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A triaxial creep model for deep coal considering temperature effect based on fractional derivative

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Abstract

A triaxial creep model for deep coal considering temperature effect based on fractional derivative is proposed for the condition of triaxial stress state. In order to study the temperature effect on creep deformation of deep coal, the thermal damage variable is established based on the Weibull distribution and continuum damage mechanics theory. The thermal damage variable is assigned to the Hooke body and Abel dashpot in order to characterize the effect of temperature on elastic modulus and viscosity coefficient. The temperature-dependent mechanical elements are connected to the creep model, and a three-dimensional creep constitutive equation based on fractional derivative is established. A creep experimental study for deep coal under the constant axial pressure and unloading confining pressure at different temperatures is carried out to characterize the creep deformation of deep coal during mining. The experimental results show that the coal sample with higher temperature has greater axial deformation, but the radial deformation does not change monotonically with the change of temperature. Moreover, the proposed triaxial creep model is validated by experimental data and the nonlinear least square method is used to determine the model parameters. It is indicated that the triaxial creep model can better describe the time-dependent deformation under the effect of temperature, especially the accelerated creep stage of creep. In addition, the sensitivity analysis of key parameters of the proposed model, especially axial stress level and creep temperature, is carried out to further verify the accuracy of the triaxial creep model.

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Abbreviations

a :

Temperature effect constant

\(\alpha\) :

Parameter of Mittag–Leffler function

\(\alpha _{1}\) :

Thermal expansion coefficient (K−1)

b :

Temperature effect constant

\(\beta\) :

Parameter of Mittag–Leffler function

\(\beta _{1}\) :

Biot coefficient

\(D^{\gamma }\) :

Fractional differential operator

\(D_{{\text{T}}}\) :

Thermal damage variable

\(D_{{\text{M}}}\) :

Mechanical damage variable

\(D_{{{\text{TM}}}}\) :

Thermal–mechanical damage variable

\(\varepsilon\) :

Strain of the Hooke body

\(\varepsilon _{1}\) :

Axial strain of microunit

\(\varepsilon (t)\) :

Strain of the Abel dashpot

\(\varepsilon _{{\text{e}}}\) :

Strain of the Hooke body

\(\varepsilon _{{{\text{ve}}}}\) :

Strain of the viscoelastic body

\(\varepsilon _{{{\text{vp}}}}\) :

Strain of the viscoplastic body

\(\varepsilon _{{\text{H}}}\) :

Strain of the Hooke body

\(\varepsilon _{{\text{A}}}\) :

Strain of the Abel dashpot

\(\varepsilon _{{ij}}^{'} \left( t \right)\) :

Total effective deviatoric strain tensor

\(\varepsilon _{{ij}}^{e}\) :

Effective deviatoric strain tensor of elastic body

\(\varepsilon _{{ij}}^{{ve}}\) :

Effective deviatoric strain tensor of viscoelastic body

\(\varepsilon _{{ij}}^{{vp}}\) :

Effective deviatoric strain tensor of viscoplastic body

E (T):

Elastic modulus at temperature T (GPa)

\(E_{0} (T)\) :

Elastic modulus at temperature T (GPa)

\(E_{1} (T)\) :

Elastic modulus at temperature T (GPa)

\(E_{{\alpha ,\beta }} (z)\) :

Mittag–Leffler function

\(F\) :

Yield function

\(F_{0}\) :

Initial reference value of the yield function

\(G_{0} (T)\) :

Shear modulus at temperature T (GPa)

\(G_{1} (T)\) :

Shear modulus at temperature T (GPa)

\(J_{2}\) :

Second invariant of deviatoric stress (MPa)

\(K(T)\) :

Bulk modulus at temperature T (GPa)

μ:

Poisson’s ratio

m :

Shape parameter

\(\eta _{0}^{\gamma }\) :

Viscosity coefficient at room temperature (GPa·hγ)

\(\eta _{{}}^{\gamma } (T)\) :

Viscosity coefficient at temperature T (GPa·hγ)

\(\eta _{1}^{\gamma } (T)\) :

Viscosity coefficient of the Abel dashpot at temperature T (GPa·hγ)

\(\eta _{2}^{\gamma } (T)\) :

Viscosity coefficient of the Abel dashpot at temperature T (GPa·hγ)

P :

Pore pressure (MPa)

\(\gamma\) :

Fractional derivative order

\(s_{{ij}}^{'}\) :

Effective deviatoric stress tensor

\(\Delta T\) :

Change from room temperature (oC)

\(T_{0}\) :

Room temperature (oC)

\(\omega\) :

Parameter of viscosity coefficient (h−1)

\(\psi\) :

Scale parameter

\(\delta _{{ij}}\) :

Kronecker symbol

\(\phi ( \cdot )\) :

Power function form

\(\sigma\) :

Stress of the Hooke body (MPa)

\(\sigma (t)\) :

Stress of the Abel dashpot (MPa)

\(\sigma _{1}\) :

Axial stress (MPa)

\(\sigma _{2}\) :

Radial stress (MPa)

\(\sigma _{3}\) :

Confining pressure (MPa)

\((\sigma _{s} )_{{\text{T}}}\) :

Yield limit with the temperature effect (MPa)

\(\sigma _{{\text{d}}}\) :

Stress of the Abel dashpot (MPa)

\(\sigma _{{\text{p}}}\) :

Stress of the fractional element (MPa)

\(\sigma _{{\text{m}}}^{'}\) :

Effective spherical stress tensor

\(\sigma _{{ij}}^{'}\) :

Effective stress tensor

References

  1. Cai TT, Feng ZC, Jiang YL, Zhao D, Zhou D, Zhang XQ (2018) Seepage evolution in coal creep under different temperatures and different stresses. Chin J Rock Mech Eng 37(S2):3898-3904C (in Chinese)

    Google Scholar 

  2. Chen L, Wang CP, Liu JF, Liu YM, Liu J, Su R, Wang J (2014) A damage-mechanism-based creep model considering temperature effect in granite. Mech Res Commun 56:76–82

    Article  Google Scholar 

  3. Chen SW, Yang CH, Wang GB (2017) Evolution of thermal damage and permeability of Beishan granite. Appl Therm Eng 110:1533–1542

    Article  Google Scholar 

  4. Danesh NN, Chen Z, Aminossadati SM, Kizil MS, Pan Z, Connell LD (2016) Impact of creep on the evolution of coal permeability and gas drainage performance. J Nat Gas Sci Eng 33:469–482

    Article  Google Scholar 

  5. Danesh NN, Chen Z, Connell LD, Kizil MS, Pan Z, Aminossadati SM (2017) Characterisation of creep in coal and its impact on permeability: an experimental study. Int J Coal Geol 173:200–211

    Article  Google Scholar 

  6. Feng YY, Yang XJ, Liu JG, Chen ZQ (2021) A new fractional Nishihara-type model with creep damage considering thermal effect. Eng Fract Mech 242:107451

    Article  Google Scholar 

  7. Gao F, Xu XL, Yang XJ, Ju Y (2009) Research on thermo-visco-elastoplastic model of rock. Chin J Rock Mech Eng 28(1):74–80 (in Chinese)

    Google Scholar 

  8. Guo ZH, Nhu Hao VuP, Hussain F (2018) A laboratory study of the effect of creep and fines migration on coal permeability during single-phase flow. Int J Coal Geol 200:61–76

    Article  Google Scholar 

  9. Heap MJ, Baud P, Meredith PG (2009) Influence of temperature on brittle creep in sandstones. Geophys Res Lett 36(19):L19305

    Article  Google Scholar 

  10. Hou RH, Zhang K, Tao J, Xue XR, Chen YL (2019) A nonlinear creep damage coupled model for rock considering the effect of initial damage. Rock Mech Rock Eng 52:1275–1285

    Article  Google Scholar 

  11. Hu QZ, Feng XT, Zhou H (2009) Study of creep model of rock salt with thermal damage considered. Rock Soil Mech 30(8):2245–2248 (in Chinese)

    Google Scholar 

  12. Kang JH, Zhou FB, Liu C, Liu YK (2015) A fractional non-linear creep model for coal considering damage effect and experimental validation. Int J Non-Linear Mech 76:20–28

    Article  Google Scholar 

  13. Liao MK, Lai YM, Liu EL, Wan XS (2017) A fractional order creep constitutive model of warm frozen silt. Acta Geotech 12:377–389

    Article  Google Scholar 

  14. Liu QS, Xu XC, Yamaguchi T, Cho A (2001) Testing study on mechanical properties of the three gorges granite concerning temperature and time. Chin J Rock Mech Eng 20(5):715–719 (in Chinese)

    Google Scholar 

  15. Li BB, Yang K, Xu P, Xu J, Yuan M, Zhang M (2019) An experimental study on permeability characteristics of coal with slippage and temperature effects. J Petrol Sci Eng 175:294–302

    Article  Google Scholar 

  16. Li BB, Yang K, Ren CH, Li JH, Xu J (2019) An adsorption-permeability model of coal with slippage effect under stress and temperature coupling condition. J Nat Gas Sci Eng 71:102983

    Article  Google Scholar 

  17. Mansouri H, Ajalloeian R (2018) Mechanical behavior of salt rock under uniaxial compression and creep tests. Int J Rock Mech Min Sci 110:19–27

    Article  Google Scholar 

  18. Nagel T, Minkley W, Böttcher N, Naumov D, Görke UJ, Kolditz O (2017) Implicit numerical integration and consistent linearization of inelastic constitutive models of rock salt. Comput Struct 182:87–103

    Article  Google Scholar 

  19. Nishihara M (1952) Creep of shale and sandy-shale. J Geol Soc Jpn 58(683):373–377

    Article  Google Scholar 

  20. Noack K (1998) Control of gas emissions in underground coal mines. Int J Coal Geol 35:57–82

    Article  Google Scholar 

  21. Qi YJ, Jiang QH, Wang ZJ, Zhou CB (2012) 3D creep constitutive equation of modified Nishihara model and its parameters identification. Chin J Rock Mech Eng 31(2):347–355 (in Chinese)

    Google Scholar 

  22. Ren CH, Li BB, Xu J, Zhang Y, Li JH, Gao Z, Yu J (2020) A novel damage-based permeability model for coal in the compaction and fracturing process under different temperature conditions. Rock Mech Rock Eng 53:5697–5713

    Article  Google Scholar 

  23. Schoenball M, Sahara DP, Kohl T (2014) Time-dependent brittle creep as a mechanism for time-delayed wellbore failure. Int J Rock Mech Min Sci 70:400–406

    Article  Google Scholar 

  24. Shao JF, Chau KT, Feng XT (2006) Modeling of anisotropic damage and creep deformation in brittle rocks. Int J Rock Mech Min Sci 43:582–592

    Article  Google Scholar 

  25. Ulusay R, Hudson JA (2007) The complete ISRM suggested methods for rock characterization, testing and monitoring: 1974–2006 In: Suggested Methods Prepared by the ISRM Commission on Testing Methods. International Society for Rock Mechanics, Lisbon

  26. Wang LJ, Zhou HW, Rong TL, Ren WG (2018) Research on experimental and nonlinear creep constitutive model of coal at depth. J China Coal Soc 43(8):2196–2203 (in Chinese)

    Google Scholar 

  27. Wang JM, Zhao YS, Mao RB (2019) Impact of temperature and pressure on the characteristics of two-phase flow in coal. Fuel 253:1325–1332

    Article  Google Scholar 

  28. Wu F, Chen J, Zou QL (2018) A nonlinear creep damage model for salt rock. Int J Damage Mech 28(5):758–771

    Article  Google Scholar 

  29. Xu WY, Yang SQ, Chu WJ (2006) Nonlinear viscoelasto-plastic rheological model(Hohai model) of rock and its engineering application. Chin J Rock Mech Eng 25(3):433–447 (in Chinese)

    Google Scholar 

  30. Xu HY, Jiang XY (2017) Creep constitutive models for viscoelastic materials based on fractional derivatives. Comput Math Appl 73(6):1377–1384

    Article  MathSciNet  MATH  Google Scholar 

  31. Xu XL, Karakus M (2018) A coupled thermo-mechanical damage model for granite. Int J Rock Mech Min Sci 103:195–204

    Article  Google Scholar 

  32. Xu T, Zhou GL, Heap MJ, Zhu WC, Chen CF, Baud P (2017) The influence of temperature on time-dependent deformation and failure in granite: a mesoscale modeling approach. Rock Mech Rock Eng 50(9):2345–2364

    Article  Google Scholar 

  33. Yang SQ, Xu P, Xu T (2015) Nonlinear visco-elastic and accelerating creep model for coal under conventional triaxial compression. Geomech Geophys Geo-Energy Geo-Resour 1:109–120

    Article  Google Scholar 

  34. Yin GZ, Jiang CB, Wang JG, Xu J (2013) Combined effect of stress, pore pressure and temperature on methane permeability in anthracite coal: an experimental study. Transp Porous Media 100:1–16

    Article  Google Scholar 

  35. Zhang L, Li XC, Ren T (2020) A theoretical and experimental study of stress–strain, creep and failure mechanisms of intact coal. Rock Mech Rock Eng 53:5641–5658

    Article  Google Scholar 

  36. Zhao YL, Wang YX, Wang WJ, Wan W, Tang JZ (2017) Modeling of non-linear rheological behavior of hard rock using triaxial rheological experiment. Int J Rock Mech Min Sci 93:66–75

    Article  Google Scholar 

  37. Zhao Y, Lin BQ, Liu T, Zheng YN, Kong J, Li QZ, Song HR (2019) Flow field evolution during gas depletion considering creep deformation. J Nat Gas Sci Eng 65:45–55

    Article  Google Scholar 

  38. Zhao Y, Zhou HW, Zhong JC, Liu D (2019) Study on the relation between damage and permeability of sandstone at depth under cyclic loading. Int J Coal Sci Technol 6(4):479–492

    Article  Google Scholar 

  39. Zhou FX, Wang LY, Liu HB (2021) A fractional elasto-viscoplastic model for describing creep behavior of soft soil. Acta Geotech 16:67–76

    Article  Google Scholar 

  40. Zhou HW, Wang CP, Han BB, Duan ZQ (2011) A creep constitutive model for salt rock based on fractional derivatives. Int J Rock Mech Min Sci 48(1):116–121

    Article  Google Scholar 

  41. Zhou HW, Wang CP, Mishnaevsky L, Duan ZQ, Ding JY (2013) A fractional derivative approach to full creep regions in salt rock. Mech Time-Depend Mater 17(3):413–425

    Article  Google Scholar 

  42. Zhou HW, Wang LJ, Rong TL, Zhang L, Ren WG, Su T (2019) Creep-based permeability evolution in deep coal under unloading confining pressure. J Nat Gas Sci Eng 65:185–196

    Article  Google Scholar 

  43. Zhou HW, Zhang L, Wang XY, Rong TL, Wang LJ (2020) Effects of matrix-fracture interaction and creep deformation on permeability evolution of deep coal. Int J Rock Mech Min Sci 127:104236

    Article  Google Scholar 

Download references

Acknowledgements

The present work is supported by the National Natural Science Foundation of China (51674266, 51827901, 51904309) and the Yueqi Outstanding Scholar Program of CUMTB (2017A03). The financial supports are gratefully acknowledged.

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Authors and Affiliations

  1. School of Mechanics and Civil Engineering, China University of Mining and Technology, Beijing, 100083, China

    Lei Zhang, Xiangyu Wang, Lei Wang, Teng Su & Qing Wei

  2. School of Energy and Mining Engineering, China University of Mining and Technology, Beijing, 100083, China

    Hongwei Zhou

  3. State Key Laboratory of Coal Resources and Safe Mining, China University of Mining and Technology, Beijing, 100083, China

    Hongwei Zhou

  4. Geotechnical Institute, TU Bergakademie Freiberg, 09599, Freiberg, Germany

    Tengfei Deng

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  1. Lei Zhang
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Correspondence to Hongwei Zhou.

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Zhang, L., Zhou, H., Wang, X. et al. A triaxial creep model for deep coal considering temperature effect based on fractional derivative. Acta Geotech. 17, 1739–1751 (2022). https://doi.org/10.1007/s11440-021-01302-w

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