  In this paper, we derive upscaled equations for modelling biofilm growth in
porous media. The resulting macro-scale mathematical models consider permeable
multi-species biofilm including water flow, transport, detachment and
reactions. The biofilm is composed of extracellular polymeric substances (EPS),
water, active bacteria and dead bacteria. The free flow is described by the
Stokes and continuity equations and the water flux inside the biofilm by the
Brinkman and continuity equations. The nutrients are transported in the water
phase by convection and diffusion. This pore-scale model includes variations of
the biofilm composition and size due to reproduction of bacteria, production of
EPS, death of bacteria and shear forces. The model includes a water-biofilm
interface between the free flow and the biofilm. Homogenization techniques are
applied to obtain upscaled models in a thin channel and a tube, by
investigating the limit as the ratio of the aperture to the length
$\varepsilon$ of both geometries approaches to zero. As $\varepsilon$ gets
smaller, we obtain that the percentage of biofilm coverage area over time
predicted by the pore-scale model approaches the one obtained using the
effective equations, which shows a correspondence between both models. The two
derived porosity-permeability relations are compared to two empirical relations
from the literature. The resulting numerical computations are presented to
compare the outcome of the effective (upscaled) models for the two mentioned
geometries.
