## 2 Background Theory  
### Aspect Ratio and Crack Density  
[PERSON] [22] introduced the theory of wave propagation in porous media and defined constitutive equations for stress, strain, and elastic moduli. These variables have a linear relationship in elastic materials, as this theory assumes that the wave-propagating medium is homogeneous, isotropic, and un-cracked. In addition, pore water pressure is assumed constant in all directions. As a result, this theory has limited applicability to heterogeneous materials that may contain many cracks. This theory was then developed further to generate a new Biot-consistent theory that related explicitly to porous rocks, as shown in Figure 1[22; 23]. This Biot-consistent theory uses three dependent variables (stress, strain, and fluid pressure) to model a canonical medium in which porosity and crack density are isotropic. Equant porosity (\(\phi_{\text{ep}}\)) and crack density (\(\varepsilon\)) are then defined in terms of the shear modulus (G), as follows:  
\[\text{G}=\text{G}_{\text{s}}\Bigg{[}1-\frac{\mathcal{O}_{\text{ep}}}{1-\left( \frac{2}{15},\frac{4-5\
u}{1-\
u}\right)}-\left(\frac{32}{45}.\frac{(1-\
u)(5- \
u)}{(2-\
u)}\right)\varepsilon\Bigg{]} \tag{1}\]  
where \(\text{G}_{\text{s}}\) is the shear modulus of a grain and \(\
u\) is Poisson's ratio. An alternative technique for investigating crack development in a composite material is self-consistent theory [24; 25]. The theory is generally used to model aggregates of crystals and can quantify the macroscopic properties of two-phase solid composites. This theory assumes that a load is uniformly applied to an ellipsoidal, heterogeneous material. [PERSON] [26] expanded this theory to consider polarization of fluid pressure, allowing this self-consistent theory to be applied to  
Figure 1: Relationship between Biot-consistent and self-consistent theories. \(\text{G}_{\text{s}}\) is the shear modulus of a grain; \(\text{K}_{\text{g}}\), \(\text{K}_{\text{sk}}\), and \(\text{K}_{\text{f}}\) are the mean bulk moduli of a grain, skeleton, and fluid; \(\text{V}_{\text{P}}\) and \(\text{V}_{\text{S}}\) are compressional and shear wave velocities, respectively; \(\gamma\), \(\
u\), and E are mass density, Poisson’s ratio, and Young’s modulus, respectively; and \(\rho\) and \(\varepsilon\) are aspect ratio and crack density, respectively. cracks and pores in a specific effective medium (CPEM), such as a porous-cracked rock [13]. The value of G in CPEM is defined as follows:  
\[\text{G}=2+\left[\frac{\varnothing_{\text{ep}}\text{:}15\text{:}(1-(1-\
u)}{7-5 \
u}\right]+\rho\left[\frac{16(1-\
u)}{15\big{(}1-\frac{\
u}{2}\big{)}}+32\frac{ (1-\
u)\left[\text{E}\,\pi\frac{\varepsilon}{4(1-\
u^{2})}\left(\frac{1}{K_{ \text{f}}}-\frac{1}{K_{\text{g}}}\right)\right]}{45\left[1+\text{E}\,\pi\frac{ \varepsilon}{4(1-\
u^{2})}\left(\frac{1}{K_{\text{f}}}-\frac{1}{K_{\text{g}}} \right)\right]}\right] \tag{2}\]  
where Kf and Kg are the bulk moduli of fluids and grains, respectively; E is the Young's modulus of the material; \(\
u\) is Poisson's ratio, \(\phi_{\text{ep}}\) is equant porosity; and \(\varepsilon\) is crack density. In a cracked medium, the total porosity (\(\phi\)), as shown in Equation (3) consists of an equant porosity (\(\phi_{\text{ep}}\)) and a crack porosity (\(\phi_{\text{cp}}\)). The aspect ratio (\(\rho\)), which is the ratio of the lengths of the shortest to the longest axes of an object, is used to classify porosity. The aspect ratio is >0.1 for equant porosity and \(\leq\)0.1 for crack porosity [27].