# Acoustic Propagation Characteristics of Unsaturated Porous Media Containing CO\({}_{2}\) and Oil
## 2 Fundamental Theory
\begin{split}\mathbf{\rho}&=\left[\begin{array}{ ccc}\rho_{s}(1-\phi)&0&0\\ 0&\rho_{1}S_{1}\phi&0\\ 0&0&\rho_{2}S_{2}\phi\end{array}\right],\mathbf{M}=\left[\begin{array}{ ccc}a_{11}+N/3&a_{12}&a_{13}\\ a_{12}&a_{22}&a_{23}\\ a_{13}&a_{23}&a_{33}\end{array}\right],\mathbf{N}=\left[\begin{array}{cc}N&0 &0\\ 0&0&0\\ 0&0&0\end{array}\right],\\ \mathbf{A}&=\left[\begin{array}{ccc}-A_{11}-2A_{12}-A_{22}&A_{11}+A_{12}&A_{ 12}+A_{22}\\ A_{11}+A_{12}&-A_{11}&-A_{12}\\ A_{12}+A_{22}&-A_{12}&-A_{22}\end{array}\right],\\ \mathbf{R}&=\left[\begin{array}{ccc}-R_{11}-R_{22}-R_{12}-R_{21}&R_{11}+R_{21 }&R_{12}+R_{22}\\ R_{11}+R_{12}&-R_{11}&-R_{12}\\ R_{21}+R_{22}&-R_{21}&-R_{22}\end{array}\right],\\ &\text{where }\rho_{l}\text{ denotes the density of phase }l,l=s, 1, 2; \phi\text{ is the porosity; }S_{1}\text{ is the saturation of nonwetting fluid, }S_{2}\text{ is the saturation of wetting fluid, and }S_{2}=1-S_{1};N\text{ is the shear modulus; }A_{11}\text{ denotes the inertial coupling between solid grain and nonwetting fluid, }A_{22}\text{ denotes the inertial coupling between solid grain and wetting fluid, and }A_{12}\text{ is the inertial coupling between nonwetting fluid and wetting fluid; }R_{11}\text{ represents the viscous coupling between solid grain and nonwetting fluid, }R_{22}\text{ denotes the viscous coupling between solid grain and wetting fluid, }R_{12}\text{ and }R_{21}\text{ are the viscous coupling between nonwetting fluid and wetting fluid; }a_{ij}\text{ (}i,j=1, 2, 3\text{) denotes the elastic coefficients, which are derived from the closure relation for porosity change, the balance of mass, and the equation of state, as shown in Appendix A. We list \(A_{ij}\), \(R_{ij}\), and \(a_{ij}\) in Appendix A, which are functions of rock parameters (\(\phi\), \(\kappa\), \(K_{s}\), \(N\), \(K_{m}\), \(\rho_{s}\)) and fluid parameters (\(K_{1}\), \(K_{2}\), \(\rho_{1}\), \(\rho_{2}\), \(S_{1}\), \(S_{2}\)).  
According to the Helmholtz theorem, applying the divergence and rotational operators to the motion equation (Equation (1)), we can obtain the governing equations of compressional and shear waves:  
\[\begin{split}\mathbf{\rho}\widetilde{\mathbf{\varphi}}+\mathbf{A} \widetilde{\mathbf{\varphi}}+\mathbf{R}\dot{\mathbf{\varphi}}=(\mathbf{M}+ \mathbf{N})\
abla^{2}\mathbf{\varphi},\\ \mathbf{\rho}\widetilde{\mathbf{\psi}}+\mathbf{A}\widetilde{\mathbf{ \psi}}+\mathbf{R}\dot{\mathbf{\psi}}=\mathbf{N}\
abla^{2}\mathbf{\psi}.\end{split} \tag{3}\]  
where \(\mathbf{\varphi}=\left[\sigma^{s},\varphi^{1},\varphi^{2}\right]^{T}\), \(\varphi^{l}=\
abla\cdot\overset{\rightarrow l}{u}\) represents the potential functions of compressional waves, \(l=s, 1, 2\); \(\psi=\left[\overset{\rightarrow\rightarrow\rightarrow}{\psi}\overset{ \rightarrow\rightarrow}{\psi}\overset{\rightarrow\rightarrow}{\psi}\right]^{T}\), \(\overset{\rightarrow l}{\psi}=\
abla\overset{\rightarrow l}{u}\) represents the potential functions of shear waves, \(l=s, 1, 2\). The definition of potential functions in porous media also refers to Refs. [7; 12; 25].  
Next, the governing equations (Equation (3)) are subjected to the steady-state analysis, i.e., the single-frequency plane waves (Equation (4)) are substituted into the governing equations (Equation (3)). The final governing equations of compressional and shear waves can be written as Equation (5).  
\[\begin{split}\begin{pmatrix}\varphi^{s}\\ \varphi^{1}\\ \varphi^{2}\end{pmatrix}=\begin{pmatrix}\alpha^{s}\\ \alpha^{1}\\ \alpha^{2}\end{pmatrix}\exp\bigl{[}i\bigl{(}\mathbf{k_{p}}\cdot\mathbf{r}- \omega t\bigr{)}\bigr{]},\begin{pmatrix}\overset{\rightarrow s}{\psi}\\ \overset{\rightarrow l}{\psi}\\ \overset{\rightarrow l}{\psi}\\ \overset{\rightarrow}{\psi}\end{pmatrix}=\begin{pmatrix}\overset{ \rightarrow s}{\beta}\\ \overset{\rightarrow\rightarrow\rightarrow}{\beta}\\ \overset{\rightarrow}{\beta}\end{pmatrix}\exp[i(\mathbf{k_{s}}\cdot\mathbf{r}- \omega t)],\end{split}\end{split}