Articles | Volume 16, issue 11 
            
                
                    
            
            
            https://doi.org/10.5194/se-16-1269-2025
                    © Author(s) 2025. This work is distributed under 
the Creative Commons Attribution 4.0 License.
                the Creative Commons Attribution 4.0 License.
https://doi.org/10.5194/se-16-1269-2025
                    © Author(s) 2025. This work is distributed under 
the Creative Commons Attribution 4.0 License.
                the Creative Commons Attribution 4.0 License.
A simplified relationship between the zero-percolation threshold and fracture set properties
Shaoqun Dong
                                            National Key Laboratory of Petroleum Resources and Engineering, China University of Petroleum (Beijing), Beijing, 102249, China
                                        
                                    
                                            College of Science, China University of Petroleum, Beijing, 102249, China
                                        
                                    
                                            School of Civil, Environmental and Mining Engineering, University of Adelaide, Adelaide, 5005, Australia
                                        
                                    Lianbo Zeng
CORRESPONDING AUTHOR
                                            
                                    
                                            National Key Laboratory of Petroleum Resources and Engineering, China University of Petroleum (Beijing), Beijing, 102249, China
                                        
                                    
                                            College of Geosciences, China University of Petroleum, Beijing, 102249, China
                                        
                                    Chaoshui Xu
                                            School of Civil, Environmental and Mining Engineering, University of Adelaide, Adelaide, 5005, Australia
                                        
                                    Peter Dowd
                                            School of Civil, Environmental and Mining Engineering, University of Adelaide, Adelaide, 5005, Australia
                                        
                                    Guohao Xiong
                                            National Key Laboratory of Petroleum Resources and Engineering, China University of Petroleum (Beijing), Beijing, 102249, China
                                        
                                    
                                            College of Science, China University of Petroleum, Beijing, 102249, China
                                        
                                    Tao Wang
                                            Chinese Academy of Geological Sciences, Beijing, 100037, China
                                        
                                    Wenya Lyu
                                            National Key Laboratory of Petroleum Resources and Engineering, China University of Petroleum (Beijing), Beijing, 102249, China
                                        
                                    
                                            College of Geosciences, China University of Petroleum, Beijing, 102249, China
                                        
                                    Cited articles
                        
                        Alghalandis, Y. F., Dowd, P. A., and Xu, C.: Connectivity Field: A Measure for Characterising Fracture Networks. Math. Geosci., 47, 63-83, https://doi.org/10.1007/s11004-014-9520-7, 2015. 
                    
                
                        
                        Ali, A. and Jakobsen, M.: Seismic characterization of reservoirs with multiple fracture sets using velocity and attenuation anisotropy data, J. Appl. Geophys., 75, 590-602, doi:1 0.1016/j.jappgeo.2011.09.003, 2011. 
                    
                
                        
                        Balberg, I. and Binenbaum, N.: Computer study of the percolation threshold in a two-dimensional anisotropic system of conducting sticks, Phys. Rev. B, 28, 3799-3812, doi:10.11 03/PhysRevB.28.3799, 1983. 
                    
                
                        
                        Balberg, I., Anderson, C. H., Alexander S., and Wagner N.: Excluded volume and its relation to the onset of percolation, Phys. Rev. B, 30, 3933, https://doi.org/10.1103/PhysRevB.30.3933, 1984. 
                    
                
                        
                        Barker, J. A.: Intersection statistics and percolation criteria for fractures of mixed shapes and sizes, Comput. Geosci., 112, 47–53, https://doi.org/10.1016/j.cageo.2017.12.001, 2018. 
                    
                
                        
                        Barton, C. C. and Hsieh, P. A.: Physical and hydrologic-flow properties of fractures, United States of America, American Geophysical Union, America, https://doi.org/10.1029/FT385, 1989. 
                    
                
                        
                        Berkowitz, B.: Analysis of fracture network connectivity using percolation theory, Math. Geosci., 27, 467–483, https://doi.org/10.1007/BF02084422, 1995. 
                    
                
                        
                        Bour, O. and Davy, P.: On the connectivity of three-dimensional fault networks, Water Resour. Res., 34, 2611–2622, https://doi.org/10.1029/98WR01861, 1998. 
                    
                
                        
                        Bour, O. and Davy, P.: Connectivity of random fault networks following a power law fault length distribution, Water Resour. Res., 33, 1567–1583, https://doi.org/10.1029/96WR00433, 1997. 
                    
                
                        
                        Catapano, E., Cassé, M., and Ghibaudo, G.: Cryogenic MOSFET Subthreshold Current: From Resistive Networks to Percolation Transport in 1-D Systems, IEEE Transactions on Electron Devices, 70, 4049–4054, https://doi.org/10.1109/TED.2023.3283941, 2023. 
                    
                
                        
                        Charlaix, E., Guyon, E., and Rivier, N.: A criterion for percolation threshold in a random array of plates, Solid State Commun., 50, 999-1002, https://doi.org/10.1016/0038-1098(84)90274-6, 1984. 
                    
                
                        
                        de Dreuzy, J., Davy, P., and Bour O.: Percolation parameter and percolation-threshold estimates for three-dimensional random ellipses with widely scattered distributions of eccentricity and size, Phys. Rev. E, 62, 5948–5952, https://doi.org/10.1103/PhysRevE.62.5948, 2000. 
                    
                
                        
                        Dogan, M. O.: Extended Multiple Interacting Continua (E-MINC) Model Improvement with a K-Means Clustering Algorithm Based on an Equi-dimensional Discrete Fracture Matrix (ED-DFM) Model, Math. Geosci., https://doi.org/10.1007/s11004-023-10110-9, 2023. 
                    
                
                        
                        Dong, S., Wang, Z., and Zeng, L.: Lithology identification using kernel Fisher discriminant analysis with well logs, Geoenergy Sci. Eng., 143, 95–102, https://doi.org/10.1016/j.petrol.2016.02.017, 2016. 
                    
                
                        
                        Dong, S., Zeng, L., Dowd, P., Xu, C., and Cao, H.: A fast method for fracture intersection detection in discrete fracture networks, Comput. Geotech., 98, 205–216, https://doi.org/10.1016/j.compgeo.2018.02.005, 2018a. 
                    
                
                        
                        Dong, S., Zeng, L., Xu, C., Cao, H., Wang, S., and Lyu, W.: Some progress in reservoir fracture stochastic modeling research, Oil Geophysical Prospecting, 53, 30–51, https://doi.org/10.13810/j.cnki.issn.1000-7210.2018.03.023, 2018b. 
                    
                
                        
                        Dong, S., Zeng, L., Cao, H., Xu, C., and Wang, S.: Principle and implementation of discrete fracture network modeling controlled by fracture density, Geol. Rev., 64, 1302–1314, https://doi.org/10.16509/j.georeview.2018.05.020, 2018c. 
                    
                
                        
                        Dong, S., Wang, T., Zeng, L., Liu, K., Liang, F., Yin, Q., and Cao D.: Analysis of relationship between underground space percolation and fracture properties, Earth Science Frontiers, 3, 140–146, https://doi.org/10.13745/j.esf.sf.2019.4.22, 2019. 
                    
                
                        
                        Dong, S.: Parameters of the percolation equation in terms of fracture properties, V1, Mendeley Data [data set], 2020. 
                    
                
                        
                        Dong, S., Lyu, W., Xia, D., Wang, S., Du, X., Wang, T., Wu, Y., and Guan C.: An approach to 3D geological modeling of multi-scaled fractures in tight sandstone reservoirs, Oil and Gas Geology, 41, 627–637, https://doi.org/10.11743/ogg20200318, 2020. 
                    
                
                        
                        Dong, S., Zeng, L., Du, X., Bao, M., Lyu, W., Ji, C., and Hao J.: An intelligent prediction method of fractures in tight carbonate reservoirs, Petroleum Exploration and Development, 49, 1179–1189, https://doi.org/10.11698/PED.20220367, 2022. 
                    
                
                        
                        Dowd, P. A., Xu, C., Mardia, K. V., and Fowell, R. J.: A Comparison of Methods for the Stochastic Simulation of Rock Fractures, Math. Geosci., 39, 697–714, https://doi.org/10.1007/s11004-007-9116-6, 2007. 
                    
                
                        
                        Einstein, H. H. and Locsin, J.: Modeling rock fracture intersections and application to the Boston Area, J. Geotech. Geoenviron., 11, 1415–1421, https://doi.org/10.1061/(ASCE)GT.1943-5606.0000699, 2012. 
                    
                
                        
                        Fadakar Alghalandis, Y.: ADFNE: Open source software for discrete fracture network engineering, two and three dimensional applications, Comput. Geosci., 102, 1–11, https://doi.org/10.1016/j.cageo.2017.02.002, 2017. 
                    
                
                        
                        Huseby, O. and Thovert, J. F.: Geometry and topology of fracture systems, Journal of Physics A: Mathematical and General, 30, 1415, https://doi.org/10.1088/0305-4470/30/5/012, 1997. 
                    
                
                        
                        Jafari, A. and Babadagli, T.: A sensitivity analysis for effective parameters on 2D fracture-network permeability, SPE Reserv. Eval. Eng., 12, 455–469, 2009. 
                    
                
                        
                        Jafari, A. and Babadagli, T.: Relationship between percolation–fractal properties and permeability of 2-D fracture networks, Int. J. Rock. Mech. Min., 60, 353–362, https://doi.org/10.1016/j.ijrmms.2013.01.007, 2013. 
                    
                
                        
                        Khamforoush, M. and Shams, K.: Percolation thresholds of a group of anisotropic three-dimensional fracture networks, Physica A, 385, 407–420, https://doi.org/10.1016/j.physa.2007.07.037, 2007. 
                    
                
                        
                        Khamforoush, M., Shams, K., Thovert, J. F., and Adler, P. M.: Permeability and percolation of anisotropic three-dimensional fracture networks, Phys. Rev. E, 77, 56307, https://doi.org/10.1103/PhysRevE.77.056307, 2008. 
                    
                
                        
                        Kolyukhin, D.: Sensitivity analysis of discrete fracture network connectivity characteristics, Math. Geosci., 54, 225–241, https://doi.org/10.1007/s11004-021-09966-6, 2022. 
                    
                
                        
                        Liu, R., Zhu, T., Jiang, Y., Li, B., Yu, L., Du, Y., and Wang Y.: A predictive model correlating permeability to two-dimensional fracture network parameters, B. Eng. Geol. Environ., 78, 1589–1605, https://doi.org/10.1007/s10064-018-1231-8, 2019. 
                    
                
                        
                        Manzocchi, T.: The connectivity of two-dimensional networks of spatially correlated fractures, Water Resour. Res., 38, 1162, https://doi.org/10.1029/2000WR000180, 2002. 
                    
                
                        
                        Manzocchi, T., Walsh, D. A., Carneiro, M., and López-Cabrera, J.: Compression-based Facies Modelling, Math. Geosci., 55, 625–644, https://doi.org/10.1007/s11004-023-10048-y, 2023. 
                    
                
                        
                        Mardia, K. V., Nyirongo, V. B., Walder, A. N., Xu, C., Dowd, P. A., Fowell, R. J., and Kent J. T.: Markov Chain Monte Carlo implementation of rock fracture modelling, Math. Geosci., 39, 355–381, https://doi.org/10.1007/s11004-007-9099-3, 2007. 
                    
                
                        
                        Masihi, M. and King, P. R.: A correlated fracture network: Modeling and percolation properties, Water Resour. Res., 43, https://doi.org/10.1029/2006WR005331, 2007. 
                    
                
                        
                        McKenna, S. A., Akhriev, A., Echeverría Ciaurri, and D., Zhuk, S.: Efficient Uncertainty Quantification of Reservoir Properties for Parameter Estimation and Production Forecasting, Math. Geosci., 52, 233–251, https://doi.org/10.1007/s11004-019-09810-y, 2020. 
                    
                
                        
                        Mourzenko, V. V., Thovert, J. F., and Adler, P. M.: Macroscopic permeability of three-dimensional fracture networks with power-law size distribution, Phys. Rev. E, 69, 66307, https://doi.org/10.1103/PhysRevE.69.066307, 2004. 
                    
                
                        
                        Mourzenko, V. V., Thovert, J., and Adler, P. M.: Percolation and permeability of three dimensional fracture networks with a power law size distribution, Fractals in Engineering, 81–95, https://doi.org/10.1007/1-84628-048-6_6, 2005. 
                    
                
                        
                        Mourzenko, V. V., Thovert, J., and Adler, P. M.: Percolation and permeability of fracture networks in excavated damaged zones, Phys. Rev. E, 86, 26312, https://doi.org/10.1103/PhysRevE.86.026312, 2012. 
                    
                
                        
                        Ngia, L. S. H. and Sjoberg, J.: Efficient training of neural nets for nonlinear adaptive filtering using a recursive Levenberg-Marquardt algorithm. IEEE T. Signal. Proces., 48, 1915–1927, https://doi.org/10.1109/78.847778, 2000. 
                    
                
                        
                        Or, D., Furtak-Cole, E., Berli, M., Shillito, R., Ebrahimian, H., Vahdat-Aboueshagh, H., and McKenna S. A.: Review of wildfire modeling considering effects on land surfaces, Earth-Sci. Rev., 245, 104–569, https://doi.org/10.1016/j.earscirev.2023.104569, 2023. 
                    
                
                        
                        Robinson, P. C.: Connectivity of fracture systems-a percolation theory approach, J. Phys. A: Math. Gen., 16, 605, https://doi.org/10.1088/0305-4470/16/3/020, 1983. 
                    
                
                        
                        Shokri, A. R., Babadagli, T., and Jafari, A.: A critical analysis of the relationship between statistical- and fractal-fracture-network characteristics and effective fracture-network permeability, SPE Reserv. Eval. Eng., 19, 494–510, https://doi.org/10.2118/181743-PA, 2016. 
                    
                
                        
                        Sun, H., Radicchi, F., Kurths, J., and Bianconi, G.: The dynamic nature of percolation on networks with triadic interactions, Nat. Commun., 14, https://doi.org/10.1038/s41467-023-37019-5, 2023.  
                    
                
                        
                        Tang, H., Zhao, Y., Kang, Z., Lv, Z., Yang, D., and Wang, K.: Investigation on the Fracture-Pore Evolution and Percolation Characteristics of Oil Shale under Different Temperatures, Energies, 15, 3572, https://doi.org/10.3390/en15103572, 2022. 
                    
                
                        
                        Thovert, J. F., Mourzenko, V. V., and Adler, P. M.: Percolation in three-dimensional fracture networks for arbitrary size and shape distributions, Phys. Rev. E, 95, 42112, https://doi.org/10.1103/PhysRevE.95.042112, 2017. 
                    
                
                        
                        Walsh, D. A. and Manzocchi, T.: Connectivity in Pixel-Based Facies Models. Math. Geosci., 53, 415–435, https://doi.org/10.1007/s11004-021-09931-3, 2021. 
                    
                
                        
                        Wei, Y., Dong, Y., Yeh, T. J., Li, X., Wang, L., and Zha, Y.: Assessment of uncertainty in discrete fracture network modeling using probabilistic distribution method, Water Science and Technology, 76, 2802, https://doi.org/10.2166/wst.2017.451, 2017. 
                    
                
                        
                        Wilson, T. H.: Scale Transitions in Fracture and Active Fault Networks, Math. Geosci., 33, 591–613, https://doi.org/10.1023/A:1011096828971, 2001. 
                    
                
                        
                        Xu, C. and Dowd, P.: A new computer code for discrete fracture network modelling. Comput. Geosci., 36, 292–301, https://doi.org/10.1016/j.cageo.2009.05.012, 2010. 
                    
                
                        
                        Xu, C., Dowd, P., Mardia, K. V., and Fowell, R. J.: A connectivity index for discrete fracture networks, Math. Geosci., 38, 611–634, https://doi.org/10.1007/s11004-006-9029-9, 2007. 
                    
                
                        
                        Yao, C., Shao, Y., Yang, J., Huang, F., Hem, C., Jiang, Q., and Zhou C.: Effects of fracture density, roughness, and percolation of fracture network on heat-flow coupling in hot rock masses with embedded three-dimensional fracture network, Geothermics, 87, 101846, https://doi.org/10.1016/j.geothermics.2020.101846, 2020. 
                    
                
                        
                        Yi, Y. and Tawerghi, E.: Geometric percolation thresholds of interpenetrating plates in three-dimensional space, Phys. Rev. E, 79, 41134, https://doi.org/10.1103/PhysRevE.79.041134, 2009. 
                    
                
                        
                        Zeng, L., Gongm, L., Guan, C., Zhang, B., Wang, Q., Zeng, Q., and Lyu W.: Natural fractures and their contribution to tight gas conglomerate reservoirs: A case study in the northwestern Sichuan Basin, China, J. Petrol. Sci. Eng., 210, 110028, https://doi.org/10.1016/j.petrol.2021.110028, 2022. 
                    
                
                        
                        Zhao, W. and Hou, G.: Fracture prediction in the tight-oil reservoirs of the Triassic Yanchang Formation in the Ordos Basin, northern China, Petrol. Sci., 14, 1–23, https://doi.org/10.1007/s12182-016-0141-2, 2017. 
                    
                
                        
                        Zhao, Y., Feng, Z., Liang, W., Yang, D., Hu, Y., and Kang, T.: Investigation of fractal distribution law for the trace number of random and grouped fractures in a geological mass, Eng. Geol., 109, 224–229, https://doi.org/10.1016/j.enggeo.2009.08.002, 2009. 
                    
                
                        
                        Zhao, Y., Feng, Z., Lv, Z., Zhao, D., and Liang, W.: Percolation laws of a fractal fracture-pore double medium, Fractals, 24, 1650053, https://doi.org/10.1142/S0218348X16500535, 2016. 
                    
                
                        
                        Zhu, W., He, X., Khirevich, S., Patzek, and T. W.: Fracture sealing and its impact on the percolation of subsurface fracture networks, Geoenergy Sci. Eng., 218, 111023, https://doi.org/10.1016/j.petrol.2022.111023, 2022. 
                    
                Short summary
            Rock fracture connectivity is key for oil/gas, geothermal energy, and nuclear waste storage. This study predicts the zero-percolation threshold – when fractures connect – using simulations. A model links this threshold to fracture number, length, and orientation, enabling fast predictions. Tests on simulated and real fractures confirm its accuracy across sizes/orientations. Provides key tool for subsurface engineering.
            Rock fracture connectivity is key for oil/gas, geothermal energy, and nuclear waste storage....