## 1 Introduction  
The steady-state temperature profiles in a depth aquifer, where groundwater penetrates from the left border to a limited region where profiles develop, is strongly dependent on the groundwater velocity and its input temperature, which generally differs from the surface and bottom temperatures of the domain. This problem has been studied decades ago [1; 2; 3; 4] and more recently [5; 6; 7; 8; 9]. The horizontal groundwater flow with a constant temperature coming from an inlet border, in an extended aquifer whose surface and bottom are subjected to isothermal (Diritlech) conditions, gives rise to a temperature field within the aquifer that depends on both the depth and the horizontal location. The extension of the region in which the temperature field is developed has been recently studied and characterized dimensionally [10]. These authors provide expressions for the width of the developed region as a function of the depth of the aquifer, groundwater velocity and thermal diffusivity of the porous media, and the latter combines the thermal conductivity of the soil-fluid matrix and the specific heat of the fluid. Based on the developed length of the temperature profile, the vertical dependence of the temperature profiles can be expressed as a function of the relative horizontal location with respect to that length, while the horizontal profile throughout the development region is a function of the relative location with respect to the depth of the aquifer. Beyond this development region, the profiles are constant and do not allow estimations of water flows. This explains the non-dependence of the profiles on the global horizontal extent of the aquifer, a novel result not assumed in these scenarios by other authors who impose the added and unrealistic condition of the existence of a constant horizontal thermal gradient along the entire aquifer, resulting in different profiles throughout the domain [11; 12]. After these studies, no other works have been published on the estimation of horizontal fluxes with the hypothesis of the existence of a development region. In addition, all the temperature profiles resulting from the temperature field are very sensitive to the dimensionless group defined from the three temperatures that set the boundary conditions. An estimation of groundwater velocity in the form of an inverse problem could be carried out using the universal type-curves related to temperature profiles, but this would require a large number of abacuses, because of the many potential values of the dimensionless group related to the three boundary temperatures. The other and more general technique for the velocity estimation is that of setting a classical inverse problem protocol [13], based on the minimization of a functional that compares, in successive steps, the experimental measurements with those coming from the numerical simulation of scenarios that progressively approach the solution [14]. This technique has been successfully applied to inverse problems of similar complexity in different fields [15; 16; 17; 18]. Due to the large number of parameters involved in these problems (depth, boundary temperatures and thermal properties, as well as the experimental thermal profiles), the proposed protocol is complex and requires a sufficient number of iterations. Two functionals are defined, one for each position of the temperature measurements. After numerical iterations, the sum of these functionals converges toward a minimum value corresponding to a scenario that provides the desired estimates. This document is structured as follows. In Section 3, the physical and mathematical model of the scenario under study is described. The steps and block diagram of the applied inverse problem protocol, which uses a classical technique [13], for flow estimation are presented in Section 4 and the results are verified in Section 5. The verification includes the introduction of temperature deviations in the measurements to simulate the errors of the measuring instruments.