Mixing in Heterogeneous Media Across Spatial and Temporal Scales: From Local Non-Equilibrium to Anomalous Chemical Transport and Dynamic Uncertainty

Natural and engineered media are in general heterogeneous across scales ranging from the pore to regional and global scales. Spatial heterogeneity and temporal fluctuations are key players for the understanding of the dynamics of flow, transport and reaction processes from pore to regional scale.

The heterogeneity impact on transport has traditionally been quantified in terms of dispersion coefficients. From pore to Darcy-scale, this refers to hydrodynamic dispersion, from Darcy to regional scale, macrodispersion. Dispersion quantifies the impact of sub-scale velocity fluctuations on solute transport, in analogy to Brownian motion, in which micro scale thermal fluctuations of particle velocities are quantified through a diffusion coefficient. However, it has been ubiquitously found that the dispersion concept in general does not provide a realistic description for transport, mixing and reaction processes in heterogeneous media.

The phenomena and processes related to spatial heterogeneity can be seen in the context of mixing.

  • Mixing may be described as a process that homogenizes a heterogeneous distribution, blends different waters, decreases maximum concentrations and reduces the entropy of a concentration distribution. In heterogeneous media, mixing is driven by flow heterogeneity (see Figure 1), which can be seen as analogous to stirring action. In general, it cannot be described by a constant dispersion coefficient.
  • Transport descriptions based on advection-dispersion mechanism require the existence of a well-mixed support volume, on which concentration is characterized by a unique value (physical equilibrium). Physical equilibrium is attained through mixing on time scales that depend on the local scale mass transfer mechanism and may be characterized through the competition of advection, diffusion and dispersion. Thus the mixing scale typically evolves in time and does not necessarily coincide with the relevant support scale, as illustrated in Figure 1. This gives rise to anomalous transport behaviors that are characterized by history-dependent dynamics and multivaluedness of concentration at the support scale.

Figure 1: Concentration distribution in a steady heterogeneous flow (after Dentz and de Barros, J. Fluid. Mech., 2015).

The predictive modeling of mixing, transport and reaction requires identifying and quantifying the heterogeneity controls on the corresponding large scale processes with the aim of arriving at predictive process models that can be related to the (statistical) medium and flow characteristics.