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Analyse mathématique de modèles de diffusion en milieu poreux élastique

Abstract : Wave propagation in fluid-saturated porous media is a complex phenomenon which occurs in various domains like seismic imaging or petroleum engineering. It is described by a Biot model which couples the displacement of the structure u with the fluid pressure p and the most complete system involves coupled equations which are mixed hyperbolic-parabolic. The first equation describes the evolution of the displacement of the structure while the evolution of the fluid is related to a combination of the Darcy law with the fluid mass conservation. The coupling terms include the pressure/deformation effects, also called consolidation effects, du to the interactions between the fluid and the porous medium. A secondary consolidation term can also occur in the first equation. When it is neglected, the model is the so-called linearized thermoelasticity system and p is then a temperature. Another interesting limit case of the Biot model is when the structure density vanishes which is a quasi-static phenomenon. At last, a nonlinear Biot model is obtained introducing a nonlinear potential with a q-Laplacian in the linear elasticity potential. Existence results are proved in several cases using the Galerkin approximation method or regularization and penalization methods while Ladyzenskaja's test-functions are used for the uniqueness. Moreover, it is proved that the thermoelastic and the quasi-static systems are asymptotic models of the complete problem. We conclude by studying the long time behavior of the solutions when secondary consolidation effects are considered.
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Contributor : Patrick Saint-Macary <>
Submitted on : Tuesday, December 7, 2004 - 6:42:42 PM
Last modification on : Friday, January 15, 2021 - 9:24:31 AM
Long-term archiving on: : Monday, September 20, 2010 - 12:06:22 PM


  • HAL Id : tel-00007651, version 2



Patrick Saint-Macary. Analyse mathématique de modèles de diffusion en milieu poreux élastique. Mathématiques [math]. Université de Pau et des Pays de l'Adour, 2004. Français. ⟨tel-00007651v2⟩



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