# Modeling of Characteristics of Complex Microstructure and Heterogeneity at the Core Scale
## 2 Methods
### Pore Structure Analysis Method  
By inverting the reduction in the density of microcracks with different aspect ratios during pressurization in the stress experiment, the microcrack density and porosity distribution of microcracks in the sample can be obtained, the flow chart shown in Figure 1. \(\Gamma(\alpha)\) represents the cumulative fracture density of all microfractures whose aspect ratio is greater than \(\alpha\). To obtain the initial aspect ratio of the microcracks, it is necessary to calculate the cumulative fracture density of microcracks. The equivalent medium theory is an important tool for linking the micropore structure with the elastic properties of rock. The equivalent medium model used in this paper is the Mori-Tanaka theory [15], whose geometric parameters use coin-shaped microcracks with small vertical and horizontal ratios [46]:  
\[\frac{K_{b}}{K_{MT}}-1=\frac{16(1-\
u_{b})(1+\
u_{b})}{9(1-2\
u_{b})}\Gamma, \tag{1}\]  
\[\frac{G_{b}}{G_{MT}}-1=\frac{32(1-\
u_{b})(5-\
u_{b})}{45(2-\
u_{b})}\Gamma, \tag{2}\]  
Figure 1: Inversion workflow of the aspect ratio distribution based on the pressurization process. where \(K_{b}\), \(G_{b}\) and \(v_{b}\) represent the bulk modulus, shear modulus and Poisson's ratio of the background matrix, respectively, and \(K_{MT}\) and \(G_{MT}\) represent the bulk modulus and shear modulus after the addition of pores, respectively. In contrast to the D-Z inversion method [44], this research determines the fracture density corresponding to each group of microcracks with varying aspect ratios by using the appropriate equivalent medium model for each group of microcracks rather than a single equivalent medium model, the selection of the background bulk modulus and shear modulus are different. During pressurization, the aspect ratios of all micropores in the rock decrease by the same amount [47]. In this study, it is assumed that at the beginning of the pressurization process \(P_{1}\), the microcracks of the group with the smallest aspect ratio (\(\alpha_{1}^{0}\)) begin to close and the aspect ratio of all microcracks decreases \(\alpha_{1}^{0}\). In this stage, compared to the rock in its original state without pressure \(P(0)\), it is equivalent to taking the rock under pressure \(P_{1}\) as the matrix and adding pores with aspect ratio \(\alpha_{1}^{0}\). (1) That is, under pressure \(P_{n}\), in Equations (1) and (2), \(K_{n+1}\), \(G_{n+1}\) and \(v_{n+1}\) replace \(K_{b}\), \(G_{b}\) and \(v_{b}\), respectively, \(K_{n}\) and \(G_{n}\) replace \(K_{MT}\) and \(G_{MT}\), respectively, and (2) the microfracture density \(\Gamma_{n}\) corresponding to the group of microfractures with the smallest porosity ratio \(\alpha_{n}^{0}\) under \(P_{n}\) is obtained by the least squares method. (3) When \(P_{n+1}\) is high and all microcracks are closed, we obtain multiple groups of microcrack densities corresponding to different aspect ratios. (4) The cumulative microfracture density \(\Gamma_{i}\) under different pressures \(P_{i}\) is obtained by summing the densities of multiple groups of unclosed microfractures with closing pressures greater than \(P_{i}\) and minimum aspect ratios greater than \(\alpha_{i}^{0}\). When the sample is in the initial state without pressure, all microfractures are open and the cumulative microfracture density \(\Gamma_{0}\) is the sum of the microfracture density of all groups. (5) The crack density of the soft pores in dry samples decreases exponentially with increasing effective pressure [48]. By fitting the obtained results, the crack density under any pressure \(P\) can be obtained:  
\[\Gamma(P)=\Gamma_{0}e^{-P}/\hat{P}, \tag{3}\]  
where \(\Gamma_{0}\) is the initial crack density at zero pressure and \(\hat{P}\) is a constant on the order of magnitude of that of the pressure \(P\).