Lagrangian Particle Methods for Biological Systems - Porous Media: Applications in Biological Systems and Biotechnology - page 265

7.3 SPHs Models for Biofilm Growth
Smoothed particle hydrodynamics was introduced by Lucy (1977) and by
Gingold and Monaghan (1977) to simulate fluid dynamics in the context
of astrophysical applications. SPH models have been successfully used to
simulate a variety of multiphase flow and transport processes in porous
media including microscale unsaturated fluid flow (Tartakovsky and Meakin
2005a,b), saturated flows (Morris et al. 1997; Zhu et al. 1999), multiphase
flows (Tartakovsky and Meakin 2005c, 2006), and nonreactive and reactive
solute transport (Zhu and Fox 2001, 2002; Tartakovsky and Meakin 2005c;
Tartakovsky et al. 2009) in fractured and porous media. The SPH approach
to the simulation of biomass (Tartakovsky et al. 2009), described here, is con-
ceptually similar to the particle-based approach of Picioreanu et al. (2004) who
represented biomass as discrete particles moving continuously (not constrained
to a grid), but calculated solute transport and reactions on a separate grid
using a multigrid PDE solution approach. Because SPH uses a well-established
meshless numerical framework, it is able to directly incorporate the full range
of relevant processes (e.g., hydrodynamic flow, solute transport, and chemical
and biological reactions) into simulation of biomass growth with an arbitrary
geometry, using a single consistent formulation. That is, no secondary grids
or other methods are required for simulation of flow, transport, or reactions
(except for an underlying grid that is used to rapidly locate pairs of parti-
cles that are located near enough to each other to interact), and all pertinent
processes are simulated within the unified SPH framework. The Lagrangian
particle nature of SPH allows complex biomass-fluid and biomass-solid inter-
actions to be modeled through simple pair-wise interaction forces.
In the SPH biomass model (Tartakovsky et al. 2009), pair-wise particle-
particle interactions are used to simulate interactions within the biomass,
and interactions between biomass and fluid and between biomass and soil
grains. A model-fractured porous medium was generated by randomly insert-
ing nonoverlapping particles with radii selected randomly from a truncated
Gaussian distribution on either side of the gap between two self-ane fractal
curves representing a microfracture, and biomass was randomly distributed on
the soil grains to initialize the simulations. The injection of two solutions, the
first containing electron donors and the second containing electron acceptors,
into different halves of the domain was simulated.
This work assumes that the changes in the biomass are governed by dual
Monod kinetics (Knutson et al. 2005):

dM
dt

A
K A + A

B
K B + B −

= Yk s M k d M (7.22)

where M is the biofilm concentration; A and B are the concentrations of
electron donors and electron acceptors; Y is the yield coecient; k S is the