Freeze-sealing pipe roof method is applied in the Gongbei tunnel, which causes the ground surface uplift induced by frost heave. A frost heaving prediction approach based on the coefficient of cold expansion is proposed to simulate the ground deformation of the Gongbei tunnel. The coefficient of cold expansion in the model and the frost heaving rate from the frost heave test under the hydration condition can achieve a good correspondence making the calculation result closer to the actual engineering. The ground surface uplift along the lateral and longitudinal direction are respectively analyzed and compared with the field measured data to validate the model. The results show that a good agreement between the frost heaving prediction model and the field measured data verifies the rationality and applicability of the proposed model. The maximum uplift of the Gongbei tunnel appears at the center of the model, gradually decreasing along with the lateral and longitudinal directions. The curve in the lateral direction presents a normal distribution due to the influence of the constraint of two sides, while the one along the lateral direction shapes like a parabola with the opening downward due to the temperature field distribution. The model provides a reference for frost heaving engineering calculation.

Freeze-sealing pipe roof (FSPR) method as an innovative pre-supporting method in tunnel engineering is applied in the Gongbei tunnel of Hongkong-Zhuhai-Macau Bridge, which is the first application in the world. Engineering applications using FSPR method has been appeared in recent years with the rapid development of underground space technology. FSPR method combines pipe-roofing method and artificial ground freezing method. The definition of FSPR method is that jacking pipes with large diameter are laid out in a circle along the cross section of tunnel in advance, then the artificial ground freezing method is adopted to freeze soil between jacking pipes to prevent water from entering into tunnel during excavation. The pipe jacking mainly plays a role of load bearing, while ground freezing mainly plays a role of water sealing [

ABAQUS as a three-dimensional finite element program is employed currently to carry out the numerical study. It is well known that temperature variation can lead to heat-induced expansion in frozen soil. Here we ignore the effect of temperature on strain ratio because it is small enough compared to frost heave. We propose the coefficient of cold expansion to simulate the expansion process, which is related to temperature. The basic framework of the model is as follows.

For the isotropic homogeneous material, equation of thermal equilibrium is expressed as

One-dimensional model

For transient heat conduction without internal heat source in semi-infinite domain, the equation can be expressed as

The equation of initial condition can be represented as

The equation of boundary conditions can be represented as

On the basis of

When

The boundary conditions are as shown in

The

Hereafter, the formula of temperature change can be expressed as

According to the following statements, a new frost heaving prediction model is established. (1) The pore water in the soil becomes ice, causing the volume of the soil to expand. The moisture migration causing the frost heaving phenomenon is the main factor. The findings above are discovered by Taber and Beskow through the soil frost heaving test [

Therefore, the linear frost heaving rate can be calculated as follows:

In which,

Based on the assumptions above, the maximum frost heaving rate can be expressed as

Based on the frost heaving test of saturated soil under hydration conditions,

Two-dimensional model

For two-dimensional model, assume that

According to the above, the equation of boundary conditions can be represented as

In

In which,

Therefore, the surface frost heaving rate can be calculated as follows:

Three-dimensional model

The three-dimensional model is composed of multiple two-dimensional models along the longitudinal direction. The numerical simulation of frost heave is based on the model above.

Finite element calculation

Since the elastic modulus varies with temperature, the frost heave strain includes the elastic strain and the temperature strain.

The elastic strain increment can be calculated by Hooke's law as follows:

Based on the obtained temperature field and the concept of linear expansion coefficient, temperature strain increment can be calculated. The results are as follows:

At any period

The node force of the element

The stiffness matrix of the element

Thus, the whole equilibrium equation is as follows:

In which,

The displacement increment

To verify the validity and feasibility of the proposed frost heaving prediction model, numerical results of the ground uplift due to freeze-sealing pipe roof method is compared with the field measured data. The freeze-sealing pipe roof (FSPR) method combining pipe roofing method and artificial ground freezing method is firstly applied in the Gongbei tunnel with the length of 255 m, which is a critical link of Hongkong-Zhuhai-Macau Bridge. ^{2} with 18.8 m in width and 20.6 m in height, which is the largest single tunnel excavation in China. The tunnel is buried within soft sandy and silty clay about 4.5 m below the ground surface. The purpose of ground freezing is preventing groundwater from entering the tunnel. A freezing process will last for 50 days before excavation. However, the ground surface uplift occurs due to the frost heave. And the amount of frost heave is simulated by the frost heaving prediction model. The correspondingly outcomes are employed to validate the proposed model.

The ground deformation due to frost heave is simulated using the finite element software ABAQUS. It is assumed that the soil layers are horizontally distributed, isotropic and homogeneous, while roofing pipes and concrete are isotropic homogeneous elastic materials. Refer to the detailed investigation survey data of the Gongbei tunnel, the adjacent soil layers are combined into one layer for calculation. There are five soil layers in the model. The geometry of the model (255 m × 150 m × 60 m in length × width × height) and its cross section are respectively shown in _{0}_{f}

Category | Thickness (m) | Friction angle |
Cohesion |
Elastic modulus |
Poisson's ratio |
---|---|---|---|---|---|

Miscellaneous fill | 6.4 | 13.9 | 39.9 | 4.6 | 0.3 |

Fine sand | 7.8 | 9.2 | 13 | 14.4 | 0.3 |

Muddy silty clay | 16.9 | 26.4 | 14.3 | 20.1 | 0.3 |

Gravel clay | 10.4 | 25.3 | 12.5 | 6.5 | 0.3 |

Completely weathered bedrock | 18.5 | 19.2 | 22.8 | 4.2 | 0.3 |

Roofing pipe | —— | —— | —— | 3.0e + 4 | 0.3 |

Concrete | —— | —— | —— | 2.06e + 5 | 0.2 |

Category | Density ^{3}) |
Thermal conductivity ^{−1}⋅K^{−1}) |
Specific heat capacity ^{−1}⋅K^{−1}) |
Maximum frost heaving rate |
Coefficient of cold expansion |
---|---|---|---|---|---|

Miscellaneous fill | 1950 | 1.69 | 2558 | 1.1 | 0.00026 |

Fine sand | 1960 | 1.758 | 2680 | 1.08 | 0.00025 |

Muddy silty clay | 1980 | 1.893 | 2933 | 3.4 | 0.00079 |

Gravel clay | 1820 | 1.79 | 2778 | 3.3 | 0.00077 |

Completely weathered bedrock | 1780 | 1.887 | 2815 | 1.18 | 0.00027 |

Roofing pipe | 7850 | 45 | 460 | —— | —— |

Concrete | 2400 | 1.28 | 970 | —— | —— |

In which, the maximum frost heaving rate is obtained from a model calibration based on soil frost heave test under hydration conditions for each soil layer in laboratory. And the coefficient of cold expansion is calculated according to

In order to facilitate the analysis of ground surface uplift law, the feature sections are selected, including lateral direction and longitudinal direction. All sections are symmetrically distributed. There are 24 lateral monitoring sections (YK + 390∼YK + 630) along the longitudinal direction, where YK + 400 is the first monitoring section, 10 m from the boundary. Meanwhile, the monitoring data on the corresponding section (YK + 400, YK + 430, YK + 440) is obtained separately, shown in

The finite element model consists of soil part, pipe-roofing part, concrete part, and lining part, with a total of approximately 489016 elements, including C3D8T elements and C3D6T elements, shown in

There are 5 steps throughout the whole construction process, containing initial ground stress balance, pipe jacking, backfilling pipe with concrete, soil freezing, tunnel excavation. The ground stress is first released 40% during excavation, and then fully released after support. The main calculation process in

Temperature is one of the factors affecting the frost heave effect, which indirectly affects the soil deformation, therefore the temperature field is the basis for calculating the soil deformation. The initial temperature of the model is 28°C, and the positive temperature does not cause frost heaving. Thus, the value of the initial temperature has little effect on the frost heave calculation. After freezing process of 50 days, the thickness of the frozen soil reaches the required width of 2 m. The temperature distribution is shown in

The ground equilibrium is achieved by ABAQUS when the surface is horizontal. Then the frost heaving model is applied to predict the vertical uplift of ground surface due to frost heave.

The vertical uplift of ground surface after 50 days of freezing due to frost heave along the longitudinal direction of the tunnel is also analyzed, shown in

A good agreement is discovered between the model prediction and field measured data, except for some misfits that are due to inhomogeneous soil and local grouting that affecting heat transfer and groundwater seepage. In addition, the weight and dynamic load of the surface buildings make the monitoring data slightly smaller than the model result in some local location. The favorable agreement indicates that the proposed frost heaving prediction model can well predict the ground surface uplift due to frost heave.

The ground deformation due to FSPR method involves a complicated hydrothermal coupling process, which is difficult to accurately simulate. According to that, a frost heaving prediction approach is proposed to simulate ground deformation of the Gongbei tunnel. The proposed model deals with the frost heave calculation by combining the coefficient of cold expansion based on the calculation equation and the frost heaving test under hydration conditions. From the result analysis of prediction model and field monitoring, some conclusions are drawn:

The coefficient of cold expansion in model and the frost heaving rate from the test under the hydration condition can achieve a good correspondence that makes the calculation result closer to the actual engineering.

The numerical simulation and monitoring data of the Gongbei tunnel are employed to validate the proposed model, and a good agreement between the frost heaving prediction model and the field measured data verifies the rationality and applicability of the proposed model to predict the ground deformation.

The maximum uplift of the Gongbei tunnel appears at the center of the model, gradually decreasing along the lateral and longitudinal directions. The result in the lateral direction represents a normal distribution due to the influence of the constraint of two sides, while the one along the lateral direction shapes like a parabola with the opening downward due to the temperature field distribution.

It should be noted that the application of the proposed frost heaving prediction model to the Gongbei tunnel will be a reference for similar cases.

The author wishes to thank the high-performance computing cluster of Computer Science Department at Princeton University and the editor Dr. Lynn Wang for the careful format review.