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the Creative Commons Attribution 4.0 License.
Effects of the dynamic effective porosity on watertable fluctuations and seawater intrusion in coastal unconfined aquifers
Abstract. Watertable fluctuations and seawater intrusion are characteristic features of coastal unconfined aquifers. The dynamic effective porosity due to watertable fluctuations is analyzed and then a modified (empirical) expression is proposed for the dynamic effective porosity based on a dimensionless parameter related to the watertable fluctuation frequency. After validation with both experimental data and numerical simulations, the new expression is implemented in existing Boussinesq equations and a numerical model, allowing for examination of the effects of the dynamic effective porosity on watertable fluctuations and seawater intrusion in coastal unconfined aquifers, respectively. Results show that the Boussinesq equation accounting for the vertical flow in the saturated zone and dynamic effective porosity can accurately predict experimental dispersion relations (that all existing theories fail to predict), highlighting the importance of the dynamic effective porosity in modeling watertable fluctuations in coastal unconfined aquifers. This in turn confirms the utility of the realvalued expression of the dynamic effective porosity. An outcome is that the phase lag between the total moisture (above the watertable) and watertable height measured in laboratory experiments using vertical soil columns (1D systems) can be ignored when predicting watertable fluctuations in coastal unconfined aquifers (2D systems). A dynamic effective porosity that is, by comparison, smaller than the soil porosity leads to a reduction in vertical water exchange between the saturated and vadose zones and hence watertable waves can propagate further landward. The dynamic effective porosity further plays a critical role in simulations of seawater intrusion, since it predicts a more landward seawaterfreshwater interface and a higher position of the upper saline plume.
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 AC1: 'Comment on hess2021634', Zhaoyang Luo, 26 Feb 2022

RC1: 'Comment on hess2021634', Anonymous Referee #1, 18 Mar 2022
Firstly, I note that this is the third time I am reviewing this work which has been previously submited to and rejected from two other journal publications. My key concerns remain (see detailed comments below) and consequently I am unable to recommend publication of the submitted paper.
Abstract
li 27: "After validation with 1D experimental data and numerical simulations ..."
li 3133: "accounting for vertical flow"  the equations used are the 2nd order approximation correction for vertical flows, there is an infinite order expression that has been excluded from the analysis (cf Nielsen et al, 1997). It is anticipated that the later (more accurate) correction vertical flow effects will not yield such favourable results.
li 3538: "the phase lag can be ignored" The observations in Table 6 of Shoustari et al (2017) indicates that, for a 2DV propagating groundwater wave system, the phase lag in moisture content fluctuations is much greater than that for the watertable fluctuations. In addition, the dynamic effective porosity presented by the authors (eq 7) is derived based on a 1D sand column system so this is somewhat contradictory. I note that a reviewer in a previpous submission was also critical of this assumption.
li3841: this has long been known.
Highlights
1. the "modified expression" is the same as Pozdniakov et al. (2019). Equation 7a is the same as Pozdniakov et al. (2019), it has just been written using different notation. If you insert the authors' equations 9 and 7b into 7a you get the same equation as when you insert Pozdniakov et al's eq 12 into eq 15. The author's fitting parameter a being equivalent to 2.pi.f(l,m)/tau0 in the notation of Pozdniakov et al. (2019). Therefore the correct description of what has been done is "Here we introduce the existing formulation of Pozdniakov et al. (2019) using different notation ... "
2. Only for the 2nd order approximation, the infiniteorder correction for vertical flow effects has been omitted by the authors
Main Body
li 118119: review language. It is known and agreed upon that the unsaturated zone does affect water table fluctuations. The dynamic efective porosity is a way of paramaterising this affect. Therefore it follows that the dynamic effective porosity will affect water table fluctuations. It is the extent to which, and our ability to correctly quantify the dynamic effective porosity that remains unclear.
li 120121: I agree that using a complex effective porosity in a practical application (e.g. numerical model) is not possible but note that Cartwright et al (2006) overcame this by using the absolute value of the complex number which led to reasonable outcomes for practical use.
li 122124: The influence of water table fluctuations on salt water intrusion is significant at long time scales (e.g. tidal  Robinson et al, storm surges  Cartwright et al). At these longer time scales, the influence of the unsaturated zone (and hence the dynamic effective porosity) on water table fluctuations becomes negligible (refer to all available dispersion relation theory and even the authors results showing that their nt/ne ~ 1 for small values of Tau_w).
li 1531161: The authors acknowledge that this is a 2nd order correction for vertical flow effects, however Nielsen et al (1997) also provide an infiniteorder solution which should be included in the analysis. I note that it is included in a supplementary figure S2 but it should be added to Figure 4 with the authors nt expression in place of ne. I anticipated that this will yield a much poorer comparison with the data and will highlight the somewhat fortuitous outcome that the 2nd order solution provides a reasonable comparison.
li 181193: It is not clear to me how the equation is modified from the existing. Whilst the authors' modified dynamic effective porosity has been derived differntly the result is the same as Pozdniakov et al. (2019) albeit with a differnt notation. If you insert the authors' equations 9 and 7b into 7a you get the same equation as when you insert Pozdniakov et al's eq 12 into eq 15. The authors' fitting parameter "a" being equivalent to 2.pi.f(l,m)/tau0 in the notation of Pozdniakov et al. (2019).
li 199: are they wetting or drying curves?
li 207209: I would argue that the authors' approach is also approximate because, ultimately at the end their eq 7 is semiemprical and requires fitting to data.
li 255257: In figure 1 there is a clear departure between the curve fit and the data as nwHpsi/K increases (and nt/ne decreases) indicating poor performance where the influence of the unstaurated zone on water table fluctuations is large (ie small nt).
li 258262: the limited ability of numerical solutions to Richards' equation to reproduce the lab data when neglecting hysterisis is discussed in depth in Cartwright et al (2005)
li 288: Nielsen and Perrochet (2000a,b) first proposed the complex effective porosity concept
li 290292: It is important to clarify that, regardless of whether the system is 1D or 2DV, water table fluctuations are induced by external forcing at a boundary (ocean tides, wave, atmospheric pressure ...). Moisture content may, or may not, play a role in influencing the nature and extent of the response. Also note that the phase lag between moisture content fluctuations in the unsaturated and those in the water table, is also present in 2D systems (Shoushari et al, 2017).
li 298: cite the source of the experiments
li 342: as per my earlier comment, I anticpate that if the infiniteorder solution was used rather than the 2nd order one the comparison will be much worse. Please add these curves to your Figure 4.
li 347349: I disagree. As per my earlier comments, it is rather fortuitous that the 2nd order solution seems to do OK.
li 353356, Fig 5 and 6: For a more rigorous comparison, rather than time series, present both the amplitude and phase profiles (ie A vs x and phase lag vs x)
sec 3.3: It seems to me that the approach adopted to examine the influence of the dynamic effective porsoity (ie the link between saturated and unsaturated zones) on saltwater intrusion is fundamentally flawed. As described, SUTRA solves the variablysaturated flow equations so therefore the moisture exchange between saturated and unsaturated zones is implicitly accounted for in the governing equations. To then replace the storage coefficient with a reduced dynamic effective porosity does not make sense phsyically as you are essentially accounting for the exchange twice.
References
Cartwright, N., O. Z. Jessen and P. Nielsen (2006). "Application of a coupled groundsurface water flow model to simulate periodic groundwater flow influenced by a sloping boundary, capillarity and vertical flows." Environmental Modelling & Software 21(6): 770778.
Shoushtari, S. M. H. J., N. Cartwright, P. Perrochet and P. Nielsen (2017). "Twodimensional vertical moisturepressure dynamics above groundwater waves: Sand flume experiments and modelling." Journal of Hydrology 544: 467478.
Nielsen, P. and P. Perrochet (2000). "Watertable dynamics under capillary fringes: experiments and modelling." Advances in Water Resources 23(5): 503515.
Nielsen, P. and P. Perrochet (2000). "ERRATA: Watertable dynamics under capillary fringes: experiments and modelling [Advances in Water Resources 23 (2000) 503515]." Advances in Water Resources 23(8): 907908.
Pozdniakov, S. P., et al. (2019). "An Approximate Model for Predicting the Specific Yield Under Periodic Water Table Oscillations." Water Resources Research 55(7): 61856197.Citation: https://doi.org/10.5194/hess2021634RC1  AC2: 'Reply on RC1', Zhaoyang Luo, 18 Mar 2022

RC2: 'Comment on hess2021634', Anonymous Referee #2, 05 Jun 2022
The study is keyed to proposing an empirical expression to evaluate a dynamic effective porosity and assess its impact on the quantification of watertable fluctuations and seawater intrusion in coastal aquifers. After studying the work, I am afraid I am not in a position to recommend publication at this stage. In addition to having some doubts about the possibility that this study constitutes more than an incremental advancement, at least the way it is framed and the way the Authors present it, I do have two major concerns. When combined, these seriously question the validity of the approach and of the key results of this work.
 The Authors observe that considering vertical flow effects making use of (i) an approximated (at secondorder) formulation and (ii) a dynamic effective porosity leads to an accurate prediction of experimental dispersion relations of watertable waves. This result is in contrast with a previous analysis according to which it is shown that an infiniteorder expression (that includes the secondorder approximation presented in this study) cannot predict these results in an accurate way. In order to resolve this issue the authors should compare their results as well as the infiniteorder expression against outcomes of the Richards' equation (which accounts for vertical flow under variably saturated flow settings). It can also be noted that, in addition to the theoretical elements described above, the physical basis according to which an approximate solution should provide improved results as opposed to its exact counterpart is not clear.
 I found the approach adopted in modeling the saltwater intrusion not convincingly supported. To the extent of my knowledge, the code adopted (SUTRA) already solves variable saturated (saturatedunsaturated ) flow settings. Therefore, while a model parameter such as a dynamic effective porosity could be considered and included to account for the effects of the unsaturated zone on water table dynamics when these effects have not yet been considered (e.g., when using saturated models such as Eq. 12 of the manuscript), I strongly doubt about its use and physical implications when solving the Richards’ equation. The latter already accounts for the unsaturated zone and its impact on subsurface flow dynamics. As such, I find the approach to be inconsistent and not substantiated by robust physical bases.
Citation: https://doi.org/10.5194/hess2021634RC2 
AC3: 'Reply on RC2', Zhaoyang Luo, 07 Jun 2022
The comment was uploaded in the form of a supplement: https://hess.copernicus.org/preprints/hess2021634/hess2021634AC3supplement.pdf
Status: closed
 AC1: 'Comment on hess2021634', Zhaoyang Luo, 26 Feb 2022

RC1: 'Comment on hess2021634', Anonymous Referee #1, 18 Mar 2022
Firstly, I note that this is the third time I am reviewing this work which has been previously submited to and rejected from two other journal publications. My key concerns remain (see detailed comments below) and consequently I am unable to recommend publication of the submitted paper.
Abstract
li 27: "After validation with 1D experimental data and numerical simulations ..."
li 3133: "accounting for vertical flow"  the equations used are the 2nd order approximation correction for vertical flows, there is an infinite order expression that has been excluded from the analysis (cf Nielsen et al, 1997). It is anticipated that the later (more accurate) correction vertical flow effects will not yield such favourable results.
li 3538: "the phase lag can be ignored" The observations in Table 6 of Shoustari et al (2017) indicates that, for a 2DV propagating groundwater wave system, the phase lag in moisture content fluctuations is much greater than that for the watertable fluctuations. In addition, the dynamic effective porosity presented by the authors (eq 7) is derived based on a 1D sand column system so this is somewhat contradictory. I note that a reviewer in a previpous submission was also critical of this assumption.
li3841: this has long been known.
Highlights
1. the "modified expression" is the same as Pozdniakov et al. (2019). Equation 7a is the same as Pozdniakov et al. (2019), it has just been written using different notation. If you insert the authors' equations 9 and 7b into 7a you get the same equation as when you insert Pozdniakov et al's eq 12 into eq 15. The author's fitting parameter a being equivalent to 2.pi.f(l,m)/tau0 in the notation of Pozdniakov et al. (2019). Therefore the correct description of what has been done is "Here we introduce the existing formulation of Pozdniakov et al. (2019) using different notation ... "
2. Only for the 2nd order approximation, the infiniteorder correction for vertical flow effects has been omitted by the authors
Main Body
li 118119: review language. It is known and agreed upon that the unsaturated zone does affect water table fluctuations. The dynamic efective porosity is a way of paramaterising this affect. Therefore it follows that the dynamic effective porosity will affect water table fluctuations. It is the extent to which, and our ability to correctly quantify the dynamic effective porosity that remains unclear.
li 120121: I agree that using a complex effective porosity in a practical application (e.g. numerical model) is not possible but note that Cartwright et al (2006) overcame this by using the absolute value of the complex number which led to reasonable outcomes for practical use.
li 122124: The influence of water table fluctuations on salt water intrusion is significant at long time scales (e.g. tidal  Robinson et al, storm surges  Cartwright et al). At these longer time scales, the influence of the unsaturated zone (and hence the dynamic effective porosity) on water table fluctuations becomes negligible (refer to all available dispersion relation theory and even the authors results showing that their nt/ne ~ 1 for small values of Tau_w).
li 1531161: The authors acknowledge that this is a 2nd order correction for vertical flow effects, however Nielsen et al (1997) also provide an infiniteorder solution which should be included in the analysis. I note that it is included in a supplementary figure S2 but it should be added to Figure 4 with the authors nt expression in place of ne. I anticipated that this will yield a much poorer comparison with the data and will highlight the somewhat fortuitous outcome that the 2nd order solution provides a reasonable comparison.
li 181193: It is not clear to me how the equation is modified from the existing. Whilst the authors' modified dynamic effective porosity has been derived differntly the result is the same as Pozdniakov et al. (2019) albeit with a differnt notation. If you insert the authors' equations 9 and 7b into 7a you get the same equation as when you insert Pozdniakov et al's eq 12 into eq 15. The authors' fitting parameter "a" being equivalent to 2.pi.f(l,m)/tau0 in the notation of Pozdniakov et al. (2019).
li 199: are they wetting or drying curves?
li 207209: I would argue that the authors' approach is also approximate because, ultimately at the end their eq 7 is semiemprical and requires fitting to data.
li 255257: In figure 1 there is a clear departure between the curve fit and the data as nwHpsi/K increases (and nt/ne decreases) indicating poor performance where the influence of the unstaurated zone on water table fluctuations is large (ie small nt).
li 258262: the limited ability of numerical solutions to Richards' equation to reproduce the lab data when neglecting hysterisis is discussed in depth in Cartwright et al (2005)
li 288: Nielsen and Perrochet (2000a,b) first proposed the complex effective porosity concept
li 290292: It is important to clarify that, regardless of whether the system is 1D or 2DV, water table fluctuations are induced by external forcing at a boundary (ocean tides, wave, atmospheric pressure ...). Moisture content may, or may not, play a role in influencing the nature and extent of the response. Also note that the phase lag between moisture content fluctuations in the unsaturated and those in the water table, is also present in 2D systems (Shoushari et al, 2017).
li 298: cite the source of the experiments
li 342: as per my earlier comment, I anticpate that if the infiniteorder solution was used rather than the 2nd order one the comparison will be much worse. Please add these curves to your Figure 4.
li 347349: I disagree. As per my earlier comments, it is rather fortuitous that the 2nd order solution seems to do OK.
li 353356, Fig 5 and 6: For a more rigorous comparison, rather than time series, present both the amplitude and phase profiles (ie A vs x and phase lag vs x)
sec 3.3: It seems to me that the approach adopted to examine the influence of the dynamic effective porsoity (ie the link between saturated and unsaturated zones) on saltwater intrusion is fundamentally flawed. As described, SUTRA solves the variablysaturated flow equations so therefore the moisture exchange between saturated and unsaturated zones is implicitly accounted for in the governing equations. To then replace the storage coefficient with a reduced dynamic effective porosity does not make sense phsyically as you are essentially accounting for the exchange twice.
References
Cartwright, N., O. Z. Jessen and P. Nielsen (2006). "Application of a coupled groundsurface water flow model to simulate periodic groundwater flow influenced by a sloping boundary, capillarity and vertical flows." Environmental Modelling & Software 21(6): 770778.
Shoushtari, S. M. H. J., N. Cartwright, P. Perrochet and P. Nielsen (2017). "Twodimensional vertical moisturepressure dynamics above groundwater waves: Sand flume experiments and modelling." Journal of Hydrology 544: 467478.
Nielsen, P. and P. Perrochet (2000). "Watertable dynamics under capillary fringes: experiments and modelling." Advances in Water Resources 23(5): 503515.
Nielsen, P. and P. Perrochet (2000). "ERRATA: Watertable dynamics under capillary fringes: experiments and modelling [Advances in Water Resources 23 (2000) 503515]." Advances in Water Resources 23(8): 907908.
Pozdniakov, S. P., et al. (2019). "An Approximate Model for Predicting the Specific Yield Under Periodic Water Table Oscillations." Water Resources Research 55(7): 61856197.Citation: https://doi.org/10.5194/hess2021634RC1  AC2: 'Reply on RC1', Zhaoyang Luo, 18 Mar 2022

RC2: 'Comment on hess2021634', Anonymous Referee #2, 05 Jun 2022
The study is keyed to proposing an empirical expression to evaluate a dynamic effective porosity and assess its impact on the quantification of watertable fluctuations and seawater intrusion in coastal aquifers. After studying the work, I am afraid I am not in a position to recommend publication at this stage. In addition to having some doubts about the possibility that this study constitutes more than an incremental advancement, at least the way it is framed and the way the Authors present it, I do have two major concerns. When combined, these seriously question the validity of the approach and of the key results of this work.
 The Authors observe that considering vertical flow effects making use of (i) an approximated (at secondorder) formulation and (ii) a dynamic effective porosity leads to an accurate prediction of experimental dispersion relations of watertable waves. This result is in contrast with a previous analysis according to which it is shown that an infiniteorder expression (that includes the secondorder approximation presented in this study) cannot predict these results in an accurate way. In order to resolve this issue the authors should compare their results as well as the infiniteorder expression against outcomes of the Richards' equation (which accounts for vertical flow under variably saturated flow settings). It can also be noted that, in addition to the theoretical elements described above, the physical basis according to which an approximate solution should provide improved results as opposed to its exact counterpart is not clear.
 I found the approach adopted in modeling the saltwater intrusion not convincingly supported. To the extent of my knowledge, the code adopted (SUTRA) already solves variable saturated (saturatedunsaturated ) flow settings. Therefore, while a model parameter such as a dynamic effective porosity could be considered and included to account for the effects of the unsaturated zone on water table dynamics when these effects have not yet been considered (e.g., when using saturated models such as Eq. 12 of the manuscript), I strongly doubt about its use and physical implications when solving the Richards’ equation. The latter already accounts for the unsaturated zone and its impact on subsurface flow dynamics. As such, I find the approach to be inconsistent and not substantiated by robust physical bases.
Citation: https://doi.org/10.5194/hess2021634RC2 
AC3: 'Reply on RC2', Zhaoyang Luo, 07 Jun 2022
The comment was uploaded in the form of a supplement: https://hess.copernicus.org/preprints/hess2021634/hess2021634AC3supplement.pdf
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