Analytical Solution of the One-Dimensional Transport of Ionic Contaminants in Porous Media with Time-Varying Velocity

Round 1

Reviewer 1 Report

1.- I believe that the increase in the permeability coefficient and its relationship with the displacement of contaminants is not sufficiently substantiated and justified in the text. Perhaps these changes can respond satisfactorily for some clays such as bentonite, but they cannot be generalized. These limitations of the research should be well expressed and highlighted in the text.

2.- The article presents a significant deficit of bibliographic citations of classic works, such as C. W. Fetter and others....

Others:

- In line 81 there is a grammar error

- In line 422 there is a grammar error

Author Response

Comment 1:

I believe that the increase in the permeability coefficient and its relationship with the displacement of contaminants is not sufficiently substantiated and justified in the text. Perhaps these changes can respond satisfactorily for some clays such as bentonite, but they cannot be generalized. These limitations of the research should be well expressed and highlighted in the text.

Reply 1:

Thanks for suggestion.

Yes, the conditions assumed in this paper have certain limitations, and the analytical solutions are only calculated for the realistic case of variation in permeability coefficients under special conditions for certain barriers (e.g. bentonite barriers, attapulgite barriers), and are not applicable to all cases. I have added more detail to the limitations of the solutions in this paper. Refer to line 96-102

The relationship between the permeability coefficient and the equations set out in this paper I have included in the article. Refer to line 125-140

Comment 2:

The article presents a significant deficit of bibliographic citations of classic works, such as C. W. Fetter and others...

Reply 2:

Thanks for suggestion. I have included references to more authoritative literature in the relevant fields consulted. Refer to line 31-36 and 73-95

1. Fetter, C.W.; Boving, T.; Kreamer, D. Contaminant hydrogeology; Waveland Press: 2017.
2. Barry, D.; Sposito, G. Analytical solution of a convection‐dispersion model with time‐dependent transport coefficients. Water Resources Research 1989, 25, 2407-2416.
3. Zamani, K.; Bombardelli, F.A. Analytical solutions of nonlinear and variable-parameter transport equations for verification of numerical solvers. Environmental Fluid Mechanics 2014, 14, 711-742.
4. Guerrero, J.P.; Skaggs, T. Analytical solution for one-dimensional advection–dispersion transport equation with distance-dependent coefficients. Journal of Hydrology 2010, 390, 57-65.
5. Basha, H.; El‐Habel, F. Analytical solution of the one‐dimensional time‐dependent transport equation. Water resources research 1993, 29, 3209-3214.
6. Singh, M.K.; Ahamad, S.; Singh, V.P. Analytical solution for one-dimensional solute dispersion with time-dependent source concentration along uniform groundwater flow in a homogeneous porous formation. Journal of engineering mechanics 2012, 138, 1045-1056.
7. Yates, S. An analytical solution for one‐dimensional transport in heterogeneous porous media. Water Resources Research 1990, 26, 2331-2338.
8. 8. Kumar, A.; Jaiswal, D.K.; Kumar, N. Analytical solutions to one-dimensional advection–diffusion equation with variable coefficients in semi-infinite media. Journal of hydrology 2010, 380, 330-337.

Comment 3:

In line 81 there is a grammar error

In line 422 there is a grammar error

Reply 3:

Thanks for suggestion.

The grammatical errors in the article have been rewritten. Refer to line 108 and 440

Author Response File: Author Response.docx

Reviewer 2 Report

water-2303453 “Analytical Solution of One-Dimensional Transport of Ionic Contaminants in Porous Media with Time-Varying Velocity” Zeng Xing, Gao Tong, He Zijian and Xie Linhui

This is a very heavy article, dense with math and laboured text. It is quite interesting and does illustrate some solute transport behaviour that should be taken into account when designing phytocaps and soil barriers. The English expression is highly variable, with many capitalised letters mid-sentence, and rogue fragments of text from the Instructions to Authors.

·       In S2.1 on line 81 the 4^th assumption is repeated and should be deleted. The automatic conversion of a number in parentheses to a special character inside a circle should be reversed.

·       Equation 3 on line 94 is mis-numbered and should be Equation 2.

·       I don’t know how the authors are setting their equations, but good software will make the size of integrals and parentheses relevant to the internal fractions and coefficients in the equation.

·       The usual nomenclature for Laplace and inverse Laplace transforms uses a cursive “L”, and curly braces “{” and “}” to enclose the argument.

·       The language describing D, Do and Dd are confusing between lines 102 and 111. At line 102, “D[t] is the diffusion coefficient” that is simplified later “Do = a uo is the hydrodynamic dispersion coefficient assuming molecular diffusion is zero in this work”.

·       It is unclear what the paragraph on lines 130-134 means, or where sentences start and end.

·       It would be more consistent if the boundary conditions (6), (7) and (8) were expressed the same as in Equation (12).

·       You can leave out Equation (13), leaving in the note about “p” (or rho? on line 141) after it. Can the authors also define uppercase “P” which first appears at the end of Equation (17) then many times later.

·       The first 3 sentences of S3 can be deleted.

·       The first sentence on line 206 has no context within Figure 1. It can be seen in Figure 2 that the 10% line is shown, and this is the assumed trigger, but this information needs to go in the description for Figure 2.

·       The legend or the caption for Figure 2 must indicate that the grey lines are the results of the present work while those including molecular diffusion are red. It is not common to show different times in this type of graph, but rather show the difference at matching times. Set of lines at 1500 and 2000 days would be shown, and the shift would be clear. In the text the authors would indicate the difference in time to reach C/Co = 10% and claim it is small at 3.9% difference given the velocity assumed.

·       Wording for the three groups of simulations on lines 227-232 should be simplified to, for example, “(1) influence coefficient a varies, maximum velocity ve varies, and time to maximum velocity te is constant”. It should also be noted that uo is constant for all simulations and does not need to appear in Table 2.

·       The first paragraph below Figure 3 lines 239 to 246 is a retelling of the parameters listed in Table 2. Replace with “Figure three shows the velocity over time for the simulations in Group 1.” then join with the second paragraph “The breakthrough time when C/Co=0.1 is evaluated for each case.”

·       Please remove the tautological sub-phrase “the maximum velocity increases with the increase of velocity increase coefficient” from line 257.

·       Authors should indicate that Figure 7 is the difference in concentration between cases in Group 1 and constant flow at minimum rate uo. However, I personally see little value in Figure 7 and it’s description given the information for Figure 6.

·       All the suggestions for S4.1 apply equally to analysis of the other two groups in S4.2 and S4.3.

·       I think on line 432 the authors mean “The percentage of attapulgite in the mix was set at 10%, 30% and 60%.” The column “ratio%” in Table 3 supports this.

·       The authors have used “E-04” notation in Tables 1 and 2 but changed to “x10^-4” in Table 3.  The Instructions to Authors does not specify which is more acceptable, but the authors should be consistent in the Tables and text, e.g., line 368 has “ve=1.44E-07m/s”, line 427 has “ks in the range 0 ~ 4x10^-9m/s”, and line 444 is back to “u1=2.75E-08m/s”.

·       Line 490 has swapped the third number and “respectively”.

·       Is the line for 50mmol/L constant rate solution missing from Figure 18, or is it very close to zero and cannot be seen after 14-years?

·       The first sentence on line 511 is misleading. The analytic solution is not one-dimensional because one term is neglected, it is only simplified. It might be better expressed as “A one-dimensional analytic solution of the transport equation was obtained with the simplification that molecular diffusion was ignored.”

There are a lot of editorial style changes suggested here rather than any large wholesale changes to the article. I noticed a lot of geotechnical, geophysics, soil and rock physics journals in the references and these may be alternate publication targets for this or work based on it.

Author Response

Thank you very much, you are a very responsible reviewer and I have the utmost respect for you. I have responded to all the comments you have made and have uploaded a word file for your review.

Author Response File: Author Response.docx

Round 2

Reviewer 1 Report

I think the paper has been improved to be published