the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 18 Sep 2023
Technical Note: Analytical Solution for Well Water Response to Earth Tides in Leaky Aquifers with Storage and Compressibility in the Aquitard
Abstract. In recent years, there has been a growing interest in utilizing the groundwater response to Earth tides as a means to estimate subsurface properties. However, existing analytical models have been insufficient in accurately capturing realistic physical conditions. This study presents a new analytical solution to calculate groundwater response to Earth tide strains, including storage and compressibility of the aquitard, borehole storage and skin effects. We investigate the effects of aquifer and aquitard parameters on well water response to Earth tides at two dominant frequencies (O1 and M2) and compare our results with hydraulic parameters obtained from a pumping test. Inversion of the six hydro-geomechanical parameters from amplitude response and phase shift of both semi-diurnal and diurnal tides provides relevant information about aquifer transmissivity, storativity, well skin effect, aquitard hydraulic conductivity and diffusivity. The new model is able to reproduce previously unexplained observations of the amplitude and frequency responses. We emphasize the usefulness in developing relevant methodology to use the groundwater response to natural drivers for characterizing hydrogeological systems.
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Notice on discussion status
The requested preprint has a corresponding peer-reviewed final revised paper. You are encouraged to refer to the final revised version.
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Preprint
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The requested preprint has a corresponding peer-reviewed final revised paper. You are encouraged to refer to the final revised version.
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Journal article(s) based on this preprint
Interactive discussion
Status: closed
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RC1: 'Comment on egusphere-2023-1727', Anonymous Referee #1, 03 Nov 2023
General comments
This paper presents a new analytical solution for earth tide induced well water level fluctuations in an aquifer and an aquitard. A cylindrical coordinate system is utilized with an infinite radius while the thickness of the aquifer and aquitard are defined manually. The periodic earth tide is expressed by amplitude and angular frequency. The governing equation of the hydraulic head in the aquitard is solved first. Its solution is then used to solve the governing equation for the aquifer.
The key contribution is that the model considers the borehole skin effects in a leaky aquifer. My recommendation is major revision.
Key general comments are:
- A key concern is the applicability of the 2-D cylindrical coordinate which describes the well level fluctuation with earth tide source far away from the observation well. If this description is correct, the conceptual model can be simplified as a cylinder with a source at the outer surface and an observation well at the central line. In this case, the model is more suitable for an island or peninsula setting. While the inland observation well in Cambodia may not be this case. Please justify the insignificance of the angular coordinate in the solution.
- To better understand the physics of the water level fluctuations in response to the tide, please plot the hydraulic head distribution contour in the results section for several time snapshots to show the effect of skin effect and the leaky aquifer.
- Add more relevant references including the solutions for fractured rock aquifers.
- Please share the field data being used in this study if applicable.
Line 15-21: The key points list is not in standard form, with phrases and sentences mixed. For example, “Derivation of semi-analytical solutions for equivalent permeability in fractured multilayered porous media.” Please check the HESS articles as an example.
Line 65: Consider adding references that apply leaky aquifers in pumping tests, e.g.,
Butler and Tsou 2003 (https://doi.org/10.1029/2002WR001484),
Wen et al. 2011 (https://doi.org/10.1016/j.jhydrol.2011.01.010),
Line 76-77: Please describe what O1 and M2 represent when they appear the first time. Consider moving the ‘semi-diurnal’ and ‘diurnal tide’ in lines 91 and 92 to lines 76 and 77.
Line 108: Cite a classic reference for the governing equations (1) and (2).
Line 123: Also give a reference that defines the skin factor.
Line 137, 155, 159: “then deriving the head response in the aquifer far away from the well (h∞)”. I am confused by the expression. Is there another well at h∞ or just a hydraulic head at a specific position at r= ?
Line 162: How is equation 20 derived? Could you give more details or explanations?
Line 207: Please increase the line width of the dashed lines in Figure 2. It is difficult to tell the difference between the solid lines and dashed lines when they are well-fitted and printed in black-and-white mode.
Line 216: Could you specify ‘useful frequency band’?
Line 227: Consider adding references related to groundwater flow in fractured rock.
Citation: https://doi.org/10.5194/egusphere-2023-1727-RC1 -
AC1: 'Reply on RC1', remi valois, 08 Jan 2024
The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2023/egusphere-2023-1727/egusphere-2023-1727-AC1-supplement.pdf
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AC3: 'Reply on RC1', remi valois, 08 Jan 2024
Dear reviewer,
Thanks for your review.
Here attached the answers.
Best wishes
Citation: https://doi.org/10.5194/egusphere-2023-1727-AC3
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RC2: 'Comment on egusphere-2023-1727', Chi-Yuen Wang, 06 Dec 2023
In this Brief Note, the authors developed a more complete solution for the water level response to Earth tides in a leaky aquifer with aquitard storage and compressibility. The derivation of the solution is clearly given and easy to understand. On the other hand, there are some important points in the paper unclear to this referee, as detailed below:
First, it is unclear why the two Leaky & Storage models in Figure 2 are so very different. Both the amplitude ratio and phase difference for the model "Leaky & Storage (present study)" are functions of frequency at frequencies lower than that for the O1 tide, but both the amplitude ratio and phase difference become constant for the model "Leaky & Storage (present study') with K'=1e-14 m/s". Why are there such differences between the responses of the two models?
Second, the captions of some diagrams are too brief that made the figures unnecessarily difficult to read. For example, the caption for Figure 5 states that the results are from 'using two aquifer models' without explaining which two leaky models. The caption also did not define the crosses or circles, which kept this referee guessing. Given the authors' comparison between their model and the model of Gao et al (2020), could the circles be that for the solution of Gao et al.?
Finally, given the greater number of unknown parameters in the new solution, it is natural that the new solution may better reproduce the observed amplitude and frequency responses than previous models. The authors correctly pointed out that the solution is prone to non-uniqueness and a priori information is needed to reduce the number of unknows in the solution.
Citation: https://doi.org/10.5194/egusphere-2023-1727-RC2 - AC2: 'Reply on RC2', remi valois, 08 Jan 2024
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AC4: 'Comment on egusphere-2023-1727', remi valois, 08 Jan 2024
Dear Editors and Reviewers,
Thank you all for your time and energy to review and edit our study.
Best wishes
Citation: https://doi.org/10.5194/egusphere-2023-1727-AC4
Interactive discussion
Status: closed
-
RC1: 'Comment on egusphere-2023-1727', Anonymous Referee #1, 03 Nov 2023
General comments
This paper presents a new analytical solution for earth tide induced well water level fluctuations in an aquifer and an aquitard. A cylindrical coordinate system is utilized with an infinite radius while the thickness of the aquifer and aquitard are defined manually. The periodic earth tide is expressed by amplitude and angular frequency. The governing equation of the hydraulic head in the aquitard is solved first. Its solution is then used to solve the governing equation for the aquifer.
The key contribution is that the model considers the borehole skin effects in a leaky aquifer. My recommendation is major revision.
Key general comments are:
- A key concern is the applicability of the 2-D cylindrical coordinate which describes the well level fluctuation with earth tide source far away from the observation well. If this description is correct, the conceptual model can be simplified as a cylinder with a source at the outer surface and an observation well at the central line. In this case, the model is more suitable for an island or peninsula setting. While the inland observation well in Cambodia may not be this case. Please justify the insignificance of the angular coordinate in the solution.
- To better understand the physics of the water level fluctuations in response to the tide, please plot the hydraulic head distribution contour in the results section for several time snapshots to show the effect of skin effect and the leaky aquifer.
- Add more relevant references including the solutions for fractured rock aquifers.
- Please share the field data being used in this study if applicable.
Line 15-21: The key points list is not in standard form, with phrases and sentences mixed. For example, “Derivation of semi-analytical solutions for equivalent permeability in fractured multilayered porous media.” Please check the HESS articles as an example.
Line 65: Consider adding references that apply leaky aquifers in pumping tests, e.g.,
Butler and Tsou 2003 (https://doi.org/10.1029/2002WR001484),
Wen et al. 2011 (https://doi.org/10.1016/j.jhydrol.2011.01.010),
Line 76-77: Please describe what O1 and M2 represent when they appear the first time. Consider moving the ‘semi-diurnal’ and ‘diurnal tide’ in lines 91 and 92 to lines 76 and 77.
Line 108: Cite a classic reference for the governing equations (1) and (2).
Line 123: Also give a reference that defines the skin factor.
Line 137, 155, 159: “then deriving the head response in the aquifer far away from the well (h∞)”. I am confused by the expression. Is there another well at h∞ or just a hydraulic head at a specific position at r= ?
Line 162: How is equation 20 derived? Could you give more details or explanations?
Line 207: Please increase the line width of the dashed lines in Figure 2. It is difficult to tell the difference between the solid lines and dashed lines when they are well-fitted and printed in black-and-white mode.
Line 216: Could you specify ‘useful frequency band’?
Line 227: Consider adding references related to groundwater flow in fractured rock.
Citation: https://doi.org/10.5194/egusphere-2023-1727-RC1 -
AC1: 'Reply on RC1', remi valois, 08 Jan 2024
The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2023/egusphere-2023-1727/egusphere-2023-1727-AC1-supplement.pdf
-
AC3: 'Reply on RC1', remi valois, 08 Jan 2024
Dear reviewer,
Thanks for your review.
Here attached the answers.
Best wishes
Citation: https://doi.org/10.5194/egusphere-2023-1727-AC3
-
RC2: 'Comment on egusphere-2023-1727', Chi-Yuen Wang, 06 Dec 2023
In this Brief Note, the authors developed a more complete solution for the water level response to Earth tides in a leaky aquifer with aquitard storage and compressibility. The derivation of the solution is clearly given and easy to understand. On the other hand, there are some important points in the paper unclear to this referee, as detailed below:
First, it is unclear why the two Leaky & Storage models in Figure 2 are so very different. Both the amplitude ratio and phase difference for the model "Leaky & Storage (present study)" are functions of frequency at frequencies lower than that for the O1 tide, but both the amplitude ratio and phase difference become constant for the model "Leaky & Storage (present study') with K'=1e-14 m/s". Why are there such differences between the responses of the two models?
Second, the captions of some diagrams are too brief that made the figures unnecessarily difficult to read. For example, the caption for Figure 5 states that the results are from 'using two aquifer models' without explaining which two leaky models. The caption also did not define the crosses or circles, which kept this referee guessing. Given the authors' comparison between their model and the model of Gao et al (2020), could the circles be that for the solution of Gao et al.?
Finally, given the greater number of unknown parameters in the new solution, it is natural that the new solution may better reproduce the observed amplitude and frequency responses than previous models. The authors correctly pointed out that the solution is prone to non-uniqueness and a priori information is needed to reduce the number of unknows in the solution.
Citation: https://doi.org/10.5194/egusphere-2023-1727-RC2 - AC2: 'Reply on RC2', remi valois, 08 Jan 2024
-
AC4: 'Comment on egusphere-2023-1727', remi valois, 08 Jan 2024
Dear Editors and Reviewers,
Thank you all for your time and energy to review and edit our study.
Best wishes
Citation: https://doi.org/10.5194/egusphere-2023-1727-AC4
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Agnès Rivière
Jean-Michel Vouillamoz
Gabriel C. Rau
The requested preprint has a corresponding peer-reviewed final revised paper. You are encouraged to refer to the final revised version.
- Preprint
(978 KB) - Metadata XML