Fractional Calculus for Hydrology, Soil Science and Geomechanics

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