Novel approaches for upscaling linear advection diffusion reaction processes in porous media

Matteo Icardi (University of Nottingham)

Thursday 31st January, 2019 14:00-15:00 Maths 311B


Although the macroscopic limit of linear advection-diffusion equations has been well understood since the early developments of porous media transport theory (through, for example, homogenisation, and volume averaging), its extension to complex flow regimes is still an open question, even in presence of well-separated spatial scales. This is due to the presence of non-trivial microscopic equilibrium configurations (compared to the trivial constant leading order solution obtained by standard periodic homogenisation), or of dynamic equilibrium configurations. For example, when dealing with fast surface reactions, high microscopic gradients can develop locally. Similarly, a conservative solute in the neighbourhood of a concentration source (injection) undergoes a dynamic evolution of the local microscopic configuration in time and space before reaching the asymptotic self-similar profile. These are only two examples where the classical upscaling approaches fail, and effective macroscopic equations are often found either empirically or by resorting to generic random walk models.
In this talk, we present some (old and) new theoretical frameworks to overcome these limitations without the need of an ad-hoc calibration or stochastic particle models. These generally involve solving more auxiliary local problems, and often computing local spectral properties (eigenvalues and eigenfunctions) of the underlying transport operators. In particular we will focus on model-order reduction approach based on projections from the full microscale formulation onto low-dimensional spaces, borrowing concepts from the finite element community (such as Hi-Mod, MsFEM, and VMM), and from statistical mechanics. New macroscopic equations are obtained for one or more macroscopic quantities, that can be seen as extended homogenised formulations. Applications of these techniques to the classical 2D Taylor dispersion in a channel and other heterogeneous periodic flows will be discussed.

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