基于分数阶时间导数常Q黏弹本构关系的含黏滞流体双相VTI介质中波场数值模拟

分数阶微分算子具有描述历史依赖性和全域相关性的特质，本文利用这种特质描述双相介质固体骨架的黏弹性特征.基于Kjartansson常Q理论将含有分数阶时间导数的黏弹固体骨架各向异性本构关系与双相介质理论有机地结合起来，并引入流变学本构关系描述孔隙流体的黏滞性力学行为，提出一种新的基于分数阶时间导数常Q黏弹本构关系的含黏滞流体双相VTI模型.推导了相应的时间域波传播方程，然后对该方程进行了数值模拟.对整数阶导数采用高阶交错网格有限差分算法，对分数阶时间导数采用短时记忆中心差分算法，进行了不同相界、不同品质因子组及双层地质结构情况下该类介质中波场的数值模拟与特征分析.模拟结果表明：将含有分数阶时间导数的常Q黏弹固体骨架各向异性本构关系及孔隙流体的黏滞性本构关系引入双相介质理论是可行的，二者的结合能更好地反映地下介质的黏弹性特征，对于进一步认识波在黏弹各向异性孔隙介质中的传播机理具有重要意义，为反演和重构地下油气储层和结构奠定正演理论基础.

Numerical simulation of the wavefield in a viscous fluid-saturated two-phase VTI medium based on the constant-Q viscoelastic constitutive relation with a fractional temporal derivative

Fractional differential operators have the trait of describing historical dependence and global correlation.In this paper,we use this trait to describe the viscoelastic characteristics of a two-phase solid matrix.Based on the Kjartansson constant-Q model theory,the anisotropic constitutive relation of the viscoelastic solid skeleton with fractional temporal derivative is combined with the two-phase medium theory,and the rheological constitutive relation is introduced to describe the viscous mechanical behavior of the pore fluid.A new VTI model of a vicious fluid-saturated two-phase VTI medium based on the constant-Q viscoelastic constitutive relation with a fractional temporal derivative is proposed,and the corresponding wavefield propagation equation is deduced in the time domain.In numerical simulation, the finite difference method on a high-order staggered grid is used for the integer order derivative, and the short time memory center difference algorithm is employed to discretize the fractional order temporal derivative.Following the equations, the wavefield in this medium model is simulated for different phase boundaries, different Q values, and a two-layer structure, and then the wavefield features are analyzed.The results of numerical modeling indicate that it is feasible to introduce the constant-Q viscoelastic solid anisotropic constitutive relation containing a fractional temporal derivative and the viscous constitutive relation of the pore fluid into the two-phase medium theory, and the combination of both the relations can better reflect the viscoelastic characteristics of the underground medium.These results are of importance for further understanding the propagation mechanism in a viscoelastic anisotropic porosity medium, and provide a theoretical basis for inversion and reconstruction of underground oil and gas reservoirs and structures.