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Advances in Water Resources 30 (2007) 493–504

Analytic modeling of groundwater dynamics with anapproximate impulse response function for areal recharge

Mark Bakker a,*, Kees Maas a,b, Frans Schaars c, Jos R. von Asmuth a,b

a Water Resources Section, Faculty of Civil Engineering and Geosciences, Delft University of Technology, Delft, The Netherlandsb Kiwa Water Research, Nieuwegein, The Netherlands

c Artesia, Schoonhoven, The Netherlands

Received 20 January 2006; received in revised form 13 April 2006; accepted 17 April 2006Available online 16 June 2006

Abstract

An analytic approach is presented for the simulation of variations in the groundwater level due to temporal variations of recharge insurficial aquifers. Such variations, called groundwater dynamics, are computed through convolution of the response function due to animpulse of recharge with a measured time series of recharge. It is proposed to approximate the impulse response function with an expo-nential function of time which has two parameters that are functions of space only. These parameters are computed by setting the zerothand first temporal moments of the approximate impulse response function equal to the corresponding moments of the true impulseresponse function. The zeroth and first moments are modeled with the analytic element method. The zeroth moment may be modeledwith existing analytic elements, while new analytic elements are derived for the modeling of the first moment. Moment matching may beapplied in the same fashion with other approximate impulse response functions. It is shown that the proposed approach gives accurateresults for a circular island through comparison with an exact solution; both a step recharge function and a measured series of 10 years ofrecharge were used. The presented approach is specifically useful for modeling groundwater dynamics in aquifers with shallow ground-water tables as is demonstrated in a practical application. The analytic element method is a gridless method that allows for the preciseplacement of ditches and streams that regulate groundwater levels in such aquifers; heads may be computed analytically at any point andat any time. The presented approach may be extended to simulate the effect of other transient stresses (such as fluctuating surface waterlevels or pumping rates), and to simulate transient effects in multi-aquifer systems.� 2006 Elsevier Ltd. All rights reserved.

Keywords: Impulse response function; Convolution; Analytic element method; Groundwater dynamics

1. Introduction

The objective of this paper is to present a new approachfor the accurate modeling of fluctuations in the groundwa-ter table due to temporal variations in rainfall and evapo-transpiration. Fluctuations of the groundwater table arereferred to as groundwater dynamics. Accurate modelingof groundwater dynamics is important for the effectivemanagement of aquifers, especially aquifers with shallow

0309-1708/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.

doi:10.1016/j.advwatres.2006.04.008

* Corresponding author. Tel.: +31 15 2783714.E-mail addresses: markbak@gmail.com (M. Bakker), Kees.Maas@

kiwa.nl (K. Maas), f.schaars@artesia-water.nl (F. Schaars), Jos.von.Asmuth@kiwa.nl (J.R. von Asmuth).

groundwater tables. Models need to take into account dailychanges in rainfall and evaporation, as well as the accuratelocation of streams, canals, ditches, and drains.

In this paper it is proposed to compute the groundwaterdynamics by developing models of the response functiondue to an impulse of recharge; these models are based onphysical boundary conditions. Once the impulse responsefunction is known at a point, fluctuations of the groundwa-ter table may be computed through convolution with timeseries of recharge. It is proposed to approximate theimpulse response function with an exponential functionof time containing two parameters that are a function ofthe two spatial coordinates. The use of an approximateimpulse response function is common in the field of time

494 M. Bakker et al. / Advances in Water Resources 30 (2007) 493–504

series analysis, where they are often called transfer func-tions. The standard Box–Jenkins method uses a discreteimpulse response function and has been applied to forecasthead fluctuations at observation wells based on time seriesof recharge (e.g., [18,10]). Bierkens et al. [5] developed anempirical space–time model to estimate parameters of theirBox–Jenkins model between observation wells using geo-statistical interpolation and information on locations ofstreams and ditches. Von Asmuth et al. [21] proposed toapproximate the impulse response function with the Pear-son type III function, which is continuous through time,and obtained results for a practical case that were at leastas accurate as results obtained with the discrete impulseresponse function of the Box–Jenkins method.

The parameters of the approximate impulse responsefunction used in this paper are computed with the analyticelement method; several new analytic elements are derived.The advantage of the analytic element method is that themodel domain does not have to be discretized spatially,e.g., in rectangles or triangles. Hydrogeologic boundariescan be defined exactly where they are located without havingto adjust them to a spatial grid. The analytic element methodhas been applied to steady flow in single aquifers and multi-aquifer systems (for an overview, see [17]), and to steadyunsaturated flow (e.g. [3]). Several formulations exist fortransient flow, each with its own advantages and limitations[23,7,2]. The approach presented in this paper may beviewed as a new transient analytic element formulation.

This paper is limited to homogeneous aquifers; varia-tions in the saturated thickness of unconfined aquifers(and thus the transmissivity) are neglected. The arealrecharge is the only transient stress on the system and doesnot vary spatially. Other imposed stresses, such as specifiedhead or flux boundaries, or pumping wells, do not vary withtime. In addition, non-linear effects, such as ditches that dryup, are not considered. The presented approach may beextended to include spatially varying recharge, other tran-sient stresses, inhomogeneous aquifers, and multi-aquifersystems, but that is beyond the scope of this paper; someof these extensions are briefly discussed in Section 8.

The basic steps of the proposed approach are outlined inthe following two sections, followed by an evaluation of theperformance of the proposed approximate response func-tion through comparison with an exact solution. In Section5, analytic elements are derived for the modeling of the zer-oth and first temporal moments of the impulse responsefunction. Performance of the new analytic elements is eval-uated in Section 6, followed by a practical example in Sec-tion 7.

2. Approach

Groundwater flow in a homogeneous, isotropic, hori-zontal aquifer is governed by (e.g., [4])

r2h ¼ ST

ohot� N

Tð1Þ

where h [L] is the hydraulic head, $2 [L�2] is the two-dimensional Laplacian, t [T] is time, S [–] is the storativityof the aquifer, T [L2T�1] is the transmissivity, and N [LT�1]is the time-varying areal recharge. Eq. (1) may be appliedto unconfined aquifers when the variation of the saturatedthickness is relatively small. The aquifer is linear, such thatthe head in the aquifer may be written as a steady compo-nent plus a transient component

hðx; y; tÞ ¼ h0ðx; yÞ þ /ðx; y; tÞ ð2Þh0 [L] fulfills (1) but with the righthand side set to zero

r2h0 ¼ 0 ð3Þand / [L] fulfills (1)

r2/ ¼ ST

o/ot� N

Tð4Þ

All imposed boundary conditions are steady state. Along aboundary where h is fixed to hf, boundary conditions are

h ¼ hf ; h0 ¼ hf ; / ¼ 0 ð5ÞAlong a boundary where the normal gradient oh/on is fixedto qn, boundary conditions are

ohon¼ qn;

oh0

on¼ qn;

o/on¼ 0 ð6Þ

Along a boundary with a mixed boundary condition,boundary conditions are

ohon¼ bhþ c;

oh0

on¼ bh0 þ c;

o/on¼ b/ ð7Þ

where b and c are constants. The boundary conditions forh0 are identical to the boundary conditions for h, and thush0 represents the steady head in the aquifer in the absenceof recharge; h0 may be modeled with a standard methodand will not be discussed here further. The remainder ofthis paper concerns the modeling of /, which representsthe head fluctuation caused by the temporal variation of re-charge. The initial condition of / is equal to the initial con-dition of h � h0.

A standard technique to solve the differential equationfor / is to determine the solution for an impulse ofrecharge of unit volume, called the impulse response func-tion h(x,y, t) [–], and to obtain /(x,y, t) for an arbitraryrecharge function N(t) through convolution (Duhamel’sprinciple, e.g. [4, Section 7.5], [24, Section II.B.7]).

The impulse response function fulfills (1) where therecharge is replaced by the Dirac delta function d(t) [T�1]

r2h ¼ ST

ohot� dðtÞ

Tð8Þ

where, as before, the analysis is limited to a recharge thatdoes not vary spatially. Once h is determined, the response/(x,y, t) to any recharge N(t) may be found throughconvolution

/ðx; y; tÞ ¼Z t

�1NðsÞhðx; y; t � sÞds ð9Þ

M. Bakker et al. / Advances in Water Resources 30 (2007) 493–504 495

without the need to solve the original differential Eq. (4)again. Boundary conditions for the impulse response func-tion h are identical to the boundary conditions for /, asmay be seen through substitution of (9) for / in the bound-ary conditions (5)–(7).

The analytic solution of (8) is difficult for general cases.Here, a new analytic but approximate approach is pre-sented. It is proposed to approximate the impulse responsefunction of the recharge by the following exponentialfunction

h ¼ 0 t < 0

h ¼ Aa expð�atÞ t P 0ð10Þ

where A(x,y) and a(x,y) are functions to be determined. Itwill be shown in the following sections that the exponentialrepresentation (10) may be used to represent the true im-pulse response function with reasonable accuracy for prac-tical cases of transient recharge. The approximate impulseresponse function contains two spatial functions A and a.To achieve an accurate overall match between the approx-imate response function (10) and the true impulse responsefunction that fulfills (8), it is proposed to determine A and a

such that the zeroth and first moments of the approximateimpulse response function are equal to the correspondingmoments of the true impulse response function. Matchingtemporal moments has been applied to estimate transportparameters (e.g. [14] and references therein). Applicationof moment matching to estimate parameters of impulse re-sponse functions of groundwater head was proposed by[12] and [20]. Li et al. [13] used moments to characterizedrawdowns of pumping tests; they found that the zerothand first temporal moments were sufficient to characterizethe drawdown curves in their study.

3. Matching temporal moments

The zeroth temporal moment M0 of the impulseresponse function is defined as

M0 ¼Z 1

�1hdt ð11Þ

Integration of the differential equation for h (8) gives

r2M0 ¼ �1

Tð12Þ

where it is used that h equals zero for both t = �1 andt =1, and that the integral of d(t) equals one. Hence,the zeroth moment M0 of the impulse response functionfulfills Poisson’s differential equation (12) with a constantrighthand side. For the steady boundary conditions usedhere, boundary conditions of M0 are identical to theboundary conditions for h, and thus for /, as may be seenfrom substitution of the boundary conditions (5)–(7) for hin the definition of M0 (11).

The first temporal moment M1 of the impulse responsefunction is defined as

M1 ¼Z 1

�1thdt ð13Þ

Multiplication of both sides of (8) with t and integrationgives

r2M1 ¼ST

Z 1

�1tohot

dt �Z 1

�1

tdðtÞT

dt ð14Þ

The right integral equals zero, and the left integral may beintegrated by parts to give

r2M1 ¼ST

htj1�1 �Z 1

�1hdt

� �ð15Þ

The first term equals zero and the remaining integral is M0

(11), such that the differential equation for M1 becomes

r2M1 ¼ �ST

M0 ð16Þ

Hence, the first moment M1 of the impulse response func-tion also fulfills Poisson’s differential equation, where thezeroth moment appears on the righthand side. For the stea-dy boundary conditions used here, boundary conditions ofM1 are identical to the boundary conditions for h, and thusfor /. Li et al. [13] presented differential equations for themoments of the impulse response function of a pumpingwell, which are similar to Eqs. (12) and (16).

Eqs. (12) and (16), with specified boundary conditions,may be solved for M0(x,y) and M1(x,y); these are the zerothand first moments of the true impulse response function. Thefunctions A and a may now be found by setting M0 and M1

of the true impulse response function equal to the corre-sponding moments of the approximate impulse responsefunction. These latter two moments are obtained by substi-tuting the approximate impulse response function (10) in thedefinitions for M0 (11) and M1 (13). The lower limit of theintegral is changed to zero, as h = 0 for t < 0, so that

M0 ¼Z 1

0

Aa expð�atÞdt ¼ A

M1 ¼Z 1

0

Aat expð�atÞdt ¼ A=að17Þ

and thus

A ¼ M0

a ¼ M0=M1

ð18Þ

In statistical terms, a is the inverse of the mean of the im-pulse response function. Hence, the mean of the exact andapproximate impulse response functions are equal.

In summary, the proposed approach for the modeling ofgroundwater dynamics due to temporal variations inrecharge consists of four steps:

(1) Develop a model for M0 by solving (12) with appro-priate boundary conditions.

(2) Develop a model for M1 by solving (16) with appro-priate boundary conditions, and using the M0 fromthe fist step in the righthand side of the differentialequation.

(3) Once M0 and M1 are known, compute A and a with(18).

496 M. Bakker et al. / Advances in Water Resources 30 (2007) 493–504

(4) Compute the fluctuation of the head / for any timeseries of recharge N(t) with convolution (9), usingthe approximate impulse response function (10).

Note that with the proposed approach the solution ofthe transient problem has been reduced to the creation ofsteady models for three variables: h0, M0, and M1, wherethe solution of M1 depends on the solution of M0. In thefollowing section, the accuracy of the proposed approachis evaluated for a simple case of radial flow.

4. Example of radial flow

The objective of this example is to demonstrate applica-tion of the approach outlined in the previous section to asimple case of radial flow, and to evaluate the performanceof the proposed approximate impulse response function(10). Consider radial groundwater flow on a circular islandof radius R. On the boundary of the island, the water levelis fixed to 0, so that

/ðr ¼ R; tÞ ¼ 0 ð19ÞFor t < 0, the heads are zero everywhere

/ðr; tÞ ¼ 0 t < 0 ð20Þ

For t < 0, the recharge is zero, but for t P 0 the recharge isequal to a constant value c

NðtÞ ¼0 t < 0

c t P 0

�ð21Þ

The first step of the approach outlined in the previous sec-tion is to determine the zeroth moment M0 of the impulseresponse function. As stated, boundary conditions for M0

are the same as boundary conditions for /, and thus

M0ðRÞ ¼ 0 ð22ÞThe differential equation for M0 (12) is written in radialcoordinates

1

rd

drr

dM0

dr

� �¼ � 1

Tð23Þ

Solution of the differential equation with the specifiedboundary condition gives

M0 ¼R2 � r2

4Tð24Þ

The second step of the approach is to determine the firstmoment M1 of the impulse response function. Boundaryconditions are again the same as for /

M1ðRÞ ¼ 0 ð25Þ

Solution of the differential equation for M1 (12), with sub-stitution of the computed function for M0 (24) in the right-hand side, and the above stated boundary condition gives

M1 ¼S

4T 2

r4

16� R2r2

4þ 3R4

16

� �ð26Þ

which may be simplified to

M1 ¼SM0ð3R2 � r2Þ

16Tð27Þ

The third step is to compute A and a with Eq. (18) and thederived expressions for M0 and M1, which gives

A ¼ R2 � r2

4T

a ¼ 16T

Sð3R2 � r2Þ

ð28Þ

The fourth and last step is to determine the head fluctua-tion through convolution. Substitution of (21) for N and(10) for h in (9) gives (note that the lower limit of the inte-gral is set to 0, as N is zero for t < 0)

/ ¼Z t

0

cAa exp½�aðt � sÞ�ds ¼ cA� cA expð�atÞ ð29Þ

Finally, substitution of (28) for A and a in the previousequation gives the final expression for /(r, t)

/ ¼ cðR2 � r2Þ4T

1� exp�16Tt

Sð3R2 � r2Þ

� �� �ð30Þ

The first term on the righthand side of / (30) is the steady-state head distribution for a constant recharge on a circularisland (e.g. [6], Eq. (233.17)), which is represented exactlyby the approximate approach. The accuracy of the approx-imate solution is assessed through comparison with the ex-act transient solution, which is given by [6], Eq. (233.16),and may be written as

/ ¼ cðR2 � r2Þ4T

� 2cR2

T

X1n¼0

J0ðanr=RÞa3

nJ1ðanÞexp �a2

n

Tt

SR2

� �ð31Þ

where J0 and J1 are Bessel functions of the first kind andorder 0 and 1, respectively, and an is the nth root of J0.The dimensionless head fluctuation 4T//(cR2) is shownat five different dimensionless times in Fig. 1, where the so-lid line is the approximate solution and the dashed line theexact solution (using 21 terms in the summation). In the ex-act solution, the head is flatter at early times than in theapproximate solution. The difference is largest at the centerof the island, where the difference decreases from 18% atTt/(SR2) = 0.05, to 8% at Tt/(SR2) = 0.1, to less than 1%at Tt/(SR2) = 0.2.

The exact solution for the impulse response function is([6], Eq. (233.15))

h ¼ 2

S

X1n¼0

J0ðanr=RÞanJ1ðanÞ

exp �a2n

Tt

SR2

� �ð32Þ

The approximate (solid) and exact (dashed) impulse re-sponse functions (multiplied by S) are plotted vs. dimen-sionless time at four different positions in Fig. 2(decending lines). Here it is visible that, at the center ofthe island, the exact impulse response function is flat forearly times, while the approximate impulse response func-tion is not. Convolution of the step recharge function

Fig. 1. Dimensionless head 4T//(cR2) vs. radial distance r/R at fivedifferent times; approximate (solid) and exact (dashed) solutions.

M. Bakker et al. / Advances in Water Resources 30 (2007) 493–504 497

(21) with the approximate impulse response function yieldsthe ascending lines in Fig. 2, which are the dimensionlesshead fluctuation 4T//(cR2) vs. dimensionless time at fourdifferent positions. These curves represent the head varia-tion that would be measured in an observation well. Thematch is good at all positions.

To evaluate the head variations caused by measured,and thus wildly fluctuating, recharge, the head variationis computed at the center of the island using time seriesof daily rainfall and evapotranspiration obtained fromweather station De Bilt in The Netherlands. Recharge iscomputed as rainfall minus evapotranspiration, which isaccurate for relatively flat areas with shallow unsaturatedzones. The convolution integral (9) is evaluated analytically

Fig. 2. Impulse response S/ (descending lines) and dimensionless head 4T/solutions at four different positions (r = 0,R/4,R/2,3R/4).

by superimposing the contribution of each daily event,assuming the recharge occurs evenly during the day. Theconvolution integral is truncated at time ttr. The truncationtime is computed such that the area under the truncatedapproximate impulse response function (normalizedthrough division by M0) is a = 0.999, which gives

ttr ¼ �lnð1� aÞ

að33Þ

A plot of the daily recharge is shown for a period of 10years in the top part of Fig. 3. The head variation at thecenter of the island is shown in the bottom part, wherethe solid line represents the approximate solution,and the dashed line the exact solution. The difference be-tween the minimum and maximum heads during the 10year period is 2.53 m. The two solutions are visually almostindistinguishable, even though there is a noticeable differ-ence between the two impulse response functions at thislocation (Fig. 2). The average difference between theapproximate and exact solutions in Fig. 3 is 0.001 cm, witha standard deviation of 4.18 cm.

To evaluate the difference between the approximate andexact solutions more closely, the head variation in the year1994 is shown in Fig. 4. Here it can be seen that there isindeed a small difference between the approximate andthe exact solutions. For real aquifers, there are no exactsolutions, but there are head measurements at an observa-tion well, for example taken once a month. As an illustra-tion, the head value of the exact solution at the end of everymonth is depicted in Fig. 4 with grey dots. The approxi-mate solution (solid line) may be considered a good fit tothe monthly values. It is concluded that the approximateimpulse response function (10) gives reasonable values forreal recharge series.

/(cR2) (ascending lines) for the approximate (solid) and exact (dashed)

0

2

4

rech

arge

(cm

/d)

1990 1992 1994 1996 1998

time

-1

0

1

2

head

var

iatio

n (m

)

ApproximateExact

Fig. 3. Daily recharge (top) and head fluctuation / at the center of the island (bottom).

0

2

4

rech

arge

(cm

/d)

01 02 03 04 05 06 07 08 09 10 11 12

month of 1994

-1

0

1

2

head

var

iatio

n (m

)

ApproximateExactEnd of month

Fig. 4. Head variation at center of island in 1994. Grey dots representhead variation at the end of each month.

498 M. Bakker et al. / Advances in Water Resources 30 (2007) 493–504

5. Analytic element modeling of M0 and M1

For practical application of the proposed method, gen-eral models need to be created of the zeroth and firstmoments of the impulse response function. It is proposedto model M0 and M1 with analytic elements, which leads

to analytic expressions for A(x,y) and a(x,y). As a result,the head fluctuation /(x,y, t) may be computed analyticallyat any point (x,y) and for any time t. Alternatively, discretesolutions of M0 and M1, and thus /, may be obtained witha numerical method such as finite differences or finiteelements.

An analytic element solution consists of the superposi-tion of many analytic solutions to the differential equation[16,8,17]. Each analytic solution is called an analytic ele-ment and has one or more free parameters that are com-puted to meet specified boundary conditions. This paperdeals with aquifers where the heads are controlled bystreams, canals, or ditches with fixed water levels. Such fea-tures may be modeled with line-sinks. The free parametersof a line-sink are the coefficients in the function represent-ing the extraction rate of the line-sink. These may be com-puted, for example, to fix the value of M0 or M1 along astream.

The differential equation for M0 (12) is the standard dif-ferential equation for steady flow in a homogeneous aqui-fer with unit recharge. The analytic element solution to (12)consists of three parts: a particular solution for unitrecharge, a number of line-sinks that fulfill Laplace’s differ-ential equation, plus an arbitrary constant C. A particularsolution to (12) is given by (24), such that M0 may be writ-ten as

M0 ¼1

4TðR2 � r2Þ þ

XN ls

n¼1

fn þ C ð34Þ

where r2 = (x � x0)2 + (y � y0)2; R and (x0,y0) may be cho-sen for convenience. Note that the particular solution rep-resents circular contours around (x0,y0) with the zerocontour going through r = R. The function fn is the zerothmoment of line-sink n, and Nls is the total number of line-

M. Bakker et al. / Advances in Water Resources 30 (2007) 493–504 499

sinks. The extraction rate of line-sink n is chosen to varyalong the line-sink as a polynomial of order Pn. The generalexpression for the M0 of line-sink n may be written as

fn ¼XP n

p¼0

an;pun;p ð35Þ

where an,p is coefficient number p of line-sink n, and un,p isthe zeroth moment for line-sink n with an extraction ratethat varies as Dp, where D is the coordinate along theline-sink. A mathematical expression for a line-sink maybe obtained through the integration of a point sink (a well)along a line segment. The expression for the M0 of a pointsink of unit strength that fulfills Laplace’s equation is (e.g.[16])

M0 ¼1

2pTln r ð36Þ

where r is the radial distance from the well. As shown by[16], integration of (36) along a line may be carried outin the complex plane; a general expression for un,p is pre-sented in the Appendix. The coefficients an,p are computedby applying the boundary condition of M0 at controlpoints along the line-sinks. The control points are distrib-uted using the cosine rule, as proposed by [9]. The resultingsystem of linear equations is solved using a standard meth-od, for example LDU decomposition.

The first moment M1 fulfills differential equation (16)where M0 in the righthand side is given by (34)

r2M1 ¼ �S

4T 2ðR2 � r2Þ � S

T

XN ls

n¼1

fn �ST

C ð37Þ

The analytic element solution to this differential equation iswritten as a particular solution, plus an arbitrary new con-stant D, plus Mls new line-sinks that fulfill Laplace’s equa-tion and are used to meet the boundary conditions for M1

along the surface water features. A particular solution tothe first term on the righthand side is given by (26) suchthat M1 may be written as

M1 ¼S

4T 2

r4

16�R2r2

4þ 3R4

16

� �� S

T

XN ls

n¼1

F n�SCr2

8T 2þXM ls

m¼1

gmþD

ð38ÞThe functions Fn fulfill

r2F n ¼ fn ð39Þ

where fn is given by (35). The functions gm are line-sinks oforder Pm that fulfill Laplace’s equation

gm ¼XP m

p¼0

bm;pum;p ð40Þ

where bm,p is coefficient number p of line-sink m. The coef-ficients bm,p are computed by applying the boundary condi-tion of M1 at control points along the line-sinks; theresulting system of linear equations may be solved usinga standard method.

The function Fn is written as

F n ¼XP n

p¼0

an;pUn;p ð41Þ

where

r2Un;p ¼ un;p ð42Þ

An expression for un,p, the M0 of a line-sink with a strengthDp, was obtained through integration of the M0 for a point-sink (36) along a line. Similarly, an expression for Un,p, theM1 of a line-sink with a strength Dp, is obtained throughintegration of the M1 of a point-sink. The first momentof a point-sink is obtained through integration of (16) with(36) for M0, which gives

M1 ¼ �S

8pT 2ðr2 ln r � r2Þ ð43Þ

Integration of this expression along a line results in anexpression for Un,p, and is presented in the Appendix.

This completes the description of an analytic elementapproach for the modeling of the moments M0 and M1

of the impulse response function of areal recharge. Equa-tions were presented for wells and line-sinks. The momentscomputed with these elements may be used to compute thecoefficients of approximate impulse response functions, forexample the exponential function used in (10). Values ofM0 and M1 (and thus heads) may be fixed along stringsof line-sinks, which may represent rivers, streams, ditches,or boundaries of fully penetrating lakes. Other analytic ele-ments may be derived in a similar fashion, but that isbeyond the scope of this paper. In the next section, theproblem of Section 4 will be solved again, but this timewith the derived analytic elements.

6. Radial flow example with analytic elements

Consider the same case of a circular island as discussedin Section 4. The objective of this example is to demon-strate that the moments M0 and M1 may be modeled accu-rately with the analytic elements derived in the previoussection. For this specific problem, the constants in the par-ticular solution (Eqs. (34) and (38)) may be chosen as(x0,y0) = (0,0) and R = 1000, so that an exact solution isobtained without the need for line-sinks. To test the useof line-sinks, the constants are chosen differently as(x0,y0) = (1000,0) and R = 2200, resulting in the contourlines of M0 shown in Fig. 5. Twenty line-sinks of equallength and second order are used to set the value of bothM0 and M1 to zero along the boundary of the island; thevertices are chosen such that the area of the island is pre-served. The analytic element solution is compared to theexact solution (Eqs. (24) and (26)). Contours of M0 andM1 are shown in Fig. 6. The lower half of each contour plot(below the dotted line) represents the exact solution. Theupper half is the analytic element solution. M0 varies from0 on the boundary to 500 at the center of the island

Fig. 5. Contour plot of particular solution for M0 on the island (island isgrey); radius of the island is R = 1000 m.

500 M. Bakker et al. / Advances in Water Resources 30 (2007) 493–504

(contour interval is 50), while M1 varies from 0 on theboundary to 37,500 at the center (contour interval is3750). The analytic element solution produces accurateresults. At the center of the island, the relative differencebetween the analytic element solution and the exact solu-tion is 0.03% for M0 and 0.2% for M1. The accuracy maybe increased when more shorter line-sinks and/or line-sinksof higher order are used to represent the circular boundarybetter. When the boundary of the island is modeled with 40line-sinks of order 2, rather than 20, the relative differenceat the center of the island decreases to 0.013% for M0 and0.05% for M1. When using 40 line-sinks of order 8, the rel-

Fig. 6. Contour plots of M0 and M1. Upper half of each plot is analytic eleme3750 for M1.

ative difference decreases further to 0.007% for M0 and0.014% for M1.

The contour plots in Fig. 6 are the zeroth and firstmoments of the true impulse response function. The coeffi-cients A and a of the approximate impulse response func-tion are computed with (18). The value of A is identical tothe value of M0 (Fig. 6). Computation of a is more difficult,as it is equal to the ratio of M0 and M1. On the boundary ofthe island, both M0 and M1 are equal to zero, while theirratio a = M0/M1 is finite. In the exact solution of Section4, the limit of this ratio could be worked out mathematically(28), but this is not possible for the general case in a numer-ical model. Hence, it is not possible to compute a right nextto the boundary of the island, but both M0 and M1 can becomputed, of course. In practice, this does not represent alimitation. After all, streams have a width while the line-sinks have a zero width, and furthermore, there is no prac-tical interest to forecast groundwater dynamics right next toa stream or ditch that controls the water table.

To be able to compute a accurately in the vicinity of ahead-specified boundary such as a stream or ditch, M0

and M1 need to be computed accurately, and thus theboundary conditions of M0 and M1 need to be met accu-rately. The contour plots of M0 and M1 in Fig. 6 are notaccurate enough to compute a near the boundary of theisland. This is illustrated in the left part of Fig. 7 (20line-sinks of order 2), where a is inaccurate near the bound-ary of the island. The accuracy may be improved byincreasing the order of the line-sinks, such that the bound-ary condition of M0 and M1 is met more accurately. Asolution with 20 line-sinks of order 8 gives accurate resultsfor a (Fig. 7, right side). It is noted that a varies over asmall range only: from 0.0133 at the center to 0.02 at theboundary.

nt solution, lower half is exact solution. Contour interval is 50 for M0 and

Fig. 7. Contour plot of parameter a. Upper half of each plot is analytic element solution, lower half is exact solution. Boundary modeled with 20 line-sinksof order 2 (left) and order 8 (right). Contour levels from 0.013 to 0.02 with interval 0.007.

M. Bakker et al. / Advances in Water Resources 30 (2007) 493–504 501

7. Practical example

The objective of this final example is to demonstrate thatthe new analytic elements may be applied to a practicalproblem with real geometries of surface water features.The method is applied to the agricultural area near thetown of ‘de Zilk’ in The Netherlands, which contains anirregular pattern of ditches. The surficial aquifer consistsof dune sand to a depth of about 10 m, which is ideally sui-

Fig. 8. Color image of parameters A (left) and a (right); A = 0 along the ditchscale is shown in right plot.

ted for the production of flower bulbs such as tulips andhyacinths. These bulbs are extremely sensitive to the depthof the groundwater table, and thus the groundwater table iscontrolled by a maze of ditches; the agricultural plots areshown in Fig. 8, where the ditches (black lines) are clearlyvisible. The ditches are modeled with line-sinks. Colorimages of the parameters A and a are shown in Fig. 8.Color coding of A is truncated at 24 for visualizationpurposes; values larger than 24 occur on the edge of the

es; no model results in bottom right hand corner of plots (black); spatial

-2

0

4

8

12

rech

arge

(m

m/d

)

0 20 40 60 80

time (days)

-2

0

2

4

6

8

Hea

d va

riatio

n (c

m)

123

Fig. 9. Recharge (top) and head variation (bottom) at three points shownin Fig. 8.

502 M. Bakker et al. / Advances in Water Resources 30 (2007) 493–504

model, which is outside the area of interest, and is coloredblack.

As an example, the values of A and a are computed atpoints 1, 2, and 3 (Fig. 8). At point 1: A = 23.8 anda = 0.176; at point 2: A = 3.64 and a = 1.03; at point 3:A = 10.9 and a = 0.233. Parameter A controls the areabelow the impulse response function and thus the magni-tude of the head variation. Parameter a controls the lengthof the tail of the impulse response function. Betweenditches that are farther apart, the value of A is larger (lar-ger magnitude of the head variation) and the value of a issmaller (longer tail). Conversely, between ditches that arecloser together, the value of A is smaller and the value ofa is larger. The head variation at points 1, 2, and 3(Fig. 8) is computed for a 40 day period of measuredrecharge, followed by a 40 day period of zero recharge(Fig. 9). As may be expected from the values of A and a,the largest head variation and longest tail occurs at point1, followed by point 3, and point 2.

8. Conclusions and discussion

A new analytic method was presented for the simulationof groundwater dynamics due to temporal variations ofrecharge in surficial aquifers. Head variations were com-puted through convolution of a measured recharge serieswith an approximate but analytic impulse response func-tion. The approximate impulse response function (10) con-tains two parameters: A(x,y) and a(x,y). Values of A and a

were computed by setting the moments M0 and M1 of the

approximate impulse response function equal to the corre-sponding moments of the exact impulse response function.The moments were modeled with analytic elements; newanalytic element equations were developed for wells andline-sinks to model M1. It was shown that the proposedapproach produces accurate results through comparisonwith the exact solution for a circular island. Applicationto an agricultural area with a number of ditches illustratedthe two main advantages of the proposed method: (1) headvariations can be computed at any point and any time(except for in the close vicinity of the ditch), even for awildly varying recharge series, and (2) a solution isobtained without the need for the specification of a compu-tational grid, allowing for the accurate (and easy) place-ment of ditches and other hydrologic boundaries in themodel. The proposed methodology constitutes a new tran-sient analytic element formulation.

A number of approximations were made in the presenta-tion of the approach. Most of the approximations weremade to limit the size of the paper and don’t represent lim-itations to the approach. Variations in the saturated thick-ness due to unconfined conditions may be taken intoaccount by formulating the approach in terms of a dis-charge potential [16]. Line-doublet elements for M1 maybe derived in the same fashion as line-sinks, and may beapplied to model polygonal areas with different transmis-sivity values. Line-doublets may be combined with line-sinks to model polygonal areas with different rechargecalled area-sinks [16]. Analytic elements to model M1

may be derived for flow in multi-aquifer systems followingthe approach of [1]. In the presented examples, rechargewas calculated as rainfall minus evapotranspiration, whichis accurate for fairly flat and permeable land surfaces withshallow unsaturated zones. In cases of sloping topographyand variable land use, recharge may be corrected using astandard method, for example the curve number method(e.g., [15]). The unsaturated zone may delay the arrival ofrecharge at the groundwater table; this effect may beincluded through a delay factor, as was done by [19].Another effect of the unsaturated zone is dispersion ofthe recharge, which was studied by [11].

At positions in the aquifer where the groundwater reactsvery quickly, the proposed approximate impulse responsefunction may not be accurate, and a different impulseresponse function may need to be used. A promising alter-native is the Pearson Type III function, as used successfullyfor time series analysis of groundwater levels by [21]. ThePearson Type III function contains three parameters whichrequires the modeling of an additional moment of theimpulse response function. The presented approach maybe extended to other stresses such as fluctuating surfacewater levels or variable discharges of pumping stations,although this may also require a different approximateimpulse response function. The analytic elements presentedin this paper are used to model M0 and M1, which may beused to estimate the parameters of any impulse responsefunction, not just the function used in this paper. The sec-

M. Bakker et al. / Advances in Water Resources 30 (2007) 493–504 503

ond moment M2 may be modeled using a similar approach.Techniques for the inclusion of non-linear effects have beendeveloped for the time series analysis program Menyanthes[22], which is based on the method presented in [21]. Incor-poration of non-linear effects in the current approachforms part of future research.

Acknowledgements

Mark Bakker is on sabbatical from the Department ofBiological and Agricultural Engineering of the Universityof Georgia, Athens, GA. Sabbatical funding was obtainedfrom the TU Delft Grants program. The authors thank EdVeling and Theo Olsthoorn for their suggestions and sup-port. Development of the model discussed in the finalexample was funded by the Amsterdam Water Supply.

Appendix

First, a mathematical expression is presented for up, aline-sink with strength Dp that fulfills Laplace’s equation(the subscript n is dropped for convenience). A line-sinkis obtained through integration of a point-sink. The zerothmoment for a point-sink (36) is written as the real part of acomplex function

M0 ¼1

2pTR lnðz� dÞ ð44Þ

where z = x + iy is the complex coordinate, andd = xw + iyw is the location of the well. The function up

is written as the real part of the complex function xp

up ¼ Rxp ð45Þ

where xp is given in [16] as

xp ¼L

4pT

Z 1

�1

Dp lnðZ � DÞdD ð46Þ

where L is the length of the line-sink, and Z and D are localcomplex coordinates

Z ¼z� 1

2ðz1 þ z2Þ

12ðz2 � z1Þ

; D ¼d� 1

2ðz1 þ z2Þ

12ðz2 � z1Þ

ð47Þ

where z1 and z2 are the end points of the line-sink. Follow-ing [16], integration by parts of (46) gives

xp ¼L

4pT ðp þ 1Þ

�lnðZ � 1Þ � ð�1Þpþ1 lnðZ þ 1Þ

þZ 1

�1

Dpþ1

Z � DdD

�ð48Þ

The remaining integral represents a line-dipole and is givenby [16, Eq. (25.52)] as

Z 1

�1

Dpþ1

Z � DdD ¼ �Zpþ1 ln

Z � 1

Z þ 1� 2

X½1þp=2�

n¼1

Zp�2nþ2

2n� 1ð49Þ

Second, a mathematical expression is presented for Up,the first moment of a line-sink with strength Dp. The first

moment of a well (43) is written as the real part of a com-plex function

M1 ¼ �S

8pT 2R ðz� dÞð�z� �dÞ½lnðz� dÞ � 1��

ð50Þ

where it is used that r2 ¼ ðz� dÞð�z� �dÞ. Up is written as thereal part of the complex function Xp

Up ¼ RXp ð51Þwhere

Xp ¼ �S

8pT 2

Z 1

�1

DpL3

8ðZ � DÞð�Z � DÞ½lnðZ � DÞ � 1�dD

ð52Þwhere it is used that D is a point along the real axis andthus D ¼ �D. Rearrangement gives

Xp ¼ �SL2

16TL

4pT

Z 1

�1

½Dpþ2 � ðZ þ �ZÞDpþ1 þ Z�ZDp�

� ½lnðZ � DÞ � 1�dD ð53Þ

The integral may be written as

Xp ¼ �SL2

16T½Kpþ2 � ðZ þ �ZÞKpþ1 þ Z�ZKp� ð54Þ

where Kp may be expressed in terms of the integral xp (48)and a new integral

Kp ¼ xp �L

4pT

Z 1

�1

Dp dD ð55Þ

where

Z 1

�1

Dp dD ¼ 1� ð�1Þpþ1

p þ 1ð56Þ

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