Multifractal Analysis of Geological Data Using a Moving Window Dynamical Approach

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The article effectively expounds on the background of using fractal geometry to analyze geological data. It emphasizes the limitations of traditional geometric modeling in dealing with the complex characteristics of well logging data, as well as the important applications of fractal analysis in reservoir characterization and other aspects. The article proposes a multifractal analysis based on the moving window dynamical approach.

 

Comment 1: Please explain the content of lines 3-4 in the text: "However, calculating fractal dimensions for an entire geological profile can mask significant local fluctuations that may contribute to a more comprehensive characterization of the data." Is this statement accurate? Why can the fractal dimension of a local area reflect the characteristics of the overall data?

 

Comment 2: Further discuss the latest advancements in multifractal analysis within the field of geological data. Also, explore the specific connections and differences between these advancements and the method proposed in this article.

 

Comment 3: Elaborate in detail on the specific and unique advantages that the multifractal analysis based on the moving window dynamical approach has over traditional methods in practical applications such as reservoir characterization and oil and gas exploration decision-making.

 

Comment 4: In lines 71, why is the Weierstrass-Mandelbrot function used to generate artificial data for verification? There is a lack of more detailed explanation. It is recommended to supplement these key pieces of information in the introduction.

Comment 5: Regarding Figure 1, the text states, "The elastic spring is bent due to: (a) the action of a moment, (b) the action of a horizontal force or (c) the action of a vertical force, as initial conditions." However, from the three images in Figure 1, it is impossible to see any differences in the Koch triadic curves of the simple oscillators. Please provide an explanation.

Comment 6: In line 126, for a dataset (xi, yi), the statement "xi represents the independent variable and yi the dependent variable" is incorrect.

Comment 7: In line 137, it doesn't specifically state where the advantages of the method in this paper lie when compared with other methods.

Comment 8:  For Figure4, only two typical function cases with fractal dimensions of 1.3 and 1.7 are presented in the article. It is recommended to add tests with data of different fractal dimensions to more comprehensively verify the effectiveness of the method for data of varying degrees of complexity.

 

Comment 9: During the validation stage, the manuscript provided detailed calculation results of fractal dimensions, including data such as the mean values and standard deviations. However, it fails to specify which statistical analysis tools were used for data processing, which may affect the reproducibility of the study. It is recommended to supplement the tools and software used for data processing in the paper.

Comment 10: The authors assert that the employment of multiple windows can significantly enhance the quality of results and mitigate standard deviations. Supporting data are furnished in Tables 1 and 2. However, the validation of this purported advantage is confined to specific artificial signals. Critically, there is an absence of comparative analysis with other well - established fractal dimension calculation methodologies. To address this gap, it is recommended to conduct comprehensive comparative experiments incorporating traditional approaches, such as the Box Counting method and the Hurst method. Through such comparisons, the proposed method’s superiority in terms of accuracy, stability, and robustness can be effectively demonstrated.

Comment 11:  In sections 4.1-4.3, it is stated for all three wells that the porosity exhibits a monofractal characteristic. Please provide an explanation as to whether the porosity does not possess any multifractal characteristics at all. If there are no multifractal characteristics, why is this logging parameter being used?

 

Comment 12: The manuscript conducts a fractal analysis of the porosity and permeability data of three oil wells, clearly presenting the variation of the fractal dimension with depth. However, it merely describes the fractal characteristics without delving deeply into the underlying geological mechanisms. For example, when it is mentioned that the abrupt change in the fractal dimension of permeability may be related to the change in fracture density, there is no explanation of the causes of fracture generation and distribution in combination with information such as geological structures and rock types. It is recommended that in the analysis results of each well in sections 4.1-4.3, relevant geological theories and research findings be introduced to enhance the depth and scientific nature of the analysis.

 

Comment 13: In lines 325-328, it is mentioned in the text that the fractal dimension holds significant importance in the fields of reservoir characterization, hydrocarbon exploration, and geological risk assessment. However, there is no elaboration on this aspect in the manuscript. The text only explains the relationship between the nuclear magnetic logging parameters, permeability and porosity, and the fractal dimension.

Comments on the Quality of English Language

The English could be improved to more clearly express the research.

Author Response

Dear Reviewer:  

We deeply value the comprehensive comments you provided, which are needed to perform this revision, and the valuable suggestions.  

Please kindly find below our replies to the points raised (in blue color) and the corresponding modifications we have made in the revised manuscript (in blue color italic) to accommodate all comments provided by the reviewer. 

Reviewer #1: 

The article effectively expounds on the background of using fractal geometry to analyze geological data. It emphasizes the limitations of traditional geometric modeling in dealing with the complex characteristics of well logging data, as well as the important applications of fractal analysis in reservoir characterization and other aspects. The article proposes a multifractal analysis based on the moving window dynamical approach. 

Initially, the authors would like to express their gratitude for the reviewer’s constructive criticisms. We have addressed all the comments below. 

Comment 1: Please explain the content of lines 3-4 in the text: "However, calculating fractal dimensions for an entire geological profile can mask significant local fluctuations that may contribute to a more comprehensive characterization of the data." Is this statement accurate? Why can the fractal dimension of a local area reflect the characteristics of the overall data? 

We understand the reviewer’s concern. The sentence was rewritten as: 

However, calculating a single fractal dimension for an entire geological profile provides a general overview, but it can obscure local variations. These localized fluctuations, if analyzed, can offer a more detailed and nuanced understanding of the profile's characteristics.  

Comment 2: Further discuss the latest advancements in multifractal analysis within the field of geological data. Also, explore the specific connections and differences between these advancements and the method proposed in this article. 

To address the reviewer’s comment, we have added the following paragraphs (with new references) in the introduction: 

Recent advancements in multifractal analysis have significantly enhanced the interpretation of geological data by enabling more precise characterization of spatial heterogeneity and complexity across scales. Key developments include the integration of multifractal features with machine learning algorithms for improved prediction and classification (Sun et al., 2024; Lal et al., 2024), the use of adaptive and localized methods to detect subtle geological anomalies (Zuo et al., 2016; Wang et. al., 2024), and the application of robust techniques like multifractal-detrended fluctuation analysis and wavelet-based analysis to handle noisy or non-stationary signals such as seismic data (Martínez et al, 2021; Nicolis et al., 2021). Additionally, the rise of 3D multifractal modeling, supported by increased computational capabilities, has enabled more realistic simulations of subsurface structures, benefiting tasks such as mineral prospectivity mapping, reservoir characterization, and geohazard detection (Sadeghi, 2021; Karimpouli et al., 2019). These innovations position multifractal analysis, such as the one proposed in this paper, as a powerful and evolving tool in data-driven geoscience. 

• Sun, T.; Feng, M.; Pu, W.; Liu, Y.; Chen, F.; Zhang, H.; Huang, J.; Mao, L.; Wang, Z. Fractal-Based Multi-Criteria Feature Selection to Enhance Predictive Capability of AI-Driven Mineral Prospectivity Mapping. Fractal Fract. 2024, 8, 224. https://doi.org/10.3390/fractalfract8040224  

• Lal, U., Chikkankod, A.V. & Longo, L. Fractal dimensions and machine learning for detection of Parkinson’s disease in resting-state electroencephalography. Neural Comput & Applic 36, 8257–8280 (2024). https://doi.org/10.1007/s00521-024-09521-4  

• Zuo, R., Wang, J. Fractal/multifractal modeling of geochemical data: A review. Journal of Geochemical Exploration, Volume 164, May 2016, Pages 33-41, https://doi.org/10.1016/j.gexplo.2015.04.010 

• Wang, J., Zuo, R., and Liu, Q.: Mapping geochemical anomalies by accounting for the uncertainty of mineralization-related elemental associations, Solid Earth, 15, 731–746, https://doi.org/10.5194/se-15-731-2024, 2024.  

• Martínez, J. L. M., Domínguez, I. S., Rodríguez, I. Q., Rangel F. A. H., Gómez, G. S., A modified Multifractal Detrended Fluctuation Analysis (MFDFA) approach for multifractal analysis of precipitation, Physica A: Statistical Mechanics and its Applications Volume 565, 1, 2021, https://doi.org/10.1016/j.physa.2020.125611 

• Nicolis, O., Gonzalez, C., Wavelet-based fractal and multifractal analysis for detecting mineral deposits using multispectral images taken by drones, Methods and Applications in Petroleum and Mineral Exploration and Engineering Geology, 2021, Pages 295-307, https://doi.org/10.1016/B978-0-323-85617-1.00017-5 

• Sadeghi, B., Simulated-multifractal models: A futuristic review of multifractal modeling in geochemical anomaly classification, Ore Geology Reviews, Volume 139, Part B, 2021, https://doi.org/10.1016/j.oregeorev.2021.104511 

• Karimpouli, S. and Tahmasebi, P. (2019), 3D multi-fractal analysis of porous media using 3D digital images: considerations for heterogeneity evaluation. Geophysical Prospecting, 67: 1082-1093. https://doi.org/10.1111/1365-2478.12681 

In fact, all methods used for multifractal analysis can benefit from the approach proposed in this paper. The idea is based on the segmentation of the analysis domain, using different segment sizes. Hence, aiming to increase the reliability of the results, the averages of the fractal dimensions obtained for each segment are applied. The proposed segmentation strategy enables localized application of fractal analysis methods, producing more robust and reliable results, thereby enhancing the quality of input parameters for machine learning models. 

Comment 3: Elaborate in detail on the specific and unique advantages that the multifractal analysis based on the moving window dynamical approach has over traditional methods in practical applications such as reservoir characterization and oil and gas exploration decision-making.  

Thanks for the input. We have added the following paragraph in the introduction: 

By computing multifractal parameters within a sliding window, this approach enables the identification of spatial variations in scaling behavior, allowing for a more localized assessment of heterogeneity. This is particularly relevant in subsurface formations, where geological properties often change abruptly over short distances. Unlike conventional multifractal models that provide a single set of descriptors for an entire dataset, the moving window approach captures the dynamic evolution of multifractal features, offering greater sensitivity to subtle changes in the data. In practical terms, this facilitates the determination of reservoir boundaries, the detection of stratigraphic discontinuities, and the assessment of flow unit connectivity with improved spatial resolution. Furthermore, the moving window framework can be integrated with other geophysical or petrophysical indicators, making it a flexible tool in multidisciplinary workflows aimed at reducing uncertainty in exploration and development decisions. 

Comment 4: In lines 71, why is the Weierstrass-Mandelbrot function used to generate artificial data for verification? There is a lack of more detailed explanation. It is recommended to supplement these key pieces of information in the introduction. 

We understand the reviewer’s concern. To address this, we have added the following paragraph in the introduction: 

To validate the proposed methodology, controlled artificial data are generated using Weierstrass-Mandelbrot functions [28], which are known for their fractal characteristics. The Weierstrass-Mandelbrot function is often used to generate artificial data for verification due to its inherent fractal nature, simulating real-world complexity. Its non-differentiability and self-similarity at all scales create realistic, though controlled, data sets. This allows researchers to test algorithms and methods' robustness against complex, fractal-like patterns, serving as a reliable benchmark for evaluating how well methods handle fractal characteristics. In the work by Silva et. al. [34], these functions were used as benchmarks to assess different fractal methods. 

Comment 5: Regarding Figure 1, the text states, "The elastic spring is bent due to: (a) the action of a moment, (b) the action of a horizontal force or (c) the action of a vertical force, as initial conditions." However, from the three images in Figure 1, it is impossible to see any differences in the Koch triadic curves of the simple oscillators. Please provide an explanation. 

Based on the journal's suggestions, subsection 2.1 was revised and shortened, with appropriate references addressed. Thus, Figure 1 was removed from the manuscript. 

Comment 6: In line 126, for a dataset (xi, yi), the statement "xi represents the independent variable and yi the dependent variable" is incorrect. 

Indeed. Thanks for pointing it out. We have corrected it to: 

xi represents the inputs of the problem's domain, and yi their respective images.

Comment 7: In line 137, it doesn't specifically state where the advantages of the method in this paper lie when compared with other methods. 

Indeed. We have added the following paragraph was adjusted to address this issue, in section 2.2: 

The choice to use the Dynamical Approach Method was based on the results of Silva et. al. [34], which demonstrated the superiority of this method when compared to the others analyzed. In that paper, the authors have conducted a comprehensive comparative analysis of eight fractal analysis methods, including Box Counting, Compass, Detrended Fluctuation Analysis, Dynamical Fractal Approach, Hurst, Mass, Modified Mass, and Persistence. These methods were applied to artificially generated fractal data, such as Weierstrass–Mandelbrot functions as well as natural datasets related to environmental and geophysical domains. They indicated that the Dynamical Fractal Approach consistently demonstrated the highest accuracy across different datasets.  

Comment 8: For Figure 4, only two typical function cases with fractal dimensions of 1.3 and 1.7 are presented in the article. It is recommended to add tests with data of different fractal dimensions to more comprehensively verify the effectiveness of the method for data of varying degrees of complexity. 

In fact, Figure 4 only intends to demonstrate how the fractal dimension affects the visual characteristics of the Weierstrass-Mandelbrot functions. Hence, two typical monofractal functions, one with dimension equal to 1.3 and another with dimension equal to 1.7, were plotted.  

However, throughout sections 3.1 to 3.4, we have simulated different scenarios to assess the method’s performance. In this sense, tests were conducted over a monofractal function and three multifractal functions (Cases 1 through 4), exploring a range from simpler (monofractal – case 1) to highly complex scenarios (multifractal with 8 levels of fractality – case 4).  

Comment 9: During the validation stage, the manuscript provided detailed calculation results of fractal dimensions, including data such as the mean values and standard deviations. However, it fails to specify which statistical analysis tools were used for data processing, which may affect the reproducibility of the study. It is recommended to supplement the tools and software used for data processing in the paper. 

We appreciate the reviewer’s input. We have added the following sentences in the last paragraph of Section 3: 

All calculations were implemented and performed using MATLAB software. The Weierstrass-Mandelbrot functions were generated using the MATLAB algorithm provided by Monge-Alvarez, J. (2024) [42]. No preprocessing technique was used prior to the application of the proposed methodology.  

Comment 10: The authors assert that the employment of multiple windows can significantly enhance the quality of results and mitigate standard deviations. Supporting data are furnished in Tables 1 and 2. However, the validation of this purported advantage is confined to specific artificial signals. Critically, there is an absence of comparative analysis with other well - established fractal dimension calculation methodologies. To address this gap, it is recommended to conduct comprehensive comparative experiments incorporating traditional approaches, such as the Box Counting method and the Hurst method. Through such comparisons, the proposed method’s superiority in terms of accuracy, stability, and robustness can be effectively demonstrated. 

Initially, it is important to clarify that the primary focus of the paper is not to compare different fractal dimension estimation methods (which was the focus of the work of Silva et. al. [34]). Instead, the proposed methodology is designed to be flexible and generalizable, as it can be applied in conjunction with any fractal analysis technique. Therefore, rather than evaluating the relative performance of different methods, the paper aims to demonstrate how the use of multiple windows enhances the consistency and quality of results within the chosen analytical framework. This important information was inserted in the last paragraph of the introduction:  

It is important to highlight that the primary focus of this paper is not to compare different fractal dimension estimation methods (which was the focus of the work of Silva et. al. [34] Instead, the proposed methodology is designed to be flexible and generalizable, as it can be applied in conjunction with any fractal analysis technique. Therefore, rather than evaluating the relative performance of different methods, the paper aims to demonstrate how the use of multiple windows enhances the consistency and quality of results within the chosen analytical framework.   

Comment 11:  In sections 4.1-4.3, it is stated for all three wells that the porosity exhibits a monofractal characteristic. Please provide an explanation as to whether the porosity does not possess any multifractal characteristics at all. If there are no multifractal characteristics, why is this logging parameter being used? 

The following sentence (first paragraph of section 4.1) was introduced to justify the use of porosity 

In the context of oil wells, the distinction between monofractal and multifractal porosity behavior lies in the complexity and variability of pore distribution across different scales (from pore-scale to reservoir scale). Monofractal porosity implies a consistent fractal dimension, indicating a uniform level of heterogeneity throughout the reservoir. This simplifies modeling, as the pore structure's complexity remains relatively constant. Conversely, multifractal porosity would reflect a more complex and scale-dependent heterogeneity. This would mean that the pore distribution's complexity changes with the scale of observation, making reservoir modeling more challenging but potentially more accurate in capturing the complex flow dynamics of hydrocarbons. 
 

Comment 12: The manuscript conducts a fractal analysis of the porosity and permeability data of three oil wells, clearly presenting the variation of the fractal dimension with depth. However, it merely describes the fractal characteristics without delving deeply into the underlying geological mechanisms. For example, when it is mentioned that the abrupt change in the fractal dimension of permeability may be related to the change in fracture density, there is no explanation of the causes of fracture generation and distribution in combination with information such as geological structures and rock types. It is recommended that in the analysis results of each well in sections 4.1-4.3, relevant geological theories and research findings be introduced to enhance the depth and scientific nature of the analysis. 

We have substantially rewritten Section 4, adding multiple paragraphs and Figures that answer this question. To avoid rewriting lengthy text, the authors direct the reviewer to the rewritten Section 4 for this answer.

Comment 13: In lines 325-328, it is mentioned in the text that the fractal dimension holds significant importance in the fields of reservoir characterization, hydrocarbon exploration, and geological risk assessment. However, there is no elaboration on this aspect in the manuscript. The text only explains the relationship between the nuclear magnetic logging parameters, permeability and porosity, and the fractal dimension. 

The reviewer is correct. We have opted to remove that sentence from the revised version of the manuscript. 

 

Reviewer 2 Report

Comments and Suggestions for Authors

In this paper, a fractal analysis method based on moving windows is proposed, enabling the evaluation of data multifractality through the dynamical approach method. The main research contents are:

(1) A moving window dynamic fractal method was introduced to effectively identify local fluctuations, addressing the limitations of traditional global fractal analysis.

(2) The method is verified by Weierstrass-Mandelbrot function generating artificial data.

(3) Analysis of data from three oil wells in Brazil’s Campos Basin reveals that permeability exhibits significant multifractal characteristics, while porosity aligns more closely with monofractal behavior.

(4) When applied to real-world geological data from oil wells, the proposed method identifies regions with distinct fractal dimensions, enabling a refined understanding of reservoir properties and fluid flow dynamics. This enhanced resolution provides valuable insights into the complexity and heterogeneity of geological structures. But there are some problems:

(1) Was the impact of non-stationary noise on fractal dimension calculations considered during the validation of analytical methods using synthetic data?  How can noise interference in real-world data be addressed?

(2) The well identifiers in Figures 16–18 appear to be mislabeled or inconsistent. Could you clarify this discrepancy?

(3) Figure 19 shows a significant decrease in fractal dimensions around 5700m depth. Why is this happening? What mechanisms could explain this phenomenon? Is it potentially related to stimulation measures implemented at this interval?

(4) Was the relationship between an abrupt change in fractality (e.g., Devil's Staircase) and fracture density/pore connectivity established on the basis of qualitative analysis? Could this relationship be quantitatively verified using core data or imaging logging?

(5) What was the rationale for selecting geological data from three wells? How do fractal dimension variations mechanistically link to changes in porosity, permeability, and fracture?

 

Author Response

Dear Reviewer:  

We deeply value the comprehensive comments you provided, which are needed to perform this revision, and the valuable suggestions.  

Please kindly find below our replies to the points raised (in blue color) and the corresponding modifications we have made in the revised manuscript (in blue color italic) to accommodate all comments provided by the reviewer. 

 

Reviewer #2: 

In this paper, a fractal analysis method based on moving windows is proposed, enabling the evaluation of data multifractality through the dynamical approach method. The main research contents are: 

(1) A moving window dynamic fractal method was introduced to effectively identify local fluctuations, addressing the limitations of traditional global fractal analysis. 

(2) The method is verified by Weierstrass-Mandelbrot function generating artificial data. 

(3) Analysis of data from three oil wells in Brazil’s Campos Basin reveals that permeability exhibits significant multifractal characteristics, while porosity aligns more closely with monofractal behavior. 

(4) When applied to real-world geological data from oil wells, the proposed method identifies regions with distinct fractal dimensions, enabling a refined understanding of reservoir properties and fluid flow dynamics. This enhanced resolution provides valuable insights into the complexity and heterogeneity of geological structures.  

Initially, the authors would like to express their gratitude for the reviewer’s constructive criticisms. We have addressed all the comments below. 

But there are some problems: 

(1) Was the impact of non-stationary noise on fractal dimension calculations considered during the validation of analytical methods using synthetic data?  How can noise interference in real-world data be addressed? 

Thanks for pointing that out. An analysis of the influence of noise on the proposed methodology's results is presented in Section 3.1. The following paragraphs and Figure 7 were added: 

The analysis of the influence of noise on the identification of the fractal dimension is performed using the signal presented in Figure 4 with added random noise defined by the Equation (4): 

y_r(x) = y(x) + p U(-a,+a),    (4) 

where y_r(x) is the noisy signal; p is a value between 0 and 2 %;  a  is the root mean square of y(x); and U(-a,+a) is random uniform distribution over the interval [-a, a]. 

Figures 7(a) and (b) show the variation of the identified fractal dimension for the signal y_r(x) as a function of the noise percentage p. In each graph, three series are displayed: the identified average values (in blue), the upper and lower limits of an approximate 95 % confidence interval (mean +/- 2 times the standard deviations), in green and magenta, respectively. 

As expected, increasing the magnitude of noise leads to a rise in the identified fractal dimension in both analyzed cases. This occurs because the introduction of noise amplifies the oscillatory behavior of the resulting signal, thereby increasing its fractal dimension. As a result, values reach approximately 1.52 with 2 % noise for both smaller and larger window sizes. However, it is observed that larger windows exhibit a more gradual increase in the fractal dimension, along with a smaller standard deviation, thereby enhancing the reliability of the results. 

(2) The well identifiers in Figures 16–18 appear to be mislabeled or inconsistent. Could you clarify this discrepancy? 

Thanks for pointing it out. We have corrected them.  

(3) Figure 19 shows a significant decrease in fractal dimensions around 5700m depth. Why is this happening? What mechanisms could explain this phenomenon? Is it potentially related to stimulation measures implemented at this interval?  (4) Was the relationship between an abrupt change in fractality (e.g., Devil's Staircase) and fracture density/pore connectivity established on the basis of qualitative analysis? Could this relationship be quantitatively verified using core data or imaging logging?  (5) What was the rationale for selecting geological data from three wells? How do fractal dimension variations mechanistically link to changes in porosity, permeability, and fracture? 

We have substantially rewritten Section 4, adding multiple paragraphs and Figures that answer this question. To avoid rewriting lengthy text, the authors direct the reviewer to the rewritten Section 4 for this answer.

 

 

 

Reviewer 3 Report

Comments and Suggestions for Authors

The manuscript proposed a fractal characterization method based on moving windows, quantifying the multifractal behavior in data and applying it to real geological data from oil wells to more accurately understand reservoir properties and fluid flow behavior. The content of this study aligns well with the scope of fractal fract, however, some concerns have also been raised while reading this paper. I have presented them below, hoping that my comments can be helpful in improving the publication of this paper.

1. The abstract does not provide research results. It is recommended to revise the abstract and provide quantitative statements.

2. The introduction provides a comprehensive overview of existing research, but I would like to know if this reflects a knowledge gap, or if the content of the first few paragraphs is simply 'literature review for the sake of literature review', rather than aimed at advancing the paper.

3. The author has analyzed the fractals of porosity and permeability, but the depth of analysis is not sufficient. It is hoped that the author can deepen the discussion on the relationship between porosity, permeability, and fractals.

4. The author proposed a fractal characterization method. Can this method be further promoted and how can it be better utilized to provide guidance for resource exploration and disaster prediction. This is what this paper lacks.

5. How does the research result of this paper compare with other studies, and whether there are significant differences or similarities? It is recommended that the author conduct comparative analysis.

6. The statement of the conclusion is too broad, and substantive conclusions should be given in conjunction with research results, such as quantitative expressions of the reliability of verification methods, quantitative expressions of the relationship between porosity, permeability, and fractals.

7. The expression of Dynamic Approach Method is incorrect, please confirm.

8. Adjust the format of the Figures according to the fractalfract-template, such as Figure 1 in the manuscript. The description of subheadings for Figures 2, 6, 7, 9, 11, and 13 should be placed after the main title, and subheadings should be listed for Figures 14-18.

9. Please revise the reference format according to the requirements of the fractal fract.

Author Response

Dear Reviewer:  

We deeply value the comprehensive comments you provided, which are needed to perform this revision, and the valuable suggestions.  

Please kindly find below our replies to the points raised (in blue color) and the corresponding modifications we have made in the revised manuscript (in blue color italic) to accommodate all comments provided by the reviewer. 

 

Reviewer #3: 

The manuscript proposed a fractal characterization method based on moving windows, quantifying the multifractal behavior in data and applying it to real geological data from oil wells to more accurately understand reservoir properties and fluid flow behavior. The content of this study aligns well with the scope of fractal fract, however, some concerns have also been raised while reading this paper. I have presented them below, hoping that my comments can be helpful in improving the publication of this paper. 

Initially, the authors would like to express their gratitude for the reviewer’s constructive criticisms. We have addressed all the comments below. 

1. The abstract does not provide research results. It is recommended to revise the abstract and provide quantitative statements.

Thank you. We have added the following sentence in the abstract:  

The results demonstrate that the proposed methodology effectively captures a wide range of fractal dimensions, from high to low, in artificially generated data. Moreover, when applied to geological datasets, it successfully identifies regions exhibiting distinct fractal characteristics, which may contribute to a deeper understanding of reservoir properties and fluid flow dynamics. 

2. The introduction provides a comprehensive overview of existing research, but I would like to know if this reflects a knowledge gap, or if the content of the first few paragraphs is simply 'literature review for the sake of literature review', rather than aimed at advancing the paper.

The introduction has been expanded to include 5 new paragraphs, covering several topics, including this reviewer's comment. 

3. The author has analyzed the fractals of porosity and permeability, but the depth of analysis is not sufficient. It is hoped that the author can deepen the discussion on the relationship between porosity, permeability, and fractals.

We have substantially rewritten Section 4, adding multiple paragraphs and Figures that answer this question. To avoid rewriting lengthy text, the authors direct the reviewer to the rewritten Section 4 for this answer. 

4. The author proposed a fractal characterization method. Can this method be further promoted and how can it be better utilized to provide guidance for resource exploration and disaster prediction. This is what this paper lacks.

The introduction has been expanded to include 5 new paragraphs, covering several topics, including this reviewer's comment. 

5. How does the research result of this paper compare with other studies, and whether there are significant differences or similarities? It is recommended that the author conduct comparative analysis.

Initially, it is important to clarify that the primary focus of the paper is not to compare different fractal dimension estimation methods (which was the focus of the work of Silva et. al. [34]). Instead, the proposed methodology is designed to be flexible and generalizable, as it can be applied in conjunction with any fractal analysis technique. Therefore, rather than evaluating the relative performance of different methods, the paper aims to demonstrate how the use of multiple windows enhances the consistency and quality of results within the chosen analytical framework. This important information was inserted in the last paragraph of the introduction:  

It is important to highlight that the primary focus of this paper is not to compare different fractal dimension estimation methods (which was the focus of the work of Silva et. al. [34] Instead, the proposed methodology is designed to be flexible and generalizable, as it can be applied in conjunction with any fractal analysis technique. Therefore, rather than evaluating the relative performance of different methods, the paper aims to demonstrate how the use of multiple windows enhances the consistency and quality of results within the chosen analytical framework.  

6. The statement of the conclusion is too broad, and substantive conclusions should be given in conjunction with research results, such as quantitative expressions of the reliability of verification methods, quantitative expressions of the relationship between porosity, permeability, and fractals.

We have substantially rewritten Section 4, adding multiple paragraphs that provide analyses relating the fractal dimension and the petrophysical profile. To avoid rewriting long texts, the authors direct the reviewer to the rewritten Section 4. In addition, the conclusions have been expanded with a new paragraph. 

For the first two wells evaluated, there was a good correspondence between the fractal dimension and the assessment of their respective petrophysical profiles. However, for the third well, a clear correspondence between the fractal dimension and the petrophysical characteristics was not observed. This suggests that while fractal dimension can be indicative, it should not be considered a unique parameter in the evaluation of a petrophysical profile and should instead be integrated with other petrophysical measurements and geological information to provide a more robust and reliable reservoir characterization. 

 

7. The expression of Dynamic Approach Method is incorrect, please confirm.

The expression of Dynamic Approach Method was taken from reference [25] Bevilacqua, L. & Barros, M. The inverse problem for fractal curves solved with the dynamical approach method. Chaos, Solitons; Fractals (2023), http://dx.doi.org/10.1016/j.chaos.2023.113113, which is the work that originally formulated this method.  

8. Adjust the format of the Figures according to the fractalfract-template, such as Figure 1 in the manuscript. The description of subheadings for Figures 2, 6, 7, 9, 11, and 13 should be placed after the main title, and subheadings should be listed for Figures 14-18.

Thank you for pointing this error. Figures are now according to the fractal-fract template 

9. Please revise the reference format according to the requirements of the fractal fract.

Thank you for pointing this error. References are now according to fractal fract. 

 

Reviewer 4 Report

Comments and Suggestions for Authors

This paper presented a novel fractal characterization method based on moving windows, which, in combination with the Dynamical Approach Method, effectively identifies and quantifies multi-fractal behavior within data. The methodological innovation lies in combining the dynamic fractal approach with overlapping moving windows, effectively resolving the noise sensitivity and insufficient resolution issues inherent to traditional segmented analysis. The research topic holds significant engineering application value. My concerns are listed as follow:

1. Based on the proposed new method, the authors make use of data and cases for verification. Compared with other traditional fractal methods such as Box Counting and Compass Method, what are the advantages of the proposed method? What are the advantages? It is recommended to supplement the comparison with traditional methods.

2. In the analysis of actual oil well data, the correlation between the change of fractal dimension and geological characteristics (such as fracture density) is mainly described qualitatively. It is suggested to combine core data or NMR logging results to calculate the correlation coefficient between fractal dimension and pore/fracture parameters to enhance the reliability of the conclusion.

3. In Case 3, the fractal dimensions obtained by the new method are 1.30 and 1.33 respectively, and the difference between the two is very small. It is suggested to discuss the theoretical limit of the resolution of the new method.

4. In section of 4. Multifractality of Geological data, what is the basis for setting the number of Windows (W=6) and step size (step=30)?

Author Response

Dear Reviewer:  

We deeply value the comprehensive comments you provided, which are needed to perform this revision, and the valuable suggestions.  

Please kindly find below our replies to the points raised (in blue color) and the corresponding modifications we have made in the revised manuscript (in blue color italic) to accommodate all comments provided by the reviewer. 

Reviewer #4: 

This paper presented a novel fractal characterization method based on moving windows, which, in combination with the Dynamical Approach Method, effectively identifies and quantifies multi-fractal behavior within data. The methodological innovation lies in combining the dynamic fractal approach with overlapping moving windows, effectively resolving the noise sensitivity and insufficient resolution issues inherent to traditional segmented analysis. The research topic holds significant engineering application value.  

Initially, the authors would like to express their gratitude for the reviewer’s constructive criticisms. We have addressed all the comments below. 

My concerns are listed as follows: 

1. Based on the proposed new method, the authors make use of data and cases for verification. Compared with other traditional fractal methods such as Box Counting and Compass Method, what are the advantages of the proposed method? What are the advantages? It is recommended to supplement the comparison with traditional methods.

Initially, it is important to clarify that the primary focus of the paper is not to compare different fractal dimension estimation methods (which was the focus of the work of Silva et. al. [34]). Instead, the proposed methodology is designed to be flexible and generalizable, as it can be applied in conjunction with any fractal analysis technique. Therefore, rather than evaluating the relative performance of different methods, the paper aims to demonstrate how the use of multiple windows enhances the consistency and quality of results within the chosen analytical framework. This important information was inserted on the last paragraph of the introduction: 

It is important to highlight that the primary focus of this paper is not to compare different fractal dimension estimation methods (which was the focus of the work of Silva et. al. [34] Instead, the proposed methodology is designed to be flexible and generalizable, as it can be applied in conjunction with any fractal analysis technique. Therefore, rather than evaluating the relative performance of different methods, the paper aims to demonstrate how the use of multiple windows enhances the consistency and quality of results within the chosen analytical framework.   

2. In the analysis of actual oil well data, the correlation between the change of fractal dimension and geological characteristics (such as fracture density) is mainly described qualitatively. It is suggested to combine core data or NMR logging results to calculate the correlation coefficient between fractal dimension and pore/fracture parameters to enhance the reliability of the conclusion.

We have substantially rewritten Section 4, adding multiple paragraphs that provide analyses relating the fractal dimension and the petrophysical profile. To avoid rewriting long texts, the authors direct the reviewer to the rewritten Section 4. In addition, the conclusions have been expanded with a new paragraph. 

For the first two wells evaluated, there was a good correspondence between the fractal dimension and the assessment of their respective petrophysical profiles. However, for the third well, a clear correspondence between the fractal dimension and the petrophysical characteristics was not observed. This suggests that while fractal dimension can be indicative, it should not be considered a unique parameter in the evaluation of a petrophysical profile and should instead be integrated with other petrophysical measurements and geological information to provide a more robust and reliable reservoir characterization. 

   

3. In Case 3, the fractal dimensions obtained by the new method are 1.30 and 1.33 respectively, and the difference between the two is very small. It is suggested to discuss the theoretical limit of the resolution of the new method.

We appreciate the reviewer's suggestion, and the discussion of the theoretical limitations of the presented methodology has been incorporated into the analysis of Case 1. The following paragraphs have been added: 

For Case 1:  

The goals of this initial analysis are to assess the accuracy and variability of fractal dimension estimation along the signal using both small and large window sizes, to compare this with using individual windows separately, to estimate theoretical limitations of the presented methodology, and to evaluate the influence of noise on the identification of fractal dimension. 

... 

The theoretical limit of the presented methodology is closely linked to the number and size of the adopted windows. For Case 1, Tables 1 and 2 show that the average fractal dimension has a percentage error of 0.15% when small windows are used and 0.03% for large windows, which can be considered good performance in identifying the average fractal dimension. However, it is necessary to observe the confidence interval of the results across the domain. In the case of small windows, the standard deviation obtained was 0.0063, resulting in an approximate 95% confidence interval between 1.5148 and 1.4896 (mean plus or minus two standard deviations), giving a variation range of 0.0252. For large windows, the 95% confidence interval has upper and lower limits of 1.5061 and 1.4949, respectively, resulting in a smaller variation range of 0.0112. Thus, based on the evaluated results of Case 1, it can be stated that the presented methodology has good sensitivity to detect changes in the fractal dimension greater than 0.0252 when using small windows and 0.0112 for large windows. 

For Case 2, the following paragraph was added: 

The variation ranges of the fractal dimensions obtained for small windows and large windows, excluding the transition region, were 0.0233 and 0.0115, respectively. These values are consistent with the theoretical limits evaluated within a 95% confidence interval in the previous case (0.0252 when using small windows and 0.0112 for large windows). 

For Case 3, the following paragraph was added: 

The results for Case 3 one more time confirm the analysis regarding the theoretical limitations discussed in Case 1. A greater fluctuation in results is observed for small windows around the exact values of the fractal dimensions when compared to the ones obtained with large windows. Furthermore, the range of variation for these fluctuations, excluding transition regions, is consistent with what was detected in the analysis of Case 1, specifically 0.0244 when using small windows and 0.0148 for large windows. 

 

4. In section of 4. Multifractality of Geological data, what is the basis for setting the number of Windows (W=6) and step size (step=30)?

The values for the number of windows (W = 6) and step size (step = 30) were determined based on a thorough sensitivity analysis conducted previously. This analysis aimed to identify parameter settings that balance resolution and computational efficiency while preserving the reliability of multifractal characterization. This information was strengthened by the phrase, inserted in the text (third paragraph of section 4): 

The definition of these parameters was made based on the good results achieved with validation data. 
 

Reviewer 5 Report

Comments and Suggestions for Authors

I see that the references do not have alignment according to alphabetical order, or according to their mentioning through the text. E.g., reference 23 and 24, reference 22 comes before reference 17,

And so on.

32, 29, and 37

Not mentioned throughout the text the reference [18].

Is there any difference between references 33 and 34?

Reference style is not the same .

Reference 19, in the Refeences section is missing some details.

Author Response

Dear Reviewer: 

We deeply value the comprehensive comments you provided, which are needed to perform this revision, and the valuable suggestions. 

Please kindly find below our replies to the points raised (in blue color) and the corresponding modifications we have made in the revised manuscript to accommodate all comments provided by the reviewer. 

Reviewer #5: 

I see that the references do not have alignment according to alphabetical order, or according to their mentioning through the text. E.g., reference 23 and 24, reference 22 comes before reference 17, and so on. 

The authors would like to express their gratitude for the reviewer’s comments. We have addressed all of them in the revised version. 

32, 29, and 37 

The order of the references was checked 

Not mentioned throughout the text the reference [18]. 

This reference was removed. 

Is there any difference between references 33 and 34? 

Thank you for pointing this error. One of these references was removed. 

Reference style is not the same. 

Reference style was checked. 

Reference 19, in the References section is missing some details. 

Old reference 19 (now reference 29) was adjusted.  

 

 

 

 

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

After the revision, the article provides a relatively comprehensive response to the reviewers' comments, especially showing significant improvements in explaining key concepts and supplementing the research background.

Author Response

Thank you for your consideration

Reviewer 3 Report

Comments and Suggestions for Authors

I think the current version meets the requirements for publication.

Author Response

Thank you for your consideration