Averaged Boltzmann's Kinetics for Colloidal Transport in Porous Media

Due to the stochastic nature of pore space geometry, particle velocities in a colloidal-suspension flux are also stochastically distributed. This phenomenon is captured by Boltzmann's kinetic equation. We formulate a BGK-form of Boltzmann's equation for particulate flow with a particle capture rate proportional to the particle speed and derive an exact method for the model's averaging. The averaged equation is of the form of a reaction-advection-diffusion equation with the average particle speed lower than the carrier fluid velocity. This delay, reported in numerous laboratory studies, is explained by preferential capture of fast particles. The large-scale model coefficients of delay, dispersion, and filtration are explicitly expressed via the microscale mixing length, equilibrium velocity distribution, and filtration coefficient. The properties of the averaged coefficients are discussed, and emerging dependencies between them are presented. The derived large-scale equation resolves two paradoxes of the traditional model for suspension-colloidal transport in porous media. The large-scale equation closely matches the laboratory breakthrough curves.