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Finite volume/finite element schemes for compressible two-phase flows inheterogeneous and anisotropic porous media

Abstract : The objective of this thesis is the development and the analysis of robust and consistent numerical schemes for the approximation of compressible two-phase flow models in anisotropic and heterogeneous porous media. A particular emphasis is set on the anisotropy together with the geometric complexity of the medium. The mathematical problem is given in a system of two degenerate and coupled parabolic equations whose main variables are the nonwetting saturation and the global pressure. In view of the difficulties manifested in the considered system, its cornerstone equations are approximated with two different classes of the finite volume family. The first class consists of combining finite elements and finite volumes. Based on standard assumptions on the space discretization and on the permeability tensor, a rigorous convergence analysis of the scheme is carried out thanks to classical arguments. To dispense with the underlined assumptions on the anisotropy ratio and on the mesh, the model has to be first formulated in the factional flux formulation. Moreover, the diffusive term is discretized by a Godunov-like scheme while the convective fluxes are approximated using an upwind technique. The resulting scheme preserves the physical ranges of the computed solution and satisfies the coercivity property. Hence, the convergence investigation holds. Numerical results show a satisfactory qualitative behavior of the scheme even if the medium of interest is anisotropic. The second class allows to consider more general meshes and tensors. It is about a new positive nonlinear discrete duality finite volume method. The main point is to approximate a part of the fluxes using a non standard technique. The application of this ideato a nonlinear diffusion equation yields surprising results. Indeed,not only is the discrete maximum property fulfilled but also the convergence of the scheme is established. Practically, the proposed method shows great promises since it provides a positivity-preserving and convergent scheme with optimal convergence rates.
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Submitted on : Friday, May 3, 2019 - 6:19:40 PM
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El Houssaine Quenjel. Finite volume/finite element schemes for compressible two-phase flows inheterogeneous and anisotropic porous media. General Mathematics [math.GM]. École centrale de Nantes; Université Moulay Ismaïl (Meknès, Maroc), 2018. English. ⟨NNT : 2018ECDN0059⟩. ⟨tel-02119515⟩



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