  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. 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 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 Peclet 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 P\'eclet numbers; it
also is consistent with the classical heat and mass transfer literature.
