Mixing-Limited Effective Reaction Rates in Porous Media: An Explicit Prediction from Pore-Scale Mixing

Shahram Asgari

Two solutes can share the same volume of a porous medium and still barely react, because sharing a volume is not the same as mixing. As the flow carries them, it stretches each into thin filaments that interleave through the pore space, and reaction is confined to the narrow fronts where those filaments meet. A continuum model that simply multiplies the average concentrations ignores this and overpredicts the reaction whenever the chemistry is faster than the flow can mix. We quantify that overprediction and turn it into a usable correction. Averaging the bimolecular rate law leaves a term, the reactant cross-covariance, that the standard continuum rate discards; written as a segregation intensity, it measures how incompletely the reactants are mixed within an averaging volume. We follow that quantity with a balance in which the dispersive flow continually produces fresh reactant contact while diffusion erases it, and we close the balance with a single mixing time. The effective rate constant that results is the intrinsic constant divided by one plus a mixing Damkohler number, the ratio of the time to mix to the time to react. Taking the mixing time from lamellar theory, in which the flow thins the reactant interface until diffusion can complete the mixing, makes the correction an explicit function of the Peclet and Damkohler numbers, evaluable from field or column data without any pore-scale simulation. In the fast-reaction limit it recovers the exact mixing-controlled rate of De Simoni and co-workers, and it reproduces the slower-than-expected product growth recorded in the Gramling, Harvey, and Meigs visualization experiment.