We consider a new model mixing sharp and diffuse interface approaches for seawater intrusion phenomena in free aquifers. More precisely, a phase field model is introduced in the boundary conditions on the virtual sharp interfaces. We thus include in the model the existence of diffuse transition zones but we preserve the simplified structure allowing front tracking. The three-dimensional problem then reduces to a two-dimensional model involving a strongly coupled system of partial differential equations of parabolic type describing the evolution of the depths of the two free surfaces, that is the interface between salt-and freshwater and the water table. We prove the existence of a weak solution for the model completed with initial and boundary conditions. We also prove that the depths of the two interfaces satisfy a coupled maximum principle.

A differential game modeling the noncooperative outcome of pollution in groundwater is studied. Spatio-temporal objectives are constrained by a convection-diffusion-reaction equation ruling the spread of the pollution in the aquifer, and the velocity of the flow solves an elliptic partial differential equation. The existence of a Nash equilibrium is proved using a fixed point strategy. A uniqueness result for the Nash equilibrium is also proved under some additional assumptions. Some numerical illustrations are provided.

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A scheme is proposed for the numerical study of optimal control problems related to groundwater quality management in the case of nonpoint-source pollution. The corresponding state equation is a system of coupled nonlinear partial differential equations. The approximation is given by a finite volume scheme based on a two-point flux approximation with upwind mobilities embedded in an iterative fixed point approximation. The analysis of the convergence of the scheme allows to establish existence and uniqueness results under reasonable assumptions. Numerical illustrations of the performance of the algorithm are given in realistic situations.

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