Effects of Peclet number on pore-scale mixing and channeling of a tracer and on directional advective porosity

Field tracer tests indicate that advective porosity, the quantity relating advective velocity to Darcy flux, may exhibit directional dependence. Theory (S. P. Neuman, 2005) suggests that directional variations in advective porosity could arise from incomplete mixing of an inert tracer between directional flow channels within a sampling volume of soil or rock that may be hydraulically isotropic or anisotropic, rendering advective porosity a function of Peclet number. Whereas Neuman was able to quantify the process for idealized intersecting fracture sets, his theory is too abstract to allow doing so for more complex systems of fractures or pores. We use Smoothed Particle Hydrodynamics (e.g., A. M. Tartakovsky and P. Meakin, 2005, 2006) to simulate advective-diffusive transport of a passive tracer in the interior of an idealized pore space between impermeable circular or elliptical solid grains of random size, arranged randomly within a square domain. Our numerical experiments reveal a transition from complete mixing of tracer at small Peclet numbers to the gradual evolution of pore-scale channels of tracer which allow less and less mixing as Peclet numbers grow. Advective porosity, computed as the ratio between total water flux across the domain boundaries and migration rate of the tracer center of mass, is found to diminish asymptotically from a value close to the total porosity at small Peclet numbers to a constant dominated entirely by advection at large Peclet numbers. When total porosity is kept constant, channeling is less pronounced and mixing more prominent when flow takes place parallel to the longer axes of elliptical grains as compared to when the grains are circular. On the other hand, channeling is more pronounced and mixing less so when flow takes place perpendicular to the longer elliptical grain axes. Correspondingly, the rate at which advective porosity diminishes with increasing Peclet number is slower and the asymptote is larger when flow takes place parallel to the longer axes of elliptical grains than when the grains are circular. However, the rate is faster and the asymptote smaller when flow takes place perpendicular to the longer elliptical grain axes.