Pore Structure and Permeability of Tight-Pore Sandstones: Quantitative Test of the Lattice–Boltzmann Method

Round 1

Reviewer 1 Report

The paper presents the characterization of the porous structure of tight-pore sandstones and examines the application of Lattice Boltzmann (LB) simulations of their permeability. The topic is important and interesting to the readers and appropriate for the scope of the Fluids journal. The introduction provides sufficient background and references.   The research design and the LBM methods for characterizing the porous structure of tight-pore sandstone are described. The results are presented. Overall, the paper is worthy of publication. However, some modifications need to be made, and the following comments are considered in a revised version.

1.       It is better to explain the LBM simulation procedure in more detail and discuss the reason for determining the Palabos, MRT collision operator, HBB-BC,  compared to other methods. What is the strong point of this method?

2.       I suggest adding the statement regarding the limitation of LBM simulation results because only the specific samples, assumptions, boundary conditions, and methods are used.

3.       Please explain the experiment condition, Direct simulation, and DHS simulation procedure for validation in more detail.

4.       It suggests rewriting the conclusion more scientifically and technically.

Author Response

Please find the response to your comments. Our reply in in yellow, the additions to the text are given in gray. If you want to see the changes in revision mode, file with document comparison is attached as a supplementary materials for review

The paper presents the characterization of the porous structure of tight-pore sandstones and examines the application of Lattice Boltzmann (LB) simulations of their permeability. The topic is important and interesting to the readers and appropriate for the scope of the Fluids journal. The introduction provides sufficient background and references.   The research design and the LBM methods for characterizing the porous structure of tight-pore sandstone are described. The results are presented. Overall, the paper is worthy of publication. However, some modifications need to be made, and the following comments are considered in a revised version.

1. It is better to explain the LBM simulation procedure in more detail and discuss the reason for determining the Palabos, MRT collision operator, HBB-BC,  compared to other methods. What is the strong point of this method?

         We added the following description to section 2.2("Materials and methods. Lattice Boltzmann simulations")

2. I suggest adding the statement regarding the limitation of LBM simulation results because only the specific samples, assumptions, boundary conditions, and methods are used.

         We added the following description to sections 1 and 2.2 ("Introduction" and "Materials and methods. Lattice Boltzmann simulations")

3. Please explain the experiment condition, Direct simulation, and DHS simulation procedure for validation in more detail.

         We added the following description to section 2.1("Materials and methods. The samples")

4. It suggests rewriting the conclusion more scientifically and technically.

     We agree with the reviewer and thank him for the comment. The discussion of the limited size effect has been moved to the subsection 3.2. "Pore network permeabilities". The comparison with experiments has been added to the conclusion. We tried to make the conclusion more precise and reflect the outcome of the paper. The conclusion has been rewritten as follows:

Our main conclusion is that LB with multiple relaxation times and half bounce back boundary conditions at the pore walls quantitatively predict permeabilities with a reasonable accuracy even in tight pore sandstones of low porosity. For samples of high porosity and wider pores(1S1, 1_S2, 1_S9,1_C1) LBM results showed a somewhat better agreement with the experiments than the state-of-the-art direct Navier--Stockes solvers. For tight-pore samples, the agreement between LBM and other state-of-the-art calculations is satisfactory: the difference between LB permeabilities and those obtained by DiMP simulations is of the same order as the difference between two direct simulation methods, DiMP and DHD.

    Generally LBM tends to:

• somewhat overpredict the permeability of very tight pore least permeable samples due to underestimations of fluid—wall friction caused by the application of half bounce-back boundary conditions
• somewhat underestimate the permeabilities of more permeable samples with open porosity due to a neglect of the compressibility effects

The permeability show, as expected, a strong correlation with the overall effective porosity and the pore volume distribution, in particular. However, the parameters of the network connectivity, in particular the pore throat coordination numbers, do not appear to influence permeability in the sample considered in this work. It is not clear how general this observation is, because the pore structure is related to the process of formation of geological samples and might be quite different for different type of rock formations. For some manufactured materials, the connectivity characteristics proved very important for fluid transport.

Speaking of requirements to the resolution of the digital porosity maps, this work confirms that the pore width should be larger than at least four voxels. If it is not the case, the LB solver experiences stability problems and shows poor convergence. In order to obtain a more stable simulation, the map can be scaled by the factor of two and the walls interpolated to achieve a smoother structure. What remains unclear is the role of the box size in comparison with the pore network feature size. Out geometrical analysis showed that in all samples considered the box is not much larger the topological features of the pore network: all maps showed a substantial anisotropy. For wide-pore sandstones that did not seem to create a problem as the calculated permeabilities compared well with the experimental values. For tighter samples, where the scale of the pore network geometric features is expected to be larger in comparison with the available box size, the effect of the limited map size is expected to be more severe, but only the comparison with experiments can answer how important the anisotropy effect is.

Author Response File: Author Response.docx

Reviewer 2 Report

In this article different single-phase flow permeabilities from experiments and numerical simulations from the literature are compared to LBM permeabilities that were determined in this study. The study is conducted using different types of digital rock microstructures with varying porosity and pores size (distribution).

The study itself including motivation and storyline is poor and not well organized. It remains unclear what the novelty of this study is. In addition, the manuscript should be improved concerning discussion of the results, readability, and also the written English could be smoothed and improved.

All in all, I cannot recommend the publication of this review.

Comments for author File: Comments.pdf

Here and there are typos or duplications of words. The sentences are often long and not very precise such that the main messages get lost. All in all, the quality of English language could be smoothed and improved.

Author Response

We thank the reviewer indeed for valuable comments. We did our best to revise our paper. Please find the response to your comments. Our reply in in yellow, the additions to the text are given in gray. If you want to see the changes in revision mode, file with document comparison is attached as a supplementary materials for review. We trust the paper is now publishable.

Text comments:

In this article different single-phase flow permeabilities from experiments and numerical simulations from the literature are compared to LBM permeabilities that were determined in this study. The study is conducted using different types of digital rock microstructures with varying porosity and pores size (distribution).

The study itself including motivation and storyline is poor and not well organized. It remains unclear what the novelty of this study is. In addition, the manuscript should be improved concerning discussion of the results, readability, and also the written English could be smoothed and improved. (общ форма)

All in all, I cannot recommend the publication of this review.

Comments from PDF review:

 In this article different single-phase flow permeabilities from experiments and numerical simulations from the literature are compared to LBM permeabilities that were determined in this study. The study is conducted using different types of digital rock microstructures with varying porosity and pores size (distribution).

 

The study itself including motivation and storyline is poor and not well organized. It remains unclear what the novelty of this study is. In addition, the manuscript should be improved concerning discussion of the results, readability, and also the written English could be smoothed and improved.

 

All in all, I cannot recommend the publication of this review.

More specific remarks and small comments are given below:

 

GENERAL COMMENTS:

 

1. The introduction gives a good overview of the current literature in the field of digital rock physics. It might be extended to briefly discuss which other research fields make use of LBM for permeability calculations and how the agreement between simulation results and experiments is there. More important, even at the end of the introduction it remains unclear what the remaining gaps and open research questions in this area are, how the authors motivate their study, and what the great novelty of this study is.

         We substantially changed the introduction. In particular, the following fragments were rewritten to emphasize the novelty and significance of the work:

 

The main obstacle to the routine engineering applications of LBM in fluid flow simulations is a lack of understanding of applicability in "tight" systems, that is, when the feature size (be it the pore width, fiber thickness, etc) becomes comparable with the effective voxel size. (pages 2)

 

And yet, success has been almost exclusively reported with structures of moderate to high porosity ($\varphi>$0.2) with networks of reasonably large pores (0.1-2 mm) and developed pore space~\cite{Saxena2017,Boek2010,Zambrano2021,Zambrano2018}.  The limites of the applicability of LBM in tight pore media remain unexplored and are of substantial interest,  recent progress in LBM simulations of complex fluids and the trend towards the development of tight pore hydrocarbon reservoirs with a very little of open porosity, where structure--permeability relationships may be decisive for efficient filtration. The pore structure -- permeability relationships have been studied in the literature. Attention was paid to characteristic pore length, overall volume, and tortuosity~\cite{Berg2014}; Euler characteristics~\cite{Liu2017}; connectivity and pore size distribution~\cite{Soares2017}.  But again, practically all these studies deal with more porous systems and the structure of tight sandstone collectors and the relationship between the topology and permeability remain unclear.  In this paper, we evaluate the accuracy of single phase LBM simulations in sandstone and carbonate samples of different pore sizes, digital sample sizes and resolution. We compare samples from different sources and validate the LBM simulations against experiments on permeability and direct NS solvers, considered here as the state of the art. The other goal of the paper is to relate the permeabilities to the structural features of the rock pore network, namely throat sizes and coordination numbers that characterize pore connectivity along the pore size distributions and pore network tortuosity. (page 3)

 

2. Line 101 pp.: The formulation is confusing. First, the authors state that “[their] target samples are classical tight sandstones […]. They form two groups.” Then, a few lines later they discuss a “group 3” and “group 4” which have not even been mentioned before.

 

         We have rewritten this section 2.1 (pages 3-4).

 

We start with relatively wide pore sandstones from the Imperial College of London (ICL) collection~\cite{ICL_ds}. The resolution of the 3D maps is 2.85 to 10.0 \textmu{}m /voxel, (lower than the other samples considered here) and the physical size ranges from 1.155 mm to 4.5 mm. The structures are mapped on relatively coarse grids, $300^3$ to $450^3$ voxels. For these samples, both experimental data and the Pore Network model simulation (PNM) of permeability are available from the literature. ~\cite{Raeini2014,Dong2009}. The experiments were performed in the most classical manner: by injecting fluid (water) mass flow with the pressure measured on inlet and outlet of each sample.

 

Group 2 also contains select samples from the ICL collection~\cite{ICL_ds}: two sandstones and two carbonates. For these samples, we did not find experimental data, but their permeability was extensively modeled in the literature ~\cite{Raeini2017, Suzuki2021, Muljadi2016}. In average, they are less porous and less permeable than the samples from group 1. They are selected for this study because the pore structure maps available for them are much larger (1000$^3$ cubic voxels) than for the samples from group 1, which makes the maps suitable for pore network characterization. As a benchmark for permeability calculations, we used the finite volume CFD simulations from Raeini et al.~\cite{Raeini2017} .

   

Our target samples are classical tight sandstones from the Achimov formation~\cite{Orlov2020_ds}. They form two groups, here referred to as group 3 and group 4 . For both groups, the CT images were obtained with spatial resolution 1.2 \textmu m/voxel (that is, substantially finer compared to the maps for groups 1 and 2).  For group 3, the 3D pore maps were reconstructed using an algorithm that utilizes the spatial covariance of the image in conjunction with indicator kriging to determine object edges~\cite{Orlov2021} (Fig.~\ref{fig:1_group_structure}). The use of indicator kriging makes the thresholding local and guarantees smoothness in the threshold surface. The samples of group 4 are approximately twice as large and obtained with the same spatial resolution. They were reconstructed using a combination of \("\)heavy\("\) band-pass (Fourier transform) and bilateral filters for denoising and local thresholding.  Random walk based algorithms were applied in binarization of the 3D maps of the samples in both groups 3 and 4. For both groups, we have no experimental data, but direct simulations (DHD, DiMP) were reported~\cite{Orlov2021}.

 

3. Line 115: The density and the speed of sound are given in LB units. How can it be that the product of both is dimensionless?

 

        We write additional information on unit conversion in section 2.2 (page 4)

        All simulations were held in LB units, representing themselves 2 step conversion: from physical units to dimensionless which do not depend on real physical sizes of the system; then from dimensionless to  discretized  - in order to convert to space and time grid-related units ~\cite{Latt2008ChoiceOU}.

 

    4. Line 118: What do the authors mean by “relaxation rate(s)”? Do they mean the relaxation time? Why do the authors argue that it important to apply a MRT collision operator here, although a relaxation time of 1 was used?

 

      We have corrected a misprint of "relaxation rate(s)" to "relaxation time(s)". We added the following description to sections 1(page 2) and section 2.2 (pages 4-5).

 

The relaxation time dependencies for the wall boundary conditions may cause poor stability. The multiple-relaxation-time (MRT)~\cite{DHumieres2002,Ginzbourg1994,Ginzburg2003, computation4010011} approach, the most general and advanced relaxation model as of today, has become common practice and improved the accuracy due to generalization of equation in terms of the momenta. It also allows choosing the bulk viscosity independently of the shear viscosity, thus supporting the Galilean invariance. (page 2)

 

To avoid artificial anisotropic velocity slip at the solid boundaries caused by the choice of the relaxation time(s), we applied the MRT collision operator~\cite{DHumieres2002}. Hence we obtained more precisely the  ratio between the kinematic and bulk viscosities of the flow. (page 4)

 

 The relaxation time is $\tau=1$, providing optimal stability and accuracy of the simulation according to ref.~\cite{Asinari2010}. (page 5).

 

5. Line 132: Did the authors mean “dead-end pores” or rather isolated pores? If they really meant “dead-end pores”, how were those defined? To be more specific, what criterion was used to differentiate voxels at the inlet or open end of a “dead-end pore” that were deleted from those that were not deleted?

 

        We have corrected a misprint of "dead-end pores" to "isolated pores". We added the following description to sections 3.1(page 5).

 

The porosity of the samples is estimated as the fraction of the void space in the sample map. More important is the effective porosity that only accounts for the open pores space (all isolated voids are excluded) .

 

 

6. Section 3: The results are presented in an unstructured and confusing manner. Especially in Section 3.2 there seems to be no intuitive order of how and for which samples results are discussed. For example, the results of group 3 are discussed at the beginning of Section 3.2 while the results of group 4 are not discussed at all.

 

Following the reviewers advice, we have restructured the entire discussion, with specific attention paid to sec 3.2. We (as mentioned above) renumbered the sample groups (groups 1 and 2 now refer to the literature databases). The discussion is started from the comparison with the experiment (group 1), proceeds to tighter samples from the ICL database (group 2), and finally to the Achimov sandstones (Group 3 and 4).

 

Conclusion has been rewritten. The discussion of the limited size effects has been moved from the conclusion section to section 3.2, and conclusion reflects the outcome of the paper.

 

In general, the results section has undergone extensive and deep revisions. In addition, we tried to clarify the text whenever possible.

 

      7. Line 244: The abbreviation “DiMP” has not been introduced before.

 

We introduced abbreviation "DiMP" in section 1 (page 2).

 

Recent efforts invested in the development of direct methods~\cite{Balashov2020,Pimanov2022,Verri2017}, such as Direct Hydrodynamic (DHD) simulations~\cite{Koroteev2013} and Dimp-Hydro Solver (DiMP)~\cite{Balashov2020} substantially improved flow predictions in tight pore samples at a certain cost in computational efficiency and  versatility. 

Author Response File: Author Response.docx

Round 2

Reviewer 1 Report

The paper is well-written in a revised version. The authors addressed most of my comments. I think the paper can be accepted in its present form.

 

Author Response

We thank the reviewer for his effort to make our publication better.

Reviewer 2 Report

Although the authors tried to improve their study, I still cannot recommend the publication of this article. The reasons can be summarized as follows. The study is still poor, but more importantly its structure is confusing and not well organized. The novelty or new information of this study is limited, but the extent of the text in this article is not. Both should be well balanced, i.e. the authors should condense the amount of text appropriately. In addition, the readability of this article can be still improved a lot. In this respect, a focus should be on the presentation and discussion of results which is not straight forward. Moreover, the English language should be edited at least moderately, and the many careless mistakes (e.g. wrong use of capital and small letters, missing punctuation marks, and spelling errors, inconsistent formatting of the references) should be removed and corrected.

Regarding the content, I would strongly recommend to improve the clarity of information. For example, the authors state in their conclusion that “this work confirms that the pore width should be larger than at least four voxels”. But, where do I find this information in the study? I would expect this information to be part of Table 1. But there, for most samples the mean pore diameter is missing and cannot be correlated with the resolution. And even for the samples where it is not missing the pore diameter and resolution are wrong (due to the units (!!!)). Thus, it seems that information is hidden in bulky text passages, just implicitly mentioned, or completely omitted. This is not how a publication considered to inform and help other researchers should look like. It is not reproduceable! Another example is Figure 2. The same information is shown twice. If the reader would know the resolution of each structure, both subfigures would be redundant and contain no additional information. But here, showing both subfigures even triggers a misunderstanding. From Figure 2A and also based on logic, I assume there are no pore volumes being smaller than one voxel. Thus, the graphs sharply end at 1. However, having a look at Figure 2B, it seems that all graphs further extend to the left side, which means there would be pore volumes smaller than 1. How can that be?

Those are just examples. Listing all flaws of the study or the article would be tedious. Thus, I disagree with the authors regarding “the paper is now publishable”. Instead, I would strongly recommend to reject this article. Before it can be submitted again, the full study should be restructured and rewritten extensively.

The text contains many careless mistakes (e.g. wrong use of capital and small letters, missing punctuation marks, and spelling errors, inconsistent formatting of the references), typos, and duplications of words. The sentences are often long and not very precise such that the main messages get lost. All in all, the quality of English language should be smoothed and improved.

Author Response

The reviewer’s text is given in regular text, our answers in yellow, the quotes from the paper text in blue

We are certainly thankful to the reviewer for his effort to improve our manuscript. Since the second round review does not correlate much (in terms of the recommendations except for the general “poor study” comment) with the first round review, we have to conclude that our response to the first round was satisfactory. Most of the comments of this round are, in fact, editorial.

Although the authors tried to improve their study, I still cannot recommend the publication of this article. The reasons can be summarized as follows. The study is still poor, but more importantly its structure is confusing and not well organized. The novelty or new information of this study is limited, but the extent of the text in this article is not. Both should be well balanced, i.e. the authors should condense the amount of text appropriately. In addition, the readability of this article can be still improved a lot. In this respect, a focus should be on the presentation and discussion of results which is not straight forward. Moreover, the English language should be edited at least moderately, and the many careless mistakes (e.g. wrong use of capital and small letters, missing punctuation marks, and spelling errors, inconsistent formatting of the references) should be removed and corrected.

From this paragraph we understood that the reviewer wants us to restructure the materials, but unfortunately, does not say how exactly. In round 1, [s]he at least gave particular recommendations, which we found certainly useful and the manuscript has already been restructured according to the reviewers comments of round 1. The text is not easy to contract when it targets several groups of researchers and engineers (petroleum engineers, geoscientists and fluid flow modelers).

We moved the paragraph on the convergence problems to the methods section and removed one of the paragraphs with mostly redundant info from the discussion section.

Regarding the content, I would strongly recommend to improve the clarity of information. But, where do I find this information in the study? I would expect this information to be part of Table 1. But there, for most samples the mean pore diameter is missing and cannot be correlated with the resolution.

We followed the recommendations of the reviewer, calculated the mean pore diameter and added that information to Table 1.

For example, the authors state in their conclusion that “this work confirms that the pore width should be larger than at least four voxels”.

We stated in the text that “In LB simulations of sample 3\_C we experienced problems with convergence because of substantial number of narrow pores less than 4 voxels in diameter. That is, the CT resolution is insufficient for permeability evaluation. To avoid the issue, we artificially increased the map resolution scaling the matrix twofold and interpolating the boundary regions using the nearest-neighbor method (0-order interpolation). Such a technique is quite common~\cite{Shah2016, Orlov2021}.” (page 5, section 2.2). This qualitative observation, is, in fact, reproducible.

And even for the samples where it is not missing the pore diameter and resolution are wrong (due to the units (!!!)).

This obvious typo has been corrected (the units are, of course, \mu m, rather than m).

Thus, it seems that information is hidden in bulky text passages, just implicitly mentioned, or completely omitted. This is not how a publication considered to inform and help other researchers should look like. It is not reproduceable!

This is a serious claim that should be well based if the reviewer wants it to be considered. All results in our manuscript are reproducible. We ask the reviewer to point out non-reproducible results and the reason to believe they will not reproduce.

Another example is Figure 2. The same information is shown twice. If the reader would know the resolution of each structure, both subfigures would be redundant and contain no additional information.

The reviewer is certainly correct saying that the subfigures present the same information. But why does [s]he want the reader to perform calculations in order to obtain the distributions in microns from voxels? The reviewer’s comment would have been valid some 20+ years back, when the journals were printed on paper and figures were redrawn by publishers.

But here, showing both subfigures even triggers a misunderstanding. From Figure 2A and also based on logic, I assume there are no pore volumes being smaller than one voxel. Thus, the graphs sharply end at 1. However, having a look at Figure 2B, it seems that all graphs further extend to the left side, which means there would be pore volumes smaller than 1. How can that be?

We have corrected the smoothing effect created by the plotting software and updated F2.

Those are just examples. Listing all flaws of the study or the article would be tedious. Thus, I disagree with the authors regarding “the paper is now publishable”. Instead, I would strongly recommend to reject this article. Before it can be submitted again, the full study should be restructured and rewritten extensively.

The paper has been already extensively rewritten according to the reviewers comments of round 1. This round contains mostly editorial recommendation.

The text contains many careless mistakes (e.g. wrong use of capital and small letters, missing punctuation marks, and spelling errors, inconsistent formatting of the references), typos, and duplications of words. The sentences are often long and not very precise such that the main messages get lost. All in all, the quality of English language should be smoothed and improved.

About two dozens of small corrections have been done (see the marked up version of the manuscript).