Fractional Calculus for Hydrology, Soil Science and Geomechanics An Introduction to Applications

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.

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

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.