Biofilm growth in porous media: derivation of a macroscopic model from the physics at the pore scale via homogenization
Anaïs Philippe, Christian Geindreau, Philippe Séchet, Jean Martins · RUC (Universidade Da Coruña) · 2005
This paper concerns the modelling of biofilm in porous media. A macroscopic model (at the Darcy’s scale) is derived from the physics at the pore scale by using an upscaling technique, namely the homogenisation method of multiple scale expansions. The domain of validity of this macroscopic description depends on the order of magnitude of different dimensionless numbers arising from the description at the microscopic scale. We show that at the macroscopic scale (i) the flow is described by the classical Darcy's law (ii) the mass balance of the fluid includes a source term due to the biomass growth (iii) the transport of substrate is describe by a diffusion-advection-reaction equation (iv) the macroscopic biomass balance is given by a diffusion-reaction relation. The macroscopic reactive term is similar to the microscopic one. The homogenization process also shows that all the effective coefficients (permeability, effective diffusion tensors) arising in the macroscopic description depends on the biomass fraction. A simple example is presented.