The influences of stress level, temperature, and water content on the fitted fractional orders of geomaterials

  • Yu PengEmail author
  • Jinzhou Zhao
  • Kamy Sepehrnoori
  • Yongming Li
  • Zhenglan Li
Original Research


Fractional viscoelastic constitutive equations have been extensively used for describing the creep and relaxation behaviors of geomaterials. Their value and efficiency for fitting the creep or relaxation curves of geomaterials have been validated by many researches. In order to study the physical meaning of fractional order, creep curves of shale, rock salt, and mudstone under different confining and deviatoric stresses, temperatures, and water contents have been fitted and compared. We found that samples from the same layer have similar fractional orders and they are almost constant during the steady creep. Increasing confining stress will increase the elasticity of the rock and a larger deviatoric stress will lead to a stronger viscosity. The most interesting phenomenon is that the fitted parameters will have a mutation, when the deviatoric stress acting on the sample is larger than the long-term strength of the rock. Both the temperature and water content, which is the ratio of water mass and dried simple mass, have significant impacts on the creep curves. However, the fractional order is more controlled by temperature than water content. This is because that temperature has a more significant influence on the property of the minerals and the cementing property between the mineral grains is more likely to be determined by water content.


Fractional calculus Fractional orders Stress level Temperature Water content Shale 


\(\alpha \)

fractional order (dimensionless)

\(\alpha _{i}\)

fractional order of stage \(i\) (dimensionless)

\(D^{\alpha }\)

fractional differential operator of order \(\alpha \)

\(\varepsilon \)

strain (dimensionless)

\(\varepsilon _{i}\)

strain of stage \(i\) (dimensionless)


time (s)

\(t _{i}\)

time of stage \(i\) (s)

\(\tau \)

relaxation time (s)

\(\tau _{i}\)

relaxation time of stage \(i\) (s)

\(\tau _{t}\)

an intermediate variable from 0 to 1 (s)

\(f (t)\)

certain function of time

\(\Gamma \)

gamma function

\(J _{s}\)

creep compliance of springpot (MPa−1)

\(J _{k}\)

creep compliances of fractional Kelvin model (MPa−1)

\(J _{z}\)

creep compliances of fractional standard linear solid model (MPa−1)

\(J _{i}\)

creep compliances of stage \(i\) (MPa−1)

\(\eta \)

fractional consistency coefficient (MPa⋅s)

\(E _{1}\), \(E _{2}\)

Young’s modulus (MPa)


Young’s modulus of stage \(i\) (MPa)


stress acting on the sample (MPa)

\(p _{i}\)

stress acting on the sample of stage \(i\) (MPa)

\(\theta ^{\alpha }\)

material constants (sα)

\(\beta \)

input parameter of Mittag–Leffler special function



This study was supported by the Major Program of the National Natural Science Foundation of China (51490653), Sichuan Youth Science and Technology Innovation Research Team Program (2017TD0013), and the China Scholarship Council (201708510130).


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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.State Key Laboratory of Oil and Gas Reservoir Geology and ExploitationSouthwest Petroleum UniversityChengduChina
  2. 2.Department of Petroleum and Geosystems EngineeringThe University of Texas at AustinAustinUSA

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