Stochastic analysis of biodegradation fronts in one-dimensional heterogeneous porous media

We consider a one-dimensional model biodegradation system consisting of two reaction–advection equations for nutrient and pollutant concentrations and a rate equation for biomass. The hydrodynamic dispersion is ignored. Under an explicit condition on the decay and growth rates of biomass, the system can be approximated by two component models by setting biomass kinetics to equilibrium. We derive closed form solutions for constant speed traveling fronts for the reduced two component models and compare their profiles in homogeneous media. For a spatially random velocity field, we introduce travel time and study statistics of degradation fronts via representations in terms of the travel time probability density function (pdf) and the traveling front profiles. The travel time pdf does not vary with the nutrient and pollutant concentrations and only depends on the random water velocity. The traveling front profiles are expressed analytically or semi-analytically as functions of the travel time. The problem of nonlinear transport by a random velocity reduces to two subproblems: one being nonlinear transport by a known (unit) velocity, and the other being linear (advective) transport by a random velocity. The approach is illustrated through some examples where the randomness in velocity stems from the spatial variability of porosity.