Flow Between Two Sites on a Percolation Cluster

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 {\it 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 {\it Ansatz} holds for $d=2$ and 3, and calculate the relevant scaling parameters. We also study two dynamical quantities: the minimal traveling time of a tracer particle between the wells and the length of the path corresponding to the minimal traveling time ``fastest path'', which is not identical to the shortest path. A scaling {\it 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. The scaling form is represented as the product of a power law and three exponential cutoff functions. We summarize our results in a table which contains estimates for all parameters which characterize the scaling form for the shortest path and the minimal traveling time in 2 and 3 dimensions; these parameters are the fractal dimension, the power law exponent, and the constants and exponents that characterize the exponential cutoff functions.

31 pages, 12 Figures

References in corpus (5)

• Conductivity Exponent and Backbone Dimension in 2-d Percolation
• Scaling of the distribution of shortest paths in percolation
• Exact critical exponent for the shortest-path scaling function in percolation
• Multifractal Properties of the Random Resistor Network
• Dependence of Conductance on Percolation Backbone Mass

Cited by in corpus (14)

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• Single-Phase Flow of Non-Newtonian Fluids in Porous Media
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• A Hamiltonian approach for explosive percolation
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• Unstable supercritical discontinuous percolation transitions
• Phase Diagram of Optimal Paths
• Tomographic-like reconstruction of the percolation cluster as a phase transition
• Possible Connection between the Optimal Path and Flow in Percolation Clusters
• Stochastic Renormalization Group in Percolation: I. Fluctuations and Crossover
• Multiple Invaded Consolidating Materials
• Fractal Behavior of the Shortest Path Between Two Lines in Percolation Systems
• Discrete Scale Invariance in Supercritical Percolation
• Oil displacement through a porous medium with a temperature gradient