Percolation through Voids around Toroidal Inclusions

TL;DR

This study investigates the percolation transition in fluid flow through voids around randomly placed impermeable toroidal grains, revealing how topology and grain shape influence critical porosity for flow connectivity.

Contribution

It introduces a novel dynamical infiltration method to analyze continuum percolation involving non-convex toroidal grains, a first in the field.

Findings

Critical grain concentration depends on torus geometry and orientation.

Topological transition occurs with the development of a central hole in tori.

Critical porosity converges to cylindrical values for high aspect ratio, randomly oriented grains.

Abstract

In the case of media comprised of impermeable particles, fluid flows through voids around impenetrable grains. For sufficiently low concentrations of the latter, spaces around grains join to allow transport on macroscopic scales, whereas greater impenetrable inclusion densities disrupt void networks and block macroscopic fluid flow. A critical grain concentration $\rho_{c}$ marks the percolation transition or phase boundary separating these two regimes. With a dynamical infiltration technique in which virtual tracer particles explore void spaces, we calculate critical grain concentrations for randomly placed interpenetrating impermeable toroidal inclusions; the latter consist of surfaces of revolution with circular and square cross sections. In this manner, we study for the first time continuum percolation transitions involving non-convex grains. As the radius of revolution increases relative to the length scale of the torus cross section, the tori develop a central hole, a topological transition accompanied by a cusp in the critical porosity for percolation. With a further increase in the radius of revolution, as constituent grains become more ring-like in appearance, we find that the critical porosity converges to that of high aspect ratio cylindrical counterparts only for randomly oriented grains.