A Correlated Random-Field Ising Model for Pore-Scale Hysteresis in Soil-Water Characteristic Curves
Downloads
The soil–water characteristic curve (SWCC) plays a central role in the behavior of unsaturated soils, yet explaining its hysteresis directly from pore-scale mechanisms remains challenging. The objective of this study is to investigate how pore-size heterogeneity, spatial correlations, and cooperative dynamics contribute to hysteresis in SWCCs. In this study, a correlated Random-Field Ising Model (RFIM) combined with Monte Carlo simulations is developed to represent the pore space as a two-dimensional lattice with a bimodal distribution of pore volumes and a spatially correlated disorder field. Drainage processes are simulated without parametric curve fitting, enabling direct analysis of pore-scale switching dynamics. The results show that macropore fraction, pore-size heterogeneity, and the activation parameter \beta exert a strong control on drainage behavior. Low \beta values produce smooth and nearly reversible drainage, whereas higher \beta stabilizes metastable pore configurations and yields abrupt transitions accompanied by hysteresis. The divergence between number-based and volume-based saturation serves as a useful indicator of size-selective drainage and cooperative pore-scale events. The novelty of this work lies in providing a physically grounded and statistically dynamics to macroscopic hysteresis in SWCCs, offering insights beyond traditional phenomenological or uncorrelated pore-network approaches.
Downloads
[1] Fredlund, D. G., & Anqing Xing. (1994). Equations for the soil-water characteristic curve. Canadian Geotechnical Journal, 31(4), 521–532. doi:10.1139/t94-061.
[2] Fredlund, D. G. (2006). Unsaturated Soil Mechanics in Engineering Practice. Journal of Geotechnical and Geoenvironmental Engineering, 132(3), 286–321. doi:10.1061/(asce)1090-0241(2006)132:3(286).
[3] Rahardjo, H., Kim, Y., & Satyanaga, A. (2019). Role of unsaturated soil mechanics in geotechnical engineering. International Journal of Geo-Engineering, 10(1), 1–23. doi:10.1186/s40703-019-0104-8.
[4] Chang, W. J., Chou, S. H., Huang, H. P., & Chao, C. Y. (2021). Development and verification of coupled hydro-mechanical analysis for rainfall-induced shallow landslides. Engineering Geology, 293, 106337. doi:10.1016/j.enggeo.2021.106337.
[5] Nguyen, T. S., & Likitlersuang, S. (2019). Reliability analysis of unsaturated soil slope stability under infiltration considering hydraulic and shear strength parameters. Bulletin of Engineering Geology and the Environment, 78(8), 5727–5743. doi:10.1007/s10064-019-01513-2.
[6] Vanapalli, S. K., & Oh, W. T. (2010). Mechanics of unsaturated soils for the design of foundation structures. Proceedings of the 3rd WSEAS international conference on Engineering mechanics, structures, engineering geology, 22-24 July, 2010, Corfu Island, Greece.
[7] Yilmazoglu, M. U., & Ozocak, A. (2023). Bearing Capacity of Shallow Foundations on Unsaturated Silty Soils. Applied Sciences (Switzerland), 13(3), 1308. doi:10.3390/app13031308.
[8] Onyelowe, K. C., Mojtahedi, F. F., Azizi, S., Mahdi, H. A., Sujatha, E. R., Ebid, A. M., Darzi, A. G., & Aneke, F. I. (2022). Innovative Overview of SWRC Application in Modeling Geotechnical Engineering Problems. Designs, 6(5), 69. doi:10.3390/designs6050069.
[9] Alnmr, A., Alzawi, M. O., Ray, R., Abdullah, S., & Ibraheem, J. (2024). Experimental Investigation of the Soil-Water Characteristic Curves (SWCC) of Expansive Soil: Effects of Sand Content, Initial Saturation, and Initial Dry Unit Weight. Water (Switzerland), 16(5), 627. doi:10.3390/w16050627.
[10] Pečan, U., Pintar, M., & Kastelec, D. (2023). Variability of in situ soil water retention curves under different tillage systems and growing seasons. Soil and Tillage Research, 233, 105779. doi:10.1016/j.still.2023.105779.
[11] Hohenbrink, T. L., Jackisch, C., Durner, W., Germer, K., Iden, S. C., Kreiselmeier, J., Leuther, F., Metzger, J. C., Naseri, M., & Peters, A. (2023). Soil water retention and hydraulic conductivity measured in a wide saturation range. Earth System Science Data, 15(10), 4417–4432. doi:10.5194/essd-15-4417-2023.
[12] Likos, W. J., Lu, N., & Godt, J. W. (2014). Hysteresis and Uncertainty in Soil Water-Retention Curve Parameters. Journal of Geotechnical and Geoenvironmental Engineering, 140(4), 4013050. doi:10.1061/(asce)gt.1943-5606.0001071.
[13] Kim, J., Hwang, W., & Kim, Y. (2018). Effects of hysteresis on hydro-mechanical behavior of unsaturated soil. Engineering Geology, 245, 1–9. doi:10.1016/j.enggeo.2018.08.004.
[14] Leong, E. C. (2019). Soil-water characteristic curves - Determination, estimation and application. 7th Asia-Pacific Conference on Unsaturated Soils, AP-UNSAT 2019, 7(2), 21–30. doi:10.3208/jgssp.v07.003.
[15] van Genuchten, M. T. (1980). A Closed‐form Equation for Predicting the Hydraulic Conductivity of Unsaturated Soils. Soil Science Society of America Journal, 44(5), 892–898. doi:10.2136/sssaj1980.03615995004400050002x.
[16] Vereecken, H., Weynants, M., Javaux, M., Pachepsky, Y., Schaap, M. G., & van Genuchten, M. T. (2010). Using Pedotransfer Functions to Estimate the van Genuchten–Mualem Soil Hydraulic Properties: A Review. Vadose Zone Journal, 9(4), 795–820. doi:10.2136/vzj2010.0045.
[17] Mohajerani, H., Teschemacher, S., & Casper, M. C. (2021). A comparative investigation of various pedotransfer functions and their impact on hydrological simulations. Water (Switzerland), 13(10), 1401. doi:10.3390/w13101401.
[18] Joekar-Niasar, V., & Hassanizadeh, S. M. (2012). Analysis of fundamentals of two-phase flow in porous media using dynamic pore-network models: A review. Critical Reviews in Environmental Science and Technology, 42(18), 1895–1976. doi:10.1080/10643389.2011.574101.
[19] Kanckstedt, M. A., Sheppard, A. P., & Sahimi, M. (2001). Pore network modelling of two-phase flow in porous rock: The effect of correlated heterogeneity. Advances in Water Resources, 24(3–4), 257–277. doi:10.1016/S0309-1708(00)00057-9.
[20] Zaretskiy, Y., Geiger, S., Sorbie, K., & Förster, M. (2010). Efficient flow and transport simulations in reconstructed 3D pore geometries. Advances in Water Resources, 33(12), 1508–1516. doi:10.1016/j.advwatres.2010.08.008.
[21] Nabipour, I., Mohammadzadeh-Shirazi, M., Raoof, A., & Qajar, J. (2025). A Data-Driven Approach for Efficient Prediction of Permeability of Porous Rocks by Combining Multiscale Imaging and Machine Learning. Transport in Porous Media, 152(4), 1–34. doi:10.1007/s11242-025-02167-3.
[22] Showkat, R., Jalal, F. E., & Babu, G. L. S. (2025). Estimation of Soil Water Characteristic Curve Using Machine-Learning Algorithms and Its Application in Embankment Response. Journal of Computing in Civil Engineering, 39(3), 4025012. doi:10.1061/jccee5.cpeng-6062.
[23] Albuquerque, E. A. C., de Faria Borges, L. P., Cavalcante, A. L. B., & Machado, S. L. (2022). Prediction of soil water retention curve based on physical characterization parameters using machine learning. Soils and Rocks, 45(3), 222. doi:10.28927/SR.2022.000222.
[24] Ding, X., & El-Zein, A. (2024). Predicting soil water retention curves using machine learning: A study of model architecture and input variables. Engineering Applications of Artificial Intelligence, 133, 108122. doi:10.1016/j.engappai.2024.108122.
[25] Achieng, K. O. (2019). Modelling of soil moisture retention curve using machine learning techniques: Artificial and deep neural networks vs support vector regression models. Computers and Geosciences, 133, 104320. doi:10.1016/j.cageo.2019.104320.
[26] Zhao, Y., Cui, Y., Zhou, H., Feng, X., & Huang, Z. (2017). Effects of void ratio and grain size distribution on water retention properties of compacted infilled joint soils. Soils and Foundations, 57(1), 50–59. doi:10.1016/j.sandf.2017.01.004.
[27] Kieu, M. T., & Mahler, A. (2018). A study on the relationship between matric suction and the void ratio and moisture content of a compacted unsaturated soil. Periodica Polytechnica Civil Engineering, 62(3), 709–716. doi:10.3311/PPci.11974.
[28] Sethna, J. P., Dahmen, K. A., & Myers, C. R. (2001). Crackling noise. Nature, 410(6825), 242–250. doi:10.1038/35065675.
[29] Illa, X., Rosinberg, M. L., & Vives, E. (2006). Influence of the driving mechanism on the response of systems with a thermal dynamics: The example of the random-field Ising model. Physical Review B - Condensed Matter and Materials Physics, 74(22), 224403. doi:10.1103/PhysRevB.74.224403.
[30] Xu, J., & Louge, M. Y. (2015). Statistical mechanics of unsaturated porous media. Physical Review E, 92(6). doi:10.1103/PhysRevE.92.062405.
[31] Salvat-Pujol, F., Vives, E., & Rosinberg, M. L. (2009). Hysteresis in the T=0 random-field Ising model: Beyond metastable dynamics. Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, 79(6), 61116. doi:10.1103/PhysRevE.79.061116.
[32] Illa, X., & Rosinberg, M. L. (2011). Zero-temperature random field Ising model on a Bethe lattice: Correlation functions along the hysteresis loop. Physical Review B - Condensed Matter and Materials Physics, 84(6), 64443. doi:10.1103/PhysRevB.84.064443.
[33] Rosinberg, M. L., & Munakata, T. (2009). Hysteresis and complexity in the mean-field random-field Ising model: The soft-spin version. Physical Review B - Condensed Matter and Materials Physics, 79(17), 174207. doi:10.1103/PhysRevB.79.174207.
[34] Fytas, N. G., Martín-Mayor, V., Picco, M., & Sourlas, N. (2018). Review of Recent Developments in the Random-Field Ising Model. Journal of Statistical Physics, 172(2), 665–672. doi:10.1007/s10955-018-1955-7.
[35] Gómez, A. V. (2025). A Theoretical Pore Network Model for the Soil–Water Characteristic Curve and Hysteresis in Unsaturated Soils. Civil Engineering Journal (Iran), 11(2), 763–778. doi:10.28991/CEJ-2025-011-02-021.
[36] Rojas, E., & Rojas, F. (2005). Modeling hysteresis of the soil-water characteristic curve. Soils and Foundations, 45(3), 135–145. doi:10.3208/sandf.45.3_135.
[37] Eyo, E. U., Ng’ambi, S., & Abbey, S. J. (2022). An overview of soil–water characteristic curves of stabilised soils and their influential factors. Journal of King Saud University - Engineering Sciences, 34(1), 31–45. doi:10.1016/j.jksues.2020.07.013.
[38] Zhai, Q., Rahardjo, H., & Satyanaga, A. (2018). Uncertainty in the estimation of hysteresis of soil-water characteristic curve. Environmental Geotechnics, 6(4), 204–213. doi:10.1680/jenge.17.00008.
[39] Higo, Y., & Kido, R. (2023). A microscopic interpretation of hysteresis in the water retention curve of sand. Geotechnique. doi:10.1680/jgeot.23.00084.
[40] Mooney, S. J., & Korošak, D. (2009). Using Complex Networks to Model Two‐ and Three‐Dimensional Soil Porous Architecture. Soil Science Society of America Journal, 73(4), 1094–1100. doi:10.2136/sssaj2008.0222.
[41] Mufti, S., & Das, A. (2023). Modeling unsaturated hydraulic conductivity of granular soils using a combined discrete element and pore-network approach. Acta Geotechnica, 18(2), 651–672. doi:10.1007/s11440-022-01597-3.
[42] Zhou, H., Mooney, S. J., & Peng, X. (2017). Bimodal Soil Pore Structure Investigated by a Combined Soil Water Retention Curve and X‐Ray Computed Tomography Approach. Soil Science Society of America Journal, 81(6), 1270–1278. doi:10.2136/sssaj2016.10.0338.
[43] Kutílek, M., Jendele, L., & Panayiotopoulos, K. P. (2006). The influence of uniaxial compression upon pore size distribution in bi-modal soils. Soil and Tillage Research, 86(1), 27–37. doi:10.1016/j.still.2005.02.001.
[44] Coppola, A. (2000). Unimodal and Bimodal Descriptions of Hydraulic Properties for Aggregated Soils. Soil Science Society of America Journal, 64(4), 1252–1262. doi:10.2136/sssaj2000.6441252x.
[45] dos Santos Pereira, S. A., Silva Junior, A. C., Mendes, T. A., Gitirana Junior, G. de F. N., & Alves, R. D. (2024). Prediction of Soil–Water Characteristic Curves in Bimodal Tropical Soils Using Artificial Neural Networks. Geotechnical and Geological Engineering, 42(5), 3043–3062. doi:10.1007/s10706-023-02716-x.
[46] Chan, W. K. V. (2013). Theory and applications of Monte Carlo simulations. BoD–Books on Demand, Hamburg, Germany.
[47] Zhou, J., Wang, Y., Qi, P., Ma, R., Vereecken, H., Minasny, B., & Zhang, Y. (2025). Soil Physics-Informed Neural Networks to Estimate Bimodal Soil Hydraulic Properties. Water Resources Research, 61(10), 2024 039337. doi:10.1029/2024WR039337.
[48] Hosseini, R., Kumar, K., & Delenne, J. Y. (2024). Investigating the source of hysteresis in the soil–water characteristic curve using the multiphase lattice Boltzmann method. Acta Geotechnica, 19(11), 7577–7601. doi:10.1007/s11440-024-02295-y.
[49] Wang, J. P., Liu, T. H., Wang, S. H., Luan, J. Y., & Dadda, A. (2023). Investigation of porosity variation on water retention behaviour of unsaturated granular media by using pore scale Micro-CT and lattice Boltzmann method. Journal of Hydrology, 626, 130161. doi:10.1016/j.jhydrol.2023.130161.
[50] Pessoa, T. N., Ferreira, T. R., Pires, L. F., Cooper, M., Uteau, D., Peth, S., Vaz, C. M. P., & Libardi, P. L. (2023). X-ray Microtomography for Investigating Pore Space and Its Relation to Water Retention and Conduction in Highly Weathered Soils. Agriculture (Switzerland), 13(1), 28. doi:10.3390/agriculture13010028.
[51] Pires, L. F., Auler, A. C., Roque, W. L., & Mooney, S. J. (2020). X-ray microtomography analysis of soil pore structure dynamics under wetting and drying cycles. Geoderma, 362, 114103. doi:10.1016/j.geoderma.2019.114103.
[52] Detmann, B. (2025). On several types of hysteresis phenomena appearing in porous media. Mathematics in Engineering, 7(3), 384–405. doi:10.3934/mine.2025016.
[53] Nepal, A., Hidalgo, J. J., Ortín, J., Lunati, I., & Dentz, M. (2025). Mechanisms of interface jumps, pinning and hysteresis during cyclic fluid displacements in an isolated pore. Journal of Colloid and Interface Science, 696, 137767. doi:10.1016/j.jcis.2025.137767.
[54] Lan, Y., Guo, P., Liu, Y., Wang, S., Cao, S., Zhang, J., Sun, W., Qi, D., & Ji, Q. (2024). State of the Art on Relative Permeability Hysteresis in Porous Media: Petroleum Engineering Application. Applied Sciences (Switzerland), 14(11), 4639. doi:10.3390/app14114639.
[55] Kierlik, E., Monson, P. A., Rosinberg, M. L., Sarkisov, L., & Tarjus, G. (2001). Capillary condensation in disordered porous materials: Hysteresis versus equilibrium behavior. Physical Review Letters, 87(5), 55701-1-55701–55704. doi:10.1103/PhysRevLett.87.055701.
[56] Dahmen, K. A., Sethna, J. P., Kuntz, M. C., & Perković, O. (2001). Hysteresis and avalanches: Phase transitions and critical phenomena in driven disordered systems. Journal of Magnetism and Magnetic Materials, 226–230(Part II), 1287–1292. doi:10.1016/S0304-8853(00)00749-6.
- Authors retain all copyrights. It is noticeable that authors will not be forced to sign any copyright transfer agreements.
- This work (including HTML and PDF Files) is licensed under a Creative Commons Attribution 4.0 International License.
