## 1 Introduction  
Geothermal energy is a clean and environmentally friendly renewable energy source with a wide distribution range, large reserves, and long duration [1, 2, 3]. The high temperature rock mass buried 3-10 km underground is characterized by low permeability and low porosity [4]. In order to extract thermal energy from such high temperature rock mass, several countries (led by the United States) proposed the use of artificial hydraulic fracturing, which led to the development of reservoir-transforming hydro-shearing techniques to establish fracture network channels and improve the heat transfer capacity [5]. This geothermal project is often referred to as an enhanced geothermal system (EGS). The fracture surface formed by hydraulic fracturing is typically rough, which directly affects the heat transfer performance of the working medium (water or CO\({}_{2}\)) [6]. The key to extracting geothermal resources by EGS is the flow and heat exchange process of working fluid in fractures of high temperature rock mass [7]. However, when the fluid begins to flow from the injection well into the fracture, the initial direction of flow often has an angle equal to the direction of the rough fracture surface, which inevitably affects the heat exchange effect. The accurate determination of the convection heat transfer coefficient between the fluid and fracture is crucial to the optimization of the reservoir transformation and productivity predictions [8]. In addition, a clear heat transfer law within the fractures is of great significance to the establishment of EGS heat recovery models [9]. Numerous studies have attempted to quantify the roughness of fracture surfaces. For example, [PERSON] [10] proposed the concept of the wave angle and established the relationship between wave angle and fracture morphology. [PERSON] [11] proposed the concept of fractal geometry and developed a fractal dimension framework to describe the geometrical characteristics of fracture rough surfaces. [PERSON] [12] used the joint roughness coefficient (\(J\)RC) to characterize the section geometry of fracture surfaces to determine 10 fracture types with an approximate length of 10 cm. [PERSON] [13] demonstrated the fractal dimension to be the measured values of the joint roughness coefficients. Experimental and numerical simulations have demonstrated the fluid flow and heat transfer processes in fractures to be affected by many factors, including the aperture, roughness, type of fluid, injection flow rate, initial temperature of rock, and rock thermophysical parameters [14]. Early scholars used the plate model to describe the heat transfer characteristics in fracture. However, [PERSON] and [PERSON] [15; 16] pointed out that the parallel plate theory may cause flow estimation errors of 1-2 orders of magnitude. In an experimental study on the seepage heat transfer in a single fracture of granite, [PERSON] and [PERSON] [17] revealed the single fracture with a certain roughness to have a better heat transfer performance than the smooth-plate fracture. Similarly, the two-dimensional coupled heat transfer model of a single fracture established by [PERSON] [18] also demonstrated a strong correlation between the local convection heat transfer coefficient and the fracture profile. [PERSON] [19] established a self-affine rough fracture coupling model that assumes a reduced heat transfer efficiency in a rough fracture with constant temperature due to the channel effect. Research carried out by [PERSON] [20] also showed that a larger relative roughness increases the flow friction and significantly reduces the heat exchange. [PERSON] [21] investigated the influence of initial rock temperature on the heat transfer efficiency of fractures, demonstrating a positive relationship between the heat transfer rate and rock temperature. The two-dimensional single-fracture heat transfer model established by [PERSON] [22] shows that the effect of fracture surface roughness on heat transfer intensity decreases with an increasing injection flow rate.