## 3 Dimensionless Groups of the Problem and Verification
### The Thermal Problem  
To obtain the dependencies in the thermal problem, it is necessary to define a region of thermal influence. This region is set using a thermal length (\(\mathrm{I_{T}}\)), which is defined as the longitude in which the thermal changes due to changes in water velocity are appreciable. For example, at a depth \(\mathrm{y=H/2}\), using the temperature interval defined as the difference between the temperature at the center of the river and that at the left edge of the domain, the value \(\mathrm{I_{T}}\) guarantees that 95% of the steady temperature is reached. It is important to note that the total length (L), in which a constant h value is imposed, is far away from the river. By introducing \(\mathrm{x^{\prime}=\frac{x}{\mathrm{I_{T}}}}\) and \(\mathrm{y^{\prime}=\frac{y}{H}}\) (previously deduced) into governing Equation (6), deleting the derivative factors, and averaging, the resulting terms are \(\frac{\mathrm{k_{m,x}}}{\mathrm{I^{2}}}\), \(\frac{\mathrm{k_{m,y}}}{\mathrm{I^{2}}}\), \(\frac{\mathrm{\rho_{ew}\mathrm{e_{c,w}}\mathrm{K_{x}}\mathrm{K_{y}}\left( \mathrm{h_{2}-h_{1}}\right)}}{\mathrm{I^{2}}}\), and \(\frac{\mathrm{\rho_{ew}\mathrm{c_{ew}}\mathrm{K_{y}}\mathrm{K_{y}}\left( \mathrm{h_{2}-h_{1}}\right)}}{\mathrm{H^{2}}}\). Assuming \(\mathrm{k_{m,x}=k_{m,y}=k_{m}}\), the independent quotients of these terms give rise to the following dimensionless groups:  
\[\pi_{3}=\frac{\mathrm{I_{T}}}{\mathrm{H}};\ \ \ \ \ \pi_{4}=\frac{\mathrm{K_{x}}}{ \mathrm{K_{y}}};\ \ \ \ \ \pi_{5}=\frac{\mathrm{k_{m}}}{\mathrm{\rho_{e,w} \mathrm{c_{ew}}\mathrm{K_{y}}\mathrm{K_{y}}\left(\mathrm{h_{2}-h_{1}}\right)}} \tag{18}\]  
Figure 3: Flow iso-lines, case 5, Table 2.  
A new monomial emerges due to the imposed boundary conditions \(\pi_{6}=\frac{\mathrm{I_{r}^{2}}}{\mathrm{H^{2}}}\frac{\mathrm{K_{x}}}{\mathrm{K_{y}}}\). As \(\mathrm{K_{x}}/\mathrm{K_{y}}\) already exists as a dimensionless group, \(\pi_{6}\) can be simplified to \(\frac{\mathrm{I_{r}}}{\mathrm{H}}\). The pi theorem is applied:  
\[\frac{\mathrm{I_{r}}}{\mathrm{H}}=\mathrm{Y_{3}}(\pi_{4},\ \pi_{5},\ \pi_{6})= \mathrm{f_{3}}\left(\frac{\mathrm{K_{x}}}{\mathrm{K_{y}}},\ \frac{\mathrm{k_{m}}}{\mathrm{e_{e,w}}\mathrm{e_{w}} \mathrm{K_{y}}(\mathrm{h_{2}}-\mathrm{h_{1}})},\frac{\mathrm{I_{r}}}{\mathrm{ H}}\right) \tag{19}\]  
Table 3 shows seven cases selected to verify solution (18). Compared to case 12, in case 13 the total depth of the aquifer, the length of the river, and the thermal length are multiplied by 0.5, keeping \(\pi_{3}\) constant. In case 14, the vertical and horizontal hydraulic conductivity and thermal conductivity are doubled, while \(\mathrm{I_{r}}\) and \(\mathrm{H}\) are halved to obtain the same thermal length (\(\mathrm{I_{r}}\)) as that in case 13, as expected. In case 15, the thermal conductivity and \(\Delta\)h are duplicated, keeping \(\pi_{5}\) constant and, as expected, obtaining the same value of \(\mathrm{I_{T}}\) as that in case 12. In case 16, the hydraulic conductivities are quadrupled, the length of the river and the total depth of the aquifer are tripled, the value of \(\Delta\)h is doubled, and the thermal conductivity of the water-soil matrix is multiplied by eight. As expected, the thermal characteristic length is multiplied by three because the aquifer depth tripled and the monomials \(\pi_{4}\), \(\pi_{5}\), and \(\pi_{6}\) retained the same values. Finally, in case 17, both \(\mathrm{k_{m}}\) and \(\mathrm{\rho_{e,w}}\mathrm{e_{e,w}}\) are duplicated, obtaining the same \(\mathrm{I_{T}}\) as that in case 12. For all the cases (scenarios 12 to 17), \(\pi_{3}\)= 2.163, \(\pi_{4}\)= 1.0, \(\pi_{5}\)= 0.05, and \(\pi_{6}\)= 0.125.