A Nonlinear Creep-damage Constitutive Model of Mudstone Based on the Fractional Calculus Theory

During the flood development in an oil field, the creep characteristic of mudstone is one of the important factors causing casing damage. In this study, based on the theory of fractional order differential and taking into account the creep damage evolution rules, a fractional nonlinear creep-damage model is proposed to reflect the instantaneous deformation in loading processes and the accelerated creep stage of mudstone. After assuming that the relationships between strain and time can be expressed by the exponential functions, the creep-damage constitutive model to describe the relationship between stress and strain stress of mudstone is established under the condition of accelerated strain rate loading. Furthermore, uniaxial creep tests and conventional triaxial compression tests were carried out to validate the proposed model. It is found that the fractional nonlinear creep-damage model can effectively describe the characteristics of the three stages of mudstone creep; moreover, this model can reduce the number of components and the parameters. Both the fractional nonlinear creep model and the creep-damage constitutive model have a high fitting relationship with the test results. Also, the initial elastic modulus and confining pressure are in a good linear relationship. Finally, the parametric sensitivity analysis of the theoretical model is carried out. The correctness and applicability of models were confirmed from three respects of the derivation process, test results, and the theoretical analysis.