## 1 Introduction  
Roughness affects the rock geometry, mechanics, and transport characteristics within fault zones [1]. In studying fractured seepage, the roughness of a fractured rock surface represents the degree of fracture in the surface structure and explains the mechanics and seepage characteristics. To date, three main types of methods have been used to describe fractured roughness: the raised height characterization method, the joint roughness coefficient (JRC) characterization method, and the fractional dimension characterization method [2]. In recent research, [PERSON] et al. [3] studied the impact of fracture roughness on solute transport, examining the flow rate and velocity against the fractal dimension. Meanwhile, [PERSON] et al. [4] leveraged the \(JRC\) to quantify the effect of roughness on seepage and derived expressions for this relationship.  
A roughness correction coefficient \(C\) is added to formulas for the cubic method to reflect the influence of roughness. From the \"Barton Ten Rules\" [5] put forward by [PERSON] and [PERSON] to the fraction dimension representation method developed later [6, 7], many researchers have formulated their own revised cubic laws and found different roughness correction coefficient values. The basic elements of a fractured rock mass network are a single fracture and its various parameters, such as its roughness, filling, and degree of fractured surface contact, which directly affect the fractured network and thus the permeability of the whole fractured rock mass [8; 9]. Therefore, as a coefficient that describes the fractured roughness, the roughness correction coefficient is indispensable within a corrected formula for the cubic law. Many scholars have devised formulas for calculating roughness [2; 10]. The roughness correction coefficient C varies within these but tends to be constant.  
[PERSON] et al. [11] used a tensor form to represent the equivalent permeability coefficient for a fractured rock mass, obtaining two-dimensional and three-dimensional tensor representations of permeability. Equally, by studying a seepage model, [PERSON] et al. [12] found that the seepage rates differ in fractured shale in three directions and are about three times higher in the vertical or horizontal direction. These findings led us to wonder whether the roughness correction coefficient should also be directional in a corrected formula for the cubic law and whether it can be expressed in a corresponding tensor form if so.  
To this end, this study used a high-precision 3D scanner to scan the surfaces of natural rock samples to obtain accurate rock surface data. In combining these data with high-precision data from CT scans of rough fractured rocks published online, a rough fractured model was generated, and after processing, a single fractured roughness correction coefficient was calculated. The rough fractured rock was divided into top and bottom surfaces. After processing the data, the spatial position coordinates for each point were read, and the gap width was calculated. The proportions of the largest square that could be taken from a single slit surface were recorded at a scale of 100% and then continuously reduced (from 100% to 10% in 10 groups), and the gap width and the spatial coordinate data were intercepted and read to study the effect of the sample size on the roughness correction coefficient. Rotating samples of the same size at a given angle, we intercepted the spatial coordinate data for different angle gap widths and fractured surfaces and calculated the change in the roughness correction coefficient in different directions to study the directional characteristics. Then, we put the results of calculating the correction coefficient at each angle into the polar coordinate system and applied elliptic fitting to observe the overall characteristics of directionality. According to the laws of changes in direction and scale, a relevant law for fractured roughness was obtained.