The present article utilises persistent homology (PH) to estimate the permeability of a single fracture (12 by 45 cm length) in sandstone from high-resolution scans of the fracture surfaces. Results are compared to permeability estimations from numerical simulations and from measurements using an air permeameter. Results are interesting because they validate the use of PH as a method for permeability estimation. However, there are significant issues that I believe should be addressed before this work could be considered for publication. The manuscript states that its goal is to address the influence of roughness on fracture permeability, but no parameters of the fracture roughness are provided. I strongly recommend including a roughness quantification for the fracture in the paper (e.g. roughness exponent from power spectral analysis). In addition, the method of PH should be described more rigorously, and the discussion makes some questionable comparisons. Please find below my detailed comments.

I have made some language suggestions that the authors can decide whether to incorporate or not, but the manuscript would greatly benefit from thorough language editing.

Abstract

L11: the topological method of persistent homology

L13: the permeability of fracture networks

L14: delete on

Introduction

L32: to study how fluid flow is influenced...

L35: cheap,

L43: References on hydro-mechanical coupling that you might consider: Nara et al. 2011 (Tectonophysics), Perez-Flores et al. 2017 (Journal of Structural Geology), Kluge et al. 2021 (JGR: Solid Earth), Stanton-Yonge et al. 2023 (JGR: Solid Earth).

L47: I suggest re-phrasing to: Flow-through experiments allow to investigate direct fluid flow through fractures. The flow distribution and preferred flow paths can be predicted by replicating the fracture geometry in transparent materials.

L49: Besides flow-through experiments, air permeameters can be used to determine the permeability of fractures…

L50: Air permeameters allow…

L51-53: Please re-phrase, it is unclear to me what you mean by ‘edge’ (fracture edge?, outcrop edge?).

L64: Delete however.

L67: without experiments or numerical simulations.

L70-71: I recommend reframing the wording to adopt a more formal and technical style.

L72: from big data.

L74: Consider re-phrasing to: ‘TDA is an analysis method that focuses on the structure of data within the field of algebraic topology, demonstrating particular strengths in handling data types such as images, complex structures, and networks’.

L75: what do you mean by ‘robust to noise’?

L76: delete most.

L77-78: I suggest including a more detailed and technical explanation of persistent homology, rather than ‘capturing how holes appear and disappear’.

L78: delete already.

L79: and biology

L80: consider re-phrasing to: In geosciences, this approach has only been applied in the past decade to characterize porous rock and to determine their permeability.

L82: define acronym (PH) in line 76.

L84-87: In these small-scale 85 (millimeter to centimeter scale) studies, the effect of fracture roughness was either found to be not crucial for flow behavior (specific reference?), or was not particularly investigated (specific reference?).

L88: delete hence

L89: to estimate its permeability. The focus is on the influence of fracture roughness on the flow behaviour..

L90: Please re-phrase and explain on what you are trying to test the influence fracture roughness resolution. I suggest: ‘In addition, in order to investigate the influence of fracture roughness resolution on (fracture permeability estimations?)…’

Methods

L98: with a length of

L99: Re-phrase to: ‘Previous studies have already characterised relevant hydro-mechanical properties of Flechtinger sandstone such as…’.

Please indicate (at least) the porosity.

L103: the range of 0.1-10mD is extremely wide. Are these results from different confining pressure conditions? Please specify. Also, a matrix permeability of 10mD (and of 1) is quite significant and does not allow this rock to be considered ‘almost impermeable’. You could justify that matrix permeability is negligible when compared to fracture permeability if the latter is several orders of magnitude higher than matrix permeability. If so, please specify the magnitude of fracture permeability.

L104: Please specify what Hale et al. (2020) and Hale and Blum (2022) found in relation to the fracture permeability of this same fracture.

L122: was then determined by applying...

L137: The assumption of impermeable matrix needs a better justification.

L171: have parallel plate geometries…

L172: I would mention that this relationship is based on the Cubic Law for flow through an open fracture and cite e.g., Lomize (1951), Zimmermann and Bovardsson (1996), Witherspoon et al. (1980), etc.

L174-175: The channel aperture h_i is the critical parameter for this study. I suggest providing a better explanation of how h_i is calculated from d_i (isn’t d a time parameter?) and delta (I believe that delta hasn’t been defined till now).

Section 2.4. I suggest mentioning the results by Hale et al. 2022 in the introduction. Because no new results from air permeability were generated by this study (if I understand correctly, if not, please be clearer), I recommend removing this section from methods.

L198: Apart from experimental…

L208-209: report results in the following section.

L219: Although it is true that the data shows that kx>ky for the three datasets, for the two lower resolution datasets the difference between kx and ky can be considered negligible. For me, the interesting result here is that the higher resolution dataset gives a higher anisotropy in permeability.

L221: Detailed examination of the individual fracture surfaces and the matched fracture shows that…

L222: This is first time that this barite vein is mentioned. I suggest mentioning it when describing the sample (section 1.2).

L231-233: I find this sentence confusing. Why would the permeability results from the different resolution datasets fit each other?

L261: PH doesn’t overestimate permeability with respect to simulations for almost half of the results. I don’t think that this is a pertinent observation.

L262: I wouldn’t call 10^-11 a ‘low’ permeability. Consider changing to ‘relatively lower permeabilities’.

L268: results from this study do not ‘fit’ those of fracture networks but have permeability values that are closer to fractures compared to those from porous rocks (which is hardly surprising).

L271: Again, they are not ‘all overestimated’.

L272: I don’t think that you can conclude that there is only a minor influence of roughness on the PH analysis only because your permeability results are on the same range as those from Suzuki et al. (2021). First, you don’t provide roughness measurements of the fracture sample, and therefore there are no quantitative parameters that allow to assess whether this fracture is rough or smooth. In fact, you call it ‘rough’ in the title and in line 270, and then ‘relatively smooth’ in line 274. I strongly recommend including a roughness quantification for this fracture in the paper. I don’t know if PH allows to determine a roughness estimation, but your high-resolution scans would allow you to, for example, follow a workflow such as that by Candela et al. 2012, and determine the roughness exponent H from a power spectral analysis. There are many works that have used this approach, which would allow you to assess comparatively how rough/smooth is this fracture.

Second, the permeability of fractures has been shown to have fall within a very wide range of magnitudes (e.g. Walsh 1981, Kranz et al. 1979, Iwai 1976, Nara et al. 2011, and many others). That your permeability results happen to fall within the same range as those from Suzuki et al. (2021) seems coincidental, considering that you are comparing different scales and flow paths (fracture networks in a 5 cm length sample vs a single fracture of 12 by 45 cm length). I therefore don’t think that you can derive meaningful conclusions from this comparison.

I also don’t understand why you discuss your results in the context of the Cubic Law assumption of parallel plates (L273-274), when PH analysis obviously considers the topography of the fracture surfaces.

Before presenting my comments, I would like to point out that my experience relevant for reviewing this manuscript is in topology-based methods for estimating fracture network connectivity, and geological analysis of fracture networks. Perhaps experience in geomechanical simulations of fracture networks is also useful. I am, however, no expert specifically in Persistent Homology (PH), or in laboratory or numerical flow simulations for fracture permeability estimation. Thus, for some of the questions or comments, an explanation might suffice.

Summary and general comments

The manuscript presents a novel topology-based method, Persistent Homology (PH), to estimate single fracture permeability from high-resolution imagery. The study demonstrates this application of PH using a sandstone sample and compares the resulting permeability estimates to values obtained with established methods for permeability measurements (air permeameter and numerical simulations). The main result of the study is that it validates the proposed use of PH, as it provides similar values for permeability as the established methods. The presented data and results are certainly interesting and of fair to good scientific value.

However, there are several issues in the presented manuscript that require major revisions before the publication of the study would seem justified. From my point of view, the main issues are:

• There is a strong mismatch between the stated focus of the study and the presented results. The authors write that “The focus is on the influence of roughness of the fracture surfaces on the flow behavior and the determination of the permeability distribution across a natural bedding plane fracture.” Yet, the manuscript contains no quantitative data on roughness or any other comparison of permeability between fractures of varying roughness that would allow the intended assessment. Permeability distribution is also limited to a very brief presentation of permeability anisotropy, which is also not satisfactory if this is the focus of the study.
• In part, the comparisons made with other studies in the results and discussion sections are difficult to justify, or they are not explained well enough. For instance, I had trouble seeing how absolute permeability values of a single fracture of a real sandstone can or should be compared to permeabilities of fracture networks in an unspecified material (probably synthetic). The same applies to the comparison with trends of PH-derived permeability of porous media.
• The authors conclusions on overestimation of permeability when using PH do not seem to convincingly match their own data. Note that is not necessarily a bad thing, because the presented permeability estimates match those from the other methods rather well.
• One interesting and perhaps critical feature in the sample, a barite vein, is qualitatively interpreted to have important influence on the fracture permeability. However, this feature is not mentioned in the sample description, nor is it described quantitatively.
• Overall, I think that the rich high-resolution image data would allow for much more nuanced and quantitative analysis and discussion. The quality of the study would greatly benefit from including quantification of the fracture’s properties (roughness, spatial variability) and combining that data with PH-derived permeability.

In addition to the scientific content, the authors might consider performing extensive language editing for conciseness and more clarity.

Provided that the authors address both general and specific comments in a thorough manner, I do believe that the study has the potential to make a valuable contribution to the state of the art in the field of fractured rocks and fracture flow.

Specific comments:

L41: rephrase. Emprical methods may be simplistic and prone to errors, but are typically also based on scientific results. They just try to put those into simple relations for fast practical application.

L51: do you mean spatial distribution? this sentence is a bit unclear

L59: Please clarify: What is the issue with testing different confining pressures in the lab?

L61: remove "it has to be considered"

L64: remove “However”

L70/71: In my view, this is an inaccurate statement and not of much use for this study anyway, since AI is not used for comparison. Please remove or rephrase (e.g., along the lines of: "adequate use of AI requires deep technical understanding, rigorous testing and sufficient amounts of training data")

L72: remove "crucial"

L76: "leading": do you mean "used"?

L85: did the studies demonstrate that roughness was not important, or did they neglect the effects? Rephrase to clarify, as it seems that this is one of the motivations for this study.

L89/90: This is a major issue, as the aims stated here do not match the presented results later on. Neither the influence of roughness on the flow behaviour nor the permeability distribution is not shown in any quantitative way. Roughness is not quantified at all, and a single fracture does not allow one to examine the effect of roughness on fracture permeability – one needs a comparison between fractures of different roughness if the stated goals are to be reached. Permeability distribution is mainly addressed as x/y anisotropy, which is also insufficient if one wants to *focus* on this.

L102/103: Two comments here: 1) if previous studies characterized fracture permeability of this rock, this is also very much of interest. In general, please expand a bit on the sample - e.g., the authors mention a barite vein later on which seems to have an important effect on the results - this needs to be mentioned here. 2) 10 mD is substantial permeability. In general, quite a large range. Please rephrase

L136: “no filling or sealing”: Later on, the authors show that there is a barite vein with potentially strong impact on the results. That does not seem to fit with this statement.

L137: see above comment on permeabilites. The stated range of 0.1-10 mD cannot justify this assumption in my view. Would violating this assumption have an impact on the results?

L156: rock “skeleton” – rephrase to “matrix” ?

L171/172: From the text, I understand this such that each channel is represented by its smallest aperture. If this the representation of "roughness" in the analysis (i.e., using the smallest aperture and applying a parallel plate model), I would encourage the authors to discuss this "roughness" in comparison to other methods.

In general, this relates to the comment made for the goal, or focus, of the study as presented in the introduction (impact of roughness on flow behavior and permeability distribution across the fracture). While I can at least imagine how the latter could be assessed, it is unclear to me how roughness is characterized based on the presented method. This would correlate with the lack of quantitative roughness analysis in the results.

L176: please specifiy the parameters di and delta. If I understand correctly, this should be smallest aperture image resolution?

L178/179: please add explanations for the meaning of each colour to the caption.

L198: replace „despite“ with „in addition to”

L203: please specify (1) the value of this parameter, and (2) how the authors chose it. This choice would seem to be critical for the resulting fracture permeability

L205: Please add definitions of all parameters in equation (2)

L208: Please briefly state how these values are obtained (cubic law I suppose)

L211: This paragraph lacks the promised evaluation of the impact of roughness on flow behaviour and permeability distribution (this is limited to a very brief description and discussion of permeability anisotropy)

L221-223: See comment on sample description - this is the first time that the barite vein is mentioned. The authors should expand on this analysis and the description of the vein. For instance, it is not explained whether this vein is a solid mineralization (effectively blocking void space), or aids to hold void space in the fracture open (providing flow paths).

As long as no quantitative data are used to support the impact of the vein, this remains rather qualitative speculation in my view (not unjustified perhaps, but not very well supported). For instance, the authors could exclude the part with the vein from the analysis and compare the results.

This approach could improve the study by delivering an analysis of permeability distribution that was declared as a key focus of this work in the introduction.

L229: remove “Nevertheless”

L229/230: if the ratio is indeed about 1.0-1.1, the whole anisotropy discussion seems a bit overemphasized, as kx and ky are essentially the same. In this regard, I would invite the authors to add a discussion of potential scale effects. One interesting observation is for instance that the 50 µm resolution data delivers the highest permeability value in x-direction, but the lowest in y-direction.

Could there be a bias introduced by the sample dimensions in combination with the vein covering almost all of the x-dimension, but only a small fraction of the y-dimension?

L237: Figure 4 basically contains a table along with a detailed view of the sample. This should be divided into a table (toward the result section) and the figure (could alternatively be moved to the sample description/method section)

L251: I am a bit unsure if the few data points here suffice to make such a general conclusion, even though the explanation given seems valid.

L261/262: I have trouble understanding this statement. Is the "previous study" Suzuki's study of 2021?

In this case, the finding seems to be of little use, because it compares permeability values of (1) a fracture network and (2) a single rough fracture.  If the comparison is made to previous studies investigating the same fracture (or at least sample/rock type), this does not become clear. From browsing Suzuki (2021), it seems that they investigated synthetic fracture networks – this would make a direct comparison of results to this study very problematic and not very useful. Please explain and revise.

L261-263: I am unsure if this conclusion can be made based on the data, especially given the small number of data points and the variability of over/underestimation with resolution. Also, is it actually PH itself that leads to overestimation, or that authors are locally employing parallel-plate/cubic law for permeability estimation?

L264: which other results do the authors mean?

L268-271: Is the difference between this study and Suzuki (2021) mainly the number of fractures? This would make it a bit more plausible to compare the results - although we still do not know any details on the materials that are being compared here. Please clarify.

L272: I am doubtful about this statement, as the authors provide no data on roughness, and previously stated that the studies of Suzuki did not thoroughly investigate the effect of surface roughness on flow properties.

L273: I suggest to reconsider this conclusion and rephrase more precisely. If I understand correctly, the method explicitly takes local fracture topography into account and locally (!) applies a cubic law for each channel. To me, this is not the same as applying the cubic law to a single fracture, whether it is rough or not.

I do agree that the results of the study suggest that estimates of permeability can be achieved using PH that are comparable to established methods. However,  the presented conclusion cannot be drawn from the presented data.

L274: replace “in large agreement” with “largely in agreement” or “overall in good agreement”

L284: except for use of PH to estimate permeability, please provide more information on why the results of porous and fracture rocks should be compared here. Are the results even comparable? I have not read the studies on porous rocks, but would assume that they estimate permeability in a very different way (not cubic law)?

L288: “Previously”: Please clarify by rephrasing. Do you mean "in previous sections of this manuscript", or in "previous studies"?

L306: This seems indeed to be the main conclusion which can be drawn from manuscript in its current form. And this is valuable.

L307: Consider rephrasing, because this sounds like PH results are less valid outside of the stated range. However, I see no data here to support such a constraint. To me, the main point seems to be that PH works well for single fracture permeability estimation (see previous comment for L306).

L308: see previous comments on conclusions about overestimation (L261-263)

L309: delete “However”

L312/313: Is this cheap in comparison to other methods? Could you provide a cost estimate to support this regarding the cost of imaging equipment vs. more conventional methods? In the discussion, it seemed like 100 µm resolution was a good compromise to get good results. Is this achievable with a cheap method like SfM or a cheap laser scanner to get such resolutions? If not, perhaps it is not so cheap after all.

L326: remove “however”.