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.

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.

Order from chaos is simultaneously a mantra of physics and a reality in biology. Physicist Norman Packard suggested that life developed and thrives at the edge of chaos. Questions remain, however, as to how much practical knowledge of biology can be traced to existing physical principles, and how much physics has to change in order to address the complexity of biology. Phil Anderson, a physics Nobel laureate, contributed to popularizing a new notion of the end of ""reductionism."" In this view, it is necessary to abandon the quest of reducing complex behavior to known physical results, and to identify emergent behaviors and principles. In the present book, however, we have sought physical rules that can underlie the behavior of biota as well as the geochemistry of soil development. We looked for fundamental principles, such as the dominance of water flow paths with the least cumulative resistance, that could maintain their relevance across a wide range of spatial and temporal scales, together with the appropriate description of solute transport associated with such flow paths. Thus, ultimately, we address both nutrient and water transport limitations of processes from chemical weathering to vascular plant growth. The physical principles guiding our effort are established in different, but related concepts and fields of research, so that in fact our book applies reductionist techniques guided by analogy. The fact that fundamental traits extend across biotic and abiotic processes, i.e., the same fluid flow rate is relevant to both, but that distinctions in topology of the connected paths lead to dramatic differences in growth rates, helps unite the study of these nominally different disciplines of geochemistry and geobiology within the same framework. It has been our goal in writing this book to share the excitement of learning, and one of the most exciting portions to us has been the ability to bring some order to the question of the extent to which soils can facilitate plant growth, and what limitations on plant sizes, metabolism, occurrence, and correlations can be formulated thereby. While we bring order to the soil constraints on growth , we also generate some uncertainties in the scaling relationships of plant growth and metabolism. Although we have made an first attempt to incorporate edaphic constraints into allometric scaling, this is but an initial foray into the forest.

Order from chaos is simultaneously a mantra of physics and a reality in biology. Physicist Norman Packard suggested that life developed and thrives at the edge of chaos. Questions remain, however, as to how much practical knowledge of biology can be traced to existing physical principles, and how much physics has to change in order to address the complexity of biology. Phil Anderson, a physics Nobel laureate, contributed to popularizing a new notion of the end of "reductionism." In this view, it is necessary to abandon the quest of reducing complex behavior to known physical results, and to identify emergent behaviors and principles. In the present book, however, we have sought physical rules that can underlie the behavior of biota as well as the geochemistry of soil development. We looked for fundamental principles, such as the dominance of water flow paths with the least cumulative resistance, that could maintain their relevance across a wide range of spatial and temporal scales, together with the appropriate description of solute transport associated with such flow paths. Thus, ultimately, we address both nutrient and water transport limitations of processes from chemical weathering to vascular plant growth. The physical principles guiding our effort are established in different, but related concepts and fields of research, so that in fact our book applies reductionist techniques guided by analogy. The fact that fundamental traits extend across biotic and abiotic processes, i.e., the same fluid flow rate is relevant to both, but that distinctions in topology of the connected paths lead to dramatic differences in growth rates, helps unite the study of these nominally different disciplines of geochemistry and geobiology within the same framework. It has been our goal in writing this book to share the excitement of learning, and one of the most exciting portions to us has been the ability to bring some order to the question of the extent to which soils can facilitate plant growth, and what limitations on plant sizes, metabolism, occurrence, and correlations can be formulated thereby. While we bring order to the soil constraints on growth , we also generate some uncertainties in the scaling relationships of plant growth and metabolism. Although we have made an first attempt to incorporate edaphic constraints into allometric scaling, this is but an initial foray into the forest.

Percolation theory describes the effects of the connectivity of microscopic or small-scale elements of a complex medium to its macroscopic or large-scale properties. It also describes the conditions under which there may be a continuously connected path of local elements across the medium. The point at which the path is formed is called the percolation threshold. Percolation theory also predicts that many macroscopic properties of complex media follow universal power laws near the percolation threshold that are independent of many microscopic features of such media. There are many applications of percolation theory across the natural sciences, from porous materials, to composite solids, complex networks, and biological systems. This book presents the essential elements of percolation theory, covers the problem of calculating the exponents that characterize the power laws that the percolation quantities follow near the percolation threshold, provides a clear description of the geometry of percolation clusters of the connected paths, and addresses several variations of percolation theory. In particular, bootstrap percolation, explosive percolation, and invasion percolation are featured, which expand the range of natural systems to which percolation may be applicable. In addition, coverage includes several important applications of percolation theory to a range of phenomena, ranging from electrical conductivity, thermopower, the Hall effect, and photoconductivity of disordered semiconductors, to flow, transport and reaction in porous media, geochemistry, biology, and ecology.