# Modeling of Characteristics of Complex Microstructure and Heterogeneity at the Core Scale
## 3 Results
### Pore Structure Parameter Analysis  
[PERSON] [48] proposed the following empirical formula for the elastic modulus and pressure:  
\[\frac{1}{K_{d}(P)}=\left(\frac{1}{K_{d}(0)}-\frac{1}{K_{h}}\right)e^{-\frac{p}{P} }+\frac{1}{K_{h}}, \tag{43}\]  
\[\frac{1}{G_{d}(P)}=\left(\frac{1}{G_{d}(0)}-\frac{1}{G_{h}}\right)e^{-\frac{p}{P} }+\frac{1}{G_{h}}, \tag{44}\]  
where \(K_{d}\) and \(G_{d}\) represent the elastic modulus of the dry core; \(K_{d}(0)\) and \(G_{d}(0)\) are the initial moduli of the dry rock at zero pressure; and \(K_{h}\) and \(G_{h}\) are obtained by fitting the measured results of the dry core and represent the elastic modulus when all the micropores are closed under high pressure (shown by the black arrow), as shown in Figure 10.  
In samples 24, 25 and 132, the modulus increases greatly when the pressure is below 15 MPa, indicating that there are many microcracks with small aspect ratios and many microcracks with large aspect ratios in these two samples. When the pressure is slightly increased, many microcracks are closed, and the modulus increases rapidly. After reaching 15 MPa, most of the microcracks are closed, and the modulus decreases with increasing pressure. With the closure of a few unclosed microcracks with large aspect ratios, the high-pressure modulus \(K_{h}\) is finally reached. In contrast, for samples 26 and 30, the modulus increases more gradually with pressure, which means that the sample has less variation in the number of microcracks with different aspect ratios. Such cores are generally denser than cores containing microfractures with large differences in aspect ratios. Notably, in Figure 10d, the curve fitting error is much larger, and the modulus before 30 MPa is nearly linear with increasing pressure. Because there are large vertical cracks in sample 30, the pressure increases until the cracks close, and the microcracks are not the main cause of the modulus change. Figure 10f shows a comprehensive comparison of the bulk modulus change at 0 MPa for several samples, where samples 24, 25 and 132 show the same pattern of change, i.e., a sharp increase in the modulus in the low-pressure range, which tends to flatten at high pressures. The rates of change increase in the order 25, 24, and 132. In contrast, the modulus changes of samples 26 and 30 are more moderate. Figure 10: Measured (hollow point) and fitted (curve) bulk moduli \(K_{d}\) and shear moduli \(G_{d}\) of samples (**a**) 24, (**b**) 25, (**c**) 26, (**d**) 30, and (**e**) 132 and (**f**) comprehensive comparison of \(K_{d}\) with respect to pressure. By using the inversion process of the pore aspect ratio described in the Methods section, we obtained the cumulative microfracture density distribution \(\tilde{\tau}\) of five cores under effective pressures of 0 MPa, 5 MPa, 10 MPa, 15 MPa, 20 MPa, 25 MPa, 30 MPa, 35 MPa, and 400 MPa and performed a comprehensive comparison for all the samples at 0 MPa, as shown in Figure 11.  
As shown in Figure 11, the cumulative crack density decreases with increasing pore aspect ratio. \(\alpha\) represents the aspect ratio of the smallest microcrack in the sample, and the cumulative crack density \(\tau\) is the sum of the densities of microcracks with aspect ratios greater than or equal to \(\alpha\). Therefore, the cumulative densities of microcracks \(\tilde{\tau}\) that are larger than \(\alpha\) decrease as the smallest aspect ratio \(\alpha\) increases. With increasing pressure, the \(\tau\) curve moves toward the direction of a smaller aspect ratio. As mentioned previously, under pressure \(P\), the reduction in the aspect ratio of all microcracks is the same, indicating that as microcracks with smaller aspect ratios close, microcracks with larger aspect ratios are also compressed, leading to a decrease in aspect ratio. Under a high effective pressure, the microcracks with initially larger aspect ratios become microcracks with smaller aspect ratios until \(P\to P_{h}\), at which point all the microcracks close.