Flow through porous media at the percolation transition

We study low-Reynolds-number fluid flow through a two-dimensional porous medium modeled as a Lorentz gas. Using extensive finite-element simulations we fully resolve the flow fields for packing fractions approaching the percolation threshold. Near the percolation transition, we find a power-law scaling of the flow rate versus the packing fraction with an exponent of <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mrow><a:mo>≈</a:mo><a:mn>5</a:mn><a:mo>/</a:mo><a:mn>2</a:mn></a:mrow></a:math>, which was predicted earlier by mapping the macroscopic flow to a discrete flow network [B. I. Halperin , ]. Importantly, we observe a rounding of the scaling behavior at small system sizes, which can be rationalized via a finite-size scaling ansatz. Finally, we show that the distribution of the kinetic energy exhibits a power-law scaling over several decades at small energies, originating from collections of self-similar, viscous eddies in the dead-end channels. Our results lay the foundation for unraveling critical behavior of complex fluids omnipresent in biological and geophysical systems.