## 1 Introduction  
With the continuous advancement of global oil and gas exploration and development into deeper geological formations, ultra-deep reservoirs have become a focal point of attention. Since the 1960s, significant progress has been made in deep reservoir drilling in North America, with successful commercial exploitation in regions such as the Mississippi Basin, the Gulf of Mexico, and the North Sea, where the maximum well depth has reached 10,960 m [1]. Russia has achieved major breakthroughs in the South Kara Sea, discovering world-class oil and gas fields such as the Pobeda Olifield and the Marshala Zhukova Gas Field. In addition, a deep exploratory well with a total depth of 14,600 m has been successfully drilled in the Sakhalin region [2]. Furthermore, China has made remarkable progress in ultra-deep oil and gas exploration in the Tarim and Sichuan Basins, with a significant increase in the number of wells drilled to depths exceeding 8000 m [3]. With the continuous increase in drilling depths, rocks are subjected to complex geological environments of high in situ stress and elevated temperatures, resulting in mechanical behaviors that differ significantly from those in shallow formations [4; 5]. During ultra-deep well drilling, a series of wellbore instability incidents, such as stuck pipe, lost circulation, and wellbore collapse, frequently occur and are showing an increasing trend [6; 7]. Currently, there is a lack of systematic and in-depth studies on the characterization of the mechanical properties of ultra-deep reservoir rocks. Therefore, investigating the deformation and failure mechanisms of ultra-deep formations and understanding the influence of temperature and confining pressure on rock strength parameters are essential for providing the fundamental data and a reliable strength evaluation model required for wellbore stability risk assessment during ultra-deep drilling operations. Many scholars have studied the variations in rock mechanical characteristics under different confining pressure conditions based on triaxial compression tests, revealing the impact of rock mechanical properties on drilling, hydraulic fracturing, and other engineering applications [8; 9]. In the ultra-deep high-temperature and high-pressure environment, the transition of rock behavior from brittleness to ductility becomes more pronounced. Understanding the mechanisms underlying the changes in rock mechanical properties in ultra-deep formations is key to addressing wellbore instability issues during drilling in ultra-deep strata [10; 11]. [PERSON] et al. [12] found that the mechanical behavior of granite is influenced by reservoir depth and temperature under in situ stress and temperature conditions, with depth being the primary controlling factor. [PERSON] et al. [13] investigated the effects of temperature-pressure coupling on the porosity, permeability, and mechanical properties of deep shale, revealing that high temperature significantly affects stress-sensitive pore permeability, increases rock plasticity, and exacerbates the compression of pores under high confining pressure. [PERSON] et al. [14] identified the characteristics of brittle and ductile deformation in organic-rich shale and uncovered the conditions and mechanisms for the transition from brittleness to ductility under deep high-temperature and high-confining pressure conditions. Rock strength theory primarily focuses on the relationship between the stress state and rock strength parameters, aiming to define the ultimate capacity of rocks to resist deformation and failure and to develop models for determining rock strength [15; 16]. Many scholars have conducted in-depth research in this area and proposed numerous rock failure models and criteria. Several classical rock failure criteria include the Mohr-Coulomb (MC) criterion, Drucker-Prager (DP) criterion, and Griffith criterion [17; 18; 19]. Among these, the MC failure criterion assumes a linear relationship between shear and normal stress on the failure surface.