. hydraulic conductivity (K); the Darcian travel time (\(t_{s}\)) has been also plotted.  
Here, \(h\) is a scalar function, twice differentiable in space; based on Equation (9) for the graphical solution of the 2D flow net, some hydraulic parameters of the Caposele groundwater flow were estimated. The simulation focused on the flow mesh of groundwater, on which the vectors analysis was performed. Then, the Darcian velocity was computed, considering the hydraulic conductivity obtained by model calibration and the relative hydraulic head (Figure 8). To assess the vertical and horizontal contribution of groundwater flow to spring discharge, a geometrical approach was followed; its graphical results are described in Figure 12.  
For each flow mesh at a distance of up to about 1 km from the spring zone, the velocity vectors \(v_{\chi}\) and \(v_{\Upsilon}\) and their relative discharges \(Q_{\chi}\) and \(Q_{\Upsilon}\) were estimated. The main results obtained from the vector analysis are duly shown in Table 2, pointing out the geometrical and analytical parameters obtained by the flow net analysis carried out along the 2D vertical plane of the Caposele spring zone. In particular, deeper flow meshes of groundwater highlight a predominance of the vertical velocity vectors \(v_{\Upsilon}\) on \(v_{\chi}\), and consequently the vertical component of the discharge \(Q_{\Upsilon}\) is predominant along deeper flow meshes. The ascendant flow detected near the spring zone is a hydraulic phenomenon common also to other springs of southern Italy [41; 47].