This book provides a complete and self-contained overview of the theoretical aspects and applications of fractional calculus-based models in soil physics and hydrology, as well as poroelastic properties of porous media. With the comprehensive and clear evidence of the practical implications of fractional calculus and the benefits of this theory for the current doctrine, this book represents a remarkable piece of literature that will certainly be a fundamental reference for those interested in theoretical studies and applications of fractional calculus in hydrology. , earth science and related subjects. This book focuses on the development of calculus-based mathematical models and their applications in hydrology, soil science and fluid mechanics in porous media.

Hydrology, soil science, flow mechanics in porous media, and other branches of geoscience and environmental science are among the fields where various mathematical models based on fractional calculus are widely used. Currently, the reality is that extensive reports on the development of fractional calculus and its applications in the form of fractional partial differential equations (fPDE) and fractional integral equations (fIE) in these fields appear in many journals, some scattered. in a limited number of books. Throughout the length of the book, I have also cited more than 900 references listed in the bibliography, which include both important historical and contemporary contributions to the field of fractional calculus and related topics.

To this end, I hope that the contents of this book will be of interest to senior undergraduates and postgraduates for their studies and useful to scientists, engineers and practitioners interested in the background, theory and applications of fractional calculus-based models in related fields. I am extremely grateful to Professor Francesco Mainardi of the University of Bologna, Italy, who was awarded the Mittag-Leffler Prize at FDA'14 held in Catania, Italy, for his pioneering contribution to the application of the Mittag-Leffler function in problems of fractional calculus .

Kozeny, Carman and Other Early Models for Hydraulic 64 Conductivity

Recent Modified Fractal Kozeny-Carman Models for 65 Hydraulic Conductivity

Fractal Relationships between Water Content and 66 Matrix Potential

Scales of Interest: Space-Time Scales in Hydrology, Soil 74 Science and Porous Media Flow

Experimental and Physical Investigation of Anomalous 90 Processes

Explanation and Justification for Anomalous Water Flow and 91 Solute Transport in Porous Media

## Theory, Models and Parameters for Water Flow and Solute 92 Transport in Porous Media

Connections between the Scaling Parameters, Fractals and the 99 Orders of Fractional Derivatives

## Relationships and Differences between Anomalous Diffusion and 103 Scale-dependent and Time-dependent Transport Processes

## Variable-order Fractional Derivatives and Related fPDEs 108

More Generic fPDEs of Functional Order 110

Fractional Partial Differential Equations for Water Movement in Soils 113

Mass-time and Space-time FPDEs for Anomalous Water 123 Flow in Soils

Distributed-order fPDEs for Water Flow in Soils with 126 Mobile and Immobile Zones

Exchange of Water between Mobile and Immobile Zones 141

## Equations of Infiltration Derived from Fractional Calculus 145 with the Concentration Boundary Condition

Water Exchange at the Interface when the Initial Water 163 Content is Known

Example of Solutions for Water Movement in a Soil of Finite Depth 165

## Fractional Partial Differential Equations for Solute Transport in Soils 168

Non-equilibrium Solute Transport in Non-swelling 170 Soils with Immobile Pores

Relationship between Flux and Residential Solute 171 Concentrations

Concurrent Water Flow and Solute Transport in Swelling Soils 173 Contents xiii

## Fractional Partial Differential Equations for Anomalous Solute 175 Transport in Soils

## The fPDE and Its Solution for Solute Exchange between 186 Mobile and Immobile Zones

The fPDE for Solute Exchange between Mobile and 186 Immobile Zones

Exact Solution for Solute Exchange at the Interface when 187 the Initial Rate of Change in Solute Concentration is Known

Exact Solution for Solute Exchange at the Interface when 187 the Initial Solute Concentration is Known

Space-fractional PDE for Flux-Residential Solute 190 Relationships

Distributed-order Time-fractional Advection for 190 Flux-Residential Solute Relationships

Solutions of the fPDEs for Solute Transport in Swelling 193 Soils and Other Issues

Anomalous Rainfall-Runoff-Infiltration Relations with a 203 Constant Rainfall Intensity on a Dry Surface

Fractional Kinematic Wave Models for Overland Flow and 205 Solute Transport on Hillslopes

Rainfall-Infiltration-Runoff Relations on Convergent and 207 Divergent Hillslopes

## Solute Transport by Runoff on Hillslopes 208

Mathematical Models for Flow in Topographically Random 209 Hillslope-Channel Networks

The Topographical Model and Its Applications in Streamflow 211 and Flood Routing

Water Advance-Infiltration-Evaporation during 212 Border-check Irrigation

Solutions of Water Advance-Anomalous Infiltration during 213 Border-check Irrigation

Fractional Partial Differential Equations for Groundwater Flow 218

Unified Equation for Groundwater Flow 223

## Radial Flow and Hydraulics of Wells in Confined and 225 Unconfined Aquifers

*Introduction to Radial Flow in Aquifers 225Contents xv**Radial Flow in Homogeneous Confined Aquifers 226 5.3 The Unified Equation for Radial Flow in Confined and 227**Equations of Flow in Aquifers on a Sloping Impervious 228 Base, in Cylindrical Coordinates**Useful Transformations for Solutions of the Equation for 231 Radial Flow*

Earth tides and barometric effects on groundwater 233 6.1 Effects of inland earth tides, atmospheric tides and 233.

Different Forms of Fractional Boussinesq Equations and 237 Their Implications

Dimensions of the Parameters in Fractional Boussinesq 239 Equations

## Example: Solutions of fPDEs for Groundwater Flow in Aquifers 256 Subject to Boundary Conditions of the First Kind

## Basic Concepts Regarding Poroviscoelastic Materials, and 287 Relationships between Them

Linear, Non-linear and Semi-linear Poroelasticity and 292 Poroviscoelasticity

Special Forms of Fractional Models for Viscoelasticity and 299 Poroviscoelasticity

Physical Interpretation of Fractional Models with Memory 299

## Overview

The Navier-Stokes equations (NSE), discussed in Chapter 3, are the fundamental equations governing the flow of fluids, including the flow of water in porous media. NSEs adjust the viscoelasticity of the porous medium, or the compressibility and compressibility of water, according to the extent of water movement in the soil and aquifers. High-order flow hydrodynamics known as Burnett hydrodynamics (Burnett is an example that can explain several physical mechanisms when NSEs cease to apply.

The processes in which the high-order hydrodynamics operate while the NSEs fail include phenomena such as absorption and propagation of sound in liquids, dynamics of swarms of particles, structure of different profiles in shock waves at large Mach numbers, Couette flows, in continuum transition flows which occurs around space vehicles, and flows in microchannels (García-Colín et al. 2008). 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 which is the study of relationships between velocities, acceleration and the forces exerted are moved through or on fluids (Dauherty et al. 1989). In science, water-related fields include fluid mechanics, hydraulics, hydrology, hydrodynamics, meteorology, oceanography, marine science, agricultural science, and soil science with water as a key element.

Water has been a subject for extensive publications in various formats, and the numerous properties and aspects of water are mentioned in many monographs on hydrology and hydraulics and their sub-disciplines, such as groundwater hydrology. Deformation is another aspect of soils and aquifers, as their physical properties have a significant impact on civil engineering infrastructures and geological materials. In particular, soil mechanics or geomechanics is concerned with the swelling properties of the soil when the water content changes and the mutual changes in water pressure due to deformation in the soil.

Many reports in soil mechanics (geomechanics) can be found alongside publications in hydrology and hydraulics. Many reports in hydrology, hydraulics, and soil science dealing with soils and aquifers generally neglect the key issues of deformation and stress-strain relationships, leaving the discussion to soil mechanics. The separation of soil mechanics and soil physics since the 1930s (Philip 1974) has discouraged hydrology and soil science from integrating with geomechanics, making these fields apparently unrelated, although groundwater hydrologists and soil scientists deal with compressive-elastic soils and aquifers—central topics in soil mechanics.

An attempt was made to build a weak bridge between hydrology and geomechanics in limited literature, such as the works of Wang (2000), which dealt with 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 a, b), and Biot thermoelasticity (Biot 1941, 1956c). However, Wang's work does not discuss any aspect of the NSEs that control water flow in poroelastic and thermoelastic media, thus eliminating important hydrological elements. Compte (1997) has shown that environmental processes such as solute transport, sediment transport and groundwater flow, etc. can be better modeled with fPDEs; groundwater flow/seepage by He (1998); and solute transport in groundwater by Lenormand (1992) and Benson (1998).

Objectives of This Book

Fundamentals of mathematics in Chapter 2, dealing with concepts commonly used in fractional calculus for models and quantitative methods in hydrology

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

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

The remaining Chapters present topics related to water flow and solute transport in unsaturated soils (Chapters 5, 6 and 7), overland flow (Chapter 8), in saturated

## A Brief Description of Key Concepts

### Evolution of Mathematical Models in Hydrology, Hydraulics, Soil Science, and Geomechanics Based on Fractional Calculus

Mathematical models in applied areas related to water in porous media have been introduced by Darcy (1856), using a differential equation as the flux of water or its velocity in porous media, and by Boussinesq (1904), who used a range of PDEs for water movement in unconfined aquifers. The concept of poroelasticity since Terzaghi (1923) and Biot began the investigation of porous media of geological origin in the 20th century. However, fractional calculus was not specifically applied to porous media until after 1974, when fractional calculus was relaunched, marked by three events: publication of the first monograph on fractional calculus by Oldham and Spanier (1974); the first doctorate awarded to Bertram Ross on the subject of fractional calculus; and the First International Conference on Fractional Calculus and its Applications, organized by Bertram Ross and held at the University of Newhaven, Connecticut in June 1974 (Miller et al. 1994).

The terminology of fractional arithmetic, as recorded in history, was the result of a question raised by the mathematician Marquis de L'Hôpital to Gottfried Wilhelm Leibniz in 1695 about the meaning of the differentiation. The slow evolution of fractional calculus from 1695 to 1974 led many mathematicians and scientists in different fields to apply fractional calculus to investigate various processes and phenomena. However, a complete set of fPDEs readily applicable to water flow and solute transport in soils, aquifers and nearby.

As will become clear from Chapter 5 onwards, the order of a fractional derivative or fractional integration can be any function, rather than being limited to a fraction. It is therefore logical to say that fractional calculus is a general form of calculus, with classical integer calculus as a special case.

Application of Fractional Calculus in Water Flow and Related Processes 7 soils, the material coordinate is valid for vertically deforming soils by neglecting the horizontal expansion of the soil due to an increase in the water content of swelling soils. This simplification allows deformation to be analyzed with void ratio as the variable that depends on the moisture content in the soil. Similar to the development of ideas for elasticity and deformation, more complicated factors such as the thermal effect and time-dependent processes or processes with memory, etc., can also be introduced.

As shown by Mainardi and Mainardi and Spada (2011), linear elasticities can be quantitatively related to reciprocal relationships observed in the field (Hsieh 1996) and used to estimate aquifer parameters (Burbey 2001).

Hydrology, Hydraulics, Soil Science, Geomechanics

## Notation in the Book

Application of the fractional conservation of mass to gas flow diffusion equation in heterogeneous porous media. Asymptotic behavior of the solution of the spatially dependent variable order fractional diffusion equation: ultraslow anomalous aggregation. Application of the nonlocal Darcy law to the propagation of nonlinear thermoelastic waves in fluid-saturated porous media.

Hydraulic conductivity, rate and order of the derivative of partial dispersion in a highly heterogeneous system. An estimation of the instantaneous unit hydrograph derived from the theory of topologically random channel networks. On the physical meaning of the dispersion equation and its solutions for various initial and boundary conditions.