# Acoustic Propagation Characteristics of Unsaturated Porous Media Containing CO\({}_{2}\) and Oil
## 4 Fluid Parameters Model Correction  
### Density Correction for the CO\({}_{2}\)-Oil Mixture  
[PERSON] developed an empirical model for the density of the gas-oil mixture [30], by introducing the effective density of the gas. On the basis of [PERSON]'s work, we proposed a density model for the CO\({}_{2}\)-oil mixture, through the ideal mixing rule and the effective density of CO\({}_{2}\) (Saryazdi [31]). The model is expressed as follows:  
\[\rho_{o,CO2}=\frac{\rho_{0}+W_{CO2}}{1+W_{CO2}/\rho_{a,CO2}}+\Delta\rho_{P,T}+ \Delta\rho_{X_{CO2}}, \tag{12}\]  
where \(\rho_{a,CO2}\) denotes the effective density of CO\({}_{2}\); \(W_{CO2}\) denotes the dissolved mass of CO\({}_{2}\); \(\Delta\rho_{P,T}\) and \(\Delta\rho_{X_{CO2}}\) are temperature-pressure and solubility correction factors, respectively. The above parameters can be written as [30; 31]  
Figure 5: The sensitivities of velocities of P1 and S to CO\({}_{2}\) saturation, CO\({}_{2}\) saturation of 0.2, 0.4, 0.6, 0.8 are considered: (**a**) P1 velocity; (**b**) S velocity; (**c**) P1 attenuation; (**d**) S attenuation.  
\[\begin{array}{ll}\rho_{a,\text{CO2}}&=2.3349\exp[-0.003157(T+273.15)]\cdot\exp\{4.12\times 10^{-4}P+\\ &[-0.3233+0.001897(T+273.15)]\big{[}1-\exp\big{(}-1.37\times 10^{-2}P\big{)}]\big{\}},\\ W_{\text{CO2}}&=0.001223*GOR*G,\\ \Delta\rho_{P,T}&=\Delta\rho_{P}-\Delta\rho_{T}+b_{00}+b_{01}T+b_{02}T^{2}+(b _{10}+b_{11}T)P,\\ \Delta\rho_{X_{\text{CO2}}}&=b_{20}+b_{21}X_{\text{CO2}},\end{array} \tag{13}\]  
where \(P\) is the reservoir pressure; \(T\) is the reservoir temperature; \(\Delta\rho_{P}\) and \(\Delta\rho_{T}\) are given in Han [30]; \(b_{ij}(i,j=0,1,2)\) is the fitting parameter, obtained from the experimental data in Calabrese [32] and given in Table 2; \(X_{\text{CO2}}\) denotes the mole fraction of CO2 in crude oil.  
Figure 6 depicts the measured values (Calabrese [32]) of density in the CO2-oil mixture, compared with the density of [PERSON]'s model [24] and the modified model in this paper. The correlation coefficient of [PERSON]'s model is 0.7548, and the modified model is 0.9807. As shown in Figure 6, the predicted densities resulting from the modified model are generally consistent with the experimental data. The modified model achieved better application results.  
To verify the applicability of the modified model, we compare the predicted densities with measured densities in different oil samples, as shown in Figure 7. Due to the differences in oil samples, environment, and other factors, the model prediction results may be slight deviations from the experimental data. In this case, we could fit a correction for \(b_{ij}\) by Table 2 and the experimental data. The solid lines in Figure 7 are the results of the corrected model. In summary, compared with the density model of [PERSON]'s model, the modified model for live oil's density has a more efficient physical meaning and considers the effect of CO2 solubility. The modified model can be obtained by simple coefficient correction for a certain oil sample, which has better applicability.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \(b_{ij}\) & \(j=0\) & \(j=1\) & \(j=2\) \\ \hline \(i=0\) & \(4.034\times 10^{-3}\) & \(1.319\times 10^{-4}\) & \(-5.814\times 10^{-7}\) \\ \(i=1\) & \(-7.016\times 10^{-5}\) & \(-1.286\times 10^{-6}\) & \\ \(i=2\) & \(-5.85\times 10^{-4}\) & \(2.477\times 10^{-2}\) & \\ \hline \hline \end{tabular}
\end{table}
Table 2: Fitting correction coefficients of temperature–pressure and solubility correction factors in Equation (13).  
Figure 6: Comparison of model results with experimental data (Calabrese [32]), \(X_{\text{CO2}}=0.2\) (model fractions): (**a**) [PERSON]’s model; (**b**) the modified model.