Adapting particle methods to model the dynamics of concentration gradients and chemical reactivity under advective diffusive transport conditions
Résumé
Particle methods are frequently used to model transport processes in porous and fractured media. When transport by advection largely dominates dispersion and diffusion processes, particle methods offer a relevant alternative to Eulerian methods. Even though recent developments in Eulerian methods have reduced numerical diffusion issues, particle methods remain broadly appropriate to low dispersion conditions. Particles efficiently adapt to flow structures and to local diffusive and dispersive conditions. They can integrate exchanges between high and low flow zones with potentially sharp interfaces (e.g. fracture/matrice) as well as chemical interactions with minerals and biofilms. Because particles naturally resolve the multi-scale diversity of the transport conditions, they are frequently used to model processes that emerge from their collective behavior. It is the case of the upscaled dispersion also called macro-dispersion that results from the differential influence of deterministic velocity correlations and stochastic dispersive/diffusive processes. Particle methods become more involved when the process of interest is driven by local interactions between particles as it is typically the case for reactions between solute species (homogeneous reactions). As reactivity is often nonlinearly sensitive to solute concentrations, particle methods have progressively evolved from random walk methods where independent particles are tracked following a Fokker-Planck equation to meshfree methods with interacting particles carrying concentration properties. Such methods provide continuous estimates of the concentration field that can be coupled numerically to chemical reactivity like any other Eulerian transport method. Particle methods are relevant to model the chemical control of reactivity expressed in terms of concentrations but not the physical control expressed in terms of concentration gradients. In fact, reactivity is physically controlled by the diffusive mixing of solutes of different chemical concentrations, as mixing two waters with equilibrated solute concentrations generally results in an out-of-equilibrium solution. High concentration gradients thus promote diffusion, mixing and reactivity. Particle methods are optimized to model concentration fields and not concentration gradients. Whether independent or interacting, particles become sparser in the medium because of spreading and dilution following the overall decrease of concentrations. Particles move apart and, after some time, can no longer resolve the spatial variations of the velocity field, challenging any approximation of concentration gradients. While particle reseeding is commonly used for concentrations, we propose here a complementary strategy based on solving directly for concentration gradients instead of deriving concentration gradients from any approximation of the concentration field. We develop this strategy on the classical advection-diffusion equation. From the transport equation of concentrations, we derive the transport equation of concentration gradients and show how both differ. We propose adapted particle methods to solve the concentration gradients and compare their performances on concentration gradients. We finally show how these methods can be combined to compute reactivity rates when controlled both by concentrations and concentration gradients.
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