## 1 Introduction
The theoretical framework of the GRC has been significantly extended by incorporating groundwater seepage forces, which are particularly relevant for tunnels below the water table [23]. Further advancements have been made in understanding GRC under reinforced conditions, where multiple configurations of plastic zones wereintroduced to model the mechanical response of tunnels in the presence of grouted zones, offering a versatile tool for evaluating reinforcement effectiveness [24]. A comprehensive solution to the GRC problem has been proposed by considering excavation damaged zones (EDZ), intermediate principal stresses, and strain-softening, enabling accurate predictions of stress and displacement fields around tunnels in medium-quality rock masses, where EDZ effects are pronounced [25]. Additionally, simplified methods for modelling the re-profiling response of squeezed tunnels using GRC have been introduced, linking re-profiled geometry to consistent stress and strain paths and recommending finite strain approaches for high-deformation scenarios due to the limitations of small strain theory [26]. A novel numerical algorithm for constructing GRCs of circular openings in strain-softening rock masses, utilizing a self-similarity method and the generalized Hoek-Brown failure criterion, provides a rigorous solution for predicting ground response in tunnels under varying internal support pressures [27]. Despite these advancements, research on GRC characteristics in extremely deep tunnels, such as those at depths exceeding 2000 m, remains limited. While substantial work has been done on typical tunnel conditions, including the effects of strain-softening, damage zones, and rock strength [25,28], and the influence of factors such as blocked steel sets [21], the specific behavior of the GRC in ultra-deep tunnels under high in situ stress conditions remains underexplored. Thus, further investigation into the GRCs of the surrounding rock under these extreme conditions is essential to refine tunnel design and ensure stability in deeper, more challenging environments. The GRC, which defines the relationship between support pressure and tunnel deformation, has been extensively used to predict the required support pressures and deformation behaviors in tunnels. Different models, including elastic-plastic, brittle-plastic, and strain-softening models, have been employed to represent the rock mass behavior surrounding tunnels [29,30]. These models are crucial for determining the optimal support strategy and dimensioning the tunnel support elements. Recent studies have focused on the behavior of yielding steel ribs under high deformation [22]. However, most of the existing research has not yet addressed water diversion tunnels buried deeper than 1000 m. In high-altitude, deep-buried water diversion tunnels, tunnels are often subjected to harsh conditions such as great burial depths, high ground stresses, and high internal and external water pressures. Currently, there are few similar projects available for reference in the construction of tunnels under such extreme burial depths, high external water pressures, and high ground stress conditions, and research in this area is scarce. Although a series of support design optimization measures have been successfully implemented in the construction of ultra-deep water diversion tunnels, such as the Jinping II deep-buried water diversion tunnel (with a maximum burial depth of 2525 m, maximum ground stress of 69.94 MPa, and maximum external water pressure of 10.22 MPa) [31], after several years of operation, many defects such as lining cracks and spalling still occurred, which severely threatens the integrity and stability of the tunnel, putting the safe operation of the tunnel at serious risk. It can be concluded that current support design methods do not fully capture the complex rock-support interactions under these high-stress conditions.