Author Contributions
Conceptualization, Y.L. and J.H.; methodology, P.L.; software, P.L. and P.Q.; validation, P.Q. and P.L.; formal analysis, P.Q.; investigation, P.Q.; resources, Y.L.; data curation, P.Q.; writing—original draft preparation, P.Q.; writing—review and editing, P.Q.; visualization, P.L.; supervision, Y.L.; project administration, Y.L.; funding acquisition, J.H. All authors have read and agreed to the published version of the manuscript.
Figure 1.
Schematic illustrations of adverse effects of seepage flow and spatial variability of soil thermal conductivity on AGF for tunnel excavation.
Figure 1.
Schematic illustrations of adverse effects of seepage flow and spatial variability of soil thermal conductivity on AGF for tunnel excavation.
Figure 2.
Relation curves during freezing. (a)Unfrozen water content against temperature; (b) relative permeability against unfrozen water content.
Figure 2.
Relation curves during freezing. (a)Unfrozen water content against temperature; (b) relative permeability against unfrozen water content.
Figure 3.
Schematic illustrations of (a) simulated model of AGF; (b) monitoring points.
Figure 3.
Schematic illustrations of (a) simulated model of AGF; (b) monitoring points.
Figure 4.
(a) A typical realization of soil thermal conductivity random field (μλs = 2.9 W/(m·K) and COV = 0.25); (b) mesh size and temperature field at 10 days when the seepage velocity is 0.5 m/d, and the model’s soil thermal conductivity is shown in (a).
Figure 4.
(a) A typical realization of soil thermal conductivity random field (μλs = 2.9 W/(m·K) and COV = 0.25); (b) mesh size and temperature field at 10 days when the seepage velocity is 0.5 m/d, and the model’s soil thermal conductivity is shown in (a).
Figure 5.
(
a) Schematic illustration of Pimentel et al. [
42]’s laboratory model; (
b) mesh size of validation model and temperature field at 40 h under 1.5 m/d seepage flow.
Figure 5.
(
a) Schematic illustration of Pimentel et al. [
42]’s laboratory model; (
b) mesh size of validation model and temperature field at 40 h under 1.5 m/d seepage flow.
Figure 6.
(a) Thermal boundary conditions of freeze pipes, and comparison between simulation and experimental results along (b) ML1, and (c) ML2 under 1.5 m/d seepage flow.
Figure 6.
(a) Thermal boundary conditions of freeze pipes, and comparison between simulation and experimental results along (b) ML1, and (c) ML2 under 1.5 m/d seepage flow.
Figure 7.
Development of the temperature field of reference groups, four typical freezing durations (i.e., 10 days, 30 days, 50 days and 70 days) are selected.
Figure 7.
Development of the temperature field of reference groups, four typical freezing durations (i.e., 10 days, 30 days, 50 days and 70 days) are selected.
Figure 8.
Temperature curves of horizontal monitoring points under different velocities of seepage: (a) 0 m/d, (b) 0.5 m/d, (c) 1 m/d, and (d) 1.5 m/d.
Figure 8.
Temperature curves of horizontal monitoring points under different velocities of seepage: (a) 0 m/d, (b) 0.5 m/d, (c) 1 m/d, and (d) 1.5 m/d.
Figure 9.
Temperature comparison of horizontal monitoring points under different velocities of seepage: (a) 0 m/d, (b) 0.5 m/d, (c) 1 m/d, and (d) 1.5 m/d. Four typical freezing durations (i.e., 10 days, 30 days, 50 days and 70 days) are selected.
Figure 9.
Temperature comparison of horizontal monitoring points under different velocities of seepage: (a) 0 m/d, (b) 0.5 m/d, (c) 1 m/d, and (d) 1.5 m/d. Four typical freezing durations (i.e., 10 days, 30 days, 50 days and 70 days) are selected.
Figure 10.
Schematic illustrations of the seepage field with the seepage velocity of 0.5 m/d: (a) after 5 days; (b) after 15 days.
Figure 10.
Schematic illustrations of the seepage field with the seepage velocity of 0.5 m/d: (a) after 5 days; (b) after 15 days.
Figure 11.
Influence of seepage flow on the formation of frozen wall between adjacent freeze pipes with different seepage velocities. Four typical freezing durations (i.e., 3 days, 6 days, 9 days and 12 days) are selected.
Figure 11.
Influence of seepage flow on the formation of frozen wall between adjacent freeze pipes with different seepage velocities. Four typical freezing durations (i.e., 3 days, 6 days, 9 days and 12 days) are selected.
Figure 12.
Illustrations of the effective frozen zone after 80 days under different velocities of seepage (i.e., 0 m/d, 0.5 m/d, 1 m/d, and 1.5 m/d).
Figure 12.
Illustrations of the effective frozen zone after 80 days under different velocities of seepage (i.e., 0 m/d, 0.5 m/d, 1 m/d, and 1.5 m/d).
Figure 13.
Temperature curves of points A and D with five different soil thermal conductivities: (a) point A, no seepage; (b) point D, no seepage; (c) point A, 0.5 m/d; (d) point D, 0.5 m/d. A target temperature is set at −3 °C.
Figure 13.
Temperature curves of points A and D with five different soil thermal conductivities: (a) point A, no seepage; (b) point D, no seepage; (c) point A, 0.5 m/d; (d) point D, 0.5 m/d. A target temperature is set at −3 °C.
Figure 14.
Temperature curves of five horizontal monitoring points where λs possess spatial variability (μλs = 2.9 W/(m·K) and COV = 0.25) under different velocities of seepage. Black curves are deterministic results in reference cases where λs is constant at 2.9 W/(m·K).
Figure 14.
Temperature curves of five horizontal monitoring points where λs possess spatial variability (μλs = 2.9 W/(m·K) and COV = 0.25) under different velocities of seepage. Black curves are deterministic results in reference cases where λs is constant at 2.9 W/(m·K).
Figure 15.
Temperature of Points A and D at three typical freezing durations (i.e., 20 days, 40 days, and 60 days) under different seepage velocities: (a) point A, 20 d; (b) point A, 40 d; (c) point A, 60d; (d) point D, 20 d; (e) point D, 40 d; (f) point D, 60 d. The red dashed line signifies the target temperature (−3 °C).
Figure 15.
Temperature of Points A and D at three typical freezing durations (i.e., 20 days, 40 days, and 60 days) under different seepage velocities: (a) point A, 20 d; (b) point A, 40 d; (c) point A, 60d; (d) point D, 20 d; (e) point D, 40 d; (f) point D, 60 d. The red dashed line signifies the target temperature (−3 °C).
Figure 16.
Histograms of the formation time of the desired frozen wall under different velocities of seepage: (a) 0 m/d; (b) 0.5 m/d; (c) 1 m/d. The red dashed lines represent deterministic results, and the black dashed lines are the mean values of Monte-Carlo simulations.
Figure 16.
Histograms of the formation time of the desired frozen wall under different velocities of seepage: (a) 0 m/d; (b) 0.5 m/d; (c) 1 m/d. The red dashed lines represent deterministic results, and the black dashed lines are the mean values of Monte-Carlo simulations.
Figure 17.
Scatter plots of the formation time of desired frozen wall under different velocities of seepage. Black line are mean values of Monte-Carlo simulations.
Figure 17.
Scatter plots of the formation time of desired frozen wall under different velocities of seepage. Black line are mean values of Monte-Carlo simulations.
Figure 18.
Additional freezing time due to seepage flow and spatial variation in soil thermal conductivity. (RFEA = random finite element analysis).
Figure 18.
Additional freezing time due to seepage flow and spatial variation in soil thermal conductivity. (RFEA = random finite element analysis).
Table 1.
Coordinates of temperature monitoring points.
Table 1.
Coordinates of temperature monitoring points.
Point No. | Coordinate (m) |
---|
x (Horizontal) | y (Vertical) |
---|
A | −6.9 | 0 |
B | −5.4 | 0 |
C (Origin) | 0 | 0 |
D | 5.4 | 0 |
E | 6.9 | 0 |
O1 | −4.88 | 4.88 |
I1 | −3.82 | 3.82 |
O2 | 0 | 6.9 |
I2 | 0 | 5.4 |
O3 | 4.88 | 4.88 |
I3 | 3.82 | 3.82 |
O4 | 4.88 | −4.88 |
I4 | 3.82 | −3.82 |
O5 | 0 | −6.9 |
I5 | 0 | −5.4 |
O6 | −4.88 | −4.88 |
I6 | −3.82 | −3.82 |
Table 2.
Material parameters involved in the hydrothermal coupling model.
Table 2.
Material parameters involved in the hydrothermal coupling model.
Description | Symbol | Value | Unit |
---|
Density of soil | ρs | 1900 | kg·m−3 |
Density of water | ρw | 1000 | kg·m−3 |
Density of ice | ρi | 910 | kg·m−3 |
Thermal conductivity of soil solid | λs | 2.9 | W·m−1 K−1 |
Thermal conductivity of water | λw | 0.6 | W·m−1 K−1 |
Thermal conductivity of ice | λi | 2.4 | W·m−1 K−1 |
Heat capacity of soil solid | Cs | 1300 | J·kg−1 K−1 |
Heat capacity of water | Cw | 4200 | J·kg−1 K−1 |
Heat capacity of ice | Ci | 2100 | J·kg−1 K−1 |
Material constant | m | 0.7 | / |
Porosity | n | 0.4 | / |
Material constant | w | 0.5 | °C |
Material constant | z | 0.7 | / |
Intrinsic permeability | κ | 1 × 10−11 | m2 |
Dynamic viscosity (at Tini) | μw | 0.001 | Pa·s |
Freezing point | T0 | 0 | °C |
Initial temperature | Tini | 18 | °C |
Latent heat | Lf | 334 | kJ·kg−1 |
Gravity acceleration | g | 9.8 | m·s−2 |
Table 3.
Cooling plan of the AGF program.
Table 3.
Cooling plan of the AGF program.
Time (Day) | 0 | 1 | 5 | 10 | 15 | 20 | 40 | 80 |
---|
Temperature (°C) | 18 | 0 | −10 | −20 | −25 | −30 | −30 | −30 |
Table 4.
Errors of numerical analysis to experimental test under 1.5 m/d seepage flow.
Table 4.
Errors of numerical analysis to experimental test under 1.5 m/d seepage flow.
Typical Duration | 0 h | 1 h | 5 h | 20 h | 40 h |
---|
Etot along ML1 (%) | 0.87 | 6.59 | 14.9 | 15.58 | 19.85 |
Etot along ML2 (%) | 0.59 | 2.75 | 21.21 | 29.18 | 37.32 |
Table 5.
Temperature statistics of horizontal monitoring points at the end of simulation.
Table 5.
Temperature statistics of horizontal monitoring points at the end of simulation.
Point | Seepage Velocity (m/d) | Deterministic Value (°C) | Average (°C) | Standard Deviation (°C) | Range (°C) |
---|
A | 0 | −16.93 | −17.17 | 0.55 | 2.36 |
0.5 | −12.15 | −12.14 | 0.76 | 3.39 |
1 | −4.54 | −5.58 | 1.65 | 7.49 |
1.5 | −2.33 | −2.07 | 2.17 | 12.61 |
B | 0 | −20.65 | −21.09 | 0.41 | 1.89 |
0.5 | −23.59 | −23.52 | 0.26 | 1.21 |
1 | −24.85 | −24.76 | 0.25 | 1.17 |
1.5 | −25.22 | −25.27 | 0.24 | 1.26 |
C | 0 | 12.2 | 12.27 | 0.83 | 3.03 |
0.5 | 9 | 8.96 | 0.62 | 2.9 |
1 | 3.07 | 3.08 | 0.39 | 1.84 |
1.5 | 2.85 | 3.09 | 0.47 | 2.04 |
D | 0 | −20.72 | −21.07 | 0.49 | 2.25 |
0.5 | −18.69 | −18.52 | 0.78 | 3.98 |
1 | −16.12 | −15.9 | 1.69 | 8.92 |
1.5 | −12.87 | −10.31 | 3.44 | 15.92 |
E | 0 | −16.73 | −17.27 | 0.53 | 2.16 |
0.5 | −20.37 | −20.3 | 0.22 | 1.83 |
1 | −21.09 | −21.15 | 0.18 | 0.84 |
1.5 | −21.71 | −21.89 | 0.22 | 1.35 |