the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 17 Jun 2025
A Simplified Relationship Between the Zero-percolation Threshold and Fracture Set Properties
Abstract. Percolation analysis is an efficient way of evaluating the connectivity of discrete fracture networks. Except for very simple cases, it is not feasible to use analytical approaches to find the percolation threshold of a discrete fracture network. The most commonly used percolation threshold corresponds to the occurrence of percolation on average for the set of parameters (p50), which is not adequate for applications in which a high confidence in the percolation threshold is required. This study investigates the direct relationships between the percolation threshold at low probability (p0, named as zero-percolation threshold) and the properties of fracture networks with one set of fractures (fractures with similar orientations) in two-demensional domains. A generalized non-linear multivariate relationship between p0 and fracture network parameters is established based on connectivity assessments of a significant number of numerical simulations of fracture networks. A feature of this relationship is the invariant shape of marginal relationships. A comparison study with an analytical solution and applications in both synthetic and real fracture networks show that the derived relationship performs well in fracture networks of different sizes and orientations. A significant benefit of this relationship is that, when an analytical solution is not available, it can provide fast and reliable connectivity statistics of fracture networks based only on fracture parameters.
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RC1: 'Comment on egusphere-2025-2440', Anonymous Referee #1, 23 Jun 2025
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This paper presents a systematic investigation into the derivation of simplified, direct relationships between the zero-percolation threshold (p0) and key fracture network parameters in discrete fracture networks (DFNs). Through extensive numerical experiments based on Monte Carlo simulations and rigorous non-linear multivariate regression, the authors succeed in constructing functional expressions that relate p0 to fracture number, orientation concentration (κ), and angular deviation (Δ), considering both exponential and lognormal distributions for fracture length. The derived equations are validated against numerical simulations and analytical benchmarks, demonstrating high predictive accuracy and robustness across a range of DFN configurations and spatial scales. It is clearly structured, methodologically sound, and contributes meaningfully to the fracture network modeling community, particularly in contexts where probabilistic guarantees are essential, such as nuclear waste repository site selection and unconventional reservoir modeling. Nonetheless, a few minor issues merit revision to further improve clarity, rigor, and presentation. A minor revision is suggested.
1.Several variables and equations are introduced early (e.g., μ, κ, Δ, p₀, L̅ₜ) without immediate intuitive explanation. Consider including a summarized table of symbols and parameters, especially for readers unfamiliar with von Mises distribution or DFN modeling conventions.
2.Machine learning offers a promising approach for capturing the complex relationship between fracture network properties and percolation behavior. The analytical formulation proposed in this work provides a strong foundation for integrating such physical insights into physics-informed machine learning frameworks. It is therefore recommended that the authors include a brief discussion on potential future directions, particularly the combination of their derived equations with emerging machine learning techniques, such as Kolmogorov–Arnold Networks (KAN), to enhance model generalizability and predictive capability.
3.Please ensure all citations are formatted uniformly (e.g., "Yi, Taverghi, 2009" vs. "Yi & Taverghi, 2009").
4. Equations are referenced inconsistently. A consistent format throughout will aid readability.
5. While the paper effectively presents a simplified relationship, a brief yet dedicated discussion on the inherent limitations of this simplification would further enhance the manuscript's academic rigor. Furthermore, outlining potential avenues for future research that could expand upon this simplified framework would significantly add to the paper's impact and forward-looking perspective."Citation: https://doi.org/10.5194/egusphere-2025-2440-RC1 -
RC2: 'Comment on egusphere-2025-2440', Anonymous Referee #2, 16 Jul 2025
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The authors implemented a comprehensive study on the zero-percolation threshold (p0) in two-dimensional discrete fracture networks (DFNs), focusing on the direct relationship between percolation thresholds and fracture network properties (e.g., fracture number, length, orientation). The authors employ extensive numerical simulations and a non-linear multivariate fitting approach to derive predictive equations for DFNs with exponential and lognormal fracture length distributions. The work is well-motivated, methodologically sound, and addresses a gap in connectivity analysis for applications requiring high-confidence impermeability. Overall Minor revision is recommended. However, several issues require clarification or revision to strengthen the manuscript.
(1) The term "zero-percolation threshold" (p0) is introduced as a critical threshold for impermeability. How does p0 differ from p50 in real-world scenarios? Discuss how engineers should select p0 versus p50 thresholds for different applications and the trade-offs involved.
(2)The caption for Figure 5 states, "Each point is obtained by 400 times simulations, and in other words, one pair of (n, L, P) can be obtained via 20 MC simulations, and each of MC simulation is repeated 20 times and then averaged." This is confusing. Please rephrase to clarify the simulation process. For example, was a set of 20 DFN realizations generated to calculate a single probability P, and was this entire process repeated 20 times and averaged to get the final plotted point? A clearer description of the simulation hierarchy is needed here and in the caption for Figure 6a.
(3) The study establishes a relationship for a threshold fracture length, Lt. When discussing the exponential distribution, the text indicates that the average length is Lbar=1/λ. It is implied, but not explicitly stated, that Lt is this average length. This should be made explicit. More importantly, this definition is missing for the lognormal distribution (Sections 3.3-3.4). Please clarify what property of the lognormal distribution Lt represents (e.g., the mean, median, etc.).
(3) The study commendably simplifies the problem by focusing on a single set of fractures. In the conclusion or discussion section, it would be beneficial to add a brief comment on the limitations of this assumption and how the proposed relationship might be extended in future work to handle DFNs with multiple fracture sets, which are common in real-world geological settings.
(4) The manuscript is generally well-written, but there are minor issues with grammar and tense. For example, in the last sentence of the introduction, "The verification of the derived equations for zero-percolation will undergo a comprehensive series of tests in Section 4" should be in the past tense (e.g., "was verified" or "is verified") as the paper is reporting completed work. A thorough proofread to correct such minor issues is recommended.Citation: https://doi.org/10.5194/egusphere-2025-2440-RC2
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