Fractal-Fractional Creep Constitutive Model for Shale under Triaxial Stress States

The long-term rheological behavior of shale critically influences wellbore stability and hydraulic fracture conductivity in unconventional reservoirs. Traditional integer-order constitutive models, such as the Nishihara model, frequently fail to accurately capture the complex nonlinear creep characteristics of shale, particularly the rapid transient decay during primary creep and the accelerated deformation during tertiary creep. To address these limitations, this study proposes a creep constitutive model based on the Fractal-Fractional Atangana-Baleanu-Caputo (FF-ABC) derivative. The viscoelastic deformation is described by a Fractal-Fractional Maxwell element, which integrates the memory effects of fractional calculus with the multi-scale structural characteristics of fractal geometry; the viscoplastic deformation is characterized by a fractal power-law damage model activated upon reaching a critical time threshold. Analytical solutions for triaxial stress states are derived using the fractal-fractional integral operator and the elastic-viscoelastic correspondence principle. The triaxial formulation explicitly incorporates the deviatoric stress tensor, enabling prediction of creep strain under variable confining pressures. Model validity is verified against experimental data from triaxial compression tests on shale samples. Results demonstrate that the proposed FF-ABC model achieves substantially higher fitting accuracy ($R^2 > 0.98$) compared to the classical Nishihara model and standard fractional derivative models. Sensitivity analysis indicates that fractal order $\beta_1$ governs the initial transient response, fractional order $\alpha_1$ controls long-term viscoelastic flow, and damage parameter $\beta_2$ determines tertiary creep acceleration. This study establishes a robust, physically meaningful theoretical framework for predicting time-dependent shale deformation in deep subsurface engineering applications.