The concept of self-organized criticality was introduced to explain the behaviour of the sandpile model. In this model, particles are randomly dropped onto a square grid of boxes. When a box accumulates four particles they are redistributed to the four adjacent boxes or lost off the edge of the grid. Redistributions can lead to further instabilities with the possibility of more particles being lost from the grid, contributing to the size of each ‘avalanche’. These model ‘avalanches’ satisfied a power-law frequency‐area distribution with a slope near unity. Other cellular-automata models, including the slider-block and forest-fire models, are also said to exhibit self-organized critical behaviour. It has been argued that earthquakes, landslides, forest fires, and species extinctions are examples of self-organized criticality in nature. In addition, wars and stock market crashes have been associated with this behaviour. The forest-fire model is particularly interesting in terms of its relation to the critical-point behaviour of the sitepercolation model. In the basic forest-fire model, trees are randomly planted on a grid of points. Periodically in time, sparks are randomly dropped on the grid. If a spark drops on a tree, that tree and adjacent trees burn in a model fire. The fires are the ‘avalanches’ and they are found to satisfy power-law frequency‐area distributions with slopes near unity. This forest-fire model is closely related to the site-percolation model, that exhibits critical behaviour. In the forest-fire model there is an inverse cascade of trees from small clusters to large clusters, trees are lost primarily from model fires that destroy the largest clusters. This quasi steady-state cascade gives a power-law frequency‐area distribution for both clusters of trees and smaller fires. The site-percolation model is equivalent to the forest-fire model without fires. In this case there is a transient cascade of trees from small to large clusters and a power-law distribution is found only at a critical density of trees.

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Self-organized criticality

In this paper, theoretical and experimental approaches to flow, hydrodynamic dispersion, and miscible and immiscible displacement processes in reservoir rocks are reviewed and discussed. Both macroscopically homogeneous and heterogeneous rocks are considered. The latter are characterized by large-scale spatial variations and correlations in their effective properties and include rocks that may be characterized by several distinct degrees of porosity, a well-known example of which is a fractured rock with two degrees of porosity---those of the pores and of the fractures. First, the diagenetic processes that give rise to the present reservoir rocks are discussed and a few geometrical models of such processes are described. Then, measurement and characterization of important properties, such as pore-size distribution, pore-space topology, and pore surface roughness, and morphological properties of fracture networks are discussed. It is shown that fractal and percolation concepts play important roles in the characterization of rocks, from the smallest length scale at the pore level to the largest length scales at the fracture and fault scales. Next, various structural models of homogeneous and heterogeneous rock are discussed, and theoretical and computer simulation approaches to flow, dispersion, and displacement in such systems are reviewed. Two different modeling approaches to these phenomena are compared. The first approach is based on the classical equations of transport supplemented with constitutive equations describing the transport and other important coefficients and parameters. These are called the continuum models. The second approach is based on network models of pore space and fractured rocks; it models the phenomena at the smallest scale, a pore or fracture, and then employs large-scale simulation and modern concepts of the statistical physics of disordered systems, such as scaling and universality, to obtain the macroscopic properties of the system. The fundamental roles of the interconnectivity of the rock and its wetting properties in dispersion and two-phase flows, and those of microscopic and macroscopic heterogeneities in miscible displacements are emphasized. Two important conceptual advances for modeling fractured rocks and studying flow phenomena in porous media are also discussed. The first, based on cellular automata, can in principle be used for computing macroscopic properties of flow phenomena in any porous medium, regardless of the complexity of its structure. The second, simulated annealing, borrowed from optimization processes and the statistical mechanics of spin glasses, is used for finding the optimum structure of a fractured reservoir that honors a limited amount of experimental data.

Flow phenomena in rocks : from continuum models to fractals, percolation, cellular automata, and simulated annealing

心臓病診療プラクティス 16. 心不全を癒す

From the Publisher:
Why do similar patterns and forms appear in nature in settings that seem to bear no relation to one another? The Windblown ripples of desert sand follow a sinuous course that resembles the stripes of a zebra or a marine fish In the trellis-like shells of microscopic sea creatures we see the same geometry as in the bubble walls of foam, Forks of lightning mirror the branches of a river network or a tree
This book explains why these are not coincidences Nature commonly weaves its tapestry without any master plan or blueprint Instead, these designs build themselves by self-organization The interactions between the component parts -- whether they be grains of sand, molecules or living cells -- give rise to spontaneous patterns that are at the same time complex and beautiful Many of these patterns are universal, recurring again and again in the natural order: spirals, spots, stripes, branches, honeycombs Philip Ball conducts a profusely illustrated tour of this gallery, and reveals the secrets of how nature's patterns are made

The Self-Made Tapestry: Pattern Formation in Nature

Granular materials are ubiquitous in our daily lives. While they have been the subject of intensive engineering research for centuries, in the last two decades granular matter has attracted significant attention from physicists. Yet despite major efforts by many groups, the theoretical description of granular systems remains largely a plethora of different, often contradictory concepts and approaches. Various theoretical models have emerged for describing the onset of collective behavior and pattern formation in granular matter. This review surveys a number of situations in which nontrivial patterns emerge in granular systems, elucidates important distinctions between these phenomena and similar ones occurring in continuum fluids, and describes general principles and models of pattern formation in complex systems that have been successfully applied to granular systems.

/pdf/patterns-and-collective-behavior-in-granular-media-2t8b6sr2ja.pdf

Patterns and collective behavior in granular media: Theoretical concepts

THE idea of self-organized criticality1(SOC) is commonly illustrated conceptually with avalanches in a pile of sand grains. The grains are dropped onto a pile one by one, and the pile ultimately reaches a stationary 'critical' state in which its slope fluctuates about a constant angle of repose, with each new grain being capable of inducing an avalanche on any of the relevant size scales. Some numerical models of sand-pile dynamics do show SOC1–8, hut the behaviour of real sand piles remains ambiguous9–18. Here we describe experiments on a granular system—a pile of rice—in which the dynamics exhibit self-organized critical behaviour in one case (for grains with a large aspect ratio) but not in another (for less elongated grains). These results show that SOC is not as 'universal' and insensitive to the details of a system as was initially supposed1, but that instead its occurrence depends on the detailed mechanism of energy dissipation.

Avalanche dynamics in a pile of rice

We have studied experimentally transport properties in a slowly driven granular system which recently was shown to display self-organized criticality [Frette et al., Nature (London) 379, 49 (1996)]. Tracer particles were added to a pile and their transit times measured. The distribution of transit times is a constant with a crossover to a decaying power law. The average transport velocity decreases with system size. This is due to an increase in the active zone depth with system size. The relaxation processes generate coherently moving regions of grains mixed with convection. This picture is supported by considering transport in a 1D cellular automaton modeling the experiment.

http://www.cmth.ph.ic.ac.uk/people/k.christensen/papers/published/prl77_1996.pdf

Tracer Dispersion in a Self-Organized Critical System.

The slow upward migration, due to buoyancy, of a nonwetting fluid through a saturated, three-dimensional porous medium has been studied experimentally, and by computer simulations. The height h of the invading fluid structure is proportional to the mass M of the invading fluid. The linear mass density S=M/h is found to scale with the dimensionless Bond number Bo (ratio of buoyancy and capillary forces) as S=A|Bo| -a , with a=0.72±0.06, consistent with theory and simulations. A noisy behavior in h(M) results from dynamical processes such as snapoffs and bursts

Buoyancy-driven fluid migration in porous media

We have studied experimentally the formation and evolution of sand ripples under water shear flow in a narrow annular channel. The amplitude of the sediment ripples vanishes at a critical value of the shear rate. Near this onset, a band of Fourier modes can be described by exponential growth functions. The growth rates vary linearly with the shear rate. On longer times the sand ripples display complex dynamics.

/pdf/sand-ripples-induced-by-water-shear-flow-in-an-annular-1faql30w6b.pdf

Vidar Frette

Papers

Avalanche dynamics in a pile of rice

Tracer Dispersion in a Self-Organized Critical System.

Buoyancy-driven fluid migration in porous media

Sand Ripples Induced by Water Shear Flow in an Annular Channel

Onset of smoldering in cotton: Effects of density