Transient Theory of Pumping Induced Depletion and Drawdown of a Stream with Finite Channel Storage

Malama, Bwalya; Lin, Ying-Fan; Yu, Hwa-Lung; Tseng, Hua-Ting; Greene, Sam

doi:https://doi.org/10.5194/hess-2022-353

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https://doi.org/10.5194/hess-2022-353

Preprints

02 Nov 2022

| 02 Nov 2022

Status: this preprint was under review for the journal HESS but the revision was not accepted.

Transient Theory of Pumping Induced Depletion and Drawdown of a Stream with Finite Channel Storage

Bwalya Malama, Ying-Fan Lin, Hwa-Lung Yu, Hua-Ting Tseng, and Sam Greene

Abstract. Mathematical models for stream depletion with stream stage decline or drawdown are developed to overcome the deficiency in existing models that typically use the constant-head (Dirichlet) or general (Robin) boundary condition and source terms at the stream-aquifer interface. Existing approaches assume a fixed stream stage during pumping, implies that the stream is an infinite water source, with depletion defined as a decrease in stream discharge. We refer to this depletion without drawdown as the ``stream depletion paradox.'' It is a glaring model limitation, ignoring the most observable adverse effect of long-term groundwater abstraction near a stream, namely stage declines that eventually lead to dry streambeds. Field data are presented to demonstrate that stream stage responds to pumping near the stream, motivating the development of an alternative theory predicts transient stream drawdown based on the concepts of finite stream storage and mass continuity at the stream-aquifer interface. Based on this alternative theory, models are developed for the cases of a non- and a fully-penetrating stream. The proposed model reduces to the fixed-stage model in the limit as stream storage becomes infinitely large and to the limiting case of confined aquifer flow with a no-flow boundary at the streambed when the stream storage vanishes. The model is applied to field observations of both aquifer and stream drawdown from tests conducted in a confined aquifer over which a shallow stream flows. Model fits and parameter estimates are obtained both aquifer and stream drawdown data. Model predicted and observed transient drawdown behavior indicate that fixed-stage models (a) underestimate late-time aquifer drawdown and (b) overestimate the available recharge from streams to pumping wells. This has significant implications for the sustainable management of water resources in hydraulically connected stream-aquifer systems with heavy groundwater abstraction.

Received: 10 Oct 2022 – Discussion started: 02 Nov 2022

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this preprint. The responsibility to include appropriate place names lies with the authors.

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Bwalya Malama, Ying-Fan Lin, Hwa-Lung Yu, Hua-Ting Tseng, and Sam Greene

Status: closed

RC1:
'Comment on hess-2022-353', Anonymous Referee #1, 01 Dec 2022

Major comments.

I have the impression the paper does not give enough credit to the body of work in which earlier analytical solutions to the flow problem discussed here were derived. The Laplace-Fourier transformation used here was adopted from Butler Jr. et al. (2001), who in turn adopted it from earlier work done in the former Soviet Union. I had to look this up to find out.

The authors should also clarify in the Introduction what this paper contributes to this existing body of work: what is the added value of their solutions? I am not implying this contribution does not exist, but I found it hard to figure out what it is, as I do not have the time to study the older papers in detail myself. It appears the answer is given in the sentence starting in line 463. It would help to move this forward.

The paper is awkwardly structured, with section 6 introducing elements that one would expect in a Materials and Methods section that is not there. On at least one occasion this leads to a reference to a flow system that was not yet discussed, which is an indicator of a poorly structured paper.

The section on the existing aquifer is too large. It is OK to use field data to test the practical use of the solutions, but the detailed description of the geological setting and the legal context distracts from the main message of the paper. So does the parameter fitting exercise, which seems more suitable for an engineer's report for local use.

The figure captions mostly are inadequate. Figures should be intelligible without having to refer to the main text, but the captions are too short to allow this. In some cases, not even all curves are properly explained.

I am not convinced the figures always focus on the most interesting aspects of the solutions, and are perhaps more driven by mathematical rather than hydrological considerations.

The sections discussing the figures should be gathered in a section 'Results and Discussion'. The section that is called 'Results' in the current version repeats the discussion in those preceding sections and can then be deleted.

There are minor and major grammar errors that sometimes hamper understanding (see detailed comments for an incomplete listing). I gave up on section 5 because I could not comprehend large parts of the text.

The paper does not always adhere to SI units.

If a revision will be submitted, the authors should place the figures in the text, not at the end. In addition, the figures not always adhere to HESS standards.

I am not sure about the elvel of novelty of the paper. If the improvement of earlier analytical solutions of the flow problem of interest is considered marginal, I may have overrated the scientific value of the paper.

Considerable rewriting is necessary in my opinion, which is why I recommend major revisions.

Minor comments are in the annotated manuscript.

Citation: https://doi.org/10.5194/hess-2022-353-RC1
- AC1:
  'Reply on RC1', Bwalya Malama, 10 Dec 2022
  
  Thank you for your review and comments on our draft manuscript. Our reply here is limited only to the two pertinent technical comments made by the reviewer. The other comments will be addresed in the full response to reviewer comments when we submit the revised manuscript, follwoing comments from the remaining reviewer.
  In response to the comment that we do not give enough credit to the body of work in which earlier analytical solutions to the flow problem, we note that the Laplace and Fourier transform techniques used in this manuscript are adopted from the textbooks of Engineering Mathematics and not directly from the works of Butler Jr. et al. (2001) . This is because they are standard methods of applied and engineering mathematics for solution of partial differential equations. We will be adding a clarifying statement to the revised manuscript citing an example standard text from which the transforms used in our work to solve flow equations may be found, viz., ``To solve the flow problem described above, the governing equation is first transformed into a dimensionless form. Details of the nondimensionalization of the governing equations and their solutions may be found in the Appendix. Laplace and Fourier-cosine transforms are applied to the dimensionless governing equations, which are then solved by standard methods for ordinary differential equations. The respective inversion formulae of the transforms are finally used to numerically obtain the applicable flow solutions in space-time. The transform and inversion formulae can be found in standard textbooks of Engineering Mathematics, and interested readers may refer to the reference text of \cite{povstenko2015linear}. Similar solution approaches have been used in the hydrogeology literature by Butler Jr. et al. (2001) and others.''
  Secondly, in response to the comment that we "should also clarify in the Introduction what this paper contributes to this existing body of work: what is the added value of their solutions?" we point you to lines 47 -- 51 of the introduction in the original manuscript where knowledge gap was identified explicitly. namely that existing models impose a fixed-stage condition via the Dirichlet or Robin boundary condition (or source function) at th stream-aquifer interface. We also state that by so doing, existing models cannot model or predict transient drawdown of the stream because stream drawdown is by definition set to zero. Additionally, we state in the orignal manuscript that existing streams inherently assume that streams have an infinite capacity to replenish the aquifer without experiencing a decline in stream stage as water is pumped from the aquifer. We address these limitations by introducing a new boundary condition using a mass balance condition in the stream channel and introducing a new model parameter (or stream property) we call the stream storage coefficient. We do this in section 2.4 of the original manuscript titled "Accounting for Stream Drawdown and Channel Storage." Next, we develop a new mathematical solution for two cases. Finally, this new mathematical solution is applied to field data, namely, (1) transient stream drawdown data and (2) aquifer drawdown data. The measurement of stream drawdown data is in itself new because existing works in the hydrogeology literature do not consider stream drawdown response to pumping; only aquifer drawdown is typically measured. In applying the model to the two data sets, we demonstrate that aquifer hydraulic parameter, stream conductance, and the newly introduced stream storage parameter, are estimable from observations of stream drawdown behavior as well as from aquifer drawdown.
  We are adding clarifying language in the revised manuscript that accentuates the new ideas of our research as well as states more explicitly the distinction between stream drawdown and stream depletion. Stream drawdwon is not addressed in existing models on stream depletion. Our work is the first to address this strange model descrepancy we call the stream depletion paradox: depletion without drawdown.
  
  Citation: https://doi.org/10.5194/hess-2022-353-AC1
  - RC3: 'Reply on AC1', Anonymous Referee #1, 02 Jan 2023
    
    Dear authors,
    These clarificationa are in themselves very useful, and I would recommend to include them in some form in the manuscript to better embed your work in the existing literature (textbooks included), and explain the added value of your work. With regards to the latter, you reply clearly states the contribution of the paper in terms of the mathematical finesse, and you explain how this relates to the hydrological reality that it represents. In your reply you start from the mathematics and then move to the hydrological reality in the field. Since HESS is a hydrological journal it would perhaps work better if you reverse the order by first explaining that existing solutions cannot handle a response of the stream level to an aquifer extracting water from it (or feeding water into it). You can then argue that you need a refined boundary condition and a stream storage coefficient representing stream properties. Then you can show how that affects the mathematical formulation of the problem, and finally how you solve the resulting problem.
    I am offering this is only a suggesttion. The reply itself is already very helpful.
    
    Citation: https://doi.org/10.5194/hess-2022-353-RC3
    
    AC4: 'Reply on RC3', Bwalya Malama, 29 Jan 2023
    
    Thank you for your suggestion. In our revised manuscript, we will add more clarity to the introduction and the mathematical formulation sections regarding our imposition of a new boundary condition at the stream-aquifer interface. As we state in the response, our contribution is imposing this new condition, defining a new stream channel property (a storage coefficient), then solving the flow problem such that the stream undergoes transient drawdown in addition to depletion.
    
    Citation: https://doi.org/10.5194/hess-2022-353-AC4
- CC1:
  'Reply on RC1', Ying-Fan Lin, 29 Dec 2022
  
  The supplement files, titled "Reply on RC1.zip," include our response to the main comments of the reviewer, found in the document titled "Reply to Referee 1.pdf," as well as our response to the minor comments, which are marked in blue in the pdf file titled "Marked manuscript R1.pdf." We hope that these materials adequately address the reviewer's feedback.
  
  Citation: https://doi.org/10.5194/hess-2022-353-CC1
  - RC2: 'Reply on CC1', Anonymous Referee #1, 02 Jan 2023
    
    Dear authors,
    Unfortunately, the zip file containing the reply seems to be corrupted. I cannot extract its files.
    
    Citation: https://doi.org/10.5194/hess-2022-353-RC2
    
    CC2: 'Reply on RC2', Ying-Fan Lin, 02 Jan 2023
    
    Dear Reviewer 1,
    We have uploaded revised versions of the files you requested: a document outlining our responses to your major comments and a revised manuscript incorporating your minor comments. We hope you are now able to access these files. If you encounter any problems with file corruption, please do not hesitate to let us know. The attached file is the revision note, titled "Reply to Referee 1.pdf".
    
    Citation: https://doi.org/10.5194/hess-2022-353-CC2
    
    RC4: 'Reply on CC2', Anonymous Referee #1, 02 Jan 2023
    
    Dear Dr. Lin,
    I can read the attached document now, thanks.
    
    Citation: https://doi.org/10.5194/hess-2022-353-RC4
    
    AC6: 'Reply on RC4', Bwalya Malama, 29 Jan 2023
    
    Thank you for letting us know.
    
    Citation: https://doi.org/10.5194/hess-2022-353-AC6
    
    CC3: 'Reply on RC2 (Marked manuscript)', Ying-Fan Lin, 02 Jan 2023
    
    Dear Reviewer 1,
    We have attached another file called "Marked manuscript R1" in response to our previous reply. Please note that we can only upload one pdf file at a time.
    
    Citation: https://doi.org/10.5194/hess-2022-353-CC3
    
    RC5: 'Reply on CC3', Anonymous Referee #1, 02 Jan 2023
    
    Dear Dr. Lin,
    I can open the manuscript file as well. But I will wait for the editor decision to see what action he wants me to take.
    
    Citation: https://doi.org/10.5194/hess-2022-353-RC5
    
    AC7: 'Reply on RC5', Bwalya Malama, 29 Jan 2023
    
    Thank you for letting us know.
    
    Citation: https://doi.org/10.5194/hess-2022-353-AC7
    
    AC5: 'Reply on RC2', Bwalya Malama, 29 Jan 2023
    
    The file issue was resolved earlier.
    
    Citation: https://doi.org/10.5194/hess-2022-353-AC5
RC6:
'Comment on hess-2022-353', Anonymous Referee #2, 08 Jan 2023

General comments

Overall, I find the paper interesting. However, the introduced model has fundamental shortcomings: in particular, it does not include any stream flow routing, which implies that all the points along the stream behave independently of one another. This is obviously unrealistic and can certainly affect the results significantly. At the minimum, the authors should much better discuss and justify the stream storage approach they have adopted (section 2.4). This is critical if they want their work to be considered relevant – so far, I have doubts.

The main contribution of the paper is not very clear: is it about an analytical solution, or is it about the implications of a poorly recognized phenomenon? In particular, the authors refer to (Zlotnik, 2004) several times, but without explaining what the difference between (Zlotnik, 2004) work and this work is. Moreover, the authors should clarify that numerical models (at least MODFLOW) do allow for simulating stream stage drawdown induced by pumping. Currently, they almost insinuate the contrary (the introduction is quite unclear about it, up to the point of being misleading).

Regarding the presentation, I find the paper well written and illustrated in general, although several grammar mistakes and typos need to be corrected. On the other hand, many things need to be clarified, calling for a significant amount of work. Please refer to the detailed comments below.

Detailed comments

L19-20: Suggest citing the nice, comprehensive USGS report by Barlow and Leake (2012), and perhaps remove some of the less relevant references given here.

L36-37: This part is not clear at all. First, I do not think Harbaugh (2005) talks about empirical hydrographs. Second, you should explicitly mention the MODFLOW packages STR1, SFR1 and SFR2, which do allow for simulating stream stage drawdown induced by pumping.

L40-41: Of what source terms are you talking about here?

L42-44: Not true with the STR1, SFR1 and SFR2 packages, as already mentioned.

L45-46: Please explain the maximum SDR concept.

L47: Rather than “because the stream flow rate is two orders or more higher than the pumping rate”, I think it should be “when the stream flow rate is two orders or more higher than the pumping rate”.

L56: This sentence appears unfinished at “Theis (1941)”.

L55-63: This is diverting from the introduction; I would suggest putting it in the Discussion section.

L68: Rather than “test the hypothesis”, I guess you mean “make the assumption”.

Figure 1b: the piezometric level should be lower than the stream level to have inflow from the stream to the aquifer!

L87-88: Where is the origin of the y axis?

L95: Are you sure of that division by π??

L99: Stream drawdown was previously noted H_r(t), why is it noted h_r(t) now?

L99-100: This would imply that the initial aquifer head is equal to the initial stream stage; isn’t it too restrictive?? This is even contradictory with the conceptual model in Figure 1b. PS: I think what you are doing is correct; but the way you write it makes it appear as if it would only work for “flat” initial conditions, whereas the superposition principle allows more than that.

Note: the same goes with the temporal variations of the boundary conditions. When in L79 you write that s_r(t) = H₀ - H_r(t), you are implying that the stream stage cannot have natural variations. But the superposition principle allows for natural variations. What you would need to do is to define drawdown as the change in stream stage due to pumping; for example, s_r(t) = H_rn(t) - H_rp(t), where H_rn(t) would represent the stream stage under natural (i.e., non-pumping) conditions, and H_rp(t) the stream stage under pumping conditions.

L101-102: These variables were already defined in L76-77; avoid repetitions.

L103: “is” should be “in”.

L113-114: Do you really need those two conditions? In other words, aren’t the continuity conditions for drawdown (L110-111) sufficient for the problem to be well-posed?

L133: I think it should be Hantush (1965), not Hantush and Jacob (1955).

L136: Same comment.

L163-164: So, in the end you are assuming a no-flow boundary at x = -W in all cases?? This would not be in line with all the rest of the paper, so is this sentence just a mistake??

L144-146: These equations would imply that s_r is a function of y (and x in the NPS case). Is this correct (previously, you wrote s_r only as a function of t, as in s_r(t))?

Furthermore, this approach implies that the stream stage can vary in each point independently of what happens upstream and downstream of that point. How reasonable is this assumption? If the stream stage decreases in one point, it should also decrease downstream of that point, even if no water if withdrawn further downstream.

L175: “ration” should be “ratio”.

L198: Why talking in terms of dimensionless time but not in dimensionless space? Be consistent.

L221-226: There is a mistake here: s_D = -γ/2 - ln(u) should be s_D = -γ/2 - ln(u)/2; and then you should get A = 1.5√α. Furthermore, rather than deriving this well-known formula for the radius of influence, you should simply refer to Bear (1979). You could also refer to Bresciani et al. (2020) for justifying your choice of radius of influence formula over other existing formula.

L230-250: How is the stream modelled in the numerical model? Does it also neglect downstream transmission of stream stage drawdown, as in the analytical model?

L254: Space missing before the parenthesis.

L253-254: Where are you evaluating the solution?

L255-256: What is the conceptual difference between the models of Hantush (1965) and Fox et al. (2002)?

L281: Indeed, the stream stage response to pumping should also depend on the stream discharge rate!! This is a fundamental aspect of the problem that has been completely ignored in the approach taken in this study.

L283: “unpmped” should be “unpumped”.

L284: I disagree: does not imply an infinite reserve of the stream.

L318: I think “the case of C_D,r” should read “the case of C_D,r = ∞".

L339: Delete “that”.

Figure 11: Show the direction of stream flow.

Figure 12: What does the 0 represent (on both axes)?

Figure 12: Why do we still see significant trends if the data have been detrended?

Figure 12: The data show that the stream stage that is furthest to the pumping well (Stenner-P1) is the most affected, which is quite weird. Give possible explanations of this outcome.

L376: Where are the recovery data?

L402: One “is” must be deleted.

References

Barlow, P.M., Leake, S.A., 2012. Streamflow depletion by wells—Understanding and managing the effects of groundwater pumping on streamflow (No. 1376), Circular.

Bear, J., 1979. Hydraulics of groundwater. McGraw-Hill, New York.

Bresciani, E., Shandilya, R.N., Kang, P.K., Lee, S., 2020. Well radius of influence and radius of investigation: What exactly are they and how to estimate them? J. Hydrol. 583, 124646. https://doi.org/10.1016/j.jhydrol.2020.124646

Harbaugh, A.W., 2005. MODFLOW-2005, The U.S. Geological Survey modular ground-water model – the ground-water flow process. U.S. Geol. Surv. Tech. Methods 6-A16.

Zlotnik, V.A., 2004. A concept of maximum stream depletion rate for leaky aquifers in alluvial valleys. Water Resour. Res. 40, W06507. https://doi.org/10.1029/2003wr002932

Citation: https://doi.org/10.5194/hess-2022-353-RC6
- AC2: 'Reply on RC6', Bwalya Malama, 23 Jan 2023
  
  Our responses to the individual comments by the reviewer are in the attached file Response-to-Reviewer-2.pdf. In response to the reviewer's comment that the contribution of our work is unclear, we strive in the revised manuscript, to clarify the main contribution of our work, which was identified explicitly in lines 47 -- 51 of the original manuscript. The contribution is about both a new analytical solution and ``the implications of a poorly recognized phenomenon,'' namely transient stream drawdown and channel storage. Our work differs from that of published works we review in the introduction specifically in our treatment of the boundary condition at the stream-aquifer interface. Whereas previous analytical works use a fixed-stream stage condition, we allow stream stage to respond to groundwater pumping; whereas all models that use the fixed-stage condition cannot predict non-zero transient stream drawdown response to pumping, our model predicts this in addition to stream depletion. To reiterate, these models are based on the fixed-stage assumption and as such, stream drawdown is by definition zero. We have developed an analytical solution that overcomes this limitation. We develop the solution by first modifying the boundary condition at the stream-aquifer interface using a mass balance condition regulated by a finite stream channel storage coefficient. We demonstrate that fixed-stage models coincide with our model in the limit as the stream channel storage coefficient approaches infinity, $C_r \rightarrow \infty$. Additionally, we also state that MODFLOW also relies on the same fixed-stage condition used by the earlier analytical models. To make our contribution clearer here in the response, we include a Figure in the attached response where we plot the predicted behavior of the old models (a) separate from our new model (b). The model of fox et al. (2002) is the general representation of all fixed-stage models for confined aquifer flow including those that account for aquifer leakage such as the model of Zlotnik (2004). The final contribution of our work is the application of the model to field measurements of transient stream drawdown, something that has never been done in the hydrogeology literature, where the focus is on only aquifer drawdown analysis. Detailed responses to individual comments can be found in the attached file as indicated earlier.
  
  Citation: https://doi.org/10.5194/hess-2022-353-AC2
- AC3: 'Reply on RC6', Bwalya Malama, 23 Jan 2023
  
  Please find attached here the file with the responses to all the individual comments of reviewer 2. In this document, our responses are in itallic and the proposed changes to the manuscript are in bold-itallic.
  
  Citation: https://doi.org/10.5194/hess-2022-353-AC3

Status: closed

RC1:
'Comment on hess-2022-353', Anonymous Referee #1, 01 Dec 2022

Major comments.

I have the impression the paper does not give enough credit to the body of work in which earlier analytical solutions to the flow problem discussed here were derived. The Laplace-Fourier transformation used here was adopted from Butler Jr. et al. (2001), who in turn adopted it from earlier work done in the former Soviet Union. I had to look this up to find out.

The authors should also clarify in the Introduction what this paper contributes to this existing body of work: what is the added value of their solutions? I am not implying this contribution does not exist, but I found it hard to figure out what it is, as I do not have the time to study the older papers in detail myself. It appears the answer is given in the sentence starting in line 463. It would help to move this forward.

The paper is awkwardly structured, with section 6 introducing elements that one would expect in a Materials and Methods section that is not there. On at least one occasion this leads to a reference to a flow system that was not yet discussed, which is an indicator of a poorly structured paper.

The section on the existing aquifer is too large. It is OK to use field data to test the practical use of the solutions, but the detailed description of the geological setting and the legal context distracts from the main message of the paper. So does the parameter fitting exercise, which seems more suitable for an engineer's report for local use.

The figure captions mostly are inadequate. Figures should be intelligible without having to refer to the main text, but the captions are too short to allow this. In some cases, not even all curves are properly explained.

I am not convinced the figures always focus on the most interesting aspects of the solutions, and are perhaps more driven by mathematical rather than hydrological considerations.

The sections discussing the figures should be gathered in a section 'Results and Discussion'. The section that is called 'Results' in the current version repeats the discussion in those preceding sections and can then be deleted.

There are minor and major grammar errors that sometimes hamper understanding (see detailed comments for an incomplete listing). I gave up on section 5 because I could not comprehend large parts of the text.

The paper does not always adhere to SI units.

If a revision will be submitted, the authors should place the figures in the text, not at the end. In addition, the figures not always adhere to HESS standards.

I am not sure about the elvel of novelty of the paper. If the improvement of earlier analytical solutions of the flow problem of interest is considered marginal, I may have overrated the scientific value of the paper.

Considerable rewriting is necessary in my opinion, which is why I recommend major revisions.

Minor comments are in the annotated manuscript.

Citation: https://doi.org/10.5194/hess-2022-353-RC1
- AC1:
  'Reply on RC1', Bwalya Malama, 10 Dec 2022
  
  Thank you for your review and comments on our draft manuscript. Our reply here is limited only to the two pertinent technical comments made by the reviewer. The other comments will be addresed in the full response to reviewer comments when we submit the revised manuscript, follwoing comments from the remaining reviewer.
  In response to the comment that we do not give enough credit to the body of work in which earlier analytical solutions to the flow problem, we note that the Laplace and Fourier transform techniques used in this manuscript are adopted from the textbooks of Engineering Mathematics and not directly from the works of Butler Jr. et al. (2001) . This is because they are standard methods of applied and engineering mathematics for solution of partial differential equations. We will be adding a clarifying statement to the revised manuscript citing an example standard text from which the transforms used in our work to solve flow equations may be found, viz., ``To solve the flow problem described above, the governing equation is first transformed into a dimensionless form. Details of the nondimensionalization of the governing equations and their solutions may be found in the Appendix. Laplace and Fourier-cosine transforms are applied to the dimensionless governing equations, which are then solved by standard methods for ordinary differential equations. The respective inversion formulae of the transforms are finally used to numerically obtain the applicable flow solutions in space-time. The transform and inversion formulae can be found in standard textbooks of Engineering Mathematics, and interested readers may refer to the reference text of \cite{povstenko2015linear}. Similar solution approaches have been used in the hydrogeology literature by Butler Jr. et al. (2001) and others.''
  Secondly, in response to the comment that we "should also clarify in the Introduction what this paper contributes to this existing body of work: what is the added value of their solutions?" we point you to lines 47 -- 51 of the introduction in the original manuscript where knowledge gap was identified explicitly. namely that existing models impose a fixed-stage condition via the Dirichlet or Robin boundary condition (or source function) at th stream-aquifer interface. We also state that by so doing, existing models cannot model or predict transient drawdown of the stream because stream drawdown is by definition set to zero. Additionally, we state in the orignal manuscript that existing streams inherently assume that streams have an infinite capacity to replenish the aquifer without experiencing a decline in stream stage as water is pumped from the aquifer. We address these limitations by introducing a new boundary condition using a mass balance condition in the stream channel and introducing a new model parameter (or stream property) we call the stream storage coefficient. We do this in section 2.4 of the original manuscript titled "Accounting for Stream Drawdown and Channel Storage." Next, we develop a new mathematical solution for two cases. Finally, this new mathematical solution is applied to field data, namely, (1) transient stream drawdown data and (2) aquifer drawdown data. The measurement of stream drawdown data is in itself new because existing works in the hydrogeology literature do not consider stream drawdown response to pumping; only aquifer drawdown is typically measured. In applying the model to the two data sets, we demonstrate that aquifer hydraulic parameter, stream conductance, and the newly introduced stream storage parameter, are estimable from observations of stream drawdown behavior as well as from aquifer drawdown.
  We are adding clarifying language in the revised manuscript that accentuates the new ideas of our research as well as states more explicitly the distinction between stream drawdown and stream depletion. Stream drawdwon is not addressed in existing models on stream depletion. Our work is the first to address this strange model descrepancy we call the stream depletion paradox: depletion without drawdown.
  
  Citation: https://doi.org/10.5194/hess-2022-353-AC1
  - RC3: 'Reply on AC1', Anonymous Referee #1, 02 Jan 2023
    
    Dear authors,
    These clarificationa are in themselves very useful, and I would recommend to include them in some form in the manuscript to better embed your work in the existing literature (textbooks included), and explain the added value of your work. With regards to the latter, you reply clearly states the contribution of the paper in terms of the mathematical finesse, and you explain how this relates to the hydrological reality that it represents. In your reply you start from the mathematics and then move to the hydrological reality in the field. Since HESS is a hydrological journal it would perhaps work better if you reverse the order by first explaining that existing solutions cannot handle a response of the stream level to an aquifer extracting water from it (or feeding water into it). You can then argue that you need a refined boundary condition and a stream storage coefficient representing stream properties. Then you can show how that affects the mathematical formulation of the problem, and finally how you solve the resulting problem.
    I am offering this is only a suggesttion. The reply itself is already very helpful.
    
    Citation: https://doi.org/10.5194/hess-2022-353-RC3
    
    AC4: 'Reply on RC3', Bwalya Malama, 29 Jan 2023
    
    Thank you for your suggestion. In our revised manuscript, we will add more clarity to the introduction and the mathematical formulation sections regarding our imposition of a new boundary condition at the stream-aquifer interface. As we state in the response, our contribution is imposing this new condition, defining a new stream channel property (a storage coefficient), then solving the flow problem such that the stream undergoes transient drawdown in addition to depletion.
    
    Citation: https://doi.org/10.5194/hess-2022-353-AC4
- CC1:
  'Reply on RC1', Ying-Fan Lin, 29 Dec 2022
  
  The supplement files, titled "Reply on RC1.zip," include our response to the main comments of the reviewer, found in the document titled "Reply to Referee 1.pdf," as well as our response to the minor comments, which are marked in blue in the pdf file titled "Marked manuscript R1.pdf." We hope that these materials adequately address the reviewer's feedback.
  
  Citation: https://doi.org/10.5194/hess-2022-353-CC1
  - RC2: 'Reply on CC1', Anonymous Referee #1, 02 Jan 2023
    
    Dear authors,
    Unfortunately, the zip file containing the reply seems to be corrupted. I cannot extract its files.
    
    Citation: https://doi.org/10.5194/hess-2022-353-RC2
    
    CC2: 'Reply on RC2', Ying-Fan Lin, 02 Jan 2023
    
    Dear Reviewer 1,
    We have uploaded revised versions of the files you requested: a document outlining our responses to your major comments and a revised manuscript incorporating your minor comments. We hope you are now able to access these files. If you encounter any problems with file corruption, please do not hesitate to let us know. The attached file is the revision note, titled "Reply to Referee 1.pdf".
    
    Citation: https://doi.org/10.5194/hess-2022-353-CC2
    
    RC4: 'Reply on CC2', Anonymous Referee #1, 02 Jan 2023
    
    Dear Dr. Lin,
    I can read the attached document now, thanks.
    
    Citation: https://doi.org/10.5194/hess-2022-353-RC4
    
    AC6: 'Reply on RC4', Bwalya Malama, 29 Jan 2023
    
    Thank you for letting us know.
    
    Citation: https://doi.org/10.5194/hess-2022-353-AC6
    
    CC3: 'Reply on RC2 (Marked manuscript)', Ying-Fan Lin, 02 Jan 2023
    
    Dear Reviewer 1,
    We have attached another file called "Marked manuscript R1" in response to our previous reply. Please note that we can only upload one pdf file at a time.
    
    Citation: https://doi.org/10.5194/hess-2022-353-CC3
    
    RC5: 'Reply on CC3', Anonymous Referee #1, 02 Jan 2023
    
    Dear Dr. Lin,
    I can open the manuscript file as well. But I will wait for the editor decision to see what action he wants me to take.
    
    Citation: https://doi.org/10.5194/hess-2022-353-RC5
    
    AC7: 'Reply on RC5', Bwalya Malama, 29 Jan 2023
    
    Thank you for letting us know.
    
    Citation: https://doi.org/10.5194/hess-2022-353-AC7
    
    AC5: 'Reply on RC2', Bwalya Malama, 29 Jan 2023
    
    The file issue was resolved earlier.
    
    Citation: https://doi.org/10.5194/hess-2022-353-AC5
RC6:
'Comment on hess-2022-353', Anonymous Referee #2, 08 Jan 2023

General comments

Overall, I find the paper interesting. However, the introduced model has fundamental shortcomings: in particular, it does not include any stream flow routing, which implies that all the points along the stream behave independently of one another. This is obviously unrealistic and can certainly affect the results significantly. At the minimum, the authors should much better discuss and justify the stream storage approach they have adopted (section 2.4). This is critical if they want their work to be considered relevant – so far, I have doubts.

The main contribution of the paper is not very clear: is it about an analytical solution, or is it about the implications of a poorly recognized phenomenon? In particular, the authors refer to (Zlotnik, 2004) several times, but without explaining what the difference between (Zlotnik, 2004) work and this work is. Moreover, the authors should clarify that numerical models (at least MODFLOW) do allow for simulating stream stage drawdown induced by pumping. Currently, they almost insinuate the contrary (the introduction is quite unclear about it, up to the point of being misleading).

Regarding the presentation, I find the paper well written and illustrated in general, although several grammar mistakes and typos need to be corrected. On the other hand, many things need to be clarified, calling for a significant amount of work. Please refer to the detailed comments below.

Detailed comments

L19-20: Suggest citing the nice, comprehensive USGS report by Barlow and Leake (2012), and perhaps remove some of the less relevant references given here.

L36-37: This part is not clear at all. First, I do not think Harbaugh (2005) talks about empirical hydrographs. Second, you should explicitly mention the MODFLOW packages STR1, SFR1 and SFR2, which do allow for simulating stream stage drawdown induced by pumping.

L40-41: Of what source terms are you talking about here?

L42-44: Not true with the STR1, SFR1 and SFR2 packages, as already mentioned.

L45-46: Please explain the maximum SDR concept.

L47: Rather than “because the stream flow rate is two orders or more higher than the pumping rate”, I think it should be “when the stream flow rate is two orders or more higher than the pumping rate”.

L56: This sentence appears unfinished at “Theis (1941)”.

L55-63: This is diverting from the introduction; I would suggest putting it in the Discussion section.

L68: Rather than “test the hypothesis”, I guess you mean “make the assumption”.

Figure 1b: the piezometric level should be lower than the stream level to have inflow from the stream to the aquifer!

L87-88: Where is the origin of the y axis?

L95: Are you sure of that division by π??

L99: Stream drawdown was previously noted H_r(t), why is it noted h_r(t) now?

L99-100: This would imply that the initial aquifer head is equal to the initial stream stage; isn’t it too restrictive?? This is even contradictory with the conceptual model in Figure 1b. PS: I think what you are doing is correct; but the way you write it makes it appear as if it would only work for “flat” initial conditions, whereas the superposition principle allows more than that.

Note: the same goes with the temporal variations of the boundary conditions. When in L79 you write that s_r(t) = H₀ - H_r(t), you are implying that the stream stage cannot have natural variations. But the superposition principle allows for natural variations. What you would need to do is to define drawdown as the change in stream stage due to pumping; for example, s_r(t) = H_rn(t) - H_rp(t), where H_rn(t) would represent the stream stage under natural (i.e., non-pumping) conditions, and H_rp(t) the stream stage under pumping conditions.

L101-102: These variables were already defined in L76-77; avoid repetitions.

L103: “is” should be “in”.

L113-114: Do you really need those two conditions? In other words, aren’t the continuity conditions for drawdown (L110-111) sufficient for the problem to be well-posed?

L133: I think it should be Hantush (1965), not Hantush and Jacob (1955).

L136: Same comment.

L163-164: So, in the end you are assuming a no-flow boundary at x = -W in all cases?? This would not be in line with all the rest of the paper, so is this sentence just a mistake??

L144-146: These equations would imply that s_r is a function of y (and x in the NPS case). Is this correct (previously, you wrote s_r only as a function of t, as in s_r(t))?

Furthermore, this approach implies that the stream stage can vary in each point independently of what happens upstream and downstream of that point. How reasonable is this assumption? If the stream stage decreases in one point, it should also decrease downstream of that point, even if no water if withdrawn further downstream.

L175: “ration” should be “ratio”.

L198: Why talking in terms of dimensionless time but not in dimensionless space? Be consistent.

L221-226: There is a mistake here: s_D = -γ/2 - ln(u) should be s_D = -γ/2 - ln(u)/2; and then you should get A = 1.5√α. Furthermore, rather than deriving this well-known formula for the radius of influence, you should simply refer to Bear (1979). You could also refer to Bresciani et al. (2020) for justifying your choice of radius of influence formula over other existing formula.

L230-250: How is the stream modelled in the numerical model? Does it also neglect downstream transmission of stream stage drawdown, as in the analytical model?

L254: Space missing before the parenthesis.

L253-254: Where are you evaluating the solution?

L255-256: What is the conceptual difference between the models of Hantush (1965) and Fox et al. (2002)?

L281: Indeed, the stream stage response to pumping should also depend on the stream discharge rate!! This is a fundamental aspect of the problem that has been completely ignored in the approach taken in this study.

L283: “unpmped” should be “unpumped”.

L284: I disagree: does not imply an infinite reserve of the stream.

L318: I think “the case of C_D,r” should read “the case of C_D,r = ∞".

L339: Delete “that”.

Figure 11: Show the direction of stream flow.

Figure 12: What does the 0 represent (on both axes)?

Figure 12: Why do we still see significant trends if the data have been detrended?

Figure 12: The data show that the stream stage that is furthest to the pumping well (Stenner-P1) is the most affected, which is quite weird. Give possible explanations of this outcome.

L376: Where are the recovery data?

L402: One “is” must be deleted.

References

Barlow, P.M., Leake, S.A., 2012. Streamflow depletion by wells—Understanding and managing the effects of groundwater pumping on streamflow (No. 1376), Circular.

Bear, J., 1979. Hydraulics of groundwater. McGraw-Hill, New York.

Bresciani, E., Shandilya, R.N., Kang, P.K., Lee, S., 2020. Well radius of influence and radius of investigation: What exactly are they and how to estimate them? J. Hydrol. 583, 124646. https://doi.org/10.1016/j.jhydrol.2020.124646

Harbaugh, A.W., 2005. MODFLOW-2005, The U.S. Geological Survey modular ground-water model – the ground-water flow process. U.S. Geol. Surv. Tech. Methods 6-A16.

Zlotnik, V.A., 2004. A concept of maximum stream depletion rate for leaky aquifers in alluvial valleys. Water Resour. Res. 40, W06507. https://doi.org/10.1029/2003wr002932

Citation: https://doi.org/10.5194/hess-2022-353-RC6
- AC2: 'Reply on RC6', Bwalya Malama, 23 Jan 2023
  
  Our responses to the individual comments by the reviewer are in the attached file Response-to-Reviewer-2.pdf. In response to the reviewer's comment that the contribution of our work is unclear, we strive in the revised manuscript, to clarify the main contribution of our work, which was identified explicitly in lines 47 -- 51 of the original manuscript. The contribution is about both a new analytical solution and ``the implications of a poorly recognized phenomenon,'' namely transient stream drawdown and channel storage. Our work differs from that of published works we review in the introduction specifically in our treatment of the boundary condition at the stream-aquifer interface. Whereas previous analytical works use a fixed-stream stage condition, we allow stream stage to respond to groundwater pumping; whereas all models that use the fixed-stage condition cannot predict non-zero transient stream drawdown response to pumping, our model predicts this in addition to stream depletion. To reiterate, these models are based on the fixed-stage assumption and as such, stream drawdown is by definition zero. We have developed an analytical solution that overcomes this limitation. We develop the solution by first modifying the boundary condition at the stream-aquifer interface using a mass balance condition regulated by a finite stream channel storage coefficient. We demonstrate that fixed-stage models coincide with our model in the limit as the stream channel storage coefficient approaches infinity, $C_r \rightarrow \infty$. Additionally, we also state that MODFLOW also relies on the same fixed-stage condition used by the earlier analytical models. To make our contribution clearer here in the response, we include a Figure in the attached response where we plot the predicted behavior of the old models (a) separate from our new model (b). The model of fox et al. (2002) is the general representation of all fixed-stage models for confined aquifer flow including those that account for aquifer leakage such as the model of Zlotnik (2004). The final contribution of our work is the application of the model to field measurements of transient stream drawdown, something that has never been done in the hydrogeology literature, where the focus is on only aquifer drawdown analysis. Detailed responses to individual comments can be found in the attached file as indicated earlier.
  
  Citation: https://doi.org/10.5194/hess-2022-353-AC2
- AC3: 'Reply on RC6', Bwalya Malama, 23 Jan 2023
  
  Please find attached here the file with the responses to all the individual comments of reviewer 2. In this document, our responses are in itallic and the proposed changes to the manuscript are in bold-itallic.
  
  Citation: https://doi.org/10.5194/hess-2022-353-AC3

Bwalya Malama, Ying-Fan Lin, Hwa-Lung Yu, Hua-Ting Tseng, and Sam Greene

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Short summary

We identify a limitation in existing models for stream depletion due to groundwater pumping. Existing models assume the stream acts as infinite source during groundwater pumping by imposing a constant head condition at a stream. We overcome this by adopting a mass conservation law and a stream storage coefficient to model stream drawdown. The model is applied to field drawdowns of stream and aquifer. The model fit the data well and give hydraulic estimates that compare well to similar sediment.


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