Skip to main content
Log in

Effect of free water on the quasi-static compression behavior of partially-saturated concrete with a fully coupled DEM/CFD approach

  • Original Report
  • Published:
Granular Matter Aims and scope Submit manuscript

Abstract

The work aims to numerically investigate the quasi-static response of partially fluid-saturated concrete under two-dimensional uniaxial compression at the mesoscale. We investigated how the impact of free pore fluid content (water and gas) affected the quasi-static strength of concrete. The totally and partially fluid-saturated concrete behavior was simulated using an improved pore-scale hydro-mechanical model based on DEM/CFD. The fluid flow concept was based on a fluid flow network made up of channels in a continuous region between discrete elements. A two-phase laminar fluid flow was postulated in partially saturated porous concrete with very low porosity. Position and volumes of pores/cracks were considered to correctly track the liquid/gas content. In both dry and wet conditions, a series of numerical simulations were performed on bonded granular specimens of a simplified spherical mesostructure that mimicked concrete. The effects of fluid saturation and fluid viscosity on concrete strength and fracture, and fluid pore pressures were investigated. It was found that each of those effects significantly impacted the hydro-mechanical behavior of concrete. Due to the rising fluid pressure in pores during initial specimen compaction under compressive loading that promoted a cracking process, the compressive strength increased as fluid saturation and fluid viscosity decreased.

Graphical abstract

DEM-CFD results for fully saturated specimen: evolution of maximum pore water pressure against vertical normal strain during uniaxial compression (from zero up to peak stress for).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21

Similar content being viewed by others

Data availability

Data will be made available on request.

References

  1. Rossi, P., Boulay, C.: Influence of free water in concrete on the cracking process. Mag. Concr. Res. 1990(42), 143–146 (1990)

    Google Scholar 

  2. Rossi, P.: Influence of cracking in presence of free water on the mechanical behavior of concrete. Mag. Concr. Res. 43, 53–57 (1991)

    Google Scholar 

  3. Li, G.: The effect of moisture content on the tensile strength properties of concrete. University of Florida, Gainesville, FL, USA (2004)

    Google Scholar 

  4. Wittmann, F.H., Sun, Z., Zhao, T.: Strength and fracture energy of concrete in seawater. In Proceedings of the 6th International Conference on Fracture Mechanics of Concrete and Concrete Structures, Catania, Italy, 17–22 June (2007)

  5. Wang, H.L., Li, Q.B.: Experiments on saturated concrete under dfferent splitting tensile rate and mechanism on strength change. Eng. Mech. 24, 105–109 (2007)

    Google Scholar 

  6. Yan, D.M., Lin, G., Wang, Z.: Research on dynamic direct tensile properties of concrete under different environments. J. Dalian Univ. Technol. 45, 416–421 (2005)

    Google Scholar 

  7. Deng, H.F., Yuan, X.F., Li, J.L.: Fracture mechanics characteristics and deterioration mechanism of sandstone under reservoir immersion interaction. J. Earth Sci. China Univ. Geosci. 39, 108–114 (2014)

    Google Scholar 

  8. Zhang, P., Sun, Z.W., Zhao, T.J.: Fracture energy and strain softening of concrete under seawater environment. J. Civ. Arch. Environ. Eng. 32, 72–77 (2010)

    Google Scholar 

  9. Reinhardt, H.W., Rossi, P., van Mier, J.G.M.: Joint investigation of concrete at high rates of loading. Mater. Struct. 23, 213–216 (1990)

    Google Scholar 

  10. Ross, C., Jerome, D.M., Tedesco, J.E.: Moisture and strain rate effects on concrete strength. ACI Mater. J. 93, 293–300 (1996)

    Google Scholar 

  11. Wang, J., Sun, K., Hu, Y., Guan, Q., Li, Q.: The mechanical properties of concrete in water environment: a review. Front. Mater. 9, 996650 (2022)

    ADS  Google Scholar 

  12. Shen, J., Xu, Q.: Effect of moisture content and porosity on compressive strength of concrete during drying at 105 °C. Constr. Build. Mater. 195, 19–27 (2019)

    Google Scholar 

  13. Zhang, G., Li, X., Li, Z.: Experimental study on static mechanical properties and moisture contents of concrete under water environment. Sustainability 11, 2962 (2019)

    Google Scholar 

  14. Yousheng, D., Weiling, Y.: Research progress on the effect of environmental water on the static strength of concrete. Adv. Sci. Technol. Water Resour. 35(4), 99–103 (2015)

    Google Scholar 

  15. Malecot, Y., Zingg, L., Briffaut, M., Baroth, J.: Influence of free water on concrete triaxial behavior: the effect of porosity. Cem. Concr. Res. 120, 207–216 (2019)

    Google Scholar 

  16. Piotrowska, E., Forquin, P.: Experimental investigation of the confined behavior of dry and wet high-strength concrete: quasi static versus dynamic loading. J. Dyn. Behav. Mater. 1(2), 191–200 (2015)

    Google Scholar 

  17. Vu, X.H., Malecot, Y., Daudeville, L., Buzaud, E.: Experimental analysis of concrete behavior under high confinement: effect of the saturation ratio. Int. J. Solids Struct. 46(5), 1105–1120 (2009)

    Google Scholar 

  18. Boxu, M., Jinyu, X., Chuanxin, L., Chao, G., Guang, P.: Effect of water content on tensile properties of cement mortar. IOP Conf. Ser.: Earth Environ. Sci. 189, 032023 (2018)

    Google Scholar 

  19. Chen, X., Huang, W., Zhou, J.: Effect of moisture content on compressive and split tensile strength of concrete. Indian J. Eng. Mater. Sci. 19, 427–435 (2012)

    Google Scholar 

  20. Zhu, F.Z., Liu, J., Li, Z.L.: Discussion on the influence of water content in concrete dynamic elastic modulus test. J. Concr. 11, 40–41 (2012)

    Google Scholar 

  21. Shoukry, S.N., William, G.W., Riad, M.Y., Downie, B. Effect of moisture and temperature on the mechanical properties of concrete. Proceedings of the SEM Annual Conference June 1–4, 2009 Albuquerque New Mexico USA, (2009)

  22. Cadoni, E., Labibes, K., Albertini, C.: Strain-rate effect on the tensile behavior of concrete at different relative humidity levels. Mater. Struct. 34, 21–26 (2001)

    Google Scholar 

  23. Wang, H.L., Li, Q.B.: Experiments of the compressive properties of dry and saturated concrete under different loading rates. J. Hydroelectr. Eng. 26, 84–89 (2007)

    Google Scholar 

  24. Fu, Q., Zhang, Z., Zhao, X., Hong, M., Guo, B., Yuan, Q., Niu, D.: Water saturation effect on the dynamic mechanical behaviour and scaling law effect on the dynamic strength of coral aggregate concrete. Cement Concr. Compos. 120, 104034 (2021)

    Google Scholar 

  25. Xu, W.B., Li, Q.B., Hu, Y.: Water content variations in the process of concrete setting. J. Hydroelectr. Eng. 36(07), 92–103 (2017)

    Google Scholar 

  26. Oshita, H., Tanabe, T.: Water migration phenomenon model in cracked concrete I: formulation. J. Eng. Mech. 126, 539–543 (2000)

    Google Scholar 

  27. Wang, H.L., Li, Q.B., Sun, X.Y., Jin, W.L.: Mesomechanism of tensile strength reduction and tension constitutive model of saturated concrete. J. Basic Sci. Eng. 16, 65–72 (2008)

    Google Scholar 

  28. Zhang, X., Chiu, Y., Hong Hao, H., Cui, J.: Free water effect on the dynamic compressive properties of mortar. Cement Concr. Compos. 118, 103933 (2021)

    Google Scholar 

  29. Weiss, T., Siegesmund, S., Kirchner, D., Sippel, J.: Insolation weathering and hygric dilatation: two competitive factors in stone degradation. Environ. Geol. 46, 402–413 (2004)

    Google Scholar 

  30. Sun, X., Wang, H., Cheng, X., Sheng, Y.: Effect of pore liquid viscosity on the dynamic compressive properties of concrete. Constr. Build. Mater. 231, 117143 (2020)

    Google Scholar 

  31. Krzaczek, M., Kozicki, J., Nitka, M., Tejchman, J.: Simulations of hydro-fracking in rock mass at meso-scale using fully coupled DEM/CFD approach. Acta Geotech. 15, 297–324 (2020)

    Google Scholar 

  32. Krzaczek, M., Nitka, M., Tejchman, J.: Effect of gas content in macropores on hydraulic fracturing in rocks using a fully coupled DEM/CFD approach. Int. J. Numer. Anal. Meth. Geomech. 45(2), 234–264 (2021)

    Google Scholar 

  33. Bolander, J.E., Eliáš, J., Cusatis, G., Nagai, K.: Discrete mechanical models of concrete fracture. Eng. Fract. Mech. 257, 108030 (2021)

    Google Scholar 

  34. Skarżyński, L., Nitka, M., Tejchman, J.: Modelling of concrete fracture at aggregate level using FEM and DEM based on x-ray µCT images of internal structure. Eng. Fract. Mech. 10(147), 13–35 (2015)

    Google Scholar 

  35. Nitka, M., Tejchman, J.: Modelling of concrete behaviour in uniaxial compression and tension with DEM. Granular Matter 217(1), 145–164 (2015)

    Google Scholar 

  36. Nitka, M., Tejchman, J.: A three-dimensional meso scale approach to concrete fracture based on combined DEM with X-ray μCT images. Cem. Concr. Res. 107, 11–29 (2018)

    Google Scholar 

  37. Suchorzewski, J., Tejchman, J., Nitka, M.: Discrete element method simulations of fracture in concrete under uniaxial compression based on its real internal structure. Int. J. Damage Mech 27(4), 578–607 (2014)

    Google Scholar 

  38. Suchorzewski, J., Tejchman, J., Nitka, M.: Experimental and numerical investigations of concrete behaviour at meso-level during quasi-static splitting tension. Theoret. Appl. Fract. Mech. 96, 720–739 (2018)

    Google Scholar 

  39. Singh, K., Menke, H., Andrew, M., et al.: Dynamics of snap-off and pore-filling events during two-phase fluid flow in permeable media. Sci. Rep. 7, 5192 (2017)

    ADS  Google Scholar 

  40. Shams, M., Singh, K., Bijeljic, B., Blunt, M.J.: Direct numerical simulation of pore-scale trapping events during capillary-dominated two-phase flow in porous media. Transp Porous Med 138, 443–458 (2021)

    MathSciNet  Google Scholar 

  41. Krzaczek, M., Nitka, M., Tejchman, J.: Modelling hydraulic and capillary-driven two-phase fluid flow in unsaturated concretes at the meso-scale with a unique coupled DEM-CFD technique. Int J Numer Anal Methods Geomech. 47(1), 23–53 (2023)

    Google Scholar 

  42. Cundall, P.: Fluid formulation for PFC2D. Itasca Consulting Group, Minneapolis, Minnesota (2000)

    Google Scholar 

  43. Hazzard, J.F., Young, R.P., Oates, J.S.: Numerical modeling of seismicity induced by fluid injection in a fractured reservoir. Proceedings of the 5th North American Rock Mechanics Symposium, Miningand Tunnel Innovation and Opportunity, Toronto, Canada, 7–10 July 2002, pp. 1023–1030 (2002)

  44. Al-Busaidi, A., Hazzard, J.F., Young, R.P.: Distinct element modeling of hydraulically fractured distinct element modeling of hydraulically fractured lac du bonnet granite. J. Geophys. Res. 110, 06302 (2005)

    ADS  Google Scholar 

  45. Yoon, J.S., Zang, A., Stephansson, O.: Numerical investigation on optimized stimulation of intact and naturally fractured deep geothermal reservoirs using hydro-mechanical coupled discrete particles joints model. Geothermic 52, 165–184 (2014)

    ADS  Google Scholar 

  46. Shimizu, H., Murata, S., Ishida, T.: The distinct element analysis for hydraulic fracturing in hard rock considering fluid viscosity and particle size distribution. Int J Rock Mech Mining Sci. 48, 712–727 (2011)

    Google Scholar 

  47. Ma, X., Zhou, T., Zou, Y.: Experimental and numerical study of hydraulic fracture geometry in shale formations with complex geologic conditions. J. Struct. Geol. 98, 53–66 (2017)

    ADS  Google Scholar 

  48. Liu, G., Sun, W., Lowinger, S.M., Zhang, Z., Huang, M., Peng, J.: Coupled flow network and discrete element modeling of injected-induced crack propagation and coalescence in brittle rock. Acta Geotech. 14(3), 843–869 (2019)

    Google Scholar 

  49. Zhang, G., Li, M., Gutierrez, M.: Numerical simulation of proppant distribution in hydraulic fractures in horizontal wells. J Nat Gas Sci Eng. 48, 157–168 (2017)

    Google Scholar 

  50. Xiao-Dong, N., Zhu, C., Wang, Y.: Hydro-mechanical analysis of hydraulic fracturing based on an improved DEM-CFD coupling model at micro-level. J. Comput. Theor. Nanosci. 12(9), 2691–2700 (2015)

    Google Scholar 

  51. Zeng, J., Li, H., Zhang, D.: Numerical simulation of proppant transport in hydraulic fracture with the upscaling CFD-DEM method. J Nat Gas Sci Eng. 33, 264–277 (2016)

    Google Scholar 

  52. Zhang, G., Sun, S., Chao, K., et al.: Investigation of the nucleation, propagation and coalescence of hydraulic fractures in glutenite reservoirs using a coupled fluid flow-DEM approach. Powder Technol. 354, 301–313 (2019)

    Google Scholar 

  53. Lathama, J.P., Munjiz, A., Mindel, J., et al.: Modelling of massive particulates for breakwater engineering using coupled FEM/DEM and CFD. Particuology 6, 572–583 (2008)

    Google Scholar 

  54. Chareyre, B., Cortis, A., Catalano, E., Barthélemy, F.: Pore-scale modeling of viscous flow and induced forces in dense sphere packings. Transp. Porous Media 94(2), 595–615 (2012)

    MathSciNet  Google Scholar 

  55. Catalano, E., Chareyre, B., Barthélémy, F.: Pore-scale modeling of fluid-particles interaction and emerging poromechanical effects. Int. J. Numer. Anal. Meth. Geomech. 238, 51–71 (2014)

    Google Scholar 

  56. Papachristos, E., Scholtès, L., Donzé, F.V., Chareyre, B.: Intensity and volumetric characterizations of hydraulically driven fractures by hydro-mechanical simulations. Int. J. Rock Mech. Min. Sci. 93, 163–178 (2017)

    Google Scholar 

  57. Caulk, R., Sholtès, L., Krzaczek, M., Chareyre, B.: A pore-scale thermo–hydro-mechanical model for particulate systems. Comput. Methods Appl. Mech. Eng. 372, 113292 (2020)

    ADS  MathSciNet  Google Scholar 

  58. Bolander, J.E., Berton, S.: Simulation of shrinkage induced cracking in cement composite overlays. Cement Concr. Compos. 26, 861–871 (2004)

    Google Scholar 

  59. Grassl, P., Bolander, J.: Three-dimensional network model for coupling of fracture and mass transport in quasi-brittle geomaterials. Materials 9, 782 (2016)

    ADS  Google Scholar 

  60. Luković, M., Šavija, B., Schlangen, E., Ye, G., van Breugel, K.: A 3D lattice modelling study of drying shrinkage damage in concrete repair systems. Materials 9(2016), 575 (2016)

    ADS  Google Scholar 

  61. Athanasiadis, I., Wheeler, S.J., Grassl, P.: Hydro-mechanical network modelling of particulate composites. Int. J. Solids Struct. 130–131(2018), 49–60 (2018)

    Google Scholar 

  62. Eliáš, J., Cusatis, G.: Homogenization of discrete mesoscale model of concrete for coupled mass transport and mechanics by asymptotic expansion. J. Mech. Phys. Solids 167, 105010 (2022)

    MathSciNet  Google Scholar 

  63. Mašek, J., Květoň, J., Eliáš, J.: Adaptive discretization refinement for discrete models of coupled mechanics and mass transport in concreto. Constr. Build. Mater. 395, 132243 (2023)

    Google Scholar 

  64. Forquin, P., Sallier, L., Pontiroli, C.: A numerical study on the influence of free water content on the ballistic performances of plain concrete targets. Mech. Mater. 89, 176–189 (2015)

    Google Scholar 

  65. Bian, H.B., Jia, Y., Pontiroli, C., Shao, J.F.: Numerical modeling of the elastoplastic damage behavior of dry and saturated concrete targets subjected to rigid projectile penetration. Int. J. Numer. Anal. Meth. Geomech. 42(2), 312–338 (2018)

    Google Scholar 

  66. Benniou, H., Accary, A., Malecot, Y., Briffaut, M., Daudeville, L.: Discrete element modeling of concrete under high stress level: influence of saturation ratio. Comput. Part. Mech. 8(1), 157–167 (2021)

    Google Scholar 

  67. Abdi, R., Krzaczek, M., Tejchman, J.: Comparative study of high-pressure fluid flow in densely packed granules using a 3D CFD model in a continuous medium and a simplified 2D DEM-CFD approach. Granular Matter 24(1), 1–25 (2022)

    Google Scholar 

  68. Abdi, R., Krzaczek, M., Tejchman, J.: Simulations of high-pressure fluid flow in a pre-cracked rock specimen composed of densely packed bonded spheres using a 3D CFD model and simplified 2D coupled CFD-DEM approach. Powder Technol. 417, 118238 (2023)

    Google Scholar 

  69. Krzaczek, M., Nitka, M., Tejchman, J.: A novel DEM-based pore-scale thermal-hydro-mechanical model for fractured non-saturated porous materials. Acta Geotech. 18, 2487–2512 (2023)

    Google Scholar 

  70. Krzaczek, M., Tejchman, J.: Hydraulic fracturing process in rocks – small-scale simulations with a novel fully coupled DEM/CFD-based thermo-hydro-mechanical approach. Eng. Fract. Mech. 289, 109424 (2023)

    Google Scholar 

  71. Kozicki, J., Donze, F.V.: A new open-source software developer for numerical simulations using discrete modeling methods. Comput. Methods Appl. Mech. Eng. 197, 4429–4443 (2008)

    ADS  Google Scholar 

  72. Šmilauer, V. et al.: Yade Documentation 3rd ed. The Yade Project, (2021). 10.5281/zenodo.5705394

  73. Cundall, P., Strack, O.: A discrete numerical model for granular assemblies. Géotechnique 29(1), 47–65 (1997)

    Google Scholar 

  74. Cundall, P., Hart, R.: Numerical modelling of discontinua. Eng. Comput. 9, 101–113 (1992)

    Google Scholar 

  75. Tomporowski, D., Nitka, M., Tejchman, J.: Application of the 3D DEM in the modelling of fractures in pre-flawed marble specimens during uniaxial compression. Eng. Fract. Mech. 277, 108978 (2023)

    Google Scholar 

  76. Widulinski, L., Tejchman, J., Kozicki, J., Leśniewska, D.: Discrete simulations of shear zone patterning in sand in earth pressure problems of a retaining wall. Int. J. Solids Struct. 48(7–8), 1191–1209 (2011)

    Google Scholar 

  77. Kozicki, J., Niedostatkiewicz, M., Tejchman, J., Mühlhaus, H.-B.: Discrete modelling results of a direct shear test for granular materials versus FE results. Granular Matter 15(5), 607–627 (2013)

    Google Scholar 

  78. Kozicki, J., Tejchman, J., Műhlhaus, H.-B.: Discrete simulations of a triaxial compression test for sand by DEM. Int. J. Num. Anal. Meth. Geomech. 38, 1923–1952 (2014)

    Google Scholar 

  79. Kozicki, J., Tejchman, J.: Relationship between vortex structures and shear localization in 3D granular specimens based on combined DEM and Helmholtz-Hodge decomposition. Granular Matter 20, 48 (2018)

    Google Scholar 

  80. Suchorzewski, J., Tejchman, J., Nitka, M., Bobinski, J.: Meso-scale analyses of size effect in brittle materials using DEM. Granular Matter 21(9), 1–19 (2019)

    Google Scholar 

  81. Nitka, M., Tejchman, J.: Meso-mechanical modelling of damage in concrete using discrete element method with porous ITZs of defined width around aggregates. Eng. Fract. Mech. 231, 107029 (2020)

    Google Scholar 

  82. Reynolds, O.: An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous, and of the law of resistances in parallel channels. Phil. Trans. Roy. Soc London 174, 935–982 (1883)

    ADS  Google Scholar 

  83. Batchelor, G.: An Introduction to Fluid Dynamics. Cambridge University Press, Cambridge (2000)

    Google Scholar 

  84. Tejchman, J., Bobinski, J.: Continous and discountinous modelling of fracture in concrete using FEM. Springer-Verlag, Berlin Heidelberg (2013)

    Google Scholar 

  85. Hökmark, H., Lönnqvist, M., Fälth, B.: Technical Report TR-10–23: THM-issues in repository rock–thermal, mechanical, thermo-mechanical and hydro-mechanical evolution of the rock at the Forsmark and Laxemar sites. SKB–Swedish Nuclear Fuel and Waste Management Co., 210; pp.26–27 (2010)

  86. Barmak, I., Gelfgat, A., Vitoshkin, H., Ullmann, A., Brauner, N.: Stability of stratified two-phase flows in horizontal channels. AIP Phys. Fluids 28, 044101 (2016)

    ADS  Google Scholar 

  87. Mathias, P.M., Naheiri, M., Oh, E.M.: A Density Correction for the Peng-Robinson Equation of State. In: Fluid Phase Equilibria, pp. 77–87. Elsevier Science Publishers B.V., Amsterdam (1989)

    Google Scholar 

  88. Peneloux, A., Rauzy, E., Freze, R.: A Consistent Correction for Redlich-Kwong-Soave Volumes. Fluid Phase Equilib. 8, 7–23 (1982)

    Google Scholar 

Download references

Acknowledgements

The present study was supported by the research project “Fracture propagation in rocks during hydro-fracking-experiments and discrete element method coupled with fluid flow and heat transport” (years 2019–2023), and financed by the National Science Centre (NCN) (UMO-2018/29/B/ST8/00255).

Author information

Authors and Affiliations

Authors

Contributions

MK: Conceptualization, Methodology, Investigation, Writing— original draft, Writing—review & editing, JT: Conceptualization, Methodology, Writing—original draft, Writing—review & editing, MN: Investigation.

Corresponding author

Correspondence to J. Tejchman.

Ethics declarations

Conflict of interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendices

Appendix A

The DEM equations are listed below:

$$\vec{F}_{n} = K_{n} U\vec{N},$$
(A1)
$$\vec{F}_{s} = \vec{F}_{s,prev} + K_{s} \Delta \vec{X}_{s} ,$$
(A2)
$$K_{n} = E_{c} \frac{{2R_{A} R_{B} }}{{R_{A} + R_{B} }}\;{\text{and}}\;K_{s} = v_{c} E_{c} \frac{{2R_{A} R_{B} }}{{R_{A} + R_{B} }},$$
(A3)
$$\vec{F}_{s} - F_{max}^{s} - \vec{F}_{n} \times {\text{tan}}\mu_{c} \le 0\;({\text{before}}\;{\text{contact}}\;{\text{ breakage}}),$$
(A4)
$$\vec{F}_{s} - \vec{F}_{n} \times {\text{tan}}\mu_{c} \le 0\;\left( {{\text{after}}\;{\text{ contact }}\;{\text{breakage}}} \right),$$
(A5)
$$F_{max}^{s} = CR^{2} \;{\text{and}}\;F_{min}^{n} = TR^{2} ,$$
(A6)
$$\vec{F}_{damp}^{k} = \vec{F}^{k} - \alpha_{d} \cdot {\text{sgn}}\left( {\vec{v}_{p}^{k} } \right)\vec{F}^{k} .$$
(A7)

where \({\overrightarrow{F}}_{n}\)—the normal contact force, U—the overlap between discrete elements, \(\overrightarrow{N}\)—the unit normal vector at the contact point, \({\overrightarrow{F}}_{s}\)—the tangential contact force, \({\overrightarrow{F}}_{s,prev}\)—the tangential contact force in the previous iteration, \({\overrightarrow{X}}_{s}\)—the relative tangential displacement increment, Kn—the normal contact stiffness, Ks—the tangential contact stiffness, Ecthe elastic modulus of the particle contact, υc—the Poisson’s ratio of particle contact, R—the particle radius, RA and RB contacting particle radii, μcthe Coulomb inter-particle friction angle, \({F}_{max}^{s}\)—the critical cohesive contact force, \({F}_{min}^{n}\)—the minimum tensile force, C—the cohesion at the contact (maximum shear stress at zero pressure), and T—the tensile strength of the contact, \({\overrightarrow{F}}_{damp}^{k}\)—the dampened contact force, \({\overrightarrow{F}}^{k}\) and \({\overrightarrow{v}}_{p}^{k}\)-kth—the components of the residual force and translational particle velocity vp and αd—the positive damping coefficient smaller than 1 (sgn(\(\cdot\)) that returns the sign of the kth component of velocity).

Appendix B

The hydraulic aperture h of the artificial channels 'S2S' is computed by a modified empirical formula developed by Hökmark et al. [85]:

$$h = \beta \left( {h_{inf} + \left( {h_{0} - h_{inf} } \right)e^{{ - 1.5 \cdot 10^{ - 7} \sigma_{n} }} } \right),$$
(B1)

where \({h}_{inf}\)—the hydraulic aperture for the infinite normal stress, \({h}_{0}\)—the hydraulic aperture for the zero normal stress, \({\sigma }_{n}\)—the effective normal stress at the particle contact and β—the aperture coefficient. The geometry of the nearby triangles has a direct bearing on the hydraulic aperture of the actual channels ‘T2T’ (Fig. 22) as

Fig. 22
figure 22

‘T2T’ channel (red colour denotes channel width h) (colour figure online)

$$h = \gamma e\cos \left( {90^{ \circ } - \omega } \right),$$
(B2)

where e is the edge length between two adjacent triangles, ω denotes the angle between the edge with the length e and the center line of the channel that connects two adjacent triangles, and γ is the reduction factor established to maintain the maximum Reynolds number Re along the main flow route at a value below the critical value for laminar flow (Re = 2100). The parameter γ was determined in parametric tests at fluid pressure up to 140 MPa.

Following Barmak et al. [86], the continuity and momentum equations are developed that are rendered dimensionless

$$div {\text{u}}_{j} = 0,$$
(B3)
$$\frac{{\partial {\text{u}}_{j} }}{\partial t} + \left( {{\text{u}}_{j} \cdot \nabla } \right){\text{u}}_{j} = - \frac{{\rho_{q} }}{{r\rho_{j} }}\nabla p_{j} + \frac{1}{{{\text{Re}}_{p} }}\frac{{\rho_{q} }}{{r\rho_{j} }}\frac{{m\mu_{j} }}{{\mu_{q} }} \cdot {\text{u}}_{j} ,$$
(B4)

where \({\text{u}}_{j}=\left({u}_{j},{v}_{j}\right)\) and \({p}_{j}\) are the velocity and pressure of the fluid phase j, \({\rho }_{j}\) and \({\mu }_{j}\) are the corresponding density and dynamic viscosity. The Reynolds number is \({\text{Re}}_{p}={\rho }_{p}{u}_{i}{h}_{p}/{\mu }_{p}\) and the density and viscosity ratios are \(r={\rho }_{q}/{\rho }_{p}\) and \(m={\mu }_{q}/{\mu }_{p}\). The lower and upper phases in the dimensionless formulation, are located in the regions \(-{n}_{d}\le y\le 0\) and \(0\le y\le 1\), where \({n}_{d}={h}_{q}/{h}_{p}\). The velocities satisfy the no-slip boundary condition at the channel walls.

$${\text{u}}_{q} \left( {y = - n_{d} } \right) = 0\;{\text{and}}\;{\text{u}}_{p} \left( {y = 1} \right) = 0.$$
(B5)

The continuity of velocities and tangential stresses is required by the boundary conditions at the interface at y = 0 [86]

$${\text{u}}_{q} \left( {y = 0} \right) = {\text{u}}_{p} \left( {y = 0} \right)$$
(B6)

and

$$\mu_{q} \left. {\frac{{\partial u_{q} }}{\partial y}} \right|_{y = 0} = \mu_{p} \left. {\frac{{\partial u_{p} }}{\partial y}} \right|_{y = 0} .$$
(B7)

The mass flow rates \({M}_{q,x}\) and \({M}_{p,x}\) for both fluid phases are derived by solving Eqs. B3 and B4 with the boundary conditions (Eqs. B5B7), as well as the shear stress \({\tau }_{f0}\) at the channel surfaces.

The fluid in VPs (triangular cells), in contrast to the model of fluid flow in the channels, is presumed to be compressible. The discretized form of the mass conservation equation for the liquid phase is

$$\frac{{\alpha_{q,i}^{n + 1} \rho_{q,i}^{n + 1} V_{i}^{n + 1} - \alpha_{q,i}^{n} \rho_{q,i}^{n} V_{i}^{n} }}{\Delta t} + \mathop \sum \limits_{f} \left( {\rho_{q,f}^{n} U_{f}^{n} \alpha_{q,f}^{n} } \right) = 0$$
(B8)

with

$$V_{i}^{n + 1} = V_{i}^{n} + \frac{{{\text{d}}V}}{{{\text{d}}t}}\Delta t,$$
(B9)

where \({V}_{i}^{n+1}\) and \({V}_{i}^{n}\) are the volume of VPi at a time increment n + 1 and n (the third dimension is the unit dimension), respectively, f is the face (edge) number, \({U}_{f}^{n}\) denotes the volume flux through the face (m3/s), based on the average velocity in the channel, \({\alpha }_{q,f}^{n}\) is the face value of the liquid phase volume fraction (–), t is the time step (s), n denotes the time increment and i is the VP number (–). The same equations are defined for the gas phase.

VPs in contrast to the channel flow model assume that the fluid is compressible. The mass flow rates of fluid phases in channels are only calculated to estimate the mass flow rate of fluid flowing through the cell faces. The fluid pressure can exceed 70 MPa in specific situations, such as during the hydraulic fracturing process. The gas phase exceeds the critical point and becomes a supercritical fluid under these conditions. Therefore, for both fluid phases in VPs, the Peng-Robinson equation of state [69] is used to describe fluid behavior above the critical point

$$P = \frac{RT}{{\left( {V_{q/p} - b_{q/p} } \right)}} - \frac{{a_{q/p} }}{{\left( {V_{q/p}^{2} + 2b_{q/p} V_{q} - b_{q/p}^{2} } \right)}},$$
(B10)

where P is the pressure (Pa), R denotes the gas constant (R = 8314,4598 J/(kmol K)), \({V}_{q/p}\) is the molar volume of liquid (q) and gas (p) fraction (m3/kmol) and T denotes the temperature (K). The parameters in Eq. B10 are:

$$a_{q/p} \left( T \right) = a_{q/p,0} \left[ {1 + n_{q/p} \left( {1 - \left( {\frac{{T_{i}^{n} }}{{T_{q/p,c} }}} \right)^{0.5} } \right)} \right]^{2} ,$$
(B11)
$$n_{q/p} = 0.37464 + 1.54226\omega_{q/p} - 0.26992\omega_{q/p}^{2} ,$$
(B12)
$$a_{q/p,0} = a_{c,q/p} \beta \left( T \right),$$
(B13)
$$a_{c,q/p} = \frac{{0.457247R^{2} T_{q/p,c}^{2} }}{{P_{q/p,c} }},$$
(B14)
$$b_{q/p} = \frac{{0.07780RT_{q/p,c} }}{{P_{q/p,c} }},$$
(B15)
$$\beta = \left[ {1 + c_{1} \left( {1 - T_{r}^{\frac{1}{2}} } \right) + c_{2} \left( {1 - T_{r}^{\frac{1}{2}} } \right)^{2} + c_{3} \left( {1 - T_{r}^{\frac{1}{2}} } \right)^{3} } \right]^{2} ,$$
(B16)

where \({T}_{q/p,c}\) is the critical temperature of phase (K), \({P}_{q/p,c}\) denotes the critical pressure of phase (Pa), \({\omega }_{q/p}\) is the acentric factor of phase (–), and Tr denotes the reduced temperature \(\frac{T}{{T}_{c}}\). When c1 = c2 = c3 = 0, the original model is obtained. The extra factors help connect vapor pressure data from highly polar liquids like water and methanol. For most substances, Equations B11B16 provide a good fit for the vapor pressure, however predicting molar volumes for liquid phase can be very inaccurate. The forecast of saturated liquid molar quantities might deviate by l0–40% [87]. Peneloux and Rauzy [88] proposed an effective correction term

$$V_{q}^{corr} = V_{q} + s,$$
(B17)

where s is the small molar volume correction term that is component dependent; \({V}_{q}\) is the molar volume predicted by Eq. B16 and \({V}_{q}^{corr}\) refers to the corrected molar volume. The value of s is negative for higher molecular weight non-polar and essentially for all polar substances. The molar volume correction term is considered to be 0.0 m3/kmol and − 0.0034 m3/kmol for the gas phase and liquid phase (water), respectively. The Peng-Robinson equation of state has the advantage of being able to describe the behavior of supercritical fluids at extremely high fluid pressures and temperatures. For each phase, the mass conservation equation is used. The mass transfer between phases and the grid velocity is ignored when there is no internal mass source. The discretized form of the mass conservation equation for the liquid phase is

$$\frac{{\alpha }_{q,i}^{n+1}{\rho }_{q,i}^{n+1}{V}_{i}^{n+1}-{\alpha }_{q,i}^{n}{\rho }_{q,i}^{n}{V}_{i}^{n}}{\Delta t}+\sum_{f}\left({\rho }_{q,f}^{n}{U}_{f}^{n}{\alpha }_{q,f}^{n}\right)=0$$
(B18)

with

$$V_{i}^{n + 1} = V_{i}^{n} + \frac{dV}{{dt}}\Delta t,$$
(B19)

where f is the face (edge) number, \({U}_{f}^{n}\) denotes the volume flux through the face (m3/s), based on the average velocity in the channel, \({\alpha }_{q,f}^{n}\) is the face value of the fluid phase volume fraction (–), t is the time step (s), n denotes the time increment and i is the VP number (–). The explicit formulation is used instead of an iterative solution of the transport equation during each time step since the volume fraction at the current time step is directly computed from known quantities at the previous time step. Similarly, the mass conservation equation for a gas phase is introduced. The product \({\rho }_{q}{U}_{f}^{n}{\alpha }_{q,f}^{n}\) in Eq. B18 is the mass flow rate \({M}_{q,f}\) of the liquid phase flowing through the face f (edge of a triangle) of VPi. The density of the liquid phase can be calculated by solving the mass conservation equation for both phases

$$\rho_{i,q}^{n + 1} = \frac{{\alpha_{q,i}^{n} \rho_{q,i}^{n} V_{i}^{n} + \Delta t\mathop \sum \nolimits_{f} M_{q,f} }}{{\left( {V_{i}^{n} + \Delta V_{i} \Delta t} \right)\alpha_{q,i}^{n + 1} }}.$$
(B20)

The density of the gas phase can also be computed in the same way. It should be noted that the molar volume V (q/p) is related to the gas density.

$$V_{i,p}^{n + 1} = \frac{{w_{p} }}{{\rho_{i,p}^{n + 1} }},$$
(B21)

and to the liquid density

$$V_{i,q}^{n + 1} = \frac{{w_{q} }}{{\rho_{i,q}^{n + 1} }} - s.$$
(B22)

where wp and wq are molar weights of gas and liquid phases, respectively. Due to the fact that the fluid phases share the same pressure

$$\frac{{RT_{i}^{n} }}{{\left( {V_{{i,p}}^{{n + 1}} - b_{p} } \right)}} - \frac{{a_{p} }}{{\left( {V_{{i,p}}^{{n + 12}} + 2b_{p} V_{{i,p}}^{{n + 1}} - b_{p}^{2} } \right)}} = \frac{{RT_{i}^{n} }}{{\left( {V_{{i,q}}^{{n + 1}} - b_{q} } \right)}} - \frac{{a_{q} }}{{\left( {V_{{i,q}}^{{n + 12}} + 2b_{q} V_{{i,q}}^{{n + 1}} - b_{q}^{2} } \right)}},$$
(B23)

the fluid phase fractions are computed. Inserting Eq. B21 for the gas phase and Eq. B22 for the liquid phase into Eq. B23, a polynomial equation is obtained with respect to the liquid fraction \({\alpha }_{q,i}^{n+1}\). The gas-phase fraction is computed as \({\alpha }_{p,i}^{n+1}=1-{\alpha }_{q,i}^{n+1}\). Equation B10 is used to calculate the new pressure \({P}_{i}^{n+1}\) in VPi. The explicit formulation is utilized instead of an iterative solution of the transport equation during each time step.

Because of the passage of a viscous fluid, there is shear stress along the channel's edge. The shear stress profile in the fluid is triangular for immovable parallel plates with no-slip boundary conditions (zero velocity). The shear stress \({\tau }_{f0}\) at the interface between liquid and gas is

$$\tau_{f0} = \frac{h}{2} \frac{{P_{i}^{n} - P_{j}^{n} }}{L}.$$
(B25)

The fluid pressures in VPs and ‘S2S’ channels are converted into the forces \({\overrightarrow{F}}_{P,j}\) and \({\overrightarrow{F}}_{S,j}\) acting on spheres. For simplicity, the fluid pressure forces acting on the sphere are calculated by assuming that the fluid–solid contact area is multiplied by the pressure in the cell (\({\overrightarrow{F}}_{P,j}\)) or channel (\({\overrightarrow{F}}_{S,j}\)). The contact area is calculated as a section of the cylinder's surface with a height equal to the diameter of the sphere, and not as a section of the sphere's surface. This simplifies the computational algorithm and only slightly overestimates the forces

$$\vec{F}_{P,j} = - P_{i} \vec{n}A_{k} ,$$
(B26)

where \(\overrightarrow{n}\)—the unit vector normal to the discretized sphere’s edge, \({P}_{i}\)—the pressure in VP, i—the VP number, j—the sphere number, and \({A}_{k}\)—the contact area between the fluid in VPi and sphere

$$A_{k} = 2r_{j} e_{k} ,$$
(B27)

with \({r}_{j}\)—the sphere radius and \({e}_{k}\)—the sphere edge length. The shear stresses are finally converted into the forces acting on spheres as

$$\vec{F}_{S,j} = \tau_{f0,i} \vec{I}A_{k} \;{\text{with}}\;A_{k} = 2r_{j} L_{k} ,$$
(B28)

where \(\overrightarrow{I}\)—the unit vector parallel to the channel wall and oriented in the fluid flow direction, \({\tau }_{f0,i}\)—the shear stress in the channel, i—the channel number, j—the sphere number and \({A}_{k}\)—the contact area between the channel and sphere, and \({L}_{k}\)—the channel length.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Krzaczek, M., Tejchman, J. & Nitka, M. Effect of free water on the quasi-static compression behavior of partially-saturated concrete with a fully coupled DEM/CFD approach. Granular Matter 26, 38 (2024). https://doi.org/10.1007/s10035-024-01409-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10035-024-01409-3

Keywords

Navigation