Directory UMM :Data Elmu:jurnal:A:Advances In Water Resources:Vol22.Issue6.1999:

Periodic forcing in composite aquifers

Michael G. Trefry

*

Centre for Groundwater Studies, CSIRO Land and Water, Private Bag, PO Wembley, WA 6014, Australia

(Received 26 February 1998; revised 7 August 1998; accepted 19 August 1998)

Observations of periodic components of measured heads have long been used to estimate aquifer di€usivities. The estimations are often made using well-known solutions of linear di€erential equations for the propagation of sinusoidal boundary ¯uctuations through homogeneous one-dimensional aquifers. Recent ®eld data has indicated several instances where the homogeneous aquifer solu-tions give inconsistent estimates of aquifer di€usivity from measurements of tidal lag and attenuation. This paper presents new algebraic solutions for tidal prop-agation in spatially heterogeneous one-dimensional aquifers. By building on ex-isting solutions for homogeneous aquifers, comprehensive solutions are presented for composite aquifers comprising of arbitrary (®nite) numbers of contiguous homogeneous sub-aquifers and subject to sinusoidal linear boundary conditions. Both Cartesian and radial coordinate systems are considered. Properties of the solutions, including rapid phase shifting and attenuation e€ects, are discussed and their practical relevance noted. Consequent modal dispersive e€ects on tidal waveforms are also examined via tidal constituent analysis. It is demonstrated that, for multi-constituent tidal forcings, measured peak heights of head oscilla-tions can seem to increase, and phase lags seem to decrease, with distance from the forcing boundary unless constituents are separated and considered in isola-tion. Ó _{1999 Elsevier Science Limited. All rights reserved}

Key words: groundwater, ¯ow, periodic, sinusoidal, tidal, composite, interface, lag, attenuation, dispersion.

1 INTRODUCTION

The reliable determination of aquifer hydrological properties is a central, yet problematic, component of any practical hydrological exercise. Traditional pum-ping tests are common tools for gathering information on hydraulic conductivity and storativity of aquifers over length scales of the order of tens of meters or less. Where present, periodic ¯uctuations in monitored heads a€ord valuable determinations of these aquifer param-eters that are independent of pumping exercises. For example, tidal ¯uctuations in sea or river water levels can induce signi®cant head ¯uctuations far into neigh-bouring aquifers2,5,10,22,24±26, as well as in the near-®eld11±13,27. Similarly, the periodic nature of climatic forcings are also of interest in characterising aquifer properties14±20. For simple tidal (water level) forcings,

relating the head and tidal ¯uctuations to determine hydraulic conductivity and storativity integrates these parameters between the tidally forced boundary and the monitoring bore, often at a length scale much greater than that achievable with single (or many) pumping tests. In this way, tidal analyses are often used to esti-mate quickly average aquifer properties based on simple tidal propagation models for homogeneous aquifers.

In the 1950's Jacob9 and Ferris6,7 derived algebraic solutions for tidal propagation in a homogeneous semi-in®nite one-dimensional aquifer, facilitating tidal esti-mations of hydraulic conductivity and storativity. These solutions have been used widely in practical studies; recent examples are Serfes22, Erskine5, Millham and Howes10, and Trefry and Johnston26. The early solutions have been extended to a variety of one- and two-dimensional systems24, including layered aquifer systems, and aquifers forced by ¯uctuating evapotran-spiration or pumping. A short review of the various approaches is given by Townley25, who, in the same

Ó1999 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0309-1708/99/$ ± see front matter

PII: S 0 3 0 9 - 1 7 0 8 ( 9 8 ) 0 0 0 3 7 - 2

*

E-mail: mike.trefry@per.clw.csiro.au.

paper, established a general analytical solution for pe-riodic propagation in homogeneous one-dimensional aquifers subject to linear boundary conditions.

The solutions are normally expressed in terms of lag and eciency functions, which relate aquifer properties to di€erences in phases and amplitudes of ¯uctuations (with respect to the forcing conditions) as functions of position in the aquifer, respectively. Thus, the Ferris6,7 semi-in®nite aquifer result yields a linear relationship between phase lag and distance from a ¯uctuating head boundary condition, and an exponential relationship between amplitude attenuation and distance. These re-lationships are parameterised by the aquifer di€usivity, T/S, where Tis the transmissivity [L2_Tÿ1_{] andS is the} storage coecient [ÿ], and by the ¯uctuation period, P [T]. Analogous results for ®nite aquifers are given by Townley25. Thus measurements of tidal lags and e-ciencies in an aquifer provide direct estimates of the aquifer di€usivity. Unfortunately, it is common for the lag-based and eciency-based estimates of di€usivity to be at odds with each other, even when the lag and e-ciency data are measured at the same time in the same monitoring bore11,22,26. In particular, di€usivity esti-mates from eciency data were found to tally closer with pumping test results than were lag-based estimates. Attempts to reconcile these di€erences by numerical simulation of the aquifer dynamics were also fruit-less22,26, and aquifer heterogeneity and con®nement status were invoked as possible explanations for the discrepancies, although detailed information on these factors was lacking.

This paper attempts to address the spatial heteroge-neity question by seeking algebraic solutions to periodic forcing problems in arbitrary composite one-dimen-sional aquifers. No attempt is made to solve the problem of periodic forcing in aquifers of higher dimensionality, where interference of pressure waves generated at dif-ferent points on the boundary becomes signi®cant. Similarly, the simplifying assumption that each of the composite sub-aquifers is internally homogeneous is made, thereby eliminating the need for statistical models of random heterogeneity. The method is to extend Townley's treatment to aquifers of speci®c composite structure, i.e. systems of homogeneous sub-aquifers ar-ranged contiguously, with ¯ow matching conditions applying at sub-aquifer interfaces. The following section details the mathematical approach and solution. Prop-erties of the solution are then exhibited, followed by a discussion of the dispersive nature of the propagation and the consequent distortions of forcing waveforms. The algorithms presented in this paper are exact, re-quiring only the solution of algebraic matrix systems and evaluation of complex functions. These solutions, whilst attainable through conventional means, are best performed using computer algebraic techniques. The computer algebra package Mathematica29 was used throughout this work.

2 PERIODIC FLOW IN COMPOSITE ONE-DIMENSIONAL AQUIFERS

Attention is focused on the propagation of periodic forcings through one-dimensional aquifer systems. In analogy with a periodic heat ¯ow analysis of Carslaw and Jaeger3, Townley25 derived a general solution for periodic ¯ow in homogeneous aquifers subject to sinu-soidal boundary conditions. His method, reproduced below in brief, is extended here to aquifers with dis-continuities in material properties, i.e. interfaces. The reader is referred to Townley's paper for full details of his method. For convenience, Townley's conventions and notation are adhered to throughout this paper where possible.

2.1 Townleys method for homogeneous 1D Cartesian aquifers

In Cartesian coordinates, the aquifer ¯ow equation is

Soh

o_t ˆT o2_h

o_x2‡R; …1†

where h(x,t) is the head, and R(x,t) is a distributed recharge term. If hand Rcan be separated into steady and periodic components, viz.

h…x;t† ˆhs…x† ‡hp…x†exp…ixt† …2†

R…x;t† ˆRs…x† ‡Rp…x†exp…ixt† …3† wherexˆ2p/Pis the angular frequency of the ¯uctua-tion, hs and hp are the steady and periodic head com-ponents, and Rs and Rp are the steady and periodic recharge components, respectively. Where recharge pe-riodicities are di€erent to the head pepe-riodicities, as in diurnal head forcings14 and seasonal recharge forc-ings16,20, the linearity of the equations allows separate periodic head and periodic recharge solutions to be added. Physical heads and recharge rates are assured in eqns (2) and (3) by taking real parts of the complex products hp exp(ixt) and Rp exp(ixt), respectively. Substituting eqns (2) and (3) into eqn (1) leads to the separated forms:

Td

2_h s

dx2 ‡Rsˆ0; …4†

Td

2_h p

dx2 ÿixShp‡Rpˆ0: …5† Thus Townley shows that, under the assumptions of periodic forcing, the linear ¯ow partial di€erential equation (1) decouples into two ordinary di€erential equations for the steady and periodic components. The relevant solutions are:

hsˆ ÿ

Rs 2Tx

2_‡C

1x‡C2; …6†

hpˆD1coshax‡D2sinhax‡

Rp

wherea2_{ˆixS/Tˆ2piS/TP, andC1,C2,D1, andD} 2are integration constants to be ®xed by application of boundary conditions. eqns (6) and (7) are fundamental solutions to the periodic forcing problem. Townley then de®nes a solution domain extending fromxˆ0 toxˆL, and lists relationships between the integration constants arising from imposing Dirichlet, Neumann, or Cauchy boundary conditions on eqns (6) and (7). These rela-tionships allow theCandDintegration constants to be ®xed for a given physical problem; the relationships are summarised here in Table 1.

Townley's method is completed for a given physical problem by choosing appropriate boundary conditions from Table 1, solving simultaneously for the C and D coecients using the listed relationships, and reconsti-tuting the solution according to eqns (6), (7) and (2).

2.2 Composite 1D Cartesian aquifers

Attention is now directed to the heterogeneous com-posite aquifer case. Consider a ®nite aquifer consisting of N contiguous sub-aquifers, as depicted in Fig. 1. Each sub-aquifer, referenced by the indexj, is assumed to be a homogeneous unit and is bounded either by two

sub-aquifer interfaces (with sub-aquifers jÿ1 and j+ 1), or by one sub-aquifer interface and one problem boundary condition (ifjˆ1 orN). Let the coordinate of the interface between sub-aquifers j and j+ 1 be de-noted byxj. TheNÿ1 interface coordinates {xj} range

between 0 andL, and need not be regularly spaced. The ¯ow equation for sub-aquiferjis

Sjohj

o_t ˆTj o2_h

j

o_x2 ‡R; …8†

where the recharge termRis assumed to be equal for all sub-aquifers, and Tj and Sj are the sub-aquifer

trans-missivity and storage coecient. Writing

hj…x;t† ˆhjs…x† ‡h

j

p…x†exp…ixt†; …9† leads to:

Tj

d2hj

s

dx2 ‡Rsˆ0; …10†

Tj

d2hj

p

dx2 ÿixSjh

j

p‡Rpˆ0 …11†

which have fundamental solutions:

hj

sˆ ÿ

Rs 2Tj

x2‡C₁jx‡C₂j; …12†

hj_pˆDj₁coshajx‡Dj₂sinhajx‡ Rp ixSj

; …13†

where aj2ˆ2piSj/TjP. The overall solution for the

composite aquifer is given by theNsets of eqns (12) and (13) for the N sub-aquifers. There are 2N integration constants Cj₁, Cj₂ associated with the steady sub-aquifer solutions, and 2N integration constants Dj₁, Dj₂ associ-ated with the periodic sub-aquifer solutions. Application of problem boundary conditions to sub-aquifers 1 andN will supply two steady and two periodic relationships for the integration constants. The remaining degrees of freedom in the fundamental solutions are ®xed by im-posing conditions for continuity of head and Darcy ¯ux at each of the Nÿ1 sub-aquifer interfaces. For the steady components, these interface matching conditions are:

hj

s…xj† ˆhjs‡1…xj†; …14† Table 1. Boundary condition relationships for Cartesian aquifers from Townley25,expressed in terms of steady and periodic components of the prescribed values

Boundary condition Steady component Periodic component

Dirichlet (Prescribed head) Hsˆ ÿ

Rs 2Tn

2

‡C1n‡C2 HpˆD1coshan‡D2sinhan‡_iR_xp_S

Neumann (Prescribed ¯ux) n…n†QsˆRsnÿTC1 n…n†Qpˆ ÿaTD1sinhanÿaTD2coshan

Cauchy (Mixed) RsnÿTC1ˆn…n†A Gs‡₂R_Tsn2ÿC1nÿC2

ÿaTD1sinhanÿaTD2coshanˆn…n†A

GpÿD1coshanÿD2sinhanÿ_iR_xp_S

h i

These components areHsandHp(Dirichlet),QsandQp(Neumann), andGsandGp(Cauchy). The boundary coordinate variablen has values 0 orL;n(0)ˆ1 andn(L)ˆ ÿ1.Ais a conductance parameter for the Cauchy condition.

Fig. 1.Schematic and coordinate system for a one-dimensional

Tjdh

and for the periodic components:

hj_p…xj† ˆhj_p‡1…xj†; …16†

Physical solutions are gained by using the matching conditions at all sub-aquifer interfaces and the appro-priate problem boundary conditions to solve simulta-neously for theCandDconstants. The linearity of both the boundary condition relationships and the interface conditions, together with the decoupling of the steady and periodic components, means that the simultaneous solution reduces to two standard matrix problems ± one for the steady component and one for the periodic component. This matrix method does not depend on the use of Darcy ¯ux interface conditions (as used in eqns (14)±(17)). Any interface conditions that are linear functions ofhj

p or its ®rst derivative are suitable for use with the matrix method, including, for example, leakage conditions23. In this work attention is limited to the Darcy ¯ux interface conditions above.

Consider the steady component of an N-composite aquifer system. The N fundamental solutions are parameterised by the C₁j and Cj₂ constants. Table 1 shows that the steady boundary conditions (for sub-aquifers 1 and N) reduce to linear relationships of the form

lsC1‡msC2ˆ/s: …18†

For example, a steady Neumann condition at the boundary xˆn reduces to eqn (18) provided ls_ˆT, ms_{ˆ0, and /}s_ˆR

s n ÿn(n) Qs. Identi®cations of ls_,ms_,/s _{coecients appropriate for various boundary} conditions are easily established by reference to Table 1. Using eqn (12) in eqns (14) and (15) yields analogous linear relationships for theCconstants arising from the interface conditions:

Thus, solving for the Cconstants involves 2Nlinear equations, two from boundary condition equations (eqn (18)) for sub-aquifers 1 and N, and 2Nÿ2 from the interface conditions (eqns (19) and (20)) for the Nÿ1 sub-aquifer interfaces. These equations de®ne the steady component matrix equation

MsCˆVs; …21† vector of all degrees of freedom of the steady component solution, VsTˆ …/s₁;Rsx21…Tÿ

N† is a vector of residual constants

from the linear eqns (18)±(20), and the steady solution coecient matrix,Ms_{, is given by}

Ms_ˆ

In these de®nitions, T _{denotes the transposition} oper-ator, and fls

dary condition coecients for sub-aquifers 1 and N, respectively. Solution of eqn (21) determines the inte-gration constants Cj₁, Cj₂ that, in turn, determine the steady head components hj

s in all sub-aquifers, and hence enable the calculation of the complete steady component solution hs for the overall composite aquifer.

The matrix formulation for the periodic component of the solution follows similarly. The boundary condi-tions contribute relacondi-tionships between theDintegration constants in sub-aquifers 1 and N that can be written as

lp_D

1‡mpD2ˆ/p: …23†

Inserting the fundamental periodic solution of eqn (13) into eqns (16) and (17) yields the following interface relations for the sub-aquifer periodic components:

Dj₁coshajxj‡Dj2sinhajxj‡

The boundary condition and interface relations then de®ne the periodic component matrix equation

MpDˆVp …26† vector of all degrees of freedom of the periodic com-ponent solution, VpTˆ …/p1;Rp…S2ÿ1ÿS1ÿ1†=ix;0;. . .; Rp…Sÿ1

j‡1ÿSjÿ1†=ix;0;. . .;/

p

N† is a vector of residual

Solution of eqn (26) is sucient to determine the sub-aquifer periodic componentshj

p, and thence the overall composite aquifer periodic componenthp.

The previous discussion presents an algorithm for the calculation of exact solutions for head distribu-tions in composite aquifers with periodic forcing. The ®nite composition numberNis left unspeci®ed, leading to generalised de®nitions of matrix and vector quan-tities involved in the solution. It is useful to emphasise the ease of the approach by means of a simple ex-ample.

2.3 Example A. Dirichlet forcing in a twin-composite Cartesian aquifer with zero recharge

Consider an aquifer bounded at xˆ0 by a no-¯ow boundary, at xˆL by the Dirichlet condition h(L,t)ˆHs+Hp exp(2pit/P), and with Rˆ0. Let there be a single aquifer discontinuity atx1. Forx6x1letT1 andS1apply; elsewhere letT2andS2 apply. It remains to construct and solve the steady and periodic compo-nent matrix systems. The steady compocompo-nent matrix system is

which has solutionC1 1ˆC steady head component in the composite aquifer is ¯at. The periodic component matrix system is

0 a1T1 0 0

The solution vector to eqn (29) is

The overall periodic component is then expressed as

hp…x† ˆ D

The solution accounts for both composite and ho-mogeneous aquifers. Hoho-mogeneous conditions are gained by settingT1ˆT2andS1ˆS2. TheDcoecients then reduce toD1₁ˆD2

1ˆHp=cosha1LandD12ˆD 2 2ˆ0, recovering Townley's solution for his Example 125.

2.4 Composite 1D radial aquifers

following Townley25, for sinusoidal forcings the tran-sient radial ¯ow equation

Soh

decouples to the steady and periodic component equa-tions:

For the composite aquifer case, these component equations admit fundamental solutions, for sub-aquifer j, of the forms

whereI0andK0are modi®ed Bessel functions 1

of order zero, and theEandFcoecients are degrees of freedom to be ®xed by applying boundary and interface condi-tions. Table 2 lists boundary condition relationships appropriate for the radial problem25.

Interface matching conditions on head and Darcy ¯ux continuity are applied to the steady and periodic solution components. For the steady components, the conditions are:

and for the periodic components:

hj_p…rj† ˆhj_p‡1…rj†; …40†

whererjis the radial interface coordinate of interfacej.

Noting the derivative identities I₀0ˆI1 and K00 ˆ ÿK1, the interface conditions become, for the steady compo-nents:

and for the periodic components:

F₁jI0…ajrj† ‡F2jK0…ajrj† ‡

The matrix systems for solving for the E and F co-ecients then follow as before. For brevity, their de®-nitions for a general composite aquifer are not given here. However, the forms for a twin-composite are listed explicitly in the following example.

2.5 Example B. Dirichlet forcing in a twin-composite radial aquifer with non-zero recharge

Consider a circular aquifer bounded at rˆ0 by a no-¯ow boundary, at rˆL by the Dirichlet condition h(L,t)ˆHs +Hp exp(2pit/P), and with RˆRs +Rp exp(2pit/P). Let there be a single aquifer discontinuity at r1. Forr6r1letT1andS1apply; elsewhere letT2andS2 apply. From eqns (42)±(45), and from Table 2, matrix systems for determining the steady coecientsEand the periodic coecientsFare, respectively:

1 0 0 0

Table 2. Boundary condition relationships for radial aquifers from Townley25, expressed in terms of steady and periodic components of the prescribed values

Boundary condition Steady component Periodic component

Dirichlet (Prescribed head) Hsˆ ÿ₄R_Tsq2‡E1 ln q‡E2 HpˆF1I0…aq† ‡F2K0…aq† ‡_iR_xp_S Neumann (Prescribed ¯ux) n…q†VsˆpRsq2ÿ2pTE1 n…q†Vpˆ ÿ2paTF1qI1…aq† ‡2paTF2qK1…aq†

Cauchy (Mixed) Rs

2qÿ

0 1 0 0

The steady and periodic coecient solution vectors are:

E₁1 which is in agreement with Townley's Example 5 for a homogeneous radial aquifer25.

3 PROPERTIES OF THE COMPOSITE SOLUTIONS

In the preceding section exact matrix algorithms for generating algebraic expressions for periodic heads in composite aquifers were presented. However the com-plexity of the resulting expressions, even for simple composite aquifers, can distract attention from the un-derlying physics. In this section the properties of the composite solutions are explored graphically.

Fig. 2(a) shows a three-dimensional plot of the peri-odic component of head versusxandtfor a Cartesian aquifer with Dirichlet forcing at xˆL, and a no-¯ow condition atxˆ0. The aquifer is composed of two sub-aquifers, one (sub-aquifer 1) with lower di€usivity than the other (sub-aquifer 2). Near the Dirichlet boundary the high di€usivity of sub-aquifer 2 leads to rapid propagation and low attenuation of the ¯uctuating sig-nal. After passing the sub-aquifer interface the signal is rapidly attenuated and lagged in sub-aquifer 1. Fig. 2(b) shows a similar plot for an aquifer composed of three sub-aquifers. This case corresponds to a uniform aquifer of high di€usivity (sub-aquifers 1 and 3) interrupted by a thin zone of low di€usivity (sub-aquifer 2). The plot

highlights how sub-aquifer 2 acts as a rapid phase-shifter and attenuator for the propagating signal. Fig. 2 gives appealing pictorial representations of the charac-teristics of propagation in thex-tphase space. However it is often more useful to concentrate on speci®c modal transfer properties.

The Jacob and Ferris solutions for in®nite homoge-neous aquifers with Dirichlet forcing are commonly expressed in terms of the lags,s, and attenuations, a, of ¯uctuations measured in the aquifer with respect to the forcing signal. That is:

aˆexp ÿx

where x, in this case, measures the distance from the forced boundary, andPˆ2p/x. Thus, for a given peri-odic mode P, the amplitude of ¯uctuation declines ex-ponentially with distance, whilst the phase lag increases linearly. Townley's solutions for ®nite aquifers essen-tially parameterize these results byL, the aquifer length. Following Townley, the piecewise attenuation coe-cient in a composite aquifer with Dirichlet forcing is given by: This phase lag can be mapped from radians to units of ¯uctuation periods by the transformation:

U…Dsh† ˆ

1ÿDsh

2p for 0<Dsh6p;

ÿDsh

2p for ÿp<Dsh60: "

…54†

Fig. 3(a) shows the variation of the attenuation co-ecienta(expressed asjhpj=jHpjon a logarithmic scale) with distance in the example triple-composite Cartesian aquifer described in Fig. 2(b). The e€ects of the two sub-aquifer interfaces are visible as changes in slope of the curve as the aquifer properties change from high dif-fusivity to low di€usivity and back to high di€usivity. The associated curve for the lag functionUis shown in Fig. 3(b). Again, changes in slope of the lag function are evident at the interfaces.

Clearly, aquifer heterogeneities hold the potential to induce sharp changes in attenuation and rapid phase shifts of periodic modes as functions of distance in the aquifer. Such changes can lead to signi®cant errors in the interpretation of tidal signals measured at

bore-holes in aquifers incorrectly assumed to be homoge-neous. For example, ®tting eqns (50) and (51) to a single-well analysis of attenuation and lag ignores the possibility of composite structure in the aquifer. The e€ect of this can also be seen in Fig. 3, where single measurements of attenuation and lag (indicated by dots) can de®ne e€ective homogeneous aquifer di€u-sivities (indicated by dashed lines) that are signi®cantly di€erent to the actual di€usivities of the three aquifers. Another tidal measurement elsewhere in sub-aquifer 1 would yield a di€erent sub-aquifer di€usivity again, although pumping tests at all points within sub-aquifer 1 would give, in principle, correct and identical values forT and S.

For twin-composite aquifers, the nature of the lag and attenuation functions give rise to general rules of thumb for tidal analysis. Fig. 4 shows schematic at-tenuation and lag curves (solid lines) for twin-com-posite aquifers. Solid dots indicate single point measurements, which can be used with homogeneous models to estimate e€ective homogeneous aquifer dif-fusivities (dashed lines). The diagrams show that if the homogeneous di€usivities are signi®cantly lower than the actual di€usivity (as determined by a pumping test) at the measuring point, then an intervening sub-aquifer of low di€usivity is indicated (Fig. 4(a),(b)). If the ho-mogeneous di€usivities are signi®cantly higher than the local di€usivity, then an intervening sub-aquifer of high

Fig. 2.Phase space (xÿt) plot of periodic component of head for two composite aquifers subject to Dirichlet forcing at

xˆL. Part (a) shows a twin-composite aquifer with L2_S 1/

T1Pˆ50/9,L2S2/T2Pˆ1/24,T1/T2ˆ3/20, (T1/S1)/(T2/S2)ˆ3/ 400,x1/Lˆ4/5. Part (b) shows a triple-composite aquifer with

L2_S

1/T1PˆL2S3/T3Pˆ1/40, L2S2/T2Pˆ50, T1/T2ˆT3/

T2ˆ200, (T1/S1)/(T2/S2)ˆ(T3/S3)/(T2/S2)ˆ2000, x1/Lˆ3/4,

x2/Lˆ4/5.

Fig. 3.Attenuation (a) and lag (b) functions (solid lines) for the propagating mode and triple-composite aquifer described in Fig. 2(b). Dashed lines show the schematic functions esti-mated from homogeneous (Ferris or Townley) models based

di€usivity is indicated (Fig. 4(c),(d)). Where aquifer homogeneity is in doubt, a useful test, though by no means de®nitive, is to use two measuring bores. If ap-plying homogeneous models to the two data sets yields incompatible estimates of aquifer di€usivity, then composite structure (or other heterogeneity) can be inferred and more sites in the aquifer must be moni-tored.

Unfortunately, this test can be unreliable. Consider Fig. 5, which shows schematic attenuation and lag functions for a notional triple-composite aquifer. The e€ects of sub-aquifers 2 and 3 sum to give attenuations and lags at interface 1 equal to those that would apply if the aquifer was homogeneous with the properties of sub-aquifer 1 throughout. In this special case, measurements at all points in sub-aquifer 1 would give consistent es-timates of aquifer properties based on homogeneous models, despite the strongly composite structure else-where in the aquifer.

In general, the true structure and properties of a composite aquifer can be determined from measure-ments of periodic components only if each sub-aquifer is sampled, i.e. measurements in one sub-aquifer cannot accurately determine the properties of the remaining sub-aquifers. The interface ¯ux condition eqn (17) means that the dimensionless groupL2_{S/TP, whereLis} identi®ed with the sub-aquifer length, noted by Townley is no longer sucient to describe the sub-aquifer dy-namics. Enough measurements must be taken in each sub-aquifer to ®xSi,Ti and the interface coordinatexi.

This may notionally be accomplished by measuring at-tenuations and lags at two points in each sub-aquifer, and ®tting theSi, Ti, and xivalues by an inverse

tech-nique using the algebraic expressions derived in this work. However the described method is somewhat heuristic, relying on an already good understanding of the aquifer structure. The advantage lies in that

mea-suring ¯uctuations of head at dispersed sites in an aquifer is much easier than conducting several pumping tests to determine S and T, and then inferring the in-terface coordinates.

Fig. 5. Schematic attenuation (a) and lag (b) functions for a notional triple-composite aquifer subject to Dirichlet forcing at xˆL. The combined e€ects of sub-aquifer 2 (high di€usi-vity) and sub-aquifer 3 (low di€usidi€usi-vity) lead to a situation where single point measurements (black dots) give homoge-neous model estimates of di€usivity (dashed lines) that are

4 DISPERSIVE MODAL PROPAGATION IN AQUIFERS

The above theories consider the response of homoge-neous and composite aquifers to single mode forcings, however through the linearity of the dynamical equa-tions more complicated forcings can also be accounted for. In this section the physical characteristics of the propagation of multi-sinusoidal periodic forcings will be discussed in terms of a simple tidal example, although similar concepts will apply to other multi-sinusoidal boundary conditions, e.g. arising from seasonal re-charge, barometric variations, cyclic pumping etc.

Tidal waveforms are commonly classi®ed into four general types ± diurnal, semi-diurnal, mixed, and double tides21. Diurnal tides have, on average, one high and one low tide each day; semi-diurnal tides have two such tides per day with comparable amplitudes. Mixed and double tides are more complex waveforms. Each tidal type can be decomposed into many sinusoidal constituents4, whose amplitudes, frequencies and phases are functions of geographic location. The optimal separation of tidal constituents is a ®eld of continuing interest8,28, however for the present discussion it is sucient to note that even simple diurnal and semi-diurnal tides may contain many constituent modes densely clustered in frequency.

In the context of oscillating modes, propagation is said to bedispersiveif the phase velocity is a function of mode frequency, as is true for the periodic solutions above. This dispersion is a simple consequence of the aquifer ¯ow equations and is not to be confused with the more familiar hydrodynamic dispersion arising from soil matrix properties. As a consequence of modal disper-sion, oscillating waveforms arising from multiple sinu-soidal components will change in shape as they propagate through even homogeneous aquifers2. Aqui-fers with composite structure will lead to still more complicated evolutions of waveform with distance. Change in the waveform, X, through dispersive propa-gation will occur when the waveform is comprised of two or more sinusoidal constituents

XˆX

M

mˆ1

Amsin…xmt‡u_m†; …55†

where Am is the amplitude, xm the angular frequency

andumthe phase of constituent min theM-constituent

waveform. By a suitable choice of the time coordinatet, u1 can be set to zero; the remaining um are then the

relative phases of the sinusoidal constituents with re-spect to constituent 1. As explained above, the ampli-tudes and phases of the constituents of Xare functions of both the penetration distance into the aquifer, and of xm. In this light, the attenuation and lag functions in the

earlier section represent the characteristic dispersion relations for the aquifer. These functions permit the explicit calculation of the dispersive e€ect of a composite

aquifer on arbitrary periodic boundary forcings. Fig. 6 plots the logarithmic attenuation and lag functions for the triple-composite aquifer of Fig. 2(b) for di€erent modal periodsP. The low di€usivity zone (sub-aquifer 2) causes strong dispersion, separating modes of di€erent periods in phase and amplitude space. Noting that frequency is inversely proportional to period, it can readily be seen that aquifers tend to act as low-pass dispersive ®lters.

With these dispersion functions available the evolu-tion of a multi-constituent waveform may be mapped as it propagates through an aquifer. Such a calculation is performed below for the triple-composite example. However, because of the large number of tidal constit-uents,M, necessary to describe actual tidal signals, the dispersive e€ects are illustrated in the following by a simple idealised waveform.

Consider a mixed tidal ¯uctuation in water level providing a Dirichlet condition to a composite Cartesian aquifer. Let the tidal ¯uctuation be described by three constituent sinusoids with parameters {Am}ˆ{1, 1, 1}

(m), {Pm}ˆ{2p/xm}ˆ{1, 0.9, 0.6} (days) and

{um}ˆ{0, p/10, 3p/2}. Fig. 7 shows the resulting tidal

¯uctuations measured at three points in the triple-com-posite aquifer of Fig. 2(b). The solid curve represents the boundary condition determined by the three con-stituents (i.e. at xˆL). The dashed curve is the tidal

Fig. 6.Plots of attenuation (a) and lag (b) functions for the triple-composite aquifer of Fig. 2(b). Functions are plotted for di€erent values of periodP(days), as indicated by ®gures on each curve. The line for in®nite period corresponds to the zero

¯uctuation as measured at a point in sub-aquifer 1 close to sub-aquifer 2 (the low di€usivity zone), while the dotted curve is measured at the opposite boundary of the aquifer (xˆ0). The dispersion e€ects greatly alter the waveforms. For example, in some time ranges the dashed peaks precede the dotted peaks (as might naively be expected), however in other ranges the order of the peaks is reversed, e.g. betweentˆ0 and 1, and between tˆ2 and 3 days. Similarly, the peak heights do not obey simple rules. At some times the dotted peaks are higher than the nearby dashed peaks (i.e. peak heights can apparently increase with distance from the periodic boundary), and correlation with preceding boundary condition peak heights is generally poor. From Fig. 2(b) it is seen that the no-¯ow boundary condition has little e€ect on the propagation of the waveform in sub-aquifer 1. However this is speci®c to the values ofT1,S1,Pmand

x1used. In general it is possible to choose combinations of the values of these parameters for which the boun-dary condition can discriminate between the modes of a multi-constituent signal.

Clearly, dispersive e€ects on multi-constituent forc-ings can lead to the observation of complex and possibly unexpected waveforms throughout the aquifer. For this reason it is unwise to attempt to correlate peaks and troughs simplistically as described above. It is possible that the dispersive e€ects could nullify some peaks in a forcing waveform, or could produce peaks in a mea-sured waveform where none are present in the forcing waveform. In such cases it is essential to employ Fourier decomposition techniques to identify individual con-stituents and apply the propagation analyses to each constituent in isolation. Propagation analysis of each constituent may then allow inferences to be drawn on the composite nature (or otherwise) of the aquifer. The presence of multi-constituent forcings can be viewed in a

positive light. Through simple Fourier decomposition, such forcings may provide multiple independent esti-mates of aquifer properties from a single point mea-surement of ¯uctuating head.

5 CONCLUSIONS

Using interface continuity and Darcy ¯ux matching conditions, previous solutions for the propagation of sinusoidal modes in homogeneous aquifers have been extended to one-dimensional aquifers with composite heterogeneities. The solutions are gained via algebraic matrix systems, the dimensionality of which is deter-mined by the number of aquifer medium interfaces (discontinuities) present. Although exact solution of the matrix systems rapidly becomes laborious as the num-ber of interfaces increases, modern computer algebraic techniques lead to fast and reliable evaluation of closed algebraic forms for the composite solutions.

The composite solutions show that even narrow sub-aquifers can signi®cantly in¯uence tidal propagation characteristics. This has rami®cations for ®eld studies where single- or few-point measurements of lags and attenuations are used with simple Ferris or Townley solutions to estimate aquifer di€usivity. In doing so, care must be taken to demonstrate coherence of estimated di€usivities throughout the aquifer, e.g. through more sampling points. Even so, heterogeneities may still be missed, especially near the banks of water bodies where tidal and biochemical e€ects may lead to local aquifer properties that are signi®cantly di€erent to those in the body of the aquifer.

As noted in earlier work, tidal propagation in aqui-fers is a dispersive process. Thus, tidal waveforms evolve with propagation distance in homogeneous aquifers. Composite aquifers lead to still more complex changes in waveform. Peak heights and lags of multi-constituent waveforms are strongly dependent on the details of aquifer properties and are not necessarily amenable to the simple rules of thumb arising from homogeneous models. However, by applying homogeneous or com-posite methods to individual constituents determined from Fourier decompositions, it is theoretically possible to predict waveform shapes at arbitrary locations in the aquifer. Mechanisms of dispersive propagation in com-posite aquifers may provide explanations for observed tidal attenuations and lags that are seemingly at odds with local aquifer di€usivity and with each other.

Finally it is stressed that all the solutions presented in this paper are exact and in closed form, and can be evaluated numerically or symbolically and analysed with ease using modern mathematical software. Thus the solutions provide an accessible framework for using measurements of aquifer ¯uctuations to determine the underlying aquifer properties and structure. More so-phisticated techniques are necessary for aquifer systems

Fig. 7.Evolution of a triple-constituent tidal signal propagat-ing through the triple-composite aquifer of Fig. 2(b). The solid curve represents the tidal boundary signal (xˆL), the dashed curve represents the signal measured just past thex1 interface (xˆ7L/10), and the dotted curve represents the signal at the no-¯ow boundary (xˆ0). Simple trends in attenuation and lag with distance do not necessarily apply to complex waveforms

where the one-dimensional and homogeneous composite assumptions do not apply, e.g. layered systems or ran-domly heterogenous aquifers.

ACKNOWLEDGEMENTS

The author wishes to thank A. J. Smith (CSIRO Land and Water) for valuable discussions. This work was part funded by the South Australian Environment Protection Authority, BP Australia Ltd., Minenco and CRA.

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