Technical Note/

Where Do Periodic Variations in the Dischargeof a Well Become Negligible?by Mark Bakker1

AbstractA rule of thumb is presented to determine where variations in the discharge of a pumping well have

a significant influence on the flow in an aquifer. The rule of thumb relates the period of the variation of thedischarge to the distance from the well beyond which the transient effect on the flow in the aquifer is insig-nificant. For example, when an irrigation well pumps intermittently during the growing season, the rulemay be applied to determine the distance from the well beyond which flow in the aquifer can be simulatedwith an average discharge during the growing season; the distance from the well beyond which flow can besimulated with a steady, yearly averaged discharge can also be computed.

IntroductionThe objective of this note is to determine at what dis-

tance away from a pumping well with a variable dischargethe flow may be simulated accurately by a well with anaverage, constant discharge. This question arises, for ex-ample, when the effect of an irrigation well on the flow ina nearby stream is investigated. In a typical operation, thewell may pump for 12 to 24 h roughly every couple ofdays during the 3- to 4-months-long growing season. Whenthe stream depletion due to the well is computed, it mustbe decided whether all details of the variable discharge ofthe well need to be taken into account or whether a wellwith an average, constant discharge during the growingseason results in the same stream depletion.

Haitjema (1995, p. 292) presented a rule of thumb todetermine whether a steady-state ground water model ofaverage wet (winter) or dry (summer) conditions is a reason-able representation of the actual, transient, periodic process.He arrived at his rule through the comparison of steady-state solutions with transient solutions for one-dimensional

flow in aquifers with sinusoidally varying recharge or headboundary conditions. In this note, an answer to the question‘‘Where do periodic variations in the discharge of a wellbecome negligible?’’ is sought through comparison of theflow due to a well with a constant discharge and a well withvariable discharge. A rule of thumb will be presented,which may be applied to practical situations. The mathe-matical derivation is given first, followed by a discussion ofthe practical implications and an example.

A Well with a Cosine-ShapedDischarge Function

Consider flow in a homogeneous, isotropic uncon-fined aquifer. The radial discharge Qs,r (the vertically inte-grated specific discharge) of a steady well with a constantdischarge Q0 is given by (e.g., Strack 1989)

Qs;r ¼ 2Q0

2p1

rð1Þ

where r is the radial distance from the well. This steadyflow is compared to the flow caused by a well with a dis-charge that has the same average Q0 but varies periodi-cally as a cosine between 0 and 2Q0 (solid line, Figure 1)

QðtÞ ¼ Q0 2 Q0 cosð2pt=TÞ ð2Þ

1Department of Biological and Agricultural Engineering,University of Georgia, Athens, GA 30602; fax (706) 542-8806;mbakker@engr.uga.edu

Received April 2005, accepted September 2005.Copyright ª 2006 The Author(s)Journal compilationª 2006 National Ground Water Association.doi: 10.1111/j.1745-6584.2005.00159.x

478 Vol. 44, No. 3—GROUND WATER—May–June 2006 (pages 478–482)

where t is time and T is the time period of the fluctuation;such a well is referred to as a cosine well here. The radialflow Qc,r for a cosine well consists of the contributionof a steady well with discharge Q0 minus a well withdischarge Q0cos(2pt/T). The latter is obtained throughlinearization of the governing Boussinesq equation and isgiven by, e.g., Streltsova (1988); it is written here in theform given by Bakker (2004)

Qc;r ¼ Qs;r 1Q0

2pk´½K1ðr

ffiffii

p=kÞ

ffiffii

pexpð2pit=TÞ� ð3Þ

where ´ stands for the real part of a complex function,K1 is the modified Bessel function of the second kind andfirst order, i is the imaginary unit, and k is called the char-acteristic length, defined as

k ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikhT=ð2pnÞ

qð4Þ

where k is the hydraulic conductivity, h is an average sat-urated thickness of the aquifer, and n is the effectiveporosity (the storativity of the aquifer). Note that the solu-tion (Equation 3) has no initial condition, as a cosine wellis pumping with discharge (Equation 2) forever.

The relative difference d between the radial flowcaused by the cosine well and the flow caused by theconstant-discharge well is defined as

d ¼ Qc;r 2 Qs;r

Qs;rð5Þ

Substitution of Equation 1 for Qs,r and Equation 3 forQc,r gives

d ¼ 2r

k´½K1ðr

ffiffii

p=kÞ

ffiffii

pexpð2pit=TÞ� ð6Þ

This equation may be simplified by writing the complexBessel function in polar form, which gives

d ¼ Aðr=kÞcosð2pt=T 1 aÞ ð7Þ

where the amplitude A is

Aðr=kÞ ¼ r

kjK1ðr

ffiffii

p=kÞj ð8Þ

and a is the phase shift; since a does not play a role in thecurrent analysis, it is not discussed further.

The amplitude A of d is plotted vs. r/k in Figure 2.The amplitude of d represents the maximum relative devi-ation from steady flow. At a distance of 6k from a well,the value of d is 0.046, and thus the flow caused bya cosine well deviates from the flow caused by a steady-state well by at most 4.6%. At a distance of 8k this devia-tion is 1.3%, and at 10k the deviation is 0.3%. Hence,from a practical standpoint, the radial flow due toa cosine well may be represented accurately by a steadywell beyond a distance of 6k from the well.

This conclusion may be extended to the computationof the inflow from a fully penetrating, straight streamwith a constant head. When the well is located at least ata distance 6k from the stream, the inflow from the streammay be approximated as steady. This may be seen as fol-lows. Consider a well located at a distance d froma stream (Figure 3). The flow field for this case may beobtained through the addition of an image well to thesolution. The steady component Qs,x of the discharge vec-tor in the x-direction is (Strack 1989, sec. 7)

Qs;x ¼ 2Q0

2px1 d

r211

Q0

2px 2 d

r22ð9Þ

where r1 is the distance from the well and r2 is the dis-tance from the image well. At the closest point on thestream (x ¼ y ¼ 0), Equation 9 simplifies to

Qs;xð0; 0Þ ¼ 2Qs;rðr ¼ dÞ ð10Þ

Figure 1. Discharge function for a cosine well (solid) anda block well (dashed).

Figure 2. Amplitude A of relative deviation d of cosine wellfrom steady well vs. r/k.

M. Bakker GROUND WATER 44, no. 3: 478–482 479

Similarly, for a cosine well, the component Qc,x ofthe discharge vector at the closest point on the stream is

Qc;xð0; 0Þ ¼ 2Qc;rðr ¼ dÞ ð11Þ

Substitution of these values in d ¼ (Qc,x – Qs,x)/Qs,x

gives Equation 7 evaluated at r ¼ d and thus leads to thesame conclusion regarding the amplitude of the relativedeviation from steady flow. Note that the previous analy-sis concerned the point on the stream closest to the well.All other points are farther away, and thus the deviationfrom steady inflow d(r ¼ d) is an upper bound; the devia-tion from steady inflow at other points along the stream issmaller.

A Well with a Block-Shaped Discharge FunctionIn reality, pumping schedules of irrigation or munici-

pal wells are more accurately represented by a block-shaped discharge function, where the well pumps at aconstant rate QB for a time �t during every period T(dashed line, Figure 1). Such a well is referred to here asa block well. The discharge of a block well may be repre-sented by the following Fourier series (e.g., Churchill andBrown 1987)

QðtÞ ¼ Q0 2Q0T

�t

XNj¼1

2

jpsin

�jp

�1 2

�t

T

��cos

�2jptT

�

ð12Þ

where Q0 ¼ QB�t/T is the time-averaged discharge. Itwas shown in the previous section that cosine variationswith a period T are insignificant beyond a distance of 6k.As k is a function of the period, it may be concluded thatat a large enough distance from the block well, the headand flow may be modeled by a cosine well with a dis-charge that includes only the first term of the summation(i.e., the largest period) in Equation 12, which gives

QðtÞ ’ Q0 2 lQ0 cosð2pt=TÞ ð13Þ

where

l ¼ 2T

p�tsin½pð1 2 �t=TÞ� ð14Þ

The multiplication factor l is plotted vs. the relativetime of pumping �t/T in Figure 4. The multiplication

factor varies from 2 for �t approaching zero to 1.27 whenthe well pumps half the time (�t ¼ T/2); Streltsova(1988) reports the same result for �t ¼ T/2.

The relative difference e between the radial flow Qb,r

of a block well and the radial flow caused by a steadywell with the average discharge is defined as

e ¼ Qb;r 2 Qs;r

Qs;rð15Þ

At a sufficiently large distance away from the well,the discharge of the block well may be represented byEquation 13, such that

e ¼ ld ¼ lAðr=kÞcosð2pt=T 1 aÞ ð16Þwhere A is given by Equation 8 and a is the phase shift.When a block well is located at a distance d froma straight stream, Equation 16 may be evaluated at r ¼d to obtain an upper bound for the deviation of the inflowalong the stream from steady flow; this result may beobtained by application of image well theory as shown inthe previous section.

Rule of ThumbThe amplitude lA of e represents the maximum rela-

tive deviation of a block well from steady flow. The mul-tiplication factor l may be estimated conservatively fromFigure 4 as l ¼ 2, independent of the ratio �t/T. Usingthis conservative estimate, the flow caused by a blockwell deviates from the flow caused by a steady well by atmost 4.9% at a distance of 7k from the well, and by atmost 0.7% at 10k.

The following rule of thumb is proposed for practicalapplication. When the discharge of a well has a periodT *, flow caused by the well may be approximated assteady at a distance L from the well (assuming that varia-tions of 4.9% are insignificant), where L is given by

well

(x,y) = (−d,0)

image well

(x,y) = (d,0)stre

am

y

x

Figure 3. Awell located at a distance d from a stream.

Figure 4. Multiplication factor l vs. fraction Dt/T of periodthat well pumps.

480 M. Bakker GROUND WATER 44, no. 3: 478–482

L ¼ 7k ’

ffiffiffiffiffiffiffiffiffiffiffiffiffi8khT�

n

sð17Þ

Vice versa, at a distance L, variations in the dis-charge of a well that have a period T * or less cause insig-nificant variations in the flow when T * fulfills thefollowing equation

T� ’L2n

8khð18Þ

Practical ApplicationTo illustrate the practical application of the rule of

thumb, consider an aquifer with the following properties:a hydraulic conductivity of k ¼ 10 m/d, an average satu-rated thickness of h ¼ 40 m, and an effective porosity ofn ¼ 0.3. An irrigation well pumps from the aquifer duringthe 4-month-long growing season (from day 100 to day220). During this period, the well pumps for half a dayevery 2.5 d with a block-shaped discharge function, fora total of 40 pumping periods. The radial flow due to thewell is simulated exactly by one steady-state well with theyearly averaged discharge plus a number of Theis wellssuch that the variation in the discharge is representedexactly through superposition in time (e.g., Strack 1989;Haitjema 1995; Fitts 2002). The variation of the radialflow is computed at three different distances from thewell (Figure 5). The radial flow in Figure 5 is normalizedthrough division by the radial flow caused by a steady-state well with the yearly averaged discharge. (The graphrepresents the flow in the third year of the pumpingschedule to reduce the effect of the initial steady-statecondition in the Theis solution.)

The solid line in Figure 5 represents the variation ofthe normalized radial flow at a distance of 150 m fromthe well. Substitution of 150 for L in Equation 18 givesT * ¼ 2.1 d. Hence, at this distance, variations of the dis-charge with a period of 2.5 d during the growing seasonstill have a significant effect on the flow, as evidenced bythe graph. The dashed line in Figure 5 represents the

variation at a distance of 300 m. Here, T * has increased to8.4 d, and the variations in the discharge during the grow-ing season (with a period of 2.5 d) are insignificant andnot visible in the graph. At this distance, the flow can besimulated accurately with a well that pumps at a constant,average discharge during the growing season. The dashed-dotted line in Figure 5 represents the variation at a dis-tance of 1950 m from the well. Here the value of T * is ~1year, and the flow can be simulated accurately witha steady-state well with a yearly averaged discharge.

It may be useful to point out that for a daily variationof the discharge (e.g., 8 h of pumping, 16 h off), the dis-tance L at which the well may be approximated as steadyis much smaller. Using the same aquifer parameters witha period of T * ¼ 1 d gives only L ¼ 103 m. Daily varia-tions of the discharge in a confined aquifer, however, areof course noticeable at much larger distances from thewell as the effective porosity in the equation for k needsto be replaced with the storativity of the confined aquifer,which is generally several orders of magnitude smallerthan the effective porosity.

ConclusionA rule of thumb was derived to determine at what

distance from a well periodic variations in the dischargeof the well have a significant influence on the flow inthe aquifer. Variations in the discharge with a periodsmaller than T * have an insignificant effect on the flowbeyond a distance L from the well, as computed withEquation 17. When the well is located at a distance L ¼d from a stream, the inflow from the stream may beapproximated as steady when periodic variations in thedischarge of the well are smaller than T * as computedwith Equation 18.

AcknowledgmentThis research was funded in part by USDA grant

2002-34393-12117. The author thanks Henk Haitjemaand two anonymous reviewers for their constructivecomments.

Figure 5. Radial flow normalized by yearly averaged flow vs. time at r = 150 m (solid), r = 300 m (dashed), and r = 1950 m(dash-dot).

M. Bakker GROUND WATER 44, no. 3: 478–482 481

ReferencesBakker, M. 2004. Transient analytic elements for periodic Du-

puit-Forchheimer flow. Advances in Water Resources 27,no. 1: 3–12.

Churchill, R.V., and J.W. Brown. 1987. Fourier Series and Bound-ary Value Problems, 4th ed. New York: McGraw-Hill.

Fitts, C.R. 2002. Groundwater Science. San Diego, California:Academic Press.

Haitjema, H.M. 1995. Analytic Element Modeling ofGroundwater Flow. San Diego, California: AcademicPress.

Strack, O.D.L. 1989. Groundwater Mechanics. EnglewoodCliffs, New Jersey: Prentice Hall. http://www.strackconsulting.com.

Streltsova, T. D. 1988. Well Testing in Heterogeneous For-mations. New York: Wiley.

482 M. Bakker GROUND WATER 44, no. 3: 478–482