The distribution of local fluxes in porous media

We study the distributions of channel openings, local fluxes, and velocities in a two-dimensional random medium of non-overlapping disks. We present theoretical arguments supported by numerical data of high precision and find scaling laws as function of the porosity. For the channel openings we observe a crossover to a highly correlated regime at small porosities. The distribution of velocities through these channels scales with the square of the porosity. The fluxes turn out to be the convolution of velocity and channel width corrected by a geometrical factor. Furthermore, while the distribution of velocities follows a Gaussian, the fluxes are distributed according to a stretched exponential with exponent 1/2. Finally, our scaling analysis allows to express the tortuosity and pore shape factors from the Kozeny-Carman equation as direct average properties from microscopic quantities related to the geometry as well as the flow through the disordered porous medium.