A Comparative Study of Viscoelastic Rheological Models Using Finite-Difference Method and an Evaluation of the Seismic Attenuation in the Búzios Field Data

Abstract

Viscoelastic seismic modeling is essential to incorporate intrinsic attenuation effects in the wave propagation. Attenuation and dispersion can be quantified in terms of the quality factor Q. The consideration of these phenomena is crucial to improve the seismic processing and imaging, mainly in very deep reservoirs, having a thick salt layer above it. This work aims to evaluate the seismic attenuation effects through the 3D finite-difference viscoelastic seismic modeling using the three classical rheological models - Maxwell (M), Kelvin-Voigt (KV) and Standard Linear Solid (SLS). These viscoelastic models are analyzed comparatively considering the seismic attributes: waveform, amplitude and phase spectra. In addition, seismic attenuation of the real seismic data from Búzios field are evaluated through a frequency analysis. Synthetic seismograms of each rheological model are generated for a multi-layered geological model considering two approaches of Q distribution: a constant Q for the entire model and different Q values per layer. The amplitude spectra of real data confirmed that the shallowest post-salt package is more dissipative than deeper regions. The analysis of synthetic data shows that KV and SLS models proved to be effective to mimic seismic wave propagation at Búzios field. Thus, the KV model can be a good approach to simulate seismic waves in anelastic media due to its lower computational cost when compared to SLS. Our results also suggest that a Q distribution per layer can yield a seismic attenuation behavior similar to that observed in the seismic data of the Búzios field.

Appendix A Analytical Solution for Homogeneous Viscoelastic Media

The viscoelastic analytical solution is obtained using the correspondence principle, which take into account the elastic solution (Carcione, 2015). The solution presented here is for a force acting in the positive z-direction and an homogeneous medium. Pilant (1979) derives the equation, and its simplification can be found in Gosselin-Cliche and Giroux (2014):

$$\begin{aligned} {{\textbf {W}}} =&\displaystyle \frac{F(\omega )}{4 \pi \rho R^{5} \omega ^{2}} \left\{ \begin{array}{c} xz\left[ \begin{array}{c} (R^{2}k_{p}^{2} - 3 - 3iRk_{p})e^{-ik_{p}R} +\\ (3 + 3iRk_{s} - R^{2}k_{s}^{2})e^{-ik_{s}R} \end{array} \right] \hat{{{\textbf {i}}}} + \\ \\ yz \left[ \begin{array}{c} (R^{2}k_{p}^{2} - 3 - 3iRk_{p})e^{-ik_{p}R} +\\ (3 + 3iRk_{s} - R^{2}k_{s}^{2})e^{-ik_{s}R} \end{array} \right] \hat{{{\textbf {j}}}} +\\ \\ \left[ \begin{array}{c} (x^{2} + y^{2} - 2z^{2})(e^{-ik_{p}R} - e^{-ik_{s}R} ) +\\ (z^2 R^2 k_{p}^{2} + i((x^{2} + y^{2} - 2z^{2})Rk_{p})e^{-ik_{p}R} + \\ ((x^{2} + y^{2})R^2 k_{s}^{2} - i(x^{2} + y^{2} - 2z^{2})Rk_{s})e^{-ik_{s}R} \end{array} \right] \hat{{{\textbf {k}}}} \end{array} \right\} \ \ \ \end{aligned}$$

(A.1)

The correspondence principle acts by substituting viscoelastic (complex) wavenumbers $k_{p}$ and $k_{s}$, contained in the complex modulus. Thus, the complex modulus of the Maxwell (Eq. 8), Kelvin-Voigt (Eq. 17) and SLS (Eq. 22) model are used in order to simulate each analytical solution.