Numerical Approximation of the In Situ Combustion Model Using the Nonlinear Mixed Complementarity Method

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

This paper presents a numerical method for solving an in-situ combustion model using a nonlinear mixed complementarity approach. While the work demonstrates some interesting mathematical developments, there are several significant concerns that need to be addressed as follows:

 

Lines 15-21: Why is there repetition of text in these lines? This appears to be a copy-paste error that wasn't caught during review.

Lines 25-27: How does this "simple" in-situ combustion model adequately represent real-world conditions, given that it only considers two nonlinear parabolic differential equations?

Lines 44-48: Why are so many simplifying assumptions made (negligible porosity changes, constant gas velocity, negligible heat loss, etc.)? How do these affect the model's real-world applicability?

Lines 55-56: Why is the reaction ratio Wr assumed to have μf = μg = μ0 = 1? What is the justification for this significant simplification?

Lines 66-68: Why was the Crank-Nicolson method specifically chosen for spatial derivatives? Were other numerical methods considered and compared?

Lines 69-72: How are the boundary conditions justified? What is the physical meaning of these particular Dirichlet and Neumann conditions?

Lines 106-111: Why was such a small error tolerance (10^-8) chosen? How was this value determined?

Lines 124-128: The FDA-MNCP method takes twice as long as FDA-NCP for 50 points - why should we use it if it's computationally more expensive?

Lines 141-146: Why was only h=1/50 used for the detailed error analysis? Wouldn't testing more mesh sizes provide better validation?

Lines 150-154: When the errors are "very similar" up to 3 decimal places, what is the practical advantage of using the more computationally intensive FDA-MNCP method?

Lines 156-159: How can the authors claim this method "can be applied to parabolic problems" in general when they only tested it on one specific case?

Lines 168-170: The computational advantage only appears with increased discretization points - why isn't this limitation more prominently discussed in the abstract?

While the paper presents an interesting approach to solving in-situ combustion models, significant revisions are required before it meets the standards for publication. The authors need to address the fundamental issues of validation, comparison, and practical applicability while providing more rigorous justification for their methodological choices.

Author Response

Dear Reviewer,

Dear reviewer,
Thank you very much for your comments and suggestions that greatly improve our work. Dear reviewer,
Thank you very much for your comments and suggestions that greatly improve our work. I attach the pdf with the improvements made and some comments.

Lines 25-27: How does this "simple" in-situ combustion model adequately represent real-world conditions, given that it only considers two nonlinear parabolic differential equations? 

Response: The model studied in this article was proposed by[Chapiro 1] It is assumed based on your experience that: Only a small part of the available space is occupied by the fuel, so that changes of porosity in the reaction are negligible,  that the temperature of solid and gas is the same (local thermal equilibrium). It is also assumed that heat losses are neglected, which is reasonable for in-situ combustion in field conditions. Finally also assume that pressure variations are small compared to prevailing pressure.

Lines 44-48: Why are so many simplifying assumptions made (negligible porosity changes, constant gas velocity, negligible heat loss, etc.)? How do these affect the model's real-world applicability?

Response: In[Chapiro], in the introduction, it is mentioned that the models previously obtained usually explores strong nonlinearity of the Arrhenius factor in the reaction rate, which allows neglecting the reaction rate as soon as the temperature decreases . This method is valid provided that most of the reaction occurs at the highest temperaturas One of the consequences of this assumption is that oxygen is not consumed completely at the highest temperatures and its breakthrough becomes possible. In such a case the oxygen gets in contact with fuel downstream of the fast reaction zone, leading to slow reaction in the downstream zone. This is the main reason for neglecting reaction at low temperatures, e.g., in laboratory experiments. However, for field applications there are low-temperature oxidation reactions, which are relatively fast; as well as heat losses are very small. This allows the simplifications made to be carried out

Lines 55-56: Why is the reaction ratio Wr assumed to have μf = μg = μ0 = 1? What is the justification for this significant simplification?

Response: In the combustión reaction, µf moles of immobile fuel react with µo moles of oxygen and generate µg moles of gaseous products and, possibly, unreactive solid products. For simplicity, we consider the case µf= µo=µg=1as, e.g., in the reactionC+O2→CO2.

Lines 66-68: Why was the Crank-Nicolson method specifically chosen for spatial derivatives? Were other numerical methods considered and compared?

Response: The Crank-Nicolson method was chosen based on the recommended literature on its computational advantages for reaction, convection and diffusion models. As a reference we cite the book: "

Mayers, D.; Morton, K. Numerical Solution of Partial Differential Equations Cambridge, UK.: Cambridge University Press, 2005.

Lines 106-111: Why was such a small error tolerance (10^-8) chosen? How was this value determined?

Response: Tolerance can vary and 10^-8 was chosen to test the method.            

 

Lines 124-128: The FDA-MNCP method takes twice as long as FDA-NCP for 50 points - why should we use it if it's computationally more expensive?

 Response: For 50 points, the FDA-MNCP method is more computationally expensive, but as the number of points in the mesh increases, we observe that the cost decreases. When there are 200 points, it is reduced by approximately 40% and with 400, by approximately 80%.

 

Lines 141-146: Why was only h=1/50 used for the detailed error analysis? Wouldn't testing more mesh sizes provide better validation?w

Response: There is no specific reason for the choice of h in our simulations, they were made with other values ​​without difference in the results.

 

Lines 150-154: When the errors are "very similar" up to 3 decimal places, what is the practical advantage of using the more computationally intensive FDA-MNCP method?

Response: In Tables 6 and 7 we see that the relative errors Eh, Eh/2, Eh/4 for θ, in the FDA-MNCP and FDA-NCP methods, are very similar since they are equal in the first three decimals, that is, the error does not increase but the computational cost is reduced as the points in the discretization increase.

"

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

The abstract is not quite logically structured: it is necessary to outline the problem at the very beginning. Only then can specific solutions be proposed. At the very end, the prospects for their further application should be outlined.

There is no literature review. Research by other authors in this area is practically not mentioned. The advantages and disadvantages of other methods, their limitations, which require detailed study and additional research, are not shown.

The last article cited by the authors dates back to 2013. More than ten years have passed. Since then, other authors have conducted extensive analyses of various modeling and research methods and possible applications in the industrial sector.  www.sciencedirect.com  alone contains more than 35,319 results on the topic "partial differential equations of the parabolic type". The authors also touch upon another fairly popular "In-situ combustion" (94,511 results), Crank-Nicolson method (12,029 results), finite difference technique (744,751 results), mixed complementarity problem (22,788 results). The authors could have justified the need for modeling of "In-situ combustion" in more detail. Then they could have focused on the well-known physical and mathematical methods and their limitations. Subsequently, they could have studied in more detail the models they propose.

Only calculations are given, but there is not enough discussion of the results. The calculations will not be of interest to readers.

The conclusions provide links to tables and graphs from the main text. The conclusions should be presented as a separate element of the article, where the authors summarize their achievements or further prospects for the development of methods

The article requires significant correction by a native speaker. Contains a number of errors, which, in general, complicate the adequate perception of the material

L.1 «INSITU» correctly in-situ

Abstract:

"for a in-situ combustion model" → "for an in-situ combustion model"

“has the advantage of provide” → “has the advantage of providing”

"which can be rewritten in the form of mixed complementarity also we do a comparison with the FDA-NCP method."

L. 15 “several mathematical models” → "Several"

L17 "the case several mathematical models" → "the case of several mathematical models"

L.20-24 Some sentences are repeated in meaning

L. 25 "the model treated in [4]" → "the model discussed in [4]"

L.31-33 "the sequence of feasible points generated is contained in a feasible region and we will verify that the directions obtained are feasible and descending"

L 58 "we do as [1]" → "we follow the approach outlined in [1]"

L62 “is a escalated activation”

L 95 Algoritmo 3.1 (implementation of FDA-MNCP.)" → Algorithm 3.1: Implementation of FDA-MNCP

Figure 5. Difference between DA-NCP and FDA-NCP methods. Maybe FDA-MNCP and FDA-NCP?

L 129-130 “In Table 3, we can also observe that the computational process time used by the FDA-MNCP method is slightly greater than the time of the FDA-NCP method as the number of iterations.” The sentence appears to be unfinished.

L.132 “1FDA-NCP”

L. 149 “show the relative errors for θ and η usando h = 1/50”.

Comments on the Quality of English Language

L.1 «INSITU» correctly in-situ

Abstract:

"for a in-situ combustion model" → "for an in-situ combustion model"

“has the advantage of provide” → “has the advantage of providing”

"which can be rewritten in the form of mixed complementarity also we do a comparison with the FDA-NCP method."

L. 15 “several mathematical models” → "Several"

L17 "the case several mathematical models" → "the case of several mathematical models"

L.20-24 Some sentences are repeated in meaning

L. 25 "the model treated in [4]" → "the model discussed in [4]"

L.31-33 "the sequence of feasible points generated is contained in a feasible region and we will verify that the directions obtained are feasible and descending"

L 58 "we do as [1]" → "we follow the approach outlined in [1]"

L62 “is a escalated activation”

L 95 Algoritmo 3.1 (implementation of FDA-MNCP.)" → Algorithm 3.1: Implementation of FDA-MNCP

Figure 5. Difference between DA-NCP and FDA-NCP methods. Maybe FDA-MNCP and FDA-NCP?

L 129-130 “In Table 3, we can also observe that the computational process time used by the FDA-MNCP method is slightly greater than the time of the FDA-NCP method as the number of iterations.” The sentence appears to be unfinished.

L.132 “1FDA-NCP”

L. 149 “show the relative errors for θ and η usando h = 1/50”.

Author Response

Dear Reviewer,

I am very grateful for the suggestions and comments that greatly improve our work. I attach the pdf with the improved comments.
Kind regards,
The authors

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

Reviewing comments for manuscript entitled_ NUMERICAL APPROXIMATION OF THE INSITU COMBUSTION MODEL USING THE NONLINEAR MIXED COMPLEMENTARITY METHOD

 

Research overview:

This research introduces a novel numerical method for approximating solutions to in-situ combustion models, through that the authors develop a nonlinear mixed complementarity method (FDA-MNCP).

This approach offers global convergence advantages over traditional finite difference and Newton's methods, which typically provide only local convergence.

The study reformulates the in-situ combustion model into a mixed complementarity framework and compares the FDA-MNCP method with the FDA-NCP method.

 

Overall, the Research shows Innovative Numerical Approach, Enhanced Convergence Properties, and Comprehensive Error Analysis.

 

Authors should consider the following for the potential improvement of their research:

1. Broader Applicability Assessment: While the method shows promise for the specific in-situ combustion model studied, evaluating its performance across a wider range of combustion scenarios and more complex models would strengthen its generalizability.
2. Computational Efficiency Optimization: The study notes that the FDA-MNCP method can be computationally intensive, especially with finer spatial discretizations. Exploring strategies to enhance computational efficiency, such as algorithmic optimizations or parallel computing techniques, could be beneficial.
3. Comparative Analysis with Alternative Methods: Conducting a more extensive comparison with other contemporary numerical methods beyond the FDA-NCP approach would provide a clearer understanding of the FDA-MNCP method's relative advantages and potential limitations
4. Almost all graphs are presented in high quality representation and resolution, however, for clarity in black / white printing, figures containing more than one curve should use different line styles such as solid line, dashed-lines, and center-dashed-lines. Examples are:
   1. Figures 1 to 4 should use different line markers
   2. Figures 5 to 7 should use different line styles.

Author Response

Dear Reviewer,

I am very grateful for the suggestions and comments that greatly improve our work. We are improving your observations with future work, removing some restrictions made in this work and adding steam injection.
Kind regards,
The authors

Author Response File: Author Response.pdf

Round 2

Reviewer 2 Report

Comments and Suggestions for Authors

The authors haven't provided point-by-point reply to my comments. Instead of responding to specific comments, they simply attached the article itself, without their comments on what was improved and where, which comments were addressed and which were not. 

Only minor changes were made to the article. A more responsible attitude of authors towards the presentation of article material is needed.

These issues still need to be addressed:

- The abstract is not quite logically structured still.

- Research by other authors in this area is practically not mentioned. The advantages and disadvantages of other methods, their limitations, which require detailed study and additional research, are not shown.

- The last article cited by the authors still dates back to 2013.

- The authors could have justified the need for modeling of "In-situ combustion" in more detail. Then they could have focused on the well-known physical and mathematical methods and their limitations. Subsequently, they could have studied in more detail the models they propose.

- Only calculations are given, but there is not enough discussion of the results. The calculations will not be of interest to readers.

- The conclusions provide links to tables and graphs from the main text. The conclusions should be presented as a separate element of the article, where the authors summarize their achievements or further prospects for the development of methods

- The article requires correction by a native speaker.

Comments on the Quality of English Language

eg." for an simple in-situ"

Author Response

Dear Reviewer,  

Regarding your observations, we have discussed them and would like to provide the following comments. We hope this clarifies our work, and we remain open to further improvements.

The abstract is not quite logically structured still.

Response: 

We consider that the scheme is logical and follows the following sequences:

1. Introduction and description of the problem .

• In this work, we study a numerical method to approximate the exact solution of a simple in-situ combustion model.

2. Methodology and advantages.

• To achieve this, we use the mixed nonlinear complementarity method (MNCP), a variation of the Newton method for solving nonlinear systems, incorporating a single Hadamard product in its formulation. The method is based on an implicit finite difference scheme and a mixed nonlinear complementarity algorithm (FDA-MNCP). One of its main advantages is that it ensures global convergence, unlike the finite difference method and the Newton method, which only guarantee local convergence.

3. Application and results.

• We apply this theory to an in-situ combustion model, reformulating it in terms of mixed complementarity. Additionally, we compare it with the FDA-NCP method, demonstrating that FDA-MNCP is more computationally efficient when the spatial discretization is refined.

But we look forward to your suggestions to work on them.

 

Research by other authors in this area is practically not mentioned. The advantages and disadvantages of other methods, their limitations, which require detailed study and additional research, are not shown.

Response: 

The main bibliography we used was:

4. RAMIREZ GUTIERREZ, ANGEL. Application of the nonlinear complementarity method for the study of in situ oxygen combustion (Master's Thesis in Mathematics). Universidade Federal de Juis de Fora, Juiz de Fora, Brasil, 2013.
5. G CHAPIRO, AA MAILYBAEV, AJ SOUZA, D MARCHESIN, J BRUINING. Asymptotic approximation of long-time solution for low-temperature filtration combustion, Computational geosciences 16, 799-808, 2012.
6. CHAPIRO, G. ; MAZORCHE, S. R. ; HERSKOVITS, J. ; ROCHE, J. R. . Solution of the Non-linear Parabolic Problems using Nonlinear Complementarity Algorithm, 2010.
7. Chapiro, G., Ramírez G., A. E., Herskovits, J., Mazorche, S. R., and Pereira, W. S. Numerical solution of a class of moving boundary problems with a nonlinear complementarity approach. Journal of Optimization Theory and Applications, 168(2):534–550 (2016).

The authors discuss the advantages of using a complementarity formulation and its applications to the in-situ model. For example, it ensures global convergence, unlike the finite difference method and the Newton method, which only guarantee local convergence. To the best of our knowledge, based on our research, we have not found any updated literature addressing the in-situ combustion problem using FDA-MNCP with a single Hadamard product, which constitutes the contribution of our work, improving computational time without worsening the approximation. 

 

The authors could have justified the need for modeling of "In-situ combustion" in more detail. Then they could have focused on the well-known physical and mathematical methods and their limitations. Subsequently, they could have studied in more detail the models they propose.

Response: 

This was included and improved in Section 2:

One of the major challenges in extracting fuel from a reservoir is its high viscosity. To reduce viscosity, techniques such as steam injection or in-situ combustion are applied. In-situ combustion is an enhanced oil recovery method that involves injecting an oxidizing agent, such as air or oxygen, directly into the reservoir to initiate a combustion reaction. The heat generated by this process reduces the oil’s viscosity, improving its flow toward production wells. In this work, we study a simple model for in-situ combustion described in Chapiro(2016). However, obtaining analytical solutions is impossible, and using the finite difference method in time presents difficulties due to the presence of shock waves. To overcome this problem, the system is reformulated as a mixed nonlinear complementarity problem (FDA–MNCP) in time and as a finite difference method in space.

Only calculations are given, but there is not enough discussion of the results. The calculations will not be of interest to readers.

Response: 

In the images and tables, we provided comments emphasizing that the approximations obtained by the FDA-NCP and FDA-MNCP methods are practically the same, without drawing any conclusions there. However, we forgot to mention that, when formulating the problem as a mixed complementarity problem, we used only one Hadamard product, unlike the existing literature, which uses two. This results in computational savings, as reflected in Tables 2, 3, 4, and 5. This was explicitly stated in the conclusions. We improved computational time without worsening the approximation.

The article requires correction by a native speaker.

It was done.

 

1. The following improvements were also made:

• Lines 44-48: It is explained why so many simplifications are made (insignificant changes in porosity, constant gas velocity, negligible heat loss, etc.), mentioning that due to the strong nonlinearity of the Arrhenius factor in the reaction rate, it is possible to neglect the reaction rate as soon as the temperature decreases.
• Lines 124-128: It is observed that the FDA-MNCP method takes twice as long as FDA-NCP for 50 points. Why should we use it if it is computationally more expensive? We are more explicit in stating that as the number of points in the mesh increases, the computational cost decreases...among others.

As a team, we discussed your observations and concerns. We hope this makes things clearer, and we are available to improve, clarify, and, if necessary, restructure.

Best regards.

Author Response File: Author Response.pdf

Round 3

Reviewer 2 Report

Comments and Suggestions for Authors

The authors have provided point-by-point response to my comments. Therefore, the article may be considered by the editorial board for further publication.