## 4 Mathematical Simulations of Biofilm Evolution in Porous Media  
As described in the second section of this review, the formation of biofilms is a complex process that involves microbial interactions with both the fluid flow and the medium's surface. The evolution of biofilms and its surroundings are influenced by many factors such as the nutrient limitations induced by microbial metabolism and shear stress induced by fluid flow. Except for laboratory experiments cultivating biofilm in modeled porous media, it is necessary to establish mathematical models considering these impacts to elucidate the underlying mechanisms and predict the behavior of these complex processes.  
Models describing biofilm development in porous media typically include descriptions of the porous media examined, the fluid hydrodynamic, mass transport and the accumulation and decay of biofilm. It is therefore necessary to propose several hypotheses (such as the uniformity of the biofilm matrix, flow condition, and properties of the porous media) to simplify models according to different research purposes (output). Pore-scale models usually originate from the actual structures of porous media including the microfluidic flow cell [80; 81; 82; 83; 84], glass pore network models [52; 85], and others [86; 87; 88; 89; 90; 91]. Fluid flow and mass transport are modeled using continuous strategies based on Darcy's law and conservation laws.  
Models describing the evolution of biofilms are the key to solving problems involving fluid hydrodynamics and mass transfer in bioclogged porous media. Figure 6 provides a schematic for the modeling strategies of biofilm development in a pore. Strategies for modeling biomass evolution are divided into three types: individual-based biomass models (IbMs), discrete biomass-based models (DbMs), and continuum biomass-based models (CbMs) [92]. CbMs use differential equations to express biomass expansion. There are no internal structures (e.g., pores and holes) in the modeled biomass. DbMs use rectangular elements (cubes or squares) moving in the finite number of lattice directions to represent biomass behavior. The pattern of biomass growth is defined based on a series of simple rules. Biomass spread process is discrete in time, space, and state. DbMs can simulate the impacts of fluid flow (detachment, captured by narrow pores, etc.). IbMs use individual solid spheres to represent biomass. Each sphere has a variable volume and a variable mass. The biomass evolution is defined explicitly with a set of rules that mimic the behavior of individual bacterial cells, and is continuous in time and space. More details with respect to modeling rules can be found in other reviews [92; 93; 94].  
Several models have been used to investigate mechanisms relating to biofilm development in fluid bioreactors. The morphology of biofilm [95; 96; 97], biofilm detachment [98; 99; 100], and removal of contaminants [101; 102] can be well simulated regardless of being in 2D or 3D as well as the mass transfer between biofilm and fluid [103; 104]. However, biofilm development in a porous media involves interactions between the biomass, porous media, and fluid flow, which adds complexity and difficulty to the calculation. Two-dimensional models thus constitute most developed models of the biofouling porous system. If the interaction between biofilm development and fluid dynamics in porous media is studied on the microscale, a 2D model could be selected. A 3D model is more practical on a larger scale at the cost of more computational time, especially when the degree of biofouling, porosity, and water conductivity of the entire system are considered.  
Model strategies presented here focus on solving problems as follows: (1) effects of biofilm formation on permeability; (2) interactions between fluid flow and biofilm development; (3) multispecies biofilm formation. Subsequently, several examples are introduced.