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We present an effective stochastic advection-diffusion-reaction (SADR) modelthat explains incomplete mixing typically observed in transport withbimolecular reactions. Unlike traditional advection-dispersion-reaction models,the SADR model describes mechanical and diffusive mixing as two separateprocesses. In the SADR model, mechanical mixing is driven by random advectivevelocity with the variance given by the coefficient of mechanical dispersion.The diffusive mixing is modeled as a Fickian diffusion with the effectivediffusion coefficient. We demonstrate that the sum of the two coefficients isequal to the dispersion coefficients, but only the effective diffusioncoefficient contributes to the mixing-controlled reactions, indicating thatsuch systems do not get fully mixed at the Representative Elementary Volumescale where the deterministic equations and dispersion coefficient are defined.We use the experimental results of Gramling et al. \cite{Gramling} to show thatfor transport and bimolecular reactions in porous media, the SADR model issignificantly more accurate than the traditional dispersion model, whichoverestimates the concentration of the reaction product by as much as 60\%.

Persistent incomplete mixing in reactive flows