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Advances in Water Resources Vol. 22, No. 6, pp 645±656, 1999
Ó 1999 Elsevier Science Ltd
Printed in Great Britain. All rights reserved
0309-1708/99/$ ± see front matter

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 solutions give inconsistent estimates of aquifer di€usivity from measurements of tidal
lag and attenuation. This paper presents new algebraic solutions for tidal propagation in spatially heterogeneous one-dimensional aquifers. By building on existing 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 oscillations can seem to increase, and phase lags seem to decrease, with distance from
the forcing boundary unless constituents are separated and considered in isolation. Ó 1999 Elsevier Science Limited. All rights reserved
Key words: groundwater, ¯ow, periodic, sinusoidal, tidal, composite, interface,
lag, attenuation, dispersion.

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 estimate 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 semiin®nite one-dimensional aquifer, facilitating tidal estimations 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 evapotranspiration or pumping. A short review of the various
approaches is given by Townley25, who, in the same

1 INTRODUCTION
The reliable determination of aquifer hydrological
properties is a central, yet problematic, component of
any practical hydrological exercise. Traditional pumping 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 parameters that are independent of pumping exercises. For
example, tidal ¯uctuations in sea or river water levels
can induce signi®cant head ¯uctuations far into neighbouring 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,

*

E-mail: mike.trefry@per.clw.csiro.au.
645

646

M. G. Trefry

paper, established a general analytical solution for periodic 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 relationships are parameterised by the aquifer di€usivity,
T/S, where T is the transmissivity [L2 Tÿ1 ] and S 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 estimates 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 fruitless22,26, 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 heterogeneity question by seeking algebraic solutions to periodic
forcing problems in arbitrary composite one-dimensional aquifers. No attempt is made to solve the problem
of periodic forcing in aquifers of higher dimensionality,
where interference of pressure waves generated at different 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 arranged contiguously, with ¯ow matching conditions
applying at sub-aquifer interfaces. The following section
details the mathematical approach and solution. Properties 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, requiring 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 sinusoidal boundary conditions. His method, reproduced

below in brief, is extended here to aquifers with discontinuities 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
oh
o2 h
ˆ T 2 ‡ R;
…1†
ot
ox
where h(x,t) is the head, and R(x,t) is a distributed
recharge term. If h and R can be separated into steady
and periodic components, viz.
S

h…x; t† ˆ hs …x† ‡ hp …x† exp …ixt†

…2†

R…x; t† ˆ Rs …x† ‡ Rp …x† exp…ixt†

…3†

where x ˆ 2p/P is the angular frequency of the ¯uctuation, hs and hp are the steady and periodic head components, and Rs and Rp are the steady and periodic
recharge components, respectively. Where recharge periodicities are di€erent to the head periodicities, as in
diurnal head forcings14 and seasonal recharge forcings16,20, 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:
T

d 2 hs
‡ Rs ˆ 0;

dx2

…4†

d 2 hp
ÿ ixShp ‡ Rp ˆ 0:
…5†
dx2
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:
T

hs ˆ ÿ

Rs 2
x ‡ C1 x ‡ C2 ;
2T

hp ˆ D1 cosh ax ‡ D2 sinh ax ‡

…6†
Rp
;
ixS

…7†

647

Periodic Forcing in Composite Aquifers

Table 1. Boundary condition relationships for Cartesian aquifers from Townley 25, expressed in terms of steady and periodic components
of the prescribed values
Boundary condition
Dirichlet (Prescribed head)
Neumann (Prescribed ¯ux)
Cauchy (Mixed)

Steady component

Periodic component

Rs
Hs ˆ ÿ n2 ‡ C1 n ‡ C2
2T
n…n†Qs ˆ Rs n ÿ TC1


Rs 2
Rs n ÿ TC1 ˆ n…n†A  Gs ‡ 2T
n ÿ C1 n ÿ C2

R

Hp ˆ D1 cosh an ‡ D2 sinhan ‡ ixSp
n…n†Qp ˆ ÿaTD1 sinh an ÿ aTD2 cosh an
ÿaTD

h 1 sinh an ÿ aTD2 cosh an ˆ n…n†A i
R
 Gp ÿ D1 coshan ÿ D2 sinh an ÿ ixSp

These components are Hs and Hp (Dirichlet), Qs and Qp (Neumann), and Gs and Gp (Cauchy). The boundary coordinate variable n
has values 0 or L; n(0) ˆ 1 and n(L) ˆ ÿ1. A is a conductance parameter for the Cauchy condition.

where a2 ˆ ixS/T ˆ 2piS/TP, and C1 , C2 , D1 , and D2 are
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 from x ˆ 0 to x ˆ L,
and lists relationships between the integration constants
arising from imposing Dirichlet, Neumann, or Cauchy
boundary conditions on eqns (6) and (7). These relationships allow the C and D integration 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 reconstituting the solution according to eqns (6), (7) and (2).
2.2 Composite 1D Cartesian aquifers
Attention is now directed to the heterogeneous composite aquifer case. Consider a ®nite aquifer consisting
of N contiguous sub-aquifers, as depicted in Fig. 1.
Each sub-aquifer, referenced by the index j, is assumed
to be a homogeneous unit and is bounded either by two

sub-aquifer interfaces (with sub-aquifers j ÿ 1 and
j + 1), or by one sub-aquifer interface and one problem
boundary condition (if j ˆ 1 or N). Let the coordinate of
the interface between sub-aquifers j and j + 1 be denoted by xj . The N ÿ 1 interface coordinates {xj } range
between 0 and L, and need not be regularly spaced. The
¯ow equation for sub-aquifer j is
Sj

ohj
o2 hj
ˆ Tj 2 ‡ R;
ot
ox

where the recharge term R is assumed to be equal for all
sub-aquifers, and Tj and Sj are the sub-aquifer transmissivity and storage coecient. Writing
hj …x; t† ˆ hjs …x† ‡ hjp …x† exp …ixt†;

…9†

leads to:
Tj

d2 hjs
‡ Rs ˆ 0;
dx2

…10†

d2 hjp
ÿ ixSj hjp ‡ Rp ˆ 0
dx2
which have fundamental solutions:
Tj

hjs ˆ ÿ

Rs 2
x ‡ C1j x ‡ C2j ;
2Tj

hjp ˆ Dj1 cosh aj x ‡ Dj2 sinh aj x ‡

Fig. 1. Schematic and coordinate system for a one-dimensional
N-composite aquifer in Cartesian coordinates. Interfaces between neighboring sub-aquifers are indicated by dashed vertical lines. Aquifer boundaries are located at x ˆ 0 and x ˆ L.

…8†

…11†

…12†
Rp
;
ixSj

…13†

where aj 2 ˆ 2piSj /Tj P. The overall solution for the
composite aquifer is given by the N sets of eqns (12) and
(13) for the N sub-aquifers. There are 2N integration
constants C1j , C2j associated with the steady sub-aquifer
solutions, and 2N integration constants Dj1 , Dj2 associated with the periodic sub-aquifer solutions. Application
of problem boundary conditions to sub-aquifers 1 and N
will supply two steady and two periodic relationships for
the integration constants. The remaining degrees of
freedom in the fundamental solutions are ®xed by imposing conditions for continuity of head and Darcy ¯ux
at each of the N ÿ 1 sub-aquifer interfaces. For the
steady components, these interface matching conditions
are:
hjs …xj † ˆ hsj‡1 …xj †;

…14†

648

M. G. Trefry

Tj



dhjs 
dhsj‡1 
ˆ
T
j‡1
dx xˆxj
dx xˆxj

…15†

and for the periodic components:
hjp …xj † ˆ hj‡1
p …xj †;

dhjp 

Tj
dx xˆxj


dhpj‡1 
ˆ Tj‡1

dx 

…16†
:

…17†

xˆxj

Physical solutions are gained by using the matching
conditions at all sub-aquifer interfaces and the appropriate problem boundary conditions to solve simultaneously for the C and D constants. 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 of hjp 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 C2j constants. Table 1
shows that the steady boundary conditions (for subaquifers 1 and N) reduce to linear relationships of the
form
s

s

s

l C1 ‡ m C2 ˆ / :

…18†

For example, a steady Neumann condition at the
boundary x ˆ n reduces to eqn (18) provided ls ˆ T,
ms ˆ 0, and /s ˆ Rs n ÿ n(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 the C constants arising from the
interface conditions:
C1j xj ‡ C2j ‡

Rs 2
Rs 2
xj ˆ C1j‡1 xj ‡ C2j‡1 ‡
x;
2Tj
2Tj‡1 j

Tj C1j ˆ Tj‡1 C1j‡1 :

…19†
…20†

Thus, solving for the C constants involves 2N linear
equations, two from boundary condition equations
(eqn (18)) for sub-aquifers 1 and N, and 2N ÿ 2 from
the interface conditions (eqns (19) and (20)) for the
N ÿ 1 sub-aquifer interfaces. These equations de®ne the
steady component matrix equation
Ms C ˆ Vs ;

…21†

where CT ˆ …C11 ; C21 ; C12 ; C22 ; . . . ; C1j ; C2j ; . . . ; C1N ; C2N † is a
vector of all degrees of freedom of the steady component
ÿ1
solution, VsT ˆ …/s1 ; Rs x21 …T2ÿ1 ÿ T1ÿ1 †=2; 0; . . . ; Rs x2j …Tj‡1
s
ÿ1
ÿTj †=2; 0; . . . ; /N † is a vector of residual constants
from the linear eqns (18)±(20), and the steady solution
coecient matrix, Ms , is given by
ls1
B x1
B
B
B T1
B
B
B0
B
B
B
Ms ˆ B
B
B
B
B ..
B .
B
B
B
B
@
0

ms1

0

0

...

1

ÿx1

ÿ1

0

ÿT2

0
..

.

0

xj
Tj

1
0

ÿxj
ÿTj‡1

ÿ1
0
..

.
xN ÿ1

1

ÿxN ÿ1

TN ÿ1

0
0

ÿTN
lsN

...

1

C
C
C
C
C
.. C
. C
C
C
C
C:
C
C
C
C
0 C
C
ÿ1 C
C
C
0 A
msN

…22†

In these de®nitions, T denotes the transposition operator, and fls1 ; ms1 ; /s1 g and flsN ; msN ; /sN g are the boundary condition coecients for sub-aquifers 1 and N,
respectively. Solution of eqn (21) determines the integration constants C1j , C2j that, in turn, determine the
steady head components hjs 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 conditions contribute relationships between the D integration
constants in sub-aquifers 1 and N that can be written
as
lp D1 ‡ mp D2 ˆ /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:
Dj1 cosh aj xj ‡ Dj2 sinh aj xj ‡

Rp
ixSj

ˆ D1j‡1 cosh aj‡1 xj ‡ D2j‡1 sinh aj‡1 xj ‡

Rp
ixSj‡1

…24†

aj Tj …Dj1 sinh aj xj ‡ Dj2 cosh aj xj †
ˆ aj‡1 Tj‡1 …D1j‡1 sinh aj‡1 xj ‡ D2j‡1 cosh aj‡1 xj †:

…25†

The boundary condition and interface relations then
de®ne the periodic component matrix equation
Mp D ˆ V p

…26†

where DT ˆ …D11 ; D12 ; D21 ; D22 ; . . . ; Dj1 ; Dj2 ; . . . ; DN1 ; DN2 † is a
vector of all degrees of freedom of the periodic component solution, VpT ˆ …/p1 ; Rp …S2ÿ1 ÿ S1ÿ1 †=ix; 0; . . . ;
ÿ1
ÿ Sjÿ1 †=ix; 0; . . . ; /pN † is a vector of residual
Rp …Sj‡1
constants from the linear eqns (23)±(25), and the periodic solution coecient matrix, Mp , is given by

649

Periodic Forcing in Composite Aquifers
0

B
B
B
B
B
B
Mp ˆ B
B
B
B
B
B
@

lp1

mp1

0
..

0

0

...
.

..
.

coshaj xj

sinh aj xj

ÿ cosh aj‡1 xj

ÿ sinh aj‡1 xj

aj Tj sinh aj xj

aj Tj cosh aj xj

ÿaj‡1 Tj‡1 sinh aj‡1 xj

ÿaj‡1 Tj‡1 cosh aj‡1 xj

..
.

..
0

0

...

0

.
lpN

mpN

1

C
C
C
C
C
C
C:
C
C
C
C
C
A

…27†

Solution of eqn (26) is sucient to determine the subaquifer periodic components hjp , and thence the overall
composite aquifer periodic component hp .
The previous discussion presents an algorithm for
the calculation of exact solutions for head distributions in composite aquifers with periodic forcing. The
®nite composition number N is left unspeci®ed, leading
to generalised de®nitions of matrix and vector quantities involved in the solution. It is useful to emphasise
the ease of the approach by means of a simple example.

0

0
cosh a1 x1

B
B
B
@ a1 T1 sinha1 x1

a1 T1 cosh a1 x1
0
0
0 11 0 1
0
D1
B D1 C B 0 C
B C B C
 B 22 C ˆ B C:
@ D1 A @ 0 A
D22

0
ÿ cosh a2 x1

a1 T1
sinha1 x1

ÿa2 T2 sinh a2 x1
cosh a2 L

0
ÿ sinha2 x1

1

C
C
C
ÿa2 T2 cosh a2 x1 A
sinha2 L

…29†

Hp

The solution vector to eqn (29) is

0
1
2a2 T2 Hp
1
D11
…a1 T1 ‡a2 T2 † cosh ‰a2 L‡x1 …a1 ÿa2 †Šÿ…a1 T1 ÿa2 T2 † cosh ‰a2 Lÿx1 …a1 ‡a2 †Š
B
C
B D1 C B
C
0
B 2C B
C
B 2Cˆ B
C:
Hp fa2 T2 cosh …a1 x1 † cosh …a2 x1 †ÿa1 T1 sinh …a1 x1 † sinh …a2 x1 †g
@ D1 A B fa2 T2 cosh …a1 x1 † cosh ‰a2 …Lÿx1 †Š‡a1 T1 sinh …a1 x1 † sinh ‰a2 …Lÿx1 †Šg C
@
A
Hp fa1 T1 sinh …a1 x1 † cosh …a2 x1 †ÿa2 T2 cosh …a1 x1 † sinh …a2 x1 †g
D22
0

…30†

fa2 T2 cosh …a1 x1 † cosh ‰a2 …Lÿx1 †Š‡a1 T1 sinh …a1 x1 † sinh ‰a2 …Lÿx1 †Šg

2.3 Example A. Dirichlet forcing in a twin-composite
Cartesian aquifer with zero recharge
Consider an aquifer bounded at x ˆ 0 by a no-¯ow
boundary, at x ˆ L by the Dirichlet condition
h(L,t) ˆ Hs + Hp exp(2pit/P), and with R ˆ 0. Let there
be a single aquifer discontinuity at x1 . For x 6 x1 let T1
and S1 apply; elsewhere let T2 and S2 apply. It remains
to construct and solve the steady and periodic component matrix systems. The steady component matrix
system is
ÿT1
B x
B 1
B
@ T1
0

0

0
1

0
ÿx1

0 ÿT2
0
L

10 1 1 0 1
0
0
C1
B
B
C
1C
ÿ1 CB C2 C B 0 C
C
CB C ˆ B C
0 A@ C12 A @ 0 A
1

C22

…28†

Hs

which has solution C11 ˆ C12 ˆ 0; C21 ˆ C22 ˆ Hs . Thus the
steady head component in the composite aquifer is ¯at.
The periodic component matrix system is

The overall periodic component is then expressed as
"
for 0 6 x 6 x1 ;
D11 cosh a1 x
hp …x† ˆ
2
2
D1 cosh a2 x ‡ D2 sinh a2 x
for x1 6 x 6 L;
…31†
where
s
2piSj
aj ˆ
Tj P

for j ˆ 1; 2:

…32†

The solution accounts for both composite and homogeneous aquifers. Homogeneous conditions are
gained by setting T1 ˆ T2 and S1 ˆ S2 . The D coecients
then reduce to D11 ˆ D21 ˆ Hp = cosha1 L and D12 ˆ D22 ˆ 0,
recovering Townley's solution for his Example 125.
2.4 Composite 1D radial aquifers
The composite method can also be applied to cases
where the aquifer is divided into radial sub-aquifers with
co-axial symmetry. The development of the corresponding matrix systems is sketched brie¯y here. Again

650

M. G. Trefry

following Townley25, for sinusoidal forcings the transient radial ¯ow equation


oh T o
oh
S ˆ
‡ R;
…33†
r
ot
r or
or
decouples to the steady and periodic component equations:


T d
dhs
‡ Rs ˆ 0;
r
…34†
dr
r dr


T d
dhp
ÿ ixShp ‡ Rp ˆ 0:
r
…35†
dr
r dr

ÿ

…42†
Tj E1j ˆ Tj‡1 E1j‡1 ;

Rs 2
r ‡ E1j ln r ‡ E2j ;
4Tj

…36†

Rp
;
…37†
ixSj
where I0 and K0 are modi®ed Bessel functions1 of order
zero, and the E and F coecients are degrees of freedom
to be ®xed by applying boundary and interface conditions. 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:
hjp ˆ F1j I0 …aj r† ‡ F2j K0 …aj r† ‡

hjs …rj † ˆ hsj‡1 …rj †;


dhj 
dhj‡1 
2prj Tj s 
ˆ 2prj Tj‡1 s  :
dr rˆrj
dr rˆrj

…39†

hjp …rj †

…40†

…38†

F1j I0 …aj rj † ‡ F2j K0 …aj rj † ‡

ˆ



dhjp 
dhpj‡1 

2prj Tj
ˆ 2prj Tj‡1

dr rˆrj
dr 

;

Rp
ixSj

ˆ F1j‡1 I0 …aj‡1 rj † ‡ F2j‡1 K0 …aj‡1 rj † ‡

Rp
;
ixSj‡1

…44†

aj F1j I1 …aj rj † ÿ aj F2j K1 …aj rj †
ˆ aj‡1 F1j‡1 I1 …aj‡1 rj † ÿ aj‡1 F2j‡1 K1 …aj‡1 rj †:

…45†

The matrix systems for solving for the E and F coe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 r ˆ 0 by a no¯ow boundary, at r ˆ L by the Dirichlet condition
h(L,t) ˆ Hs + Hp exp(2pit/P), and with R ˆ Rs + Rp
exp(2pit/P). Let there be a single aquifer discontinuity at
r1 . For r 6 r1 let T1 and S1 apply; elsewhere let T2 and S2
apply. From eqns (42)±(45), and from Table 2, matrix
systems for determining the steady coecients E and the
periodic coecients F are, respectively:

and for the periodic components:
hj‡1
p …rj †;

…43†

and for the periodic components:

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

Rs 2
Rs 2
rj ‡ E1j ln rj ‡ E2j ˆ ÿ
r ‡ E1j‡1 ln rj ‡ E2j‡1 ;
4Tj
4Tj‡1 j

…41†

rˆrj

1

0

0

B
B ln
B
BT
@ 1

1

ÿ ln r1

0

ÿT2

0
0

lnL

0

where rj is the radial interface coordinate of interface j.
Noting the derivative identities I00 ˆ I1 and K00 ˆ ÿK1 ,
the interface conditions become, for the steady components:

0

B
B
ˆB
B
@

0

0

Rs r12
4

10

E11

1

CB C
ÿ1 CB E21 C
CB C
B C
0 C
A@ E12 A
E22

1

1

…T1ÿ1 ÿ T2ÿ1 † C
C
C;
C
0
A
2
Rs L
Hs ‡ 4T2

…46†

Table 2. Boundary condition relationships for radial aquifers from Townley 25, expressed in terms of steady and periodic components of
the prescribed values
Boundary condition

Steady component

Periodic component

Dirichlet (Prescribed head)

Rs 2
q ‡ E1 ln q ‡ E2
Hs ˆ ÿ 4T

Hp ˆ F1 I0 …aq† ‡ F2 K0 …aq† ‡ ixSp

Neumann (Prescribed ¯ux)

n…q†Vs ˆ pRs q2 ÿ 2pTE1


Rs
Rs 2
q ÿ Tq E1 ˆ n…q†A  Gs ‡ 4T
q ÿ E1 ln qÿE2
2

n…q†Vp ˆ ÿ2paTF1 qI1 …aq† ‡ 2paTF2 qK1 …aq†

Cauchy (Mixed)

R

ÿaTF
h 1 I1 …aq† ‡ aTF2 K1 …aq† ˆ n…q†Ai
R
 Gp ÿF1 I0 …aq† ÿ F2 K0 …aq† ÿ ixSp

These components are Hs and Hp (Dirichlet), Vs and Vp (Neumann), and Gs and Gp (Cauchy). The boundary coordinate variable q
has values r0 or L; n(r0 ) ˆ 1 and n(L) ˆ ÿ1. If r0 ˆ 0, then only a no-¯ow Neumann condition is appropriate, yielding the two
constraints E1 ˆ 0 and F2 ˆ 0. A is a boundary conductance parameter for the Cauchy condition.

651

Periodic Forcing in Composite Aquifers
0

0
I0 …a1 r1 †

1
K0 …a1 r1 †

0
ÿI0 …a2 r1 †

B
B
B
@ a1 T1 I1 …a1 r1 † ÿa1 T1 K1 …a1 r1 † ÿa2 T2 I1 …a2 r1 †
0
0
I0 …a2 L†
0 11 0
1
0
F1
B F 1 C B Rp …S ÿ1 ÿ S ÿ1 † C
1
B C B ix 2
C
 B 22 C ˆ B
C:
0
@ F1 A @
A
R
p
2
ÿ
H
F2
p
ixS2

1
0
ÿK0 …a2 r1 † C
C
C
a2 T2 K1 …a2 r1 † A
K0 …a2 L†

…47†

The steady and periodic coecient solution vectors are:
0 11 0
1
0
E1
2
2
B E1 C B Rs r1 …T2 ÿ T1 † ‡ Hs ‡ L Rs C
B 2 C B 4T1 T2
4T2 C
…48†
B 2Cˆ B
C;
0
@ E1 A @
A
2
Hs ‡ L4TR2s
E22

0

F11

1

0

B
BF1C B
B 2C B
B 2Cˆ B
@ F1 A B
@
F22

highlights how sub-aquifer 2 acts as a rapid phaseshifter and attenuator for the propagating signal. Fig. 2
gives appealing pictorial representations of the characteristics of propagation in the x-t phase space. However
it is often more useful to concentrate on speci®c modal
transfer properties.
The Jacob and Ferris solutions for in®nite homogeneous 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:
r!
pS
;
a ˆ exp ÿx
PT

T2 fS1 ‰iRp ‡xS2 Hp Šÿia2 r1 Rp ‰S1 ÿS2 Š‰I1 …a2 r2 †K0 …a2 L†‡I0 …a2 L†K1 …a2 r1 †Šg
xr1 S1 S2 fa1 T1 I1 …a1 r1 †‰I0 …a2 L†K0 …a2 r1 †ÿI0 …a2 r1 †K0 …a2 L†Š‡a2 T2 I0 …a1 r1 †‰I1 …a2 r1 †K0 …a2 L†‡I0 …a2 L†K1 …a2 r1 †Šg

0
ifa1 T1 I1 …a1 r1 †‰Rp …S1 ÿS2 †K0 …a2 L†ÿS1 …Rp ÿixS2 Hp †K0 …a2 r1 †Šÿa2 S1 T2 …Rp ÿixS2 Hp †I0 …a1 r1 †K1 …a2 r1 †g
xS1 S2 fa1 T1 I1 …a1 r1 †‰I0 …a2 r1 †K0 …a2 L†ÿI0 …a2 L†K0 …a2 r1 †Šÿa2 T2 I0 …a1 r1 †‰I1 …a2 r1 †K0 …a2 L†‡I0 …a2 L†K1 …a2 r1 †Šg
ifa1 T1 Rp …S1 ÿS2 †I0 …a2 L†I1 …a1 r1 †‡S1 …R0 ÿixS2 Hp †‰a2 T2 I0 …a1 r1 †I1 …a2 r1 †ÿa1 T1 I1 …a1 r1 †I0 …a2 r1 †Šg
xS1 S2 fa1 T1 I1 …a1 r1 †‰ÿI0 …a2 r1 †K0 …a2 L†‡I0 …a2 L†K0 …a2 r1 †Š‡a2 T2 I0 …a1 r1 †‰I1 …a2 r1 †K0 …a2 L†‡I0 …a2 L†K1 …a2 r1 †Šg

Setting T1 ˆ T2 , S1 ˆ S2 , a1 ˆ a2 , Hs ˆ Hp ˆ 0 in
eqns (48) and (49) yields E21 ˆ E22 ˆ L2 Rs =4T1 ; E11 ˆ
ˆ F22 ˆ 0,
E12 ˆ 0; F11 ˆ F12 ˆ Rp =…ixS1 I0 …a1 L††; F21
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 complexity of the resulting expressions, even for simple
composite aquifers, can distract attention from the underlying physics. In this section the properties of the
composite solutions are explored graphically.
Fig. 2(a) shows a three-dimensional plot of the periodic component of head versus x and t for a Cartesian
aquifer with Dirichlet forcing at x ˆ L, and a no-¯ow
condition at x ˆ 0. The aquifer is composed of two subaquifers, 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 signal. 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

…50†

1

C
C
C
C:
C
A

r
PS
;
sˆx
4pT

…49†

…51†

where x, in this case, measures the distance from the
forced boundary, and P ˆ 2p/x. Thus, for a given periodic mode P, the amplitude of ¯uctuation declines exponentially with distance, whilst the phase lag increases
linearly. Townley's solutions for ®nite aquifers essentially parameterize these results by L, the aquifer length.
Following Townley, the piecewise attenuation coecient in a composite aquifer with Dirichlet forcing is
given by:
2 1
jhp j
for 0 6 x < x1 ;
6 jHp j
j
jhp j 6
jhp j
ˆ 6 jH
…52†
aˆ
for xjÿ1 6 x < xj ;
j
jHp j 6
4 Np
jhp j
for xN ÿ1 6 x < L;
jHp j
where |x + iy| ˆ (x2 + y2 )1=2 is the modulus function.
The associated phase lag, Dsh , is
Dsh ˆ arg…hp † ÿ arg …Hp †
 1
2
h
arg Hpp
for 0 6 x < x1 ;
6
 j
6
hp
ˆ6
for xjÿ1 6 x < xj ;
6 arg Hp
4
 N
h
for xN ÿ1 6 x < L;
arg Hpp

…53†

where arg(x + iy) ˆ tanÿ1 y/x is the argument function.
This phase lag can be mapped from radians to units of
¯uctuation periods by the transformation:

652

M. G. Trefry

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 estimated from homogeneous (Ferris or Townley) models based
on single point measurements (black dots).

Fig. 2. Phase space (x ÿ t) plot of periodic component of head
for two composite aquifers subject to Dirichlet forcing at
x ˆ L. Part (a) shows a twin-composite aquifer with L2 S1 /
T1 P ˆ 50/9, L2 S2 /T2 P ˆ 1/24, T1 /T2 ˆ 3/20, (T1 /S1 )/(T2 /S2 ) ˆ 3/
400, x1 /L ˆ 4/5. Part (b) shows a triple-composite aquifer with
L2 S2 /T2 P ˆ 50,
T1 /T2 ˆ T3 /
L2 S1 /T1 P ˆ L2 S3 /T3 P ˆ 1/40,
T2 ˆ 200, (T1 /S1 )/(T2 /S2 ) ˆ (T3 /S3 )/(T2 /S2 ) ˆ 2000, x1 /L ˆ 3/4,
x2 /L ˆ 4/5.

U…Dsh † ˆ

"

h
1 ÿ Ds
2p

for 0 < Dsh 6 p;

h
ÿ Ds
2p

for ÿ p < Dsh 6 0:

…54†

Fig. 3(a) shows the variation of the attenuation coecient a (expressed as jhp j=jHp j on a logarithmic scale)
with distance in the example triple-composite Cartesian
aquifer described in Fig. 2(b). The e€ects of the two subaquifer interfaces are visible as changes in slope of the
curve as the aquifer properties change from high diffusivity to low di€usivity and back to high di€usivity.
The associated curve for the lag function U is 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 homogeneous. 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€usivities (indicated by dashed lines) that are signi®cantly
di€erent to the actual di€usivities of the three subaquifers. Another tidal measurement elsewhere in subaquifer 1 would yield a di€erent aquifer di€usivity
again, although pumping tests at all points within subaquifer 1 would give, in principle, correct and identical
values for T 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 attenuation and lag curves (solid lines) for twin-composite aquifers. Solid dots indicate single point
measurements, which can be used with homogeneous
models to estimate e€ective homogeneous aquifer diffusivities (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 homogeneous di€usivities are signi®cantly higher than the
local di€usivity, then an intervening sub-aquifer of high

Periodic Forcing in Composite Aquifers

653

Fig. 4. Schematic attenuation (a), (c) and lag (b), (d) functions for twin-composite aquifers subject to Dirichlet forcing at x ˆ L. The
e€ects of low and high di€usivity aquifer zones on the propagation characteristics are shown. Dashed lines show schematic attenuation and lag functions estimated from homogeneous (Ferris or Townley) models based on single point measurements (dots).

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 applying 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 monitored.
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 subaquifer 1 throughout. In this special case, measurements
at all points in sub-aquifer 1 would give consistent estimates of aquifer properties based on homogeneous
models, despite the strongly composite structure elsewhere in the aquifer.
In general, the true structure and properties of a
composite aquifer can be determined from measurements 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 group L2 S/TP, where L is
identi®ed with the sub-aquifer length, noted by Townley
is no longer sucient to describe the sub-aquifer dynamics. Enough measurements must be taken in each
sub-aquifer to ®x Si , Ti and the interface coordinate xi .
This may notionally be accomplished by measuring attenuations and lags at two points in each sub-aquifer,
and ®tting the Si , Ti , and xi values by an inverse technique 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-

Fig. 5. Schematic attenuation (a) and lag (b) functions for a
notional triple-composite aquifer subject to Dirichlet forcing
at x ˆ L. The combined e€ects of sub-aquifer 2 (high di€usivity) and sub-aquifer 3 (low di€usivity) lead to a situation
where single point measurements (black dots) give homogeneous model estimates of di€usivity (dashed lines) that are
congruent with the di€usivity of sub-aquifer 1.

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 interface coordinates.

654

M. G. Trefry

4 DISPERSIVE MODAL PROPAGATION IN
AQUIFERS
The above theories consider the response of homogeneous and composite aquifers to single mode forcings,
however through the linearity of the dynamical equations 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 recharge, 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 be dispersive if 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 dispersion, oscillating waveforms arising from multiple sinusoidal components will change in shape as they
propagate through even homogeneous aquifers2. Aquifers with composite structure will lead to still more
complicated evolutions of waveform with distance.
Change in the waveform, X, through dispersive propagation will occur when the waveform is comprised of
two or more sinusoidal constituents
Xˆ

M
X
Am sin …xm t ‡ um †;

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 periods P. 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 evolution 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 constituents, 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-composite aquifer of Fig. 2(b). The solid curve represents
the boundary condition determined by the three constituents (i.e. at x ˆ L). The dashed curve is the tidal

…55†

mˆ1

where Am is the amplitude, xm the angular frequency
and um the phase of constituent m in the M-constituent
waveform. By a suitable choice of the time coordinate t,
u1 can be set to zero; the remaining um are then the
relative phases of the sinusoidal constituents with respect to constituent 1. As explained above, the amplitudes and phases of the constituents of X are 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

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 period P (days), as indicated by ®gures on
each curve. The line for in®nite period corresponds to the zero
frequency (steady component) limit.

Periodic Forcing in Composite Aquifers

655

positive light. Through simple Fourier decomposition,
such forcings may provide multiple independent estimates of aquifer properties from a single point measurement of ¯uctuating head.

5 CONCLUSIONS

Fig. 7. Evolution of a triple-constituent tidal signal propagating through the triple-composite aquifer of Fig. 2(b). The solid
curve represents the tidal boundary signal (x ˆ L), the dashed
curve represents the signal measured just past the x1 interface
(x ˆ 7L/10), and the dotted curve represents the signal at the
no-¯ow boundary (x ˆ 0). Simple trends in attenuation and lag
with distance do not necessarily apply to complex waveforms
propagating through aquifers.

¯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 (x ˆ 0). 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. between t ˆ 0 and 1, and between
t ˆ 2 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 of T1 , S1 , Pm and
x1 used. In general it is possible to choose combinations
of the values of these parameters for which the boundary condition can discriminate between the modes of a
multi-constituent signal.
Clearly, dispersive e€ects on multi-constituent forcings 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 measured waveform where none are present in the forcing
waveform. In such cases it is essential to employ Fourier
decomposition techniques to identify individual constituents 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

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 determined by the number of aquifer medium interfaces
(discontinuities) present. Although exact solution of the
matrix systems rapidly becomes laborious as the number 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 subaquifers 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 aquifers 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 composite 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 composite 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 sophisticated techniques are necessary for aquifer systems

656

M. G. Trefry

where the one-dimensional and homogeneous composite
assumptions do not apply, e.g. layered systems or randomly 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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