  We study the kinetics of infiltration in which contaminant particles, which
are suspended in a flowing carrier fluid, penetrate a porous medium. The
progress of the ``invader'' particles is impeded by their trapping on active
``defender'' sites which are on the surfaces of the medium. As the defenders
are used up, the invader penetrates further and ultimately breaks through. We
study this process in the regime where the particles are much smaller than the
pores so that the permeability change due to trapping is negligible. We develop
a family of microscopic models of increasing realism to determine the
propagation velocity of the invasion front, as well as the shapes of the
invader and defender profiles. The predictions of our model agree qualitatively
with experimental results on breakthrough times and the time dependence of the
invader concentration at the output. Our results also provide practical
guidelines for improving the design of deep bed filters in which infiltration
is the primary separation mechanism.

Statistical Mechanics;Soft Condensed Matter