Skip to main content
SpringerLink
Log in

An effective thermal conductivity model for unsaturated compacted bentonites with consideration of bimodal shape of pore size distribution

  • Article
  • Published:
Science China Technological Sciences Aims and scope Submit manuscript

Abstract

An effective thermal conductivity model was proposed for unsaturated compacted bentonites with consideration of the bimodal shape of pore size distribution curves. The pores of soils were grouped into two dominant pore size modes corresponding to the intra- and inter-particle pores, and were simulated with randomly distributed spheroidal inclusions of different aspect ratios. With the assumption of preferential invasion of the wetting fluid (water) into pores of smaller sizes and by virtue of the analytical solution to the inhomogeneous inclusion problem in heat conduction, the model was developed using the Mori-Tanaka (MT), Ponte Castañeda-Willis (PCW) and self-consistent (SC) homogenization approaches for different considerations of the interactions between pores and the solid phase. The proposed model is functions of the thermal conductivities of the solid, liquid and gas phases, porosity, the degree of saturation, the aspect ratios of pores and/or soil particles, and the fraction of the smaller group of pores. The proposed model was validated against five sets of laboratory measurement data on the MX-80, FEBEX, Kunigel-V1 and GMZ01 bentonites, showing a good agreement between the model predictions and the laboratory measurements. The responses of the model with respect to the geometries of pores and solid particles were examined. Compared to series-parallel structural models, the proposed model may overall exhibit better performance if proper homogenization schemes are adopted, but as an advantage, the model has clearer physical mechanisms and a smaller number of parameters.

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

Similar content being viewed by others

References

  1. Tong F, Jing L, Zimmerman R W. An effective thermal conductivity model of geological porous media for coupled thermo-hydro-mechanical systems with multiphase flow. Int J Rock Mech Min Sci, 2009, 46: 1358–1369

    Article  Google Scholar 

  2. Chen Y, Zhou S, Hu R, et al. Estimating effective thermal conductivity of unsaturated bentonites with consideration of coupled thermo-hydro-mechanical effects. Int J Heat Mass Transfer, 2014, 72: 656–667

    Article  Google Scholar 

  3. Chen Y, Zhou C, Jing L. Modeling coupled THM processes of geological porous media with multiphase flow: Theory and validation against laboratory and field scale experiments. Comput Geotech, 2009, 36: 1308–1329

    Article  Google Scholar 

  4. Rutqvist J, Zheng L, Chen F, et al. Modeling of coupled thermohydro-mechanical processes with links to geochemistry associated with bentonite-backfilled repository tunnels in clay formations. Rock Mech Rock Eng, 2014, 47: 167–186

    Article  Google Scholar 

  5. Kahr G, Müller-Vonmoos M. Wärmeleitfähigkeit von Bentonit MX80 und von Montigel nach der Heizdrahtmethode. Schweiz: Nagra, 1982. NTB 82-06

    Google Scholar 

  6. Knutsson S. On the thermal conductivity and thermal diffusivity of highly compacted bentonite. Swedish Nuclear Fuel and Waste Management Co.. Stockholm: SKB, 1983. 83–72

    Google Scholar 

  7. Madsen F T. Clay mineralogical investigations related to nuclear waste disposal. Clay Miner, 1998, 33: 109–129

    Article  Google Scholar 

  8. Villar M V. Caracterización termo-hidro-mecánica de una bentonita de Cabo de Gata. Dissertation of Doctoral Degree. Madrid: Universidad Complutense de Madrid. 2000, 396

    Google Scholar 

  9. Ould-Lahoucine C, Sakashita H, Kumada T. Measurement of thermal conductivity of buffer materials and evaluation of existing correlations predicting it. Nucl Eng Des, 2002, 216: 1–11

    Article  Google Scholar 

  10. Lloret A, Romero E, Villar M V. FEBEX II Project Final report on thermo-hydro-mechanical laboratory tests. Empresa Nacional de Residuos Radiactivos. Madrid: Publicación técnica. 2004, 10: 11–165

    Google Scholar 

  11. Lloret A, Villar M V. Advances on the knowledge of the thermohydro-mechanical behaviour of heavily compacted “FEBEX” bentonite. Phys Chem Earth, Parts A/B/C, 2007, 32: 701–715

    Article  Google Scholar 

  12. Liu Y, Cai M, Wang J. Thermal properties of buffer material for high-level radioactive waste disposal. Chin J Rock Mech Eng, 2007, 26: 3891–3896

    Google Scholar 

  13. Tang A M, Cui Y J, Le T T. A study on the thermal conductivity of compacted bentonites. Appl Clay Sci, 2008, 41: 181–189

    Article  Google Scholar 

  14. Ye W M, Chen Y G, Chen B, et al. Advances on the knowledge of the buffer/backfill properties of heavily-compacted GMZ bentonite. Eng Geol, 2010, 116: 12–20

    Article  Google Scholar 

  15. Yong R N. Overview of modeling of clay microstructure and interactions for prediction of waste isolation barrier performance. Eng Geol, 1999, 54: 83–91

    Article  Google Scholar 

  16. Ye W M, Wang Q, Pan H, et al. Thermal conductivity of compacted GMZ01 bentonite (in Chinese). Chin J Geotech Eng, 2010, 32: 821–826

    Google Scholar 

  17. Ye W M, Qian L X, Chen B, et al. Characteristics of Micro-structure of densely compacted Gaomiaozi bentonite (in Chinese). J Tongji Univ (Nat Sci), 2009, 37: 31–35

    Google Scholar 

  18. Holzer L, Münch B, Rizzi M, et al. 3D-microstructure analysis of hydrated bentonite with cryo-stabilized pore water. Appl Clay Sci, 2010, 47: 330–342

    Article  Google Scholar 

  19. Tomioka S, Kozaki T, Takamatsu H, et al. Analysis of microstructural images of dry and water-saturated compacted bentonite samples observed with X-ray micro CT. Appl Clay Sci, 2010, 47: 65–71

    Article  Google Scholar 

  20. Saba S, Delage P, Lenoir N, et al. Further insight into the microstructure of compacted bentonite-sand mixture. Eng Geol, 2014, 168: 141–148

    Article  Google Scholar 

  21. Ichikawa Y, Kawamura K, Fujii N, et al. Microstructure and micro/macro-diffusion behavior of tritium in bentonite. Appl Clay Sci, 2004, 26: 75–90

    Article  Google Scholar 

  22. Jiang P, Xiang H, Xu R. Theoretical and experimental study of the thermal conductivity of nanoporous media. Sci China Tech Sci, 2012, 55: 2140–2147

    Article  Google Scholar 

  23. Johansen O. Thermal conductivity of soils. Dissertation of Masteral Degree. Norway: University of Trondheim. 1975, 236

    Google Scholar 

  24. Ewen J, Thomas H R. The thermal probe-a new method and its use on an unsaturated sand. Géotechnique, 1987, 37: 91–105

    Article  Google Scholar 

  25. Sakashita H, Kumada T. Heat transfer model for predicting thermal conductivity of highly compacted bentonite. J At Energy Soc Jpn, 1998, 40: 235–240

    Article  Google Scholar 

  26. Côté J, Konrad J M. A generalized thermal conductivity model for soils and construction materials. Can Geotech J, 2005, 42: 443–458

    Article  Google Scholar 

  27. Cheng P, Hsu C T. The effective stagnant thermal conductivity of porous media with periodic structures. J Porous Media, 1999, 2: 19–38

    Article  MATH  Google Scholar 

  28. Wang J, Carson J K, North M F, et al. A new structural model of effective thermal conductivity for heterogeneous materials with co-continuous phases. Int J Heat Mass Transfer, 2008, 51: 2389–2397

    Article  Google Scholar 

  29. Tarnawski V R, Leong W H. A series-parallel model for estimating the thermal conductivity of unsaturated soils. Int J Thermophys, 2012, 33: 1191–1218

    Article  Google Scholar 

  30. Gong L, Wang Y, Cheng X, et al. A novel effective medium theory for modelling the thermal conductivity of porous materials. Int J Heat Mass Transfer, 2014, 68: 295–298

    Article  Google Scholar 

  31. Fricke H. A mathematical treatment of the electric conductivity and capacity of disperse systems I. The electric conductivity of a suspension of homogeneous spheroids. Phys Rev, 1924, 24: 575

    Article  Google Scholar 

  32. Brailsford A D, Major K G. Thermal conductivity of aggregates of several phases, including porous materials. Br J Appl Phys, 1964, 15: 313

    Article  Google Scholar 

  33. Zimmerman R W. Thermal conductivity of fluid-saturated rocks. J Pet Sci Eng, 1989, 3: 219–227

    Article  Google Scholar 

  34. Shafiro B, Kachanov M. Anisotropic effective conductivity of materials with non-randomly oriented inclusions of diverse ellipsoidal shapes. J Appl Phys, 2000, 87: 8561–8569

    Article  Google Scholar 

  35. Sevostianov I. Thermal conductivity of a material containing cracks of arbitrary shape. Int J Eng Sci, 2006, 44: 513–528

    Article  Google Scholar 

  36. Gruescu C, Giraud A, Homand F, et al. Effective thermal conductivity of partially saturated porous rocks. Int J Solids Struct, 2007, 44: 811–833

    Article  MATH  Google Scholar 

  37. Giraud A, Gruescu C, Do D P, et al. Effective thermal conductivity of transversely isotropic media with arbitrary oriented ellipsoidal inhomogeneities. Int J Solids Struct, 2007, 44: 2627–2647

    Article  MATH  Google Scholar 

  38. Bary B. Estimation of poromechanical and thermal conductivity properties of unsaturated isotropically microcracked cement pastes. Int J Numer Anal Meth Geomech, 2011, 35: 1560–1586

    Article  Google Scholar 

  39. Chen Y, Li D, Jiang Q, et al. Micromechanical analysis of anisotropic damage and its influence on effective thermal conductivity in brittle rocks. Int J Rock Mech Min Sci, 2012, 50: 102–116

    Article  Google Scholar 

  40. Hashin Z. Assessment of the self consistent scheme approximation: conductivity of particulate composites. J Compos Mater, 1968, 2: 284–300

    Article  Google Scholar 

  41. Lee Y M, Yang R B, Gau S S. A generalized self-consistent method for calculation of effective thermal conductivity of composites with interfacial contact conductance. Int Comm Heat Mass Transfer, 2006, 33: 142–150

    Article  Google Scholar 

  42. Mori T, Tanaka K. Averages stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Metall, 1973, 21: 571–574

    Article  Google Scholar 

  43. Hashin Z. The differential scheme and its application to cracked materials. J Mech Phys Solids, 1988, 36: 719–734

    Article  MATH  MathSciNet  Google Scholar 

  44. Castañeda P P, Willis J R. The effect of spatial distribution on the behavior of composite materials and cracked media. J Mech Phys Solids, 1995, 43: 1919–1951

    Article  MATH  MathSciNet  Google Scholar 

  45. Zheng Q S, Du D X. An explicit and universally applicable estimate for the effective properties of multiphase composite which accounts for inclusion distribution. J Mech Phys Solids, 2001, 49: 2765–2788

    Article  MATH  Google Scholar 

  46. Chen Y F, Zhou S, Hu R, et al. A homogenization-based model for estimating effective thermal conductivity of unsaturated compacted bentonites. Int J Heat Mass Transfer, 2015, doi: 10.1016/j.ijheatmasstransfer.2014.12.053

    Google Scholar 

  47. Zhou C, Li K, Pang X. Effect of crack density and connectivity on the permeability of microcracked solids. Mech Mater, 2011, 43: 969–978

    Article  Google Scholar 

  48. Carson J K, Sekhon J P. Simple determination of the thermal conductivity of the solid phase of particulate materials. Int Comm Heat Mass Transfer, 2010, 37: 1226–1229

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, Wuhan, 430072, China

    YiFeng Chen, Min Wang, Song Zhou, Ran Hu & ChuangBing Zhou

  2. Key Laboratory of Rock Mechanics in Hydraulic Structural Engineering (Ministry of Education), Wuhan University, Wuhan, 430072, China

    YiFeng Chen, Min Wang, Song Zhou, Ran Hu & ChuangBing Zhou

  3. School of Civil Engineering and Architecture, Nanchang University, Nanchang, 330031, China

    ChuangBing Zhou

Authors
  1. YiFeng Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Min Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Song Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Ran Hu
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. ChuangBing Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to YiFeng Chen.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chen, Y., Wang, M., Zhou, S. et al. An effective thermal conductivity model for unsaturated compacted bentonites with consideration of bimodal shape of pore size distribution. Sci. China Technol. Sci. 58, 369–380 (2015). https://doi.org/10.1007/s11431-014-5738-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11431-014-5738-3

Keywords

Search

Navigation