Hydrodynamics of Time-Periodic Groundwater Flow: Diffusion Waves in Porous Media

By Joe S. Depner and Todd C. Rasmussen

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.

Read an interview with the editors to find out more:
https://eos.org/editors-vox/a-foundation-for-modeling-time-periodic-groundwater-flow

• Language: English
• Publisher: Wiley
• Release date: Dec 2, 2016
• ISBN: 9781119133971

Book preview

Part I

Introduction

1

Introduction

Abstractly, one can consider time‐periodic groundwater flow to be the response of a physical system to a stimulus or excitation. The groundwater system consists of a subsurface porous medium and a resident pore fluid such as fresh or saline water, oil, air, or natural gas. The stimulus is some kind of time‐periodic forcing, and the response is the time‐periodic variation of hydraulic head and specific discharge. Thus, a possible alternative title for this book is Introduction to the Theory of Periodically Forced Groundwater Systems.

1.1. TERMINOLOGY

To clearly articulate ideas about periodic flow, it will be useful to first clarify and standardize some basic related terminology.

For this book, we define functions of practical interest as those mathematical functions that are capable of representing real physical phenomena. We will assume that every function of practical interest is either periodic or aperiodic.

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

Thus, periodicity is a type of global translational symmetry. Synonyms for the term periodic include the terms fully periodic, purely periodic, and strictly periodic.

We can represent every periodic function of time as the sum of one or more distinct harmonic constituents (also frequency components or modes), each of which is a purely sinusoidal function of time, wherein the constituent frequencies are rational multiples of one another. This description coincides with the classical Fourier series representation of a periodic function.

An aperiodic function (also nonperiodic function) is any function that is not periodic. Aperiodic functions include an important class of functions that are closely related to the periodic functions—the almost‐periodic functions. An almost‐periodic function (also, quasiperiodic function) is a function composed of (i.e., formed by summing) two or more harmonic constituents, at least two of which have frequencies that are not rational multiples of one another. A simple example of an almost‐periodic function having two distinct harmonic constituents is

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

Throughout this book we generally use the term periodic function to represent that broad class of functions that includes both the strictly periodic functions and the almost‐periodic functions. What these functions have in common is that we can represent both types by generalized Fourier series. Similarly, we use the term nonperiodic function to represent the class of functions that are neither strictly periodic nor almost periodic.

In the literature readers may encounter numerous terms that have meanings related to concepts of periodicity. For instance, some authors use the terms cyclic (or cyclical) and rhythmic as synonyms for periodic. In some contexts the terms oscillating (or oscillatory) and undulating also can have meanings similar to that of the term periodic; in other contexts these terms might be used to describe types of variation that are more irregular. Similarly, the term fluctuating is commonly used to describe variations that are less regular or less predictable than those described by the term periodic; in fact, the term fluctuating is frequently used to describe variations controlled by random processes. Other terms that may reflect temporally periodic flow include alternating (e.g., alternating flow; see Stewart et al. [1961]), pulsatile (also, pulsating, pulsed, pulsing), and reciprocating.

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

1.2. PERIODIC FORCING

Time‐periodic flow occurs in a groundwater system only if the system undergoes periodic forcing. Based on system geometry, periodic forcing can be of two basic types, which can occur either alone or in combination. In boundary forcing, a system boundary is subject to time‐periodic conditions. An example is the time variation of hydraulic head at an aquifer’s seaward boundary. In internal forcing, an internal water source/sink is time periodic. An example is time‐periodic water injection/pumping at an injection well.

We can classify periodic forcing of groundwater systems according to various additional criteria as well. In summary, criteria for the classification of periodic forcing include 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 versus periodic mechanical loading on a horizontal upper boundary).

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

1.3. POTENTIAL AREAS OF APPLICATION

The following is a summary of potential areas of application 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 the number of applications to grow with time.

Atmospheric Pressure Natural. Diurnal, annual (seasonal), etc. Mechanical effect of barometric forcing on the upper surface of the capillary fringe, on the upper surface of a confining unit, or on the free surface within a well. Here the theory is used both to understand aquifer response to atmospheric forcing and to investigate aquifer hydrologic properties. 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]. Recently the theory has been used to assess the effectiveness of subsurface energy resource exploitation efforts [e.g., Burbey and Zhang, 2010].

Infiltration/Recharge Mass flow (hydraulic) effect at recharge boundaries. Here the theory is used to model the effects of periodic recharge cycles on groundwater systems.

Artificial Recharge Seasonal and other cycles. Examples: Latinopoulos [1984, 1985].

Natural Infiltration/Recharge Associated with seasonal cycles of precipitation and evapotranspiration. Examples: Latinopoulos [1984], Maddock and Vionnet [1998], Rasmussen and Mote [2007].

Plant Water Uptake/Transpiration Mass flow effect at or near the water table. Seasonal and diurnal cycles. Here the theory is used for modeling the interaction of the biosphere with groundwater systems. Examples: Butler et al. [2007], Kruseman and de Ridder [2000], Lautz [2008a,b].

Tides Natural. Multiple periods, from semidiurnal to monthly and longer. Both hydraulic and mechanical effects.

Earth Tides The theory is used to infer aquifer and 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 effectiveness of subsurface energy resource exploitation efforts [e.g., Burbey and Zhang, 2010], and to understand geyser eruption timing [e.g., Rinehart, 1972].

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

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

Field (Pumping) Tests Hydraulic effect at face of well. The testing is conducted in situ. Examples: Black and 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 of material sample. The testing is conducted on material samples in the laboratory. Examples: Adachi and Detournay [1997], Bernabé et al. [2006], Fischer [1992], Kranz et al. [1990], Rigord et al. [1993], Song and Renner [2006, 2007].

Periodic Groundwater Pumping/Injection Artificial. Variable period(s). Hydraulic effect at face of well(s). Here the theory could be used to design subsurface environmental remediation systems [e.g., Zawadzki et al., 2002], to design aquifer recharge systems [e.g., Latinopoulos, 1984, 1985], or to evaluate hydraulic connectivity in functioning geothermal well fields [e.g., Becker and Guiltinan, 2010; Yano et al., 2000].

This list represents only a sample of the available literature.

1.4. CHAPTER SUMMARY

In this chapter we introduced basic terminology on the time behavior of periodically forced groundwater systems and proposed criteria for the classification of time‐periodic forcing. We also briefly listed some potential areas of application for the theory of time‐periodic groundwater flow. The list illustrates that time‐periodic groundwater flow is relevant across multiple fields:

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

Engineering fields: civil (environmental and geotechnical) and energy (geothermal and petroleum).

Part II

Problem Definition

In this part we describe the conceptual, mathematical basis of time‐periodic groundwater flow within the framework of the classical boundary value problem (BVP). This part lays the foundation for subsequent parts.

2

Initial Boundary Value Problem for Hydraulic Head

Consider a confined and fully saturated groundwater system defined on a particular space domain. We wish to define an initial boundary value problem (IBVP) to formally describe the physical behavior of this system. The IBVP 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

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 set in N‐dimensional space, where N can be 1, 2, or 3 depending on the particular situation. Let x denote the N‐dimensional vector of space coordinates. Then the space coordinates of points in D satisfy

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

We can think of D as the N‐dimensional region enclosed by the boundary Γ. The domain and its boundary are mutually disjoint sets:

We assume that we can represent the geometry of the space domain equivalently by specifying a dimensionless vector field, , that satisfies the following conditions:

is directed outwardly perpendicular to the boundary at all points that lie on the boundary and

vanishes at all points that do not lie on the boundary.

Thus, is N dimensional. In addition, is normalized so that it satisfies

This definition and notation are convenient because they allow us to compactly represent the geometry of the space domain using the parameter , which is also convenient for the expression of a flux boundary condition (see Section 2.4).

2.2. GOVERNING EQUATION

We assume that groundwater generally is homogeneous and therefore incompressible except insofar as its low but nonzero compressibility contributes to the storage capacity of porous media. Then consideration of groundwater mass conservation leads to the following continuity equation [see Freeze and Cherry, 1979]:

(2.1)

where

denotes the N‐dimensional divergence operator (dimensions: );

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 deforming porous medium, as a function of position and time (dimensions: );

u(x, t) is the volumetric strength of internal water sources/sinks as a function of position and time (dimensions: ); and

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

We define the hydraulic head as

(2.2)

where ρ w(p) denotes water density as a function of pressure, p 0 denotes a reference pressure, and g is the acceleration of gravity. Consequently, variations in hydraulic head are related to variations in elevation and pressure as

(2.3)

The constitutive relation known as Darcy’s law describes the relationship between the specific discharge and the hydraulic gradient ( ). The generalized Darcy’s law is

(2.4)

where

K(x) is the medium’s hydraulic conductivity (tensor) as a function of position (dimensions: ) and

is the gradient operator (dimensions: ).

See Bear [1972] for a discussion of the generalized Darcy’s law. The hydraulic gradient is a dimensionless, vector function of position and time. Section 2.5.1 discusses the hydraulic conductivity.

Substituting the right‐hand side (RHS) of (2.4) for q (r) in (2.1) yields the equation governing the transient flow of homogeneous groundwater through a fully saturated, elastically deformable, porous medium:

(2.5)

The groundwater flow equation (2.5) is the governing equation for hydraulic head. All of the quantities in (2.5) are real valued. Equation (2.5) is linear in h and its derivatives and is a second‐order, nonhomogeneous partial differential equation (PDE). Its coefficients (K and S s) are spatially variable but time invariant. We can write this equation more compactly as

(2.6)

where we define L h [ ], the homogeneous, linear, second‐order, differential operator, as

(2.7)

2.3. INITIAL CONDITION

We assume that the hydraulic head satisfies the following IC equation:

(2.8)

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

2.4. BOUNDARY CONDITIONS

We assume that the hydraulic head satisfies the following general mixed BC equation:

(2.9)

where

The BC coefficient functions ch(x) and cq(x) and the boundary value function ψ(xt) are known (specified) at every point on the boundary;

ch(x) has dimensions , cq(x) has dimensions , and ψ(x, 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 (c h, c q, K, and ) that are spatially variable but time invariant.

The mixed BC formulation (2.9) can accommodate problems for which the hydraulic head satisfies any one of the following three particular types of boundary condition at different locations on the boundary:

specified hydraulic head (also Dirichlet, essential, or first type);

specified volumetric flux (also Neumann, natural, or second type); and

impedance (also Robin or third type).

Numerous types of boundary conditions occur in the definition of groundwater flow BVPs, but these three probably are the simplest and most commonly used types. Other notable types of boundary conditions include Cauchy (i.e., spatially coincident and simultaneous application of Dirichlet and Neumann conditions) and moving boundary conditions, including nonlinear moving boundary conditions. The mixed BC formulation (2.9) is not capable of representing Cauchy or moving boundary conditions. However, it is sufficiently general to describe the boundary conditions for a wide variety of groundwater flow problems, and it is linear in the hydraulic head h and its derivatives.

We assume that we can partition the domain boundary into three mutually disjoint subsets; i.e.,

where

and where the coefficient functions satisfy

(2.10a)

(2.10b)

(2.10c)

Notice that no boundary points exist for which both c h and c q vanish.

Considering each boundary subset (i.e., Γ h, Γ q, and Γ h q) separately leads to the following explicit form for the boundary condition:

(2.11)

Thus, the BC equation (2.9) simultaneously represents the following:

Dirichlet boundary condition on subset Γ h;

Neumann boundary condition on subset Γ q; and

Robin boundary condition on subset Γ h q.

We can express the BC equation (2.9) even more compactly as

(2.12)

where we define W [ ] as the homogeneous, linear, first‐order, differential operator

(2.13)

2.5. OTHER PARAMETERS

To apply the governing, IC, and BC equations to any particular problem, all of the quantities (variables and constants) in those equations, both known and unknown, must be defined. In the case of our hydraulic head IBVP, we assume that the only unknown quantity is the hydraulic head h; we assume that all other quantities are known. In the context of the IBVP, we refer to the known quantities as parameters. The parameters are listed below, with the space domain on which each must be specified:

Geometry of the space domain, represented by

Material hydrologic properties of the porous medium:

Hydraulic conductivity,

Specific storage,

Internal source/sink strength,

Initial‐value function,

BC coefficient functions, ch(x) and

Boundary value function,

For this particular IBVP all of the parameters are space or space‐time fields.

2.5.1. Hydraulic Conductivity

For the remainder of this book we assume that K is symmetric and positive definite throughout the space domain. Nye [1957] and Bear [1972] present theoretical arguments for the symmetry of K based on the Onsager‐Casimir reciprocal relations [e.g., see Onsager, 1931a, b, and Casimir, 1945]. Note 2.1 presents rationale for the positive definiteness of K.

Note 2.1 Positive Definiteness of Hydraulic Conductivity Tensor

We expect that for real porous media whenever the hydraulic gradient is nonzero the component of the specific discharge in the direction of the fluid driving force (i.e., opposite the hydraulic gradient) is positive:

That is, the specific discharge is not perpendicular to, or opposite, . Using Darcy’s law (2.4) we can express this as

This must be satisfied for all nonzero hydraulic gradient values, so we require the following:

(2.14)

Using Einstein notation, we can also write this as

(2.15)

Inequalities (2.14) and (2.15) are mathematically equivalent; both state that the hydraulic conductivity tensor K is positive definite throughout the space domain.Δ

Because K is real valued, symmetric, and positive definite, it has real‐valued, positive eigenvalues. Hence, K is nonsingular. The inverse of K is the reciprocal hydraulic conductivity tensor, .

2.5.2. Specific Storage

Freeze and Cherry [1979] give the specific storage as

where

α is the bulk compressibility of the porous medium;

ϕ is the porosity; and

βw is the isothermal compressibility of water.

If we allow for the possibility that the bulk compressibility and/or the porosity depend on position, then the specific storage generally depends on position.

For the remainder of this book we assume that throughout the space domain. The rationale for this assumption is as follows. If fluid flow occurs at any point x in D, then both the solid skeleton of the porous medium and the resident fluid (i.e., groundwater) make positive, albeit possibly very small, relative contributions to the specific storage at x via their respective bulk compressibilities.

2.6. SPECIAL CASES

2.6.1. Ideal Media

An ideal medium is one that is homogeneous and isotropic with respect to hydraulic conductivity and homogeneous with respect to specific storage. We represent an ideal medium mathematically as follows:

(2.16a)

(2.16b)

Here I denotes the identity tensor, i.e.,

where δ m p is the Kronecker delta:

(2.17)

and both and .

Consequently, the hydraulic diffusivity, which we define as

(2.18)

is a real, positive constant.

2.6.2. Source‐Free Domains

When the space domain is free of internal sources and sinks, we have

2.7. CHAPTER SUMMARY

In this chapter we derived a general IBVP to describe the transient flow of homogeneous groundwater through a multidimensional, nonhomogeneous, anisotropic, elastically deformable, saturated porous medium whose domain boundary is motionless. We described the major elements of the IBVP, including the space‐time domain, governing equation, initial condition, boundary conditions, and other parameters. The IBVP governing and BC equations are jointly linear in the following variables:

hydraulic head, h(x, t), and its derivatives;

internal source/sink strength, u(x, t);

initial‐value function, ϕ(x); and

boundary‐value function, ψ(x, t).

3

Hydraulic Head Components and Their IBVPs

In Chapter 2 we described an IBVP for hydraulic head h in a saturated medium with stationary boundaries, and we showed that the IBVP is jointly linear with respect to the hydraulic head and its derivatives, the source/sink strength u, the initial value ϕ, and the boundary value ψ. In this chapter, we exploit this linearity to resolve the hydraulic head into multiple components based on their general behavior over time, and we describe the general IBVPs corresponding to these components.

3.1. STEADY AND TRANSIENT COMPONENTS

Assume that we can resolve the hydraulic head, source/sink strength, and boundary value, respectively, into steady and transient components as

(3.1a)

(3.1b)

(3.1c)

Here the bar (i.e., ‒) and tilde (i.e., ~) accents denote the steady and transient components, respectively. The steady components are those for which there is no time variation:

(3.2a)

(3.2b)

(3.2c)

We also assume that we can resolve the initial value into steady and transient components:

3.2. STEADY AND TRANSIENT HYDRAULIC HEAD BVPs

We assume that the steady and transient hydraulic head components independently satisfy corresponding BVPs, which we describe below. Appendix A.1 presents the rationale for this assumption. The steady component of the hydraulic head satisfies the following BVP:

(3.3a)

(3.3b)

where the operators L h and W were defined by (2.7) and (2.13), respectively. This BVP has no corresponding IC. The transient component of the hydraulic head satisfies the following IBVP:

(3.4a)

(3.4b)

(3.4c)

3.3. NONPERIODIC AND PERIODIC TRANSIENT COMPONENTS

We will assume that we can resolve the transient components of the hydraulic head, source/sink strength, and boundary value, respectively, into components that are nonperiodic and time periodic as

(3.5a)

(3.5b)

(3.5c)

Here the hat (i.e.,^) and ring (i.e., °) accents denote the nonperiodic transient and time‐periodic transient components, respectively. Each time‐periodic component represents that part of the transient component that we can represent globally in time by a trigonometric series. Conversely, each nonperiodic transient component represents that part of the transient component that we cannot represent globally in time by a trigonometric series.

Although the nonperiodic transient components may be oscillatory in time, they are neither strictly periodic nor almost periodic. For example, the nonperiodic transient component might be represented by a sum of functions that decay exponentially with time, possibly at different rates. Such functions may include exponentially decaying (damped) sinusoids.

In contrast, the time‐periodic transient components are either strictly periodic or almost periodic in time. Although they do not converge to fixed values as time approaches infinity, they do have finite upper and lower bounds because the alternative is not physically meaningful.

We also assume that we can resolve the initial value for the transient component into nonperiodic and time‐periodic components:

(3.6)

A warning. Terminology is not uniform across different scientific and engineering disciplines. In some analogous contexts, such as electrical conduction in continuous media, authors sometimes use other terms (e.g., steady state or steady periodic) as synonyms for the term periodic transient.

3.4. NONPERIODIC AND PERIODIC TRANSIENT HYDRAULIC HEAD BVPS

We assume that the nonperiodic and time‐periodic transient hydraulic head components satisfy corresponding BVPs, which we describe below. Appendix A.2 presents the rationale for this assumption. The nonperiodic transient component satisfies the following IBVP:

The time‐periodic transient component satisfies the following IBVP equations:

(3.7a)

(3.7b)

(3.7c)

3.5. CHAPTER SUMMARY

In this chapter we showed how, conceptually, one could use superposition to resolve the hydraulic head, source/sink strength, initial value, and boundary value into steady, nonperiodic transient, and time‐periodic transient components. Combining those results leads to the following decompositions:

(3.8)

Similarly, we can resolve the hydraulic gradient and specific‐discharge vectors into their corresponding steady, nonperiodic transient, and time‐periodic transient components:

(3.9)

where

(3.10a)

(3.10b)

(3.10c)

We also defined a BVP corresponding to each of the steady, nonperiodic transient, and time‐periodic transient hydraulic head components. While in principle we can represent each of the time‐periodic transient components of hydraulic head, source/sink strength, and boundary value globally in time by a nontrivial trigonometric series, this is not true of the corresponding nonperiodic transient and steady components.

4

Periodic Transient Components

In Chapter 3 we saw that the periodic transient components of hydraulic head, source/sink strength, and boundary value are periodic or almost periodic in time, so that we can represent them globally in time by trigonometric series (i.e., sums of harmonic constituents). In this chapter we give expressions for the trigonometric series and the harmonic constituents. For each constituent we describe the relevant parameters (frequency, amplitude, and phase) and the relationships between them. We also express the hydraulic gradient and specific‐discharge vector fields as trigonometric series and relate their harmonic constituents to those of the hydraulic head.

4.1. TRIGONOMETRIC SERIES REPRESENTATION

We assume that we can represent the time‐periodic transient components of hydraulic head, source strength, and boundary value globally in time by trigonometric series as

(4.1a)

(4.1b)

(4.1c)

where the corresponding harmonic constituents are

(4.2a)

(4.2b)

(4.2c)

We refer to (4.2) as the rectangular form for the harmonic constituents. We define the harmonic constituent parameters as follows:

m is the harmonic constituent index (integer) (dimensionless) and

ωm is the angular frequency (also circular frequency) of the mth harmonic constituent (i.e., the constituent frequency) (dimensions: , units: rad s ).

We can express the harmonic constituents in polar form also, as

(4.3a)

(4.3b)

(4.3c)

where

M(x ; ωm) is the amplitude (also amplitude function) of the mth harmonic constituent as a function of position:

Mh(x; ωm) has dimensions L,

Mu(x; ωm) has dimensions , and

Mψ(x ; ωm) is dimensionless;

θ(x ; ωm) is the phase (also phase function) of the mth harmonic constituent as a function of position (dimensionless, units: rad); and

is the corresponding unit‐amplitude constituent of the mth harmonic constituent as a function of position (dimensionless).

The A and B coefficient functions are related to the amplitude and phase functions as

(4.4a)

(4.4b)

(4.4c)

and

(4.5a)

(4.5b)

(4.5c)

Equations (4.4) and (4.5) follow from (4.2), (4.3), and trigonometric identity (B.2a).

Solving (4.4) and (4.5) for the amplitudes yields

(4.6a)

(4.6b)

(4.6c)

Equations (4.4) and (4.5) yield

(4.7a)

(4.7b)

(4.7c)

Exercise 4.1 Invariance under Time Shift: Rectangular Form Coefficients

Problem: Consider a uniform translation of the time coordinate:

(4.8)

where Δt is a constant time shift and the breve accent (i.e., ˇ) over a variable indicates the variable is expressed relative to the transformed time coordinate, . Derive expressions showing how the rectangular form amplitude functions, A h and B h, transform under a uniform time shift. Are A h and B h invariant under a uniform time shift?

Solution: Solving (4.8) for t and substituting the result for t in (4.2a) lead to

Using trigonometric identities (B.2b) and (B.2d), we can write this as

Grouping terms gives

Noting that , we can express this as

(4.9)

where we define the transformed amplitudes as

(4.10a)

(4.10b)

Thus, the rectangular form amplitudes A h and B h are not invariant under a time shift.□

Exercise 4.2 Invariance under Uniform Time Shift: Amplitude Function

Problem: Derive an equation that shows how the amplitude function, M h, transforms under a uniform time shift. Is M h invariant under a uniform time shift?

Solution: The amplitude function for the transformed harmonic constituent (4.9) is [compare equation (4.6a)]

Substituting the RHSs of (4.10) for the corresponding terms in the previous equation and using trigonometric identity (B.1) to simplify the resulting expression yield

(4.11)

Thus, the amplitude function M h is invariant under a uniform time shift.□

Exercise 4.3 Invariance under Uniform Time Shift: Phase Function

Problem: Derive an equation that shows how the phase function, θ h, transforms under a uniform time shift. Is θ h invariant under a uniform time shift?

Solution: Substituting the RHSs of (4.4a) and (4.5a) for A h and B h, respectively, in (4.10) yields

Using trigonometric identities (B.2b) and (B.2d), respectively, we can write these as

(4.12a)

(4.12b)

We also require

(4.13a)

(4.13b)

Combining (4.12), (4.13), and (4.11) gives

Simultaneous solution of these requires

To ensure that the phase is single valued, we must fix the value of n; we arbitrarily set , giving

(4.14)

Thus, the phase function θ h is not invariant under a uniform time shift.□

Exercises 4.1, 4.2, and 4.3 illustrate an important difference between the polar and rectangular forms with regard to uniform time shifts. With the polar form, the amplitude is invariant and the phase undergoes a shift of . The effect of a uniform time shift on the rectangular form amplitudes A h and B h is less intuitive.

4.2. HARMONIC CONSTITUENT PARAMETERS

4.2.1. Amplitude

The harmonic constituent amplitudes are, by definition, real valued and nonnegative:

(4.15a)

(4.15b)

(4.15c)

This assumption does not reduce the generality of our results, because we can represent any harmonic constituent with a negative amplitude by an equivalent constituent with a positive amplitude and a phase shift of π rad:

This is based on the angle‐sum relation for the cosine (see trigonometric identity B.2a with ).

Any time‐independent terms (i.e., harmonic constituents for which ) in the trigonometric series in (4.1) vanish,

(4.16a)

(4.16b)

(4.16c)

because, by definition, such terms would contribute to the corresponding steady component rather than the transient component.

4.2.2. Phase

Consider a general phase function, θ, which could represent θ h, θ u, or θ ψ. The phase function is real valued at every point in its domain. We will assume that, with few exceptions, the phase function is spatially continuous on its domain. In addition, we expect that the phase function is bounded on any finite space domain; in contrast, on an infinite domain it may be unbounded.

The phase function is undefined at the zero points of the corresponding amplitude function (i.e., at those points where ); hence, the domain of the phase function does not include the zero points of the amplitude function. That is, the domain of the phase function is discontinuous at such points. Strictly, the phase function is neither continuous nor discontinuous at these points.

Because the inverse‐tangent function (arctan) is multiple valued, equations (4.7) by themselves are insufficient to unambiguously define the phase functions on their respective domains. Generally one must also impose the requirement that the phase function is spatially continuous on its domain. Then each of equations (4.7) leads to an equation of the form

(4.17)

where the value of n is determined by the requirement that the phase function is spatially continuous and arctan denotes the principal‐value inverse‐tangent function, i.e.,

4.2.3 Frequency

We assume that the angular frequencies are unique:

(4.18)

where m and n are positive integers. This assumption does not reduce the generality of our results because we can represent the sum of any two constituents having identical frequencies (and arbitrary amplitudes and phases) as a single equivalent constituent having the same