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Study of the Factors Influencing Diffusive Tortuosity Based on Pore-Scale SPH Simulation of Granular Soil

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Abstract

Tortuosity quantitatively reflects the complexity of porous media. For the diffusion process in porous media, diffusive tortuosity is inevitably an important research topic. A simulation experimental method is proposed to study diffusive tortuosity in granular soil based on 3D pore-scale simulation by smoothed particle hydrodynamics (SPH). On the basis of the simulation results, the relationships between diffusive tortuosity and microstructural parameters are discussed. Simulation experiments are implemented on 3D granular soil columns generated by PFC software and soil layers formed by periodic expansion of the soil columns. The accuracy of the pore-scale SPH diffusion model is verified by the analytic solution for the particular case of a pure water column. The results show that the dimensionality does affect the diffusive tortuosity. 2D profiles cannot represent the original 3D medium in terms of the diffusion characteristics. Within the limited range of the variation of porosity during soil compression, the relationship between porosity and diffusive tortuosity can be considered to be linear. In addition, for the case of uniform granule size, diffusive tortuosity is almost linearly related to the specific surface area. Tortuosity values for soil columns are larger than those for the soil layer, which denotes that the effect of the sidewall on diffusion cannot be overlooked for a centimeter-scale soil column.

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Abbreviations

c :

Mass concentration of species in the solution

C :

Total mass concentration of the species based on the total volume of soil

c 0 :

Initial mass concentration of species in the solution

D :

Effective diffusion coefficient in porous media

D 0 :

Diffusion coefficient in pure water (usually molecular diffusion coefficient)

f :

Field function

h :

The smoothing length of the kernel function

i :

Serial number of objective particle

j :

Serial number of particle i’s adjacent particles

J :

Diffusion flux through the representative volume of porous media

k :

Summation series number of the analytic solution

L :

Length of porous media

L e :

The average length of diffusive pathway

m :

Mass of particle

n :

Porosity of porous media

N :

The total number of particles within the influence area of particle i

r* :

General term representing chemical or biological reactions

r :

Particle distance

R :

Ratio of r to h

R f :

Structural parameter (ratio of D0 to D)

S :

Specific surface area

t :

Time

T v :

Dimensionless time, Tv = D0t/L2

x, x′, xi, xj :

Distance vector of particle

x :

x axe of Cartesian coordinates

y :

y axe of Cartesian coordinates

z :

z axe of Cartesian coordinates, direction of macroscopic concentration gradient

Z :

Characteristic length, Z = z/L

W :

Kernel function

\(\nabla\) :

Vector gradient operator

α d :

Normalized constant in kernel function

λ :

Rate constant

τ :

Tortuosity factor, τ = (Le/L)2

Δx, Δy, Δz :

Particle spacing

ρ :

Density of particle

Ω :

Integral domain

References

  • Adler, P.M., Jacquin, C.G., Thovert, J.F.: The formation factor of reconstructed porous media. Water Resour. Res. 28, 1571–1576 (1992)

    Article  Google Scholar 

  • Akanni, K.A., Evans, J.W., Abramson, I.S.: Effective transport coefficients in heterogeneous media. Chem. Eng. Sci. 42, 1945–1954 (1987)

    Article  Google Scholar 

  • Allen, R., Sun, S.: Computing and comparing effective properties for flow and transport in computer-generated porous media. Geofluids 2017, 1–24 (2017)

    Article  Google Scholar 

  • Bai, B., Rao, D., Xu, T., Chen, P.: SPH–FDM boundary for the analysis of thermal process in homogeneous media with a discontinuous interface. Int. J. Heat Mass Transf. 117, 517–526 (2018)

    Article  Google Scholar 

  • Bai, B., Rao, D., Chang, T., Guo, Z.: A nonlinear attachment–detachment model with adsorption hysteresis for suspension-colloidal transport in porous media. J. Hydrol. 578(11), 124080 (2019)

    Article  Google Scholar 

  • Bayesteh, H., Mirghasemi, A.A.: Numerical simulation of porosity and tortuosity effect on the permeability in clay: microstructural approach. Soils Found. 55, 1158–1170 (2015)

    Article  Google Scholar 

  • Bezzar, A., Ghomari, F.: Monitoring of pollutant diffusion into clay liners by electrical methods. Transp. Porous Media 97, 147–159 (2013)

    Article  Google Scholar 

  • Boudreau, B.P.: The diffusive tortuosity of fine-grained unlithified sediments. Geochim. Cosmochim. Acta 60, 3139–3142 (1996)

    Article  Google Scholar 

  • Carslaw, H., Jaeger, J.: Conduction of Heat in Solids. Oxford University Press, Oxford (1959)

    Google Scholar 

  • Cleary, P.W.: Modelling confined multi-material heat and mass flows using SPH. Appl. Math. Model. 22, 981–993 (1998)

    Article  Google Scholar 

  • Cleary, P.W., Monaghan, J.J.: Conduction modelling using smoothed particle hydrodynamics. J. Comput. Phys. 148, 227–264 (1999)

    Article  Google Scholar 

  • Cundall, P.A., Strack, O.D.L.: A discrete numerical model for granular assemblies. Géotechnique 29, 47–65 (1979)

    Article  Google Scholar 

  • Davarzani, H., Marcoux, M., Costeseque, P., et al.: Experimental measurement of the effective diffusion and thermodiffusion coefficients for binary gas mixture in porous media. Chem. Eng. Sci. 65, 5092–5104 (2010)

    Article  Google Scholar 

  • Elwinger, F., Pourmand, P., István, F.: Diffusive transport in pores. Tortuosity and molecular interaction with the pore wall. J. Phys. Chem. C 121, 13757–13764 (2017)

    Article  Google Scholar 

  • Esmaili, M.A., Nikseresht, A.H.: Neumann and Robin boundary conditions for heat conduction modeling using smoothed particle hydrodynamics. Comput. Phys. Commun. 198, 1–11 (2016)

    Article  Google Scholar 

  • Ghanbarian, B., Hunt, A.G., Ewing, R.P., et al.: Tortuosity in porous media: a critical review. Soil Sci. Soc. Am. J. 77, 1461–1477 (2013)

    Article  Google Scholar 

  • Ghassemi, A., Pak, A.: Pore scale study of permeability and tortuosity for flow through particulate media using Lattice Boltzmann method. Int. J. Numer. Anal. Methods Geomech. 35, 886–901 (2011)

    Article  Google Scholar 

  • Li, Z., Dong, M.: Experimental study of diffusive tortuosity of liquid-saturated consolidated porous media. Ind. Eng. Chem. Res. 49, 6231–6237 (2010)

    Article  Google Scholar 

  • Liu, M.B., Liu, G.R.: Smoothed particle hydrodynamics (SPH): an overview and recent developments. Arch. Comput. Methods Eng. 17, 25–76 (2010)

    Article  Google Scholar 

  • Monaghan, J.J.: Simulating free surface flows with SPH. J. Comput. Phys. 110, 399–406 (1994)

    Article  Google Scholar 

  • Niya, S.M.R., Selvadurai, A.P.S.: A statistical correlation between permeability, porosity, tortuosity and conductance. Transp. Porous Media 121, 741–752 (2018)

    Article  Google Scholar 

  • Olsen, H.W.: Hydraulic flow through saturated clays. Clays Clay Miner. 11, 131–161 (1962)

    Article  Google Scholar 

  • Price, D.J.: Smoothed particle hydrodynamics and magnetohydrodynamics. J. Comput. Phys. 231, 759–794 (2012)

    Article  Google Scholar 

  • Ryan, E.M., Tartakovsky, A.M., Amon, C.: Pore-scale modeling of competitive adsorption in porous media. J. Contam. Hydrol. 120–121, 56–78 (2011)

    Article  Google Scholar 

  • Saomoto, H., Katagiri, J.: Direct comparison of hydraulic tortuosity and electric tortuosity based on finite element analysis. Theor. Appl. Mech. Lett. 5, 177–180 (2015)

    Article  Google Scholar 

  • Scholes, O.N., Clayton, S.A., Hoadley, A.F.A., et al.: Permeability anisotropy due to consolidation of compressible porous media. Transp. Porous Media 68, 365–387 (2007)

    Article  Google Scholar 

  • Shackelford, C.D.: Laboratory diffusion testing for waste disposal—a review. J. Contam. Hydrol. 7, 177–217 (1991)

    Article  Google Scholar 

  • Shen, L., Chen, Z.: Critical review of the impact of tortuosity on diffusion. Chem. Eng. Sci. 62, 3748–3755 (2007)

    Article  Google Scholar 

  • Tartakovsky, A.M., Trask, N., Pan, K., et al.: Smoothed particle hydrodynamics and its applications for multiphase flow and reactive transport in porous media. Comput. Geosci. 20, 807–834 (2016)

    Article  Google Scholar 

  • Trinh, S., Arce, P., Locke, B.R.: Effective diffusivities of point-like molecules in isotropic porous media by Monte Carlo simulation. Transp. Porous Media 38, 241–259 (2000)

    Article  Google Scholar 

  • Valdés-Parada, F.J., Porter, M.L., Wood, B.D.: The role of tortuosity in upscaling. Transp. Porous Media 88, 1–30 (2011)

    Article  Google Scholar 

  • Wang, P.: Lattice Boltzmann simulation of permeability and tortuosity for flow through dense porous media. Math. Probl. Eng. 2014, 1–7 (2014)

    Google Scholar 

  • Yang, X., Mehmani, Y., Perkins, W.A., et al.: Intercomparison of 3D pore-scale flow and solute transport simulation methods. Adv. Water Resour. 95, 176–189 (2016)

    Article  Google Scholar 

  • Zheng, J.H., Soe, M.M., Zhang, C., et al.: Numerical wave flume with improved smoothed particle hydrodynamics. J. Hydrodyn. 22, 773–781 (2010)

    Article  Google Scholar 

  • Zhu, Y., Fox, P.J.: Simulation of pore-scale dispersion in periodic porous media using smoothed particle hydrodynamics. J. Comput. Phys. 182, 622–645 (2002)

    Article  Google Scholar 

Download references

Acknowledgements

This research was funded by the National Natural Science Foundation of China (51678043, 51878035), and Beijing Natural Science Foundation (8182046). The authors would like to thank the reviewers and editors for their constructive comments.

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  1. School of Civil Engineering, Beijing Jiaotong University, Beijing, 100044, People’s Republic of China

    Dengyu Rao & Bing Bai

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  1. Dengyu Rao
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  2. Bing Bai
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Correspondence to Bing Bai.

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Rao, D., Bai, B. Study of the Factors Influencing Diffusive Tortuosity Based on Pore-Scale SPH Simulation of Granular Soil. Transp Porous Med 132, 333–353 (2020). https://doi.org/10.1007/s11242-020-01394-0

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  • DOI: https://doi.org/10.1007/s11242-020-01394-0

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