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.
Table of Contents:
Application of Fractional Calculus in Water Flow and Related Processes
Overview
Objectives of this book
A brief description of key concepts
Notation in the book
Mathematical Preliminaries
Introduction
Integral transforms
Asymptotic analysis
Special Functions
Fundamental solution, Green function, delta functions and generalized functions
Fractional integration and fractional differentiation
Summary
Essential Properties of Soils and Aquifers as Porous Media
Introduction: Soils and aquifers as porous media
Descriptive concepts and definitions of soils and aquifers
Fundamental equations of flow in soils and aquifers
Applicability of Darcy’s law
Traditional and new parameters for hydraulic properties
Similarity, scales, models and measurements
Other forces coupled with the flow of fluids in porous media
Heterogeneities and isotropy
Summary
Transition from Classic Diffusion to Anomalous Diffusion– The evolution of concepts and ideas
Introduction
The inception of models based on fractional calculus in geoscience and related fields
Theory, models and parameters for water flow and solute transport in porous media
Relationships and differences between anomalous diffusion and scale-dependent and time-dependent transport processes
Dimensions of the parameters in fPDEs
Variable-order fractional derivatives and related fPDEs
Summary
Fractional Partial Differential Equations for Water Movement in Soils
Introduction
Integer calculus-based models for water flow in soils
Fractional calculus-based models for water movement in soils
Conservation of mass in the context of fPDEs
fPDEs for coupled water movement, energy transfer, gas flow and solute transport in porous media
Functional-order fractional partial differential equations
Exchange of water between mobile and immobile zones
Summary
Applications of Fractional Partial Differential Equations to Infiltration and Water Movement in Soils
Introduction
Background and connections between different equations of infiltration
Equations of infiltration derived from fractional calculus with the concentration boundary condition
Infiltration into soils on hillslopes
Infiltration equations derived from an fPDE with a given flux on the soil surface
Water exchange between large and small pores
Example of solutions for water movement in a soil of finite depth
Summary
Fractional Differential Equations for Solute Transport in Soils
Introduction
Solute transport in non-swelling soils
Concurrent water flow and solute transport in swelling soils
Fractional Partial Differential Equations for Anomalous Solute Transport in Soils
Dimensions of the parameters in multi-term fPDEs
Functional-order fPDEs
The fPDE and its solution for solute exchange between mobile and immobile zones
Fractional flux-residential solute concentration relationships during anomalous transport
Applications of fPDEs for coupled solute transport in swelling and non-swelling soils
Summary
Hydraulics of Anomalous Flow on Hillslopes, in Catchment Networks and Irrigated Fields
Introduction
Rainfall-infiltration-runoff relations on a planar hillslope
Rainfall-infiltration-runoff relations on convergent and divergent hillslopes
Solute transport by runoff on hillslopes
Related topics
Streamflow through catchment networks
Anomalous flow during irrigation
Summary
Fractional Partial Differential Equations for Groundwater Flow
Introduction
Governing equations for isothermal groundwater flow in confined aquifers
Governing equation for groundwater flow in unconfined aquifers
Unified concepts and equations for groundwater flow in confined and unconfined aquifers
Radial flow and hydraulics of wells in confined and unconfined aquifers
Earth tides and barometric effects on groundwater
Other factors related to model construction for groundwater flow
fPDEs for isothermal groundwater flow in unconfined aquifers
fPDEs for isothermal groundwater flow in confined aquifers
Distributed-order fPDEs in Cartesian coordinates
fPDEs for hydraulics of anomalous radial flow in wells on a horizontal base
Exchange of water between mobile and immobile zones
Example: Solutions of fPDEs for groundwater flow in aquifers subject to boundary conditions of the first kind
Groundwater flow as a multiphase flow
Summary
Fractional Partial Differential Equations for Solute Transport in Groundwater
Introduction
fPDE-based models for solute transport in different dimensions
Fractional conservation of mass
Symmetrical fADE for solute transport
fPDEs for reactive solute transport with sink and source terms
fPDEs of distributed order for solute transport in aquifers
Solute transfer between mobile and immobile zones
fPDEs for flux and residential solute relationships
fPDEs of distributed order and their asymptotic solutions
Radial anomalous solute transport in groundwater
Functional-order fPDEs
Multi-dimensional symmetrical fPDEs with variable and functional orders
Tempered anomalous solute transport
Summary
Fractional Partial Differential Equations, Poroviscoelastic Media and Geomechanics
Introduction
Basic concepts regarding poroviscoelastic materials, and relationships between them
Approaches to viscoelastic materials with linear elasticity
Fractional calculus-based models for linear viscoelasticity and poroviscoelasticity
Summary
Bibliography
About the Author :
Dr. Su is Adjunct Professor at James Cook University, Australia and Guest Professor at Ningxia University, China. He was previously Guest Professor at several universities in China. He received a PhD at the Australian National University, MSc at the Institute of Soil and Water Conservation, the Chinese Academy of Sciences, and BSc at the College of Agricultural Science, Ningxia University. His research interests span several fields including hydrology, environmental modelling and applications of fractional calculus, which have evolved while working in Australia, China and New Zealand.
