Evolution of pore-scale concentration PDFs and estimation of transverse dispersion from numerical porous media column experiments

Knowing local concentration distributions is important for transport and mixing, particularly in porous media, yet a comprehensive understanding of them remains a challenge. Computing advancements have enabled high-resolution pore-scale simulations, offering an unprecedented opportunity for in-depth investigation of mixing. In this study we use simulation data to examine concentration distributions at the pore scale in the context of longitudinal (pseudo-one-dimensional) solute transport through a porous column. These distributions arise in a single column from heterogeneous flow at the pore-scale, which gets averaged out when upscaled and are not with reference to statistics across multiple random realizations. To measure these distributions, we first devise a semi-analytical approach to estimate the mean effective transport velocity profile for a non-uniform Darcy-scale fluid velocity, which unavoidably occurs due to the presence of lateral boundaries. This development allows sampling micro-scale concentrations over a moving surface that possesses a well defined Darcy-scale mean concentration, enabling empirical computation of the local concentration distribution. As an added benefit we find that our approach allows for the estimation of transverse dispersion coefficients, which is not typical in traditional column experiments. The implemented approach can estimate it via inverse modeling, and it agrees closely with previously published experimental data across the range of Peclet numbers we studied. We found that the measured pore-scale concentration probability density functions are best represented by a beta distribution, thus validating this longstanding hypothesis with direct evidence. Furthermore, we propose a model to describe the temporal and spatial evolution of the local concentration pdf, as well as its Péclet number dependence.