Power law for the permeability in a two-dimensional disordered porous medium

We report numerical calculations of the permeability as a function of porosity for a two-dimensional disordered porous medium. This medium is modeled using the well known “Swiss Cheese” model. The fluid is simulated using a cellular automata algorithm. We find that for relatively high porosities the permeability decays exponentially with the density of obstacles. As a consequence of this exponential behavior, a power-law dependence of the permeability as a function of the porosity is obtained, for this model system. We find that the power-law exponent is given by the ratio between two characteristic scales. One scale is given by the inverse of the area of one obstacle, and is approximately equal to the density of obstacles necessary to reach the percolation threshold. The other scale is equal to the average change in the density of obstacles necessary for the permeability to be reduced to about 1/e of its original value.