Boltzmann's colloidal transport in porous media with velocity-dependent capture probability

Abstract

Mathematical modeling of suspension-colloidal-nano transport in porous media at different scales has long been a fascinating topic of fluid mechanics. In this study, we discuss the multi-pore scale, where Boltzmann's approach of distributed velocities is valid, and average (homogenize) the micro-scale equation up to the core scale. The focus is on the filtration function (particle capture probability per unity trajectory length) that highly depends on the carrier fluid velocity. We develop a modified form of the Boltzmann equation for micro-scale particle capture and diffusion. An equivalent sink term is introduced into the kinetic equation instead of non-zero initial data, resulting in the solution of an operator equation in the Fourier space and an exact homogenization. The upper scale transport equation is obtained in closed form. The upscaled model contains the dimensionless delay number and large-scale dispersion and filtration coefficients. The explicit formulas for the large-scale model coefficients are derived in terms of the micro-scale parameters for any arbitrary velocity-dependent filtration function. We focus on three micro-scale models for the velocity-dependent particle capture rate corresponding to various retention mechanisms, i.e., straining, attachment, and inertial capture. The explicit formulas for large-scale transport coefficients reveal their typical dependencies of velocity and the micro-scale parameters. Treatment of several laboratory tests reveals close match with the modeling-based predictions.