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

Abstract

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 artifacts, 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.

Evolutive Models for the Geometry and Heat Conductivity of an Intumescent EVA-ATH Composite during Its Thermal Degradation

“…In particular, the method for the characterization of the gaseous inclusions shape ( Appendix A ) is original. The tensorial distance ( Appendix B ) used to assess the quality of the conductivity predictions is also a new contribution, mentioned in the literature only in the recent related works [ 26 , 40 ].…”

“…Their choice is not obvious, and it can affect the predicted effective tensor. A detailed study of the impact of various kinds of boundary conditions on the numerical solution was conducted and presented in [ 26 ]. In short words, in the practical range of local conductivity contrast, the predictions based on the periodicity conditions applied for the results presented hereafter cannot differ by more than 2% from those that would result from other reasonable choices of boundary condition in [S1].…”

“…Therefore, we measure the difference of and by the distance and its normalized dimensionless counterpart defined by Note that is a distance in the mathematical sense, i.e., a symmetric positive-definite function satisfying the triangle inequality. It was introduced in [ 26 , 40 ] where its properties are detailed. is the maximal relative deviation (combining magnitude and direction differences) of the fluxes predicted by and when applied to the same gradient.…”

On the influence of boundary conditions when determining transport coefficients from digital images of heterogeneous media.

“…The same question was addressed in the recent contribution [1], where four tomographic images of materials of various natures were used for a thorough assessment of the BC's influence. The present paper proceeds along the same lines and also considers synthetic media, which makes a systematic investigation of geometrical parameters possible and allows to formulate the phenomenological observations of [1] in a more general and quantitative way. In order to be reasonably self-contained, many definitions of mathematical and conceptual tools are repeated here but some elements are skipped for concision.…”

“…Sensitivity to boundary conditions (BC's) is a well known issue when determining macroscopic transport coefficients by numerical upscaling, i.e., by solving governing equations in a finite domain where the required fields of the local properties are provided, either for instance by X-ray tomographic imaging for core-scale samples [1], or from geophysical characterization techniques on the field-scale [2]. In the first case, with a full knowledge of the microscopic geometry, the upscaling of Stokes flow equations for the determination of a permeability tensor can be addressed (Stokes → Darcy), as well as any conduction or diffusion transport processus.…”