Viscoelastic Strains of Palaeozoic Shales under the Burger’s Model Description

Round 1

Reviewer 1 Report

This paper conducted rheological experimental analysis on shale samples collected from the Baltic Basin and found that the Burger’s model can be used to describe the creep of the target sample, and the model parameters were identified. Meanwhile, it was found that the creep related with volumetric change can be omitted when analyzing shale creep. However, it should be noted that there are the following issues to be addressed:

1. In the description of the abstract, there is a lack of clarity regarding the innovation of this article. Please modify the abstract to make this section clearer.

2. In Section 6, the following sentence is mentioned “This hypothesis in the creep calculations was accepted by some researchers and discarded by others”. Please provide a brief overview of the reasons.

3. The display of Figures 6 and 7 may be a bit poor, as the large dot annotations and dense arrangement make the results not clear. Please adjust the pictures to make the results of four different rock strata samples clearer.

4. It is also recommended to revise the conclusion section to make the research conclusions more prominent. Suggest writing the “further comments and assumptions related with the creep mechanism” mentioned in the conclusion in another paragraph to make it clearer and more reasonable.

5. Some references have the year in bold, while others do not. Please pay attention to the citation list format again. Moreover, some relevant work should be added:

(1) Direct numerical simulation of gas-liquid drag-reducing cavity flow by the VOSET method. Polymers, 2019, 11(4): 596.

(2) An integrated model with stable numerical methods for fractured underground gas storage. Journal of Cleaner Production, 2023, 393, 136268.

(3) Mixed convection of non-Newtonian nanofluid in an enclosure using Buongiorno’s mathematical model. Int. J. Heat Mass Transf. 2017, 108, 1481–1500.

(4) A globally mass-conservative method for dual-continuum gas reservoir simulation. Journal of Natural Gas Science and Engineering, 2018, 53: 301-316.

(5) Mixed convection of Al2O3-water nanofluid in a double lid-driven square cavity with a solid inner insert using Buongiorno’s two-phase model. Int. J. Heat Mass Transf. 2018, 119, 939–961.

(6) On accelerated flows of a viscoelastic fluid with the fractional Burgers’ model. Nonlinear Analysis: Real World Applications, Volume 10, Issue 4, 2009, Pages 2286-2296.

Author Response

Dear reviewer, thank you for your thoughtful remarks and comment. 

1. An original approach and analysis results are presented here, which allow the separation and monitoring of shear and volume creep effects, and on this basis, determining the significance of the contribution of volume creep in the entire creep process. A relatively simple methodology for determining the parameters of the Burgers model using this division is presented. The original value of the article is also the test results themselves and the parameter values of the analyzed model for triaxial creep of shales, which are not numerous in the literature.

2. Thank you for this comment. This wording in the article is inappropriate and imprecise. It has been changed to the following: „Such assumptions in the creep calculations was accepted by some researchers [36, 46] and discarded by others [41, 47].

3. Corrected according to the comment.

4. Dear reviewer. We agree that the issue you raise here is extremely important, we see it too. The conclusions included in the summary of our article are guidelines for further research rather than conclusions resulting from the presented results. Determining the creep mechanism is a very difficult task. This mechanism varies significantly depending on the brittleness or ductility of the rock. In addition to research specially planned for this purpose, appropriate measurement techniques should also be used to x-ray samples to determine changes in the rock microstructure. In our research, tracking such changes was not planned, hence the conclusions are only very general. In our opinion, this is a topic for a separate publication. If you allow, we will not expand this article with an additional chapter, because it is too little to discuss this complex and broad issue in the context of the studied rocks, and we do not have the research results for this. We will only expand the conclusions by adding a comment on the probable ductile creep mechanism in our case.

5. Corrected according to the comment

   One publication from the suggested list was used in the literature review and added to the list of publications.

Reviewer 2 Report

This work presents.

The article presents the results of creep studies of Palaeozoic shales from the Baltic Basin in which the exploitation of shale gas in Poland was planned. Knowledge of instantaneous and long-term properties investigated in triaxial stress conditions is important from the point of view of exploitation techniques related to hydraulic fracturing. Rheological phenomena also play an important role in the analysis of the initial stress in shales, the knowledge of which is indispensable in the hydraulic fracturing process. The tests were carried out on samples representing four siltstone-claystone lithostratigraphic units occurring in the Baltic Basin. The studies and analyses were aimed at determining the character of creep in shales, selection of the appropriate rheological model for the analyzed rocks, and determining the threshold of the linear creep under triaxial compression conditions. Out of the several analyzed models, Burger’s model was selected to describe creep in the studied gas shales. The investigations were performed at various loading levels in relation to the triaxial strength of the shales. Depending on the load, at its low values up to 0.7 (s1-s3)max, creep had a determined character and did not show features of progressive creep. The linear creep threshold was also analyzed in this range. The loading level of 0.7 (s1-s3)max was the limit of linear creep. Exceeding this load resulted in the loss of the linear character of creep, which in consequence lead to the subsequent third creep phase ending with rock damage. Parameters of the Burger’s model for gas shales from the Baltic Basin (northern Poland) have been identified. There are significant differences in the behavior of shales depending on the lithostratigraphic unit from which the samples were collected. The mineral composition of the shales also influenced their behavior.

1.      Redraw figure 6, 7,8, 9 and 11 in origin pro or some other software with nice representation and at least 300 dpi

2.      Novelty of this work?

3.      How much improvement in your case by comparing creep test with standard and literature make a comparison table and prove with citation.

 

 

Author Response

Dear reviewer, thank you for your thoughtful remarks and comment. 

1. Corrected as possible according to the comment.

2. An original approach and analysis results are presented here, which allow the separation and monitoring of shear and volume creep effects, and on this basis, determining the significance of the contribution of volume creep in the entire creep process. A relatively simple methodology for determining the parameters of the Burgers model using this division is presented. The original value of the article is also the test results themselves and the parameter values of the analyzed model for triaxial creep of shales, which are not numerous in the literature.

3. In our studies of shale creep, one confining pressure was applied, related to the depth of the shale gas deposit, and the Burgers model was used to describe the creep in describing shear and volume changes. The research focused mainly on determining the parameters of the adopted model and demonstrating the different creep of individual shale formations. These assumptions limit the scope and possibility of comparing the obtained results with those published in the literature. However, in the article, after the presentation of the obtained results, a discussion was added in the context of those previously obtained in the literature on shales from the Baltic basin and others. However, it was not decided to tabulate and compare the results obtained with those in the literature.

Reviewer 3 Report

The paper under review is devoted to a study of the rheological behavior of  shales from the  Baltic Basin in Poland. This study is important for prediction of the long-time deformations of rocks when hydraulic fracturing is used for enhancing gas production. The authors used the simplest Burgers model for describing creep of the matter. The theoretical part of the work is correct asnd the study in whole is interesting and useful.

However, I have some comments to the experimental part as listed below

 

1.    It is necessary to explain in more details why the stress state in three dimensional

2.    I am not sure that the choice of the height to diameter ratio of 2 is optimal for such experiments. Please, explain it.

3.    Please, remove inches as a non-system unit.

4.    Do you receive data really in a linear viscoelastic regime? This requires experimental proof.

5.    There is no words about experimental errors, spread of experimental data, confidence limits and all other statistical estimation that are necessary for such rather delicious experiments like yours.

6.    In according to point 5, it has no sense to present values in Table 1 with great number of digitals. I am sure that you can never obtain the results with accuracy better than 10%. Then only two digitals have real value. Please, reconsider Table 1 taking into account Comments 5 and 6. Real authors’ results show that this is an important issue.

7.     The list of reference seems to me too long for such an article.

Author Response

Dear reviewer, thank you for your thoughtful remarks and comment.

1. The tests were performed in a triaxial state because the results were intended to be used in work and analyzes related to shale gas extraction. The rocks and deposit lie at a certain depth, to which the value of the ambient test pressure Pconf = 50 MPa was associated. Rock creep under uniaxial loading differs from that realized under triaxial loading.

2. Test samples were prepared for laboratory tests in accordance with the standard ASTM D 4543-01. Standard Practices for Preparing Rock Core Specimens and Determining Dimensional and Shape Tolerances. In Annual Book of ASTM Standards; ASTM International: West Conshohocken, PA, USA, 2001.

3. Corrected according to the comment. 

4. The loading level of 0.7 (s1-s3)max was the limit of linear creep. This, of course, does not mean that the behavior of rocks in this respect was exclusively viscoelastic. The creep mechanism and the type of rock being tested provide important information here. It is known that at the micro level, all rocks are heterogeneous and discontinuous, and the instantaneous loading of such a medium causes permanent deformations as a result of micro slips and the propagation of micro cracks, even at low load levels. Macroscopically obtained strains are permanent and we treat them as plastic. This behavior can be observed, for example, on load and unload loops, but the values of permanent deformations are then small and can be neglected. However, at higher load levels they may assume higher values. Our tests did not analyze this type of phenomena due to much higher values of creep strains over a long period of time. According to the authors, up to the limit of linear creep, the viscoelastic model can be used to describe the macroscopic creep of samples. Only when the load threshold of linear creep is exceeded, the use of viscoplastic model is required.

5. The basis for estimating the parameters of the Burgers model equations separately for shear and volume creep (Table 1) were the creep characteristics of samples at different load levels (e.g. Fig. 2b). For this purpose, the professional statistical package Statistica and non-linear regression analysis using the least squares method were used. Due to the reviewer's comment, in Table 2 the coefficient of determination R² has been added to the results obtained for each creep test, determining the quality of the estimation of the parameters of the Burgers model equations and the correctness of fitting the model to the experimental test results. Due to the digital measurement of displacement and force (MTS system), the errors associated with the measurements themselves in experiments are small. A certain scatter in the experimental results can be observed in connection with the process of applying force and the implementation of the servo control process of the testing machine (visible in Fig. 9a). However, the greatest variability of the results obtained is caused by the heterogeneity of the test material, because the shale samples came from the drill core. No preliminary selection of research material was carried out here. This is clearly visible in Fig. 6, 7, where a large scatter of the obtained model parameters for individual formations is observed. Therefore, it was not decided to provide functional dependencies for the results presented in Fig. 7 and Fig. 8. The statistical parameters of the analyzes obtained for individual samples are satisfactory. In the article, where the results are presented (Fig. 6, 7), appropriate comments about the spread of results and their sources have been added.

6. The data in Table 1 are presented with an accuracy of two decimal places. When presenting the results in Table 1, attention was paid to the units of parameters and they were adopted in such a form so as not to present the results in the form of large numbers, and at the same time with sufficient accuracy for further analyzes (eq. 21, eq.22, Fig. 8).

7. The literature was reviewed and redundant citations were removed.

Round 2

Reviewer 2 Report

The response is not satisfactory  I do not recommend this paper in its present form. The reviewer did not prepare the response well in serious mode and not all followed the critical comments. 

Author Response

Please find our response on your comments. We compare our creep test results with those in the literature.

Author Response File: Author Response.pdf

Reviewer 3 Report

I accept all authors’ comments, except presenting data in table 1.

 Your answer about two decimal digitals is incorrect, since you present up to 6 significant numbers. It means that you guarantee the presented value with accuracy of 0.001% that is senseless. I strongly recommend presenting the figures in accordance with the confidence limitl of the measurement accuracy.  This is necessary for publication the article.

Author Response

The data in the table have been corrected. The confidence bounds of 0.95 is also marked in the figures.

Author Response File: Author Response.pdf

Round 3

Reviewer 2 Report

I think now its looking fine.