Simulation-based approach for the optimization of ground freezing in tunneling

Artificial ground freezing is used in tunnelling for temporary ground improvement primarily to control groundwater flow and provide excavation support. The principle of ground freezing is based on freeze pipes bored into the ground where a coolant flows through the freeze pipes. Eventually, the ground freezing process converts pore water into ice by withdrawing the heat from the soil. In tunneling, artificial ground freezing is applied to form a closed arch frozen ground around the tunnel. Under the presence of high seepage flow, the formation time of the frozen arch is delayed. In some cases, it cannot be formed, leading to high construction costs or unsafe temporary frozen ground support. This study proposes a strategy for systematically reducing the freezing time through an optimal design arrangement of the freeze pipes. The strategy is based on the combination of the numerical modeling of the ground freezing process using a multifield finite element model in conjunction with a surrogate model to enable real-time prediction.


Introduction
Artificial ground freezing is frequently used to make soil watertight to control rapid groundwater flow during the construction of cross passages of tunnels or mining tunnels.Ground freezing technology involves the drilling of pipes into the ground, installation of refrigeration plants, monitoring of the temperatures on the frozen soil body, estimation of the time necessary to freeze the soil to satisfy the structural design.One key factor for successfully performing ground freezing is the pipe layout design.This study aims to determine an optimal pipe arrangement under project-specific boundary conditions.This requires finding a balance between feasibility for application in actual engineering projects and fast evaluation of several layouts.Accordingly, an automated ground freezing model generation was developed, creating several pipe layouts used to generate several simulation models.The simulation model is a multiphase thermo-hydromechanical finite element computational model for soil freezing-thawing boundary value problems [1].In addition, the finite element model generation using the surrogate modeling approach based on the combination of proper orthogonal decomposition and radial basis functions (POD-RBF).Finally, the optimal layout was determined from the real-time evaluations of possible layout candidates using the surrogate POD-RBF model.

Automatic design framework of ground freezing pipe layout for water flow control in tunneling applications
The proposed approach is based on the concept of automatic model generation tailored for parametric analysis of artificial ground freezing problems in tunneling.The automatic model generation involves the flexible generation of multiple freeze pipe layout, rapid mesh generation of each layout, and design-oriented numerical simulation of the frozen soil body formation.Figure 1 shows a schematic example of the model generation procedure.The original pipe layout, encompassing rectangular, circular, and New Austrian tunneling method (NATM) arrangements, along with the domain geometry, finite element mesh, and computed temperature distribution were included in the analysis.

Design concepts of freezing pipe layout and frozen body thickness
The frozen soil body must reach an average design temperature to establish whether the frozen body has been formed.The average design temperature is typically defined based on the strength and stiffness at a specific temperature that the frozen body must achieve to withstand sustained loads and avoid creep failure during the ground support period.The mechanical behavior of frozen soils is complex because the strength of the frozen soil changes with the temperature, ice content, and loading rate [2].
In this computational design framework, the reference pipe layout is used to create outer and inner boundaries to delimitate the size of the frozen body.Figure 2 illustrates the outer and inner boundary domains of the frozen soil body.The simulator extracts the finite elements contained in the defined boundaries and computes the average temperature of the elements at each time step.Subsequently, the average computed temperature of the frozen body is compared against the designed frozen body temperature.Figure 2 shows an example of the computational modeling based on the control of the average design temperature frozen soil body.

Surrogate modeling
Proper orthogonal decomposition (POD) is a widely used technique for representing high dimensional data using a subspace (a lower dimensional space), expressed by a basis of elements containing the primary characteristics of the given data.Combined with the radial basis functions (RBF), the POD-RBF method has been successfully employed as to generate surrogate models to enable real-time predictions to support different tasks in mechanized tunneling, such as interactive tunnel track design [3] and simulation aiding the advancement of tunnel boring machines [4].This study used the POD-RBF approach to establish a surrogate model for the real-time predictions of temperature fields corresponding to different possible pipe layouts in artificial ground freezing during the design stage of a tunnel project.
The first step in creating a POD basis is to obtain a collection of M snapshots of possible system solutions (temperature) by varying the input parameters (pipe layout).The N rows of a snapshot contain output values (temperature values at all points of the field).The collected matrix T is called the snapshot matrix with M columns and N rows.The POD basis of the matrix T is expressed as a linear transformation of the snapshot matrix and an associated matrix V : The POD basis vectors are obtained from solving an eigenvalue problem of the covariance matrix of the snapshot matrix T. Depending on the desired amount of energy retained between the original and reduced datasets, the truncated POD basis vector Φ can be formed by selecting a few initial POD modes.With respect to an arbitrary layout of the pipes (i.e., input parameters), the approximation of the output temperature field is obtained as follows: were ‫ܤ‬ represents the unknown coefficient matrix to be solved by the POD-RBF model and ‫ܨ‬ denotes the sampling input (i.e., pipe layout).It should be noted that the full matrix F is generated using an RBF function as the interpolation function to build a relationship between samples of the snapshot matrix.

Case study: parametric modeling of soil freezing for ground seepage control in tunneling
This section presents a numerical case study illustrating the proposed computational approach.The numerical case study focuses on the modeling of soil freezing for water flow control before NATM tunnel construction.For the numerical case study, a freeze pipe layout following the geometrical shape of NATM tunnel is selected.

Conclusions
A computational design framework was developed to automatically generate finite element models tailored for parametric analysis and design of artificial ground freezing problems in tunneling.The proposed framework was used to generate random pipe layouts starting from an initial reference pipe layout.Finally, a numerical case study was presented, focusing on the design of artificial ground freezing for water flow control before NATM tunnel construction.In the future extension of this work, we intend to determine a set of pipe layout samples that can significantly reduce the time to reach the thermal design of the frozen soil body.

Figure 1 .
Figure 1.Model generation and analysis of different pipe layouts for soil freezing modeling.

Figure 2 .
Figure 2. Computational design approach of frozen soil body.The white boundaries represent the desired frozen soil body.Left: The frozen soil body is designed to achieve the average design temperature.Right: state of the frozen soil body ice saturation thickness.

Figure 3 .
Figure 3. Random sample generation of freeze pipe layout.The red dots represent the reference configuration of the pipe layout.The light blue dots represent the randomly generated positions of the pipes.3.1.Application of soil freezing for ground seepage control before NATM tunnel constructionFigure4shows the model setup of the numerical case study.In this case study, the temperature distribution, ice formation, and time of the formation of the frozen soil body were computed for the reference pipe layout from figure3.The computation design finishes once the average frozen soil body has reached T_average = −10 ∘ C with a desired thickness of 1.5 m.This section presents only the results of the reference pipe configuration.In the numerical case study, the soil freezing was modeled under a high seepage flow of 1.5 m/d.The numerical results indicated that the frozen soil body satisfied the average temperature design requirement at 115 d.In addition, at 115 d, the ice formation and streamlines of figure4indicated that the frozen soil body is watertight.

Figure 4 .
Figure 4. Model setup of NATM freeze pipe layout arrangement and the corresponding temperature distributions and ice formation at different time instants.The boundary of the desired frozen soil body and seepage streamlines are depicted in black and white lines, respectively.