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Effect of mesoscale internal structure on effective thermal conductivity of anisotropic geomaterials

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Abstract

Geomaterials tend to be stratified due to the geological process, and their properties are often anisotropic. In this study, the quartet structure generation set (QSGS) method is adopted to reconstruct the mesoscale internal structure of anisotropic geomaterials. In addition, the fractal theory and several morphological parameters are employed to describe the structural features of anisotropic geomaterials. The effective thermal properties parallel and perpendicular to the layered structure are evaluated via the finite element method coupled with Monte Carlo simulations. A representative volume element of anisotropic geomaterials is determined by the homogenization method. The importance of morphological parameters is ranked by Pearson correlation coefficient. Results indicate that the porosity and arrangements of solid fabric have significant influences on thermal conductivity and its anisotropy. Prediction models for assessing the orthogonal thermal conductivities and anisotropic ratio are established, and their performances are benchmarked against the finite element analysis results. The results obtained from this work can provide a reference for the investigation of anisotropy of geomaterials properties and a link between effective thermal properties and features of the internal structure.

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Abbreviations

A :

Area of each solid cluster

a :

Major axis of an ellipse

b :

Minor axis of an ellipse

C :

Boundary length of a solid cluster

C d :

Distribution probability of solid core

D e :

Equivalent diameter of a solid cluster

D f :

Fractal dimension

D m :

Mass fractal dimension

D s :

Surface fractal dimension

d :

Length of a grid cell

F max :

Maximum Feret diameter

F min :

Minimum Feret diameter

k :

Thermal conductivity

k a :

Thermal conductivity of air

k eff :

Effective thermal conductivity

k s :

Thermal conductivity of solid

k x :

Thermal conductivity along x-direction

k y :

Thermal conductivity along y-direction

k 0 :

Geometric mean value of ks and ka

L m :

Length of an object at macroscale

L :

Length of an object at mesoscale

m :

Number of data points of each variable

M(ε):

Measurement of a fractal subject in fractal theory

N :

Shape function

n :

Porosity

P i :

Growth probability of solid along i-direction (i = 1, 2, 3, 4, 5, 6, 7, 8)

q :

Heat flux density

r k :

Anisotropy ratio of thermal conductivity

T :

Temperature

x i :

ith data points of variable x

\(\overline{x}\) :

Mean values of variable x

\(\overline{{x_{\max ,i} }}\) :

Average solid length along x-direction

y i :

ith data points of variable y

\(\overline{y}\) :

Mean values of variable y

\(\overline{{y_{\max ,j} }}\) :

Average solid length along y-direction

Y :

Ratio of P1 to P2

Y 1 :

Average value of the aspect ratio of solid cluster

Y 2 :

Anisotropy of solid cluster

ε :

Length scale in fractal theory

ρ :

Value of Pearson correlation coefficient

Γ:

Boundary of simulated domain

Ω:

Simulated domain

FEM:

Finite element method

LSM:

Least square method

MCS:

Monte Carlo simulation

PCC:

Pearson correlation coefficient

QSGS:

Quartet structure generation set

RMSE:

Root mean squared error

RVE:

Representative volume element

References

  1. Abuel-Naga HM, Bergado DT (2005) Thermal conductivity and fabric anisotropy of saturated clays. Geotech Eng 36(2):99–108

    Google Scholar 

  2. Andra (2005) Synthesis of an evaluation of the feasibility of a geological repository in an argillaceous formation. In: Dossier 2005 Argile, National Agency for Radioactive Waste Management, Chatenay-Malabry, France

  3. Buntebarth G (2004) Bestimmung thermophysikalischer Eigenschaften an Opalinustonproben. Geophysikalisch-Technisches Büro, Clausthal-Zellerfeld

    Google Scholar 

  4. Čermák V, Rybach L (1982) Thermal conductivity and specific heat of minerals and rocks. In: Angenheister G (ed) Landolt-Börnstein: Numerical Data and Functional Relationships in Science and Technology, vol V/1a. Springer, Berlin, pp 305–343

    Google Scholar 

  5. Dao LQ, Delage P, Tang AM, Cui YJ, Pereira JM, Li XL, Sillen X (2014) Anisotropic thermal conductivity of natural Boom Clay. Appl Clay Sci 101:282–287

    Article  Google Scholar 

  6. Davis MG, Chapman DS, Van Wagoner TM, Armstrong PA (2007) Thermal conductivity anisotropy of metasedimentary and igneous rocks. J Geophys Res: Solid Earth 112:B05216

    Google Scholar 

  7. Deming D (1994) Estimation of the thermal conductivity anisotropy of rock with application to the determination of terrestrial heat flow. J Geophys Res: Solid Earth 99(B11):22087–22091

    Article  Google Scholar 

  8. Goldsmid H, Bowley A (1960) Thermal conduction in mica along the planes of cleavage. Nature 187:864–865

    Article  Google Scholar 

  9. Greene MS, Liu Y, Chen W, Liu WK (2011) Computational uncertainty analysis in multiresolution materials via stochastic constitutive theory. Comput Methods Appl Mech Eng 200(1–4):309–325

    Article  MathSciNet  Google Scholar 

  10. Grubbe K, Haenel R, Zoth G (1983) Determination of the vertical component of thermal conductivity by line source methods. Zentralblatt für Geologie und Paläontologie Teil 1(1–2):49–56

    Google Scholar 

  11. He H, Zhao Y, Dyck MF, Si B, Jin H, Lv J, Wang J (2017) A modified normalized model for predicting effective soil thermal conductivity. Acta Geotech 12(6):1281–1300

    Article  Google Scholar 

  12. Hill R (1963) Elastic properties of reinforced solids: some theoretical principles. J Mech Phys Solids 11(5):357–372

    Article  Google Scholar 

  13. Huai X, Wang W, Li Z (2007) Analysis of the effective thermal conductivity of fractal porous media. Appl Therm Eng 27(17–18):2815–2821

    Article  Google Scholar 

  14. Jorand R, Vogt C, Marquart G, Clauser C (2013) Effective thermal conductivity of heterogeneous rocks from laboratory experiments and numerical modeling. J Geophys Res: Solid Earth 118(10):5225–5235

    Article  Google Scholar 

  15. Kappelmeyer O, Haenel R (1974) Geothermics with special reference to application. Berlin Gebrueder Borntraeger Geoexploration Monographs Series 4. Gebruder Borntraeger, Berlin, p 31

    Google Scholar 

  16. Li KQ, Li DQ, Chen DH, Gu SX, Liu Y (2021) A generalized model for effective thermal conductivity of soils considering porosity and mineral composition. Acta Geotech 16(11):3455–3466

    Article  Google Scholar 

  17. Li KQ, Li DQ, Li PT, Liu Y (2019) Meso-mechanical investigations on the overall elastic properties of multiphase construction materials using finite element method. Constr Build Mater 228:116727

    Article  Google Scholar 

  18. Li KQ, Li DQ, Liu Y (2020) Meso-scale investigations on the effective thermal conductivity of multiphase materials using the finite element method. Int J Heat Mass Transfer 151:119383

    Article  Google Scholar 

  19. Liu Y, Li KQ, Li DQ, Tang XS, Gu SX (2022) Coupled thermal–hydraulic modeling of artificial ground freezing with uncertainties in pipe inclination and thermal conductivity. Acta Geotech 17(1): 257–274.

    Article  Google Scholar 

  20. Lu Y, Ye WM, Wang Q, Zhu YH, Chen YG, Chen B (2020) Investigation on anisotropic thermal conductivity of compacted GMZ bentonite. Bull Eng Geol Environ 79(3):1153–1162

    Article  Google Scholar 

  21. Mandelbrot BB, Mandelbrot BB (1982) The fractal geometry of nature, vol 1. WH freeman, New York

    MATH  Google Scholar 

  22. Midttømme K, Roaldset E (1998) The effect of grain size on thermal conductivity of quartz sands and silts. Pet Geosci 4(2):165–172

    Article  Google Scholar 

  23. Midttømme K, Roaldset E, Aagaard P (1997) Thermal conductivities of argillaceous sediments. Geol Soc Lond Eng Geol Special Publ 12(1):355–363

    Google Scholar 

  24. Midttømme K, Roaldset E, Aagaard P (1998) Thermal conductivity of selected claystones and mudstones from England. Clay Miner 33(1):131–145

    Article  Google Scholar 

  25. Mitchell JK, Soga K (2005) Fundamentals of soil behaviour, vol 3. Wiley, New York

    Google Scholar 

  26. Mügler C, Filippi M, Montarnal P, Martinez JM, Wileveau Y (2006) Determination of the thermal conductivity of opalinus clay via simulations of experiments performed at the Mont Terri underground laboratory. J Appl Geophys 58(2):112–129

    Article  Google Scholar 

  27. Ostoja-Starzewski M (2006) Material spatial randomness: from statistical to representative volume element. Probab Eng Mech 21(2):112–132

    Article  Google Scholar 

  28. Pearson K (1895) Notes on regression and inheritance in the case of two parents. Proc R Soc Lond 58:240–242

    Article  Google Scholar 

  29. Penner E (1963) Anisotropic thermal conduction in clay sediments. In: Proceeding international clay conference, vol 1. Stockholm, pp 365–376

  30. Popov YA, Pribnow DF, Sass JH, Williams CF, Burkhardt H (1999) Characterization of rock thermal conductivity by high-resolution optical scanning. Geothermics 28(2):253–276

    Article  Google Scholar 

  31. Popov Y, Tertychnyi V, Romushkevich R, Korobkov D, Pohl J (2003) Interrelations between thermal conductivity and other physical properties of rocks: experimental data. In: Kümpel HJ (ed) Thermo-hydro-mechanical coupling in fractured rock. Birkhäuser, Basel, pp 1137–1161

    Chapter  Google Scholar 

  32. Šafanda J (1995) Effect of thermal conductivity anisotropy of rocks on the subsurface temperature field. Geophys J Int 120(2):323–3639

    Article  Google Scholar 

  33. Schön JH (1996) Physical properties of rocks: fundamentals and principles of petrophysics. Elsevier, Oxford

    Google Scholar 

  34. Simmons G (1961) Anisotropic thermal conductivity. J Geophys Res 66(7):2269–2270

    Article  Google Scholar 

  35. Walton WH (1948) Feret’s statistical diameter as a measure of particle size. Nature 162(4113):329–330

    Article  Google Scholar 

  36. Wang M, Pan N (2007) Numerical analyses of effective dielectric constant of multiphase microporous media. J Appl Phys 101(11):114102

    Article  Google Scholar 

  37. Xu P (2015) A discussion on fractal models for transport physics of porous media. Fractals 23(03):1530001

    Article  Google Scholar 

  38. Yu B, Li J (2001) Some fractal characters of porous media. Fractals 9(03):365–372

    Article  Google Scholar 

  39. Zhang T, Cai G, Liu S, Puppala AJ (2017) Investigation on thermal characteristics and prediction models of soils. Int J Heat Mass Transf 106:1074–1086

    Article  Google Scholar 

  40. Zhang F, Cui YJ, Zeng L, Conil N (2019) Anisotropic features of natural Teguline clay. Eng Geol 261:105275

    Article  Google Scholar 

Download references

Acknowledgements

This research is supported by the National Natural Science Foundation of China (Grant Nos. 51879203; 52079099).

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Authors and Affiliations

  1. State Key Laboratory of Water Resources and Hydropower Engineering Science, Institute of Engineering Risk and Disaster Prevention, Wuhan University, 299 Bayi Road, Wuhan, 430072, People’s Republic of China

    Kai-Qi Li, Zhuang Miao, Dian-Qing Li & Yong Liu

  2. Key Laboratory of Rock Mechanics in Hydraulic Structural Engineering of the Ministry of Education, Wuhan University, 299 Bayi Road, Wuhan, 430072, People’s Republic of China

    Kai-Qi Li, Zhuang Miao, Dian-Qing Li & Yong Liu

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  1. Kai-Qi Li
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Correspondence to Yong Liu.

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Li, KQ., Miao, Z., Li, DQ. et al. Effect of mesoscale internal structure on effective thermal conductivity of anisotropic geomaterials. Acta Geotech. 17, 3553–3566 (2022). https://doi.org/10.1007/s11440-022-01458-z

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  • DOI: https://doi.org/10.1007/s11440-022-01458-z

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