On the Influence of Boundary Conditions when Determining Transport Coefficients from Finite Samples of Porous Media: Assessment for Tomographic Images of Real Materials

摘要

The increasing access to 3d digital images of porous media provides an ideal avenue for the determination of their transport properties, by solving the governing equations in their actual microscale geometry and evaluating the tensor coefficient that relates the mean flux and driving gradient. However, the first and puzzling question along the way is the choice of the conditions to be imposed for this resolution at the boundaries of the sample. This methodological issue is explored here with the purpose of quantifying the influence of the boundary conditions (BC) in relation with the parameters of the system (porosity, characteristic length scale of the microstructure, ratio of the phase conductivities), assessing the level of confidence associated with the predictions, devising criteria to anticipate the risk of serious artefacts, and if possible proposing ways to limit them. Although the terminology of thermal transfer is used, the developments apply to the upscaling of any transport property governed by a diffusion equation, including thermal or electrical conduction, mass diffusion or Darcy flow. Quantitative indicators are introduced for a rigorous individual or comparative assessment of conductivity tensors, and they are used in the analysis of the results of extensive calculations based on four tomographic images of various kinds of porous materials, with a broad range of conductivity contrasts, and various kinds of BC's. Ultimately, practical criteria are proposed for the a priori and a posteriori detection of at-risk situations, and a self-diagnosing protocol is proposed to screen out the influence of the BC's, when this is possible.