Colloidal transport in anisotropic porous media: Kinetic equation and its upscaling

Abstract

We discuss three-dimensional suspension-colloidal-nano transport in anisotropic porous media. The Boltzmann kinetic equation captures stochastic particle velocity distribution on the micro-scale due to the stochastic nature of the pore-space geometry. We express the anisotropy by tensorial micro-scale transport coefficients and a direction-dependent velocity distribution. The Bhatnagar–Gross–Krook (BGK) form of Boltzmann’s equation for particle transport with a particle capture rate proportional to the particle velocity is formulated. We introduce an equivalent sink term into the kinetic equation instead of non-zero initial data, yielding an evolution system not requiring Cauchy’s data. Exact probabilistic homogenisation is performed by solution of a functional operator equation in Hilbert space of Fourier images. The homogenised macro-scale model has the form of a three-dimensional advective–diffusive–reactive equation, where the particle flow in all directions is slower than the carrier fluid speed. This delay is the collective result of particle flow and capture and is explained by preferential capture of faster particles. Closed expressions of upscaled transport coefficients as a function of microscale parameters are explicit. For the particular case of quasi-one-dimensional flow, where the average transverse flux is zero, we derive the upscaled equation and investigate the continuous and pulse colloidal injections.