## 1 Introduction  
Based on the artificial ground freezing technique, advanced refrigeration methods are applied to transform water-rich geological strata into solidified, frozen earth, thereby combining the load-bearing capabilities of the ground and protecting subterranean infrastructure from groundwater infiltration. This approach represents a \"green\" construction method that can be applied to complex engineering conditions [1, 2]. Lately, in China, the development of coal has gradually transitioned to the western region due in large part to the abundant coalfield exploitation in Western China. The region has a straightforward geological composition and shallowly embedded coal seams, making it ideal for the application of inclined-shaft freezing techniques for coal extraction [3]. Currently, despite the extensive research into shaft freezing sinking [4; 5; 6; 7], the design principles related to the freezing sinking of inclined shafts remain unrefined. Due to the influence of the inclination angle in inclined shafts, the freezing process is more complex than that observed in vertical shafts. Therefore, directly employing vertical-shaft freezing strategies may compromise the safety and cost-efficiency of inclined-shaft freezing projects. Hence, numerous studies have investigated freezing-sinking techniques for inclined shafts. For example, aiming at the safety problem of a shaft lining in the freezing-sinking process of an inclined shaft, through a large amount of measured data, [PERSON] et al. [8] analyzed the temporal change in the strain of the shaft lining in the freezing section and optimized the freezing construction method of the inclined shaft. [PERSON] et al. [9] introduced a comprehensive method for evaluating the stress and displacement interactions between an irregular frozen wall and the adjacent ground, accounting for their mutual influences in the context of irregularities in inclined shafts. The complex freezing temperature field around inclined shafts requires thorough consideration. [PERSON] et al. [10] conducted an inclined-shaft freezing model test based on the similarity theory and examined the evolution characteristics of the freezing temperature field. Inspired by the vertical-shaft freezing theory, [PERSON] et al. [11] deduced a calculation model for an inclined-shaft frozen wall; they analyzed and verified the relationship between the frozen wall thickness and the inclination angle of the shaft axis under deep and shallow burial conditions through examples. [PERSON] et al. [12] developed a more accurate numerical calculation method for inclined-shaft freezing by combining the double-model method with the enhanced overlapping element method. [PERSON] et al. [13] devised a plane finite element calculation model that coupled the temperature and stress fields, considering shaft freezing, excavation, and thawing processes. They investigated the thermal change law of the frozen wall and the mechanical properties of the shaft wall during the defrosting process. [PERSON] et al. [14; 15] combined physical and mechanical tests, field measurements, and finite element numerical simulations to analyze the mechanical properties and temperature-field evolution of inclined-shaft frozen walls when traversing water-rich sand layers. [PERSON] et al. [16] established a large-scale three-dimensional freezing model test bench and studied the temperature field distribution law of the frozen wall of the inclined shaft under the two freezing methods of vertical hole and inclined hole. [PERSON] et al. [17] performed a similarity model test in conjunction with an actual inclined-shaft project, and the evolution law of the temperature field around inclined shafts under freezing conditions was analyzed. In summary, extensive investigations have been conducted on the freezing process of inclined shafts, primarily focusing on subterranean excavation segments. However, the freezing temperature field in the open-excavation portion of inclined shafts has rarely been analyzed.