Frequency domain water table fluctuations reveal recharge in fractured aquifers depends on both intense and seasonal rainfall and unsaturated zone thickness
 ^{1}Univ Rennes, CNRS, Geosciences Rennes  UMR 6118, F35000 Rennes, France
 ^{2}INRAE, UR Riverly, F69625 Villeurbanne, France
 ^{1}Univ Rennes, CNRS, Geosciences Rennes  UMR 6118, F35000 Rennes, France
 ^{2}INRAE, UR Riverly, F69625 Villeurbanne, France
Abstract. 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 WTF method, consists in inferring recharge directly from the water table fluctuations (WTF) observed in boreholes. However, the WTF method neglects the impact of lateral groundwater redistribution in the aquifer, i.e. assumes that all the WTF 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 inpulse of recharge and mitigates the link between WTF and recharge. First, we calibrated a 1D analytical groundwater model to estimate hydrodynamic parameters at each borehole. These parameters were defined from the WTF recorded for several years, independently of prescribed potential recharge. Second, calibrated models are reversed analytically in the frequency domain to estimate recharge fluctuations (RF) at weekly to monthly scales from the observed WTF. Models were tested on two twin sites with 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 to generate recharge. While all rainfall contribute to recharge, the ratio of recharge to rainfall minus potential evapotranspiration is frequency dependent, varying between 20–30 % at periods <10 days, 30–50 % at monthly scale, and reaching 75 % at seasonal time scales. We further show that the unsaturated zone thickness controls the intensity and timing of RF. Overall, this approach contributes to better assess recharge and enable to improve the representation of groundwater systems within hydrological models. In spite of the heterogeneous nature of aquifers, parameters controlling WTF can be inferred from WTF time series.
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Luca Guillaumot et al.
Status: open (until 30 Sep 2022)

RC1: 'Comment on hess2022201', Ty P. A. Ferre, 04 Aug 2022
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I enjoyed reading this paper. It was a pleasure to see that the famous Ploemeur site continues to contribute fundamental insights! The approach is both novel and thoughtful. The authors do an excellent job of presenting their findings and also addressing potential limiting assumptions that underly the analyses. I am convinced that they have demonstrated that their approach offers a useful way to understand how precipitation and evapotranspiration are modified within the vadose zone to produce recharge. My only suggestion is that the authors may help readers to see the underlying approach and to understand the significance of the work with a couple of schematic diagrams and some added background explanations. For example, given that the method is heavily reliant on frequency domain analyses, it may be useful to show the power spectrum of recharge at the two sites and to describe it in terms tha a general reader can appreciate. Later, this could help to explain, again at a more intuitive level, how the method filters the effects of pumping and leads to the conclusions regarding the contribution of PET to recharge at different frequencies. I would also have appreciated a paragraph to explain the coherence and transfer function results. Can the authors help the average reader to make sense of these so that they can appreciate the results that follow? Finally, I would recommend that the authors provide some thoughts on the applicability of the method to a broader range of sites. Is it, for any reason, limited to fractured rock settings? Does it require a highly instrumented site with a long record? What practical benefits might other researchers and practitioners realize if they apply this approach at their site? All of this is simply aimed at broadening the impact and readership of the work ... the underlying science was a pleasure to read!
Best
Ty Ferre

AC1: 'Reply on RC1', Luca Guillaumot, 27 Sep 2022
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Dear reviewer, thank you for your positive feedback. We addressed your different comments below.
Sincerely.
Luca Guillaumot, on behalf of all coauthors.
My only suggestion is that the authors may help readers to see the underlying approach and to understand the significance of the work with a couple of schematic diagrams and some added background explanations. For example, given that the method is heavily reliant on frequency domain analyses, it may be useful to show the power spectrum of recharge at the two sites and to describe it in terms that a general reader can appreciate.
Thank you for this suggestion. We plotted the power spectrum of recharge (see attachment) obtained from soil models and from groundwater levels in Ploemeur and Guidel. First, we observe a similar pattern between the three signals: (1) there is a peak at T = 365 days, (2) recharge amplitude increases with the time period (T). In addition, we observe that the amplitude is higher at high frequency in Guidel compared to Ploemeur. We added this figure along with the previous description to the Supplementary file. However, we think this figure is a bit redundant with Figure 9 showing the transfer function response (recharge/(precipPET)) in the frequency domain. As suggested, we will give more explanations concerning the frequency domain analyses to help readers.
I would also have appreciated a paragraph to explain the coherence and transfer function results. Can the authors help the average reader to make sense of these so that they can appreciate the results that follow?
Coherence and transfer function are introduced in section 2.6 “How unsaturated zone transforms precipitation into recharge”. Then, associated results are presented in section 5.3 “The unsaturated zone and recharge fluxes”. Finally, these results are discussed in sections 6.2.1 and 6.3.1.
Section 2.6 will be corrected in the revised manuscript to both describe and « popularize » equation 7 and 8. That is why, we will add some explanations about frequency domain analysis in this part: “These functions allow to infer the role of the soil and more generally of the unsaturated zone.” (line 211212) will be replaced by “These transfer functions comparing flux coming in vs out of the unsaturated zone allow to infer its role in the recharge dynamics. Switching to the frequency domain offers the additional advantage to visualise how precipitation is converted into recharge at each frequency”. In addition, sentences from section 6.2 will move to 2.6 (cf. reviewer 1).
Following your remark, and because Figure 9 constitutes a main result of our study, we will better explain coherence and transfer function results in the revised manuscript: “On figure 9, the coherence and transfer functions (Eq. 7 and 8) between P − PET fluctuations and RF inform on the efficiency of the transformation of rainfall events into recharge. These functions therefore illustrate the unsaturated zone response to rainfall in frequency domain. In particular the transfer function can be seen as a proxy of rainfall efficiency to generate recharge. From Figure 9, results can be summarised as follows: recharge estimated from soil models and recharge estimated from WTF have similar longterm behavior, recharge estimated from soil models is too sensitive to rainfall at shortterm, recharge estimated from WTF is more sensitive to shortterm events on the natural site compared to the pumped site” (instead of lines 382383).
Then, more explanations and discussions are given in section 5.3, 6.2.1 and 6.3.1.
Later, this could help to explain, again at a more intuitive level, how the method filters the effects of pumping and leads to the conclusions regarding the contribution of PET to recharge at different frequencies.
Pumping fluctuations are already included as a boundary condition of the model (in x=0) so that the effect of pumping is taken into account (see line 104107 and 253254). In addition, we will provide more steps when developing the analytical groundwater model in Appendix 1, so that the pumping boundary condition will appear clearly. We will also mention explicitly this point in section ‘Defining the 1D flow model for each field site’.
Finally, I would recommend that the authors provide some thoughts on the applicability of the method to a broader range of sites. Is it, for any reason, limited to fractured rock settings? Does it require a highly instrumented site with a long record? What practical benefits might other researchers and practitioners realize if they apply this approach at their site? All of this is simply aimed at broadening the impact and readership of the work ... the underlying science was a pleasure to read!
Thank you for this comment. The applicability of the method is an important point. We are convinced that the method can be employed to other regions with different aquifer complexity provided that aquifer response can be approximated by a Dupuit equation. It would be very interesting to test it in karstic system where heterogeneity is more pronounced and Dupuit equation more critical. Due to the fixed 1D model structure, a critical aspect is that the method will be not relevant for boreholes located in areas where the water table pattern changes across seasons (see line 460470).
The method requires longterm water table records and a first guess of weekly or monthly recharge rates which can be roughly estimated from precipitation and temperature data (line 493497). In order to inverse recharge it is more suitable to use high frequency water table records (daily). Finally, benefits are twice : estimating aquiferscale characteristic time from a single point and estimating recharge (then its relationship with other variables as we did it by comparing recharge to PPET on two sites).
Following your comment, the last part of the abstract will be revised as follows in the revised manuscript: “Overall, this approach contributes to better assess recharge and enable to improve the representation of groundwater systems within hydrological models. In spite of the heterogeneous nature of aquifers, parameters controlling WTF can be inferred from WTF time series making confident that the method can be deployed in different geological contexts where longterm water table records are available”.
Moreover, the last sentence of the conclusion will be revised as follows : “This method could be applied in several parts of the world where GW levels time series are available over long time scales. In this study, the method is applied in crystalline contexts that display fractured aquifers, highly heterogeneous which is challenging. Thus, similar approaches could be deployed in different geological contexts. 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, precipitations and water use.”

RC3: 'Reply on AC1', Ty P. A. Ferre, 27 Sep 2022
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Thank you for your thoughtful and complete response! It is great to see more excellent work coming from your grouip!!
Best
Ty

RC3: 'Reply on AC1', Ty P. A. Ferre, 27 Sep 2022
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AC1: 'Reply on RC1', Luca Guillaumot, 27 Sep 2022
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RC2: 'Comment on hess2022201', Anonymous Referee #2, 26 Aug 2022
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Review of ‘Frequency domain water table fluctuations reveal recharge in fractured aquifers depends on both intense and seasonal rainfall and unsaturated zone thickness’ by L. Guillaumot et al. HESS2022201.
Major comment
The paper analyzes the effect of recharge on groundwater level fluctuations in aquifer wells in the frequency domain through Fourier transforms of an analytical groundwater model. The authors invert the model to derive recharge fluctuations from water level fluctuations in wells, which is quite interesting. In the model calibration phase, the mathematical efficiency of the analytical model is taken full advantage of by carrying out a massive number of model runs that explore the entire parameter space on a regular grid.
The approach is original, interesting, and wellsuited for HESS. At times, the explanation of the model and the mathematical techniques is a bit brief (see detailed comments). The method is applied to a pumped and an unpumped aquifer near the Atlantic coast of France, in a relatively humid climate.
From the fitting results in Fig. 6 it appears that the parameter identifiability would benefit from replacing the aquifer length as a fitting parameter by the characteristic time, but this is not discussed or explored in the paper.
A considerable weakness of the paper is that the model is calibrated for both aquifers, but not validated, making it difficult to assess the model performance.
Overall, the line of thought is a bit hard to follow some times because individual sections are not as focused as the can be. Throughout the paper, the clarity can be improved by thinking about what exactly the authors wish to convey to the reader and how to do so clearly. In the detailed comments I indicate where I got completely lost. I hope this will lead to a more structured, coherent paper.
Sections 5, 6, and 7 are disappointing. They lack focus and structure and do not convey the main strengths of the study. That does not mean these strengths are not there. I very much like the modelling approach and the intended use. The Results and Discussion sections need to bring that out more strongly though.
Detailed comments
The title is informative but a bit long.
L. 18: Something is missing.
You use ‘indeed’ a few times, but it is unclear to me why.
L. 41: Potential recharge suggests this to be the maximum possible recharge in analogy to ‘potential evapotranspiration’, e.g., in the absence of evapotranspiration and overland flow. It would be equal to the rainfall rate, but with damping and delay due to the flow through the unsaturated. zone. I am quite sure this is not what you mean. Perhaps use another term, like ‘plausible’ or ‘probable’. From line 88 it appears ‘instantaneous recharge’ might also be adequate.
L. 75: ‘Although simple...’ Is not the simplicity of the model the reason it has the adaptability you highlight here?
Fig. 1: I do not understand ‘backward analytical inversion’. Is there a forward analytical inversion, as this term implies? If yes, what are the differences between the two, and are there any references you can provide?
The boundary conditions in the aquifer sketch do not seem to match those described in the caption.L. 105: Both imposed heads are not allowed to vary in time, correct?
L. 120: This sentence is difficult to follow. Also, the wells are located in a 2D system (the map, if you will), while your solutions are onedimensional. Is the well distance measured along the coordinate that is represented in the 1D system? If so, than you essentially spread out a point sink (the well) over the full width W. It would be a line sink in a 2D system then. This is not to criticize this approach, just to make sure I understand it correctly. If I am correct, than ‘distance to the pumping wells’ should be changed to reflect it is the distance along a single coordinate only, not a true Euclidian distance.
L. 121125: Is that the reason you kept the boundary conditions timeinvariant? Is this a reflection of the applicability of the superposition principle?
L. 157. Apparently, I misunderstood you at L. 120. Is the aquifer length then related to the halfdistances between pumping wells in the aquifer? This needs to be explained better.
L. 158: In a similar fashion, is the aquifer length related to halfdistance between adjacent rivers?
N.B. Reading on, I see that you discuss this in section 3.2. Perhaps give some of that info here and refer for details to section 3.2. I do not find your choice for a constant head BC terribly convincing. Is it mathematically or physically necessary? If so, please point that out more clearly.
L. 157 and 158: It appears to me that if you base aquifer lengths on distances between rivers and wells, the boundary that is not the river or the well should be a noflow boundary, not a prescribed head boundary.
L. 164165 (and possible elsewhere): Units are usually not in italics, and the multiplication dot is unusual.
L. 177: ‘...along time.’ I do not understand.
L. 182: The double division signs of Q/W/L are confusing. I presume you want to divide Q by the aquifer area.
L. 183185: Unclear sentence, please rephrase.L. 203: From this sentence it is not clear how anomalies are defined.
Eq. 7: The dependency of Cxy on omega is missing. One of the Pxx should be Pyy.
Section 2.6 is so brief it is difficult to follow for me, as a reader who is not familiar with this technique. Please expand and add some references that give the basics of the methodology.
Fig. 2. Panel c has wells that are not mapped on the crosssection depicted in panel b. The dashed blue line in pane c is the phreatic level, right? Please mention this in the caption or the legend.
L. 234235: How does this observation (high sensitivity of deep fractures to recharge) affect the nature and the role of the unsaturated zone model for modifying the recharge signal?
Fig. 4. Why does the SURFEX recharge appear to increase and then fall before heavy rainfall?
Section 4: You only show the results of the calibration. A validation step is missing. This is a major weakness of the paper.
L. 332: ‘L/T appears ... in equation 2‘. So does S/T, through the characteristic time. I am not sure this argument is valid.
Fig. 6: The characteristic time is fully defined once the storage coefficient, transmissivity, and aquifer length are determined. How can it then be a fitting parameter along with the other three? (Earlier in the paper, it was not.) From Fig. 6 it would appear a good idea to use the characteristic time instead of the aquifer length as a fitting parameter and determine L from the other three.
L. 358362: This paragraph starts clear, but then you lost me. What does ‘intercompared’ mean, for instance? And how can you reduce noise amplification during backward modelling by doing something that apparently makes graphs easier to read?
L. 363: How does Fig. 8 show similarity between wells? Also, the ‘noise’ can be easily explained physically. Why is it called noise instead of fluctuations or temporal variations caused by pumping?
L. 372375: I can find neither a graph nor a table backing up these statements.
L. 380: Thornthwaite does not show anything in winter 2002.
L. 383 and 387: What are rainfall and P PET efficiency?
Section 5.3 uses terminology with which I am too unfamiliar to understand what points are being made. Throughout the paper, the English is a bit off, but here it somehow becomes so much so that I can no longer decipher the meaning of several sentences. This section needs to be thoroughly rewritten to be accessible to readers outside the immediate field of this paper, and the English needs substantial improvement.
Sections 6 and 7 require over five pages of text. That is rather long. They also introduce a large number of new references, which indicates that the paper is not well organized. The Introduction and the Methods sections should cover most of the literature needed in the paper.
L. 499502. What is the point of this paragraph?
Section 6.2.1. This is some kind of recap, mixed with a discussion. I like the evaluation of the soil models but do not really know what to do with the rest of this text. It reads like a brainstorm session.
Section 6.2.2. What is ‘linear behavior’ in this context and how does it link different phenomena? Is a ‘linear coefficient’ (which normally is a singlevalued number, not a curve), in reality the slope of a linear relationship that you have at this point not properly defined? This is an example where lax formulations obscure what I believe to be a useful message.
L. 530: I know of very few examples, other than interflow, of lateral flow in the unsaturated zone.
Section 6.3.1. This does not really arise from the Results, does it? Also, it is a bit obvious.
L. 534: This info can be moved to the site description earlier in the paper.
Section 7: Only the last two paragraphs are conclusions, the rest is more of a summary.
Appendix A. You do not present a formal Fourier transform of the governing equation and the boundary conditions, which would perhaps make it easier to understand the line of thought and you choice to make the prescribed heads, but not the prescribed flux, timeinvariant.
I can follow most of the development, but nevertheless would like to have more references and a few more steps in the derivations. I am not intimately familiar with the techniques you use.
You never clarify how many terms of the infinite sums are required to achieve convergence, or what criterion you used to define convergence.
L. 585: ‘...of the aquifer’ There is a symbol I do not know after ‘aquifer’. Typo?
Eq. (A2): The only timedependent variables appear to be R(t) and h(x,t). Is that correct?
L. 593: The steadystate part of Eq. (A1) is that the second derivative of h with respect to x equals a constant recharge rate, correct? In that case, should C3 equal the mean recharge rate?
L. 600: Mixedtype (Cauchy BC) are not permitted?
Eq. (A9): Are the first three terms the parabolic groundwater level found for steady flow towards drains modeled as fully penetrating ditches?
It is not clear to me how you arrive at Eqs. (A10) and (A11).
L. 630633: This is not a real test, is it? You modeled numerically the simplified setup that permitted the analytical solution. This used to be done when in the 1970s and 1980s to test the accuracy of numerical models but was never intended to evaluate the analytical models.
Supporting information: If you have a radial aquifer which is pumped in the middle, does that not lead to a singularity at the center of the radius of the well is zero?

AC2: 'Reply on RC2', Luca Guillaumot, 27 Sep 2022
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Dear reviewer,
thank you for your interest in our manuscript. We answered below to the main comments. The revised manuscript will address all these points as well as minor comments.
Sincerely.
Luca Guillaumot, on behalf of all coauthors.
From the fitting results in Fig. 6 it appears that the parameter identifiability would benefit from replacing the aquifer length as a fitting parameter by the characteristic time, but this is not discussed or explored in the paper.
The model contains three parameters (transmissivity, porosity and length). These parameters have a physical meaning and can be compared to values from the field. Once we inverted parameters, indeed we chose to highlight the characteristic time (a combination of the three parameters) as we found it is well constrained by observed groundwater level fluctuations and it has also a physical meaning, allowing to summarize the behavior of the aquifer (line 111112). One of the reasons explaining that we did not replace aquifer length by characteristic time during parameter inversion is that characteristic time ranges over more than five orders of magnitude given the plausible ranges of the three parameters. Finally, the aquifer length can not be removed from the analytical model (equation 2) or this would require either to fix the aquifer length or to introduce additional parameter combinations as a parameter. As suggested, we will discuss this point.
A considerable weakness of the paper is that the model is calibrated for both aquifers, but not validated, making it difficult to assess the model performance.
Indeed, we did not split observations for calibration and validation periods. Note we highlighted in Supplementary the impact of the size of the study period on storage coefficient estimates. This shows that the studied period should be as long as possible when estimating parameters. However, following your remark, we run the calibration over half the period from 1996 to 2004, instead of from 1996 to 2012, for borehole F19 located on the Ploemeur site. Results are illustrated by the figures in attachment comparing parameter estimates and simulated water table fluctuations. They will be included in the revised Supplementary. Estimated parameters appear very similar when obtained over the first half of the period or over the whole period. Consequently similar water table fluctuations are obtained. To complete, results in term of parameters could be slightly altered in function of the studied time windows.
Our approach aims to estimate the informative content of the water table timefluctuations in terms of parameters and recharge rather than to provide predictions. Our approach shows that water table fluctuations recorded in different boreholes contain the same aquiferscale properties in spite of the aquifer heterogeneity. Moreover, we tested all possible combinations of parameters within extended ranges of potential parameter values with the finest systematic sampling of the parameter space. Thus, for these models we will not find any other better simulations than those we obtained. We argue that the more data we have the less equifinality issues occur.
Overall, the line of thought is a bit hard to follow some times because individual sections are not as focused as they can be. Throughout the paper, the clarity can be improved by thinking about what exactly the authors wish to convey to the reader and how to do so clearly. In the detailed comments I indicate where I got completely lost. I hope this will lead to a more structured, coherent paper.
Thank you for your remark. First, following your detailed comments, the revised manuscript will clarify several points of the methodology and more details will be provided regarding the development of the analytical model, including more substeps and references in Appendix 1 describing model geometry, boundary conditions and analytical resolution.
Sections 5, 6, and 7 are disappointing. They lack focus and structure and do not convey the main strengths of the study. That does not mean these strengths are not there. I very much like the modelling approach and the intended use. The Results and Discussion sections need to bring that out more strongly though.
Thanks for your comment. We understood that this main comment relates to several detailed comments (mainly the two comments below):
 Section 5.3 uses terminology with which I am too unfamiliar to understand what points are being made. Throughout the paper, the English is a bit off, but here it somehow becomes so much so that I can no longer decipher the meaning of several sentences. This section needs to be thoroughly rewritten to be accessible to readers outside the immediate field of this paper, and the English needs substantial improvement.
 Sections 6 and 7 require over five pages of text. That is rather long. They also introduce a large number of new references, which indicates that the paper is not well organized. The Introduction and the Methods sections should cover most of the literature needed in the paper.
The whole section 5 describes results about recharge estimates. Section 6 hosts discussion, while section 7 constitutes the conclusion.
Section 5.3 is very important, as underlined by reviewer 1, the revised manuscript will make it more accessible. We chose to analyze recharge fluctuations in frequency domain to highlight the different behaviours of recharge from soil models and obtained from groundwater levels in function of the time scale. The main advantage is to summarize the recharge signals for each frequency rather than comparing each recharge event individually and at different timesteps. As you suggested, we will revise our sentences carefully in this part.
The first part of Section 6 (section 6.1.1) contains additional references mainly in order to compare estimated parameters with the literature. They are very specific to the studied site and they do not need to be presented before.
We will take into account your remarks in order to reduce sections 6 and 7 and make them more fluuid for the readers. See below the main modifications that will appear in the revised manuscript :
 We acknowledge that section 6.1.3 ‘Limitations’ is long (25% of section 6) and will be reduced a lot in order to avoid losing readers. Indeed, this part introduces new references.
 While section 6.1 discusses methodology, sections 6.2 and 6.3 deals with processes understanding in Ploemeur and Guidel. In particular, the transformation of infiltration into recharge through the unsaturated zone and the impact of pumping. Indeed, the beginning of section 6.2 (line 499502) is not appropriate here. We will modify it with a better explanation in section 6.2.1. Following your minor comments, we agree that section 6.2.1 repeats too much results from section 5.3. One part of section 6.2.1 will move to section 2.6 (Methods) and will help to clarify the method. Then, we will gather sections 6.2.1 and sections 6.2.2.
 Finally, the first part of the conclusion (lines 551 to 555) will be reduced. As suggested, we will also take care to avoid summarising too much in conclusion.
We hope that this will help to clarify the manuscript.
The title is informative but a bit long.
We will propose : "Frequency domain water table fluctuations reveal impacts of intense rainfall and vadose zone thickness on recharge".

AC2: 'Reply on RC2', Luca Guillaumot, 27 Sep 2022
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