Skip to main content
SpringerLink
Log in

Volume-of-fluid-based implementation for the spatiotemporal variation of viscosity in grouting process: transport time tracking approach

  • Research Paper
  • Published:
Acta Geotechnica Aims and scope Submit manuscript

Abstract

Due to the spatiotemporal variation of viscosity, the simulation of grouting processes is quite challenging. In this work, based on the volume-of-fluid method, the transport time tracking approach is developed to model the spatiotemporal variation of viscosity in grouting processes. Firstly, a partial differential equation of transport time is proposed and applied to obtain the resided time of fluid clusters in the domain. Meanwhile, in order to address the variable viscosity of the slurry phase, the dependence of viscosity on transport time is associated. Secondly, an artificial convection term is introduced in the equation of transport time to capture a sharp interface. This algorithm is implemented using the open-source CFD code OpenFOAM. Subsequently, three different test cases are launched to verify our algorithm and implementation. By using the combination of these numerical treatments, the transport time can be solved in the slurry phase with limited numerical diffusion. Finally, the approach is further compared with the previous experiment. The results show that not only can our algorithm solve the transport time correctly, but also the pressure data agree well with the experiment. It can be seen that this approach could be further applied to the simulation of other fields such as thrombosis, ventilation, and radionuclide migration problems.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

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

Similar content being viewed by others

Data availability

All data, models, and code generated are used during the study that appears in the submitted article.

References

  1. Alizadeh MR, Fatemi M (2021) Mechanistic study of the effects of dynamic fluid/fluid and fluid/rock interactions during immiscible displacement of oil in porous media by low salinity water: Direct numerical simulation. J Mol Liq 322:114544

    Article  Google Scholar 

  2. Ashgriz N, Poo J (1991) Flair: Flux line-segment model for advection and interface reconstruction. J Comput Phys 93(2):449–468

    Article  Google Scholar 

  3. Berger M, Aftosmis M, Muman S (2005) Analysis of slope limiters on irregular grids. In: 43rd AIAA Aerospace Sciences Meeting and Exhibit, p 490

  4. Brackbill JU, Kothe DB, Zemach C (1992) A continuum method for modeling surface tension. J Comput Phys 100(2):335–354

    Article  MathSciNet  Google Scholar 

  5. Chen T, Zhang L, Zhang D (2014) An fem/vof hybrid formulation for fracture grouting modelling. Comput Geotech 58:14–27

    Article  Google Scholar 

  6. Chern IL, Glimm J, McBryan O, Plohr B, Yaniv S (1986) Front tracking for gas dynamics. J Comput Phys 62(1):83–110

    Article  MathSciNet  Google Scholar 

  7. Claassen CM, Islam S, Peters E, Deen NG, Kuipers J, Baltussen MW (2020) An improved subgrid scale model for front-tracking based simulations of mass transfer from bubbles. AIChE J 66(4):16889

    Article  Google Scholar 

  8. Cui W, Tang Q, Song H (2020) Washout resistance evaluation of fast-setting cement-based grouts considering time-varying viscosity using cfd simulation. Constr Build Mater 242:117959

    Article  Google Scholar 

  9. Deising D, Marschall H, Bothe D (2016) A unified single-field model framework for volume-of-fluid simulations of interfacial species transfer applied to bubbly flows. Chem Eng Sci 139:173–195

    Article  Google Scholar 

  10. Dopazo C (1977) On conditioned averages for intermittent turbulent flows. J Fluid Mech 81(3):433–438

    Article  Google Scholar 

  11. El Tani M (2012) Grouting rock fractures with cement grout. Rock Mech Rock Eng 45(4):547–561

    Article  Google Scholar 

  12. Engberg RF, Wegener M, Kenig EY (2014) The influence of Marangoni convection on fluid dynamics of oscillating single rising droplets. Chem Eng Sci 117:114–124

    Article  Google Scholar 

  13. Francois MM, Cummins SJ, Dendy ED, Kothe DB, Sicilian JM, Williams MW (2006) A balanced-force algorithm for continuous and sharp interfacial surface tension models within a volume tracking framework. J Comput Phys 213(1):141–173

    Article  Google Scholar 

  14. Ganapathy H, Al-Hajri E, Ohadi M (2013) Mass transfer characteristics of gas-liquid absorption during Taylor flow in mini/microchannel reactors. Chem Eng Sci 101:69–80

    Article  Google Scholar 

  15. Ganguli A, Kenig E (2011) A CFD-based approach to the interfacial mass transfer at free gas-liquid interfaces. Chem Eng Sci 66(14):3301–3308

    Article  Google Scholar 

  16. Gjennestad MA, Munkejord ST (2015) Modelling of heat transport in two-phase flow and of mass transfer between phases using the level-set method. Energy Procedia 64:53–62

    Article  Google Scholar 

  17. Habla F, Dietsche L, Hinrichsen K-O (2013) Modeling and simulation of conditionally volume averaged viscoelastic two-phase flows. AIChE J 59(10):3914–3927

    Article  Google Scholar 

  18. Haroun Y, Legendre D, Raynal L (2010) Volume of fluid method for interfacial reactive mass transfer: application to stable liquid film. Chem Eng Sci 65(10):2896–2909

    Article  Google Scholar 

  19. Hartmann D, Meinke M, Schröder W (2008) Differential equation based constrained reinitialization for level set methods. J Comput Phys 227(14):6821–6845

    Article  MathSciNet  Google Scholar 

  20. Hill S, Deising D, Acher T, Klein H, Bothe D, Marschall H (2018) Boundedness-preserving implicit correction of mesh-induced errors for vof based heat and mass transfer. J Comput Phys 352:285–300

    Article  MathSciNet  Google Scholar 

  21. Hirt CW, Nichols BD (1981) Volume of fluid (vof) method for the dynamics of free boundaries. J Comput Phys 39(1):201–225

    Article  Google Scholar 

  22. Huang C, Zhang D, Shi Y, Si Y, Huang B (2018) Coupled finite particle method with a modified particle shifting technology. Int J Numer Methods Eng 113(2):179–207

    Article  MathSciNet  Google Scholar 

  23. Irfan M, Muradoglu M (2017) A front tracking method for direct numerical simulation of evaporation process in a multiphase system. J Comput Phys 337:132–153

    Article  MathSciNet  Google Scholar 

  24. Issa RI (1986) Solution of the implicitly discretised fluid flow equations by operator-splitting. J Comput Phys 62(1):40–65

    Article  MathSciNet  Google Scholar 

  25. Jacqmin D (1999) Calculation of two-phase Navier–Stokes flows using phase-field modeling. J Comput Phys 155(1):96–127

    Article  MathSciNet  Google Scholar 

  26. Jasak H (1996) Error analysis and estimation for the finite volume method with applications to fluid flows

  27. Jasak H (2003) A consistent derivation of the sea-ice model using conditional averaging. Citeseer

  28. Jin Q, Bu Z, Pan D, Li H, Li Z, Zhang Y (2021) An integrated evaluation method for the grouting effect in karst areas. KSCE J Civ Eng 25(8):3186–3197

    Article  Google Scholar 

  29. Julien M, Cyprien S (2018) A new compressive scheme to simulate species transfer across fluid interfaces using the volume-of-fluid method. Chem Eng Sci 190:405–418

    Article  Google Scholar 

  30. Kim JS, Lee IM, Jang JH, Choi H (2009) Groutability of cement-based grout with consideration of viscosity and filtration phenomenon. Int. J. Numer. Anal. Met. 33(16):1771–1797

    Article  Google Scholar 

  31. Li D, Christian H (2017) Simulation of bubbly flows with special numerical treatments of the semi-conservative and fully conservative two-fluid model. Chem Eng Sci 174:25–39

    Article  Google Scholar 

  32. Li R, Zhong W (2023) A robust and efficient component-wise weno scheme for euler equations. Appl Math Comput 438:127583

    MathSciNet  Google Scholar 

  33. Li D, Li Z, Gao Z (2018) Compressibility induced bubble size variation in bubble column reactors: Simulations by the cfd-pbe. Chinese J. Chem. Eng. 26(10):2009–2013

    Article  MathSciNet  Google Scholar 

  34. Li JQ, Fan TH, Taniguchi T, Zhang B (2018) Phase-field modeling on laser melting of a metallic powder. Int J Heat Mass Transf 117:412–424

    Article  Google Scholar 

  35. Li D, Marchisio D, Hasse C, Lucas D (2019) Comparison of Eulerian QBMM and classical Eulerian-Eulerian method for the simulation of polydisperse bubbly flows. AIChE J 65:16732

    Article  Google Scholar 

  36. Li S, Pan D, Xu Z, Lin P, Zhang Y (2020) Numerical simulation of dynamic water grouting using quick-setting slurry in rock fracture: the sequential diffusion and solidification (sds) method. Comput Geotech 122:103497

    Article  Google Scholar 

  37. Li D, Marchisio D, Hasse C, Lucas D (2020) twowaygpbefoam: an open-source eulerian qbmm solver for monokinetic bubbly flows. Comput Phys Commun 250:107036

    Article  Google Scholar 

  38. Li D, Wei Y, Marchisio D (2021) Qeefoam: a Quasi-Eulerian-eulerian model for polydisperse turbulent gas-liquid flows. implementation in OpenFOAM, verification and validation. Int J Multiphase Flow 136:103544

    Article  MathSciNet  Google Scholar 

  39. Liang Y, Sui W, Qi J (2019) Experimental investigation on chemical grouting of inclined fracture to control sand and water flow. Tunn. Undergr. Sp. Tech. 83:82–90

    Article  Google Scholar 

  40. Liu B, Sang H, Liu Q, Kang Y, Pan Y, Lu C, Zhang C (2020) New algorithm for simulating grout diffusion and migration in fractured rock masses. Int J Geomech 20(3):04019188

    Article  Google Scholar 

  41. Liu Y, Ye H, Zhang H, Zheng Y (2020) Coupling lattice Boltzmann and material point method for fluid-solid interaction problems involving massive deformation. Int J Numer Methods Eng 121(24):5546–5567

    Article  MathSciNet  Google Scholar 

  42. Liu B, Sang H, Liu Q, Liu H, Pan Y, Kang Y (2021) Laboratory study on diffusion and migration of grout in rock mass fracture network. Int J Geomech 21(1):04020242

    Article  Google Scholar 

  43. López J, Gómez P, Hernández J (2010) A volume of fluid approach for crystal growth simulation. J Comput Phys 229(19):6663–6672

    Article  Google Scholar 

  44. Maes J, Soulaine C (2020) A unified single-field volume-of-fluid-based formulation for multi-component interfacial transfer with local volume changes. J Comput Phys 402:109024

    Article  MathSciNet  Google Scholar 

  45. Marschall H (2011) Towards the numerical simulation of multi-scale two-phase flows. PhD thesis, Technische Universität München

  46. Marschall H, Hinterberger K, Schüler C, Habla F, Hinrichsen O (2012) Numerical simulation of species transfer across fluid interfaces in free-surface flows using openfoam. Chem Eng Sci 78:111–127

    Article  Google Scholar 

  47. Mohajerani S, Baghbanan A, Bagherpour R, Hashemolhosseini H (2015) Grout penetration in fractured rock mass using a new developed explicit algorithm. Int J Rock Mech Min 80:412–417

    Article  Google Scholar 

  48. Mu W, Li L, Yang T, Yu G, Han Y (2019) Numerical investigation on a grouting mechanism with slurry-rock coupling and shear displacement in a single rough fracture. B. Eng. Geol. Environ. 78(8):6159–6177

    Article  Google Scholar 

  49. Muzaferija S (1999) A two-fluid navier-stokes solver to simulate water entry. In: Proceedings of 22nd symposium on naval architecture, 1999, pp 638–651. National Academy Press

  50. Nieves-Remacha MJ, Yang L, Jensen KF (2015) Openfoam computational fluid dynamic simulations of two-phase flow and mass transfer in an advanced-flow reactor. Ind Eng Chem Res 54(26):6649–6659

    Article  Google Scholar 

  51. Noh WF, Woodward P (1976) Slic (simple line interface calculation). In: Proceedings of the fifth international conference on numerical methods in fluid dynamics June 28–July 2, 1976 Twente University, Enschede, Springer, pp 330–340

  52. Osher S, Sethian JA (1988) Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations. J Comput Phys 79(1):12–49

    Article  MathSciNet  Google Scholar 

  53. Pan X, Chu J, Yang Y, Cheng L (2020) A new biogrouting method for fine to coarse sand. Acta Geotech 15(1):1–16

    Article  Google Scholar 

  54. Raeini AQ, Blunt MJ, Bijeljic B (2012) Modelling two-phase flow in porous media at the pore scale using the volume-of-fluid method. J Comput Phys 231(17):5653–5668

    Article  MathSciNet  Google Scholar 

  55. Ren Y-X, Zhang H et al (2003) A characteristic-wise hybrid compact-weno scheme for solving hyperbolic conservation laws. J Comput Phys 192(2):365–386

    Article  MathSciNet  Google Scholar 

  56. Roghair I, Annaland MVS, Kuipers J (2016) An improved front-tracking technique for the simulation of mass transfer in dense bubbly flows. Chem Eng Sci 152:351–369

    Article  Google Scholar 

  57. Rusche H (2003) Computational fluid dynamics of dispersed two-phase flows at high phase fractions. PhD thesis, Imperial College London (University of London)

  58. Scardovelli R, Zaleski S (1999) Direct numerical simulation of free-surface and interfacial flow. Annu Rev Fluid Mech 31(1):567–603

    Article  MathSciNet  Google Scholar 

  59. Sethian JA (1999) Level set methods and fast marching methods: evolving interfaces in computational geometry, fluid mechanics, computer vision, and materials science, vol 3. Cambridge University Press, Cambridge

    Google Scholar 

  60. Shams M, Raeini AQ, Blunt MJ, Bijeljic B (2018) A numerical model of two-phase flow at the micro-scale using the volume-of-fluid method. J Comput Phys 357:159–182

    Article  MathSciNet  Google Scholar 

  61. Smuda M, Kummer F (2022) On a marching level-set method for extended discontinuous Galerkin methods for incompressible two-phase flows: application to two-dimensional settings. Int J Numer Methods Eng 123(1):197–225

    Article  MathSciNet  Google Scholar 

  62. Sun Y, Beckermann C (2007) Sharp interface tracking using the phase-field equation. J Comput Phys 220(2):626–653

    Article  MathSciNet  Google Scholar 

  63. Thiyyakkandi S, McVay M, Bloomquist D, Lai P (2014) Experimental study, numerical modeling of and axial prediction approach to base grouted drilled shafts in cohesionless soils. Acta Geotech 9(3):439–454

    Article  Google Scholar 

  64. Ubbink O, Issa R (1999) A method for capturing sharp fluid interfaces on arbitrary meshes. J Comput Phys 153(1):26–50

    Article  MathSciNet  Google Scholar 

  65. Vukčević V (2016) Numerical modelling of coupled potential and viscous flow for marine applications. University of Zagreb, Zagreb

    Google Scholar 

  66. Wang Q, Ye X, Wang S, Sloan SW, Sheng D (2018) Use of photo-based 3d photogrammetry in analysing the results of laboratory pressure grouting tests. Acta Geotech 13(5):1129–1140

    Article  Google Scholar 

  67. Wang Y, Yang P, Li Z, Wu S, Zhao Z (2020) Experimental-numerical investigation on grout diffusion and washout in rough rock fractures under flowing water. Comput Geotech 126:103717

    Article  Google Scholar 

  68. Weller HG (2008) A new approach to vof-based interface capturing methods for incompressible and compressible flow. OpenCFD Ltd., Report TR/HGW 4:35

  69. Weller H et al. (1993) The development of a new flame area combustion model using conditional averaging. Thermo-fluids section report TF 9307

  70. Wu C, Chu J, Cheng L, Wu S (2019) Biogrouting of aggregates using premixed injection method with or without ph adjustment. J Mater Civ Eng 31(9):06019008

    Article  Google Scholar 

  71. Wu C, Chu J, Wu S, Guo W (2019) Quantifying the permeability reduction of biogrouted rock fracture. Rock Mech Rock Eng 52(3):947–954

    Article  Google Scholar 

  72. Wu S, Li B, Chu J (2020) Large-scale model tests of biogrouting for sand and rock. Proc. InsT. Civ. Eng.-Gr. 1–10

  73. Xiao F, Honma Y, Kono T (2005) A simple algebraic interface capturing scheme using hyperbolic tangent function. Int J Numer Method Fluid 48(9):1023–1040

    Article  Google Scholar 

  74. Xiaoshuai W, Yuxin Z (2015) A high-resolution hybrid scheme for hyperbolic conservation laws. Int. J. Numer. Meth. Fl. 78(3):162–187

    Article  MathSciNet  Google Scholar 

  75. Yang C, Mao Z (2005) Numerical simulation of interphase mass transfer with the level set approach. Chem Eng Sci 60(10):2643–2660

    Article  Google Scholar 

  76. Yang L, Nieves-Remacha MJ, Jensen KF (2017) Simulations and analysis of multiphase transport and reaction in segmented flow microreactors. Chem Eng Sci 169:106–116

    Article  Google Scholar 

  77. Yang J, Yin Z-Y, Liu Y-J, Laouafa F (2022) Multiphysics modelling of backfill grouting in sandy soils during tbm tunnelling. Acta Geotech., 1–19

  78. Youngs DL (1982) Time-dependent multi-material flow with large fluid distortion. Numer Method Fluids Dyn

  79. Zhang Q, Zhang L, Liu R, Li S, Zhang Q (2017) Grouting mechanism of quick setting slurry in rock fissure with consideration of viscosity variation with space. Tunn. Undergr. Sp. Tech. 70:262–273

    Article  Google Scholar 

  80. Zhang D-M, Liu Z-S, Wang R-L, Zhang D-M (2019) Influence of grouting on rehabilitation of an over-deformed operating shield tunnel lining in soft clay. Acta Geotech 14(4):1227–1247

    Article  Google Scholar 

  81. Zhang E, Xu Y, Fei Y, Shen X, Huang L (2021) Influence of the dominant fracture and slurry viscosity on the slurry diffusion law in fractured aquifers. Int. J. Rock Mech. Min. 141(1):104731

    Article  Google Scholar 

  82. Zhao Y, Zhao C, Xu Z, Xu H (2016) Modeling metal foam enhanced phase change heat transfer in thermal energy storage by using phase field method. Int J Heat Mass Transf 99:170–181

    Article  Google Scholar 

  83. Zhou Z, Du X, Wang S, Cai X, Chen L (2019) Micromechanism of the diffusion of cement-based grouts in porous media under two hydraulic operating conditions: constant flow rate and constant pressure. Acta Geotech 14(3):825–841

    Article  Google Scholar 

  84. Zou L, Håkansson U, Cvetkovic V (2018) Two-phase cement grout propagation in homogeneous water-saturated rock fractures. Int J Rock Mech Min 106:243–249

    Article  Google Scholar 

Download references

Acknowledgements

We would like to acknowledge the support from the National Science Fund for Excellent Young Scholars (Grant No.: 52022053), the Science Fund for Distinguished Young Scholars of Shandong Province (Grant No.: ZR201910270116), the National Natural Science Foundation of China (Grant No.: 52109129).

Author information

Authors and Affiliations

  1. Geotechnical and Structural Engineering Research Center, Shandong University, Jinan, 250-061, Shandong, China

    Yi-Chi Zhang, Dong-Dong Pan & Zhen-Hao Xu

  2. State Key Laboratory of Advanced Metallurgy, University of Science and Technology Beijing, Beijing, China

    Dong-Yue Li

  3. DYFLUID Ltd, Beijing, China

    Dong-Yue Li

Authors
  1. Yi-Chi Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dong-Dong Pan
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Dong-Yue Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Zhen-Hao Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhen-Hao Xu.

Additional information

Publisher's Note

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

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

Zhang, YC., Pan, DD., Li, DY. et al. Volume-of-fluid-based implementation for the spatiotemporal variation of viscosity in grouting process: transport time tracking approach. Acta Geotech. 19, 1929–1942 (2024). https://doi.org/10.1007/s11440-023-01982-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11440-023-01982-6

Keywords

Search

Navigation