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.PACS numbers: 47.55. Mh, 47.15.Gf Fluid flow through a porous medium is of importance in many practical situations ranging from oil recovery to chemical reactors and has been studied experimentally and theoretically for a long time [1,2]. Due to disorder, porous media display many interesting properties that are however difficult to handle even numerically. One important feature is the presence of heterogeneities in the flux intensities due the varying channel widths. They are crucial to understand stagnation, filtering, dispersion and tracer diffusion. These are subjects of much practical interest in medicine, chemical engineering and geology and on which a vast literature is available [3].Many stochastic models for disordered porous media have been proposed and used to describe the above mentioned effects. One of the most successful is the so-called q-model for force distributions in random packings [4] in which a scalar fluid is transfered downwards from layer to layer. Although the distribution of local flux intensities should be the basis for any quantitative evolution of these stochastic models, detailed studies of them at the pore level are still lacking.The traditional approach for the investigation of singlephase fluid flow at low Reynolds number in disordered porous media is to characterize the system in terms of Darcy's law [1,3], which assumes that a macroscopic index, the permeability K, relates the average fluid velocity V through the pores with the pressure drop ∆P measured across the system,where L is the length of the sample in the flow direction and µ is the viscosity of the fluid. In fact, the permeability reflects the complex interplay between porous structure and fluid flow, where local aspects of the pore space * Formerly at Institute for Computer Physics, University of Stuttgart.morphology and the relevant mechanisms of momentum transfer should be adequately considered. In previous studies [5,6,7,8,9,10,11], computational simulations based on detailed models of pore geometry and fluid flow have been used to predict permeability ...