the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Investigating rough single fracture permeabilities with persistent homology
Anna Suzuki
Togo Hasumi
Philipp Blum
Abstract. The permeability of rock fractures is a crucial parameter for flow processes in the subsurface. In the last decades different methods were developed to investigate on permeability in fractures, such as flow through experiments, numerical flow simulations or empirical equations. In recent years, the topological method persistent homology was also used to estimate permeability in fracture networks and porous rocks, but not for rough single fractures. Hence, we apply persistent homology analysis on a decimeter-scale, rough sandstone bedding joint. To investigate on the influence of roughness, three different data sets are created to perform the analysis: (1) 200 µm resolution, (2) 100 µm resolution and (3) 50 µm resolution. All estimated permeabilities were then compared to values derived by experimental air permeameter measurements and numerical flow simulation. The results reveal that persistent homology analysis is able to estimate the permeability of a single fracture even if it tends to slightly overestimate permeabilities compared to conventional methods. Previous studies using porous media showed the same overestimation trend. Furthermore, expenditure of time for persistent homology analysis as well as air permeameter measurements and numerical flow simulation was compared which showed that persistent homology analysis can be also an acceptable alternative for conventional methods in this regard.
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Marco Fuchs et al.
Status: open (until 20 Oct 2023)
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RC1: 'Comment on egusphere-2023-1855', Anonymous Referee #1, 22 Sep 2023
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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 hi is the critical parameter for this study. I suggest providing a better explanation of how hi is calculated from di (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.
Citation: https://doi.org/10.5194/egusphere-2023-1855-RC1
Marco Fuchs et al.
Marco Fuchs et al.
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