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We compare the ability of various continuum-scale models to reproduce the key features of a transport setting associated with a bimolecular reaction taking place in the fluid phase and numerically simulated at the pore-scale level in a disordered porous medium. We start by considering a continuum-scale formulation which results from formal upscaling of this reactive transport process by means of volume averaging. The resulting (upscaled) continuum-scale system of equations includes nonlocal integro-differential terms and the effective parameters embedded in the model are quantified directly through computed pore-scale fluid velocity and pore space geometry attributes. The results obtained through this predictive model formulation are then compared against those provided by available effective continuum models which require calibration through parameter estimation. Our analysis considers two models recently proposed in the literature which are designed to embed incomplete mixing arising from the presence of fast reactions under advection-dominated transport conditions. We show that best estimates of the parameters of these two models heavily depend on the type of data employed for model calibration. Our upscaled nonlocal formulation enables us to reproduce most of the critical features observed through pore-scale simulation without any model calibration. As such, our results clearly show that embedding into a continuum-scale model the information content associated with pore-scale geometrical features and fluid velocity yields improved interpretation of typically available continuum-scale transport observations.

SUMMARY The aim of this paper is to propose an effective anisotropic mesh adaptation procedure for the solution of the shallow water equations. The hyperbolic partial differential equation system is solved via the streamline diffusion FEM, suitably modified by a shock‐capturing correction. The proposed adaptation procedure relies on a recovery‐based error estimator. In particular, we look for an anisotropic error estimator able to select not only the size but also the shape and the orientation of the mesh elements, with the aim of optimizing the computational advantages yielded by a standard isotropic mesh adaptation strategy. The robustness of the proposed estimator is assessed when either a single physical quantity, meaningful for the problem at hand, or a combination of all the shallow water system components drives the adaptation procedure. For this purpose, steady shallow water problems as well as unsteady configurations are considered. Copyright © 2011 John Wiley & Sons, Ltd.