Seismic wave equation formulated by generalized viscoelasticity in fluid-saturated porous media

Biot’s theory of poroelasticity describes seismic waves propagating through fluid-saturated porous media, so-called two-phase media. The classic Biot’s theory of poroelasticity considers the wave dissipation mechanism being the friction of relative motion between the fluid in the pores and the solid rock skeleton. However, within the seismic frequency band, the friction has a major influence only on the slow P-wave and has an insignificant influence on the fast P-wave. In order to represent the intrinsic viscoelasticity of the solid skeleton, we incorporate a generalized viscoelastic wave equation into Biot’s theory for the fluid-saturated porous media. The generalized equation which unifies the pure elastic and viscoelastic cases is constituted by a single viscoelastic parameter, presented as the fractional order of the wavefield derivative in the compact form of the wave equation. The generalized equation that includes the viscoelasticity appropriately describes the dissipation characteristics of the fast P-wave. Plane-wave analysis and numerical solutions of the proposed wave equation reveal that (1) the viscoelasticity in the solid skeleton causes the energy attenuation on the fast P-wave and the slow P-wave at the same order of magnitude, and (2) the generalized viscoelastic wave equation effectively describes the dissipation effect of the waves propagating through the fluid-saturated porous media.

Viscoelastic Wave Propagation Simulation Using New Spatial Variable-Order Fractional Laplacians

ABSTRACT Time-domain constant-Q (CQ) viscoelastic wave equations have been derived to efficiently model Q, but are known to break down in accuracy in describing CQ attenuation at low Q. In view of this, a new time-domain viscoelastic wave equation for modeling wave propagation in anelastic medium is evaluated based on Kjartansson’s CQ model to improve the accuracy in describing CQ attenuation at low Q. We use an approximate frequency-domain viscoelastic wave equation to replace the accurate frequency-domain viscoelastic wave equation. Then, a new time-domain wave equation is derived by converting the approximate viscoelastic wave equation from the frequency domain to the time domain. The newly derived viscoelastic wave equation consists of several Laplacian differential operators with variable fractional order. We use an arbitrary-order Taylor series expansion (TSE) to approximate the derived mixed domain fractional Laplacian operators, and realize the decoupling of the wavenumber and fractional order. Then, the proposed viscoelastic wave equation can be solved directly using the staggered-grid pseudospectral method (SGPSM). We evaluate the precision of the new viscoelastic wave equation by comparing the numerical solutions with the analytical solutions in homogeneous medium. Theoretical curve analysis and numerical results indicate that the proposed fractional viscoelastic wave equation has higher precision in describing CQ attenuation than that of the traditional fractional viscoelastic wave equation, especially for cases that P-wave quality factor QP is less than 10, and S-wave quality factor QS is less than 8. Furthermore, we use two numerical examples to verify the effectiveness of the TSE SGPSM in heterogeneous media. The discussion shows that the advantage of using our fractional viscoelastic wave equation over the traditional fractional viscoelastic wave equation is the higher precision in describing CQ attenuation at different frequency.