A Network Model for Diffusion in Media with Partially Resolvable Pore Space Characteristics

A microstructure-informed meso-scale model for diffusion of foreign species in porous media is proposed. The model is intended for media where the pore geometry data acquired experimentally represent a fraction of total porosity. A cellular complex, with a cell representing the average pore neighbourhood, is used to generate 3D graphs of sites at cell centres and bonds between neighbouring cells. The novel interpretation of pore systems as graphs allows for clear separation between topology (here connectedness) and physics (here diffusion) in the mathematical formulation of transport. Further, it allows for easy introduction of dynamics into the system, i.e., local changes in topology due to other physical mechanisms, such as micro-cracking or blockage of pores. A mapping between microstructure features and graph elements is used for model construction. The mapping is based on data for clays, where the experimentally resolved pore system comprises isolated elongated pores of preferred orientation with a large volume fraction of unresolved pores. Both the resolved and the hidden systems are accounted for. The graph geometry is described by a principal length, the cell size in the preferred orientation, and a secondary length, the cell size out of preferred orientation. This is considered as a representation of mineralogical heterogeneity of clays. Analysis on graphs, a specialisation of the discrete exterior calculus, is used to obtain connectivity and diffusivity properties of formed networks. Since the experimental data are not sufficient to determine the principal length, upper and lower limits are determined from the limited information. Effects of the principal cell size between limits and of the secondary cell size are studied. The results are within the range of experimentally measured macroscopic (bulk) diffusivity for the material studied, including anisotropic diffusion coefficients. The variation of calculated diffusivity coefficients with principal and secondary lengths provides an explanation for the variability in experimentally measured coefficients across different clays.