A Semi-Analytical Model of Contaminant Transport in Barrier Systems with Arbitrary Numbers of Layers

Round 1

Reviewer 1 Report

I suggest the following changes and improvements:

1.     The authors need to improve the Abstract section considering the important findings and underscore the scientific value added to your manuscript in your abstract.

2.     The current structure of the introduction is not well written. The authors need to improve. Additionally, the last part needs to be revised considering the main theme/objectives and findings of the research.

3.     Please avoid too short paragraphs throughout the manuscript.

4.     The novelty and research significance of this work should be stated clearly in the introduction section.

5.     Figure 2 and 11 legend should provide enough information.

6.     The author needs to add solid future direction and suggestions in the conclusion section.

7.     The number of references needs to be extended to support the data/statement.

Author Response

Comments and Suggestions for Authors

I suggest the following changes and improvements:

1. The authors need to improve the Abstract section considering the important findings and underscore the scientific value added to your manuscript in your abstract.

The abstract was reformulated to summarize some aspects that were relevant in the simulations carried out, for example:

• Line 17-19: The analyses showed the importance of choosing the most appropriate construction system based on the location and availability of materials;
• Line 19-20: Considering toluene contamination, geomembranes with molecular diffusion coefficient (DGM) greater than 10⁻¹³ m² s⁻¹ exhibited similar efficiency when compared with compacted clay liners (60 cm thick);
• Line 20-22: Results showed that the liner system may have the same efficiency changing soil liner (60 cm thick) for a geosynthetic clay liner (1 cm thick).

2. The current structure of the introduction is not well written. The authors need to improve. Additionally, the last part needs to be revised considering the main theme/objectives and findings of the research.

The introduction was reformulated with new references (item 7), we think the structure was improved showing the importance of the theme. For example:

• Line 36-38: Derived from rainfall, surface drainage, and decomposition of MSW, landfill leachate contains high concentrations of dissolved organic matter (DOM), inorganic salts, heavy metals, and xenobiotic organic compounds;
• Line 38-41: If the landfill is not well lined with impermeable material or the lining fails due to geological conditions, landfill leachate will enter the groundwater as a contaminant plume, which is one of the most significant environmental problems of landfills;
• Line 45-48: In all cases, the materials used must have certain technical characteristics, such as durability, adequate resistance and low hydraulic conductivity. Hydraulic conductivity (k) de-scribes the capacity of a porous medium to transmit a given liquid and is a function of both the medium and the liquid;
• Line 60-64: CCLs have advantages such as good damping capacity and cheapness, as well as disadvantages such as high swelling and shrinkage [16]. According to the United States Environmental Protection Agency (US EPA), the minimum thickness of CCLs should be 60 cm and the hydraulic conductivity of compacted liners should not be more than 1x10^-7 cm/s;
• Line 65-70: A GCL is a relatively thin layer of processed clay (typically bentonite) either bonded to a geomembrane or fixed between two sheets of geotextile. A geotextile is a woven or nonwoven sheet material less impervious to liquid than a geomembrane, but more resistant to penetration damage. GCLs are 5 to 10 mm thick and their hydraulic conductivity typically ranges from 5x10^-12 to 5x10^-11 m/s and the reason for such low hydraulic conductivity is because the bentonite;

 Furthermore, the last paragraphs were modified to (Line 96-125):

(…)

The semi analytical model in this research may serve as a fundamental design tool that helps evaluate the potential contamination of groundwater and environmental dam-age caused to leachate in sanitary landfills. The model described in this paper is not only semi-analytical, but it also incorporates the complexity of multi-layered systems, recognizing the importance of thin layers as geomembranes, the influence of partition coefficients, and the criticality of mass flow calculations. As shown a long the research, the model agrees well with other models in the literature, which includes those by Chen et al. [36] and Rowe and Booker [37].

In the simulations were predicted a low-permeability material overlaid with a geomembrane because in the installation of the geomembrane is recommended to prepare the foundation area. The prepared foundation must be smooth and free of projections pre-venting damage to the liner, thus, even with the execution of the geomembrane is common to have a CCL or SL to prevent any damage. Moreover, in the simulations considering the liner system with GM and molecular diffusion coefficient (DGM) greater than 10−13 m2 s−1 exhibited similar efficiency when compared with CCL (60 cm thick), for toluene contamination. Otherwise, DGM with values less than 10-14 m2 s−1 proved to be an efficient liner system containing the toluene front. Hence, obtaining DGM with adequate precision is necessary to achieve the non-pollution of soil and groundwater. Considering a real case scenario in a sanitary landfill in Brazil, the simulations for the current liner system of the structure showed that the benzene takes 50 years to reach the bottom of the liner. However, the life span of the landfill is just 13 years, showing the efficiency of the liner system and the applicability of the model.

Given the evolving complexities and challenges in environmental geotechnics, particularly concerning sanitary landfills, the need for more refined and versatile modeling techniques is urgent. Thus, the proposed model provides a more integrated understanding and prediction of any contaminant (dissolved organic matter, inorganic salts, heavy metals, and xenobiotic organic compounds) behavior as an essential tool for contemporary geotechnical engineers. This original perspective aims to elevate the standards of landfill liner design, making it more adaptable and effective in safeguarding our environment.

3. Please avoid too short paragraphs throughout the manuscript.

The paper was reviewed and the short paragraphs were avoided. Furthermore, it is important to mention that the article was reviewed by an expert (Language Editing Services, MDPI) with knowledge in the area and checked the structure, clarity of expression and specific terminology on the subject. We attached to this response the document certifying the review.

4. The novelty and research significance of this work should be stated clearly in the introduction section.

New paragraphs were introduced in the introduction topic aiming to make clear the importance of models as the one created in this research, for example:

• Line 30-33: In most countries, sanitary landfill is still the primary approach to the disposal of MSW, which can generate pollution in soil, air, surface water, and groundwater because of the presence of leachate;
• Line 33-35: The contamination of areas around solid urban waste dumps is a global challenge for the maintenance of environmental quality in large urban centers in developing countries;
• Line 36-38: Derived from rainfall, surface drainage, and decomposition of MSW, landfill leachate contains high concentrations of dissolved organic matter (DOM), inorganic salts, heavy metals, and xenobiotic organic compounds;
• Line 38-41: If the landfill is not well lined with impermeable material or the lining fails due to geological conditions, landfill leachate will enter the groundwater as a contaminant plume, which is one of the most significant environmental problems of landfills;
• Line 96-98: The semi analytical model in this research may serve as a fundamental design tool that helps evaluate the potential contamination of groundwater and environmental dam-age caused to leachate in sanitary landfills;
• Line 98-103: The model described in this paper is not only semi-analytical, but it also incorporates the intricacies of multi-layered systems, recognizing the importance of thin layers as geomembranes, the influence of partition coefficients, and the criticality of mass flow calculations. As shown a long the research, the model agrees well with other models in the literature, which includes those by Chen et al. [31] and Rowe and Booker [32].

5. Figure 2 and 11 legend should provide enough information.

• Legend of Figure 2 changed to:

Line 207: Figure 2. Comparison of the semi-analytical model and the numerical model of Chen et al. [21]

• Figure 11 was erased because of a problem with copyright

6. The author needs to add solid future direction and suggestions in the conclusion section.

• The following paragraph was introduced into the conclusion topic:

Line 369-374: Aiming a more comprehensive application, future research should investigate the applicability of the solution comparing with laboratory results analyzing multilayers of materials. Furthermore, for a holistic consideration of the contaminant transport phenomenon, it is recommended the implementation of the material saturation (CCL and SL) and temperature in the model.

7. The number of references needs to be extended to support the data/statement.

The following references were introduced into the text

Line 444-446: Wijekoon, P.; Koliyabandara, P. A.; Cooray, A. T.; Lam, S. S.; Athapattu, B. C. L.; Vithanage, M. (2022). Progress and prospects in mitigation of landfill leachate pollution: Risk, pollution potential, treatment and challenges. Journal of Hazardous Materials, 421, 126627.

Line 451-452: Bernardo, B.; Candeias, C.; Rocha, F. Characterization of the Dynamics of Leachate Contamination Plumes in the Surroundings of the Hulene-BWaste Dump in Maputo, Mozambique. Environments 2022, 9, 19.

Line 453-454: Christensen, T. H.; Kjeldsen, P.; Bjerg, P. L.; Jensen, D. L.;Christensen, J. B.; Baun, A.; Albrechtsen, H. J.; Heron, C.Biogeochemistry of landfill leachate plumes.Appl. Geochem.2001,16(7−8), 659−718

Line 455-456: Jiang, Y.; Li, R.; Yang, Y.; Yu, M.; Xi, B.; Li, M.; Xu, Z.; Gao, S.; Yang, C. Migration and evolution of dissolved organic matter in landfill leachate-contaminated groundwater plume. Resources Conservation and Recycling 2019, 151, 104463.

Line 460-461: Özçoban, M. S.; Acarer, S.; Tüfekci, N. Effect of solid waste landfill leachate contaminants on hydraulic conductivity of landfill liners. Water Sci Technol. 2022; 85 (5): 1581–1599.

Line 467-468: U.S. EPA. 1989. Requirements for Hazardous Waste Landfill Design, Construction, and Closure. EPA625-4-89-022. Prepared by Daniel, D.E., and P.M. Estornell for Office of Research and Development, Washington, DC. August.

Line 472-473: Uma Shankar, M. & Muthukumar, M. 2017 Comprehensive review of geosynthetic clay liner and compacted clay liner. IOP Conference Series: Materials Science and Engineering 263 (3).

Line 474-475: Visentin, C., Zanella, P., Kronhardt, B. K., da Silva Trentin, A. W., Braun, A. B. & Thomé, A. 2019 Use of geosynthetic clay liner as a waterproofing barrier in sanitary landfills. Journal of Urban and Environmental Engineering 13 (1), 115–124.

Line 476-477: Giroud, J., Badu-Tweneboah, K. & Soderman, K. L. 1998 Comparison of leachate flow through compacted clay liners and geosynthetic clay liners in landfill liner systems. Geosynthetics International 4, 391–431.

Line 478: U.S. EPA. 2001. Geosynthetic Clay Liners Used in Municipal Solid Waste Landfills. EPA530-F-97-002. Washington, DC. December.

Line 488-489: Ozelim, L.C.S.M.; Paz, Y.P.L.; Cunha, L.S.; Cavalcante, A.L.B. Enhanced landfill’s characterization by using an alternative ana-lytical model for diffusion tests. Environ. Monit. Assess. 2021, 193, 739.

Line 490-491: Xie H.J.; Thomas, H.R.; Chen, Y.M.; Sedighi M.; Tang, X.W. Diffusion of organic contaminants in triple-layer composite liners – an analytical modelling approach. Acta Geotech. 2015;10:255–62.

Line 492-493: Peng, M.-Q.; Feng, S.-J.; Chen, H.-X.; Chen, Z.-L. An analytical solution for organic pollutant diffusion in a triple-layer composite liner considering the coupling influence of thermal diffusion. Computers and Geotechnics. 2021. 137, 104283.

Line 494-495: Qiu, J.; He, Y.; Song, D.; Tong, J. Analytical solution for solute transport in a triple liner under non-isothermal condition. Geo-synthetics International. 2022. 30. 364-381

Line 496-497: Lin, H.; Huang, W.; Wang, L.; Liu, Z. Transport of Organic Contaminants in Composite Vertical Cut-Off Wall with Defective HDPE Geomembrane. Polymers. 2023, 15, 3031.

Line 513-514: Vieira, G.M.D.; Lemos, M.A.C.; Cavalcante, A.L.B.; Casagrande, M.D.T. Effectiveness of Polyvinylidene Fluoride Fibers (PVDF) in the Diffusion and Adsorption Processes of Atrazine in a Sandy Soil. Sustainability. 2023, 15, 11729.

Reviewer 2 Report

1. Overview and general recommendation:

I enjoyed reviewing the manuscript of “A Semi-Analytical Model of Contaminant Transport in Barrier Systems with Arbitrary Numbers of Layers”. The authors proposed a semi-analytical model of contaminant transport, which is valid for the advection–dispersion–reaction equation and compatible with zero- and first-order terms. They verified the model by comparing with other models from the literature, performed parametric analyses under various conditions, and lastly applied this model to a real sanitary landfill in Brazil.

Overall, the paper is well written. I just have few comments. Therefore, I suggest a minor revision.

2. Detailed comments:

(1) Please add line numbers, which can make it easier for reviewers to add comments.

(2) Pages 6-7. You specified the contaminants that these parameters correspond to in Table 2. So which contaminant are the parameters in Table 1 with respect to?

Author Response

(1) Please add line numbers, which can make it easier for reviewers to add comments.

Line numbers were added to the file.

(2) Pages 6-7. You specified the contaminants that these parameters correspond to in Table 2. So which contaminant are the parameters in Table 1 with respect to?

• Tabel 1 is related to toluene contaminant. In lines 196-198 were add the following text:

Line 206-208: “The parameters of the CCL were taken from Foose et al. [21], and the parameters of the SL were obtained from Rowe and Booker [32]. All parameters are listed in Table 1 and are related to toluene contaminant.”

Reviewer 3 Report

This is an interesting work associated with a semi-analytical model of contaminant transport in barrier systems with arbitrary numbers of layers. The manuscript was well organized and I suggested a major revision with the following comments:

1. In introduction (Paragraph 3), please explain why low-permeability soil layers overlaid with a geomembrane was chosen as the way for constructing liners in this study, and what are the advantages of this way compared with other ways? And it is recommended that the introduction be further consolidated to form fewer paragraphs. For example, paragraphs 3-5 can form a complete paragraph.

2. In introduction (Paragraph 6), the authors proposed that several solutions described in the literature disregard the advective term, it is suggested that several literatures should be cited to illustrate this point.

3.  In introduction, first-order kinetics is the most-used model and the first-order term effects on the mobility of organic contaminants. Please explain whether the influence of multiple factors on the kinetic constant is considered in the study. The reference (Proc. Natl. Acad. Sci. U.S.A. 2023, 120, e2304552120) may be helpful for this section and is suggested to be cited.

4. A semi-analytical model is mentioned in the manuscript to describe the convection-dispersion-response equation for pollutants. What are the advantages of this semi-analytic model over numerical models? How does it maintain continuity in the space and time domains?

5. In Figure 2, the expression in the lower left corner should be further confirmed. Please explain the meaning of AL.

6. In 3.2, it is recommended to summarize the conclusions of Figures 3 and Table 2, and explain the reasons for selecting acetone and benzene as contaminants to validate the proposed model.

7. It is well known that pollutants in landfills include organic pollutants and inorganic pollutants. In this study, organic pollutants were used to verify the applicability of the model. Can the model be applied to inorganic pollutants in leachate?

8. Please explain whether this model takes into account the interaction of landfill liners with contaminants, such as adsorption.

Author Response

1. In introduction (Paragraph 3), please explain why low-permeability soil layers overlaid with a geomembrane was chosen as the way for constructing liners in this study, and what are the advantages of this way compared with other ways? And it is recommended that the introduction be further consolidated to form fewer paragraphs. For example, paragraphs 3-5 can form a complete paragraph.

The following paragraph was added to the paper:

• Line 104-108: In the simulations were predicted a low-permeability material overlaid with a geomembrane because in the installation of the geomembrane is recommended to prepare the foundation area. The prepared foundation must be smooth and free of projections pre-venting damage to the liner, thus, even with the execution of the geomembrane is common to have a CCL or SL to prevent any damage

Furthermore, the introduction was revised adding new references and some paragraphs were consolidated as suggested;

2. In introduction (Paragraph 6), the authors proposed that several solutions described in the literature disregard the advective term, it is suggested that several literatures should be cited to illustrate this point.

The following references [25, 26, 27, 28, 29] were added to the article. These articles are related to solutions that disregard the advective term.

• Line 488-489: Ozelim, L.C.S.M.; Paz, Y.P.L.; Cunha, L.S.; Cavalcante, A.L.B. Enhanced landfill’s characterization by using an alternative ana-lytical model for diffusion tests. Environ. Monit. Assess. 2021, 193, 739.
• Line 490-491: Xie H.J.; Thomas, H.R.; Chen, Y.M.; Sedighi M.; Tang, X.W. Diffusion of organic contaminants in triple-layer composite liners – an analytical modelling approach. Acta Geotech. 2015;10:255–62.
• Line 492-493: Peng, M.-Q.; Feng, S.-J.; Chen, H.-X.; Chen, Z.-L. An analytical solution for organic pollutant diffusion in a triple-layer composite liner considering the coupling influence of thermal diffusion. Computers and Geotechnics. 2021. 137, 104283.
• Line 494-495: Qiu, J.; He, Y.; Song, D.; Tong, J. Analytical solution for solute transport in a triple liner under non-isothermal condition. Geo-synthetics International. 2022. 30. 364-381
• Line 496-497: Lin, H.; Huang, W.; Wang, L.; Liu, Z. Transport of Organic Contaminants in Composite Vertical Cut-Off Wall with Defective HDPE Geomembrane. Polymers. 2023, 15, 3031.

3. In introduction, first-order kinetics is the most-used model and the first-order term effects on the mobility of organic contaminants. Please explain whether the influence of multiple factors on the kinetic constant is considered in the study. The reference (Proc. Natl. Acad. Sci. U.S.A. 2023, 120, e2304552120) may be helpful for this section and is suggested to be cited.

The authors appreciate the suggestion in the mentioned article, however, the solution developed in this research can only be applied to first order kinetics with a constant parameter (sorption distribution coefficient, Kd). Therefore, the authors are grateful for the recommendation, but the mentioned paper was not used once it could not be applied to the solution developed.

4. A semi-analytical model is mentioned in the manuscript to describe the convection-dispersion-response equation for pollutants. What are the advantages of this semi-analytic model over numerical models? How does it maintain continuity in the space and time domains?

• A semi-analytical model is a mathematical model that combines elements of both analytical and numerical techniques to solve complex problems, particularly in the fields of science and engineering;
• Analytical/ semi analytical models provide exact, closed-form solutions to problems when they can be expressed mathematically. This precision is particularly valuable in situations where an exact solution is required, such as in engineering or physics.
• Analytical models often require fewer computational resources (e.g., CPU power, memory) than numerical methods, which is advantageous in resource-constrained environments or situations where computational efficiency is critical.
• Analytical models can be computationally efficient, especially for simple and well-defined problems. They don't require iterative numerical methods, which can be time-consuming, especially for complex systems.
• Numerical models often involve discretizing continuous problems into discrete grids or elements, which can introduce errors. Analytical models avoid these discretization errors by working directly with continuous mathematical equations.

5. In Figure 2, the expression in the lower left corner should be further confirmed. Please explain the meaning of AL.

• The Figure 2 was modified. The correct abbreviation is SL

6. In 3.2, it is recommended to summarize the conclusions of Figures 3 and Table 2, and explain the reasons for selecting acetone and benzene as contaminants to validate the proposed model.

• The reasons to selecting acetone and benzene in the validation of the proposed model was added to the paper as following:

Line 220-222: Benzene and acetone were selected to represent hydrophobic and hydrophilic organic compounds, respectively, once these contaminants are usually found in leachate compound [36].

7. It is well known that pollutants in landfills include organic pollutants and inorganic pollutants. In this study, organic pollutants were used to verify the applicability of the model. Can the model be applied to inorganic pollutants in leachate?

• The model can be extended and applied to any type of contaminant since the physical-mathematical phenomena are the same. In the introduction, a paragraph was added about the possibility of applying it to any contaminant present in the leachate.

Line 120-123: Thus, the proposed model provides a more integrated understanding and prediction of any contaminant (dissolved organic matter, inorganic salts, heavy metals, and xenobiotic organic compounds) behavior as an essential tool for contemporary geotechnical engineers

8. Please explain whether this model takes into account the interaction of landfill liners with contaminants, such as adsorption.

Equation 2 represents the retardation factor which takes into account the sorption distribution coefficient. The sorption distribution coefficient Kd is the term that takes into account adsorption-desorption of the interaction between the soil and contaminant. The Kd values ​​are obtained through experiments and the values ​​used here are referenced in other works that were properly cited in this research. Furthermore, in the text was introduced a mention of the effect of sorption in the mobility of solutes:

Line 151-153: In geotechnical engineering, absorption and adsorption (referred to as sorption) can affect the concentration and mobility of solutes [33].

Reviewer 4 Report

I have evaluated the present work from an environmental pollution point of view, but I am not aware of equations derivation. My comments are listed below:

1-      Title “Brasília Sanitary Landfill case study” should be mentioned in the title.

2-      Replace the following words with another one; “hefty” and “underscore” in the abstract, plume, “final cover and into”, “intricacies”, “holistic”,  “crucial”, and “fresh” in the introduction,

3-      Where are the numbering of pages?

4-      Citations should be added to equations that were taken from other studies (equations 1-25, please check).

5-      A schematic diagram to simplify the experimental work should be mentioned in the methods and materials section (describe all experiments in detail).

6-      What is “the harmonic mean”, equation no.5?

7-      How authors could validate his model with other simulations (3. Results, 3.1. Model Validation)? I think the influential factors may be different. Please clarify.

8-      A full description should be added in the caption of Fig. 2.

9-      Details information should be added to Figure No. 11.

10-  Citation should be added to Table 6. The same tendency with all Tables and Figures that involve materials from other research.

11-  What do you mean by “The open-source model”?

Minor editing of the English language required

Author Response

1-      Title “Brasília Sanitary Landfill case study” should be mentioned in the title.

• The main focus of the paper is the development of the semi-analytical solution. The authors believe that all the results generated (validation, parametric analysis and application to the real case) are ways of showing the efficiency of the model. This way, we understood the suggestion, however, we did not add it to the title so that the paper would focus on the semi-analytical solution and not on the application of the real case.

2-      Replace the following words with another one; “hefty” and “underscore” in the abstract, plume, “final cover and into”, “intricacies”, “holistic”,  “crucial”, and “fresh” in the introduction,

• All the words mentioned were changed or erased in the paper.
  • Hefty to substantial;
  • Underscore to indicate;
  • Contaminant plume to contaminant front;
  • final cover to cover layer
  • “into” to “to the surrounding”
  • Intricacies to complexity
  • Holistic to integrated
  • Crucial to essential
  • Fresh to original

3-      Where are the numbering of pages?

Line numbers were added to the file.

4-      Citations should be added to equations that were taken from other studies (equations 1-25, please check).

The citations were added

5-      A schematic diagram to simplify the experimental work should be mentioned in the methods and materials section (describe all experiments in detail).

• No experimental tests were carried out in this research. All data available in this paper were obtained from other researches. The approaches of obtaining these values ​​or the motivation for choosing these values ​​can be consulted in each research individually.

6-      What is “the harmonic mean”, equation no.5?

• The harmonic mean is a type of average used in mathematics and statistics, which is especially useful when dealing with rates, ratios, and inversely proportional quantities. It is defined as the reciprocal of the arithmetic mean of the reciprocals of a set of numbers.

7-      How authors could validate his model with other simulations (3. Results, 3.1. Model Validation)? I think the influential factors may be different. Please clarify.

The developed model can be validated with other contaminants and situations if experimental test data is available. If the parameters necessary to apply the model are not available in the laboratory test, all parameters can be obtained through retro analysis based on the contaminant front data. If data is available simulations can also be carried out (same as simulations carried out in this research). For real cases application of the developed model, it is necessary to obtain all parameters for the specific contaminant, which can be obtained in the literature.

8-      A full description should be added in the caption of Fig. 2.

• Legend of Figure 2 changed to:

Line 207: Figure 2. Comparison of the semi-analytical model and the numerical model of Chen et al. [21]

9-      Details information should be added to Figure No. 11.

• Figure 11 was erased because of a problem with copyright

10-  Citation should be added to Table 6. The same tendency with all Tables and Figures that involve materials from other research.

• Citation was added to all tables

11-  What do you mean by “The open-source model”?

• The open-source model means that the codes generate in this research can be requested for any person, thus, the person by herself can inspect, modify, and enhance the code.

Reviewer 5 Report

Dear Authors,

Thank you for your manuscript submission “A Semi-Analytical Model of Contaminant Transport in Barrier Systems with Arbitrary Numbers of Layers.” The manuscript built a model to stimulate the contaminant transport across various layers in a landfill. Here are my specific comments:

-          Most of the references are out of date. More updated references should be added

-          Abstract should be rewritten as it does not summarize key points from the study. More quantitative results should be indicated in the abstract

-          Why is semi-analytical model used for this study? What are the advantages over other models?

-          There are too many equations in the main context of mathematical modeling section. Removing some equations to appendix should be considered

-          Are there any assumptions for the equations?

-          What factors may influence the accuracy of this model? How do you include these factors in the results?

-          Temperature is one of the most critical factors affecting the transport of contaminant in a landfill. Did you evaluate this factor in your model? If yes, can you please show the results?

-          Are there any limitations for the method in this study?

-          Future research is missing

N/A

Author Response

1. Most of the references are out of date. More updated references should be added

The following references were introduced into the text:

Line 444-446: Wijekoon, P.; Koliyabandara, P. A.; Cooray, A. T.; Lam, S. S.; Athapattu, B. C. L.; Vithanage, M. (2022). Progress and prospects in mitigation of landfill leachate pollution: Risk, pollution potential, treatment and challenges. Journal of Hazardous Materials, 421, 126627.

Line 451-452: Bernardo, B.; Candeias, C.; Rocha, F. Characterization of the Dynamics of Leachate Contamination Plumes in the Surroundings of the Hulene-BWaste Dump in Maputo, Mozambique. Environments 2022, 9, 19.

Line 453-454: Christensen, T. H.; Kjeldsen, P.; Bjerg, P. L.; Jensen, D. L.;Christensen, J. B.; Baun, A.; Albrechtsen, H. J.; Heron, C.Biogeochemistry of landfill leachate plumes.Appl. Geochem.2001,16(7−8), 659−718

Line 455-456: Jiang, Y.; Li, R.; Yang, Y.; Yu, M.; Xi, B.; Li, M.; Xu, Z.; Gao, S.; Yang, C. Migration and evolution of dissolved organic matter in landfill leachate-contaminated groundwater plume. Resources Conservation and Recycling 2019, 151, 104463.

Line 460-461: Özçoban, M. S.; Acarer, S.; Tüfekci, N. Effect of solid waste landfill leachate contaminants on hydraulic conductivity of landfill liners. Water Sci Technol. 2022; 85 (5): 1581–1599.

Line 467-468: U.S. EPA. 1989. Requirements for Hazardous Waste Landfill Design, Construction, and Closure. EPA625-4-89-022. Prepared by Daniel, D.E., and P.M. Estornell for Office of Research and Development, Washington, DC. August.

Line 472-473: Uma Shankar, M. & Muthukumar, M. 2017 Comprehensive review of geosynthetic clay liner and compacted clay liner. IOP Conference Series: Materials Science and Engineering 263 (3).

Line 474-475: Visentin, C., Zanella, P., Kronhardt, B. K., da Silva Trentin, A. W., Braun, A. B. & Thomé, A. 2019 Use of geosynthetic clay liner as a waterproofing barrier in sanitary landfills. Journal of Urban and Environmental Engineering 13 (1), 115–124.

Line 476-477: Giroud, J., Badu-Tweneboah, K. & Soderman, K. L. 1998 Comparison of leachate flow through compacted clay liners and geosynthetic clay liners in landfill liner systems. Geosynthetics International 4, 391–431.

Line 478: U.S. EPA. 2001. Geosynthetic Clay Liners Used in Municipal Solid Waste Landfills. EPA530-F-97-002. Washington, DC. December.

Line 488-489: Ozelim, L.C.S.M.; Paz, Y.P.L.; Cunha, L.S.; Cavalcante, A.L.B. Enhanced landfill’s characterization by using an alternative ana-lytical model for diffusion tests. Environ. Monit. Assess. 2021, 193, 739.

Line 490-491: Xie H.J.; Thomas, H.R.; Chen, Y.M.; Sedighi M.; Tang, X.W. Diffusion of organic contaminants in triple-layer composite liners – an analytical modelling approach. Acta Geotech. 2015;10:255–62.

Line 492-493: Peng, M.-Q.; Feng, S.-J.; Chen, H.-X.; Chen, Z.-L. An analytical solution for organic pollutant diffusion in a triple-layer composite liner considering the coupling influence of thermal diffusion. Computers and Geotechnics. 2021. 137, 104283.

Line 494-495: Qiu, J.; He, Y.; Song, D.; Tong, J. Analytical solution for solute transport in a triple liner under non-isothermal condition. Geo-synthetics International. 2022. 30. 364-381

Line 496-497: Lin, H.; Huang, W.; Wang, L.; Liu, Z. Transport of Organic Contaminants in Composite Vertical Cut-Off Wall with Defective HDPE Geomembrane. Polymers. 2023, 15, 3031.

Line 513-514: Vieira, G.M.D.; Lemos, M.A.C.; Cavalcante, A.L.B.; Casagrande, M.D.T. Effectiveness of Polyvinylidene Fluoride Fibers (PVDF) in the Diffusion and Adsorption Processes of Atrazine in a Sandy Soil. Sustainability. 2023, 15, 11729.

2. Abstract should be rewritten as it does not summarize key points from the study. More quantitative results should be indicated in the abstract

• The following text was introduced to the abstract:

Line 17-22: The analyses showed the importance of choosing the most appropriate construction system based on the location and availability of materials. Considering toluene contamination, GM molecular diffusion coefficient (DGM) greater than 10⁻¹³ m² s⁻¹ exhibited similar efficiency when compared with CCL (60 cm thick). Otherwise, results showed that the liner system may have the same efficiency changing SL (60 cm thick) for a GCL (1 cm thick).

3. Why is semi-analytical model used for this study? What are the advantages over other models?

• Analytical models provide exact, closed-form solutions to problems when they can be expressed mathematically. This precision is particularly valuable in situations where an exact solution is required, such as in engineering or physics.
• Analytical models often require fewer computational resources (e.g., CPU power, memory) than numerical methods, which is advantageous in resource-constrained environments or situations where computational efficiency is critical.
• Analytical models can be computationally efficient, especially for simple and well-defined problems. They don't require iterative numerical methods, which can be time-consuming, especially for complex systems.
• Numerical models often involve discretizing continuous problems into discrete grids or elements, which can introduce errors. Analytical models avoid these discretization errors by working directly with continuous mathematical equations.

4. There are too many equations in the main context of mathematical modeling section. Removing some equations to appendix should be considered

• Some equations were taken from the mathematical modeling item and putted into the appendix section

5. Are there any assumptions for the equations?

The following paragraph was added to the text:

Line 131-135: The assumptions of the model proposed are: i) initial concentrations within layers may vary in each material; ii) top surface concentration is constant or an arbitrary function of time; iii) interface concentrations and mass fluxes are continuous between layers and iv) the bottom boundary condition implies that the concentration derivative is zero (Neumann boundary condition).

6. What factors may influence the accuracy of this model? How do you include these factors in the results?

All parameters influence the validation of the model and its application to real cases. Therefore, it is extremely important that these values ​​are obtained through experimental tests. Furthermore, in the case of contaminant head, it is important to design the system for a typical head found in landfills, since these values ​​can be obtained using piezometers installed in the landfill.

7. Temperature is one of the most critical factors affecting the transport of contaminant in a landfill. Did you evaluate this factor in your model? If yes, can you please show the results?

We appreciate the comment on the importance of considering the effect of temperature on contaminant transport. The authors agree on its importance and how it can affect the transport of contaminants. The equation solved (Equation 1) in this research did not take into account temperature variation either in space or time. Another way to take the effect of temperature into account would be for each parameter individually (Dm, v, Rm, etc.). Neither of the two approaches were used in this research. However, the authors think, beyond the effects of temperature, the use of different materials for liner system in sanitary landfills has a greater impact on the transport of contaminants and this factor was taken into account in the model presented. Furthermore, the authors consider that to solve Equation 1 coupled with thermodynamic effects, numerical modeling would be necessary, in addition to experimental data to validate the model.

The following paragraph was introduced into the conclusion topic:

• Line 369-374: Aiming a more comprehensive application, future research should investigate the applicability of the solution comparing with laboratory results analyzing multilayers of materials. Furthermore, for a holistic consideration of the contaminant transport phenomenon, it is recommended the implementation of the material saturation (CCL and SL) and temperature in the model.

8. Are there any limitations for the method in this study?

• Analytical models also have limitations and are not suitable for all problems. They rely on simplifying assumptions and may not be applicable to highly complex or nonlinear systems. Numerical models, on the other hand, are more versatile in handling complex problems but come with their own advantages and disadvantages.
• In practice, the choice between analytical and numerical models depends on the nature of the problem, the availability of data, computational resources, and the trade-off between accuracy and computational efficiency. Often, a combination of both approaches may be used to exploit the strengths of each method in solving complex, real-world problems.

9. Future research is missing

• The following paragraph was introduced into the conclusion topic:

Line 369-374: Aiming a more comprehensive application, future research should investigate the applicability of the solution comparing with laboratory results analyzing multilayers of materials. Furthermore, for a holistic consideration of the contaminant transport phenomenon, it is recommended the implementation of the material saturation (CCL and SL) and temperature in the model.

Round 2

Reviewer 3 Report

The comments have been carefully revised by authors and the manuscript has been improved. I recommend the publication of the manuscript.

Author Response

Dear Reviewer 1,

I sincerely appreciate your time and effort in reviewing our manuscript. We are grateful for recognizing the improvements made to the document and for recommending its publication. Your positive assessment and support of our work are highly valued. We hope that our study will make a significant contribution to the field of environmental geotechnics.

Sincerely,

Reviewer 4 Report

Based on the author's response “No experimental tests were carried out in this research. All data available in this paper were obtained from other research. The approaches of obtaining these values ​​or the motivation for choosing these values ​​can be consulted in each research individually”.

Also, I can notice from section “2. Mathematical Modeling” equations from 1 to 22, were taken from other studies.

Accordingly, from my point of view, the present work presents weak innovation, and can not be accepted for publication in Sustainability.

 Minor editing of the English language required

Author Response

Dear Reviewer 2,

Thank you for your comments and critical analysis of our manuscript. I would like to clarify some points to highlight the innovation and value of our work:

1. Semi-Analytical Modeling: While the equations used in our model are derived from previous studies, the way we integrate and apply them to a multi-layered system with thin layers such as geomembranes is innovative. This approach offers a more comprehensive insight into contaminant behavior in sanitary landfills.

2. Complexity of the Multi-Layered System: Our model not only employs existing equations but adapts them to consider the complexity and interaction between various barrier layers, which is a significant contribution to the field.

3. Practical Application and Relevance: The application of our model to a real case study, along with parametric analyses, demonstrates its practical relevance and capability to predict the efficiency of different barrier systems.

We understand the importance of innovation in research published in your journal and believe our work meets this criterion. Therefore, we request that you reconsider your assessment, taking into account the innovative aspects and practical applicability of our model.

Sincerely,

Reviewer 5 Report

Dear Authors,

Thank you for your manuscript submission. All of my concerns have been carefully considered. In my opinion, the revised manuscript meets the criteria to publish in Sustainability. Therefore, I recommend to accept this manuscript.

It is good

Author Response

Dear Reviewer 3,

We are very grateful for your time and careful consideration of our manuscript. We are encouraged by your remarks and the recommendation for publication. Your positive evaluation reaffirms the relevance and potential impact of our study in the field of environmental geotechnics and landfill design. Thank you for supporting our work.

Sincerely,

Round 3

Reviewer 4 Report

It was difficult for me to understand the author's response. This may be because I am not familiar with equation derivation (I have evaluated the present work from an Environmental pollution perspective, as I mentioned in my first report)

However, based on the previous response of the authors "No experimental tests were carried out in this research. All data available in this paper were obtained from other research. The approaches of obtaining these values ​​or the motivation for choosing these values ​​can be consulted in each research individually”.

Also, I can notice from section “2. Mathematical Modeling” that the last equations (no. 21 and 22) were taken from other studies,  [18, 23, 24] & [37,39], lines 196 and 201, respectively.

So, I can not recommend this work for publication in Sustainability.

Minor editing of the English language required.

Author Response

The authors sincerely thank the reviewer for the comment and apologize for not making it clear in the last response the innovation of this research in the authors’ opinions. Furthermore, the following answer better explains why there are equations that were also used in other papers.

In short, the advection-dispersion-reaction equation - ADRE (Equation 1) is the physical-mathematical approach to obtaining the advancement of contaminants in a porous medium. Ogata and Banks (1961) were one of the first researchers to develop an analytical solution to this problem. ADRE is an equation that requires an initial condition and two boundary conditions so that it can be solved for the one-dimensional case. An upper boundary condition that represents the beginning of the liner (in this research being related to the leachate generated in the landfill), a lower one, at the end of the liner (related to the variation of the contaminant at the end of the liner) and an initial condition representing whether the material was previously contaminated or not. The research by Ogata and Banks (1961) considers a homogeneous medium, which for laboratory conditions would be ideal to obtain the transport parameters (molecular diffusion coefficient, Darcy velocity) of certain contaminants in a single material. However, due to computational advances and the need to better represent field conditions, analytical/semi-analytical solutions that consider heterogeneous media have become extremely necessary in recent decades. Thus, in order to be able to solve ADRE (Equation 1) for heterogeneous media, continuity conditions between each of the materials are necessary in addition to the initial and boundary conditions mentioned here.

Most of the research that focuses on developing new analytical and numerical solutions uses the equations mentioned in topic 2 (Mathematical Modeling). Therefore, the innovation of this research is not the equations presented in topic 2, as they are all fundamental equations that are necessary to solve ADRE. For example, equation 21 is a lower boundary condition used in most papers in the literature on contaminant transport in porous media. Equation 22 is the general equation for calculating flow at any point and is also commonly used in research to quantify contaminant flow.

Thus, it is important to mention that the innovation of this research is using these equations to solve equation 1. Appendix A contains the solution proposed in this research considering equation 7 as the initial condition, equations 8 or 9 as an upper boundary condition, equation 21 as a lower boundary condition. For the continuity equations, equations 10, 11 and 12 were used when there is no geomembrane. In the case of a geomembrane in the liner system, equations 15 and 16 were used (and better developed in equations 17 to 20). Equations 2 and 3 represent the consideration of sorption in the transport of contaminants and equations 4, 5 and 6 are ways of considering the transport parameters of the materials to be involved in the simulations.

In lines 98-101, the innovation of the solution developed here is briefly discussed and the authors are not aware of any other research involving a semi-analytical solution that has incorporated several factors (multi-layered systems, recognizing the importance of thin layers as geomembranes, the influence of partition coefficients, and the criticality of mass flow calculations) to predict the transport of contaminants in liners systems of sanitary landfills.

In lines 131-135, the authors explain the assumptions of the model, thus, i) initial concentrations within layers may vary in each material; ii) top surface (upper boundary condition) concentration is constant or an arbitrary function of time; iii) interface concentrations and mass fluxes are continuous between layers and iv) the bottom boundary condition implies that the concentration derivative is zero (Neumann boundary condition). In line 142 is explained that the Mathematical Modeling topic contains the governing equations and boundary conditions used in this paper. In lines 197-199 is mentioned that based on the initial and boundary conditions and the continuity of the interfaces, the solution of the Equation for a multi-layer system can be proposed and the solution is presented in Appendix A.

Therefore, the authors believe that in this answer we were clearer about the innovation of this research and how fundamental equations (1 - 22) helped in the development of the solution proposed here.