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In the upscaling from pore- to continuum (Darcy) scale, reaction anddeposition phenomena at the solid-liquid interface of a porous medium have tobe represented by macroscopic reaction source terms. The effective rates can becomputed, in the case of periodic media, from three-dimensional microscopicsimulations of the periodic cell. Several computational and semi-analyticalmodels have been studied in the field of colloid filtration to describe thisproblem. They often rely on effective deposition rates defined by simple linearreaction ODEs, neglecting the advection-diffusion interplay, and assuming slowreactions. Therefore, when these rates are inserted into general macroscopictransport equations, they can lead to conceptual inconsistencies and,therefore, often qualitatively wrong results. In this work, we study theupscaling of Brownian deposition on face-centred spherical arrangements using alinear effective reaction rate, defined by volume averaging, and a macroscopicadvection-diffusion-reaction equation. The case of partial deposition, definedby an attachment probability, is studied and the limit of ideal deposition isretrieved as a particular case. We make use of a particularly convenientcomputational setup that allows the direct computation of the asymptoticstationary value of effective rates. This allows to drastically reduce thecomputational domain down to the scale of the single repeating periodic unit:the savings are ever more noticeable in the case of higher Peclet numbers, whenlarger physical times are needed to reach the asymptotic regime, and thus,equivalently, a much larger computational domain and simulation time would beneeded in a traditional setup. We show how this new definition of depositionrate is more robust and extendable to the whole range of P\'eclet numbers; italso is consistent with the classical heat and mass transfer literature.

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