L.M.P.A
Laboratoire de Mathématiques Pures et Appliquées
Joseph Liouville

# Analyse mathématique de modèles d'intrusion marine dans les aquifères côtiers

### Sujet

The theme of this thesis is the analysis of mathematical models describing saltwater intrusion in coastal aquifers. The simplicity of sharp interface approach is chosen : there is no mass transfer between fresh water and salt water (respectively between the saturated zone and the area dry). We compensate the mathematical difficulty of the analysis of free interfaces by a vertical averaging process allowing us to reduce the 3D problem to system of pde's defined on a 2D domain $\Omega$. A second model is obtained by combining the approach of 'sharp interface' in that with 'diffuse interface' ; this approach is derived from the theory introduced by Allen-Cahn, using phase functions to describe the phenomena of transition between fresh water and salt water (respectively the saturated and unsaturated areas). The 3D problem is then reduced to a strongly coupled system of quasi-linear parabolic equations in the unconfined case describing the evolution of the DEPTHS of two free surfaces and elliptical-parabolic equations in the case of confined aquifer, the unknowns being the depth of salt water/fresh water interface and the fresh water hydraulic head. In the first part of the thesis, the results of global in time existence are demonstrated showing that the sharp-diffuse interface approach is more relevant since it allows to establish a mor physical maximum principle (more precisely a hierarchy between the two free surfaces). In contrast, in the case of confined aquifer, we show that both approach leads to similar results. In the second part of the thesis, we prove the uniqueness of the solution in the non-degenerate case. The proof is based on a regularity result of the gradient of the solution in the space $\text{Lr}(\Omega\Gamma)$, $r > 2$, $(\Omega\,T = (0,T) \times \Omega)$. Then we are interest in a problem of identification of hydraulic conductivities in the unsteady case. This problem is formulated by an optimization problem whose cost function measures the squared difference between experimental hydraulic heads and those given by the model.