Article,
Flow between two sites on a percolation cluster
PHYSICAL REVIEW E, 62 (6, B): 8270-8281 (2000)
DOI: 10.1103/PhysRevE.62.8270
Abstract
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.
