## 1 Introduction  
With the continuous growth of global energy demand, the development and utilization of clean energy have become a crucial part of China's energy strategy [1, 2]. Hot dry rock (HDR) geothermal energy is a clean, renewable energy source with great development potential [3, 4, 5]. In the actual development of HDR geothermal resources, fractures formed by natural or artificial fracturing in reservoirs are not only the primary channels for fluid storage and migration but also the critical media for heat transfer and exchange. The temporal and spatial evolution characteristics of heat transfer and distribution in fracturedrock mass are complex. Therefore, it is urgent to study the heat transfer characteristics within fractured rock mass, which is of great significance for geothermal development. Since the 1990s, researchers have explored heat transfer mechanisms in fractured rocks through both experimental and simulation studies [6; 7; 8], with a predominant focus on single-fracture systems. Extensive research has demonstrated that heat transfer between flowing fluids and fractured rock masses can be described using either local thermal equilibrium (LTE) or local thermal non-equilibrium (LTNE) theories. The LTE theory assumes that the fluid and the fracture walls share the same temperature, an assumption that, while suitable for numerical simulations, deviates from real-world conditions. In contrast, the LTNE theory posits a temperature difference between the fluid and the fracture surface, with heat exchange governed by [PERSON]'s law of cooling [9; 10; 11; 12]. More recently, researchers have advanced the study of heat transfer in fractured rock masses by developing a range of mathematical models [13; 14; 15], often in conjunction with numerical simulation techniques. Specifically, [PERSON] et al. [16] defined connectivity coefficients to quantify fracture network connectivity. They investigated the effects of fracture connectivity, discontinuity, and injection-production pressure difference on EGS extraction performance, verifying that the connected area ratio (Rs) serves as an effective indicator of fracture network connectivity levels. [PERSON] et al. [17] employed the finite element method to study flow and heat transfer processes in fracture networks, incorporating stress-induced anisotropic aperture fields. Their analysis evaluated how different stress regimes influence thermal-fluid transport, identifying nonlinear normal deformation as the primary factor governing flow and heat transfer. [PERSON] et al. [18] proposed a dynamic fracture aperture model accounting for thermal stresses, implementing this methodology in EGS simulations to conduct parameter sensitivity analysis of mass-heat transfer processes. Their results demonstrate that fracture geometry and dip angle significantly impact mass and heat transfer within reservoirs. [PERSON] et al. [19] introduced a method for determining the representative elementary volume (REV) of three-dimensional thermal conductivity tensors and heat transfer in fractured rock masses, examining the influence of key fracture parameters (density, disc diameter, and thermal contact resistance) on heat transfer characteristics. [PERSON] et al. [20] developed a two-dimensional model to investigate thermal energy transfer in fracture-rock matrix systems, evaluating the spatiotemporal temperature distribution. Their findings reveal that lateral heat conduction within the rock matrix is a critical parameter governing the spatiotemporal temperature evolution in both fractures and rock matrices. In studies examining the influence of rock fractures on fluid flow and heat transfer, scholars typically focus on factors like fracture quantity [21], surface roughness [22; 23; 24], and contact ratio [25; 26], among others. Additionally, in a pioneering study in 2000, [PERSON] [27] introduced the concept of utilizing supercritical CO\({}_{2}\) as a heat transfer fluid for geothermal exploitation.