Groundwater recharge is difficult to estimate, especially in fractured aquifers, because of the spatial variability of the soil properties and because of the lack of data at basin scale. A relevant method, known as the water table fluctuation (WTF) method, consists in inferring recharge directly from the WTFs observed in boreholes. However, the WTF method neglects the impact of lateral groundwater redistribution in the aquifer; i.e., it assumes that all the WTFs are attributable to recharge. In this study, we developed the WTF approach in the frequency domain to better consider groundwater lateral flow, which quickly redistributes the impulse of recharge and mitigates the link between WTFs and recharge. First, we calibrated a 1D analytical groundwater model to estimate hydrodynamic parameters at each borehole. These parameters were defined from the WTFs recorded for several years, independently of prescribed potential recharge. Second, calibrated models are reversed analytically in the frequency domain to estimate recharge fluctuations (RFs) at weekly to monthly scales from the observed WTFs. Models were tested on two twin sites with a similar climate, fractured aquifer and land use but different hydrogeologic settings: one has been operated as a pumping site for the last 25 years (Ploemeur, France), while the second has not been perturbed by pumping (Guidel). Results confirm the important role of rainfall temporal distribution in generating recharge. While all rainfall contributes to recharge, the ratio of recharge to rainfall minus potential evapotranspiration is frequency-dependent, varying between 20 %–30 % at periods

Increasing anthropogenic and climate pressures on water resources call for a better understanding of the way water is transiently stored and the way it flows in the subsurface

Recharge, as the main water inflow feeding GW, is critical for the proper management of GW systems. GW recharge is defined as the water percolating from the last unsaturated horizon down to the water table and is therefore broadly inaccessible to direct observations

Modeling the recharge is therefore a relevant method to estimate it. Several recharge models have been developed

Oppositely, GW recharge can also be estimated “from below”. GW levels in boreholes are indeed the most direct observation to characterize aquifers behavior. The water table fluctuation (WTF) method has thus been used to provide vertical recharge estimates from GW-level variations

While strong attention has been put on mean annual recharge estimations, the characterization of recharge fluctuations (RFs) over time, at short to long timescales, remains critical and has been investigated less. This has been highlighted as one of the 23 unsolved problems in hydrology

In this study, we propose to quantify the RFs by developing a novel method able to quantify the GW recharge from below. The main objectives of this work are thus threefold: (1) deciphering the respective impact of GW lateral flow and recharge on GW-level fluctuations in heterogeneous aquifers, (2) estimating RFs over a 20-year period and (3) studying how rainfall distribution and unsaturated zone thickness controls GW recharge. To do that, we develop the WTF method in the frequency domain to decompose GW-level fluctuations into GW lateral flow and RFs (Sect. 1). While GW systems are heterogeneous in nature, we propose to represent them by an equivalent homogeneous 1D Dupuit model (Fig.

The method developed in this study relies on several steps, as illustrated by Fig.

As developed below, we focus on water table and recharge anomalies, here called “fluctuations”. These fluctuations are obtained by removing the mean value from input data (potential recharge and pumping rates used during Step 1) over the study period. Consequently, simulated WTFs obtained during Step 1 will be compared to observed fluctuations. Finally, RFs are obtained from observed WTFs (Step 2) and will be compared to potential RFs.

In the next sections, we describe (1) the analytical GW flow model deployed to estimate recharge from observed WTFs, (2) the soil models used to estimate potential recharge as input to the GW model, (3) the GW parameter calibration strategy and (4) the analytically inverted GW model. Finally, we also present how inverted recharge is analyzed with rainfall at the two study sites.

Considering a homogeneous and confined aquifer, and assuming that the vertical component of the GW flow can be neglected (Dupuit assumption), the GW flow equation can be described by a diffusion equation (Eq.

Solving Eq. (

Equations (

Equations (

The GW model described previously is driven by GW recharge

The first – and simplest – soil model, called “Thornthwaite”, is based on the representation of the unsaturated zone as a simple reservoir accumulating rainwater and satisfying potential evapotranspiration (calculated using the Thornthwaite method) while water is available. When the reservoir is full, excess water recharges GW

The second soil model is derived from the GR4J hydrological model

The third potential recharge estimate is provided by the SURFEX modeling platform. SURFEX is composed of a spatially distributed land surface model (Interaction between Soil Biosphere and Atmosphere, ISBA) that simulates water and energy fluxes at the interface between the atmosphere and the surface (soil, vegetation and snow)

Schematic of Ploemeur and Guidel sites.

The forward model is now fully defined. The first step of our approach consists in defining geometric (

In order to explore the informative content of the observed WTFs to define geometric and hydrodynamic parameters, our strategy consists in defining equiprobable parameter sets testing sensitivity to imposed recharge rates (Sect. 2.3). Analytical GW models are computationally efficient. Thus, the whole parameter space (

For each observation well, modeled WTFs are evaluated against root mean square error (RMSE) divided by the standard deviation of observed WTFs (called normalized RMSE, nRMSE) to favor comparison among the different observation wells:

In a last step, GW recharge (

A similar analytical solution is obtained for the case without pumping from Eq. (

This new approach has been evaluated with a numerical MODFLOW model (see Appendix B). The evaluation consists of providing daily recharge rates (Thornthwaite model) to a numerical model equivalent to the analytical 1D model used for the Guidel case. The resulting WTFs at

Finally, rainfall fluctuations obtained from climate data and RFs obtained by inverting the best GW models are analyzed at both sites in time and frequency domains. The classical approach consists in defining statistics on the distribution and intensity (e.g., number of days without rain, cumulative sorted rainfall) but does not often yield satisfactory results. Considering the relationship between

In order to focus on the transformation of rainfall into recharge, we compute both the coherence and the transfer function

The model ability to estimate RFs is tested on the Ploemeur–Guidel hydrogeological observatory (

Crystalline rocks are generally considered impermeable. However, several examples show high-yielding aquifers, which are mostly explained by the presence of fractures and weathered porous structures

At the Ploemeur site, more than 25 wells have been monitoring groundwater levels since 1991. These wells are generally

The Guidel site

Conceptually, each monitoring well intercepts a flow line between two distant boundary conditions. For the pumping site (Ploemeur site), the coordinate

For the natural case study (Guidel site), each monitoring well is part of one hillslope bounded by a river at

Observed GW-level variations in boreholes at the Ploemeur pumping site

While first GW-level data at Ploemeur date back to 1991, we focus our analysis on the 1996–2017 period to avoid potential transient responses to the pumping setup. At the Guidel site, data are available from 2009 to 2017. Water levels are recorded at minute to daily time steps and decimated to daily timescales for our analysis. Hydraulic heads measured in boreholes are corrected from atmospheric pressure variations.

In Ploemeur, GW levels are relatively deep due to the pumping (depth

Conversely, transient variations in response to the rainfall and water cycle vary significantly among boreholes in Guidel (lower graph in Fig.

A national weather station (METEO-FRANCE) is located in between the two study sites. It provides daily precipitation and Penman–Monteith potential evapotranspiration (PET) estimates. Both are used to generate potential recharge from Thornthwaite and GR4J soil models (next section), while climate data used by SURFEX are derived from large-scale climate simulations. Within the studied period (1996–2017), annual precipitation ranges from 600 to 1100 mm yr

Within the studied period, mean potential recharge rates derived from Thornthwaite, GR4J and SURFEX models are respectively 242, 320 and 246 mm yr

Recharge typically occurs from December to March. Modeled potential recharge rates, as simulated by three different soil models, remain highly variable (Fig.

This section describes results obtained by applying and calibrating the 1D GW model to the two study sites (Step 1 in Fig.

This part synthesizes results of the parameter space exploration for the Ploemeur and Guidel sites. Observed and modeled WTFs are compared at different boreholes (see borehole locations in Fig.

Overall, the 1D GW model seems satisfactory when comparing observed and best modeled WTFs for the Ploemeur site (Fig.

Comparison between best modeled and observed water table fluctuations at borehole F9

The best RMSE for each borehole is increasing with decreasing distance to the pumping zone:

All parameters are not equally well determined (Fig.

Evolution of the minimal normalized RMSE for Ploemeur wells as a function of model parameters: storage coefficient

Interestingly, storativity is well constrained as

Estimated characteristic time, a combination of the three calibration parameters (Eq.

Similarly to Ploemeur, the analytical GW model manages to adequately describe WTFs at Guidel (Fig.

In the previous section, we showed that a simple GW model that neglects aquifer heterogeneity can reproduce observed WTFs well. An important result is that estimated hydrodynamic and geometric parameters are independent of prescribed potential recharge models (Fig.

Note that transmissivity is not well defined from temporal fluctuations. Mean water table in each borehole is impacted by heterogeneity (Fig.

The analytical GW model appears as a low-pass filter in Eqs. (

Figure

Monthly recharge fluctuations estimated from the groundwater analytical model at different boreholes at the Ploemeur (in red) and Guidel (blue) sites, propagating model parameter uncertainties. Shaded areas represent GW model uncertainty and variability between the different observation wells. Black line represents potential recharge fluctuations (anomalies compared to the mean value) from the Thornthwaite model.

Coherence

On average, the Thornthwaite model overestimates analytically estimated RFs by 10 % to 20 %, though, for a few events, analytically estimated RFs can be larger. Based on these results, we can assert that recharge events greater than 25 mm per month can be detected with this method, as highlighted by single events that occurred in winter 2002 and 2005. Overall, RFs estimated from above and RFs performed with the GW analysis agree well at seasonal to long-term timescales. The main differences appear at the monthly scale or at shorter timescales. Therefore, here we compared RFs at a weekly timescale (not shown). At the Ploemeur site (respectively Guidel), the Thornthwaite model overestimates RF temporal variability obtained analytically from well F7 (respectively well PSR1) by 40 % (respectively 20 %), while both GR4J and SURFEX models fall within 5 %–6 %. In terms of the succession of recharge events, the correlation is 0.55, 0.58 and 0.65 for Thornthwaite, GR4J and SURFEX respectively, at the Ploemeur site. In general, Thornthwaite potential recharge events are

In Fig.

Inverted RFs at both study sites (in blue and green in Fig.

Soil models generally fail to describe

Figure

Overall, RFs estimated from GW levels can be described as a fraction of potential RFs using a linear regression when integrated at annual time step, but significant deviations exist in terms of amplitude and variability. The Thornthwaite RF equals 92 % of wet-season

Here, we originally inferred GW properties based on WTFs measured in boreholes. These fluctuations generally bear typical frequencies from climatic and anthropic boundary conditions (hourly rainfall event, small diurnal variations due to evapotranspiration fall during the night, seasonal human water consumption, climatic cycles, etc.). They are also linked to the characteristic time of the system, and consequently to spatial scale. A first outcome of this work is that a simple physically based GW model can explain WTFs at the scale of the piezometric network, although local geological heterogeneities can play an important role, as shown in the steady-state case in Appendix C or by daily to monthly pumping tests

Aquifer characteristic time (L

Here, storage coefficient values (

In this work, transmissivity is estimated in the range

The steady-state approach shows the difficulty of obtaining relevant aquifer-scale information from mean GW levels because of local heterogeneities, incomplete sampling of the GW system and high sensitivity to model assumptions (Appendix C). Thus, heterogeneity largely impacts the ability to estimate mean GW recharge rate from GW levels. We found that WTFs observed in boreholes contain the overall aquifer response for observation periods around and larger than the aquifer characteristic time. RFs generate lateral GW flow that links different GW-level observations. For this reason, a single well contains information on aquifer-scale recharge, as underlined in the WTF approach. However, the WTF method alone has some limitations. We show that the well position within the GW flow system is as important as storativity to define recharge (geometric term in Eq.

The main assumptions of the GW model are (1) the 1D lateral flow structure, (2) homogeneous GW parameters and (3) uniformly distributed recharge. Regarding the 1D assumption on Ploemeur, pumping controls aquifer behavior so that the 1D assumption is valid over the system except close to the pumping wells. Pumping has generated a GW flow structure more or less constantly for 25 years. The water table does not interact with the surface. In this case, aquifer characteristic time is perfectly constrained and not borehole-dependent. At Guidel, GW intercepts the ground locally. Therefore, the flow structure is mainly driven by topography and is more complex, as highlighted by the different WTF patterns (Fig.

We assumed that the heterogeneous system could be described by equivalent hydrodynamic properties. Previous works have highlighted that the behavior of complex aquifers could be described by equivalent homogeneous models when focusing on specific spatial and temporal scales

The assumption of uniform recharge might be seen as unrealistic considering that local topographic and geological structures can favor exchanges between surface and groundwater

Finally, note that the forward and backward models can be run at any – and different – time steps. The heart of the model is in the frequency domain, so that the first step consists in computing a Fourier transform to define amplitudes over a series of cosine functions. The number of frequencies is limited by the WTF sampling (Nyquist frequency). The recomposition in the temporal domain requires the cosine functions to be summed again, but all frequencies do not need to be used, and the temporal sampling can be adapted. Thus, applying the method requires time series of water levels at appropriate time steps to meet study objectives. To reduce computing time, parameter calibration can be done at bigger time steps; thus the potential recharge time series can be provided at this bigger time step. Here, we did it at a weekly time step, while RFs were computed from daily WTFs. We highlight that potential recharge can be a rough estimate or a first guess of RFs.

The dependence of GW recharge to rainfall intensity and distribution throughout the year has been documented in several studies

Based on RFs estimated from WTFs, we highlight that the frequency-dependent relationship between

The proposed approach allowed for computation of both RFs and associated uncertainties at seasonal timescales to reinvestigate the relationship between wet season

The comparison between the Ploemeur and Guidel sites offers the opportunity to gain insights into the role of the unsaturated zone. Pumping thickens the unsaturated zone, so that potential recharge under the soil is first buffered in the deep unsaturated zone before generating GW recharge. We can infer that the unsaturated zone plays an inertial role by storing water and filtering out high-frequency variability. This is confirmed when looking at frequency-dependent time lags between

Inferred aquifer parameters slightly differ between the two neighboring sites, although they are located in the same geological context. On average, storage coefficients are larger and transmissivities smaller at the natural site (Guidel) than at the pumping site (Ploemeur). One interpretation of this result is that the weathered zone contributes more to GW flow at the Guidel site. Indeed, the weathered zone (0–20 m depth), known to be more porous and less permeable, should impact GW flow more when GW levels are closer to the surface. Conversely, at the Ploemeur site, the “deep” fractured aquifer controls flow as GW levels are all

The GW models developed here have several advantages. They manage to reproduce GW-level fluctuations (or anomalies) in heterogeneous aquifers well with three physical parameters and a limited execution time. We showed that GW-level fluctuations observed in one borehole contain aquifer-scale information at timescales equivalent to or larger than the aquifer characteristic time, while time-averaged groundwater levels are sensitive to heterogeneity. Therefore, the impact of local heterogeneities is smoothed out so that the aquifer-scale equivalent characteristic time and storage coefficient are reachable with limited dependence on prescribed potential recharge. The developed GW models are also invertible analytically to recover groundwater recharge fluctuations from observed GW-level fluctuation time series. A key novelty of this approach is developing the WTF method in the frequency domain. Note the method can be used to infer mean annual recharge value; however, we argue this is less relevant in heterogeneous aquifers.

The approach was tested at two neighboring sites, one that has been pumped for 25 years. First, recharge estimated from GW levels is coherent among each borehole, at each site. Second, the response to rainfall is more important when the unsaturated zone thickness is small for timescales

A large uncertainty in hydrological modeling lies in the fact that GW recharge can be derived from oversimplified conceptual soil models. Such an approach as described here gives hope that the GW heterogeneity issue could be overcome in hydrological models by defining the equivalent basin response with a similar frequency-domain analytical model. A similar approach has been designed to model streamflow variations in a bigger basin

In this study, the method is applied in crystalline contexts that display fractured aquifers, that are highly heterogeneous, which is challenging. Thus, similar approaches could be deployed in different geological contexts where GW-level time series are available over long timescales. In particular, it could be very interesting to test it in karstic aquifers. This method constitutes a useful alternative to study GW flows and recharge processes and their sensitivity to imposed boundary conditions, namely, precipitation and water use.

The 1D diffusivity equation (also called GW flow equation), under the Dupuit assumption, and considering a non-leaky confined aquifer of uniform thickness or that the variations in the phreatic level are negligible compared to the aquifer thickness, can be written as

In addition, the steady-state part of Eq. (

In Eq. (

Scheme of the analytical 1D groundwater model. Hydraulic head is variable along the

The first model configuration described in Fig.

at

at

at

at

at

at

The second model configuration described in Fig.

at

at

at

at

at

at

Analytical solutions presented previously constitute a computationally much faster method than numerical models to represent hydraulic heads. They also offer the possibility to analytically recompute recharge rate

As developed before, we will separate the steady state and the transient state. From Eqs. (

Following

GW-level fluctuations from the analytical and numerical models are compared. The imposed recharge was computed from the Thornthwaite soil model. To mimic the steady state of the analytical model, the initial condition in the numerical model is defined by applying a mean recharge rate. As illustrated in Fig.

Comparison of water table fluctuations, at

In a second time, we performed a numerical test to estimate the ability of the analytic approach to estimate RFs. This experience is based on a comparison with the ModFlow model described previously. At the end of the numerical simulation, the hydraulic head at

Although the analytical model can be run at the same time step as the well data, RF estimates are affected by (1) numerical oscillations linked to the discrete frequency-domain computations over a finite length time series and (2) the amplification of high-frequency GW head variations, including observation errors. Estimated RFs can be integrated over time to avoid these spurious oscillations. Overall, an integration time larger than a few original time steps is sufficient to ensure an estimation of the recharged volume at a 99 % level (Fig.

Impact of integration time on recharge volume, determination coefficient

Impact of uncertainties on the estimation of characteristic time

This test also gives an idea of parameter uncertainties. We can see how RFs estimated from GW levels are influenced in terms of timing and mean amplitude. Parameter (

We explored the stationary part of Eq. (

Mean groundwater levels observed at the Ploemeur pumping site (blue curve), shown on a SE–NW cross section. The dashed red curve represents the best 1D steady-state analytical model, which has the lowest least-square difference between data and the model.

In addition, we investigated this steady-state issue using a numerical homogeneous 2D MODFLOW model. The actual 10 m resolution topography was imposed as upper-boundary conditions (DRAIN PACKAGE), and pumping wells were set up at their actual position in the domain with their respective pumping rates. The advantage of such a model was to set a realistic geometry and to remove geometric parameters like model length

Finally, the information content of mean GW levels is limited by (1) incomplete sampling within the observation network, considering the punctual nature of borehole data with respect to local heterogeneities in recharge and hydrodynamic properties, blurring the evolution of hydraulic head in space (Fig.

Data and MATLAB models developed for this study, as well as models with different configurations, are available at

The supplement related to this article is available online at:

LG and NL collected and preprocessed the data. LG and LL developed models and methodology. LG and LL post-processed simulated data. LG, LL, OB and JM interpreted results. LG prepared the manuscript with contributions from LL, OB and JM.

The contact author has declared that none of the authors has any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work is part of the ANR project EQUIPEX CRITEX (grant ANR-11-EQPX-0011) and relies on the Ploemeur Critical Zone Observatory

This research has been supported by Université de Rennes 1.

This paper was edited by Gerrit H. de Rooij and reviewed by Ty P. A. Ferre and one anonymous referee.