Flow between two sites on a percolation cluster

We study the flow of fluid in porous media in dimensions d=2 and 3. The med ium is modeled by bond percolation on a lattice of L-d sites, while the flo w front is modeled by tracer particles driven by a pressure difference betw een two fixed sites (''wells'') separated by Euclidean distance r. We inves tigate the distribution function of the shortest path connecting the two si tes, 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 f or d=2 and 3. Further, we study two dynamical quantities: (i) the minimal t raveling 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 probabi lity p. We find that the scaling form for the distribution functions for th ese 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 paramete rs are the fractal dimension, the power law exponent, and the constants and exponents that characterize the exponential cutoff functions.