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Geophysical Monograph 224

Hydrodynamics of Time-PeriodicGroundwater Flow

Diffusion Waves in Porous Media

Joe S. DepnerTodd C. Rasmussen

This Work is a co-publication between the American Geophysical Union and John Wiley & Sons, Inc.

This Work is a co-publication between the American Geophysical Union and John Wiley & Sons, Inc.

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9781119133940

Cover image: The image presents the water-level response in an aquifer due to periodic excitation of four groundwater wells atdifferent amplitudes and frequencies.

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CONTENTS

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii

Part I: Introduction 1

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Part II: Problem Definition 7

2 Initial Boundary Value Problem for Hydraulic Head . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Hydraulic Head Components and Their IBVPs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4 Periodic Transient Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

5 BVP for Harmonic Constituents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

6 Polar Form of Space BVP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

7 Complex-Variable Form of Space BVP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

8 Comparison of Space BVP Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

Part III: Elementary Examples 45

9 Examples: 1D Flow in Ideal Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

10 Examples: 1D Flow in Exponential Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

11 Examples: 1D Flow in Power Law Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

12 Examples: 2D and 3D Flow in Ideal Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

13 Examples: Uniform-Gradient Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

Part IV: Essential Concepts 121

14 Attenuation, Delay, and Gradient Collinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

15 Time Variation of Specific-Discharge Constituent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

Part V: Stationary Points 149

16 Stationary Points: Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

17 Stationary Points: Amplitude and Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

18 Flow Stagnation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

v

vi CONTENTS

Part VI: Wave Propagation 181

19 Harmonic, Hydraulic Head Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

20 Wave Distortion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

21 Waves in One Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

22 Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

Part VII: Energy Transport 231

23 Mechanical Energy of Groundwater . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233

24 Mechanical Energy: Time Averages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

25 Mechanical Energy of Single-Constituent Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

Part VIII: Conclusion 261

26 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263

Part IX: Appendices 269

A Hydraulic Head Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271

B Useful Results from Trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273

C Linear Transformation of Space Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275

D Complex Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281

E Kelvin Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295

PREFACE

Goal and Purpose

Our goal in writing this book is to present a clearand accessible mathematical introduction to the basictheory of time-periodic groundwater flow. Understand-ing the basic theory is essential for those who seek acomprehensive knowledge of groundwater hydraulics andgroundwater hydrology. In addition, the basic theory hasan aesthetic beauty that readers can learn to appreciateand thereby enjoy.

Intended Audience

We intend this book to be used primarily for self-directed study by advanced undergraduate and graduatestudents and by working scientists and engineers in theearth and environmental sciences. This book is suitable forwell-prepared readers who either (a) are new to the field ofperiodic groundwater flow and seek a formal introductionto the theory or (b) were introduced to the field in the dis-tant past and wish to renew their knowledge and enrichtheir understanding. Additionally, we hope that this bookwill be a useful resource for educators.

The mathematical framework for time-periodic ground-water flow is structurally equivalent to that of time-periodic diffusion. Therefore, some of the theory pre-sented in this book may be relevant to time-periodicphenomena encountered in fields other than groundwa-ter flow, like electrical conduction, thermal conduction,and molecular diffusion. Consequently, we expect thatstudents and professionals in these other fields also willfind parts of this book useful.

Prerequisites

We assume that the reader has completed universitycourses in multivariable calculus, linear algebra, and sub-surface fluid dynamics (e.g., groundwater hydraulics).Also, the reader should have a basic familiarity withcomplex variables, Fourier series, and partial differentialequations (PDEs). Readers do not need to know contourintegration in the complex plane or Green functions.

Approach

Our development is quantitative. We emphasize prob-lem definition and problem understanding, rather than

problem solution techniques, because we believe theformer are fundamental prerequisites of the latterand because solution techniques have been describedexhaustively by other authors (e.g., Carslaw and Jaeger[1986], Özisik [1989], Hermance [1998], Bruggeman [1999],Mandelis [2001]).

Much of the information presented here could begleaned from reading articles in peer-reviewed scientificpublications such as those listed in the Bibliography. How-ever, one would have to read many such articles, which typ-ically present only terse descriptions of the mathematicaldevelopment. This book is more explicit to accommodatethe needs of those who are new to the field of peri-odic groundwater flow. It shows more intermediate stepsso that readers can follow the logic of the development,understand the mathematical context, and recognize thelimitations of the approach.

Scope

Assumptions

The scope of this book is limited to time-periodic flowsof homogeneous fluids through fully saturated, elasticallydeformable, porous media in which Darcy’s law is sat-isfied. Within this scope, we have attempted to presentthe basic theory in a general form so that the results arewidely applicable. To that end we make the following basicassumptions, among others:

• The relevant space domain is N-dimensional, whereN can be 1, 2, or 3.

• The porous medium is macroscopically nonhomo-geneous (i.e., spatially nonuniform) with respect to mate-rial hydrologic properties. That is, the medium’s hydraulicconductivity and specific storage are functions of thespace coordinates.

• In multidimensional cases, the porous mediumgenerally is nonuniformly anisotropic with respect tohydraulic conductivity. That is, the problem under con-sideration cannot be transformed to one in which theanisotropic medium is replaced by an equivalent, macro-scopically isotropic one simply by linearly transformingthe space coordinates.

• The periodic component of the forcing need not bestrictly periodic; it may be almost periodic (see Section 1.1for a discussion of relevant terminology).

vii

viii PREFACE

With the exception of some illustrative examples andexercises, which we clearly identify, we adhere to theseassumptions throughout this book.

Organization

This book consists of the following parts:• Part I (Introduction, Chapter 1) introduces basic

terminology, proposes criteria for the classification oftime-periodic forcing, and lists potential areas of appli-cation for the theory of time-periodic groundwaterflow.

• Part II (Problem Definition, Chapters 2–8) describesthe conceptual, mathematical basis of periodic groundwa-ter flow within the framework of the classical boundary-value problem (BVP). It lays the foundation for subse-quent parts.

• Part III (Elementary Examples, Chapters 9–13)presents examples of elementary solutions of thecomplex-variable form of the space BVP.

• Part IV (Essential Concepts, Chapters 14 and 15)explores some basic concepts of periodic flow, such asattenuation, delay, and local time variation of the specificdischarge.

• Part V (Stationary Points, Chapters 16–18) exam-ines the existence and nature of stationary points of thehydraulic head amplitude and phase functions and theirrelation to flow stagnation.

• Part VI (Wave Propagation, Chapters 19–22)presents a conceptualization of periodic groundwaterflow as propagation of spatially attenuated (damped),traveling diffusion waves, i.e., harmonic, hydraulic headwaves.

• Part VII (Energy Transport, Chapters 23–25)explores the transport of fluid mechanical energy byperiodic groundwater flow under isothermal conditions.

• Part VIII (Conclusion, Chapter 26) briefly summa-rizes the results obtained in the preceding chapters, unre-solved issues, and limitations of the book.

Suggested Use

The chapters are meant to be read sequentially withineach part.

We believe that all readers should begin by study-ing Parts I and II. This material forms the core ofthe subject and is prerequisite for learning about themore advanced topics presented in subsequent parts.After studying Part II, readers should at least browseParts III and IV to familiarize themselves with their scope.The reader’s subsequent course of action depends on

individual preference. Those who have both the inter-est and sufficient time should read all of the remainingparts sequentially. Those who are pressed for time orwhose interests are more limited may study a combina-tion of Parts V–VII. Lastly, all readers will want to readPart VIII.

We have embedded more than 360 exercises (see Listof Exercises) in the text and included the solutions fornearly all. Each exercise is numbered and accompaniedby a title that briefly summarizes its topic. The exer-cises are intended to reinforce the ideas presented inthe text and in many cases are essential elements ofthe theoretical development. Exercises typically empha-size abstract reasoning, requiring symbolic manipulationrather than numerical computation. Believing that mostreaders will be more familiar with the material in theearlier parts than that in the later parts, we have placedmore exercises in the later parts. Ideally, less advancedreaders should attempt to complete every exercise theyencounter. More advanced readers and those pressed fortime should, at minimum, carefully read each exercise andthe accompanying solution to maintain the flow of thepresentation.

Usability

The electronic version of this book employs the follow-ing features for reader convenience:

• All book components (parts, chapters, sections, sub-sections, appendices) listed in the Contents are digitally“bookmarked.” This allows the reader to navigate to thebeginning of any such component, from any point inthe book, via hyperlink. To activate a bookmark hyper-link, click on the corresponding label in the bookmark’snavigation panel.

• In-line hyperlinks are used extensively. These includereferences to the following items:

– Worldwide website uniform resource locators(URLs)

– Specific book components: parts, chapters, sections,subsections, appendices, etc.

– Specific content features: equations, examples, exer-cises, figures, notes, tables

– Page references in the keyword indexHyperlinked references, both in the table of contentsand elsewhere, appear as blue-colored text. To activatean in-line hyperlink, click on the corresponding blue-colored text.

• Important terms appear in italics to draw the reader’sattention.

PREFACE ix

• Examples, exercises, and notes appear with translu-cent shaded backgrounds colored blue violet, yellow, andgreen, respectively, to help the reader quickly recognizethem.

Website and Contacts

Readers are invited to help improve the quality of thisbook by reporting errors and suggesting changes.

For details, visit the companion website at hydrol-ogy.uga.edu/periodic/.

Joe S. DepnerSeattle, Washington

Todd C. RasmussenAthens, Georgia

NOTATION

Latin Symbols

Symbol Description

A Coefficient of cosine term in Fourier series; real component of complex amplitude; dimensionlesscoefficient in frequency response function for one-dimensional flow

A Coefficient matrix of linear differential operatoradj Adjugate matrixArctan Principal value of inverse-tangent functionarctan Arctangent (inverse-tangent) functionarg Argument of complex numberB Coefficient of sine term in Fourier series; imaginary component of complex amplitude; dimensionless

coefficient in frequency response function for one-dimensional flowb Coefficient vector of linear differential operatorBC Boundary conditionber , bei Kelvin functions, b-type, order ( R)BVP Boundary value problemC Matrix of linear transformation of space coordinatesc Constant; propagation speed of harmonic, traveling wavec Eigenvector of FRF for uniform-gradient flow in exponential mediach Coefficient of hydraulic head term in boundary condition equation; propagation speed of hydraulic

head constituent wavec n Propagation speed of nth-component wave (n integer, n 0)cq Coefficient of specific-discharge term in boundary condition equationconst Constantcos Cosine functioncosh Hyperbolic cosine function1D One dimensional2D Two dimensional3D Three dimensionald Ordinary differentiation operatordB Decibel(s)det Determinant of matrixD N-dimensional space domainD0 m Zero set of hydraulic head constituent amplitude Mh x; mD+ m Cozero set of hydraulic head constituent amplitude Mh x; mE Fluid mechanical energye Void ratioe Euler’s number (also, Napier’s constant), the mathematical constant, e = 2.7182818em Unit basis vector for Cartesian coordinate xm (m = 1, 2, 3)exp Exponentiation operator, i.e., exp = eF Complex amplitudeF n Frequency response eigenfunctionFRE Frequency response eigenfunctionFRF Frequency response functiong Acceleration of gravityG Space domainG r , G x Frequency response function

xi

xii NOTATION

Symbol Description

GD Dimensionless frequency response functionG n Frequency response eigenfunctionH f x Hessian matrix of the function f xH Complex-valued harmonic constituent of periodic transient component of hydraulic headh Hydraulic headh Harmonic constituent of periodic transient component of hydraulic headI Identity tensorI0 Modified Bessel function of the first kind, order zeroIBVP Initial boundary value problemIC Initial conditionIm Imaginary part of complex numberi Imaginary unit (i.e., i = 1)i0 Modified spherical Bessel function of the first kind, order zero

Ip In-phase component of harmonic constituentJ Fluid mechanical energy flux density (vector)K Hydraulic conductivity (tensor) of porous mediumK Hydraulic conductivity (scalar) of porous mediumK0 Modified Bessel function of the second kind, order zeroK 1 2 Modified Bessel function of the second kind, order 1 2k Wave numberk n Wave number for nth-component wave (n integer, n 0)k Wave vectork n Wave vector for nth-component wave (n integer, n 0)k0 Modified spherical Bessel function of the second kind, order zeroker , kei Kelvin functions, k-type, order ( R)kg KilogramL Length (dimension)L n Penetration depth for nth-component wave (n integer, n 0)L Linear differential operatorLh Linear differential operatorLM Linear differential operatorL Linear differential operatorLHS Left-hand side (of equation or inequality)l Length of one-dimensional space domainlim Limitln Natural (base-e) logarithmlog10 Base-10 logarithmM Mass (dimension)Mh Amplitude function for hydraulic head harmonic constituentM n Amplitude function for nth-component wave (n integer, n 0)Mu Amplitude function for source term harmonic constituentM Amplitude function for boundary condition harmonic constituentM Kelvin modulus function, b-type, order ( R)m Harmonic constituent index (m integer, m 0)m Metermax Maximum valuemaxt Maximum value with respect to timeME Mechanical energymin Minimum valueN Newton

NOTATION xiii

Symbol Description

N Dimension of space domain (N = 1, 2, 3)N m Number of component waves for the mth-harmonic constituent (N integer, N 0)N Kelvin modulus function, k-type, ordern Unit vector outwardly perpendicular to space domain boundary

n Component wave index (n integer, n 0)O Order ofODE Ordinary differential equationp Fluid pressurep n

m Dimensionless exponent of FRF for power law media (m, n integer; m, n 0)PDE Partial differential equationQ Complex-valued, harmonic constituent (vector) of periodic transient component of groundwater

specific dischargeQ Quality factorQ0 Complex amplitude of point source

Q0 Complex amplitude of line source or plane sourceq Specific discharge (vector)q Harmonic constituent (vector) of periodic transient component of groundwater specific discharge

Qu Quadrature component of harmonic constituentR Set of all real numbersR Dimensionless radial space coordinaterad RadianrD Dimensionless envelope of specific-discharge constituentRe Real part of complex numberRHS Right-hand side (of equation or inequality)S Control surfaces Seconds Wave travel distances n Travel distance for component wave (n integer, n 0)sh Unit ray path vector of hydraulic head constituent waves n Unit ray path vector of nth-component wave (n integer, n 0)sech Hyperbolic secant functionsign Sign functionsin Sine functionsinh Hyperbolic sine functionSs Specific storage of porous mediumT Travel time; fluid temperaturet Time (dimension or independent variable)Tm Period of mth-harmonic constituent (m integer, m 0)T n Travel time of nth-component wave (n integer, n 0)tr Trace of matrix

T Matrix transposeU Complex-valued harmonic constituent of periodic transient component of groundwater volumetric

source strengthu Harmonic constituent of periodic transient component of groundwater volumetric source strengthu Vector of unit length pointing in the direction of K MhURL Uniform resource locatorV Control volumev Nominal seepage velocity; phase velocity of traveling wavevh Phase velocity of hydraulic head constituent wavev Vector of unit length pointing in the direction of K h

xiv NOTATION

Symbol Description

v n Phase velocity of nth-component wave (n integer, n 0)W Linear differential operator for boundary condition equationX Dimensionless space coordinatex N-dimensional space coordinate vectorx Unit basis vector for Cartesian space coordinate x (also, e1)x Cartesian space coordinate (also, x1)xm Cartesian space coordinate (m = 1, 2, 3)Yh Logarithm of amplitude of hydraulic head harmonic constituentZ Set of all integersz Complex variable

Greek Symbols

Symbol Description

Bulk compressibility of porous mediumm Constituent parameter [see equation ( 5.31)] (m integer, m 0)

Coefficient of order-zero term in generalized wave equationw Isothermal compressibility of liquid water

Boundary of space domainh Subset of the boundary for which the pertinent boundary condition is of the Dirichlet typehq Subset of the boundary for which the pertinent boundary condition is of the Robin typen

m Dimensionless eigenvalue (m, n integer; m, n 0)n Ray path geometric divergence for nth-component wave (n integer, n 0)

q Subset of the boundary for which the pertinent boundary condition is of the Neumann type+ m Boundary of subregion D+ m

Coefficient of order-one term in generalized wave equationh Spatial attenuation scale for hydraulic head harmonic constituentmn Kronecker delta (sometimes referred to as Kronecker’s delta) (m, n integer; m, n 0)n Spatial attenuation scale for nth-component wave (n integer, n 0)

Fluid mechanical energy densityijk Three-dimensional Levi-Civita symbolh Space derivative of hydraulic head harmonic constituent phase function in one-dimensional flow

n Space derivative of nth-component wave phase function in one-dimensional flow (n integer, n 0)Local, per-volume, fluid mechanical energy dissipation rate

m Dimensionless parameter used for plotting dimensionless FRF solutions (m integer, m 0)D Phase function for dimensionless hydraulic head frequency response functionh Phase function for hydraulic head harmonic constituent

n Phase function for nth-component wave (n integer, n 0)u Phase function for source term harmonic constituent

Phase function for boundary condition harmonic constituentKelvin phase function, b-type, order ( R)Hydraulic diffusivityLogarithm of bulk compressibility

1m , 2

m Dimensionless eigenvalues for mth constituent (m integer, m 0)Wavelength

m, 1m , 2

m Eigenvalues for mth constituent (m integer, m 0)n Local wavelength of nth-component wave (n integer, n 0)

Reciprocal length scale for spatial variation of natural logarithm of hydraulic conductivity or specificstorage

NOTATION xv

Symbol Description

Gradient (vector) of natural logarithm of hydraulic conductivity or specific storageDimensionless length scale of hydraulic conductivity or specific storage; order of Bessel function orKelvin functionWave phase

h Wave phase for hydraulic head constituent waven Wave phase for nth-component wave (n integer, n 0)

The mathematical constant, = 3.14159265s Volumetric mass density of solid phase of porous mediumw Volumetric mass density of groundwater

SummationLocal, per-volume rate of delivery of fluid mechanical energy by internal source(s)

e Effective stressInitial-value function for hydraulic head; porosityKelvin phase function, k-type, order ( R)

e Effective porosityComplex-valued harmonic constituent of periodic transient component of boundary value functionBoundary value functionHarmonic constituent of periodic transient component of boundary value functionAngular frequency

m Angular frequency of mth-harmonic constituent (m integer, m 0)

Other Symbols

Symbol Description

(bar accent) Steady component(tilde accent) Transient component(hat accent) Nonperiodic transient component(ring accent) Time-periodic transient component(breve accent) Transformed variable

* (asterisk) Complex conjugate0 Zero vector

1 Reciprocal; matrix inversedv Integration with respect to the variable v

InfinityPartial differentiation operatorN-dimensional spatial gradient operator (vector)N-dimensional spatial divergence operator (scalar)

2 N-dimensional Laplacian operator (scalar)4 Fourth-order, N-dimensional Laplacian operator (scalar)

Positive square root= Is equal to

Is approximately equal toIs less thanIs much less thanIs greater thanIs much greater thanIs defined asIs an element of the setIs a subset of the set

xvi NOTATION

Symbol Description

Intersection of setsUnion of sets

∅ Empty setFor all

: Such that: Matrix double inner product

Such thatAbsolute value of real number; modulus of complex number; magnitude (L2-norm) of n-dimensionalvectorInner (scalar) product on finite-dimensional vector spaceCross (vector) productApproachesImpliesInner product on infinite-dimensional vector space; time averageSet

ACKNOWLEDGMENTS

The authors thank the following people:• Gary Streile, for thoroughly reviewing draft versions

of Chapters 1–3. His helpful comments led to significantimprovements in those and other chapters.

• Several anonymous reviewers for their helpful sug-gestions.

• Hans Weinberger, for his permission to excerpt fromthe book Maximum Principles in Differential Equations[see Protter and Weinberger, 1999].

• Richard Koch, and many others too numerousto mention individually here, who have contributedto the development and maintenance of TeXShop(www.texshop.org)—a noncommercial TEX previewer forMac OS X. Prepublication drafts of this document weretypeset using TeXShop.

xvii

Part IIntroduction

1Introduction

Abstractly, one can consider time-periodic groundwaterflow to be the response of a physical system to a stim-ulus or excitation. The groundwater system consists ofa subsurface porous medium and a resident pore fluidsuch as fresh or saline water, oil, air, or natural gas. Thestimulus is some kind of time-periodic forcing, and theresponse is the time-periodic variation of hydraulic headand specific discharge. Thus, a possible alternative title forthis book is “Introduction to the Theory of PeriodicallyForced Groundwater Systems.”

1.1. TERMINOLOGY

To clearly articulate ideas about periodic flow, it will beuseful to first clarify and standardize some basic relatedterminology.

For this book, we define functions of practical interestas those mathematical functions that are capable of rep-resenting real physical phenomena. We will assume thatevery function of practical interest is either periodic oraperiodic.

If f t is a periodic function of time, then there exists anonzero real number (period) T for which

f t + T = f t t R.

Thus, periodicity is a type of global translational symme-try. Synonyms for the term periodic include the terms fullyperiodic, purely periodic, and strictly periodic.

We can represent every periodic function of time asthe sum of one or more distinct harmonic constituents(also frequency components or modes), each of which isa purely sinusoidal function of time, wherein the con-stituent frequencies are rational multiples of one another.This description coincides with the classical Fourier seriesrepresentation of a periodic function.

An aperiodic function (also nonperiodic function) is anyfunction that is not periodic. Aperiodic functions includean important class of functions that are closely related tothe periodic functions—the almost-periodic functions. Analmost-periodic function (also, quasiperiodic function) isa function composed of (i.e., formed by summing) two ormore harmonic constituents, at least two of which havefrequencies that are not rational multiples of one another.A simple example of an almost-periodic function havingtwo distinct harmonic constituents is

f t = 3 sin 2t + 7 cos 3 2t t R.

Equivalently, an almost-periodic function is an aperiodicfunction that we can represent as the sum of two or moreperiodic functions and thus as a generalized Fourier series.

Throughout this book we generally use the term peri-odic function to represent that broad class of functionsthat includes both the strictly periodic functions and thealmost-periodic functions. What these functions have incommon is that we can represent both types by general-ized Fourier series. Similarly, we use the term nonperiodicfunction to represent the class of functions that are neitherstrictly periodic nor almost periodic.

In the literature readers may encounter numerous termsthat have meanings related to concepts of periodicity. Forinstance, some authors use the terms cyclic (or cyclical)and rhythmic as synonyms for periodic. In some contextsthe terms oscillating (or oscillatory) and undulating also canhave meanings similar to that of the term periodic; in othercontexts these terms might be used to describe types ofvariation that are more irregular. Similarly, the term fluc-tuating is commonly used to describe variations that areless regular or less predictable than those described by theterm periodic; in fact, the term fluctuating is frequently usedto describe variations controlled by random processes.

Hydrodynamics of Time-Periodic Groundwater Flow: Diffusion Waves in Porous Media, Geophysical Monograph 224,First Edition. Joe S. Depner and Todd C. Rasmussen.© 2017 American Geophysical Union. Published 2017 by John Wiley & Sons, Inc.

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4 HYDRODYNAMICS OF TIME-PERIODIC GROUNDWATER FLOW

Other terms that may reflect temporally periodic flowinclude alternating (e.g., alternating flow; see Stewart etal. [1961]), pulsatile (also, pulsating, pulsed, pulsing), andreciprocating.

To distinguish between periodicity in time and periodic-ity in space, some authors use the terms time periodic (alsotemporally periodic or steady periodic) and space periodic(or spatially periodic). In the literature, the term periodicmedia generally refers to (porous) media that are spatiallyperiodic with respect to material properties.

1.2. PERIODIC FORCING

Time-periodic flow occurs in a groundwater system onlyif the system undergoes periodic forcing. Based on sys-tem geometry, periodic forcing can be of two basic types,which can occur either alone or in combination. In bound-ary forcing, a system boundary is subject to time-periodicconditions. An example is the time variation of hydraulichead at an aquifer’s seaward boundary. In internal forc-ing, an internal water source/sink is time periodic. Anexample is time-periodic water injection/pumping at aninjection well.

We can classify periodic forcing of groundwater sys-tems according to various additional criteria as well. Insummary, criteria for the classification of periodic forcinginclude the following:Origin Natural versus artificial.Frequency High frequency (short period) versus low

frequency (long period).Geometry Boundary versus internal.Periodicity Purely periodic versus almost periodic.Other Hydraulic versus nonhydraulic (e.g., periodic

water pressurization at a vertical boundary versusperiodic mechanical loading on a horizontal upperboundary).

This book addresses both purely periodic and almost-periodic forcing. This choice is largely a matter ofconvenience—we can represent the time behavior of bothtypes mathematically using general trigonometric series.

1.3. POTENTIAL AREAS OF APPLICATION

The following is a summary of potential areas of appli-cation for the theory of time-periodic groundwater flow.While this summary is broad, it is not comprehensive;there likely are additional applications, and we expect thenumber of applications to grow with time.Atmospheric Pressure Natural. Diurnal, annual (sea-

sonal), etc. Mechanical effect of barometric forcingon the upper surface of the capillary fringe, onthe upper surface of a confining unit, or on the

free surface within a well. Here the theory is usedboth to understand aquifer response to atmosphericforcing and to investigate aquifer hydrologic prop-erties. Examples: Furbish [1991], Hanson [1980],Hobbs and Fourie [2000], Merritt [2004], Neeper[2001, 2002, 2003], Rasmussen and Crawford [1997],Rinehart [1972], Ritzi et al. [1991], Rojstaczer [1988],Rojstaczer and Agnew [1989], Rojstaczer and Riley[1990, 1992], Seo [2001], Toll and Rasmussen [2007],van der Kamp and Gale [1983], Weeks [1979]. Recentlythe theory has been used to assess the effectiveness ofsubsurface energy resource exploitation efforts [e.g.,Burbey and Zhang, 2010].

Infiltration/Recharge Mass flow (hydraulic) effect atrecharge boundaries. Here the theory is used to modelthe effects of periodic recharge cycles on groundwatersystems.

Artificial Recharge Seasonal and other cycles. Exam-ples: Latinopoulos [1984, 1985].

Natural Infiltration/Recharge Associated with sea-sonal cycles of precipitation and evapotran-spiration. Examples: Latinopoulos [1984],Maddock and Vionnet [1998], Rasmussen andMote [2007].

Plant Water Uptake/Transpiration Mass flow effect at ornear the water table. Seasonal and diurnal cycles. Herethe theory is used for modeling the interaction of thebiosphere with groundwater systems. Examples: But-ler et al. [2007], Kruseman and de Ridder [2000], Lautz[2008a,b].

Tides Natural. Multiple periods, from semidiurnal tomonthly and longer. Both hydraulic and mechanicaleffects.

Earth Tides The theory is used to infer aquiferand petroleum reservoir physical properties[e.g., Bredehoeft, 1967; Chang and Firoozabadi,2000; Cutillo and Bredehoeft, 2011; Hsieh et al.,1987, 1988; Kümpel et al., 1999; Marine, 1975;Morland and Donaldson, 1984; Narasimhan et al.1984; Ritzi et al., 1991], to assess the effectivenessof subsurface energy resource exploitation efforts[e.g., Burbey and Zhang, 2010], and to understandgeyser eruption timing [e.g., Rinehart, 1972].

Ocean Tides The theory is used to infer aquiferhydraulic properties [e.g., Carr and van der Kamp,1969; Erskine, 1991; Ferris, 1951; Jacob, 1950; Jhaet al., 2008; Trefry and Bekele, 2004; Trefry andJohnston, 1998], to correct nonsinusoidal hydraulictest results for tidal influence [e.g., Chapuis et al.,2006; Trefry and Johnston, 1998], to assess ground-water fluxes in coastal aquifers [e.g., Serfes, 1991],and to assess groundwater–surface water fluxes incoastal environments [e.g., Burnett et al., 2006;Taniguchi, 2002].

INTRODUCTION 5

Sinusoidal Hydraulic Tests Artificial. Variable period(s).The theory is used to design and interpret the resultsof tests to infer material hydraulic properties.

Field (Pumping) Tests Hydraulic effect at face of well.The testing is conducted in situ. Examples: Blackand Kipp [1981], Cardiff et al. [2013], Hvorslev[1951], Mehnert et al. [1999], Rasmussen et al.[2003], Renner and Messar [2006].

Laboratory Tests Hydraulic effect at opposite faces ofmaterial sample. The testing is conducted on mate-rial samples in the laboratory. Examples: Adachiand Detournay [1997], Bernabé et al. [2006], Fischer[1992], Kranz et al. [1990], Rigord et al. [1993], Songand Renner [2006, 2007].

Periodic Groundwater Pumping/Injection Artificial. Vari-able period(s). Hydraulic effect at face of well(s).Here the theory could be used to design subsurfaceenvironmental remediation systems [e.g., Zawadzkiet al., 2002], to design aquifer recharge systems [e.g.,

Latinopoulos, 1984, 1985], or to evaluate hydraulicconnectivity in functioning geothermal well fields[e.g., Becker and Guiltinan, 2010; Yano et al., 2000].

This list represents only a sample of the availableliterature.

1.4. CHAPTER SUMMARY

In this chapter we introduced basic terminology onthe time behavior of periodically forced groundwatersystems and proposed criteria for the classification oftime-periodic forcing. We also briefly listed some poten-tial areas of application for the theory of time-periodicgroundwater flow. The list illustrates that time-periodicgroundwater flow is relevant across multiple fields:

Earth sciences: geophysics, groundwater hydrology andhydrogeology, and oceanography.

Engineering fields: civil (environmental and geotechni-cal) and energy (geothermal and petroleum).

Part IIProblem Definition

In this part we describe the conceptual, mathemati-cal basis of time-periodic groundwater flow within the

framework of the classical boundary value problem(BVP). This part lays the foundation for subsequent parts.

Hydrodynamics of Time-Periodic Groundwater Flow: Diffusion Waves in Porous Media, Geophysical Monograph 224,First Edition. Joe S. Depner and Todd C. Rasmussen.© 2017 American Geophysical Union. Published 2017 by John Wiley & Sons, Inc.

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2Initial Boundary Value Problem for Hydraulic Head

Consider a confined and fully saturated groundwatersystem defined on a particular space domain. We wishto define an initial boundary value problem (IBVP) to for-mally describe the physical behavior of this system. TheIBVP consists of the following elements:

• Space-time domain• Governing equation• Initial condition (IC)• Boundary condition (BC)• Other parameters

The following sections discuss each of these elements.

2.1. SPACE-TIME DOMAIN

We define the time domain of the BVP as

t R, t 0,

where t is time and R denotes the set of real numbers.We will assume that the space domain of the BVP,

which we will denote D, is an open, connected setin N-dimensional space, where N can be 1, 2, or 3depending on the particular situation. Let x denote theN-dimensional vector of space coordinates. Then thespace coordinates of points in D satisfy

x D.

Let denote the (closed) set of points that lie on theboundary of the space domain D. We call the domainboundary. The space coordinates of the boundary pointssatisfy

x .

We can think of D as the N-dimensional region enclosedby the boundary . The domain and its boundary aremutually disjoint sets:

D = ∅.

We assume that we can represent the geometry of thespace domain equivalently by specifying a dimensionlessvector field, n x , that satisfies the following conditions:

n x is directed outwardly perpendicular to the bound-ary at all points that lie on the boundary and

n x vanishes at all points that do not lie on theboundary.

Thus, n is N dimensional. In addition, n is normalized sothat it satisfies

n x =1, x0, x .

This definition and notation are convenient because theyallow us to compactly represent the geometry of the spacedomain using the parameter n x , which is also conve-nient for the expression of a flux boundary condition (seeSection 2.4).

2.2. GOVERNING EQUATION

We assume that groundwater generally is homogeneousand therefore incompressible except insofar as its low butnonzero compressibility contributes to the storage capac-ity of porous media. Then consideration of groundwa-ter mass conservation leads to the following continuityequation [see Freeze and Cherry, 1979]:

q r x, t + u x, t = Ss xh x, t

tt 0, x D

(2.1)

Hydrodynamics of Time-Periodic Groundwater Flow: Diffusion Waves in Porous Media, Geophysical Monograph 224,First Edition. Joe S. Depner and Todd C. Rasmussen.© 2017 American Geophysical Union. Published 2017 by John Wiley & Sons, Inc.

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10 HYDRODYNAMICS OF TIME-PERIODIC GROUNDWATER FLOW

wheredenotes the N-dimensional divergence operator

(dimensions: L 1);h x, t is the hydraulic head as a function of position and

time (dimensions: L);q r x, t is the specific discharge (vector), as observed in

a frame moving with the solid matrix of the deform-ing porous medium, as a function of position and time(dimensions: L t 1);

u x, t is the volumetric strength of internal watersources/sinks as a function of position and time (dimen-sions: t 1); and

Ss x is the volumetric specific storage of the porousmedium as a function of position (dimensions: L 1).Section 2.5.2 discusses the specific storage.

We define the hydraulic head as

h x, t x3 +

p

p0

du

w u g(2.2)

where w p denotes water density as a function of pres-sure, p0 denotes a reference pressure, and g is the accelera-tion of gravity. Consequently, variations in hydraulic headare related to variations in elevation and pressure as

dh = dx3 +dp

w p g. (2.3)

The constitutive relation known as Darcy’s lawdescribes the relationship between the specific dischargeand the hydraulic gradient ( h). The generalized Darcy’slaw is

q r x, t = K x h x, t (2.4)

whereK x is the medium’s hydraulic conductivity (tensor) as a

function of position (dimensions: Lt 1) andis the gradient operator (dimensions: L 1).

See Bear [1972] for a discussion of the generalized Darcy’slaw. The hydraulic gradient is a dimensionless, vectorfunction of position and time. Section 2.5.1 discusses thehydraulic conductivity.

Substituting the right-hand side (RHS) of (2.4) for q r

in (2.1) yields the equation governing the transient flowof homogeneous groundwater through a fully saturated,elastically deformable, porous medium:

K x h x, t + u x, t

= Ss xh x, t

tt 0, x D. (2.5)

The groundwater flow equation (2.5) is the governing equa-tion for hydraulic head. All of the quantities in (2.5) arereal valued. Equation (2.5) is linear in h and its deriva-tives and is a second-order, nonhomogeneous partial dif-ferential equation (PDE). Its coefficients (K and Ss) are

spatially variable but time invariant. We can write thisequation more compactly as

Lh h x, t + u x, t = 0 t 0, x D (2.6)

where we define Lh , the homogeneous, linear, second-order, differential operator, as

Lh f K x f Ss xft

. (2.7)

2.3. INITIAL CONDITION

We assume that the hydraulic head satisfies the follow-ing IC equation:

h x, 0 = x x D (2.8)

where the initial-value function, x , is known (specified)at every point in the space domain. The IC equation (2.8)is linear in h, nonhomogeneous, and with constant coeffi-cient [i.e., the implied “1” immediately preceding the h onthe left-hand side (LHS) of the equation].

2.4. BOUNDARY CONDITIONS

We assume that the hydraulic head satisfies the follow-ing general mixed BC equation:

ch x h x, t + cq x K x h x, t n x

= x, t t 0, x(2.9)

whereThe BC coefficient functions ch x and cq x and the

boundary value function xt are known (specified) atevery point on the boundary;

ch x has dimensions L 1, cq x has dimensions L 1t, andx, t is dimensionless; and

all of the quantities in the BC equation are real valued.The BC equation (2.9) is linear in h and its derivatives,

nonhomogeneous, and with coefficients (ch, cq, K, and n)that are spatially variable but time invariant.

The mixed BC formulation (2.9) can accommodateproblems for which the hydraulic head satisfies any oneof the following three particular types of boundary con-dition at different locations on the boundary:

• specified hydraulic head (also Dirichlet, essential, orfirst type);

• specified volumetric flux (also Neumann, natural, orsecond type); and

• impedance (also Robin or third type).Numerous types of boundary conditions occur in the defi-nition of groundwater flow BVPs, but these three probablyare the simplest and most commonly used types. Othernotable types of boundary conditions include Cauchy(i.e., spatially coincident and simultaneous application ofDirichlet and Neumann conditions) and moving boundary