A review of some homogenization results for transport equations in porous media

 In this talk we shall discuss some old and new results on the homogenization of convection-diffusion-reaction equations in periodic porous media. Assuming that the porous medium is rigid and saturated, and that the flow velocity is given, we focus on the transport equations of single ou multiple species. Our main scaling assumption is that the local P\'{e}clet and Damk\"{o}hler numbers are of order one, i.e., that all terms of convection, diffusion and reaction are of the same order of magnitude at the microscopic pore scale. Our goal is to obtain effective or macroscopic models, the coefficients of which can be evaluated through local cell problems. One key mathematical tool is the notion of two-scale asymptotic expansion with drift and its rigorous counterpart fo two-scale convergence with drift. We shall illustrate our approach with numerical results on the dependence of the effective dispersion tensor with respect to the P\'{e}clet number. 

This is a summary of joint works with R. Brizzi, H. Hutridurga, A. Mikelic, I. Pankratova and A. Piatnitski.