Heterogeneous porous media as multiscale structures for maximum flow access

Abstract

Natural porous structures are heterogeneous with multiple scales that are distributed nonuniformly. Few large pores (fissures, channels, and cracks) are accompanied by numerous finer channels. Can this type of flow architecture be attributed to a principle of maximization of global flow access? Features similar to those of multiscale porous structures are exhibited by tree-shaped flow structures. Trees have been deduced from the maximization of flow access between a point and a volume, a point and an area, and a point and a curve (e.g., circle). In this paper we invoke the same principle and consider fundamentally the question of how to bathe with minimal flow resistance a microchannel structure that globally behaves as a porous medium. We develop completely multiscale configurations that guide the flow from one side of the porous structure to the other (line to line and plane to plane) and show analytically the advantages of tree structures over the usual stacks of parallel microchannels. The “porous medium” that has tree-shaped labyrinths is heterogeneous, with multiple scales that are distributed nonuniformly. These features justify comparisons with the design of natural porous structures.