Artificial Ground Freezing (AGF) represents a widely adopted auxiliary technology utilized to mitigate groundwater infiltration and ensure the stability of excavation faces in underground construction endeavors. Notably, the hydrodynamic condition stands as the primary contributor to the non-uniformity observed in the freezing curtain. However, directly assessing the hydrodynamic condition during the construction of AGF poses a formidable challenge. In this study an moisture-heat model was initially formulated, incorporating two boundary treatment methodologies, to quantify temperature variations throughout the AGF under different hydrodynamic conditions. Given the inherent uncertainties associated with hydrodynamic conditions, a novel approach grounded in optimization theory (MHO) was proposed and integrated with the moisture-heat model. This methodology aims to ascertain the hydrodynamic condition within AGF by minimizing the summation of squared differences between calculated and monitored temperatures at selected, typical measurement points throughout the entire freezing. The proposed method was numerically resolved and subsequently validated through rigorous laboratory tests conducted by fellow researchers. The results indicate that the methodology presented in this paper offers more accurate predictions of hydrodynamic conditions; the comparison between calculated and monitored temperatures under optimized hydrodynamic conditions exhibits a significantly closer alignment than that obtained when solely considering horizontal hydrodynamic conditions.
Introduction
AGF technique has been acknowledged for its dependable sealing ability, constructional flexibility, cost- effectiveness, and robust adaptability. Within the realm of geotechnical engineering, particularly for tunnels traversing water-rich strata, the AGF technique, primarily comprising brine freezing and liquid nitrogen freezing, has been effectively utilized in soil reinforcement for the main tunnels, cross passages, rescue projects, and end wells . Notably, the construction of the Gongbei Tunnel in China incorporated this technique .
Currently, the design of the freezing tube layout is predominantly carried out under hydrostatic conditions. However, soil, as a porous medium consisting of particles of varying sizes, poses particular challenges. The development of a significant hydraulic gradient within the soil, whether induced by artificial precipitation or natural phenomena, can result in heightened groundwater flow through the pores . This, in turn, may elevate the average temperature of the frozen soil mass and potentially lead to the persistence of an open freezing curtain. For example, in the former Soviet Union in 1955, the interval between Forest and Valor Square stations on Line 1 of the St. Petersburg metro was situated in proximity to an underground river, which had a replenishment effect on the groundwater layer. The high groundwater flow during the AGF construction could dissipate the majority of the cold energy, leading to the non-closure of the freezing curtain and subsequent sand inrush . Comparable engineering challenges attributed to hydrodynamic conditions were also encountered in Shenzhen in 2003, Tianjin in 2006, Shanghai in 2007, Xiamen in 2012, and the Nantong subway project in 2020. Detailed descriptions of these cases can be found in the relevant literature .
From a macroscopic perspective, the hydrodynamic factors influencing the distribution of the freezing curtain primarily encompass the velocity and direction of groundwater flow. Under these hydrodynamic conditions, an increase in groundwater velocities exacerbates the non-uniform distribution of the freezing curtain, leading to a thinner curtain upstream compared to downstream. This phenomenon can be attributed to the transfer of cold energy from upstream to downstream, a process driven by heat convection, which represents a highly complex multi-field coupling mechanism, especially when considering the thermal properties incorporated in the governing equations. Traditionally, groundwater flow has been presumed to be horizontal. In specific scenarios, such as when impermeable layers are present above and below the excavation site, this assumption holds true. However, it is important to note that not all flows are horizontal, and the direction of flow can be highly unpredictable. Recent research has highlighted the significant impact of flow direction on geotechnical engineering , yet the investigation into its influence on the formation of the freezing curtain remains inadequate. Additionally, the diameter and spacing of the freezing tubes have a substantial impact on the development of the freezing curtain , albeit this aspect lies outside the scope of the present study.
The formation of a freezing curtain under hydrodynamic conditions constitutes a sophisticated thermophysical process, encompassing internal heat sources, phase transitions, moving boundaries, and intricate boundary conditions. Currently, the primary methodologies employed to assess the evolution of the freezing curtain within such hydrodynamic contexts involve theoretical analysis, numerical modeling, and scaled experimental approaches. In the realm of theoretical research, considerable endeavors have been directed towards the development of analytical models that delineate the steady-state temperature field of typical freezing tube configurations under hydrostatic conditions . To the author's limited knowledge, only a handful of studies have taken into account the hydrodynamic conditions within a limited number of freezing tubes , primarily due to the highly nonlinear nature of the governing equations. Consequently, two numerical methodologies, which account for the hydrothermal properties of soil, have been harnessed to predict temperature variations: the apparent method and the enthalpy-porosity method . Both methods are proficient in simulating the formation of a freezing curtain under horizontal groundwater flow conditions. Nonetheless, the direction of groundwater flow has not been incorporated into the numerical models, as the boundary conditions pose a significant challenge in simulations. A comparable situation is evident in model test studies, which have only simulated conditions with horizontal groundwater flow .
In light of these limitations, it is crucial to propose a methodology for determining the hydrodynamic conditions. During AGF construction, it seems impractical to directly ascertain the magnitude and direction of groundwater flow within the frozen zones. Therefore, this study introduces an optimization approach to calculate these two variables. Presently, optimization techniques are primarily categorized into two groups: analytic gradient methods (such as SNOPT, IPOPT, and MMA) and approximate gradient algorithms (including Nelder Mead, COBYLA, and BOBYQA). In comparison to analytic gradient methods, the approximate gradient approach, renowned for its adaptability, has been extensively employed in geotechnical engineering . Additionally, the approximate gradient algorithm is particularly suited for problems involving variable geometries.
In this study, a moisture-heat coupling model, which takes into account heat conduction and convection under various hydrodynamic conditions, was first established in Section 2, based on the coupling model proposed by Li et al.. Subsequently, in Section 3, the formation process of the freezing curtain, achieved through the linear arrangement of three tubes, was investigated under different velocities and directions of groundwater flow. Furthermore, in Section 4, a MHO method was proposed to determine the hydrodynamic conditions during AGF and was validated through experiments conducted by Pimentel et al. Finally, some significant conclusions were drawn in Section 5.
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