# Modeling of Characteristics of Complex Microstructure and Heterogeneity at the Core Scale
## 2 Methods
### Squit Flow Theory Considering Different Aspect Ratios  
Assuming that a rock is composed of stiff pores and microfractures (compliant pores), the total porosity of the rock is represented as \(\phi=\phi_{s}+\phi_{c}\approx\phi_{s}\), where \(\phi_{s}\) represents the porosity of the stiff pores, \(\phi_{c}\) represents the total porosity of the microfractures, and \(\phi_{c}\ll\phi_{s}\). The bulk modulus and shear modulus of the rock mineral matrix are represented as \(K_{g}\) and \(\mu_{g}\), respectively, while the dry skeleton elastic moduli are represented as \(K_{d}\) and \(\mu_{d}\). The permeability of the rock is represented as \(\kappa\). The bulk modulus of the modified frame \(K(P)\) and \(\mu(P)\) can be expressed as [45]  
\[\frac{1}{K(P,\omega)}=\frac{1}{K_{h}}+\frac{1}{\left(\frac{1}{K_{d}(P)}-\frac {1}{K_{h}}\right)^{-1}+\frac{1}{\phi_{c}(P)}\left(\frac{1}{K_{f}^{*}(\omega)} -\frac{1}{K_{g}}\right)^{-1}}, \tag{10}\]  
\[\frac{1}{\mu(P)}=\frac{1}{\mu_{d}(P)}-\frac{4}{15}\bigg{(}\frac{1}{K_{d}(P)}- \frac{1}{K(P)}\bigg{)}, \tag{11}\]  
where \(\phi_{c}(P)\) represents the microcrack porosity under the current effective pressure \(P\) and \(\eta\) and \(K_{f}^{*}(\omega)\) are the viscosity and bulk modulus of the pore fluid, respectively. \(K_{h}\) represents the high-pressure bulk modulus of the dry rock when all microcracks are closed. In this study, the pores obtained from the thin sections present different aspect ratios, and considering only the stiff and compliant pores is insufficient. Therefore, the improved squirt-flow model adopted in this study considers not only the squirt flow effect of the closure of compliant pores on stiff pores but also the squirt effect between compliant pores with various aspect ratios. According to the pressurization method introduced in the previous section, compared to when the pressure is zero, microcracks with an aspect ratio of \(\alpha_{1}\) close in the rock at time \(P_{1}\), where \(\alpha_{i}(P)\) is the minimum aspect ratio of microcracks under pressure \(P\). When ultrasonic waves approach before the ultrasonic wave peak arrives, the soft cracks with the minimum aspect ratio close first, and as the ultrasonic wave arrives, the microcracks close in order of increasing aspect ratio. When the rock contains only one kind of pore with aspect ratio \(\alpha_{c}\), the volume compression coefficient \(C_{p}\) of the microfracture satisfies [47]  
\[C_{p}(P)=\frac{1/K_{d}(P)-1/K_{h}}{\phi_{c}(P)}. \tag{12}\]  
The pressure gradient between the microcracks with the minimum aspect ratio \(\alpha_{1}\) and those with larger aspect ratio is generated under the action of waves, forming a local flow,  
\[\frac{1}{K_{1}}=\frac{1}{K_{2}}+\frac{\phi_{1}(p_{1})}{\frac{1}{C_{1}(p_{1})} +\left(\frac{1}{K_{f}^{*}(\omega)}-\frac{1}{K_{g}}\right)^{-1}}, \tag{13}\]  
\[C_{1}(p_{1})=\frac{1/K_{1}(p_{1})-1/K_{h}}{\phi_{1}(p_{1})}, \tag{14}\]  
where \(\phi_{1}(p_{1})\), \(\alpha_{1}(p_{1})\), and \(C_{1}(p_{1})\) are the porosity, aspect ratio, and volume compressibility coefficient, respectively, of the first group of microcracks with aspect ratio \(\alpha_{1}\). \(p_{1}\) is the closing pressure of this group of microcracks.