## 1 Introduction and Background  
The process of recharge/discharge between aquifers and surface water bodies, which is known in the scientific literature as surface water-groundwater interaction (SW-GW) [1, 2, 3, 4], is a topic of great interest in hydrogeology. In the context of the relationship between global warming and aquifer depletion [5], the management of groundwater resources to supply populations and ensure agricultural exploitation is of relevance. The estimation of recharge flows towards the aquifer or their discharge towards surface water bodies is not a simple issue; to the best of our knowledge, this issue has not been accurately addressed from a quantitative theoretical and dimensional point of view. The protocol used to study this interaction involves the application of the standard inverse problem, in which the input data are the aquifer temperature-depth profiles resulting from the coupling of water flow and heat transport; the latter is caused by the temperature conditions imposed at the aquifer boundaries [6]. One-dimensional groundwater flow and transport in problems in which there is no SW-GW interaction (only vertical or horizontal flow) have also been addressed, and analytical solutions have been deduced for temperature profiles in both cases, even for harmonic temperature conditions on the surface [7; 8; 9; 10]. [PERSON] and [PERSON] [8] provided type curves for the estimation of vertical groundwater velocity in 1D scenarios, while [PERSON] and [PERSON] [10] presented an equivalent [1; 2; 3; 4] solution for the same problem. On the other hand, [PERSON] [9] provided a solution for vertical velocity estimation, extending its application to transient scenarios. Scenarios with oblique flows have also been investigated and have provided analytical solutions [11]; in these cases, an unrealistic hypothesis is assumed, such as the imposition of a constant horizontal temperature gradient in the domain. However, most of the studies focused on qualitative rather than quantitative aspects, such as determining recharge and/or discharge regions within a domain [12; 13]. For instance, [PERSON] et al. [14] carried out a complete interpretation of horizontal-vertical flow components from temperature-depth profiles in a complex anthropized scenario subjected to artificial recharge in the coastal aquifer of Agua Amarga. The conclusions, although supported by a numerical simulation of the mechanical problem, did not consider the quantitative aspects of mechanical and thermal interaction. Regarding the study of simultaneous heat and fluid flow in two-dimensional scenarios, the works of [PERSON] and [PERSON] [15] and [PERSON] and [PERSON] [16] can be highlighted, as can the contributions of [PERSON] and [PERSON] [17] and [PERSON] et al. [18], whose work focused specifically on rivers. Using VS2 DH software, [PERSON] and [PERSON] [15] addressed the solution of the inverse problem for the calculation of flows in semi-arid deserts based on the physical properties of the sediment [19]. Over the period of a year, they recorded monthly temperatures at different depths under the stream or on the nearby riversides. The detection of high infiltration zones was proposed as a measure that could be used to suggest artificial recharge operations. [PERSON] and [PERSON] [16] studied a model consisting of an aquifer of negligible thickness confined between two impermeable regions with a constant horizontal flow of water. The surface temperature conditions ensured the continuous coupling between flow and heat transport. Transient temperature-depth profiles were studied to estimate horizontal groundwater velocity. [PERSON] and [PERSON] [17] monitored sediment and streambed temperature to obtain data for the qualitative detection of vertical flows in small rivers. This allowed them to identify gains and losses in volume along the stream. [PERSON] et al. [18] investigated an SW-GW interaction scenario, applying heat as a tracer to quantify the recharge volume from the Motril-Salobrena aquifer to the Guadalfeo River.