A robust upscaling of the effective particle deposition rate in porous media

Abstract

In the upscaling from pore-to continuum (Darcy) scale, reaction and deposition phenomena at the solid-liquid interface of a porous medium have to be represented by macroscopic reaction source terms. The effective rates can be computed, in the case of periodic media, from three-dimensional microscopic simulations of the periodic cell. Several computational and semi-analytical models have been studied in the field of colloid filtration to describe this problem. They often rely on effective deposition rates defined by simple linear reaction ODEs, neglecting the advection-diffusion interplay, and assuming slow reactions (or large Péclet numbers). Therefore, when these rates are inserted into general macroscopic transport equations, they can lead to conceptual inconsistencies and, therefore, often qualitatively wrong results. In this work, we study the upscaling of Brownian deposition on face-centred cubic (FCC) spherical arrangements using a linear effective reaction rate, defined by volume averaging, and a macroscopic advection-diffusion-reaction equation. The case of partial deposition, defined by an attachment probability, is studied and the limit of ideal deposition is retrieved as a particular case. We make use of a particularly convenient computational setup that allows the direct computation of the asymptotic stationary value of effective rates. This allows to drastically reduce the computational domain down to the scale of the single repeating periodic unit: the savings are ever more noticeable in the case of higher Péclet numbers, when larger physical times are needed to reach the asymptotic regime, and thus, equivalently, a much larger computational domain and simulation time would be needed in a traditional setup. We show how this new definition of deposition rate is more robust and extendable to the whole range of Péclet numbers; it also is consistent with the classical heat and mass transfer literature.

Transport and Deposition of Large Aspect Ratio Prolate and Oblate Spheroidal Nanoparticles in Cross Flow

“…A modification of the diffusion coefficient for the porous medium is then necessary using the appropriate tortuosity. Finally, the occurrence of deposition in the porous medium should also be modeled, including a reaction term, see Boccardio et al [16].…”

Colloid Transport in Porous Media: A Review of Classical Mechanisms and Emerging Topics

“…There are two primary components to modern CFT formulations: 1) a mechanistic model that solves a force/torque balance to estimate the trajectory of a colloid near a collector, and 2) a correlation equation that summarizes the results of the mechanistic model. These results are then upscaled to [40,[96][97][98]. For a granular medium idealized as a packed bed of Happel Sphere in Cells with diameter , is estimated via Eq.…”

Upscaling Flow and Transport Processes

“…There are a number of issues facing these models; first of all they are most often based on a single idealised geometrical model representing the porous medium, thus failing to grasp the pore-scale complexities and heterogeneity and its effect on particle filtration. Another conceptual hurdle in the application of these models is the difficult translation of the obtained efficiency parameter η D into an effective macro-scale reaction term employable in a macroscopic transport equation [115], and to understand its dependence on the flow parameters and its inseparable connection with the effective dispersion and velocity.…”