Effective Microbial Kinetics under Pore-Scale Transport Limitation: A Biofilm Effectiveness-Factor Theory

Shahram Asgari 

Reactive-transport models of porous media usually evaluate microbial reactions with Monod kinetics and rate constants measured on well-mixed laboratory cultures, in which every cell is exposed to the same substrate concentration. In natural sediments and aquifers this assumption fails at the pore scale, because the cells are not suspended but reside in biofilms and aggregates attached to the grains, into which the limiting substrate must diffuse while the cells consume it. The interior of a biofilm is therefore starved relative to its surface, and the rate that a continuum model should assign to the average pore-water concentration falls below the laboratory Monod rate. Here we ask how large this reduction is, what controls it, and whether it can be captured in a single correction that a continuum model can apply. We reduce the steady reaction-diffusion problem inside the film to dimensionless form, which leaves one controlling group, a Thiele modulus equal to the square root of a Damkohler number that measures how strongly diffusion limits the reaction. Integrating the governing equation once gives the reduction explicitly, as a biofilm effectiveness factor: it equals the classical hyperbolic-tangent law when the surface is dilute and a half-power, partial-penetration law when it is saturated, and the uptake of a thick biofilm follows a law whose apparent reaction order slides continuously from one to one-half. A direct consequence is that the half-saturation constant fitted to such a community exceeds its true value, and we quantify the excess: the apparent constant is 1.2 times the intrinsic one at a Thiele modulus of one, 15 times at ten, and approaches one eighth of the Damkohler number when diffusion dominates. Multiplied by the biofilm surface area per unit volume, the effectiveness factor becomes a drop-in rate correction for continuum reactive-transport models, accurate to better than a percent provided the substrate is drawn down by less than about a fifth across a grid cell, and provided the external boundary layer is not itself limiting. Applied to the anaerobic oxidation of methane in marine sediments, the same formulas act as a screening test, and on current estimates of aggregate size and rate they indicate that such aggregates are reaction-limited and that their measured kinetics are close to intrinsic.