Fractals

Concepts and Applications in Geosciences

Behzad Ghanbarian, Allen G. Hunt

• Publisher: CRC Press
• Year: 2017
• ISBN:
  9781351648301
• Edition: 1
• Topic: Matematica
• Subtopic: Geometria

This book provides theoretical concepts and applications of fractals and multifractals to a broad range of audiences from various scientific communities, such as petroleum, chemical, civil and environmental engineering, atmospheric research, and hydrology. In the first chapter, we introduce fractals and multifractals from physics and math viewpoints. We then discuss theory and practical applications in detail. In what follows, in chapter 2, fragmentation process is modeled using fractals. Fragmentation is the breaking of aggregates into smaller pieces or fragments, a typical phenomenon in nature. In chapter 3, the advantages and disadvantages of two- and three-phase fractal models are discussed in detail. These two kinds of approach have been widely applied in the literature to model different characteristics of natural phenomena. In chapter 4, two- and three-phase fractal techniques are used to develop capillary pressure curve models, which characterize pore-size distribution of porous media. Percolation theory provides a theoretical framework to model flow and transport in disordered networks and systems. Therefore, following chapter 4, in chapter 5 the fractal basis of percolation theory and its applications in surface and subsurface hydrology are discussed. In chapter 6, fracture networks are shown to be modeled using fractal approaches. Chapter 7 provides different applications of fractals and multifractals to petrophysics and relevant area in petroleum engineering. In chapter 8, we introduce the practical advantages of fractals and multifractals in geostatistics at large scales, which have broad applications in stochastic hydrology and hydrogeology. Multifractals have been also widely applied to model atmospheric characteristics, such as precipitation, temperature, and cloud shape. In chapter 9, these kinds of properties are addressed using multifractals. At watershed scales, river networks have been shown to follow fractal behavior. Therefore, the applications of fractals are addressed in chapter 10. Time series analysis has been under investigations for several decades in physics, hydrology, atmospheric research, civil engineering, and water resources. In chapter 11, we therefore, provide fractal, multifractal, multifractal detrended fluctuation analyses, which can be used to study temporal characterization of a phenomenon, such as flow discharge at a specific location of a river. Chapter 12 addresses signals and again time series using a novel fractal Fourier analysis. In chapter 13, we discuss constructal theory, which has a perspective opposite to fractal theories, and is based on optimizationof diffusive exchange. In the case of river drainages, for example, the constructal approach begins at the divide and generates headwater streams first, rather than starting from the fundamental drainage pattern.

	CHAPTER
	1

An Introduction to Multifractals and Scale Symmetry Groups

Daniel Schertzer* and Ioulia Tchiguirinskaia

Hydrology Meteorology and Complexity (HMCo), Ecole des Ponts ParisTech, U. Paris-Est, 6–8 av. B. Pascal, Cité Descartes, 77455 Marne-la-Vallée, France

1. Introduction and Motivations

The complexity of geosciences has inspired the development of many innovative concepts and techniques. For instance, the extreme variability of geophysical fields over wide ranges of space-time scales, particularly their intermittency, has been a key driver towards the development of multifractals. In turn, this development has significantly improved our understanding and modeling capacities of our complex environment as a whole, i.e. not only its physico-ecological component from its smallest scales (e.g. micro-turbulence) to its largest scales (e.g. climate (Lovejoy and Schertzer 2013), astrophysics (Sylos Labini et al. 1998)), but also its socio-economical component (e.g. city dynamics (Murcio et al. 2015, Dupuy 2016)). This introduction intends to demonstrate that recent theoretical developments on multifractals, particularly on their formalism, will make them even more useful and indispensable for geophysics.

In general, a multifractal is not a geometric set, but a map from a space-time domain X onto a codomain $\tilde{X}$ that is a vector space or a manifold, displaying structures at all scales. Furthermore, a multifractal is not necessarily a pointwise function, but is often a (mathematical) measure or a generalized function. In fact, multifractal formalisms have made manageable, by physicists and engineers, mathematical objects like stochastic multi-singular measures, which were earlier mathematical curiosities. Multifractals are indeed very broad generalizations of the geometrical fractals whose field is a set indicator function with a binary codomain $\tilde{X}=\{0,1\}$, therefore a kind of degenerate case for multifractal fields.

The property to have structures at all scales is trivially scale invariant since it does not depend on the scale of observation. This points out that a multifractal field can also be defined as being invariant under a given scale transform, which defines a symmetry of this field. A precise definition of this invariance and of the corresponding scale transform, which could either be deterministic or stochastic (e.g. involves only equality in probability distribution or other statistical equivalences), isotropic or not, will be given below. However, the above rather intuitive definition already shows that multifractals are not only quite general, but also quite fundamental. Indeed, symmetry principles are the building blocks of physics and of many other disciplines (Weyl 1952, Zee 1986). Scale symmetry is in fact an element of the extended Galilean invariance. Unfortunately, the maximum attention in mechanics, especially in point mechanics, has been given to the space shifts between two (Galilean) frameworks that differ only by a constant relative velocity or by a given rotation that defines the pure Galilean group. But with continuous mechanics, it has broadened to other transforms such as scale dilations. Following the Buckingam Π-theorem (Buckingham 1914, Buckingham 1915, Sonin 2004), the key role of the scale symmetry in fluid mechanics was demonstrated by Sedov (1972), using many applications. More recently, Galilean invariance was used to assess subgrid models in turbulence (...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Dedication
• Table of Contents
• Preface
• 1. An Introduction to Multifractals and Scale Symmetry Groups
• 2. Fractal Capillary Pressure Curve Models
• 3. Two- and Three-phase Fractal Models of Porous Media: A Multiscale Approach
• 4. The Mathematics of Complexity in the Study of Soil and Granular-porous Media
• 5. The Fractals of Percolation Theory in the Geosciences
• 6. Nonlocal Models for Transport in Fractal Media
• 7. Multifractals and Geostatistics
• 8. Why the Warming Can’t be Natural: The Nonlinear Geophysics of Climate Closure
• 9. Fractals and Multifractals in Geophysical Time Series
• 10. Multi-fractal Random Walk and Its Application in Petro-physical Quantities
• 11. Combining Fractals and Multifractals to Model Geoscience Records
• 12. Use of Constructal Theory in Modeling in the Geosciences
• Index