Taylor dispersion in heterogeneous porous media: Extended method of moments, theory, and modelling with two-relaxation-times lattice Boltzmann scheme

Abstract

This article describes a generalization of the method of moments, called extended method of moments (EMM), for dispersion in periodic structures composed of impermeable or permeable porous inclusions. Prescribing pre-computed steady state velocity field in a single periodic cell, the EMM sequentially solves specific linear stationary advection-diffusion equations and restores any-order moments of the resident time distribution or the averaged concentration distribution. Like the pioneering Brenner's method, the EMM recovers mean seepage velocity and Taylor dispersion coefficient as the first two terms of the perturbative expansion. We consider two types of dispersion: spatial dispersion, i.e., spread of initially narrow pulse of concentration, and temporal dispersion, where different portions of the solute have different residence times inside the system. While the first (mean velocity) and the second (Taylor dispersion coefficient) moments coincide for both problems, the higher moments are different. Our perturbative approach allows to link them through simple analytical expressions. Although the relative importance of the higher moments decays downstream, they manifest the non-Gaussian behaviour of the breakthrough curves, especially if the solute can diffuse into less porous phase. The EMM quantifies two principal effects of bi-modality, as the appearance of sharp peaks and elongated tails of the distributions. In addition, the moments can be used for the numerical reconstruction of the corresponding distribution, avoiding time-consuming computations of solute transition through heterogeneous media. As illustration, solutions for Taylor dispersion, skewness, and kurtosis in Poiseuille flow and open/impermeable stratified systems, both in rectangular and cylindrical channels, power-law duct flows, shallow channels, and Darcy flow in parallel porous layers are obtained in closed analytical form for the entire range of Péclet numbers. The high-order moments and reconstructed profiles are compared to their predictions from the advection-diffusion equation for averaged concentration, based on the same averaged seepage velocity and Taylor dispersion coefficient. In parallel, we construct Lattice-Boltzmann equation (LBE) two-relaxation-times scheme to simulate transport of a passive scalar directly in heterogeneous media specified by discontinuous porosity distribution. We focus our numerical analysis and assessment on (i) truncation corrections, because of their impact on the moments, (ii) stability, since we show that stable Darcy velocity amplitude reduces with the porosity, and (iii) interface accuracy which is found to play the crucial role. The task is twofold: the LBE supports the EMM predictions, while the EMM provides non-trivial benchmarks for the numerical schemes.