## 1 Introduction and Background
A model was simulated using VS2 DHI to consider both saturated and unsaturated conditions [20]. The SW-GW thermo-mechanical interaction is a complex problem since groundwater flow, which results from hydraulic potentials on the boundaries that are independent of the thermal problem, is not constant at each point. As one of the authors studying this problem in specific scenarios and in a qualitative way, [PERSON]. [21] used data collected in different locations at the same depth; the depth was sufficient to avoid daily temperature oscillations. He estimated groundwater flows as a function of temperature through an empirical expression. [PERSON] et al. [22] mapped streambed temperatures at various depths along a 60 m reach to identify spatial patterns of groundwater discharge, as well as to quantify water fluxes at the stream-aquifer interface, by applying the 1D equation of Bredehoeft and Papadopulos [7] and using a least square optimization method. They combined a large-scale assessment of aquifer contamination with a high-resolution survey of groundwater discharge zones to estimate contaminant mass fluxes between the aquifer and the stream. [PERSON] et al. [23] analyzed a synthetic test case involving a two-dimensional cross-section perpendicular to a losing stream; they considered the convective-conductive heat transport flow within the aquifer and the heat exchange with an underlying aquitard and the soil surface. They plotted the solution of the temperature amplitude over the entire domain and demonstrated that there was a region affected by the recharge temperature of the river water. This study investigates the dimensionless characterization of the 2D problem, which could be classified as a pattern in which an aquifer of anisotropic hydraulic conductivity is recharged with water from a river or surface water body. The objective of this study is to obtain new analytical solutions for the SW-GW interaction problem and to graphically express these solutions in the form of type curves. The solution for the mechanical (hydraulic) problem will allow the flow to be directly obtained without the need to resort to numerical or analytical calculations. The solution for the thermal problem is the novelty and main contribution of the procedure. The applicability of these type curves will depend on the parameters for which the values are known. For instance, if the mechanical problem to be estimated involves the recharge flow or the anisotropic hydraulic conductivity, then the temperature-depth profiles, along with the geometric parameters, thermal parameters, and hydraulic gradient, can be used to calculate the values of the components of hydraulic conductivity and the infiltration flow rate into the aquifer. The manipulation of the governing equations and boundary conditions is aimed at obtaining dimensionless expressions. The relationships between them are ultimately represented in the form of type curves. The procedure, which is represented in Figure 1, follows a protocol in which (1) the conceptual and mathematical models are defined, (2) the dimensionless groups are obtained using discriminated and normalized non-dimensionalization, (3) the analytical solutions are obtained by applying the pi theorem [24], (4) the solutions are verified with numerical simulations, and (5) the solutions are represented graphically (type curves). It is important to note that the discriminated non-dimensionalization process of the governing equations used to obtain the lumped parameters that govern the solutions is applied for the first time to the SW-GW interaction. The technique has been successfully applied to other complex problems in porous media [25; 26; 27], in which reliable dimensionless groups were compared to experimental cases or benchmark problems.