Upscaling of transport processes in porous media with biofilms in non-equilibrium conditions

In this work, we derive two different Darcy-scale models for the transport of biodegradable solutes in porous media containing a microbial biomass that developed under the form of a biofilm. Biofilms are composed of bacterial populations and extra cellular polymeric substances, and grow attached to a solid–fluid interface, e.g. the pore walls of a water-saturated porous medium. We begin with the pore-scale description of mass transport, mass transfer between phases (fluid phase – generally water – and biofilm phase) and biologically-mediated reactions. Then, two limit cases of non-equilibrium transport are identified and characterized. Finally, these processes are upscaled using the method of volume averaging to obtain two different macroscale mass balance models. The models are derived for specific cases of non-equilibrium reactive transport (i.e., spatial concentration gradients may exist in one or both phases), for which mechanisms of mass transfer or reaction kinetics limit the rate of biodegradation. In both cases, a one-equation model can be developed even if non-equilibrium conditions exist. The validity domains of the two considered transport models (the Reaction-Rate Limited Consumption model – RRLC model – and the Mass Transfer Limited Consumption model – MTLC model) are established in terms of reactive and hydrodynamic conditions of transport (Damköhler number and Péclet number). The RRLC model is found to be consistent with direct numerical simulation (DNS) results at high Péclet and Damköhler numbers, while the applicability of the MTLC model is limited to high Damköhler numbers but low Péclet numbers.