An improved time series approach for estimating groundwaterrecharge from groundwater level fluctuations

M. O. Cuthbert1

Received 27 August 2009; revised 5 April 2010; accepted 15 April 2010; published 10 September 2010.

[1] An analytical solution to a linearized Boussinesq equation is extended to develop anexpression for groundwater drainage using estimations of aquifer parameters. This is thenused to develop an improved water table fluctuation (WTF) technique for estimatinggroundwater recharge. The resulting method extends the standard WTF technique bymaking it applicable, as long as aquifer properties for the area are relatively well known, inareas with smoothly varying water tables and is not reliant on precipitation data. Themethod is validated against numerical simulations and a case study from a catchmentwhere recharge is “known” a priori using other means. The approach may also be invertedto provide initial estimates of aquifer parameters in areas where recharge can be reliablyestimated by other methods.

Citation: Cuthbert, M. O. (2010), An improved time series approach for estimating groundwater recharge from groundwaterlevel fluctuations, Water Resour. Res., 46, W09515, doi:10.1029/2009WR008572.

1. Introduction

[2] Techniques for estimating recharge based ongroundwater level (h [L]) fluctuations are amongst the mostcommonly applied methods [Healy and Cook, 2002].Although relevant analytical solutions to various Boussinesqequations [Bear, 1972] have been derived for this purpose[Su, 1994; Rai et al., 2006], configurations useful forpractical application to real hydrogeological situations arenormally intractable. Hence, alternative approaches arecommonly adopted. Assuming that changes in groundwaterlevel in an aquifer through time (t [T]) are controlled by thebalance of recharge (q [LT−1]) to and net groundwaterdrainage (D [LT−1]) away from a given observation point,recharge may be calculated as follows:

q ¼ SY@h

@tþ D; ð1Þ

where SY is the specific yield [−]. This method (often termedthe water table fluctuation (WTF) method) assumes thatwater level changes are not due to other factors such asentrapped air or barometric fluctuations. Its main limitationsare difficulties of defining and estimating specific yield andaccounting for the drainage term (D) robustly. As infiltratingrainfall moves through the unsaturated zone, the timing andmagnitude of the water table response may vary greatlydepending on antecedent conditions, evapotranspirativedemand, hydraulic properties, and depth to water table. Anincreasing lag and attenuation of the rainfall signal withincreasing depth to water table is often observed. Thus, inshallow water table situations, q may be much larger than Dduring a recharge event and so the error in recharge esti-

mation due to error in D may be negligible. Where depth tothe water table is greater and the water table varies moresmoothly, it becomes more important to account more rig-orously for D in order to minimize the error in estimating q.[3] Crosbie et al. [2005] offer an improved WTF method

(for instance, by giving a more rigorous expression for SY)but takes an empirical approach to accounting for D, whichmay therefore limit its applicability to areas of shallow watertable. The method also requires a detailed rainfall time seriesfor the analysis. More recently, Park and Parker [2008]included an expression for D in their model for water tablefluctuations in response to precipitation which is linked tothe aquifer properties. However, the model assumes thatrecharge is a fixed proportion of rainfall, which is unrealisticin temperate areas where there is a marked seasonality inrecharge due mainly to changes in evapotranspiration ratherthan rainfall. Thus, the challenge of developing an appro-priate expression for predicting D, which was posed by Healyand Cook [2002], is still open.[4] This paper builds on the existing literature and pre-

sents an improvement of the WTF technique by developingan expression for the groundwater drainage componentbased on estimations of aquifer parameters. The resultingtechnique extends the standard WTF method by making itapplicable, as long as aquifer properties for the area arerelatively well known, in areas with deep water tables and/orwhere groundwater levels may be smoothly varying.

2. Estimating Drainage (D) for a ParticularLocation in a Catchment

[5] In order to improve the WTF method, it is desirable toreliably estimate drainage (D). To investigate how D mayvary spatially and temporally across a catchment, considerthe case of an ideal homogeneous, horizontal aquiferbounded at one end (x = 0) by a no‐flow boundary re-presenting a flow divide and at the other (x = L) by a riverassumed to be a constant head boundary (Figure 1).

1School of Geography, Earth and Environmental Sciences,University of Birmingham, Birmingham, UK.

Copyright 2010 by the American Geophysical Union.0043‐1397/10/2009WR008572

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Although idealized, the situation is typical of many aquifersystems.[6] A one‐dimensional Boussinesq equation of ground-

water flow for an aquifer receiving homogeneous rechargethat follows a sinusoidal pattern (with an average value qa)can be given as

@

@xKH

@H

@x

� �¼ SY

@H

@t� q tð Þ; ð2Þ

with

q tð Þ ¼ qa 1� cos!tð Þ; ð3Þ

where K is hydraulic conductivity [L T−1], SY is specificyield [−], w is the angular frequency [T−1], H(x, t) is satu-rated aquifer thickness [L], t is time [T], x is distance [L],and q is groundwater recharge [L T−1].[7] Generalizing H to h (groundwater head above ordi-

nary datum) and if changes in transmissivity due to fluc-tuations in groundwater levels are assumed to be negligible,equation (2) may be linearized as follows:

T@2h

@x2¼ SY

@h

@t� q tð Þ; ð4Þ

where T is transmissivity [L2 T−1] and h(x, t) is groundwaterhead [L].[8] Given the boundary conditions

@h 0; tð Þ@x

¼ 0

h L; tð Þ ¼ 0;

Erskine and Papaioannou [1997] find a solution to equation (4)by first assuming that a quasi steady state solution exists(i.e., a solution periodic in time). They then derive thesolution explicitly as a complex function which is the sum ofperiodic and steady state head variations. The derivation isgiven byErskine and Papaioannou [1997] and is not repeated

here. The combined result, correcting for the sign of thefirst term as mistakenly printed in their paper, is given by

h x; tð Þ ¼ Re � qaei!t

Si!1� cosh�x

cosh�L

� �� qa2T

x2 � L2� �� �

; ð5Þ

where l2 = i!SYT .

[9] The second component of the right‐hand side (RHS)of equation (5) is a well‐known steady state solution, whichdefines the general form of the water table. The first sinu-soidal, transient term on the RHS of equation (5) has beenplotted in Figure 2 for a set of arbitrary aquifer parameters(T = 10 m2/d, SY = 0.02, L = 1000 m) using an annualrecharge oscillation (qa = 0.0003 m/d). As would be ex-pected, maximum head changes are seen nearest the no‐flowboundary with smaller fluctuations occurring toward theriver as the constant head boundary exerts greater control.[10] Taking Erskine and Papaioannou’s [1997] analysis

further by comparing equations (1) and (4), it is clear that

D ¼ �T@2h

@x2: ð6Þ

By substituting equation (5) into equation (6), we cantherefore derive the following expression for D,

D x; tð Þ ¼ Re �qaei!t cosh�x

cosh�L

� �þ qa

� �: ð7Þ

This is a periodic function oscillating around qa withamplitude A, where

A ¼ qacosh�x

cosh�L

� ���������: ð8Þ

Figure 3 plots D for various values of x using the same set ofparameters used in Figure 2. Since head variations are at amaximum away from the influence of the river and therecharge input is constant across the model domain, we seethat the variation in D is smallest away from the river.[11] For a range of values of aquifer diffusivity

( TSY [L2 T−1]) and L, equation (8) has been used to define the

value of x/L below which A becomes a small percentage(arbitrarily chosen as 5%) of D, and the results are sum-marized in Table 1. Thus, for a wide range of values ofaquifer diffusivity, for locations sufficiently far away fromthe river, A is small and D ≈ qa in all but the smallest ofcatchments.[12] Since the recharge signal is periodic, the temporally

averaged head h is given by the steady state component ofequation (5),

h ¼ qa2T

L2 � x2� �

: ð9Þ

This may be simply rearranged to give an approximationfor D, which improves as the distance away from the riverincreases,

D � qa ¼ 2hT

L2 � x2: ð10Þ

For thinner aquifers where T may vary significantly with h,the equivalent transient Boussinesq equation (equation (2))is not known to be tractable analytically. However, given

Figure 1. Idealized aquifer receiving sinusoidal recharge(after Erskine and Papaioannou [1997], with permissionfrom Elsevier).

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the boundary conditions @h 0;tð Þ@x = 0 and H(L,t) = b, the

steady state solution, equivalent to equation (9) for thelinearized problem, can be shown to be given by

H2 ¼ b2 þ qa

KL2 � x2� �

; ð11Þ

where H is the temporally averaged saturated aquiferthickness [L]. This is a well‐known solution which fol-lows, for example, from considerations of flow to hori-zontal drains under constant recharge [e.g., Hiscock, 2005,p. 61].

[13] We may thus expect that an equivalent approxima-tion may be made for D as follows:

D � qa ¼K H

2 � b2 L2 � x2

: ð12Þ

The applicability of this expression is confirmed numeri-cally below.[14] From this idealized case, we may infer that in many

analogous real catchments, away from the influence ofconnected surface water bodies, since the recharge signal

Figure 2. Time variant components of recharge and head against time for various values of x (withT = 10 m2/d, SY = 0.02, L = 1000 m, qa = 0.0003 m/d).

Figure 3. Plot of recharge and drainage against time for various values of x (with T = 10 m2/d, SY = 0.02,L = 1000 m, qa = 0.0003 m/d).

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will include long‐term climatic signals in addition to annualsignals, a reasonable approximation for qa and therefore Dmay be found using equation (10) or equation (12) for h orH equal to the annual running mean of the groundwaterlevel above the river level or aquifer base, respectively. Asdefined by equation (8), as x or aquifer diffusivity increase,and/or L decreases, the temporal variation in D is likely toincrease, and hence, it becomes much more difficult toderive a generalized expression for estimating D.

3. Methodology for Estimating Recharge FromTime Series of Groundwater Levels

[15] The following methodology is proposed for esti-mating groundwater recharge using data for a given obser-vation well.[16] Step 1: develop a simple conceptual model of the

catchment in which the well is located. This should includean estimate of the main hydrogeological boundaries such asgroundwater divides and drainage features; the likely con-trols on recharge due to variations in climate variables,superficial deposits, and the thickness of the unsaturatedzone; and estimations of aquifer storage and transmissivity.[17] Step 2: check that the monitoring well to be used for

analysis is sufficiently far from a drainage outlet. Equation (8)and/or Table 1 may be used to check that A � D for theaquifer/catchment in question. If the well is too near adrainage outlet, then the proposed method is likely to bemisleading. In such a case, if the water table is relativelyshallow (less than a few meters) the degree of smoothing inthe recharge signal due to incident rainfall at the water tablemay be minimal. If rainfall data are also available, then anapproach such as that proposed byCrosbie et al. [2005] couldbe implemented.[18] Step 3: check that the time series of groundwater

level data is at a sufficiently small temporal resolution toresolve the likely variations in WTF. The best way ofachieving this will vary from catchment to catchment andmay be possible through consideration of high frequencytemporal groundwater level data, if they are available in thecatchment of interest as is the case in the worked exampledescribed below. If not, but if rainfall data are available orthe typical magnitude and variability of the incident rainfallfor the area is known, then this could be done, for example,using a simple 1‐D variably saturated flow model such asHydrus [Simunek et al., 2005], which is freely available.[19] Step 4: assess whether the well will be representative

of the catchment as a whole. Variations in recharge mayoccur across a catchment because of variations in, forexample, climatic variables, superficial geology or vegeta-tion on a range of length scales. However, the proposed

method for calculating recharge assumes that the rechargedistribution is homogeneous. An individual well will showwater level responses which spatially and temporally inte-grate pressure responses to recharge from across the widercatchment. The extent to which WTFs in a particular loca-tion may be controlled by the local rate of recharge versusrecharge in the wider area will be primarily controlled bythe aquifer diffusivity and the length scale over which therecharge variation occurs. Hence, it is important that thelikely spatial and temporal variability in recharge acrossthe catchment is considered in order to assess whether thewell used for analysis may be representative of the catch-ment. One way of making such an assessment is given in theworked example from Shropshire below.[20] Barometric responses may be seen in wells situated

where the aquifer is confined by low permeability depositsand, if significantly large, water levels should be correctedfor such effects before the proposed analysis is used.Clearly, this is only feasible if atmospheric pressure read-ings are available, for instance using the method proposedby Gonthier [2007], and wells thus situated may not alwaysbe appropriate for the WTF methodology.[21] Wells used for analysis should also be sufficiently

distant from pumping wells to ensure that water levels arenot significantly lowered by pumping. In the case of long‐term constant rate pumping, if quasi steady state conditionshave been attained, then the lowered water level in the wellwould, through equation (10) or equation (12), be expectedto underestimate D and hence q. Time variant pumping rateswill lead to changes in groundwater levels with time and so,with reference to equation (13), periods of increasedpumping rate and therefore drawdown will lead to lowerestimates of recharge, whereas periods of decreased pump-ing and therefore recovery will lead to higher estimates.Thus, the effects of superficial geology and groundwaterabstraction may further restrict the choice of wells appro-priate for this methodology.[22] Step 5: calculate D from equations (10) or (12) and

then q from equation (1). For example, if the aquifer haslarge saturated thickness and changes of T with h arethought to be negligible, equation (10) is applicable. Com-bining equations (10) and (1) enables recharge to be cal-culated (for example, this can be simply implemented in aspreadsheet) for each time step (Dt) as follows:

qt ¼SY ht � h t�Dtð Þ

� �Dt

þ 2hT

L2 � x2ð Þ ; ð13Þ

where ht and h(t ‐ Dt) are the groundwater levels (relative tothe river level, h0) [L] at time t and t ‐ Dt [T], respectively.[23] A value for L for a 3‐D catchment can be estimated

using the “equivalent aquifer length” calculation describedby Erskine and Papaioannou [1997], which represents theaverage distance between river and groundwater divide.Simply dividing half the catchment area by the total riverlength works well for simple geometrical aquifer and riverconfigurations [Erskine and Papaioannou, 1997].[24] Step 6: check the sensitivity of the derived recharge

values to the estimated values of L, K, b, and SY to makesure the uncertainty in the method is acceptably small. Insome areas, the constraint on the aquifer parameters maybe poor and lead to large uncertainties in the estimatedrecharge, in which case other methods should be sought.

Table 1. Values of x/L Below Which A < 0.05D for a RangeValues for L and Aquifer Diffusivitya

Diffusivity (m2/d) L = 1000 m L = 10,000 m L = 100,000 m

1 0.968 0.997 1.00010 0.898 0.990 0.999100 0.677 0.968 0.9971000 – 0.898 0.99010,000 – 0.677 0.968100,000 – – 0.898

aA dash denotes A > 0.05D for all values of x.

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[25] Step 7: where possible, estimate recharge by at leastone other method as a means of verification. Because of theuncertainties involved in all recharge estimation methods, itis best practice to corroborate findings by more than onemethod [Scanlon et al., 2002].

4. Testing the Wider Applicability of the SolutionUsing Numerical Models

[26] The appointed form of the groundwater flow equa-tion given by equation (4) is for one‐dimensional flowthrough a homogeneous aquifer with a constant saturatedthickness, horizontal base, and spatially constant, time variantsinusoidal recharge. Although these assumptions will neverbe fullymet in real aquifer systems, they have allowed a directanalytical solution for the current problem to be found,suggesting a relatively simple approach for estimating D.[27] In reality, more complex situations will be encoun-

tered so a series of numerical models were developed to testthe applicability of the analytical solution in such situations.A single layered forward model was created in MODFLOW[McDonald and Harbaugh, 1988] for several scenariosdescribed below. For all scenarios, a 10 km domain at a 100mgrid spacing was bounded at one end by a no‐flow boundaryand at the other by a constant head boundary (Figure 1). Asteady state model was first run to create the initial conditionsfor a 10 year transient run setup with daily stress periods usedto input a sinusoidal recharge signal (qbase) at the upperboundary with an amplitude and average both equal to 3 ×10−4 m d−1. A head change criterion of 0.001 m was used forconvergence, and a set of synthetic groundwater observa-tions was output for the same location for each model run, atx = 2500 m, and used to estimate recharge using the modifiedwater table fluctuation methodology proposed above.[28] The various scenarios and results are now described

in turn.[29] Scenario a: a baseline test to check the analytical

solution against a numerical model. The baseline parametersare as follows: SY = 0.1, L = 10,000 m, T = 400 m2 d−1. Asshown in Figure 4a, the recharge input to the numericalmodel exactly matches that derived using the methodologybased on the analytical solution.[30] Scenario b: an aquifer where T may vary signifi-

cantly with h. Scenario a was altered so that the saturatedaquifer thickness at the river, b = 100 m and K = 4 m d−1

and therefore the minimum T = 400 m2 d−1. The result (b1)is presented in Figure 4a, indicating that although themagnitude of the resulting sinusoidal recharge has the cor-rect magnitude, the average value is underestimated byaround 13%. It is noted that the minimum predictedrecharge, rather than being zero, is a negative number sinceequation (10) has underestimated the value of D. Hence, therecharge estimation method was then rerun using equation(12) instead of equation (10) and the resulting outputexactly matches the input recharge used for the numericalsimulations, confirming the applicability of this equation.[31] Scenario c: Scenario b was altered to test the case of

an aquifer with an inclined base. The aquifer base was set to100 m depth below the river with increasing elevation under arange of gradients toward the no‐flow boundary. The pre-dicted recharge becomes increasingly out of phase concom-itant with increases in the amplitude of the recharge and itsaverage value. For the case where the gradient is equal to

0.01, the maximum tested (Figure 4a), the errors in theaverage and amplitude of the recharge are 9.8% and 2.5%,respectively, with a phase shift of around 10 days. This in-dicates that, for catchments with this type of geometry, themethod may produce significant errors due to the assump-tions included in the analytical approach, in addition to anydue to parameter uncertainty, and the method should be usedwith caution.[32] Scenario d: Scenario a was altered for the case of a

“patchy” variability in the amplitude of the recharge signal,for instance, in the case where discontinuous low perme-ability deposits overlie an aquifer. Alternating 1000 m pat-ches of low recharge (0.1 × qbase) and high recharge (1.9 ×qbase) were used to maintain an average recharge equal toqbase. The location x = 2500 m, from which the syntheticgroundwater heads used for estimating recharge werederived, is in this case situated at the center of a lowrecharge patch. Two variations were run. In the first (d1),the low recharge patches were set to coincide with confiningconditions in the aquifer limiting SY to the value of theconfined storage coefficient (1 × 10−4) in these areas, withthe high recharge areas left with a value of SY equal tospecific yield (0.1). The value of SY used for the rechargeestimation was therefore the average storage coefficient forthe catchment of 0.05005. A second variation (d2) was runin which the whole domain was left unconfined with SY =0.1. The results are shown in Figure 4b indicating that bothruns underestimate the recharge by around 8% and showvarying degrees of phase shift and attenuation. These resultsare reasonable given that the synthetic groundwater ob-servations, output at a position situated in the center of apatch of low recharge, would be expected to integrate thelower recharge contributions in the vicinity of the observa-tion point with the high recharge signal from areas furtheraway, with the difference in the timing of the signals due tothe variation in storage coefficient, and thus aquifer diffu-sivity, used for the two runs. It is notable that d1 shows amuch better agreement in the timing and magnitude of therecharge series in the case where the confining conditionsexist beneath the patches of low recharge. These resultsseem to indicate that, even where recharge is very variable,reasonable results may be obtained using the proposedmethod if the diffusivity of the aquifer is high enough.[33] Scenario e: Scenario a was altered for the case of an

arbitrary, 10 year, daily, nonsinusoidal recharge input. Therecharge input shown in Figure 5a was used to generate thesynthetic hydrograph also shown in Figure 5a. The method-ology was then applied to this synthetic hydrograph to gen-erate an estimated time series of recharge. A regression of theinput recharge and the estimated recharge is shown in Figure5b, indicating excellent agreement to within 0.1% error onaverage.[34] The result of these various numerical tests indicate

that, though the methodology is based on an analyticalsolution for the case of a highly simplified idealized aquifer,the errors associated with other model structure or rechargeinputs may not necessarily cause unacceptable error in therecharge estimated using the proposed methodology.

5. Parameter Sensitivity of the Baseline Scenario

[35] A parameter sensitivity analysis of the method wasalso carried out against the baseline scenario a parameter set

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described above. Each of the parameters L, T, SY werevaried one by one with the other parameters being heldconstant and an estimated time series of recharge was thenderived using equation (13). Since the appropriate value ofriver base elevation may also be, to some extent, uncertain, h0was also included in the sensitivity analysis with the baselinevalue set at 100 m above an arbitrary datum. Each estimatedrecharge time series was compared to the baseline case.[36] The comparison of the sensitivity runs with the

baseline case, both in terms of the average recharge and theamplitude of the recharge time series is summarized in

Table 2. All results were found to be sinusoidal and in phasewith the baseline case. As is expected from the form ofequation (13), increases in T or decreases in L or h0 lead toincreases in average recharge, respectively (and vice versa),but no change in the amplitude of the sinusoid. Thesechanges are proportional for T but more sensitive for L andh0. In contrast, changes in SY lead proportionally to changesin recharge amplitude but have no affect on the long termaverage recharge. This result is, in part, an artefact of thesinusoidal input, since the average change in head will beequal to zero over a whole number of sine wave periods, and

Figure 4. Input and estimated recharge output from numerical model tests (for x = 2500 m): (a) scenariosa–c and (b) scenario d.

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the average recharge will thus entirely be controlled by thesecond term on the right‐hand side of equation (13). How-ever, the results indicate, helpfully, the control that theaquifer hydraulic conductivity and length parameters haveon the long term recharge estimates and that the storagecoefficient has on the magnitude of the recharge fluctuationsproduced by the modified WTF methodology. When themethodology is applied to a groundwater level time series

with differing beginning and end head values, then there willbe a sensitivity of the estimated long‐term recharge to SY, inproportion to the ratio of the first and second terms on theRHS of equation (13).

6. Worked Example for Shropshire, UK

[37] To give further validation, a worked example of themethod is now presented for a subcatchment of the River

Figure 5. Results of numerical model test for scenario e. (a) Nonsinusoidal recharge time series input tothe model and the resulting synthetic groundwater hydrograph used for recharge estimation. (b) Regres-sion of input and estimated recharge.

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Tern, itself a tributary of the River Severn in Shropshire,UK. This area was chosen since groundwater recharge forthe catchment is relatively well known, having been esti-mated by a variety of methods as summarized by Cuthbert[2006]. Although the potential recharge may be relativelystraightforward to estimate on the basis of climate variablesand observed patterns of land use, highly variable superficialgeology makes the estimation of actual recharge for thecatchment much more challenging. Thus, it is a useful testcase in which to apply the modified WTF method. The stepsoutlined in the methodology above are now applied to thiscontext.[38] Step 1: the study area has a variable covering of

superficial deposits in terms of both lithology and thickness,overlying a regional sandstone aquifer. It is a lowlandcatchment with a gently undulating terrain as shown by theshaded relief map in Figure 6a based on a Digital TerrainModel (DTM) at a 50 m spatial resolution [Morris andFlavin, 1990]. The terrain is generally relatively flat witha small degree of valley incision due to erosion by thePotford and Platt Brooks. Landcover 2000 data [Fuller etal., 2002] indicate that land cover is predominantly arable(55%) and grassland (37%) with a small amount of wood-land and heath (6%) and urban areas (2%).[39] The 1:50,000 superficial geology map of the area is

shown in Figure 6b. This suggests that around 70% of thearea is underlain by superficial deposits comprising a vari-able covering of glacial till, glaciofluvial deposits, andvalley alluvium.[40] The underlying Permian Bridgnorth Sandstone is

thought to be around 100 m thick and mostly representedby red to red‐brown, fine to medium grained, pebble‐free,dune bedded sandstones of aeolian origin [Cuthbert et al.,2009]. Groundwater flows southeast from a flow divide inthe northwest to the River Tern, with some discharge tothe Potford and Platt Brooks in their lower reaches. Valuesfor transmissivity of 200 m2/d, specific yield of 0.1, andconfined storage coefficient of 1 × 10−4 were assumed forthe Permo‐Triassic sandstone of Shropshire [Streetly andShepley, 2005]. Long‐term average (1970–1999) annualrainfall and potential evapotranspiration for the catchmentare around 670 and 507 mm, respectively [Cuthbert,2006].[41] Step 2: a time series of groundwater level data were

available for several observation wells in the area. Twowells, 2042 and 2086, were chosen for analysis. These aresituated very near the groundwater flow divide (x = 0.1L)and about halfway up the catchment (x = 0.5L), respectively,

and from equation (8) are expected to have negligibletemporal variation in D.[42] Step 3: on the evidence of available high temporal

resolution groundwater level data for the sandstone aquiferin other parts of Shropshire, significant smoothing of the

Table 2. Parameter Sensitivity Analysis Expressed as PercentageChange in Average and Amplitude of the Estimated RechargeFrom the Baseline

ParameterEstimatedRecharge −50% −10% Baseline +10% +50%

T average −50 −10 0 +10 50amplitude 0 0 0 0 0

SY average 0 0 0 0 0amplitude −50 −10 0 +10 50

L average +400 +25 0 −18 −57amplitude 0 0 0 0 0

h0 average 142 28 0 +28 +142amplitude 0 0 0 0 0

Figure 6. (a) Shaded relief map of the Potford BrookCatchment. (b) Superficial geology and groundwater levelcontours (January 1989) at 2 m intervals. The geologicalmapping is reproduced from 1:50,000 scale British Geolog-ical Survey Map Data (Sheet 138 Wem). British GeologicalSurvey © NERC 2010. All rights reserved. IPR/128‐27C.Some features of the maps are based on digital spatialdata licensed from the Centre for Ecology and Hydrology,© NERC. The figure includes mapping data based on Ord-nance Survey 1:50,000 maps with the permission of HMSO© Crown Copyright and/or database right 2006. Licence100017572.

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recharge signal is known to occur where unsaturated zonesare more than a few meters thick [Streetly and Shepley,2005]. Since the thickness of the unsaturated zone is 13and 4 m at observation wells 2042 and 2086, respectively,the approximately fortnightly temporal resolution of thegroundwater level data for these locations was thereforedeemed sufficient to adequately capture the true dynamicsof the water table fluctuations.[43] Step 4: effects of superficial geology and effects of

pumping. The catchment has a variable covering of super-ficial deposits including clay rich till of low permeability,which is thought to restrict significantly the amount ofrecharge entering the aquifer in some areas [Cuthbert et al.,2009, 2010]. Hence, it was necessary to assess how repre-sentative groundwater level observations from a particularwell may be for quantifying recharge across the widercatchment area by ensuring that the measured water tableresponse is not predominantly influenced by the rate ofrecharge local to the observation well. As discussed aboveand illustrated by the numerical models, the higher thediffusivity of the aquifer the more likely it is for this to bethe case.[44] To explore this for the study catchment, an analytical

model was used to describe the propagations of pressurevariations across a strip of aquifer covered by low perme-ability material due to recharge at its boundaries. The strip,through which no recharge occurs, has width 2X, boundedby regions of high recharge represented by two identicalvariable head boundaries whose heads (h [L]) vary sinu-soidally according to

h ¼ ha sin !tð Þ; ð14Þwhere w is the angular frequency [T−1], ha is the amplitudeof the oscillation [L], and t is time [T].[45] The propagation of a sinusoidal head boundary into an

infinite homogeneous medium has been given by Carslawand Jaeger [1959, p. 65] by analogy with the equivalentheat flow boundary value problem. The equivalent solution tothe linearized 1‐D groundwater flow equation (as given inequation (4) for the case q = 0) for h, a given distance l awayfrom the boundary is

h l; tð Þ ¼ hae�kl sin !t � klð Þ; ð15Þ

with k = !2�

� �0.5 and aquifer diffusivity, � = T

S [L2T−1], where S

is the storage coefficient of the aquifer [−].[46] Since the solutions of the linear flow equation can be

superposed, the method of images can be used in order toapply this well‐known solution to the case of the strip ofaquifer for which we currently desire a solution. To solve forh at the center of the strip, the set of necessary source andsink image boundaries are at ±(4n + 1)X and ±(4n + 3)X,respectively, where n is zero or any positive integer. Thusthe solution at the center of the strip is

h ¼ 2X1n¼0

hae�k 4nþ1ð ÞX sin !t � k 4nþ 1ð ÞXð Þ

h

�hae�k 4nþ3ð ÞX sin !t � k 4nþ 3ð ÞXð Þ

i: ð16Þ

We can use this approach to approximate the propagation ofhead responses on a variety of time scales from an area

receiving recharge into a strip which receives no rechargefor a given aquifer diffusivity.[47] For the study catchment the patches of low perme-

ability superficial deposits are on the scale of a few hundredmeters up to several kilometers. For the most part, wherethey are thick enough to reduce recharge significantly, suchdeposits confine the underlying sandstone aquifer. Applyingthe analytical approach using values of diffusivity for theconfined aquifer shows that fluctuations in groundwater levelwith daily, monthly, and annual periods leads to attenuationof the boundary head fluctuation by around 90%, 4% and0.0002% for X = 2500 m. Hence, it is reasonable to assumethat such patches of low permeability superficial depositsmay reduce the overall recharge to the aquifer but notsignificantly bias monthly to annual water level fluctuationsof a well placed near or even within such an area.[48] Considering the effects of pumping on the chosen

observation wells, for 8 of the 10 years of the analyzed timeperiod, groundwater abstraction from the catchment is onlya minor component of the total water balance (Figure 7).Since this small amount of pumping was from a number oflocations all greater than 500 m distance from the obser-vation wells, it is assumed that the affect of this pumping isinsignificant for the estimation of recharge by the modifiedWTF methodology, and smaller than other uncertainties, forinstance in aquifer parameters. However, during twounusually dry years, 1995 and 1996, a much larger amountof groundwater was abstracted from nearby boreholes aspart of the Shropshire Groundwater Scheme and it is ex-pected that this may lead to errors in recharge estimationduring this time and perhaps during the subsequentgroundwater recovery period.[49] Steps 5, 6, and 7: since the aquifer is more than 100 m

thick and groundwater level fluctuations are on the order of afew meters, it was assumed that the variation of T with headwas insignificant. Hence, equations (10) and (1) were used tocalculate time series of D and q, respectively. An estimationof L was made by taking the distance along an approximateflow line between the groundwater divide in the northwestand the point of discharge to the River Tern at an elevation(h0) of 55 mAOD.[50] Monthly aggregated model output is shown in Figure 7

alongside estimated recharge used for a recent regionalgroundwater modeling study of the area by the EnvironmentAgency for England and Wales (EA) [Streetly and Shepley,2005]. This EA recharge estimate, used for comparisonpurposes, was based on soil moisture balance models refinedthrough iterative application within a groundwater model,itself refined against river flow and groundwater level data.Figure 7 also shows spatially accumulated groundwaterabstraction data for the catchment.[51] The pattern and magnitude of responses compares

well with the EA recharge estimate. However, it is clear thatthe recharge variability is greater in well 2086 in compari-son to well 2042 and that the timing of peaks for these wellslies at either side of the EA estimate. This is attributed to thethicker unsaturated zone at well 2042, giving a delayed andattenuated response in the recharge signal. The averagerecharge for wells 2042 and 2086 are 110 and 127 mm/a incomparison to 114 mm/a for the EA estimate. As anticipatedabove, during dry periods, such as in 1995 and 1996, whengroundwater abstraction in the catchment was greater thanrecharge, the method yields a large negative recharge at both

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observation wells presumably due to drawdown of waterlevels in these locations.[52] Consistent with the sensitivity analysis presented

above, percentage changes in T lead to equivalent changesin average recharge. Sensitivity to changes in L is greater,with a 10% increase or decrease leading to a 16% decreaseor 21% increase in average recharge, respectively. Further-more, even large changes in SY (e.g., 50%) only lead to verysmall changes (<5%) in average recharge although themonthly amplitudes of variation become much smaller andfit less well with the EA recharge estimates in this case.

7. Conclusion

[53] By extending an analytical solution to a linearizedBoussinesq equation an expression has been developed forgroundwater drainage based on estimations of aquiferparameters. This has then been used to develop an improvedWTF technique for estimating groundwater recharge. Theresulting method extends the standard WTF technique bymaking it applicable in areas with smoothly varying watertables as long as aquifer properties for the area are relativelywell known. The method can be easily implemented in aspreadsheet and has been validated against numerical si-mulations which show that the approach may be applicablein more complex situations where the assumptions of theanalytical approach are not necessarily met. Furthermore,the approach has been demonstrated using data from a realcatchment where it produces recharge estimates whichcompare well with prior estimates made by other means.[54] The advantages of the method are that it can be

applied in areas with smoothly varying, i.e., in many cases,deep, water tables without the need for high temporal res-olution data. However, in order to produce accurate rechargeestimates the aquifer parameters need to be reasonably wellknown, and factors that significantly alter the groundwater

level distributions such as trending heterogeneity, stronglyinclined aquifer geometry or groundwater abstraction closeto available observation wells may limit the accuracy of therecharge estimations produced. The method may be mostuseful where the effects of vegetation or complex superficialgeology complicate the estimation of recharge by othermeans. Furthermore, the approach may be applicable in datasparse regions of the world where groundwater levels arerelatively cheap and straightforward to collect in comparisonwith data needed for other recharge estimation techniques.[55] An additional contribution of the analysis presented

is in demonstrating the relationships between groundwaterlevel fluctuations, aquifer properties and geometry, andrecharge. Thus, the method can be inverted to derive asimulated groundwater hydrograph for catchments in whichrecharge has already been estimated by other methods inorder to estimate likely ranges of aquifer parameters. Forexample, a useful application of such an analysis may bewithin the first “conceptualization” stage of a catchmentscale groundwater resources appraisal or modeling exercise.[56] Further development of the method would be needed

to consider the effects of anisotropy and inhomogeneity inaquifer properties and for catchments with unusual geometryand/or where vertical flows are significant. Where timeseries climate data are available, using other techniques ofrecharge estimation in combination with the method devel-oped here may enable uncertainties in recharge estimation tobe reduced for cases where aquifer properties are not wellknown.

[57] Acknowledgments. The author would like to acknowledge sup-port from the AS Charitable Trust for developing this methodology andreceipt of a Studentship under the UK Natural Environment ResearchCouncil’s Lowland Catchment Research (LOCAR) thematic program incarrying out the background research for the case study. Thanks to MartinShepley and Kevin Voyce from the Environment Agency for England and

Figure 7. Comparisons of recharge estimates for the Potford Brook Catchment (EA Rech, WTF 2086,WTF 2042) alongside observed groundwater hydrographs (GWL 2086, GWL 2042) and total groundwa-ter abstraction (GWABS).

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Wales for kindly supplying the data from Shropshire. Thanks also to TimLewis and Angelo Papaioannou for indirectly inspiring the idea behind thispaper. The manuscript benefited from comments from three anonymousreviewers and comments/discussions with Rae Mackay, Mick Riley, andJochen Voss for which I am most grateful.

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