# Formation of a Freezing Wall Around a Vertical Shaft Under Localized Freezing
###### Abstract  
A localized freezing technique was proposed as an auxiliary method for retrofitting the lining of a vertical shaft. The influence of the freezing temperature, lining thickness, slot height, and slot duration on the evolution of the freezing wall in the clay layer was analyzed using a hydro-thermal numerical model. Under the baseline conditions (stratum temperature of 24 \({}^{\circ}\)C, shaft lining thickness of 2 m, and freezing temperature of -30 \({}^{\circ}\)C), the freezing wall behind the slotting zone was 0.74 m at 90 d, 1.89 m at 180 d, 2.78 m at 270 d, and 3.48 m at 360 d. The average growth rate of the freezing wall during one year was negatively linearly correlated with the freezing temperature, lining thickness, thickness, with change rates of \(-0.00033\) m\(/\)(d\({}^{\circ}\)C) and \(-0.00262\) m\(/\)(d\(\cdot\)m), respectively. Using the thickness of the freezing wall behind the slotting zone to reach 1.2 m as the slotting criterion, a freezing duration of 123 days is required under typical operational parameters. The evolution of the freezing wall was simulated for a slotting duration of 15 d with a slot height of 0.5-2.0 m and for a slot height of 1.5 m with a slotting duration of 5-20 d. The freezing walls did not melt in both schemes and expanded outward. The research findings are significant for improving freezing methods for shaft linings. vertical shaft lining; localized freezing; clay layer; freezing wall Article  
Y. Ma, Y. The vertical shaft serves as a critical pathway for deep resource extraction [7; 8], and the disaster of shaft lining cracks has not only resulted in significant economic losses but also significantly weakened the load-bearing and water-sealing capabilities of the shaft lining [9; 10; 11]. For operational shafts constructed by using artificial ground freezing (AGF), the method of structure retrofitting is usually used to prevent and treat shaft lining disasters [11; 12]. And the freezing method is primarily used as an auxiliary construction technique for treating shaft lining disasters in vertical shafts [13; 14; 15]. In recent years, research on the freezing method has mainly focused on the application of freezing holes to circulate cold liquid to obtain freezing curtains and the derived studies on the formation laws of the frozen wall [16; 17]. [PERSON] and [PERSON] pointed out that the cooling loss caused by the temperature drop in the unfrozen soil outside the frozen soil accounts for approximately 20-30% of the total cooling capacity, which should be considered when calculating the total required cooling capacity [18; 19]. Their study divided the temperature field development process of a single-row freezing pipe into two stages: the first stage involves the freezing wall expanding around individual freezing pipes, during which the freezing temperature develops logarithmically; the second stage occurs after the freezing wall intersection, where the temperature in the frozen zone develops linearly, while that in the unfrozen zone continues to follow a logarithmic curve. [PERSON] and [PERSON] conducted extensive numerical calculations and obtained the relationship between the dimensionless heat flux density and parameters such as the Fourier number, the ratio of thermal conductivity between frozen and unfrozen soil, the Kosovich number, and the dimensionless saline water temperature [20]. It was found that the ratio of specific heat between frozen and unfrozen soil is almost irrelevant to the heat flux density. A dimensionless criterion equation for the single-pipe heat flux density was derived. [PERSON] and [PERSON] et al. [21] used numerical analysis to study the development and distribution patterns of frozen walls under different spacing backgrounds. The research results showed that the temperature of frozen walls decreased with the decrease in spacing between frozen pipes, and the frozen walls became more uniform.