1st Edition

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

By Ninghu Su Copyright 2021
    358 Pages 1 Color & 9 B/W Illustrations
    by CRC Press

    358 Pages 1 Color & 9 B/W Illustrations
    by CRC Press

    358 Pages 1 Color & 9 B/W Illustrations
    by CRC Press

    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

    Biography

    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.