This essential reference for graduate students and researchers provides a unified treatment of earthquakes and faulting as two aspects of brittle tectonics at different timescales. The intimate connection between the two is manifested in their scaling laws and populations, which evolve from fracture growth and interactions between fractures. The connection between faults and the seismicity generated is governed by the rate and state dependent friction laws - producing distinctive seismic styles of faulting and a gamut of earthquake phenomena including aftershocks, afterslip, earthquake triggering, and slow slip events. The third edition of this classic treatise presents a wealth of new topics and new observations. These include slow earthquake phenomena; friction of phyllosilicates, and at high sliding velocities; fault structures; relative roles of strong and seismogenic versus weak and creeping faults; dynamic triggering of earthquakes; oceanic earthquakes; megathrust earthquakes in subduction zones; deep earthquakes; and new observations of earthquake precursory phenomena.

The Mechanics of Earthquakes and Faulting

Estimation of land surface temperature-vegetation abundance relationship for urban heat island studies

https://www.math.utah.edu/~epshteyn/selectedpapers.html/paperHou1.pdf

A Multiscale Finite Element Method for Elliptic Problems in Composite Materials and Porous Media

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.

/pdf/self-organized-criticality-4v87dkoa15.pdf

Self-organized criticality

Characterizing flow and transport in fractured geological media: A review

Fractal Distributions in Geology, Scale Invariance, and Deterministic Chaos (D.L. Turcotte, J. Huang). Some Long Run Properties of Geophysical Records (B.B. Mandelbrot, J.R. Wallis). Some Remarks on the Numerical Estimation of Fractal Dimension (S.A. Pruess). Measuring the Dimension of Selfaffine Fractals: The Example of Rough Surfaces (S.R. Brown). A Review of the Fractal Character of Natural Fault Surfaces with Implications for Friction and the Evolution of Fault Zones (W.L. Power, T.E. Tullis). Fractals and Ocean Floor Topography: A Review and a Model (A. Malinverno). Fractal Transitions on Geological Surfaces (C.H. Scholz). Fractal Scaling of Fracture Networks (C.C. Barton). Fractal Fragmentation in Crustal Shear Zones (C.G. Sammis, S.J. Steacy). Fractal Distribution of Fault Length and Offsets: Implications on Brittle Deformation Evaluation: The Lorraine Coal Basin (T. Villemin et al.). Fractal Dynamics of Earthquakes (P. Bak, K. Chen). Mineral Crystallinity in Igneous Rocks: Fractal Method (A.D. Fowler). Fractal Structure of Electrum Dendrites in Bonanza Epithermal AuAg Deposits (J.A. Saunders, P.A. Schoenly). Appendix. Index.

Fractals in the Earth Sciences

Fractal characterization of fracture surfaces in concrete

Correlation between fractal dimensions and fracture properties in cementitious material is reported. Wedge splitting tests of specimens as much as 1.52m (5 ft) long and having 7.62cm (3-in.) maximum size aggregate were first performed to determine fracture toughness \IK\dI\dc\N and fracture energy \IG\dF\N. Subsequently, one of the split parts was mapped using a profilometer to provide detailed one-dimensional profiles. Finally, the fractal dimension of the profiles was determined by a specially developed computer program. The fracture surfaces are fractal over the measured range of scales, and the fractal dimension is independent of crack trajectory. There is a weak correlation between the fracture properties and the fractal dimensions. The implications of the fractal nature of the cracked surfaces on the fracture energy \IG\dF\N are discussed, as is the relationship between a fractal analysis and the size effect law.

Fractals, Fractures, and Size Effects in Concrete

Fracture traces exposed on three 214- to 260-m{sup 2} pavements in the same Miocene ash-flow tuff at Yucca Mountain, southwestern Nevada, have been mapped at a scale of 1:50. The maps are two-dimensional sections through the three-dimensional network of strata-bound fractures. All fractures with trace lengths greater than 0.20 m were mapped. The distribution of fracture-trace lengths is log-normal. The fractures do not exhibit well-defined sets based on orientation. Since fractal characterization of such complex fracture-trace networks may prove useful for modeling fracture flow and mechanical responses of fractured rock, an analysis of each of the three maps was done to test whether such networks are fractal. These networks proved to be fractal and the fractal dimensions (D) are tightly clustered (1.12, 1.14, 1.16) for three laterally separated pavements, even though visually the fracture networks appear quite different. The fractal analysis also indicates that the network patterns are scale independent over two orders of magnitude for trace lengths ranging from 0.20 to 25 m. 7 refs., 7 figs.

Fractal Geometry of Two-Dimensional Fracture Networks at Yucca Mountain, Southwestern Nevada

The editors of this book chose a diverse spectrum of papers written by pioneers in the field of fractals and their application to the exploration and production of hydrocarbons. The geology of the Earth`s crust is complex, chaotic, and unpredictable. Fractal geometry can quantify the spatial heterogeneity of the different geologic patterns and ultimately help improve the results of both production and exploration. To this goal the book has accomplished such an objective with diverse, well-chosen contributions from a variety of experts in the field. The book starts with a chapter introducing the basics, with a short historical foot-note by Benoit Mandelbrot, who is considered the {open_quotes}father of fractals.{close_quotes} Mandelbrot emphasized that geologic processes not only exhibit fractal properties but also are strongly connected to the economic system. This paved the way for the next three chapters that deal with the size and spatial distribution of hydrocarbon reserves and their importance in economic evaluations. The following four chapters deal with the fractal processes as related to sedimentologic, stratigraphic, and geomorphologic systems. Chapter five is an interesting one that deals with stratigraphic models and how their fractal processes can be tied with the inter-well correlation and reconstruct depositional environments. The nextmore » three chapters are concerned with porous and fractured rocks and how they affect the flow of fluids. The last two chapters (chapters 13 and 14) are of particular interest. Chapter 13 deals with the vertical vs. horizontal well-log variability and application to fractal reservoir modeling. Chapter 14 illustrates how fractal geometry brings mathematical order to geological and geophysical disorder. This is evident when dealing with geophysical modeling and inversion.« less

Christopher C. Barton

Papers

Fractals in the Earth Sciences

Fractal characterization of fracture surfaces in concrete

Fractals, Fractures, and Size Effects in Concrete

Fractal Geometry of Two-Dimensional Fracture Networks at Yucca Mountain, Southwestern Nevada

Fractals in Petroleum Geology and Earth Processes