Research On Seismic Wave Propagation Theory In Complex Single-phase And Two-phase Media

It is widely known that the recorded seismic waves carry the property information of the Earth’s interior.Therefore,providing an accurate wave propagation model and theory is very important to understand wave propagation in real formation.Numerous studies indicate that most of the Earth’s interior exhibits viscoelastic and anisotropic characteristics,which leads to complicated seismic wave behaviors,such as velocity and attenuation anisotropy,velocity dispersion and energy attenuation.The exact analysis of wave behaviors observed during seismic wave propagation in the Earth’s interior is essential to seismic imaging and inversion.The existing wave propagation theories are mainly developed in single-phase and two-phase media.Considerable attention has devoted to velocity anisotropy.Like velocity anisotropy,the study of attenuation anisotropy,however,still deserves attention as it may have a more significant impact on seismic wave behavior.For twophase media,the classic Biot and Biot-squirt(BISQ)theories significantly underestimate the velocity dispersion and attenuation in a wide frequency range.These problems cause it difficult to apply the existing wave propagation theories in complex media.To handle the aforementioned problems,this paper proposes some theories that can be used to describe wave propagation in complex single-phase and two-phase media with anisotropy and viscoelasticity,as follows:1.Starting from the constant-Q model,an attenuative vertical transversely isotropic(VTI)single-phase medium model containing both velocity and attenuation anisotropy is proposed.Based on the time fractional derivative,the corresponding constitutive relation and wave equations are derived,and the analytical expressions of complex velocity,phase velocity,energy velocity,and quality factor for quasicompression(qP),quasi-shear(qSV),and pure-shear(SH)waves,respectively,propagating in(x,z)plane with a vertical axis of symmetry are presented by using the homogeneous plane wave theory.The results of numerical examples demonstrate that the proposed model can be used to describe the wave behavior of attenuation anisotropy in a single-phase medium.2.The generalized Zener linear body is introduced into the classical BISQ model to describe the relative motion between wave-induced pore solid skeleton mineral particles,and a more complex case of orthorhombic anisotropy(ORA)are considered.The constitutive relation and wave equation are presented.Meanwhile,the accurate expressions of complex velocity,phase velocity,energy velocity and quality factor in poro-viscoelastic ORA media are derived by invoking plane wave theory.To implement the numerical simulation of 3-D multi-component wavefield,the staggered grid finite difference algorithm is used to solve the wave equation in time domain.The snapshots show the ORA and viscoelastic effects lead to significant changes in the wavefield.The comparison to the synthetic seismogram in the poroelastic model indicates that the relaxed skeleton can describe the strong attenuation of seismic waves in the seismic exploration band.Second,the variations in the velocities and the quality factor curves with propagation direction,frequency,and porosity are analyzed.Both examples present the characteristics of wave propagation in the poro-viscoelastic ORA media and validate the effectiveness of the proposed theory and equations.3.Motivated by the classic Biot model,this paper develops a novel theory about the attenuative VTI porous medium including both the anisotropy of velocity and attenuation,which incorporates the complete relaxed solid skeleton(RS)and relaxed fluid diffusion(RF)mechanisms.The proposed theory introduces two anisotropic Q matrices,which control the attenuation anisotropy caused by the RS and RF mechanisms respectively,into the relaxation functions containing the wave behavior information in poro-viscoelastic media.These functions are applied in the dynamic Darcy’s law,a rewritten constitutive relationship,and the fluid pressure considered as viscoacoustic.The resulting convolution operations can be replaced by the fractional time derivatives.Since the propagation and maximum attenuation directions of seismic waves are different in the attenuative porous media,the Christoffel and energy balance equations in the plane of symmetry are deduced by using a more general inhomogeneous plane wave theory with a different method of complex slowness vector.This method leads to the solution of a sixth-order algebraic equation to obtain the complex slowness of inhomogeneous fast and slow quasi-compressional(qP1 and qP2)and quasi-shear(qS)waves.Invoking the derived energy balance equation and the calculated complex slowness,we further present the explicit expressions of energy velocity,whereas the dissipation factors are provided by using different definitions(i.e.,the ratio of the average dissipated energy density to the average strain energy density or the average stored energy density).Our expressions for the inhomogeneous wave are degenerated to give their counterparts of the homogeneous wave.The reduced forms are identical to that presented by the existing poro-viscoelastic theory based on the approach of the complex wave vector.To consider a more complicated anisotropic symmetry,the theory of attenuative VTI porous media is further extended to attenuative ORA porous media,and a more practical 3-D case is considered.Similar to the derivation in attenuative VTI porous media,the basic relationships and accurate expressions of measurable quantities(phase velocity,energy velocity and dissipation factor)are derived in 3-D homogeneous unbounded attenuative ORA porous media.Finally,several numerical examples are shown to illustrate the proposed theory.The results verify that the proposed theory can effectively explain the wave behavior in a broad range of frequencies and demonstrate the significant effects of the degree of attenuation anisotropy and inhomogeneity on velocity dispersion and attenuation.Such a study is necessary for non-planar wavefield simulation in heterogeneous media,and is helpful to comprehensively understand the phenomenon of seismic wave propagation in complex reservoir media with anisotropy and viscoelasticity.