# Acoustic Propagation Characteristics of Unsaturated Porous Media Containing CO\({}_{2}\) and Oil
## Appendix A  
The inertial coupling coefficients are expressed as [14]  
\[ \begin{array}{l}A_{11}=-(F_{s}-1)\rho_{1}S_{1}\phi,A_{22}=-(F_{s}-1)\rho_{2} S_{2}\phi,\\ A_{12}=-0.1\sqrt{{F_{s}}^{2}\rho_{1}S_{1}\rho_{2}S_{2}\phi^{2}},\end{array}  \tag{1}\]where \(F_{s}\) is the structure factor, which can be expressed as [36]  
\[F_{s}=0.5\bigg{(}1+\frac{1}{\phi}\bigg{)}. \tag{11}\]  
The viscous coupling coefficients are expressed as [14]  
\[ \begin{array}{l}R_{11}=-\frac{1+\xi}{2\xi}\frac{\mu_{1}}{kx_{1}}{S_{1}}^{2} \phi^{2},R_{22}=-\frac{1+\xi}{2\xi}\frac{\mu_{2}}{kx_{2}}{S_{2}}^{2}\phi^{2}, \\ R_{12}=\frac{1-\xi}{2\xi}\frac{\mu_{2}}{kx_{2}}{S_{1}}{S_{2}}\phi^{2},R_{12}= \frac{1-\xi}{2\xi}\frac{\mu_{1}}{kx_{1}}{S_{1}}{S_{1}}{S_{1}}{S_{1}}{S_{1}}{S_ {1}}{S_{1}}{S_{1}}{S_{1}}{S_{1}}{S_{1}}{S_{1}}{S_{1}}{S_{1}}{S_{1}}{S_{1}}{S_{1}} \phi^{2},\end{array}  \tag{12}\]  
where \(\xi\) denotes coupling coefficient of the fluid-fluid interface, \(\xi=1\) in this paper [37]; \(\mu_{l}\) denotes the viscosity of fluid phase \(l\), \(l=1\), \(2\); \(\kappa\) denotes the absolute permeability of porous media; and \(k_{rl}\) denotes the relative permeability of fluid phase \(l\), \(l=1\), \(2\), which can be expressed as [38]  
\[k_{r1}={S_{1}}^{\eta}\Big{(}1-{S_{2}}^{1/m}\Big{)}^{2m},k_{r2}={S_{2}}^{\eta} \Big{[}1-\Big{(}1-{S_{2}}^{1/m}\Big{)}^{m}\Big{]}^{2}, \tag{13}\]  
where \(\eta\) and \(m\) are model fitting parameters, which can be obtained by experiment, \(\eta=0.5\), \(m=0.56\) in this paper [39]. With the assumption of homogeneous media, the mass balance equations of unsaturated porous media are expressed as [13]  
\[ \begin{array}{l}\frac{\partial{\rho_{s}}(1-\phi)}{\partial t}+\rho_{s}(1- \phi)\
abla\cdot\overset{\rightarrow}{V}=0,\\ \frac{\partial{\rho_{s}}({S_{1}}\phi)}{\partial t}+\rho_{l}S_{l}\phi\
abla \cdot\overset{\rightarrow}{V}=0,l=1,2,\end{array}  \tag{14}\]  
where \(\overset{\rightarrow}{V}\) denotes the vibration velocity of phase \(l\), \(l=s\), \(1\), \(2\). According to the equation of state, the mass balance equations can be written as [13]  
\[ \begin{array}{l}\frac{\partial{p_{t}}}{\partial t}=-\frac{{K_{s}}}{1-\phi} \frac{\partial{(1-\phi)}}{\partial t}-{K_{s}}\
abla\cdot\overset{\rightarrow }{V},\\ \frac{\partial{p_{l}}}{\partial t}=-\frac{{K_{s}}}{S_{1}\phi}\Big{(}\phi\frac{ \partial{S_{l}}}{\partial t}+{S_{l}}\frac{\partial{\phi}}{\partial t}\Big{)} -{K_{l}}\
abla\cdot\overset{\rightarrow}{V},l=1,2,\end{array}  \tag{15}\]  
where \(K_{l}\) refers to the bulk modulus of phase \(l\), \(l=s\), \(1\), \(2\); \(p_{s}\) is the mean principal dilatational stress in the solid grain; \(p_{1}\) and \(p_{2}\) are the pressure in nonwetting fluid and wetting fluid, respectively. In the equations of mass balance (Equation (15)), \(\partial{S_{l}}/\partial t\) can be expressed through the capillary pressure (Equation (16)), and \(\partial{\phi}/\partial t\) can be expressed through the closure relation for porosity change (Equation (17)). \[ \begin{array}{l}\frac{\partial{S_{1}}}{\partial t}=\frac{1}{p_{c}^{\prime} }\Big{(}\frac{\partial{p_{1}}}{\partial t}-\frac{\partial{p_{2}}}{\partial t }\Big{)},\frac{\partial{S_{2}}}{\partial t}=-\frac{\partial{S_{1}}}{\partial t },\\ p_{c}^{\prime}=\frac{p_{2}\xi}{mn\chi}\Big{(}{S_{2}}^{-1/m}-1\Big{)}^{-m}{S_{2}}^ {-(1/m+1)},\end{array}  \tag{16}\]  
where \(p_{c}^{\prime}\) represents the first-order derivative of capillary pressure on the nonwetting fluid saturation; \(m\), \(n\), and \(\chi\) model-fitting parameters, \(m=1-1/n\), \(n=2.28\), \(\chi=0.69\) in this paper [39]. \[\frac{\partial{\phi}}{\partial t}=\delta_{s}\
abla\cdot\overset{\rightarrow }{V}^{s}+\delta_{1}\
abla\cdot\overset{\rightarrow}{V}^{1}+\delta_{2}\
abla \cdot\overset{\rightarrow}{V}^{2}, \tag{17}\]  
where \(\delta_{s}\), \(\delta_{1}\), and \(\delta_{2}\) are the nondimensional parameters of the closure relation for porosity change, which can be obtained by the \"unjacked experiment\" and \"jacked experiment\", which can be written as [14]  
\[