the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 12 May 2026
Statistical Assessment of the Representative Elementary Area for Areal Fracture Intensity (P21) in Digital Outcrop Models
Abstract. The definition of a representative elementary volume (REV) or area (REA) for a target parameter is a fundamental step toward the upscaling of fracture network properties generated by discrete fracture network models (DFN) to an equivalent continuous medium for in situ applications in engineering geology, hydrogeology, and structural geology. The target parameter of this work is the areal fracture intensity (P21), a key metric often used as a stopping criterion in stochastic DFN simulations, that is derived directly from surface data collected at natural outcrops. We propose a novel approach to define the REA as a range bounded by a lower and an upper limit. The upper limit, often overlooked but nonetheless theorized, identifies the largest representative domain, which is crucial for optimizing computational efficiency. We evaluate the REA range based on three statistical parameters, namely: the shape, mean, and variance of the P21 distributions obtained with progressively increasing scan area sizes. Each statistical parameter is assessed by combining formal statistical tests and diagnostic plots. Within a multi-parametric framework, the method enables a detailed analysis of the statistical behaviour of the dataset, supporting informed decisions in defining the REA range. The methodology is tested on two fractured limestone outcrops with markedly different characteristics: (i) an abandoned quarry in the Murge Plateau (Puglia, Italy) and (ii) the Lilstock Benches in the southern Bristol Channel basin.
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Status: open (until 06 Jul 2026)
- CC1: 'Comment on egusphere-2026-2409', Jie Liu, 10 Jun 2026 reply
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RC1: 'Comment on egusphere-2026-2409', Jie Liu, 12 Jun 2026
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The manuscript "Statistical Assessment of the Representative Elementary Area for Areal Fracture Intensity (P21) in Digital Outcrop Models" focuses on the range (lower and upper bounds) of REA by investigating the shape, mean, and variance of P21 distributions obtained with progressively increasing scan area sizes. The authors applied the approach at two field sites with distinct fracture characteristics and conducted subsequent analyses and discussions. The methods, results, and analyses appear reasonable; however, there are three major problems with the manuscript.
I. The practical motivation for defining an upper REA bound is not well established.
The paper presents the determination of an upper bound of the REA range as a key novelty. However, the practical motivation for defining such an upper bound is not well established. The REV/REA literature has historically focused on the lower bound, largely because the engineering requirement is to identify the smallest scale at which a property is still representative. The need for a statistically defined upper bound is far less clear: a scale larger than the lower bound remains representative in principle, and grid resolution in reservoir models is typically constrained by geological boundaries (e.g., lithological contacts, fault zones) rather than by the upper limit of statistical homogeneity.
If the upper limit of REA is intended to define the maximum cell size in numerical modeling (Lines 67–72), the result for the Lilstock outcrop (0.3 m) is counterintuitive when viewed against Fig. 2b, even though it may be justifiable on purely statistical grounds. Using the maximum cell size of 0.3 m to perform numerical modeling of this area (Fig. 2b) is not reasonable from the perspective of a numerical modeller. The paper would benefit from a more critical discussion of why an upper REA bound, derived solely from P21 statistics, should be preferred over, or even considered alongside, geologically based cell-size decisions.
II. The manuscript's organizational logic needs improvement.
1) Placing a subsection entitled "Data collection" within the "Method" section is confusing.
2) In Line 161, Eq. [4] is referred to, yet Eq. [4] is not presented until Line 216, several subsections later.
3) Figures 4 to 7 are mentioned in Section 3 (Method), whereas these figures are explained in Section 4 as results.
4) The Method section should introduce the concepts and methods used in the study using purely mathematical language and expressions. The subsequent section can then describe data acquisition (including the studied sites) and implementation details, accompanied by necessary schematic figures, but without forward references to figures that belong to the Results section. By this standard, the implementation of the analyses has not been explained clearly.
5) In the paragraph from Line 324 to Line 332, the discussion order is confusing: Fig. 9C and Fig. 9D are discussed first, followed by Fig. 10, and then Fig. 9A and Fig. 9B.
III. The manuscript requires much more concise writing.
Some statements are repeated several times across different paragraphs or sections, for example, regarding the calculation of P21. The sentence in Line 175 ("P21 is not calculated using …") is unnecessary.
More broadly, the manuscript contains many unnecessarily detailed interpretations and self-explanatory expressions, which greatly reduce readability. The two paragraphs from Line 145 to Line 158 serve as a representative example:
In the first paragraph (Line 145), the rationale that random sampling ensures independence is unnecessarily elaborated for a specialist audience. The boundary-handling procedure is also described in excessive detail, including a bullet-like enumeration within a sentence and an explicit justification for allowing tangency. Furthermore, the explanation that larger scan areas lead to more frequent boundary rejections and that areas larger than the interpretation boundary cannot be placed is largely self-explanatory. This paragraph could be substantially shortened without any loss of methodological rigor.
In the second paragraph (Line 154), the comparison with grid sampling is introduced via a cumbersome, literature-review-style sentence, when a simple "Compared to grid sampling" would suffice. The subsequent justification for equal sample sizes, while correct, expands into a general discussion of statistical power and the risk of misleading comparisons, complete with a citation. A single sentence stating that equal sample sizes were maintained to ensure comparability of statistical tests would convey the same message more efficiently.
I recommend a thorough revision of these and similarly verbose passages throughout the manuscript. The focus should be on reporting what was done and why in the briefest possible terms that a domain expert would require, eliminating over-justification and redundant explanation.
IV. Miscellaneous issues
1) It is not necessary to include P21 in the title, nor in the abstract.
2) Fig. 11 could be plotted in a single panel using narrow bars.
Citation: https://doi.org/10.5194/egusphere-2026-2409-RC1
Data sets
Pontrelli quarry dataset Stefano Casiraghi and Andrea Bistacchi https://doi.org/10.5281/zenodo.17483588
Model code and software
FracArea Andrea Bistacchi and Stefano Casiraghi https://doi.org/10.5281/zenodo.19735298
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Publisher’s note: the content of this comment was removed on 12 June 2026 since the comment was posted by mistake.