## 2.1 Fault samples

As shown in Fig. 1, the study area is located in a mining area in the southwest of Shandong Province, China. The overall geological structure in the area is mainly composed of 70 faults, with a density of 11/km2. The area is dominated by large and medium-sized faults with large drop and long extension. In order to master the specific conditions of large faults in the mining area, geological exploration holes were designed and the fault zone was exposed at the depth of 673.73m-739.12m. The normal fault with a drop of 120-140m is located in the upper part of Shanxi formation. The borehole revealed that the fault length of this section is 65.39m, the core length is 51.30m, and the core recovery is 78.45%. The core color is mainly grayish black and grayish white. The lithology is mainly sandy mudstone, medium sandstone and siltstone. Locally, it contains plant debris fossils and calcite veins. The core at the depth of 689.12m-701.32m (mainly medium sandstone, including a small amount of siltstone and mudstone) is relatively complete. The core at the depth of 701.32m-712.74m is mudstone containing plant clastic fossils and a large amount of argillaceous locally, which is close to coal seam. Below the coal seam is siltstone with cracks. The fault core in this section has the characteristics of “fracture-intact-fracture”. In addition, there are great differences in the degree of fragmentation and weathering in different layers. Therefore, the fault core is divided into upper, middle and lower sections according to the sample characteristics and depth.

## 2.2 Test methods and geometric characteristic parameters

In this study, the crack morphology of fault samples was observed by plane-polarized light microscope and scanning electron microscope. Under certain pressure and temperature conditions, the blue epoxy resin is pressed into the rock. After it is cured, the cast sheet is polished, and then placed under the plane-polarized light microscope for observation. The microstructure of fault rocks was observed by Scanning Electron Microscope - Back-scatter Electron (SEM-BSE) microscopy. The phase composition is detected by XRD. After the fault samples are tested, the diffraction results are analyzed by using the standard comparison card in Jade analysis software to obtain the X-ray diffraction analysis patterns of each sample and conduct semi quantitative analysis (Nikkhah et al. 2017).

In order to analyze the geometric characteristics of cracks in fault rocks, the optical photomicrographs are imported into the Auto CAD. The cracks that can be distinguished and recognized are identified by using polylines, and a digital generalized crack network is obtained for geometric characteristics analysis.

The research on the crack characteristics can be divided into single crack and crack network. The connectivity degree of each crack has a significant impact on the permeability of the fault. The study of single crack mainly focuses on the length, opening and roughness of crack, while the study of crack network focuses on its geometric characteristic parameters. The characterization indexes of rock crack network include crack density, connectivity, fractal characteristics of crack network and so on.

Crack density

In the study of crack density, there are many different definitions of crack density. Litorowicz (2006) defined the crack density as the ratio of the total length of cracks in each image to the image area when studying cracks in concrete structures. Leung and Zimmerman (2012) defined crack density as the total number of cracks per unit area in the study of estimating hydraulic conductivity by statistical parameters of random crack network. In the effective medium theory, cracks are usually regarded as inclusions in porous materials. The crack density is usually expressed as dimensionless area density (Bristow 1960). In this paper, the crack density is calculated by this dimensionless method:

$$\rho =\frac{1}{A}\sum _{i=1}^{n}{\left(\frac{{l}_{i}}{2}\right)}^{2}$$

1

Where \(A\) is the area of the digitized image, which is 10.63 mm2, \(n\) is the total number of cracks in image area, \({l}_{i}\) is the length of cracks.

Fractal dimension

Fractal dimension is an important parameter to describe crack distribution. At present, the main method to calculate the fractal dimension is the grid covering method (Mandelbrot 1982). The natural cracks in rocks have certain self-similarity. F is a set of bounded points on the plane, which is contained in a rectangular region. The rectangular region is divided into smaller grid that side length is ε according to a certain proportion \(r\). If the number of non-null grid is \(N\left(r\right)\), the capacity dimension \({D}_{c}\) is defined:

$${D}_{c}=-\underset{r\to 0}{\text{lim}}\frac{\text{ln}N\left(r\right)}{\text{ln}r}$$

2

Crack connectivity

The topological property of crack network means that the connection structure of each crack will not change due to the change of crack length and node position. This research method for crack connectivity can avoid the influence of scale effect, because topology pays more attention to the inherent properties of crack network (Valentini et al. 2007).

According to the topological graph theory, any crack network is composed of nodes and segments. Nodes are the points where two crack intersect or pinch out, and segments are the crack connecting two nodes. As shown in Fig. 2, nodes can be divided into X-type nodes (cross), Y-type nodes (adjacent) and I-type nodes(isolated) (Sanderson and Nixon 2018). The connectivity \(f\) of crack network can be calculated from the number of nodes of the above three types.

Saevik and Nixon (2017) compared five topological graphs used to describe the connectivity of crack network and obtained the expression of connectivity after fitting most suitable for predicting the hydraulic connectivity of variable topological crack mode:

$$\left\{\begin{array}{c}\eta =\frac{4{n}_{X}+2{n}_{Y}}{4{n}_{X}+2{n}_{Y}+{n}_{I}}\\ f=max\left(\text{0,2.94}\eta -2.13\right)\end{array}\right.$$

3

where \(\eta\) is the equivalent average connection number of each crack, \(f\) is the connectivity of crack network, \({n}_{X}\),\({n}_{Y}\) an, \({n}_{I}\) are the number of X-type nodes, Y-type nodes and I-type nodes.

Nonuniformity coefficient

(Ke et al. 2011) established a virtual internal bond model based on meso damage and simulated the dynamic process of crack generation and propagation in anisotropic rocks. The results show that the higher the uniformity coefficient is, the higher the macro mechanical strength is. Xia et al. (2021) defined the quantitative calculation method of nonuniformity coefficient based on normal distribution, which is the ratio of standard deviation of Young's modulus of rock to expectation. Because the fault rock also has nonuniformity of strength distribution and randomness of mineral distribution, the concept of nonuniformity coefficient is also applicable. However, the proportion of main minerals in rocks also has a great impact on the properties of rocks. Based on the research of Xia et al. (2021), the mineral content is increased and Equation 4 is proposed to describe the nonuniformity coefficient:

$$c=\frac{\text{max}\left({E}_{i}{\omega }_{i}\right)-\text{min}\left({E}_{i}{\omega }_{i}\right)}{\frac{\sum _{i=1}^{n}{E}_{i}{\omega }_{i}}{n}}$$

4

Where \(c\) is the nonuniformity coefficient; \({E}_{i}\) is the Young's modulus of mineral, GPa; \({\omega }_{i}\) is the content of minerals in rock.