Modeling of chemo-hydromechanical behavior of unsaturated porous media: a nonlocal approach based on integral equations

  • Xiaoyu SongEmail author
  • Shashank Menon
Research Paper


Unsaturated clay is a heterogeneous porous medium consisting of three phases, namely solid soil skeleton, pore water, and pore air. It has been well recognized that the variation of the chemical property of pore fluid in clay can affect the hydromechanical behavior of this material remarkably. In this study, we formulate a non-local chemo-hydromechanical model for unsaturated clay via the constitutive correspondence principle in the state-based peridynamics—a reformulation of classical continuum mechanics using integral equations instead of partial differential equations. We numerically implement this non-local constitutive model through the implicit return mapping algorithm at the material particle level and then integrate the material subroutine into a computational peridynamics code. We conduct a series of numerical simulations of unsaturated clay samples under different chemical loading rates. The numerical results demonstrate that the proposed non-local model can capture the dramatic impact of organic chemicals on the mechanical behavior of unsaturated clay. The numerical results also show that the proposed non-local numerical model can simulate localized deformation in chemically active unsaturated clay because of the intrinsic length scale embedded in the integral equations.


Chemo-plasticity Integral equations Peridynamics Return mapping Unsaturated clay 



Support for this work was provided by the Geotechnical Engineering and Materials Program of the US National Science Foundation (NSF) under contract number CMMI-1659932 to the University of Florida. The support is gratefully acknowledged. Any opinions or positions expressed in this article are those of the authors only and do not reflect any opinions or positions of the NSF. We also thank the anonymous reviewers for their constructive reviews.


  1. 1.
    Alonso EE, Gens A, Josa A (1990) A constitutive model for partially saturated soils. Géotechnique 40(3):405–430CrossRefGoogle Scholar
  2. 2.
    Barbour SL, Fredlund DG (1989) Mechanisms of osmotic flow and volume change in clay soils. Can Geotech J 26(4):551–562CrossRefGoogle Scholar
  3. 3.
    Belytschko T, Lu YY, Gu L (1994) Element-free galerkin methods. Int J Numer Methods Eng 37(2):229–256MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bolt GH (1956) Physico-chemical analysis of the compressibility of pure clays. Géotechnique 6(2):86–93CrossRefGoogle Scholar
  5. 5.
    Bolt G, Miller R (1955) Compression studies of illite suspensions 1. Soil Sci Soc Am J 19(3):285–288CrossRefGoogle Scholar
  6. 6.
    Bolzon G, Schrefler B, Zienkiewicz O (1996) Elastoplastic soil constitutive laws generalized to partially saturated states. Géotechnique 46(2):279–289CrossRefGoogle Scholar
  7. 7.
    Borja RI (2004) Cam-clay plasticity. Part V: a mathematical framework for three-phase deformation and strain localization analyses of partially saturated porous media. Comput Methods Appl Mech Eng 193(48–51):5301–5338MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Borja RI (2006) On the mechanical energy and effective stress in saturated and unsaturated porous continua. Int J Solids Struct 43(6):1764–1786MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Borja RI (2013) Plasticity: modeling & computation. Springer Science & Business Media, New YorkCrossRefzbMATHGoogle Scholar
  10. 10.
    Borja RI, Koliji A (2009) On the effective stress in unsaturated porous continua with double porosity. J Mech Phys Solids 57(8):1182–1193CrossRefzbMATHGoogle Scholar
  11. 11.
    Borja RI, Song X, Rechenmacher AL, Abedi S, Wu W (2013) Shear band in sand with spatially varying density. J Mech Phys Solids 61(1):219–234CrossRefGoogle Scholar
  12. 12.
    Borja RI, Song X, Wu W (2013) Critical state plasticity. part vii: triggering a shear band in variably saturated porous media. Comput Methods Appl Mech Eng 261:66–82CrossRefzbMATHGoogle Scholar
  13. 13.
    Boukpeti N, Charlier R, Hueckel T (2004) Modelling contamination of clays. Elsevier Geo Eng Book Ser 2:523–528CrossRefGoogle Scholar
  14. 14.
    Bunger AP (2010) The mandelcryer effect in chemoporoelasticity. Int J Numer Anal Methods Geomech 34(14):1479–1511CrossRefzbMATHGoogle Scholar
  15. 15.
    Cao J, Jung J, Song X, Bate B (2018) On the soil water characteristic curves of poorly graded granular materials in aqueous polymer solutions. Acta Geotech 13(1):103–116CrossRefGoogle Scholar
  16. 16.
    Castellanos E, Villar M, Romero E, Lloret A, Gens A (2008) Chemical impact on the hydro-mechanical behaviour of high-density febex bentonite. Phys Chem Earth Parts A/B/C 33:S516–S526 Clays in Natural & Engineered Barriers for Radioactive Waste ConfinementCrossRefGoogle Scholar
  17. 17.
    Chen G, Gallipoli D, Ledesma A (2007) Chemo-hydro-mechanical coupled consolidation for a poroelastic clay buffer in a radioactive waste repository. Transp Porous Media 69:189–213CrossRefGoogle Scholar
  18. 18.
    Cleall P (1998) An investigation of the thermo/hydraulic/mechanical behaviour of unsaturated soils, including expansive soils. Ph.D. thesis, University of Wales Cardiff, UKGoogle Scholar
  19. 19.
    DeJong J, Soga K, Kavazanjian E, Burns S, Van Paassen L, Al Qabany A, Aydilek A, Bang S, Burbank M, Caslake LF et al (2013) Biogeochemical processes and geotechnical applications: progress, opportunities and challenges. Geotechnique 63(4):287CrossRefGoogle Scholar
  20. 20.
    Detournay E, Sarout J, Tan C, Caurel J (2005) Chemoporoelastic parameter identification of a reactive shale, vol. 125 of Solid Mechanics and its Applications, pp 125–132Google Scholar
  21. 21.
    D’Onza F, Gallipoli D, Wheeler S, Casini F, Vaunat J, Khalili N, Laloui L, Mancuso C, Mašín D, Nuth M et al (2011) Benchmark of constitutive models for unsaturated soils. Géotechnique 61(4):283–302CrossRefGoogle Scholar
  22. 22.
    Du Q, Gunzburger M, Lehoucq RB, Zhou K (2013) A nonlocal vector calculus, nonlocal volume-constrained problems, and nonlocal balance laws. Math Models Methods Appl Sci 23(03):493–540MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Fernandez F, Quigley RM (1985) Hydraulic conductivity of natural clays permeated with simple liquid hydrocarbons. Can Geotech J 22(2):205–214CrossRefGoogle Scholar
  24. 24.
    Fernandez F, Quigley RM (1988) Viscosity and dielectric constant controls on the hydraulic conductivity of clayey soils permeated with water-soluble organics. Can Geotech J 25(3):582–589CrossRefGoogle Scholar
  25. 25.
    Fernandez F, Quigley RM (1991) Controlling the destructive effects of clay-organic liquid interactions, by application of effective stresses. Can Geotech J 28(3):388–398CrossRefGoogle Scholar
  26. 26.
    Fisher R (1926) On the capillary forces in an ideal soil; correction of formulae given by wb haines. J Agric Sci 16(3):492–505CrossRefGoogle Scholar
  27. 27.
    Flanagan D, Taylor L (1987) An accurate numerical algorithm for stress integration with finite rotations. Comput Methods Appl Mech Eng 62(3):305–320CrossRefzbMATHGoogle Scholar
  28. 28.
    Foster JT, Silling SA, Chen WW (2010) Viscoplasticity using peridynamics. Int J Numer Methods Eng 81(10):1242–1258zbMATHGoogle Scholar
  29. 29.
    Gajo A, Loret B (2003) Finite element simulations of chemo-mechanical coupling in elasticplastic homoionic expansive clays. Comput Methods Appl Mech Eng 192(31):3489–3530CrossRefzbMATHGoogle Scholar
  30. 30.
    Gajo A, Loret B, Hueckel T (2002) Electro-chemo-mechanical couplings in saturated porous media: elastic–plastic behaviour of heteroionic expansive clays. Int J Solids Struct 39(16):4327–4362CrossRefzbMATHGoogle Scholar
  31. 31.
    Gallipoli D, Gens A, Sharma R, Vaunat J (2003) An elasto-plastic model for unsaturated soil incorporating the effects of suction and degree of saturation on mechanical behaviour. Géotechnique 53(1):123–136CrossRefGoogle Scholar
  32. 32.
    Ganzenmller G, Hiermaier S, May M (2015) On the similarity of meshless discretizations of peridynamics and smooth-particle hydrodynamics. Comput Struct 150:71–78CrossRefGoogle Scholar
  33. 33.
    Gens A (2010) Soil-environment interactions in geotechnical engineering. Géotechnique 60(1):3–74CrossRefGoogle Scholar
  34. 34.
    Guimarães LdN, Gens A, Olivella S (2007) Coupled thermo-hydro-mechanical and chemical analysis of expansive clay subjected to heating and hydration. Transp Porous Media 66(3):341–372CrossRefGoogle Scholar
  35. 35.
    Hueckel T (1997) Chemo-plasticity of clays subjected to stress and flow of a single contaminant. Int J Numer Anal Methods Geomech 21(1):43–72CrossRefzbMATHGoogle Scholar
  36. 36.
    Hueckel T (2002) Reactive plasticity for clays during dehydration and rehydration. Part 1: concepts and options. Int J Plast 18(3):281–312CrossRefzbMATHGoogle Scholar
  37. 37.
    Kaczmarek M (2001) Chemically induced deformation of a porous layer coupled with advectivedispersive transport. Analytical solutions. Int J Numer Anal Methods Geomech 25(8):757–770CrossRefzbMATHGoogle Scholar
  38. 38.
    Kaczmarek M, Hueckel T (1998) Chemo-mechanical consolidation of clays: analytical solutions for a linearized one-dimensional problem. Transp Porous Media 32:49–74CrossRefGoogle Scholar
  39. 39.
    Kenney T (1967) The influence of mineral composition on the residual strength of natural soils. In: Proceedings of the geotechnical conference on shear strength properties of natural soils and rocks, Oslo, Norway, vol 1, pp 123–129Google Scholar
  40. 40.
    Khalili N, Zargarbashi S (2010) Influence of hydraulic hysteresis on effective stress in unsaturated soils. Geotechnique 60(9):729–734CrossRefGoogle Scholar
  41. 41.
    Khalili N, Geiser F, Blight G (2004) Effective stress in unsaturated soils: review with new evidence. Int J Geomech 4(2):115–126CrossRefGoogle Scholar
  42. 42.
    Lehoucq RB, Silling SA (2008) Force flux and the peridynamic stress tensor. J Mech Phys Solids 56(4):1566–1577MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Lei X, Wong H, Fabbri A, Limam A, Cheng Y (2014) A thermo-chemo-electro-mechanical framework of unsaturated expansive clays. Comput Geotech 62:175–192CrossRefGoogle Scholar
  44. 44.
    Lei X, Wong H, Fabbri A, Limam A, Cheng Y (2016) A chemo-elasticplastic model for unsaturated expansive clays. Int J Solids Struct 88–89:354–378CrossRefGoogle Scholar
  45. 45.
    Li Y-C, Cleall PJ, Thomas H (2011) Multi-dimensional chemo-osmotic consolidation of clays. Comput Geotech 38(4):423–429CrossRefGoogle Scholar
  46. 46.
    Liu Z, Boukpeti N, Li X, Collin F, Radu J-P, Hueckel T, Charlier R (2005) Modelling chemo-hydro-mechanical behaviour of unsaturated clays: a feasibility study. Int J Numer Anal Methods Geomech 29(9):919–940CrossRefzbMATHGoogle Scholar
  47. 47.
    Loret B, Khalili N (2002) An effective stress elastic–plastic model for unsaturated porous media. Mech Mater 34(2):97–116CrossRefGoogle Scholar
  48. 48.
    Loret B, Hueckel T, Gajo A (2002) Chemo-mechanical coupling in saturated porous media: elasticplastic behaviour of homoionic expansive clays. Int J Solids Struct 39(10):2773–2806CrossRefzbMATHGoogle Scholar
  49. 49.
    Lu N, Godt JW, Wu DT (2010) A closed-form equation for effective stress in unsaturated soil. Water Resour Res 46(5):1–14CrossRefGoogle Scholar
  50. 50.
    Lu N, Khalili N, Nikooee E, Hassanizadeh SM (2014) Principle of effective stress in variably saturated porous media. Vadose Zone J 13(5):1–4Google Scholar
  51. 51.
    Macek RW, Silling SA (2007) Peridynamics via finite element analysis. Finite Elem Anal Des 43(15):1169–1178MathSciNetCrossRefGoogle Scholar
  52. 52.
    Madsen FT, Mitchell JK (1989) Chemical effects on clay farbric and hydraulic conductivity. Springer, Berlin, Heidelberg, pp 201–251Google Scholar
  53. 53.
    Maio CD (1996) Exposure of bentonite to salt solution: osmotic and mechanical effects. Géotechnique 46(4):695–707CrossRefGoogle Scholar
  54. 54.
    Mesri G, Olson R (1971) Mechanisms controlling the permeability of clays. Clays Clay Miner 19:151–158CrossRefGoogle Scholar
  55. 55.
    Michaels AS, Lin C (1954) Permeability of kaolinite. Ind Eng Chem 46(6):1239–1246CrossRefGoogle Scholar
  56. 56.
    Mitchell JK, Witherspoon PA, Greenberg JA. Chemico-osmotic effects in fine-grained soils. J Soil Mech Found Div Am Soc Civ Eng (United States) 99:SM4, 04 1973Google Scholar
  57. 57.
    Mitchell J K, Soga K et al (2005) Fundamentals of soil behavior, vol 3. Wiley, New YorkGoogle Scholar
  58. 58.
    Monaghan JJ (1992) Smoothed particle hydrodynamics. Ann Rev Astron Astrophys 30(1):543–574CrossRefGoogle Scholar
  59. 59.
    Moyne C, Murad MA (2002) Electro-chemo-mechanical couplings in swelling clays derived from a micro/macro-homogenization procedure. Int J Solids Struct 39(25):6159–6190CrossRefzbMATHGoogle Scholar
  60. 60.
    Nalbantoglu Z, Tuncer ER (2001) Compressibility and hydraulic conductivity of a chemically treated expansive clay. Can Geotech J 38(1):154–160Google Scholar
  61. 61.
    Nova R, Castellanza R, Tamagnini C (2003) A constitutive model for bonded geomaterials subject to mechanical and/or chemical degradation. Int J Numer Anal Methods Geomech 27(9):705–732CrossRefzbMATHGoogle Scholar
  62. 62.
    Nuth M, Laloui L (2008) Effective stress concept in unsaturated soils: clarification and validation of a unified framework. Int J Numer Anal Methods Geomech 32(7):771–801CrossRefzbMATHGoogle Scholar
  63. 63.
    O’ Donnell S T, Rittmann B E, Kavazanjian E Jr (2017) Midp: Liquefaction mitigation via microbial denitrification as a two-stage process. I: Desaturation. J Geotech Geoenviron Eng 143(12):04017094CrossRefGoogle Scholar
  64. 64.
    O’ Donnell S T, Kavazanjian E Jr, Rittmann B E (2017) Midp: Liquefaction mitigation via microbial denitrification as a two-stage process. II: Micp. J Geotech Geoenviron Eng 143(12):04017095CrossRefGoogle Scholar
  65. 65.
    Ouchi H, Katiyar A, York J, Foster JT, Sharma MM (2015) A fully coupled porous flow and geomechanics model for fluid driven cracks: a peridynamics approach. Comput Mech 55(3):561–576MathSciNetCrossRefzbMATHGoogle Scholar
  66. 66.
    Peters GP, Smith DW (2004) The influence of advective transport on coupled chemical and mechanical consolidation of clays. Mech Mater 36(5):467–486 Coupled Chemo-Mechanical PhenomenaCrossRefGoogle Scholar
  67. 67.
    Quigley RM, Fernandez F, Yanful E, Helgason T, Margaritis A, Whitby J (1987) Hydraulic conductivity of contaminated natural clay directly below a domestic landfill. Can Geotech J 24(3):377–383CrossRefGoogle Scholar
  68. 68.
    Sarout J, Detournay E (2011) Chemoporoelastic analysis and experimental validation of the pore pressure transmission test for reactive shales. Int J Rock Mech Min Sci 48(5):759–772CrossRefGoogle Scholar
  69. 69.
    Seetharam S, Thomas H, Cleall PJ (2007) Coupled thermo/hydro/chemical/mechanical model for unsaturated soilsnumerical algorithm. Int J Numer Methods Eng 70(12):1480–1511CrossRefzbMATHGoogle Scholar
  70. 70.
    Sherwood J (1993) Biot poroelasticity of a chemically active shale. Proc R Soc Lond A Math Phys Eng Sci 440(1909):365–377CrossRefzbMATHGoogle Scholar
  71. 71.
    Silling SA (2000) Reformulation of elasticity theory for discontinuities and long-range forces. J Mech Phys Solids 48(1):175–209MathSciNetCrossRefzbMATHGoogle Scholar
  72. 72.
    Silling SA, Askari E (2005) A meshfree method based on the peridynamic model of solid mechanics. Comput Struct 83(17–18):1526–1535CrossRefGoogle Scholar
  73. 73.
    Silling SA, Lehoucq R (2010) Peridynamic theory of solid mechanics. Adv Appl Mech 44:73–168CrossRefGoogle Scholar
  74. 74.
    Silling SA, Epton M, Weckner O, Xu J, Askari E (2007) Peridynamic states and constitutive modeling. J Elast 88(2):151–184MathSciNetCrossRefzbMATHGoogle Scholar
  75. 75.
    Simo J, Hughes T (1998) Computational inelasticity. Springer, New YorkzbMATHGoogle Scholar
  76. 76.
    Song X (2014) Strain localization in unsaturated porous media. Ph.D. thesis. Stanford UniversityGoogle Scholar
  77. 77.
    Song X (2017) Transient bifurcation condition of partially saturated porous media at finite strain. Int J Numer Anal Methods Geomech 41(1):135–156CrossRefGoogle Scholar
  78. 78.
    Song X, Borja RI (2014) Finite deformation and fluid flow in unsaturated soils with random heterogeneity. Vadose Zone J 13(5):1–11Google Scholar
  79. 79.
    Song X, Borja RI (2014) Mathematical framework for unsaturated flow in the finite deformation range. Int J Numer Methods Eng 97(9):658–682MathSciNetCrossRefzbMATHGoogle Scholar
  80. 80.
    Song X, Wang K, Ye M (2018) Localized failure in unsaturated soils under non-isothermal conditions. Acta Geotech 13(1):73–85CrossRefGoogle Scholar
  81. 81.
    Song X, Idinger G, Borja RI, Wu W (2012) Finite element simulation of strain localization in unsaturated soils. In: Unsaturated soils: research and applications. Springer, pp 189–195Google Scholar
  82. 82.
    Song X, Ye M, Wang K. Strain localization in a solid-water-air system with random heterogeneity via stabilized mixed finite elements. Int J Numer Methods Eng.
  83. 83.
    Sridharan A (1991) Engineering behaviour of fine grained soils: a fundamental approach. Indian Geotehn JGoogle Scholar
  84. 84.
    Sridharan A, Rao GV (1973) Mechanisms controlling volume change of saturated clays and the role of the effective stress concept. Géotechnique 23(3):359–382CrossRefGoogle Scholar
  85. 85.
    Taron J, Elsworth D (2009) Thermal-hydrologic-mechanical-chemical processes in the evolution of engineered geothermal reservoirs. Int J Rock Mech Min Sci 46(5):855–864CrossRefGoogle Scholar
  86. 86.
    Taron J, Elsworth D, Min K-B (2009) Numerical simulation of thermal-hydrologic-mechanical-chemical processes in deformable, fractured porous media. Int J Rock Mech Min Sci 46(5):842–854CrossRefGoogle Scholar
  87. 87.
    Thomas H, Cleall P (1999) Inclusion of expansive clay behaviour in coupled thermo hydraulic mechanical models. Eng Geol 54(1):93–108CrossRefGoogle Scholar
  88. 88.
    Thomas H, Cleall P (1997) Chemico-osmotic effects on the behaviour of unsaturated expansive clays. In: Geoenvironmental Engineering: Contaminated GroundGoogle Scholar
  89. 89.
    Thomas H, Cleall P, Seetharam S (2002) Numerical modelling of the thermal-hydraulic-chemical-mechanical behaviour of unsaturated clay. Environ Geomech Monte Verità, pp 125–136Google Scholar
  90. 90.
    Ulm F-J, Coussy O (1995) Modeling of thermochemomechanical couplings of concrete at early ages. J Eng Mech 121(7):785–794CrossRefGoogle Scholar
  91. 91.
    Van Genuchten MT (1980) A closed-form equation for predicting the hydraulic conductivity of unsaturated soils. Soil Sci Soc Am J 44(5):892–898CrossRefGoogle Scholar
  92. 92.
    Wheeler S, Sivakumar V (1995) An elasto-plastic critical state framework for unsaturated soil. Géotechnique 45(1):35–53CrossRefGoogle Scholar
  93. 93.
    Witteveen P, Ferrari A, Laloui L (2013) An experimental and constitutive investigation on the chemo-mechanical behaviour of a clay. Geotechnique 63(3):244CrossRefGoogle Scholar
  94. 94.
    Zhang H, Zhou L (2008) Implicit integration of a chemo-plastic constitutive model for partially saturated soils. Int J Numer Anal Methods Geomech 32(14):1715–1735CrossRefzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Department of Civil and Coastal EngineeringUniversity of FloridaGainesvilleUSA

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