Analyse mathématique d'un systéme de transport-diffusion-réaction modélisant la restauration biologique d'un milieu poreux

Abstract

In this paper, a mathernatical analysis of in-situ biorestoration is presented. Mathematical formulation of sucb process leads tía a system of non-linear partial differential equationa coupled with ordinary differential equations.First, we introduce a notion of weak solution then we prove the existence of at least ono such a solution by a linearization technique used in [8]. Positivenns and unifonn bound for tbe subsírates concentration is derived frorn the rnaximum principIe while sorne regularity propreties, for the pressure and velocity, are obtained from a local Meyers lemma [1], [12]. Next, assuming sorne regularity on the sohution, an uniqueness result is presented. Asymptotical behavior for the contarninant is also studied. RésuméDans ce travail, nona présentons une analyse rnathématique de la restauration biologique en rnilieu poreux. La modélisation d'un tel pbénom'ene conduit it un syst~rne d'équations aux dérivées partielles non-linéaires couplé it des équations différentielles ordinaires.ApÑs avoir introduit une notion de sohution faible, on montre l'existence d'au mnoins une solution de ce type par une technique de linéarisation utilisée dans [8]. La positivité et une borne uniforme pour la concentration sont déduites dú príncipe du maximum tandis qu'une propriété de régularité pour la vitesse et la preosion est

Upscaling of transport processes in porous media with biofilms in non-equilibrium conditions

“…Effective coefficients in transport equations are computed for a given geometry and a velocity field previously calculated. An example of the simple representation of the "solid/biofilm" microstructure that was adopted is represented by the two-dimensional periodic unit cell illustrated in discretized with a first order upstream scheme with anti-diffusion [41] and the dispersive part has been discretized with an implicit scheme [37]. The difficulties generated by the integrodifferential terms and the conditions of zero average in the different phases of the solution field have been overcome by a method of decomposition of variables (e.g.…”