A Full Empirical Description of Mixing and Dilution - The Fingerprint of Non-Fickian Mixing

Abstract:

Transport in porous media is frequently simulated via particle methods. These methods formulate transport as a stochastic process of particle position increments. The validity of the formulation is scale dependent. At larger scales, the current particle motion strongly depends on the previous motion. This prohibits the commonly made assumption of independent and normally distributed increments to represent dispersion at larger scales. The dependence is caused by pore-space heterogeneities like fast-flow pore throats or dead-end pores. These characteristics trigger the non-Fickian transport behavior and control the kinetics and scaling of mixing and dilution.

Based on micro-CT imaging of the pore geometry, we generate two highly realistic velocity fields by solving the Stokes equation at pore scale for two Peclet regimes. Then, a PTRW-based transport simulation is performed to represent transport thoroughly only by advection and diffusion. A modified Pollock algorithm is used to obtain the particle trajectories. Thus, we can simulate and analyze very extensively the complex (non-Fickian) mixing and dilution processes the pore scale.

We analyze the temporal evolution of particle pairs. Under non-Fickian conditions, the variance of the two-particle displacements grows non-linearly with time. We also focus on the higher-order moments of the evolving empirical probability distribution as a function of travel time and initial separation. The focus on the high order moments results in a fingerprint of non-Fickian mixing and dilution that includes the full statistics and scale dependence.

While the probability distribution of separation for initially non-separated particles describes dilution, the probability of zero-separation of initially separated particles describes mixing. Further, the probability over time for a first-time encounter of particles at zero separation is a measure for the potential reactivity of the system for mixing-limited bimolecular reactions.