## 4 Mathematical Simulations of Biofilm Evolution in Porous Media
### Effects of Biofilm Formation on Permeability  
The reduction in permeability is one of the most striking features of biofouling in porous media. The first 3D numerical pore-scale model of biofilm growth in porous media was presented by [PERSON] et al. [89] based on a lattice Boltzmann simulation platform complemented with an individual-based biofilm model (LB-IbM). The numerical model can calculate parameters such as the velocity, pressure, nutrient concentration fields, biomass distribution, and porous media's permeability. By comparing the simulation results in 2D and 3D space, it was found that pore space clogging caused by biofilm growth was slower in 3D space because of the greater connectivity between pores. The LB-IbM method was applied by the same research group to achieve optimum reduction of permeability through biofilm growth in porous media in a 2D space [88]. Critical biofilm strength was observed when the separation of biomass was considered. This study also indicates that the detachment of biofilm is of significance in the study of blockage events, which was also verified in another modeling study [86]. They further extended this simulation method to consider the permeability of biofilms and explored how the permeability of biofilms influence their growth rate in 3D space [105]. As shown in Figure 7, a higher magnitude of biofilm permeability accelerated the biomass accumulation. With the increase of biofilm permeability, the velocities of biofilm-occupied regions increased (areas within white box), but the velocities in the free pore space decreased (areas within red box).  
In addition, a modeling study showed that the permeability of the biofilm has an effect on the permeability of the porous system [91]. It was revealed that the model would underestimate the permeability of the biofouling porous system if the permeability value of biofilm was not taken into account. Furthermore, solute transport in porous media is also an important factor for permeability reduction. A 3D continuum biomass model for simulating the relationship between solute transport and biofilm growth in porous media was developed by [PERSON] et al. [36]. The pore was saturated with water and the homogenous biofilm accumulation was uniform. The results revealed that biofilm clogging occurred at the inlet region when the solute supply was insufficient, which resulted in the fast reduction of the porous media's permeability. When the solute supply was sufficient, biofilm would grow uniformly throughout the pore network. The results also indicated that there is no unique relationship between permeability and porosity in the pore network of biofilm growth.  
Figure 6: Schematic of the modeling strategies of biofilm development in a pore. Discrete biomass-based models (DbMs) can be established through cellular automaton (CA) in 2D and 3D space with discrete distribution directions and discrete displacement. Individual-based biomass models (IbMs) treat individual microbes as basic units, allowing cell variation and movement in a continuous direction and distance. In continuum biomass-based models (CbMs), the development process of biofilm growth is represented by differential equations.