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Hydrodynamics of Time-Periodic Groundwater Flow introduces the emerging topic of periodic fluctuations in groundwater. While classical hydrology has often focused on steady flow conditions, many systems display periodic behavior due to tidal, seasonal, annual, and human influences. Describing and quantifying subsurface hydraulic responses to these influences may be challenging to those who are unfamiliar with periodically forced groundwater systems. The goal of this volume is to present a clear and accessible mathematical introduction to the basic and advanced theory of time-periodic groundwater flow, which is essential for developing a comprehensive knowledge of groundwater hydraulics and groundwater hydrology.
Volume highlights include:
- Overview of time-periodic forcing of groundwater systems
- Definition of the Boundary Value Problem for harmonic systems in space and time
- Examples of 1-, 2-, and 3-dimensional flow in various media
- Attenuation, delay, and gradients, stationary points and flow stagnation
- Wave propagation and energy transport
Hydrodynamics of Time-Periodic Groundwater Flow presents numerous examples and exercises to reinforce the essential elements of the theoretical development, and thus is eminently well suited for self-directed study by undergraduate and graduate students. This volume will be a valuable resource for professionals in Earth and environmental sciences who develop groundwater models., including in the fields of groundwater hydrology, soil physics, hydrogeology, geoscience, geophysics, and geochemistry. Time-periodic phenomena are also encountered in fields other than groundwater flow, such as electronics, heat transport, and chemical diffusion. Thus, students and professionals in the field of chemistry, electronic engineering, and physics will also find this book useful.
I INTRODUCTION
1 Introduction
1.1 Terminology
1.2 Periodic Forcing
1.3 Potential Areas of Application
1.4 Chapter Summary
II PROBLEM DEFINITION
2 Initial Boundary-Value Problem for Hydraulic Head
2.1 Space-Time Domain
2.2 Governing Equation
2.3 Initial Condition
2.4 Boundary Conditions
2.5 Other Parameters
2.6 Special Cases
2.7 Chapter Summary
3 Hydraulic-Head Components and Their IBVPs
3.1 Steady and Transient Components
3.2 Steady and Transient Hydraulic-Head BVPs
3.3 Nonperiodic and Periodic Transient Components
3.4 Nonperiodic and Periodic Transient Hydraulic-Head BVPs
3.5 Chapter Summary
4 Periodic Transient Components
4.1 Trigonometric-Series Representation
4.2 Harmonic-Constituent Parameters
4.3 Vector Harmonic Constituents
4.4 Chapter Summary
5 BVP for Harmonic Constituents
5.1 Rectangular Form of Space BVP
5.2 Space-Time BVP
5.3 Special Case: Ideal Media
5.4 Chapter Summary
6 Polar Form of Space BVP
6.1 Assumptions
6.2 Governing Equations
6.3 BC Equations
6.4 Chapter Summary
7 Complex-Variable Form of Space BVP
7.1 Frequency-Response Functions
7.2 Space BVP for Hydraulic-Head FRF
7.3 Complex-Valued Harmonic Constituents
7.4 Hydraulic-Gradient and Specific-Discharge FRFs
7.5 Chapter Summary
8 Comparison of Space BVP Forms
8.1 Introduction
8.2 The Three Main Forms
8.3 Other Forms
III ELEMENTARY EXAMPLES
9 Examples: 2-D and 3-D Flow in Ideal Media
9.1 Assumptions
9.2 Governing and BC Equations
9.3 Three-Dimensional, Radial Spherical Flow
9.4 Two-Dimensional, Axisymmetric Flow
9.5 Chapter Summary
10 Examples: 1-D Flow in Ideal Media
10.1 Assumptions
10.2 Governing Equation and General Solution
10.3 Finite Domain
10.4 Semi-Infinite Domain
10.5 Discussion
10.6 Chapter Summary
11 Examples: 1-D Flow in Exponential Media
11.1 Assumptions
11.2 Governing Equation and General Solution
11.3 Finite Domain
11.4 Semi-Infinite Domain
11.5 Analogy: Time-Periodic Temperature Variation
11.6 Discussion
11.7 Chapter Summary
12 Examples: 1-D Flow in Power-Law Media
12.1 Assumptions
12.2 Governing Equation and General Solution
12.3 Generalization
12.4 Chapter Summary
13 Examples: Uniform-Gradient Flow
13.1 Assumptions
13.2 Governing Equations
13.3 Complimentary Eigenvectors and Linear Independence
13.4 Unit Gradient Vectors
13.5 One-Dimensional Flow
13.6 Two-Dimensional Flow
13.7 Three-dimensional Flow
13.8 Ideal Media
13.9 Chapter Summary
IV ESSENTIAL CONCEPTS
14 Attenuation, Delay, and Gradient Collinearity
14.1 Attenuation
14.2 Delay
14.3 Phase Shift
14.4 In-Phase and Quadrature Components
14.5 Collinearity of Amplitude and Phase Gradients
14.6 Chapter Summary
15 Time Variation of Specific-Discharge Constituent
15.1 Periodicity of Specific-Discharge Constituent
15.2 Time Extrema of Specific-Discharge Magnitude
15.3 Discharge Space and Discharge Envelope
15.4 Locally Vanishing Gradients
15.5 Collinear Amplitude and Phase Gradients
15.6 Noncollinear Amplitude and Phase Gradients
15.7 Discussion
15.8 Chapter Summary
V STATIONARY POINTS
16 Stationary Points: Basic Concepts
16.1 Elliptic Operators
16.2 Maximum Principles
16.3 Stationary Points of A h and B h
16.4 Chapter Summary
17 Stationary Points: Amplitude and Phase
17.1 Physical Significance
17.2 Governing and BC Equations
17.3 Stationary Points of the Amplitude
17.4 Stationary Points of the Phase
17.5 Chapter Summary
18 Flow Stagnation
18.1 Stagnation Points
18.2 Stagnation Sets
18.3 Chapter Summary
VI WAVE PROPAGATION
19 Hydraulic-Head Diffusion Waves
19.1 Introduction
19.2 Waves in Continuous Media
19.3 Two Wave Interpretations
19.4 Examples
19.5 Discussion: Constituent Waves Versus Component Waves
19.6 More Wave Kinematics
19.7 Chapter Summary
20 Wave Distortion
20.1 Introduction
20.2 Dispersive and Nondispersive Distortion
20.3 Frequency Dispersion and Group Velocity
20.4 Longitudinal and Transverse Gradients
20.5 Wave Attenuation
20.6 Wave Dilation/Contraction
20.7 Chapter Summary
21 Waves in One Dimension
21.1 Linearly Independent FRFs
21.2 Component-Wave Hypotheses
21.3 Uniform-Velocity Waves
21.4 Nonuniform-Velocity Waves
21.5 Chapter Summary
22 Wave Equation
22.1 Generalized Wave Equation
22.2 Interpretation
22.3 The Coefficient Functions ? and ß
22.4 Discussion
22.5 Chapter Summary
VII ENERGY TRANSPORT
23 Mechanical Energy of Ground Water
23.1 Basic Definitions
23.2 Conservation Equations
23.3 Rate of Change of Energy Density
23.4 Mechanical-Energy Flux Density
23.5 Dissipation
23.6 Chapter Summary
24 Mechanical Energy: Time Averages
24.1 Time Averages
24.2 Time Variation of Water Density and Porosity
24.3 Time-Average Energy Density and Its Rate of Change
24.4 Time-Average Energy Conservation
24.5 Time-Average Energy Flux Density
24.6 Time-Average Energy Dissipation
24.7 Chapter Summary
25 Mechanical Energy of Single-Constituent Fields
25.1 Time-Average Energy Density
25.2 Time-Average Mechanical-Energy Flux Density
25.3 Time-Average Energy Dissipation
25.4 Mechanical Power Transmission Across A Surface
25.5 Chapter Summary
VIII CONCLUSION
26 Conclusion
26.1 Overview
26.2 Unresolved Issues
26.3 Unexplored Topics
26.4 Final Note
A Hydraulic-Head Components
B Useful Results from Trigonometry
C Linear Transformation of Space Coordinates
D Complex Variables
E Kelvin Functions
References
Index
Joe Depner graduated with an M.S. from the Department of Hydrology and Water Resources at the University of Arizona in 1985. His thesis topic was Estimation of the three-dimensional anisotropic spatial covariance of log permeability using single-hole and cross-hole packer test data from fractured granites, under the direction of Professor Shlomo P. Neuman, which was subsequently published (Neuman and Depner, 1988). He has also published on the topic of periodic flow in groundwater (Depner, 2000). He has worked professionally for multiple private consulting services and for Pacific Northwest National Laboratory in Hanford, WA.
Todd Rasmussen is a Professor of Hydrology and Water Resources at the University of Georgia (UGA). He is a member of the Faculty of Water Resources, the Faculty of Engineering, and the Academy of the Environment at UGA. He is an associate editor for the Journal of Hydrology, and has been an associate editor for Water Resources Research and Hydrogeology Journal. He received his PhD from the Department of Hydrology and Water Resources, College of Engineering and Mines, at the University of Arizona in 1988. His publications focus on uid ow and contaminant transport through surface and subsurface environments, including the physical, chemical, mathematical, and statistical description and quantification of hydrologic processes. He was a co-author of the AGU Geophysical Monograph 42 (Evans et al., 2001) as well as multiple journal articles specifically related to subsurface periodic behavior (Toll and Rasmussen, 2007; Rasmussen and Mote, 2007; Rasmussen et al., 2003).
