Estimation of the Permeability of Rock Samples Obtained from the Mercury Intrusion Method Using the New Fractal Method

Round 1

Reviewer 1 Report

Dear authors,
In my opinion, your work is important, mainly when applied to shapes and tight sandstones because in this kind of pore system is, frequently, not easy to obtain adequate images enabling permeability numerical simulation.
The manuscript is ready to be published in Fractal an Fractional Journal. I would like to put only a comment: The aperture size, determined when the mercury first percolate the system, that is, not for a given mercury saturation, contains more information than a size, some information of the pore system connectivity is implicitily involved. The problem is to determine this aperture size of percolation from mercury intrusion curve of complex rocks. Perhaps, the fractal model is a summary of size distribution and also the connectivity for percolation.

Minor editing corrections: ... that that ... line 49 and ...20% saturation volume... line 50.

Author Response

Thanks for your positive feedback and we really agree with your opinion that the fractal model is a combination of the pore size distribution and the connectivity of percolation. For the editing, we changed based on your comments. Thank you very much. 

Reviewer 2 Report

In the manuscript a model to predict the permeability of the rocks is provided.

Some improvements should be done before the paper can be accepted for publication:

- avoid duplication of definitions (such as the fractal dimension D - line 75 and 79, etc.)

- Novelties of the paper should be ebtter pointed out.

- References should be improved.

Author Response

Thanks for your  great suggestions. We made the changes based on your comments.

1) avoid duplication of definitions (such as the fractal dimension D - line 75 and 79, etc.)

Response: Thanks and we revised that.

2)Novelties of the paper should be ebtter pointed out.

Response: Thanks and we added some sentences in the revised manuscirpt, 

3)Novelties of the paper should be ebtter pointed out.

Response: Thanks and we reedited the reference based on the requirement of the journal. 

Reviewer 3 Report

In the manuscript “Estimation of the permeability of the rock samples from the mercury intrusion method by using the new fractal method”, the authors present a new methodology to predict rock permeability based on data measured by mercury intrusion. More precisely, they empirically derive a prediction formula, which allows to express the rock permeability in terms of the porosity and the fractal dimension of the rock. The authors show that the prediction is best, if the fractal dimensions of small and large pores are considered separately. The fractal dimension is determined from the mercury intrusion measurements. The methodology and results are—in general—well presented. The paper elucidates the relationship between the fractal dimension of rocks and the corresponding permeability of fluid flow, which fits to the scope of fractal and fractional. However, some parts of the manuscript need further clarification prior to publication, see my comments below.

Comments:

·         The authors collect mercury intrusion measurements from different papers. Are these measurements performed in the same way? Can we assume that these values are comparable?

·         The authors claim that the new model is better than existing models from the literature. Is it possible to give a quantitative underpinning here, e.g., compare the fitting coefficients with those of the existing models (or to compute other descriptors for the goodness-of-fit)?

·         Related to the last comment: How is the fitting coefficient defined?

·         Abstract, first sentence: The authors write “Rock permeaility, defined …”. What follows is not a definition from my point of view. I suggest to write “Rock permeability quantifying the fluid flow through porous rocks is one of the most important properties.” Here one might ask, most important properties of what. This should be specified.

·         Page 1, line 40: “Kozeny” instead of “Kozney”

·         Page 1, line 41: Recently, new models for the prediction of permeability based on geometric microstructure characteristics have been derived, e.g., in [1,2]. This should be mentioned here.

·         Equation 2: The symbol for the empty set appears here, while in line 79, it is said that phi represents porosity.

References:

[1]    M. Röding, Z. Ma, and S. Torquato. Predicting permeability via statistical learning on higher-order microstructural information. Scientific Reports, 10(1):15239, 2020.

[2]     M. Neumann, O. Stenzel, F. Willot, L. Holzer, and V. Schmidt. Quantifying the influence of microstructure on effective conductivity and permeability: virtual materials testing. International Journal of Solid and Structures, 184:211–220, 2020.

Author Response

Please check the attachment

Author Response File: Author Response.docx