Dependence of conductance on percolation backbone mass

We study , the average conductance of the backbone, defined by two points separated by Euclidean distance r, of mass M(B) on two-dimensional percolation clusters at the percolation threshold. We find that with increasing M(B) and for fixed r, asymptotically decreases to a constant, in contrast with the behavior of homogeneous systems and nonrandom fractals (such as the Sierpinski gasket) in which conductance increases with increasing M(B). We explain this behavior by studying the distribution of shortest paths between the two points on clusters with a given M(B). We also study the dependence of conductance on M(B) above the percolation threshold and find that (i) slightly above p(c), the conductance first decreases and then increases with increasing M(B) and (ii) further above p(c), the conductance increases monotonically for all values of M(B), as is the case for homogeneous systems.

Flow between two sites on a percolation cluster

We study the flow of fluid in porous media in dimensions d=2 and 3. The medium is modeled by bond percolation on a lattice of L(d) sites, while the flow front is modeled by tracer particles driven by a pressure difference between two fixed sites ("wells") separated by Euclidean distance r. We investigate the distribution function of the shortest path connecting the two sites, and propose a scaling ansatz that accounts for the dependence of this distribution (i) on the size of the system L and (ii) on the bond occupancy probability p. We confirm by extensive simulations that the ansatz holds for d=2 and 3. Further, we study two dynamical quantities: (i) the minimal traveling time of a tracer particle between the wells when the total flux is constant and (ii) the minimal traveling time when the pressure difference is constant. A scaling ansatz for these dynamical quantities also includes the effect of finite system size L and off-critical bond occupation probability p. We find that the scaling form for the distribution functions for these dynamical quantities for d=2 and 3 is similar to that for the shortest path, but with different critical exponents. Our results include estimates for all parameters that characterize the scaling form for the shortest path and the minimal traveling time in two and three dimensions; these parameters are the fractal dimension, the power law exponent, and the constants and exponents that characterize the exponential cutoff functions.