Simulating degradation of organic compounds accounting for the growth of microorganisms (Monod kinetics) in a fully Lagrangian framework

Biologically mediated degradation of organic compounds in porous media is a complex mathematical problem, described by a non-linear differential equation. The organic compound gets in contact with the biomass, and an enzyme-catalysed reaction takes place. The net result is that part of the parent compound degrades into some daughter product, while some of the organic carbon is used for microbial growth. The rate of biomass growth in the presence of a limiting nutrient supply is usually modelled with the experimentally derived Monod equation, i.e., it is proportional to the actual existing biomass multiplied by a factor that is non-linear in terms of available organic matter. This non-linearity in the degradation equation implies a strong difficulty in directly implementing a numerical solution within a fully Lagrangian framework, and thus, numerical solutions have traditionally been sought in either an Eulerian, or else an Eulerian–Lagrangian framework. Here we pursue a fully Lagrangian solution to the problem. First, the Monod empirical equation is formulated as the outcome of a two-step reaction; while the approach is less general than other derivations existing in the literature based on a full understanding of the thermodynamics of the process, it allows two things: 1) providing some physical meaning to the actual parameters in the Monod equation, and more interestingly, 2) formulating a methodology for the solution of the degradation equation incorporating Monod kinetics by means of a particle tracking formulation. For the latter purpose, both reactants and biomass are represented by particles, and their location at any given time is represented by a kernel that accounts for the uncertainty in the actual physical location. By solving the reaction equation in a kernel framework, we can reproduce the Monod kinetics and, as a particular result in the case no biomass growth is allowed, the Michaelis–Menten kinetics. The methodology proposed is then successfully applied to reproduce two studies of microbially induced degradation of organic compounds in porous media, first, the observed kinetics of Pseudomonas putida F1 in batch reactors while growing on benzene, toluene and phenol, and second, the column study of carbon tetrachloride biodegradation by the denitrifying bacterium Pseudomonas Stutzeri KC.