New approach for predicting time-dependent deformation of shale rock: a modified fractional-order creep constitutive model

Abstract

Time-dependent deformation is critical for ensuring safety and efficiency in horizontal well stability and the long-term permeability of shale gas. Purely empirical methods that lack a physical background cannot reflect the mechanism of shale creep, and classical elemental models lack control over both viscous and elastic parameters, leading to poor nonlinear behavior. Moreover, an appropriate amount of time is necessary to perform time-dependent deformation testing as such testing usually takes a long time. Additionally, reports on radial creep data for shale are rare. In this study, several short-term and long-term deformation experiments were conducted on shale samples to evaluate their deformation behavior. Furthermore, a modified creep constitutive model was proposed to fit the measured data; the result was a perfect fitting. The experimental results showed that axial deformations have significant anisotropy and differential stress dependence, whereas radial deformations do not. Volumetric deformations can form three types of strain–time profiles. Creep compliance (3 h) was negatively correlated with Young’s modulus and Poisson’s ratio. Model validation showed that the shale creep deformation would be incorrectly estimated without the new model, which outperformed the classical model in the deceleration stage. Model parameters α and β, which reflect the degree of viscoelasticity in the shale, were negatively correlated with Young’s modulus and the differential stress. In addition, using the new model to predict the behavior of creep deformation through short-term (48 h) creep experiments is feasible.

Appendices

Appendix A

Please see Figs. 19, 20 and 21.

Please see Tables 3 and 4.

Fig. 19

Triaxial strength test results of experimental samples (P_c = 10 MPa)

Fig. 20

Experimental equipment deformation calibration experiment results

Fig. 21

Ternary diagram of mineral composition of shale samples

Table 3 Mineral composition and percentage of shale samples

Table 4 Relative content of clay minerals in shale samples

Appendix B

Please see Fig. 22.

Fig. 22

Fractional-order nonlinear viscoelastic shale creep model. (I) is the Hooke body, (II) is the Fractional element, (III) is the Fractional improved Kelvin-Voigt model

The nonlinear creep model of shale is composed of three parts: the Hooke body; fractional element; and fractional Kelvin-Voigt model. According to the series rule, their relationship with the total stress $\sigma$ and strain $\varepsilon$ can be expressed as follows:

$$\left\{\begin{array}{c}\varepsilon ={\varepsilon }_{e}+{\varepsilon }_{vi}+{\varepsilon }_{ve}\\ \sigma ={\sigma }_{e}={\sigma }_{vi}={\sigma }_{ve}\end{array}\right.$$

(12)

The subscript are connected in series, assuming that the strain of the Hooke body is ε_e and the stress is σ_e, the strain of the fractional element is ε_vi and the stress is σ_vi, and the strain of the fractional Kelvin-Voigt model is ε_ve and the stress is σ_ve.

• (i) The stress–strain relationship of the Hooke body is:

  $${\varepsilon }_{e}=\frac{{\sigma }_{e}}{{E}_{1}}=\frac{\sigma }{{E}_{1}}$$

  (13)

• (ii) The stress–strain relationship of the fractional element is:

  Displayed in the “Fractional element with variable viscosity coefficients” section the viscosity coefficient is a function of time obeying the relation η(t) = η₁₀·e^−ωt. Bringing in the constitutive relation of the fractional element, we obtain Eq. (14).

  $$\begin{array}{cc}\sigma \left(t\right)={\eta }_{10}\cdot {e}^{-\omega t}\cdot \frac{{d}^{\alpha }\varepsilon \left(t\right)}{{dt}^{\alpha }}& \left(0\le \alpha \le 1\right)\end{array}$$

  (14)

  Substitute the constant σ(t) = σ_vi, σ_vi (the creep stress) into Eq. (14), and we apply the Laplace transform for Eq. (14) to obtain:

  $$\varepsilon \left(s\right)=\frac{{\sigma }_{vi}}{{\eta }_{10}\left(s-\omega \right){s}^{\alpha }}$$

  (15)

  In the solution process we must apply the Laplace inverse transform of the two-parameter Mittag–Leffler function, which are defined respectively as^{57, 74}:

  $$\begin{array}{cc}{\int }_{0}^{\infty }{e}^{-st}{t}^{zk+b-1}{E}_{a,b}^{\left(k\right)}\left(\pm {pt}^{a}\right)dt=\frac{k!{s}^{a-b}}{{\left({s}^{a}\mp p\right)}^{k+1}}& \left({\text{Re}}\left(s\right)>{\left|p\right|}^{1/a}\right)\end{array}$$

  (16)

  where a, b, and k are the Mittag–Leffler function parameters. Finally, the creep equation for the fractional-order element with variable viscosity coefficient is obtained as:

  $$\begin{array}{cc}\varepsilon \left(t\right)=\frac{{\sigma }_{vi}}{{\eta }_{10}}\cdot {t}^{\alpha }{E}_{1,\alpha +1}\left(\omega t\right)& \left(0<\alpha <1\right)\end{array}$$

  (17)

• (iii) According to the parallel theory of the combined model, the stress–strain relationship of the fractional Kelvin-Voigt model can be obtained as follows:

  $$\left\{\begin{array}{c}{\varepsilon }_{ve}={\varepsilon }_{H}={\varepsilon }_{F}\\ {\sigma }_{ve}={\sigma }_{H}+{\sigma }_{F}\end{array}\right.$$

  (18)

  where ε_H and σ_H are the strains and stresses of the spring, and ε_F and σ_F are the strains and stresses of the fractional-order element in part (III). And they obey the following stress–strain relationship: σ_H = ε_H·E₂ and σ_F = η₂₀·e^−ω·(d^β ε_F)/dt^β. Substituting into Eq. (18), we obtain the third part of the constitutive equation.

  $${\sigma }_{ve}={E}_{2}{\varepsilon }_{ve}+{\eta }_{20}{e}^{-\omega }\cdot \frac{{d}^{\beta }{\varepsilon }_{ve}}{{dt}^{\beta }}$$

  (19)

Here we assume that σ_ve is a constant, and use the Laplace transform for Eq. (19) to obtain:

$${\varepsilon }_{ve}\left(s\right)=\frac{{\sigma }_{ve}\cdot {s}^{-1}}{{\eta }_{20}{e}^{-\omega }\left({s}^{\beta }+\frac{{E}_{2}}{{\eta }_{20}{e}^{-\omega }}\right)}$$

(20)

Applying the Laplace inverse transform of the two-parameter Mittag–Leffler function (Eq. (17), the creep equation for part (III) can be obtained as:

$${\varepsilon }_{ve}\left(t\right)=\frac{{\sigma }_{ve}}{{\eta }_{20}}\cdot {t}^{\beta }{E}_{\beta ,\beta +1}\left(-\frac{{E}_{2}}{{\eta }_{20}{e}^{-\omega }}{t}^{\beta }\right)$$

(21)

Finally, the nonlinear fractional shale creep model is obtained by incorporating Eq. (13), Eq. (17) and Eq. (21) into Eq. (12):

$$\varepsilon \left(t\right)=\frac{\sigma }{{E}_{1}}+\frac{\sigma }{{\eta }_{10}}\cdot {t}^{\alpha }{E}_{1,\alpha +1}\left(\omega t\right)+\frac{\sigma }{{\eta }_{20}}\cdot {t}^{\beta }{E}_{\beta ,\beta +1}\left(-\frac{{E}_{2}}{{\eta }_{20}{e}^{-\omega }}{t}^{\beta }\right)$$

(22)

Appendix C

Please see Table 5, 6 and 7.

Table 5 Creep parameter values of different samples under different stresses using fractional nonlinear models

Table 6 4# -H sample creep parameters at different time scales

Table 7 5# -H sample creep parameters at different time scales