Fractional Viscoelastic Two-phase Wavefield Simulation And Propagation Characteristics Analysis

As the exploration and development level of oil and gas reservoirs continues to deepen,the exploration targets will gradually shift to more complex oil and gas reservoirs,and the requirements for exploration accuracy will also become higher and higher.The three mechanisms that cause changes in seismic wavefields in subsurface media include pores and fluids,anisotropy,and matrix viscoelasticity.Therefore,a comprehensive consideration of the above three mechanisms to establish a more accurate subsurface media model can better describe the subsurface media.This is of important practical significance to study and analyze the attenuation characteristics of seismic wave propagation,as well as the interpretation of actual seismic data and improve the resolution of seismic data.Fractional calculus can describe the viscoelasticity of materials more succinctly.In the field of seismic wave field numerical simulation,Kjartansson first established a completely constant Q model based on mathematical model assumptions,and quantitatively describes the absorption and dispersion relations of subsurface viscoelastic media.This paper introduces the mathematical basis of fractional calculus,and Kjartansson's regular Q-fractional model.Based on Kjartansson's constant Q theory,the viscoelastic solid skeleton constitutive relation with fractional time derivative is combined with the two-phase medium theory.A new viscoelastic two-phase medium model based on the fractional time derivative of constant Q viscoelastic constitutive relation,and the corresponding time domain wave propagation equations are deduced.Biot's theory lays the foundation for the theory of wave propagation in two-phase porous media.It can well describe the attenuation and frequency dispersion of elastic waves in porous medium containing saturated fluids.It successfully predicts the existence of the second type of P wave(slow P wave).It is the first step in the theoretical study of dual-phase porous media.Therefore,in order to comprehensively and systematically study the viscoelastic two-phase porous media in depth,this paper combines the constant Q viscoelastic solid skeleton constitutive relation with fractional time derivative and the Biot theory,and derives the constant Q fractional viscoelastic isotropic Biot-type wave propagation equation and constant-Q fractional viscoelastic VTI Biot-type wave propagation equation,macroscopically understand the physical meaning of the parameters in the viscoelastic two-phase medium model.It lays a foundation for the subsequent derivation of viscoelastic two-phase medium wave propagation under more complex mechanisms.The wave propagation equations established in this paper all contain fractional differential operators.Due to the non-locality of fractional differential operators,the wavefield values of all time nodes before the current time are needed in the calculation process.This leads to the numerical calculation and storage of fractional-wave propagation equations quite large.In this paper,three methods are used to calculate the fractional-order time derivative: global memory method,short-term memory method,and adaptive memory method.Comparing the simulation accuracy,calculation time and occupied memory of the calculation methods,the advantages and disadvantages of each method are summarized.It is considered that the adaptive memory method is a compromise between the global memory method and the short-term memory method,which is a follow-up viscoelastic two-phase reformed BISQ model forward simulation and new numerical algorithm development provide methodological reference.The Biot flow mechanism describes the macroscopic flow attenuation mechanism,while the Squirt flow mechanism describes a microscopic jet flow attenuation mechanism that actually influences the propagation and attenuation of seismic waves for a coupling process.Dvorkin and Nur(1993)proposed a BISQ model that integrates both macroscopic Biot flow and microscopic jet flow.Diallo and Appel et al.proposed the reformed BISQ model which overcomes the ambiguity in the physical meaning of characteristic jet length in the BISQ model,the difficulty to measure experimentally.This model has a large practical application prospect.In this paper,the constant Q viscoelastic solid skeleton constitutive relation with fractional time derivative is combined with the reformed BISQ model,and the constant Q fractional viscoelastic reformed BISQ theory is proposed.And then,the reformed BISQ wave propagation equation of the constant Q fractional viscoelastic isotropic media and the constant Q fractional viscoelastic VTI medium are deduced.Finally,the adaptive memory method is used for the fractional differential operator to perform the numerical simulation and characteristic analysis of the wave field in the medium with different phase boundaries,different quality factor groups,and double-layer geological structure,and get some understanding of wave propagation law in viscoelastic two-phase porous media.