## 3 Dimensionless Groups of the Problem and Verification
### The Mechanical Problem
The values of the lumped parameters are \(\frac{\mathrm{K_{x}}}{\mathrm{L^{2}}}\frac{\mathrm{H^{2}}}{\mathrm{K_{y}}}=5\) and \(\frac{\mathrm{l}}{\mathrm{L}}=0.1\). In scenarios 1 and 2, the values of \(\mathrm{K_{x}}\), \(\mathrm{K_{y}}\), and \(\mathrm{h_{2}-h_{1}}\) (\(\Delta\mathrm{h}\)) are modified to keep \(\frac{\mathrm{K_{x}H^{2}}}{\mathrm{L^{2}K_{y}}}\) and \(\frac{\mathrm{l}}{\mathrm{L}}\) constant. It can be seen that the recharge flow Q and the dimensionless recharge flow \(\left(\frac{\mathrm{Q}}{(\mathrm{h_{2}-h_{1}})\sqrt{\mathrm{K_{x}K_{y}}}}\right)\) remain constant. In scenarios 3 and 4, the values of \(\mathrm{K_{x}}\), \(\mathrm{K_{y}}\) are modified, as is the geometry of the domain (H, L, and \(\mathrm{l_{r}}\)), retaining the values of the groups \(\frac{\mathrm{K_{x}H^{2}}}{\mathrm{L^{2}K_{y}}}\) and \(\frac{\mathrm{l}}{\mathrm{L}}\). As expected, the recharge flow doubles while the dimensionless discharge retains its value. The heat transport requires the separation of \(\frac{\mathrm{K_{x}}}{\mathrm{L^{2}}}\frac{\mathrm{H^{2}}}{\mathrm{K_{y}}}\) into two independent groups since the thermal conductivity is isotropic. Consequently, the thermal problem depends on \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Scenario**} & \(\mathbf{K_{x}}\) & \(\mathbf{K_{y}}\) & \(\mathbf{H}\) & \(\mathbf{l_{r}}\) & \(\mathbf{L}\) & \(\mathbf{h_{2}-h_{1}}\) & \(\mathbf{Q}\) \\  & **(m\(\cdot\mathbf{s^{-1}}\))** & **(m\(\cdot\mathbf{s^{-1}}\))** & **(m\(\cdot\mathbf{s^{-1}}\))** & **(m)** & **(m)** & **(m)** & **(m\({}^{2}\cdot\mathbf{s^{-1}}\))** \\ \hline \(1\) & \(5\times 10^{-5}\) & \(1\times 10^{-5}\) & \(10\) & \(10\) & \(100\) & \(0.1\) & \(479.9\times 10^{-9}\) \\ \(2\) & \(1\times 10^{-4}\) & \(2\times 10^{-5}\) & \(10\) & \(10\) & \(100\) & \(0.05\) & \(479.9\times 10^{-9}\) \\ \(3\) & \(1\times 10^{-4}\) & \(2\times 10^{-5}\) & \(20\) & \(20\) & \(200\) & \(0.1\) & \(959.8\times 10^{-9}\) \\ \(4\) & \(5\times 10^{-5}\) & \(1\times 10^{-5}\) & \(20\) & \(20\) & \(200\) & \(0.2\) & \(959.8\times 10^{-9}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Verification of expression (14). \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Case**} & \(\mathbf{k_{x}}\) & \(\mathbf{k_{y}}\) & \(\mathbf{H}\) & \(\mathbf{L}\) & \(\mathbf{l_{r}}\) & \(\mathbf{k_{x}H^{2}}\) \\  & **(m\(\cdot\mathbf{s^{-1}}\))** & **(m\(\cdot\mathbf{s^{-1}}\))** & **(m)** & **(m)** & **(m)** & **(m)** & **(k\({}_{y}\)L^{2}\)** \\ \hline \(5\a high number of monomials. This creates additional complexity when it comes to defining the universal set of type curves. Table 2 contains a set of cases (in which some of the physical and/or geometric parameters vary) that allow us to represent universal solutions associated with the values \(\pi_{1}=0.1\), 0.5, 1, 5, and 10. For all cases, \(\mathrm{h_{2}-h_{1}}\)= 1.00 m and \(\mathrm{\frac{h}{L}=0.10}\). Figure 3 represents the pattern of flow lines for case 5. It shows groundwater velocity in each line, deduced from expression (15). This pattern, at first glance, separates two regions within the aquifer: the first, with velocity components \(\mathrm{v_{x}}\) and \(\mathrm{v_{y}}\) that are comparable with each other, and the second, where groundwater flow is markedly horizontal. Cases 6 and 7 have a similar pattern as they keep the same value of \(\pi_{1}\) and \(\pi_{2}\), but the flow is deduced from the previous one using the expression. \[\frac{\mathrm{Q_{Case\ 5}}}{\left\{\left(\mathrm{h_{2}-h_{1}}\right)\sqrt{ \mathrm{K_{x}}\mathrm{K_{y}}}\right\}_{\text{Case\ 5}}}=\frac{\mathrm{Q_{Case\ 6}}}{ \left\{\left(\mathrm{h_{2}-h_{1}}\right)\sqrt{\mathrm{K_{x}}\mathrm{K_{y}}} \right\}_{\text{Case\ 6}}}=\frac{\mathrm{Q_{Case\ 7}}}{\left\{\left(\mathrm{h_{2}-h_{1}} \right)\sqrt{\mathrm{K_{x}}\mathrm{K_{y}}}\right\}_{\text{Case\ 7}}} \tag{17}\]