We have developed a new theory of flow statistics in heterogeneous media that uncovers a surprising universality in flow statistics in disordered media. This universality renders the theory highly predictive, as the core features are determined by universal features of the underlying network topology, and the remaining parameters can be directly related to simple quantities with clear physical meaning, such as the porosity and the mean pore throat width and its variance. This theory has significant potential to serve as a build block for the description of flow statistics, and later transport in reaction models, in different media under different conditions.
Regarding reaction in partially-saturated media, so far we have mainly studied surface reaction in two-dimensional media with time-independent flows. Surprisingly, we have found that linear reactions are virtually insensitive to the saturation degree, despite markedly different flow patterns. We interpret this in the context of previous work based on the theory of stochastic processes under restart, where we showed that surface reaction rates are limited by low velocities in the shear layer near no-slip boundaries, and that a strong impact of mixing is limited to chaotic conditions. This appears to remain true even under partial saturation, where the simultaneous presence of air and water leads to complex patterns of preferential paths and recirculation regions characterized by low velocities. Our numerical results are based on Eulerian flow simulations and Lagrangian particle tracking simulations for transport. These simulations were performed based on frozen fluid and solid phase configurations obtained from images of air-water co-injection experiments in quasi-two dimensional media, where the solid phase was composed of disordered cylindrical pillars with varying degrees of radius variability and spatial correlation. In addition to surface reactions, we are currently closely involved in the development of a reactive, partially-saturated, heterogeneous medium in the laboratory, in order to obtain new experimental data on the role of partial saturation on fluid-fluid reaction rates and front patterns. Theoretical advances on the understanding of flow distributions have also led to advances in our understanding of dispersion processes and the coupling between surface reactions and transport in saturated media.
Regarding numerical techniques, we have developed a new simulation approach for transport and mixing in heterogeneous media. This approach is based on a hyperbolic formulation that combines benefits from Eulerian and Lagrangian techniques. This deterministic framework for describing mixing and scalar transport in fluid flows captures dispersion without relying on stochastic calculus or Monte Carlo sampling. This means, for example, that like in Lagrangian methods strong advection does not pose stability issues, while at the same time solute concentrations can be computed with high accuracy at specified locations like in Eulerian methods, without requiring large numbers of sample trajectories. The approach reproduces classical mixing results and extends them to complex flow configurations, offering a promising alternative for modeling mixing across scales. The underlying framework has been published in the Journal of Fluid Mechanics and presented at various conferences.
We have also developed a particle tracking algorithm meant to be used in conjunction with the popular computational fluid dynamics software OpenFOAM. This tool, developed in the C++ language, implements advection, diffusion, and reaction process. In particular, it includes a new implementation of transport in time-dependent, two-phase flows. This implementation is based on a chemical potential formulation adapted from previous developments for phase field models. The particle tracking algorithm has also been employed to validate a new theory for the scaling laws and mechanisms governing dispersion in saturated porous media.
