84 / 2023-06-14 03:25:28

Development of a Modified Perturbation Solution to the Stefan Problem for Artificial Ground Freezing

artificial ground freezing,underground mine,mathematical model,Monte-Carlo method,stefan problem

Artificial ground freezing (AGF) has evolved as a commonly used technique for ground stabilization in permafrost-affected underground mines, tunnels, and construction projects (Alzoubi et al., 2019). Ground integrity maintenance is a vital aspect in guaranteeing the safety and stability of these installations. AGF was first used to sink a shaft in a coal mine, but it has since acquired worldwide recognition and approval as a dependable and effective ground-support system in a variety of mining, civil, and environmental projects (Harris, 1995). This technology is one of the most advantageous geotechnical-support approaches due to its compatibility with various ground types and low environmental impact. Over the last few decades, there has been an increasing interest in investigating complicated processes in AGF using experiments or mathematical modeling.



The Stefan problem employs one or more partial differential equations (PDEs) and a moving boundary condition capable of tracking the nonlinear interface during phase transition (Gupta, 2017). In the context of AGF, the Stefan problem is critical in understanding and predicting the behavior of the frozen ground. Since exact solutions are restricted to specific configurations and boundary conditions, the perturbation method (commonly employed to solve the Stefan problem) assumes a small Stefan number to establish a perturbation series. However, higher-order terms in the series may deviate from the true values, compromising the accuracy of the solution. In addition, the accuracy of existing perturbation solutions for the Stefan problem is frequently limited due to the perturbation method's requirement of a small Stefan number (Ste) (Xu et al., 2020).



To overcome this limitation, we propose a novel approach using a Monte-Carlo simulation-based correction term in conjunction with the perturbation method. The proposed correction term is a dimensionless factor derived by minimizing the difference between the zeroth-order perturbation solution and the numerical solution obtained using the enthalpy method, which has been proven to be reliable yet associated with high computational cost (Alzoubi et al., 2018). Specifically, an annulus domain with a convective boundary at the inner surface and an insulated boundary at the outer surface is simulated, and the dimensionless form of the solution is developed, for the simplicity and versatility of AGF in practice. The results demonstrate that the modified perturbation solution exhibits a significantly improved accuracy, particularly for larger Stefan numbers (Ste>0.01). The proposed solution remains accurate and reliable in a wide range of Stefan numbers (0<Ste<1) and Biot numbers (1<Bi<1000), enhancing its applicability to various AGF scenarios. The application of the proposed solution extends to predicting the transient temperature profile and non-linear interface motion during the AGF process in underground mines and construction projects. By offering a more accurate representation of ground behavior, the improved solution contributes to enhanced safety and stability assessments for AGF projects.