In this paper we consider a system of two coupled nonlinear diffusion-reaction partial differential equations (PDEs) which model the growth of biofilm and consumption of the nutrient. At the scale of interest the biofilm density is subject to a pointwise constraint, thus the biofilm PDE is framed as a parabolic variational inequality. We derive rigorous error estimates for a finite element (FE) approximation to the coupled nonlinear system and confirm experimentally that the numerical approximation converges at the predicted rate. We also show simulations in which we track the free boundary in the domains which resemble the pore scale geometry and in which we test the different modeling assumptions. KeywordsParabolic variational inequality, coupled system, biofilm growth, finite elements, error estimates, semismooth Newton solver.

<abstract><p>In this paper we describe a coupled model for flow and microbial growth as well as nutrient utilization. These processes occur within and outside the biofilm phase formed by the microbes. The primary challenge is to address the volume constraint of maximum cell density but also to allow some microbial presence outside the contiguous biofilm phase. Our model derives from the continuum analogues of the mechanism of cell shoving introduced in discrete biomass models, and in particular from the models exploiting singular diffusivity as well as from models of variational inequality type which impose explicit constraints. We blend these approaches and propose a new idea to adapt the magnitude of the diffusivity automatically so as to ensure the volume constraint without affecting the reactions; this construction can be implemented in many variants without deteriorating the overall efficiency. The second challenge is to account for the flow and transport in the bulk fluid phase adjacent to the biofilm phase. We use the Brinkman flow model with a spatially variable permeability depending on biomass amount. The fluid flow allows some advection of the nutrient within the biofilm phase as well as for the flow even when the pores are close to being plugged up. Our entire model is monolithic and computationally robust even in complex pore-scale geometries, and extends to multiple species. We provide illustrations of our model and of related approaches. The results of the model can be easily post—processed to provide Darcy scale properties of the porous medium, e.g., one can predict how the permeability changes depending on the biomass growth in many realistic scenarios.</p></abstract>

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