the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
A pulse-decay method for low permeability analyses of granular rock media
Tao Zhang
Behzad Ghanbarian
Derek Elsworth
Zhiming Lu
Abstract. Nano-darcy level permeability measurements of porous media, such as nano-porous mudrocks, are only practically feasible with gas invasion methods into granular-sized samples with short diffusion lengths and thereby reduced experimental duration; however, these methods lack rigorous solutions and standardized experimental procedures. For the first time, we resolve this by providing an integrated technique (termed as gas permeability technique) with coupled theoretical development, experimental procedures, and data interpretation workflow. Three exact mathematical solutions for transient and slightly compressible spherical flow, along with their asymptotic solutions, are developed for early- and late-time responses. Critically, one late-time solution is for an ultra-small gas-invadable volume, important for a wide range of practical usages. Developed as applicable to different sample characteristics (permeability, porosity, and mass) in relation to the storage capacity of experimental systems, these three solutions are evaluated from essential considerations of error difference between exact and approximate solutions, optimal experimental conditions, and experimental demonstration of mudstone and molecular-sieve samples. Moreover, a practical workflow of solution selection and data reduction to determine permeability is presented by considering samples with different permeability and porosity under various granular sizes. Overall, this work establishes a rigorous, theory-based, rapid, and versatile gas permeability measurement technique for tight media at sub-nano darcy levels.
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Tao Zhang et al.
Status: final response (author comments only)
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RC1: 'Comment on egusphere-2022-1471', Anonymous Referee #1, 21 Feb 2023
Please find my comments and suggestions in the attached pdf file.
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CC2: 'Reply on RC1', Tao Zhang, 27 Jun 2023
We express our gratitude for the thorough review and insightful remarks on our work, particularly a total of 42 detailed comments addressing punctuation, word choice, sentence structure, and overall organization of this manuscript. We have carefully reviewed each comment and made the corresponding revisions accordingly, which improve the presentation and clarity of the work. We also attached the revised word file here, thank you!
- Line 23 – 25: Nano-darcy level permeability measurements … are frequently conducted with gas invasion methods into granular-sized samples with short diffusion lengths and thereby reduced experimental duration;...”
- Line 70: we revised the “e.g., 254 cm in diameter” to be “consolidated cm-sized core-plug samples”
- Line 81: we revised the “confounding” effects as the “side” effects.
- Line 92: 10-60 mesh has been explained specificity to be 0.67 mm to 2.03 mm.
- Line 108: “The rest of this article is organized as follows.” has be deleted in the revised manuscript.
- Line 150: Reference error has been corrected.
- Line 161 The second reference has been deleted.
- Line 198: hold has been corrected as holds.
- Line 208: “ … helps TO select …”
- Line 213 – 215: The second explanation of Kc has been deleted.
- Line 221: delete “plotted in”
- Line 223 & Fig. 1: We have extended the x-axis to the point in the plot of 50.
- Line 249: We have rephrased the sentence as “… where Kc equals 10 or 50 are…”
- Line 259: “… a minor difference but become very close …” This sentence has been revised as “ there difference is very small especially for10”.
- 2(b): We’ve used different color schemes to enable it more visible.
- Line 291: “Happen” has been revised as “occur”.
- Line 297-298: “For example” has been revised as “This is particularly noticeable for”.
- Line 302: for “600s around for 0.1 nD”, the word of “around” has been deleted.
- Line 305: The info in the parentheses has been rephrased as “The mudrock samples we tested, with results presented in Section 5.3, exhibit low permeabilities, approximately on the order of 0.1 nD.”
- Line 327 ff: add space between equal sign and parameter/numbers. This has been corrected for the whole paragraph (from Lines 326-332).
- Line 334: The word “new” has been revised as “newly”.
- Line 341-344: We have relocated the paragraph from Lines 341-344 to the suggested position between Lines 345 and 346; thanks for the comment.
- Line 342: We’ve unified the term as mudstone.
- Line 352: We’ve rephrased as “or”.
- Line 353: We’ve rephased as “ S2 shows how”.
- Line 356: The sentence has been simplified as “ 5 shows the pressure variance with time during the experiment using sample size from 0.34 mm to 5.18 mm for sample X-1 and sample X-2.”
- Line 359: Quotation marks have been applied to “Penetration zone” here.
- Line 370: Psi has been converted to SI units in parentheses, and 50 and 200 psi is 0.345 MPa to 1.38 MPa.
- Line 397 – 401: We’ve deleted the content between Lines 397-401 of “Though the liquid permeability is not complicated by the gas slippage effect, the liquid test is difficult in achieving the flow state of Knudsen number greater than 10-3, which normally occurs in the ultra-low permeability media. Therefore, gases are chosen, and practically needed, as the testing fluid in this work.”
- Line 404: We have rephrased the “behavior for a sample size of 0.675 mm (average granular diameter)” to “behavior for sample size with an average granular diameter of 0.675 mm”.
- Line 408: About “For pressure range, argon … ”, this sentence has be revised as “In terms of pressure drop, argon exhibited the most significant decrease.”
- Line 424: For “slow equilibrium time” , we’ve changed as “slow equilibrium process”.
- Line 429: “Adsorption… (Busch et al., 2008)” ; this sentence has been deleted.
- Line 439: “would provide more analyzable data to determine the …” has been revised as “would provide more data to be analyzed for determining the permeability”.
- Line 440 – 441: We’ve added “is” and “and” in this sentence.
- Line 442: “The” has been revised as “This”.
- Table 2: “size (mm)” has been changed as “granular size (mm)”
- Line 466: The description related to the Size A has been replaced by the value of 1.27 mm.
- Line 484: We’ve deleted the crystalline rocks here.
- Line 495: We’ve added the description that 2 mm is the
- Line 527: We’ve revised sentence to improve clarity “Of the three derived solutions, one is valid during early times when the gas storage capacity approaches infinity, while the other two are late-time solutions valid when is either small or tends towards infinity.”
- Line 540: We tested each of the samples twice, and the results were demonstrated in Table 2, from the results there, we think the data demonstrated a good repeatability.
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AC1: 'Reply on RC1', Qinhong Hu, 08 Aug 2023
Thank you for your comments and review, we have carefuly revised the paper based on your suggestions, please refer the reply by Tao Zhang. Thank you.
Citation: https://doi.org/10.5194/egusphere-2022-1471-AC1
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CC2: 'Reply on RC1', Tao Zhang, 27 Jun 2023
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RC2: 'Comment on egusphere-2022-1471', Anonymous Referee #2, 25 Jul 2023
In this manuscript, Zhang et al. present an improved method of analyzing pressure pulse decay measurements to obtain permeability values in low-permeability porous media. The method builds on previous work by Cui et al (2009).
Overall the mathematical derivation is satisfactory, although it is largely identical to that offered by Cui et al. My biggest concern is that it is not clear what shortcomings from Cui the authors are trying to address. There are some vague statements about Cui et al. lacking "detailed analyses" of Kc and tau (at line 187) and lacking a discussion of "practical applications" of their work (at line 191), but a concise problem statement is lacking. So the novelty of the work is not clear to readers. I recommend revising this paragraph with a series of direct statements about specifically what was missing from Cui's work and how the present work addresses those shortcomings.
In table 2, it would be helpful to have a comparison between results of the present work and those using Cui's methods to demonstrate improvement.
Citation: https://doi.org/10.5194/egusphere-2022-1471-RC2 - CC3: 'Reply on RC2', Tao Zhang, 27 Jul 2023
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AC2: 'Reply on RC2', Qinhong Hu, 08 Aug 2023
Thank you for your review and suggestions. We have carefully considered your comments and made the revisions accordingly. Meanwhile, we have included detailed explanations from both the mathematical and experimental perspectives regarding the method we proposed. For a more comprehensive explaination of the innovation, please refer to Tao Zhang's reply. Thank you.
Citation: https://doi.org/10.5194/egusphere-2022-1471-AC2
Tao Zhang et al.
Tao Zhang et al.
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