基于分数阶导数的盐岩流变本构模型
摘要
A modeling approach to the creep behavior of natural rock salt is of special interest for many underground engineering such as underground storage of natural gas and crude oil, and a permanent repository of CO2 and radioactive and toxic wastes. Based on the definition of fractional derivative, the Abel dashpot with a constant viscosity coefficient is generalized to an Abel dashpot with a variable viscosity coefficient, referred to as a new time-dependent element. By replacing a Newtonian dashpot in the classical Nishihara model with the fractional derivative Abel dashpot of a variable viscosity coefficient, a new creep constitutive model is proposed on the basis of time-based fractional derivative. The analytic solution for the fractional derivative creep constitutive model is given theoretically. Moreover, a computer-controlled creep set-up is employed to measure creep strain of a salt rock sample under the condition of uniaxial compression, taking from the Chuwang Well No.1, Jianghan Oilfield, Hubei Province. The parameters of the fractional derivative creep model are determined by fitting to the experimental results of time-dependent deformation of salt rock. In addition, a sensitivity study for the analytic solution of the time-based fractional derivative model is carried out, showing the effects of stress level, fractional derivative order and viscosity coefficient on a creep strain of salt rock. Furthermore, it is shown that the time-based fractional derivative model in the paper can be simplified to the classical Nishihara model in the special case. It is indicated that the fractional derivative creep model proposed in the paper gives a precise description of the three stages of creep especially the tertiary stage, i.e., an unsteady state period.