  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 pressure drop with an exponent of $\approx 5/2$, which has
been predicted earlier by mapping the macroscopic flow to a discrete flow
network [Phys. Rev. Lett. 54, 1985]. 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.
