Stochastic‐Convective Transport with Nonlinear Reaction: Biodegradation With Microbial Growth

Abstract

The representation of subsurface flow and reactive transport as an ensemble of one‐dimensional stream tubes is extended to account for nonlinear biodegradation with coupled microbial growth. The Stochastic‐convective reaction (SCR) model is derived for bioreaction of a single solute by a single class of microorganisms coupled with dynamic microbial growth. A new global variable, the integral of the solute degraded per unit length of system traversed, accounts for degradation. Dimensionless scaling and the method of characteristics are used to reduce the model, written for a single convecting reactor (stream tube), to a pair of coupled nonlinear functional equations for solute concentration and microbial biomass. Existence of a solution to the stream tube system is shown, both numerical and approximate analytical approaches to the solution are given, and example computations using both methods are presented. Conditions under which the stream tube solution is “canonical,” or scalable to fit any permissible stream tube travel time function, arise from requirements for invariance (over the stream tube ensemble) of effective one‐dimensional stream tubes used to represent transport along real stream tubes in three‐dimensional space. Averaging of the stream tube solution over travel time and reaction properties representative of physical and chemical heterogeneities is described as a way to separate and upscale the processes of macrodispersion and microbiological reaction. The approach is exercised to simulate Monte Carlo average behavior of bioreactive transport in physically heterogeneous two‐dimensional media. Results show that the method captures the ensemble average large‐scale effects of the nonlinear reactions more accurately than done in the classical reactive convection‐dispersion equation (CDR), even when the appropriate scale dependent dispersion coefficient is afforded to the CDR.