Percolation properties of 3-D multiscale pore networks: how connectivity controls soil filtration processes

Quantifying the connectivity of pore networks is a key issue not only for modelling fluid flow and solute transport in porous media but also for assessing the ability of soil ecosystems to filter bacteria, viruses and any type of living microorganisms as well inert particles which pose a contamination risk. Straining is the main mechanical component of filtration processes: it is due to size effects, when a given soil retains a conveyed entity larger than the pores through which it is attempting to pass. We postulate that the range of sizes of entities which can be trapped inside soils has to be associated with the large range of scales involved in natural soil structures and that information on the pore size distribution has to be complemented by information on a critical filtration size (CFS) delimiting the transition between percolating and non percolating regimes in multiscale pore networks. We show that the mass fractal dimensions which are classically used in soil science to quantify scaling laws in observed pore size distributions can also be used to build 3-D multiscale models of pore networks exhibiting such a critical transition. We extend to the 3-D case a new theoretical approach recently developed to address the connectivity of 2-D fractal networks (Bird and Perrier, 2009). Theoretical arguments based on renormalisation functions provide insight into multi-scale connectivity and a first estimation of CFS. Numerical experiments on 3-D prefractal media confirm the qualitative theory. These results open the way towards a new methodology to estimate soil filtration efficiency from the construction of soil structural models to be calibrated on available multiscale data