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Averaging of a Singular Random Source Term in a Diffusion Convection Equation

We consider a simplified model for the radionuclides migration in an underground nuclear waste repository, based on a linear partial differential equation of diffusion convection type. This partial differential equation has a source term constituted by a large number of "local" sources spatially periodically distributed and lying on the porous domain median plane. The behavior of each source is spatially homogeneous but their time dependence is uncertain; their release curve (source emission versus space and time) parameters are random both in space and in time. Starting from the mesoscopic model described above, our aim is then to obtain by "upscaling" a model with a deterministic "averaged" source term, describing the global evolution of such a system, and to prove the convergence, estimate (under proper mixing assumptions) the rate of convergence, and characterize the asymptotic behavior of the corrector.