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Random Lattices with Fractal Bond Permeabilities

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Abstract

The macroscopic permeability of random lattices has been studied when the permeability of each link is a power law of its length with an exponent . When they are sufficiently long, the link lengths are shown to follow exponential laws which depend on the density. The macroscopic permeability is studied as a function of ; it is compared to a modified effective medium theory (EMT).

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Preface to the Second Edition Preface to the First Edition Introduction: Forest Fires, Fractal Oil Fields, and Diffusion What is percolation? Forest fires Oil fields and fractals Diffusion in disordered media Coming attractions Further reading Cluster Numbers The truth about percolation Exact solution in one dimension Small clusters and animals in d dimensions Exact solution for the Bethe lattice Towards a scaling solution for cluster numbers Scaling assumptions for cluster numbers Numerical tests Cluster numbers away from Pc Further reading Cluster Structure Is the cluster perimeter a real perimeter? Cluster radius and fractal dimension Another view on scaling The infinite cluster at the threshold Further reading Finite-size Scaling and the Renormalization Group Finite-size scaling Small cell renormalization Scaling revisited Large cell and Monte Carlo renormalization Connection to geometry Further reading Conductivity and Related Properties Conductivity of random resistor networks Internal structure of the infinite cluster Multitude of fractal dimensions on the incipient infinite cluster Multifractals Fractal models Renormalization group for internal cluster structure Continuum percolation, Swiss-cheese models and broad distributions Elastic networks Further reading Walks, Dynamics and Quantum Effects Ants in the labyrinth Probability distributions Fractons and superlocalization Hulls and external accessible perimeters Diffusion fronts Invasion percolation Further reading Application to Thermal Phase Transitions Statistical physics and the Ising model Dilute magnets at low temperatures History of droplet descriptions for fluids Droplet definition for the Ising model in zero field The trouble with Kertesz Applications Dilute magnets at finite temperatures Spin glasses Further reading Summary Numerical Techniques
Experimental study of miscible displacement fronts in rough self-affine fractures The shear strength of rock joints in theory and practice, Rock Mech
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Auradou, H., Hulin, J.-P. and Roux, S.: 2001, Experimental study of miscible displacement fronts in rough self-affine fractures, Phys. Rev. E. 63, 066306. Barton, N. and Choubey, V.: 1970, The shear strength of rock joints in theory and practice, Rock Mech. 10, 1–34.