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Hysteresis phenomenon within unsaturated granular assemblies: Capillary forces and matric suction

Nabil Younes1,2,3,4,*, Antoine Wautier3, Richard Wan4, Olivier Millet2, and François Nicot5

  • *Contact author: nabil.younes@umontpellier.fr
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Phys. Rev. E 113, 035401 – Published 2 March, 2026

DOI: https://doi.org/10.1103/1qdn-hnns

Abstract

In this paper, we present a numerical approach to simulate the condensation and evaporation processes of capillary bridges within granular materials. The formation and dynamics of capillary bridges are captured using a phase-field-based framework implemented within the Lattice Boltzmann Method (LBM) to solve the Allen-Cahn and Navier-Stokes equations, while evaporation and condensation processes accounted for local changes in position of the capillary interfaces. Our model captures the emergence of hysteresis as a result of irreversible geometric transitions—such as bridge coalescence and snap-off—without prescribing any constitutive relation between suction and saturation degrees. This change in capillary regimes arises naturally from the interface dynamics and is not solved by the LBM. In particular, we can capture and analyze the discontinuities in capillary forces, when capillary bridges merge or split within small elementary assemblies of three or four spherical particles. Having validated our numerical results for the above elementary assemblies, a polydispersed granular assembly composed of 1000 spherical grains is next addressed. Our simulations capture well-known condensation and evaporation hysteresis phenomenon while offering the possibility to inspect the underlying topology of air and water clusters for the same water saturation along different hydraulic paths.

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References (46)

  1. D. G. Fredlund and H. Rahardjo, Soil Mechanics for Unsaturated Soils (Wiley, New York, 1993).
  2. F. Xiao, J. Jing, S. Kuang, L. Yang, and A. Yu, Capillary forces on wet particles with a liquid bridge transition from convex to concave, Powder Technol. 363, 59 (2020).
  3. Y. Tang, S. Cheng, et al., Capillary forces on a small particle at a liquid-vapor interface: Theory and simulation, Phys. Rev. E 98, 032802 (2018).
  4. M. Dörmann and H.-J. Schmid, Simulation of capillary bridges between particles, Procedia Eng. 102, 14 (2015).
  5. B. Saint-Cyr, F. Radjai, J.-Y. Delenne, and P. Sornay, Cohesive granular materials composed of nonconvex particles, Phys. Rev. E 87, 052207 (2013).
  6. C. Semprebon, M. Scheel, S. Herminghaus, R. Seemann, and M. Brinkmann, Liquid morphologies and capillary forces between three spherical beads, Phys. Rev. E 94, 012907 (2016).
  7. M. Pakpour, M. Habibi, P. Møller, and D. Bonn, How to construct the perfect sandcastle, Sci. Rep. 2, 549 (2012).
  8. A. Gans, O. Pouliquen, and M. Nicolas, Cohesion-controlled granular material, Phys. Rev. E 101, 032904 (2020).
  9. V. Richefeu, M. S. El Youssoufi, and F. Radjai, Shear strength properties of wet granular materials, Phys. Rev. E 73, 051304 (2006).
  10. J. Duriez, M. Eghbalian, R. Wan, and F. Darve, The micromechanical nature of stresses in triphasic granular media with interfaces, J. Mech. Phys. Solids 99, 495 (2017).
  11. W. J. Likos and N. Lu, Hysteresis of capillary stress in unsaturated granular soil, J. Eng. Mech. 130, 646 (2004).
  12. R. Hilfer, Macroscopic capillarity and hysteresis for flow in porous media, Phys. Rev. E 73, 016307 (2006).
  13. H. Liu, Y. Ju, N. Wang, G. Xi, and Y. Zhang, Lattice Boltzmann modeling of contact angle and its hysteresis in two-phase flow with large viscosity difference, Phys. Rev. E 92, 033306 (2015).
  14. A. Poulovassilis, The effect of the entrapped air on the hysteresis curves of a porous body and on its hydraulic conductivity, Soil Science 109, 154 (1970).
  15. H. Pham, D. Fredlund, and S. L. Barbour, A practical hysteresis model for the soil–water characteristic curve for soils with negligible volume change, Géotechnique 53, 293 (2003).
  16. D. Gallipoli, A hysteretic soil-water retention model accounting for cyclic variations of suction and void ratio, Géotechnique 62, 605 (2012).
  17. H. Q. Pham, D. G. Fredlund, and S. L. Barbour, A study of hysteresis models for soil-water characteristic curves, Can. Geotech. J. 42, 1548 (2005).
  18. B. Mielniczuk, T. Hueckel, and M. S. El Youssoufi, Laplace pressure evolution and four instabilities in evaporating two-grain liquid bridges, Powder Technol. 283, 137 (2015).
  19. O. Pitois, P. Moucheront, and X. Chateau, Liquid bridge between two moving spheres: An experimental study of viscosity effects, J. Colloid Interface Sci. 231, 26 (2000).
  20. B. Mielniczuk, O. Millet, G. Gagneux, and M. S. El Youssoufi, Characterisation of pendular capillary bridges derived from experimental data using inverse problem method, Granular Matter 20, 14 (2018).
  21. H. N. G. Nguyen, O. Millet, and G. Gagneux, On the capillary bridge between spherical particles of unequal size: Analytical and experimental approaches, Continuum Mech. Thermodyn. 31, 225 (2019).
  22. H. N. G. Nguyen, O. Millet, and G. Gagneux, Liquid bridges between a sphere and a plane-classification of meniscus profiles for unknown capillary pressure, Math. Mech. Solids 24, 3042 (2019).
  23. H. N. G. Nguyen, O. Millet, C.-F. Zhao, and G. Gagneux, Theoretical and experimental study of capillary bridges between two parallel planes, Eur. J. Environ. Civ. Eng. 26, 1198 (2022).
  24. J.-P. Wang, E. Gallo, B. François, F. Gabrieli, and P. Lambert, Capillary force and rupture of funicular liquid bridges between three spherical bodies, Powder Technol. 305, 89 (2017).
  25. A. Di Renzo, G. Picarelli, and F. P. Di Maio, Numerical investigation of funicular liquid bridge interactions between spherical particles, Chem. Eng. Technol. 43, 830 (2020).
  26. M. Miot, G. Veylon, A. Wautier, P. Philippe, F. Nicot, and F. Jamin, Numerical analysis of capillary bridges and coalescence in a triplet of spheres, Granular Matter 23, 65 (2021).
  27. Z. Benseghier, O. Millet, P. Philippe, A. Wautier, N. Younes, and E. Liberge, Relevance of capillary interfaces simulation with the Shan–Chen multiphase LB model, Granular Matter 24, 82 (2022).
  28. H. Liang, J. Xu, J. Chen, H. Wang, Z. Chai, and B. Shi, Phase-field-based lattice Boltzmann modeling of large-density-ratio two-phase flows, Phys. Rev. E 97, 033309 (2018).
  29. H. Liang, H. Liu, Z. Chai, and B. Shi, Lattice Boltzmann method for contact-line motion of binary fluids with high density ratio, Phys. Rev. E 99, 063306 (2019).
  30. A. Fakhari and M. H. Rahimian, Phase-field modeling by the method of lattice Boltzmann equations, Phys. Rev. E 81, 036707 (2010).
  31. A. Fakhari and T. Lee, Multiple-relaxation-time lattice Boltzmann method for immiscible fluids at high Reynolds numbers, Phys. Rev. E 87, 023304 (2013).
  32. A. Fakhari, T. Mitchell, C. Leonardi, and D. Bolster, Improved locality of the phase-field lattice-Boltzmann model for immiscible fluids at high density ratios, Phys. Rev. E 96, 053301 (2017).
  33. N. Younes, Z. Benseghier, O. Millet, A. Wautier, F. Nicot, and R. Wan, Phase-field lattice Boltzmann model for liquid bridges and coalescence in wet granular media, Powder Technol. 411 117942 (2022).
  34. J.-P. Gras, Approche micromécanique de la capillarité dans les milieux granulaires: Rétention d'eau et comportement mécanique, Ph.D. thesis, Université Montpellier II-Sciences et Techniques du Languedoc, 2011.
  35. G. Gagneux and O. Millet, An analytical framework for evaluating the cohesion effects of coalescence between capillary bridges, Granular Matter 18, 16 (2016).
  36. S. Wang, F. Liu, J. Cui, M. Miao, and C. Pu, Experimental study on the rupture behavior of the liquid bridge between three rigid spheres, Langmuir 38, 13857 (2022).
  37. H. Safari, M. H. Rahimian, and M. Krafczyk, Extended lattice Boltzmann method for numerical simulation of thermal phase change in two-phase fluid flow, Phys. Rev. E 88, 013304 (2013).
  38. H. Safari, M. H. Rahimian, and M. Krafczyk, Consistent simulation of droplet evaporation based on the phase-field multiphase lattice Boltzmann method, Phys. Rev. E 90, 033305 (2014).
  39. R. Ledesma-Aguilar, D. Vella, and J. M. Yeomans, Lattice-Boltzmann simulations of droplet evaporation, Soft Matter 10, 8267 (2014).
  40. Q. Li, P. Zhou, and H. J. Yan, Improved thermal lattice Boltzmann model for simulation of liquid-vapor phase change, Phys. Rev. E 96, 063303 (2017).
  41. M. Sugimoto, Y. Sawada, M. Kaneda, and K. Suga, Consistent evaporation formulation for the phase-field lattice Boltzmann method, Phys. Rev. E 103, 053307 (2021).
  42. S. Schlüter, S. Berg, M. Rücker, R. Armstrong, H.-J. Vogel, R. Hilfer, and D. Wildenschild, Pore-scale displacement mechanisms as a source of hysteresis for two-phase flow in porous media, Water Resour. Res. 52, 2194 (2016).
  43. J. E. McClure, R. T. Armstrong, M. A. Berrill, S. Schlüter, S. Berg, W. G. Gray, and C. T. Miller, Geometric state function for two-fluid flow in porous media, Phys. Rev. Fluids 3, 084306 (2018).
  44. A. M. Boelens and H. A. Tchelepi, The effect of topology on phase behavior under confinement, Processes 9, 1220 (2021).
  45. N. Younes, A. Wautier, R. Wan, O. Millet, F. Nicot, and R. Bouchard, DEM-LBM coupling for partially saturated granular assemblies, Comput. Geotech. 162, 105677 (2023).
  46. T. Hueckel, B. Mielniczuk, and M. S. El Youssoufi, Adhesion-force micro-scale study of desiccating granular material, Géotechnique 70, 1133 (2020).

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Phys. Rev. E 113, 035401– Published 2 March, 2026
  • Received 28 November 2023
  • Accepted 6 February 2026
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