Analysis of approaches to the calculation of groundwater filtration flows in modeling artificial frozen walls formation

Three approaches to the calculation of groundwater filtration flows occurred during the formation of frozen walls (FW) in mine shafts under construction are considered. The first ‘simplified' approach is based on the assumption that the filtration rate depends only on the volumetric ice content. The second ‘analytical' approach implies that the frozen zone is formed and retains the shape of a circular cylinder in the course of time, therefore the filtration can be calculated from the classical analytical expressions describing the fluid flow around a circular cylinder. In the third ‘numerical' approach, the Darcy and mass balance equations are used. Owing to its formulation, the numerical approach is most accurate, but it takes more time to implement it. The paper compares the results obtained by the simplified and analytical approaches with the values of the numerical solution taken as a reference solution. The purpose of such an analysis is to find a simplified method for calculating the filtration velocity field, which will make it possible to carry out calculations most quickly and with an acceptable error relative to the reference solution. A computational domain of the frozen wall formation problem is a circular area of the ground layer with a freezing contour around the designed mine shaft. The model freezing circuit consists of eight freeze pipes. The temperature field is calculated by solving the convection-diffusion problem in the enthalpy formulation, taking into account the phase transformations of moisture. Numerical simulation is carried out in polar coordinates by means of the finite difference method. Approximation of differential equations in space is performed using the second-order central scheme, in time - the explicit first-order Euler scheme. The obtained results demonstrate that the simplified approach to the calculation of filtration can be applied only to a narrow range of freezing times and filtration rates. The analytical approach gives small differences in the results obtained when compared with the numerical approach, but the analytical approach needs to be improved in case of deviation of the geometry of the frozen zone from a circular cylinder during prolonged freezing and high groundwater filtration rates.