Fractional Calculus for Hydrology, Soil Science and Geomechanics

An Introduction to Applications

Ninghu Su

• Publisher: CRC Press
• Year: 2020
• ISBN:
  9781351032407
• Edition: 1
• Topic: Physical Sciences
• Subtopic: Geology & Earth Sciences

This book is an unique integrated treatise, on the concepts of fractional calculus as models with applications in hydrology, soil science and geomechanics. The models are primarily fractional partial differential equations (fPDEs), and in limited cases, fractional differential equations (fDEs). It develops and applies relevant fPDEs and fDEs mainly to water flow and solute transport in porous media and overland, and in some cases, to concurrent flow and energy transfer. It is an integrated resource with theory and applications for those interested in hydrology, hydraulics and fluid mechanics. The self-contained book summaries the fundamentals for porous media and essential mathematics with extensive references supporting the development of the model and applications.

Chapter 1
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Application of Fractional Calculus in Water Flow and Related Processes

This chapter begins by introducing the readers to the significance of water, which is one central topic of this book. Understanding of water is the source of knowledge, wisdom, pursuits leading to the rigorous formulation of mathematic models, and, eventually, fractional partial differential equations (fPDEs), which are central topics of this book.

It overviews the objectives of this book and stresses the systematicefforts to present the fundamentals, and models and methodologies used for investigating water flow and solute transport in porous media such as soils and aquifers. It deliberately aims to integrate models and methodologies based on new approaches with fractional calculus without the invisible boundaries which divide the traditional fields.

1. Overview

Water and its movement on land, in soils and aquifers, and in the oceans support terrestrial and marine life on the Earth. Since the very ancient times, human settlements had to either be in the vicinity of viable water sources or build water storage facilities for their survival. In the first instance, the appearance of fresh water sources, in the form of rivers or springs, was essential for humans’ survival and, notably, all the earliest civilizations were facilitated by the availability of fresh water such as along the river systems of Nile in Egypt, Tigris and Euphrates in Mesopotamia, Indus in India, and Huanghe (or Dahe, the Yellow River) in China In the second situation wherein humans lived in dry areas or needed water for special purposes, water engineering works can be traced back to 3,200 BC (Biswas 1967) across the ancient civilizations.

In addition to meeting human survival and basic needs as an essential resource and commodity, water has been the object of spiritual, mythical, mythological, religious and philosophical activities. The earliest Greek philosopher Thales (624 BC–548 BC) (Cartledge 1998) is credited with the hypothesis that water was the underlying factor behind the development of the world. The great Chinese philosopher Laozi (or Laotzu) (~ 571 BC–~ 471 BC) reiterated the virtue of water for humans with the saying, “the upmost kindness of a man is like water, being the most modest and gracious like water which nourishes all things without conflicts, ends up in the lowest positions and provides services without demand for a reward”.

To understand water and its various properties has always been of constant interest to mankind Over several millennia, scientific communities have explored water as a central topic for various purposes and utilities with Archimedes’ principle (287 BC–212 BC) as one early example. Truesdell (1953) and Darrigol (2005) documented in great detail the evolution of the discipline of hydrodynamics, dealing with the motion of ideal fluids as a highly hypothetical form of water, from Isaac Newton (1643–1727), Daniel Bernoulli (1700–1782) and Leonhard Euler (1707–1783) to George G. Stokes (1819–1903) and Burnett (1935, 1936).

The topic of water waves alone has attracted significant investigations, particularly from the 18th century (Stoker 1958) when many leading scientists and mathematicians joined the race to understand water waves, particularly, Joseph-Louis Lagrange (1736–1813), Seméon D. Poisson (1781-1840), Claude-Louis Navier (1785–1836), Augustin-Louis Cauchy (1789-1857), A.J.C. Barré de Saint-Venant (1797–1886), George G. Stokes (1819–1903), William Thomson (Lord Kelvin, 1824–1907), Joseph Valentin Boussinesq (1842–1929), Horace Lamb (1849–1934) and Jules Henri Poincaré (1854–1912) to list a few.

After centuries of developments by scientists, particularly physicists and mathematicians (Debler 1990), a set of general equations for fluid flow was finalized and named after two important contributors—C.L. Navier and G.G. Stokes. The Navier-Stokes equations (NSEs), discussed in Chapter 3, are fundamental equations that govern the flow of fluids, including that of water in porous media. The NSEs accommodate the viscoelasticity of the porous media, or poroviscoelasticity, and the compressibility of water to the extent of water movement in soils and aquifers.

Just like other fields of human knowledge, the pursuit of better solutions is never-ending. The high-order hydrodynamics of flow, known as Burnett hydrodynamics (Burnett 1935, 1936), is an example capable of explaining more physical mechanisms when the NSEs cease to be valid. The processes in which the high-order hydrodynamics work while the NSEs fail include phenomena such as absorption and dispersion of sound in fluids, dynamics of swarms of particles, structure of different profiles in shock waves at large Mach numbers, Couette flows, in continuum transition flows that appear around space vehicles, and flows in micro-channels (García-Colín et al. 2008).

Classic hydrodynamics, evolving since the 17th century, takes largely from mathematics as it deals with imaginary ideal fluids which are frictionless. Its application by experimentalists to real fluids creates the applied field of hydraulics. The empirical nature of hydraulics is limited in scope to water only. With the development of and interests in other forms of fluids in aeronautics, petroleum engineering and other areas in civil engineering, a broader field of study was developed fluid mechanics. Fluid mechanics has three branches: fluid statics which is concerned with the mechanics of fluids at rest, kinematics which deals with velocities and streamlines without considering forces or energy, and fluid dynamics for the study of relations between velocities, acceleration and the forces exerted by or upon fluids in motion (Daugherty et al. 1989).

The quest for knowledge about water has created many applied fields in the modern classification of scientific and engineering disciplines. In sciences, water-related fields include fluid mechanics, hydraulics, hydrology, hydrodynamics, meteorology, oceanography, marine science, agricultural science and soil science with water as a key element. On the other hand, water-related fields in engineering comprise hydraulic engineering, irrigation and drainage engineering, marine and coastal engineering, etc.

Water has been a topic for extensive publications in various formats, and the myriad properties and aspects of water find mention in many monographs of hydrology and hydraulics and their sub-disciplines such as groundwater hydrology/ hydraulics, surface water hydrology and soil hydrology.

Deformation is another aspect of soils and aquifers, for their physical properties have significant impact on civil engineering infrastructures and geological materials as well as on the environment. In particular, soil mechanics or geomechanics deals with the swelling properties of soil when its water content changes and the reciprocal changes in water pressure as a result of deformation in soil. Many reports in soil mechanics (geomechanics) can be found in parallel with publications in hydrology and hydraulics. Cauchy in 1822 and 1827 laid the foundation of the general theory of elasticity and its extension to mathematical physics (Love 1892). Cauchy’s work was followed by Green in 1837 and de Saint-Venant in 1844, with six components of stress and strains investigated (Love 1892). The reports on the subsidence of geological strata first conceptualized by Pratt and Johnson (1926) in oilfields, and by Geertsma (1966) and Verruijt (1969) in aquifers resulting from the extraction of groundwater are regarded as the early examples of investigation in applied areas of geoscience. The term poroelasticity first used by Geertsma (1966) is also important for addressing this specific property of porous media (Wang 2000) in civil and petroleum engineering.

Many reports in hydrology, hydraulics and soil science dealing with soils and aquifers generally ignore the key issues of deformation and stress-strain relations, leaving such discourse for soil mechanics The separation of soil mechanics and soil physics since the 1930s (Philip 1974) discouraged hydrology and soil science to integrate with geomechanics, making these fields apparently disconnected even though groundwater hydrologists and soil scientists deal with poroviscoelastic soils and aquifers—central topics in soil mechanics

A weak bridge between hydrology and geomechanics was attempted in limited literature such as the works of Wang (2000), which addressed the linear poroelasticity of porous media, covering a range of issues—geomechanics (soil mechanics), hydrogeology (groundwater hydrology), petroleum engineering, the poroelasticity theory and applications based on the works of Terzaghi (1923) and Biot (1935, 1939, 1941, 1956a, b), and Biot’s thermoelasticity (Biot 1941, 1956c). However, Wang’s work does not discuss any aspect of the NSEs which govern the flow of water in poroelastic and thermoelastic media, thus eliminating key hydrological elements.

In terms of quantitative methods, integer partial differential equations (PDEs) have enjoyed success as the central mathematical models in hydrology, soil science, hydraulics and geomechanics, etc., for over a century since Darcy (1856) embarked on the use of differential equations (DEs) for describing water flow in porous media and Boussinesq (1904) presented PDEs for groundwater flow in unconfined aquifers.

The decade around 1990 was a turning point when fractional PDEs (fPDEs) appeared as better models, with more information about environmental processes (Lenormand 1992, Zaslaysky 1992, Compte 1997). Environmental processes such as solute transport, sediment transport and groundwater flow, etc., have been shown to be better modelled with fPDEs by Compte (1997); groundwater flow/seepage by He (1998); and solute transport in groundwater by Lenormand (1992) and Benson (1998). These developments were part of the evolution since 1974 when fractional calculus was re-launched, and monographs appeared in applied mathematics and other fields of science (Oldham and Spanier 1974, Samko et al. 1987, 1993, Miller and Ross 1993, Podlubny 1999, Kilbas et al. 2006, Hilfer 2000, Mainardi 2010, Herrmann 2011, Atanacković et al. 2014, Atangana 2018).

2. Objectives of This Book

In order to eliminate the invisible boundaries between hydrology, hydraulics, soil science and geomechanics (soil mechanics), and to address the inconsistent spectrum of mathematical models based on fPDEs in these fields, this book aims to systematically present key concepts, theories, quantitative methods and ideas centered on the application of fractional calculus in hydrology, soil science, flow in porous media and geomechanics (soil mechanics). This book aims to establish frameworks of mathematical models with concepts in fractional calculus, particularly fPDEs and fractional integral equations (flEs), and stochastic methods such as the continuous-time random walks (CTRW) theory to water flow, solute transport and related processes.

Water flow on land is categorized as overlandflow, water movement in unsaturated soils as flow in unsaturated soils, and that in saturated aquifers as groundwater flow (generally classified into two types—confined and unconfined aquifers). With the aforementioned goals and classification in mind, this book presents the following materials:

1. Fundamentals of mathematics in Chapter 2, dealing with concepts commonly used in fractional calculus for models and quantitative methods in hydrology and hydraulics of water flow, solute transport on land, in soils and aquifers, and soil mechanics;

2. Essential properties of soils and aquifers in the context of porous media, in Chapter 3;

3. An overview of the historical transition from quantitative methods based on integer PDEs to fractional calculus-based approach, in Chapter 4, and

4. The remaining Chapters present topics related to water flow and solute transport in unsaturated soils (Chapters 5, 6 and 7), overlandflow (Chapter 8), in saturated aquifers (Chapters 9 and 10), and geomechanics (Chapter 11).

The topics in this book are central issues in hydrology, soil science and geomechanics, and the fundamentals, models and methodologies used for investigating water-related processes can be categorized in three parts, namely, fundamentals, traditional methods for quantification and evolving approaches with fractional calculus.

3. A Brief Description of Key Concepts

Soils and aquifers are ...

Table of contents

• Cover
• Title Page
• Copyright Page
• Foreword
• Preface
• Table of Contents
• 1. Application of Fractional Calculus in Water Flow and Related Processes
• 2. Mathematical Preliminaries
• 3. Essential Properties of Soils and Aquifers as Porous Media
• 4. Transition from Classical Diffusion to Anomalous Diffusion—The Evolution of Concepts and Ideas
• 5. Fractional Partial Differential Equations for Water Movement in Soils
• 6. Applications of Fractional Partial Differential Equations to Infiltration and Water Movement in Soils
• 7. Fractional Partial Differential Equations for Solute Transport in Soils
• 8. Hydraulics of Anomalous Flow on Hillslopes, in Catchment Networks and Irrigated Fields
• 9. Fractional Partial Differential Equations for Groundwater Flow
• 10. Fractional Partial Differential Equations for Solute Transport in Groundwater
• 11. Fractional Partial Differential Equations and their Applications to Poroviscoelastic Media and Geomechanics—Re-familiarization and Introduction for Hydrologists and Soil Scientists
• Bibliography
• Index