Mathematical and numerical modelling of seawater intrusion models in coastal aquifers

Résumé

Numerical modelling and simulation of marine intrusion in coastal aquifers have been an interesting research topic and different approaches and appropriate strategies have been developed to predict seawater intrusion. Such a problem is modelled by a coupled system of two parabolic partial differential equations. This thesis aims to develop and implement robust and accurate finite volume schemes, based on recent and efficient tools, for two (sharp and sharp–diffuse) seawater intrusion models in coastal aquifers. Moreover, we focus in this work on the numerical analysis of a finite volume method for the mixed sharp-diffuse interfaces model. The thesis is organized into six chapters. The general context of the thesis and a literature review were the subject of the first chapter. In the second one, we present a state of the art on existing approaches for modelling marine intrusion in coastal aquifers. Afterwards, we present the derivation of the sharp interface 2D model, which assumes that freshwater and saltwater are immiscible and separated by a sharp interface. Chapter 3 is devoted to the numerical resolution of this model. In this context, we have considered a fully coupled, fully implicit finite volume TPFA (Two Point Flux Approximation) method for performing numerical simulations using this approach. The numerical scheme is implemented in the DuMuX framework. Our code is validated on several 1D and 2D test cases, including two realistic cases: the Souss-Chtouka plain in Morocco and Tripoli aquifer in Lebanon. The comparison of our numerical results with others presented in the literature proves the ability of our module to predict seawater intrusion in large-scale aquifers. Then in chapter 4, we consider an extension of the sharp interface approach to take into account the dynamics of transition (mixing) zones to the so-called sharp-diffuse interfaces 2D model. The mathematical formulation, recently introduced, of this approach is briefly described. As in the previous chapter, this model is discretized by a fully implicit TPFA method and implemented in DuMuX. The proposed module is validated on several test cases and proved its capacity to deal with physical and hydrological phenomena like tidal fluctuations. Thereafter, we compared the numerical result for the 2D model with those obtained with the classical 3D model for miscible displacements. The numerical results showed that this approach yields physically realistic performance results. To predict seawater intrusion into heterogeneous and anisotropic coastal aquifers, we have considered in chapter 5 a numerical method, based on an MPFA (Multi-Point Flux Approximation) method of the sharp-diffuse interfaces model. The study of the numerical analysis of the scheme is presented. The non-negativity of the discrete solution is proved and an existence result is shown using a fixed point theorem. Based on recent compactness tools, we have proved the strong convergence of the approximate solution to the weak solution of the continuous problem. The MPFA numerical scheme is further implemented in the DuMuX framework. The accuracy and efficiency of our method in a strongly heterogeneous case are examined by performing 2D numerical simulations. Lastly, some concluding remarks and perspectives are forwarded.

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