A Closer Look: High-Resolution Pore-Scale Simulations of Solute Transport and Mixing Through Porous Media Columns

Abstract

Mixing is pivotal to conservative and reactive transport behaviors in porous media. Methods for investigating mixing processes include mathematical models, laboratory experiments and numerical simulations. The latter have been historically limited by the extreme computational resources needed for solving flow and transport at the microscopic scale within the complex pore structure of a three-dimensional porous medium, while dealing with a sufficiently large domain in order to generate meaningful emergent continuum-scale observables. We present the results of such a set of virtual column experiments, which have been conducted by taking advantage of modern high-performance computing infrastructure and Computational Fluid Dynamics software capable of massively parallel simulations. The computational approach has important advantages such as full control over the experimental conditions as well as high spatial and temporal resolution of measurements. Hydrodynamic dispersion results agree with the empirical and theoretical literature and link dispersivity to median grain size, while elucidating the impact of grain size variability on the critical Péclet number. Reactive transport results also indicate that the relative degree of incomplete mixing is related to the granular material’s mean hydraulic radius and not directly to the median grain size. When compared to a well-known laboratory experiment with similar configuration, less incomplete mixing is observed in our simulations. We offer a partial explanation for this discrepancy, by showing how an apparent nonlinear absorbance–concentration relationship may induce laboratory measurement error in the presence of local concentration fluctuations.

Article Highlights

• High-resolution numerical simulation experiments were conducted to study fluid-fluid mixing in porous (granular) media.

• Results unravel the roles of Péclet number and grain size variability on emergent conservative and reactive transport.

• Non-linear light attenuation and local concentration fluctuations could cause measurement errors in physical experiments.

Appendix: Estimating Pore Size Distributions

To estimate the distributions of inscribed pore diameters, we use an arbitrary cubic sub-sample of each medium, with dimensions $(9d_0)^3$. We first discretize the space in equal cubic voxels of size $d_0/60$ and compute, for each fluid phase voxel, its associated minimum Eulerian distance to any solid phase voxel. The map containing such values is usually referred to as Eulerian Distance transform or EDT. We then follow the morphology analysis principles of Silin and Patzek (2006), who define an individual pore as being composed of a master voxel (which holds the local maximum EDT value) and a set of connected slave voxels. A voxel with associated value ${\mathrm {EDT}}_{\mathrm{S}}$ is a slave of another voxel with ${\mathrm {EDT}}_{\mathrm {M}}$ if: (i) ${\mathrm {EDT}}_{\mathrm {M}}\ge {\mathrm {EDT}}_{\mathrm{S}}$ and (ii) $\Delta _{\mathrm {MS}}\le {\mathrm {EDT}}_{\mathrm {M}}+{\mathrm {EDT}}_{\mathrm{S}}$, where $\Delta _{\mathrm {MS}}$ is their mutual distance. A hierarchy can then be defined such that “my slave’s slave is also my slave.” The voxels that remain “unslaved” are the pores’ master voxels, and their respective ${\mathrm {EDT}}_{\rm {M}}$ values correspond to each pore’s maximum inscribed radius. Finally, the empirical distributions are fit to a generalized extreme value model, which agrees very well with the empirical data (see Fig. 2b) showing the best indicators, for both the Bayesian and Akaike information criteria, among a wide variety of tested distributions. The empirically determined pore size distribution in the monodisperse case appears to correctly reflect the geometrical limit $\phi /d_0\ge 0.22$, which corresponds to a close tetrahedral stack of four equal spheres. The comparison between Fig. 2a and Figure 2b demonstrates that the three log-normal grain size distributions, built with different mean and variance to have matching hydraulic radius, yield pore size distributions with equal mode and a smaller difference in variance. In other words, these results suggest a close relationship between the mean hydraulic radius and the typical pore size, for granular media of the kind considered here.