# Acoustic Propagation Characteristics of Unsaturated Porous Media Containing CO\({}_{2}\) and Oil
## 3 Numerical Analysis  
### Wave Modes Characteristics  
According to the results in Section 2, we calculated the propagation characteristics of acoustic waves at different CO\({}_{2}\) saturations and frequencies, including phase velocities and attenuations in the porous media. The physical parameters of the porous media are presented in Table 1. In particular, we consider the dissolution effect on the live oil parameters, given by [PERSON]'s model [23]. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Type** & **Parameters** & **Value** \\ \hline \multirow{7}{*}{Utsira sand} & grain bulk modulus, \(K_{\text{s}}\) (GPa) & 40 \({}^{1}\) \\  & frame bulk modulus, \(K_{\text{m}}\) (GPa) & 1.37 \({}^{1}\) \\ \cline{1-1}  & shear modulus, \(N\) (GPa) & 0.82 \({}^{1}\) \\ \cline{1-1}  & grain density, \(\rho_{s}\) (kg/m\({}^{3}\)) & 2600 \({}^{1}\) \\ \cline{1-1}  & porosity, \(\phi\) & 0.36 \({}^{1}\) \\ \cline{1-1}  & permeability, \(\kappa\) (D) & 1.6 \({}^{1}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Physical parameters of unsaturated porous media. We first examine the impact of CO\({}_{2}\) dissolution on the pore fluid's phase state. Following [PERSON]'s work [17], we assume that when the CO\({}_{2}\) content in the pore is greater than the maximum CO\({}_{2}\) dissolution of crude oil, there are two phase fluids in the pore (live oil and free CO\({}_{2}\)); otherwise, the pore contains one type of fluid (live oil). It can be written as [17]  
\[S_{1}^{\prime}=\left\{ \begin{array}{l}0,S_{1}<S_{c}\\ \frac{S_{1}-S_{c}}{1-S_{c}},S_{1}\geq S_{c}\end{array} \right., \tag{9}\]  
where \(S_{1}{}^{\prime}\) denotes the actual CO\({}_{2}\) saturation after CO\({}_{2}\) dissolution; \(S_{1}\) denotes the saturation of injection; \(S_{c}\) denotes the critical saturation, calculated by the maximum dissolution of crude oil, which can be written as  
\[S_{c}=\frac{\rho_{CO2s}\rho_{oil}GOR}{\rho_{CO2}\rho_{0}+\rho_{CO2s}\rho_{oil }GOR}, \tag{10}\]  
where \(\rho_{CO2s}\) and \(\rho_{CO2}\) are the density of CO\({}_{2}\) at the surface (15.6 \({}^{\circ}\)C and 1 atm) and reservoir, respectively; \(\rho_{0}\) and \(\rho_{oil}\) are the density of crude oil at the surface and reservoir, respectively; \(GOR\) is the gas-oil ratio at the standard condition, which can be calculated by Emera [28]. According to the above description, we calculated the phase velocities and attenuations as functions of frequency and CO\({}_{2}\) saturation. The results are shown in Figures 1 and 2. Firstly, the P1 and S waves propagate fairly fast and have low attenuation. Thus, the waves can be observed at the seismic exploration (10-100 Hz) frequency. In contrast, the P2 and P3 waves propagate slowly and have high attenuation, which is difficult to be observed at the seismic exploration frequency. Secondly, as frequency increases, the phase velocities of all four mode waves increase, and the attenuations of P2 and P3 decrease. That is because the effect of inter-fluid viscosity dominates in the low-frequency range. As the frequency increases, the inertial effect becomes increasingly dominant. Obviously, due to the dissolution effect, the phase velocities and attenuations of the four mode waves all change significantly at the critical saturation (42% in this case). When CO\({}_{2}\) saturation is less than critical saturation, the pore contains only one fluid. At this time, P3 has low velocity and high attenuation in the entire frequency range, without fluid-fluid interaction. When the critical saturation is exceeded, two fluids in the porous media interact with each other, resulting in a decrease in P2 velocity and an increase in P3 velocity. In addition, with the increase in CO\({}_{2}\) saturation, the transition zone between low and high frequencies of P3 moves toward high frequency, while P2 and S move toward low frequency.