An Improved Fractional Damage Creep Model for Shale and Its Application to the Long-term Fracture Conductivity Prediction

Formation creep not only reduces the effective gas flow channels but may also lead to engineering disasters, such as wellbore narrowing and collapse. Therefore, it is crucial to conduct in-depth research on the viscoelastic-plastic behavior of shale under complex conditions and its impact on the long-term fracture conductivity. This study combines statistical damage mechanics and fractional calculus theory to establish a shale fractional damage creep model (FDC model) that considers the combined effects of stress, temperature, and time. FDC model is applied to evaluate the fracture aperture, and further, a long-term fracture conductivity model (LFC model) is proposed, which accounts for proppant deformation, embedment, and rock creep. The accuracy and rationality of proposed FDC and LFC model are evaluated using experimental data published in literature. The results indicate that higher external stress accelerates the transition into the tertiary creep stage, leading to the pronounced strain growth and significant reductions in fracture conductivity. Compared to the initial condition, the conductivity decreases by two orders of magnitude after 8.50 d under closure pressure. Moreover, fracture conductivity is positively correlated with proppant particle size and negatively correlated with closure pressure. Both FDC model for characterizing viscoelastic-plastic behavior and LFC model for predicting long-term fracture conductivity demonstrate good applicability and accuracy. The study suggests that both excessively high and low proppant elastic moduli are unfavorable for maintaining stable fracture aperture. This is because, although increasing the proppant elastic modulus reduces its deformation, it simultaneously exacerbates its embedment in formation. In this study, when rock elastic modulus is 13,000 MPa, proppant with an elastic modulus ranging between 80,000 MPa and 110,000 MPa are found to be the most suitable. This research provides new insights for optimizing fracturing design and improving shale gas long-term recovery.