# Acoustic Propagation Characteristics of Unsaturated Porous Media Containing CO\({}_{2}\) and Oil
## 3 Numerical Analysis
### Sensitivity Analysis  
Numerous factors influence the phase velocities and attenuations of the four mode waves, including CO\({}_{2}\) saturation, porosity, permeability, density, and velocity of pore fluid. The sensitivity can be used to describe the effect of each parameter on the phase velocities and attenuations at different frequencies. The sensitivity can be expressed as [29]  
\[S_{p}(v)=\frac{p\partial v}{v\partial p},S_{p}\left(Q^{-1}\right)=\frac{p \partial Q^{-1}}{Q^{-1}\partial p}, \tag{11}\]  
where \(p\) denotes the media parameter; \(S_{p}\) denotes the sensitivity of \(p\); and \(v\) and \(Q^{-1}\) denote the velocity and inverse quality factor, respectively. Sensitivity greater than or less than 0 indicates that the velocities and attenuations increase or decrease with increasing parameters. The sensitivities of different parameters to the velocities and attenuations of P1 and S are shown in Figure 4. Among them, the initial value of CO\({}_{2}\) saturation is taken as 0.5, and the initial values of other parameters are taken from Table 1.  
The sensitivity of P1 and S velocities are independent of the frequency to all parameters in Figure 4. Among them, the increase in porosity increases the velocities of P1 and S, and the increase in CO\({}_{2}\) saturation decreases the velocity of P1 and increases the velocity of S. At the same time, permeability has almost no effect on the velocities of P1 and S. Moreover, the attenuations of P1 and S show the same pattern of sensitivity to pore fluid density, porosity, and permeability. Notably, the sensitivity of P1 and S attenuations to CO\({}_{2}\) saturation is higher than other parameters in Figure 4c,d, and the attenuation of P1 is more sensitive than that of S at seismic frequency. It indicates that using attenuation to monitor CO\({}_{2}\) migration is feasible. In addition, whereas the velocity of CO\({}_{2}\) and oil only  
Figure 3: Effect of CO\({}_{2}\) saturation on the velocities and attenuations of P1 and S, \(f=100\) Hz, \(P=10.3\) MPa, \(T=45\)\({}^{\circ}\)C: (**a**) phase velocities of P1 and S; (**b**) attenuations of P1 and S.  
affects the velocity and attenuation of P1 wave, the density has an impact on the velocities and attenuations of both P1 and S waves, which corresponds to the analysis in Figure 3. Obviously, the sensitivity of P1 to the pore fluid velocity is higher than that to the pore fluid density. In CGUS, CO\({}_{2}\) saturation, the ultimate inversion result of seismic monitoring, is the underlying cause of the change in acoustic propagation characteristics. Therefore, we focus on the sensitivity of CO\({}_{2}\) saturation to the acoustic response, which represents the inversion accuracy of monitoring CO\({}_{2}\) migration. Figure 5 shows the sensitivity of P1 wave and S wave phase velocities to different CO\({}_{2}\) saturations, corresponding to the change in Figure 3. With CO\({}_{2}\) saturation increases, the sensitivity of the S velocity increases, while the sensitivity of P1 velocity initially increases and then decreases. Near the critical CO\({}_{2}\) saturation, the P1 velocity is most sensitive. It indicates that the closer to the critical saturation, the more precise the inversion of P1 velocity. Further, the inversion of P1 velocity is less reliable, and the inversion of S velocity is more reliable with higher CO\({}_{2}\) saturation. Therefore, the combined inversion of P1 and S can achieve better monitoring results. Figure 4: The sensitivities of velocities and attenuations of P1 and S to different parameters, including CO\({}_{2}\) density (\(\rho_{CO2}\)) and velocity (\(v_{CO2}\)), live oil density (\(\rho_{\text{Oil}}\)) and velocity (\(v_{\text{Oil}}\)), CO\({}_{2}\) saturation (\(S_{\text{CO2}}\)), porosity (\(\phi\)), and permeability (\(\kappa\)): (**a**) P1 velocity; (**b**) S velocity; (**c**) P1 attenuation; (**d**) S attenuation. For the porous media with certain porosity, the propagation characteristics of P1 and S are related to pore fluid parameters.