GNGTS 2016 - Atti del 35° Convegno Nazionale

GNGTS 2016 S essione 3.3 595 Velocity model. We now need to compute the seismic velocities and attenuations of the partially saturated medium. To do this, we use White’s mesoscopic rock-physics theory (White, 1975), which is well described in appendix A of Carcione et al. (2012). White’s model appro- ximates the partially saturated medium by considering patches of CO 2 in an otherwise brine- saturated medium; more precisely, he considers an outer sphere of radius r 1 > r 0 saturated with brine and an inner sphere of radius r 0 saturated with gas. Therefore, saturation can be defined as S g = r 0 3 /r 1 3 . White’s mesoscopic model allows to compute the p-wave phase velocity as a function of frequency and attenuation factors in an isotropic medium. In fact, the model allows to compute the complex bulk modulus as a function of porosity, permeability, gas saturation, clay content and fluid viscosity, knowing the high frequency bulk modulus when there is no fluid flow between the patches. Then, a complex phase velocity can be computed, the real and imaginary parts are the phase p-wave velocity and attenuation factor respectively. In fact, the complex bulk modulus can be computed, knowing the high frequency bulk modulus when there is no fluid flow between the patches and the permeability of the medium. Furthermore, the density is given by an average of the densities weighted over porosity and clay content for the solid part and over saturation for the fluid components. White does not provide a mesoscopic model for shear deformations. Therefore, we assume that the complex shear mo- dulus is described by a Zener element having a peak frequency. Practically, we assume that the stiffer the medium, the higher the quality factor. Actually, this is not a bad approximation, since, as shown by Picotti et al. (2010), White’s model is consistent with Zener’s and the computed velocities differ by less than 5% in this simulation. Finally, we consider the grains forming the rocks to be a mixture of quartz and clay. The presence of the latter changes the value of the effective bulk and shear moduli of the rock. We follow Hashing and Strickman’s (1963) variational approach and take the arithmetic average of the upper and lower bounds. This model confirms the empirical evidence that p-wave velocity decreases with increasing gas partial saturation, until a threshold value (depending on the clay content) is reached. After that, fluid density effects prevail and we observe an increase in velocity. As for the S-waves, we record an increase in velocity with increasing gas saturations, In Figs. 2c and 2d, we show respectively the relaxed (at f 0 ) p- and s- wave velocities as computed with White’s model, while in (e) and (f) one can see the quality factors for p- and s- waves respectively. Seismic modeling. The synthetic seismograms are computed with a modeling code based on an isotropic and viscoelastic stress-strain relation. The equations can be found in Section 3.9 of Carcione (2015). The attenuation is described by the standard linear solid, also called the Ze- ner model, that gives relaxation and creep functions in agreement with experimental results. Simulations are run using a 315x315 staggered grid. 99 shots were computed, the shallowest of which is at -750 m with a source spacing of 5 m. An array of 315 receivers is placed in the other well, covering an entire column with a receiver spacing of 2.5 m. The dominant source frequency is set to 80 Hz, with a Ricker wavelet as source time history. The time step is set to 0.1 ms. The algorithm is based on a fourth-order Runge Kutta method to calculate the synthetic seismograms recursively in time, while spatial derivatives are computed with a Fourier pseudo- spectral method. The receivers were set to start recording when the source wavelet reaches the maximum, in order to have zero-phase. Therefore, to perform a tomographic inversion of the direct arrivals, we picked the maximum amplitudes of the first break in the seismogram. In Fig. 3 a sample shot is shown, with the evident anomalies in the velocity of the first arri- vals. The source time history is also displayed. Tomographic inversion. The last step of this work consists of a tomographic inversion of the first arrivals (direct waves) of the synthetic seismograms using CAT3D software (see, for