Fractals

An Introduction to Multifractals and Scale Symmetry Groups
Daniel Schertzer, and Ioulia Tchiguirinskaia

Fractal Capillary Pressure Curve Models
Behzad Ghanbarian and Humberto Millán

Two- and three- phase fractal models of porous media: A multiscale approach
Edith Perrier

The Mathematics of Complexity in the study of soil and granular-porous media
Miguel Ángel Martín


The Fractals of Percolation Theory in the Geosciences
A. G. Hunt, and F. Yu

Nonlocal Models for Transport in Fractal Media
Daniel O'Malley, and John H. Cushman

Multifractals and Geostatistics
A.M. Tarquis, Juan J. Martín-Sotoca, M.C. Morató, M.T. Castellanos, J. Borondo, N.R. Bird, and A. Saa-Requejo

Why the warming can't be natural: the nonlinear geophysics of climate closure
S. Lovejoy

Fractals and Multifractals in Geophysical Time Series
Mikhail I Bogachev, Naiming Yuan, and Armin Bunde

Multi-fractal Random Walk and its Application in Petro-physical Quantities
Gholamreza Jafari, Soheil Vasheghani Farahani, and Zahra Koohi Lai

Combining Fractals and Multifractals to Model Geoscience Records
Carlos E. Puente, Mahesh L. Maskey, and Bellie Sivakumar

Use of Constructal Theory in Modeling in the Geosciences
A. Hunt

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.