Simplified analytical solution for annual periodic temperature field along radial depth of lining and surrounding rock in deep-buried tunnels in cold regions
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摘要: 为了得到应用性更好的寒区深埋隧道衬砌结构和围岩沿径向温度场解析解，用以指导隧道冻害防治设计，基于叠加原理，将隧道简化为多层圆筒计算模型，在明确围岩温度影响边界确定方法的基础上，以空气年平均温度和围岩初始温度为恒温边界条件，提出了隧道衬砌结构和围岩沿径向不同深度处年平均温度解析解；将隧道简化为多层平板计算模型，采用温度简谐波传播原理，考虑衬砌混凝土、围岩热物性差异对温度简谐波传播的影响，提出了衬砌结构和围岩沿径向不同深度处年温度振幅解析解。分析结果表明：隧道衬砌结构、围岩沿径向不同深度处年平均温度和年温度振幅解析解的函数形式，与现场测试、数值模拟计算得到的隧道沿径向温度场变化规律数学表征函数形式一致；年平均温度和年温度振幅沿径向分别呈对数函数、指数函数变化，其中衬砌结构接近于线性变化；年平均温度理论解析和数值模拟计算结果差值较小，低于0.1 ℃，年温度振幅理论解析结果大于数值模拟计算结果，两者最大差值可达1.3 ℃，最大差值所在位置沿径向距衬砌表面约3 m；年温度振幅理论解析和数值模拟计算结果差值大小受空气年温度振幅和围岩导温系数影响较大，随空气年温度振幅和围岩导温系数降低，逐渐减小，在年温度振幅低于15 ℃、围岩导温系数小于1.10×10-6 m2·s-1条件下，两者差值低于1 ℃。Abstract: An analytical solution with higher applicability was expected to be obtained for the temperature field along the radial depth of the lining structure and the surrounding rock in deep-buried tunnels in cold regions, so as to guide the design of tunnel frost damage prevention. Based on the superposition principle, the tunnel was simplified as a multi-layer cylinder calculation model. After clarifying the determination method for the temperature influence boundary of the surrounding rock, the annual average air temperature and initial surrounding rock temperature were taken as constant temperature boundary conditions to propose an analytical solution for the annual average temperature along the different radial depths of the tunnel lining structure and the surrounding rock. In addition, the tunnel was simplified into a multi-layer slab calculation model. The temperature harmonic wave propagation principle was applied. The effects of thermal property differences between the lining concrete and the surrounding rock on the propagation of the temperature harmonic wave were considered. Analytical solutions of the annual temperature amplitude along the different tunnel radial depths of the lining structure and the surrounding rock were proposed. According to the analysis results, the functional form of the analytical solutions for the annual average temperature and annual temperature amplitude along the different radial depths of the tunnel lining structure and the surrounding rock is consistent with that of the mathematical characterization of the temperature field variation patterns obtained through field tests and numerical simulations. The annual average temperature and annual temperature amplitude vary logarithmically and exponentially along the radial depth, respectively, with the approximately linear variation of the lining structure. The gap between the theoretical analysis and numerical simulation results of the annual average temperature is small, which is lower than 0.1 ℃. The theoretical analysis results for the annual temperature amplitude are larger than those obtained from numerical simulation calculations, which are relatively conservative. The maximum gap between the two can reach up to 1.3 ℃, occurring at a radial distance of approximately 3 m from the lining surface. The gap between theoretical analysis and numerical simulation results for the annual temperature amplitude is most significantly influenced by the annual air temperature amplitude and the thermal conductivity of the surrounding rock. As the annual air temperature amplitude and the thermal diffusivity of the surrounding rock decrease, the gap gradually narrows. When the annual temperature amplitude is less than 15 ℃ and the thermal diffusivity of the surrounding rock is below 1.10×10-6 m2·s-1, the gap between the two remains below 1 ℃.
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图 4 隧道沿径向不同深度处温度随时间变化
Figure 4. Temperature of tunnel lining and surrounding rock varies with time at different radial depths
图 5 断面1的年平均温度和年温度振幅沿径向分布
Figure 5. Distributions of the annual mean temperature and annual temperature amplitude along the radial for section 1
图 6 断面2的年平均温度和年温度振幅沿径向分布
Figure 6. Distributions of the annual mean temperature and annual temperature amplitude along the radial for section 2
图 7 断面3的年平均温度和年温度振幅沿径向分布
Figure 7. Distributions of the annual mean temperature and annual temperature amplitude along radial for section 3
图 8 隧道衬砌结构和围岩沿径向传热的数值模拟计算模型
Figure 8. Numerical calculation model of lining and surrounding rock heat transfer along tunnel radial direction
图 9 工况1隧道拱顶沿径向不同深度处温度随时间变化
Figure 9. Temperature variation over time at different radial depths of tunnel vault in case 1
图 10 工况1的理论解析和数值模拟计算结果对比
Figure 10. Comparison of analytical and numerical simulation calculation results in case 1
图 11 年温度振幅计算最大差值随空气年温度振幅变化
Figure 11. Maximum difference of annual temperature amplitude variations with annual air temperature amplitude
图 12 年温度振幅计算最大差值随随围岩导温系数变化
Figure 12. Maximum difference of annual temperature amplitude variations with the thermal diffusivity of surrounding rock
表 1 隧道各材料热物性参数取值与范围
Table 1. Thermophysical property parameter values and ranges of various materials in tunnel
材料 密度/(kg·m-3) 比热容/[J·(kg·K)-1] 导热系数/[W·(m·K)-1] 二次衬砌模筑混凝土 2 500 920 1.74 初期支护喷射混凝土 2 300 1 000 2.23 围岩 2 400~2 800 950 1.00~5.26 
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表 2 不同工况计算参数
Table 2. Calculation parameters of different cases
工况 年平均气温/℃ 年温度振幅/℃ 围岩初始温度/℃ 围岩导热系数/[W·(m·K)-1] 围岩导温系数/(10-6 m2·s-1) 1 3 17 7 3.5 1.32 2 5 17 7 3.5 1.32 3 4 17 7 3.5 1.32 4 2 17 7 3.5 1.32 5 1 17 7 3.5 1.32 6 3 15 7 3.5 1.32 7 3 13 7 3.5 1.32 8 3 11 7 3.5 1.32 9 3 9 7 3.5 1.32 10 3 17 5 3.5 1.32 11 3 17 9 3.5 1.32 12 3 17 11 3.5 1.32 13 3 17 13 3.5 1.32 14 3 17 7 3.0 1.32 15 3 17 7 2.5 1.32 16 3 17 7 2.0 1.32 17 3 17 7 1.5 1.32 18 3 17 7 3.5 1.50 19 3 17 7 3.5 1.10 20 3 17 7 3.5 0.90 21 3 17 7 3.5 0.70
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表 3 工况1~5的理论解析和数值模拟计算结果统计
Table 3. Statistics of analytical and numerical simulation calculation results of cases 1-5
工况 年平均温度 年温度振幅 拟合函数 最大差值/℃ 拟合函数 最大差值/℃ 1 衬砌结构 理论解析 tM(r)=-4.90+4.70ln(r)
R2=1.000.02 tV(x)=15.74exp(-x/1.96)
R2=1.001.18 数值模拟 tM(r)=-5.02+4.78ln(r)
R2=0.99tV(x)=15.76exp(-x/1.60)
R2=1.00围岩 理论解析 tM(r)=-0.91+2.52ln(r)
R2=1.00tV(x)=13.27exp(-x/3.63)
R2=1.00数值模拟 tM(r)=-0.91+2.52ln(r)
R2=1.00tV(x)=12.77exp(-x/2.99)
R2=1.002 衬砌结构 理论解析 tM(r)=0.98+2.39ln(r)
R2=1.000.02 tV(x)=15.74exp(-x/1.96)
R2=1.001.18 数值模拟 tM(r)=0.98+2.39ln(r)
R2=0.99tV(x)=15.76exp(-x/1.60)
R2=1.00围岩 理论解析 tM(r)=3.05+1.26ln(r)
R2=1.00tV(x)=13.27exp(-x/3.63)
R2=1.00数值模拟 tM(r)=3.05+1.26ln(r)
R2=1.00tV(x)=12.77exp(-x/2.99)
R2=1.003 衬砌结构 理论解析 tM(r)=-1.90+3.51ln(r)
R2=1.000.03 tV(x)=15.74exp(-x/1.96)
R2=1.001.18 数值模拟 tM(r)=-2.01+3.58ln(r)
R2=0.99tV(x)=15.76exp(-x/1.60)
R2=1.00围岩 理论解析 tM(r)=1.06+1.89ln(r)
R2=1.00tV(x)=13.27exp(-x/3.63)
R2=1.00数值模拟 tM(r)=1.07+1.89ln(r)
R2=1.00tV(x)=12.77exp(-x/2.99)
R2=1.004 衬砌结构 理论解析 tM(r)=-7.94+5.91ln(r)
R2=1.000.05 tV(x)=15.74exp(-x/1.96)
R2=1.001.18 数值模拟 tM(r)=-8.03+5.97ln(r)
R2=0.99tV(x)=15.76exp(-x/1.60)
R2=1.00围岩 理论解析 tM(r)=-2.88+3.15ln(r)
R2=1.00tV(x)=13.27exp(-x/3.63)
R2=1.00数值模拟 tM(r)=-2.88+3.15ln(r)
R2=1.00tV(x)=12.77exp(-x/2.99)
R2=1.005 衬砌结构 理论解析 tM(r)=-10.92+7.09ln(r)
R2=1.000.06 tV(x)=15.74exp(-x/1.96)
R2=1.001.18 数值模拟 tM(r)=-11.03+7.17ln(r)
R2=0.99tV(x)=15.76exp(-x/1.60)
R2=1.00围岩 理论解析 tM(r)=-4.86+3.78ln(r)
R2=1.00tV(x)=13.27exp(-x/3.63)
R2=1.00数值模拟 tM(r)=-4.86+3.78ln(r)
R2=1.00tV(x)=12.77exp(-x/2.99)
R2=1.00
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表 4 工况6~9的理论解析和数值模拟计算结果统计
Table 4. Statistics of analytical and numerical simulation calculation results of cases 6-9
工况 年平均温度 年温度振幅 拟合函数 最大差值/℃ 拟合函数 最大差值/℃ 6 衬砌结构 理论解析 tM(r)=-4.90+4.70ln(r)
R2=1.000.04 tV(x)=13.88exp(-x/1.96)
R2=1.001.05 数值模拟 tM(r)=-5.03+4.70ln(r)
R2=0.99tV(x)=13.90exp(-x/1.90)
R2=1.00围岩 理论解析 tM(r)=-0.91+2.52ln(r)
R2=1.00tV(x)=11.72exp(-x/3.63)
R2=1.00数值模拟 tM(r)=-0.91+2.52ln(r)
R2=1.00tV(x)=11.27exp(-x/2.99)
R2=1.007 衬砌结构 理论解析 tM(r)=-4.90+4.70ln(r)
R2=1.000.04 tV(x)=12.03exp(-x/1.96)
R2=1.000.91 数值模拟 tM(r)=-5.02+4.78ln(r)
R2=0.99tV(x)=12.05exp(-x/1.60)
R2=1.00围岩 理论解析 tM(r)=-0.91+2.52ln(r)
R2=1.00tV(x)=10.16exp(-x/3.63)
R2=1.00数值模拟 tM(r)=-0.91+2.52ln(r)
R2=1.00tV(x)=9.77exp(-x/2.99)
R2=1.008 衬砌结构 理论解析 tM(r)=-4.90+4.70ln(r)
R2=1.000.04 tV(x)=10.18exp(-x/1.96)
R2=1.000.77 数值模拟 tM(r)=-5.02+4.78ln(r)
R2=0.99tV(x)=10.20exp(-x/1.60)
R2=1.00围岩 理论解析 tM(r)=3.50+1.12ln(r)
R2=0.94tV(x)=8.59exp(-x/3.63)
R2=1.00数值模拟 tM(r)=3.50+1.12ln(r)
R2=0.95tV(x)=8.26exp(-x/2.99)
R2=1.009 衬砌结构 理论解析 tM(r)=-4.90+4.70ln(r)
R2=1.000.04 tV(x)=8.33exp(-x/1.96)
R2=1.000.63 数值模拟 tM(r)=-5.02+4.78ln(r)
R2=0.99tV(x)=8.34exp(-x/1.60)
R2=1.00围岩 理论解析 tM(r)=-0.91+2.52ln(r)
R2=1.00tV(x)=7.03exp(-x/3.64)
R2=1.00数值模拟 tM(r)=-0.91+2.52ln(r)
R2=1.00tV(x)=6.76exp(-x/2.99)
R2=1.00
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表 5 工况10~13的理论解析和数值模拟计算结果统计
Table 5. Statistics of analytical and numerical simulation calculation results of cases 10-13
工况 年平均温度 年温度振幅 拟合函数 最大差值/℃ 拟合函数 最大差值/℃ 10 衬砌结构 理论解析 tM(r)=-1.02+2.39ln(r)
R2=1.000.02 tV(x)=15.74exp(-x/1.96)
R2=1.001.18 数值模拟 tM(r)=-1.02+2.39ln(r)
R2=0.99tV(x)=15.76exp(-x/1.60)
R2=1.00围岩 理论解析 tM(r)=1.05+1.26ln(r)
R2=1.00tV(x)=13.27exp(-x/3.63)
R2=1.00数值模拟 tM(r)=1.06+1.26ln(r)
R2=1.00tV(x)=12.77exp(-x/2.99)
R2=1.0011 衬砌结构 理论解析 tM(r)=-8.92+7.09ln(r)
R2=1.000.06 tV(x)=15.74exp(-x/1.96)
R2=1.001.18 数值模拟 tM(r)=-9.03+7.17ln(r)
R2=0.99tV(x)=15.76exp(-x/1.60)
R2=1.00围岩 理论解析 tM(r)=-2.86+3.78ln(r)
R2=1.00tV(x)=13.27exp(-x/3.63)
R2=1.00数值模拟 tM(r)=-2.86+3.78ln(r)
R2=1.00tV(x)=12.77exp(-x/2.99)
R2=1.0012 衬砌结构 理论解析 tM(r)=-12.90+9.46ln(r)
R2=1.000.07 tV(x)=15.74exp(-x/1.96)
R2=1.001.18 数值模拟 tM(r)=-13.05+9.56ln(r)
R2=0.99tV(x)=15.76exp(-x/1.60)
R2=1.00围岩 理论解析 tM(r)=-4.82+5.04ln(r)
R2=1.00tV(x)=13.27exp(-x/3.63)
R2=1.00数值模拟 tM(r)=-4.81+5.04ln(r)
R2=1.00tV(x)=12.77exp(-x/2.99)
R2=1.0013 衬砌结构 理论解析 tM(r)=-16.83+11.80ln(r)
R2=1.000.09 tV(x)=15.74exp(-x/1.96)
R2=1.001.18 数值模拟 tM(r)=-17.06+11.95ln(r)
R2=0.99tV(x)=15.76exp(-x/1.60)
R2=1.00围岩 理论解析 tM(r)=-6.77+6.30ln(r)
R2=1.00tV(x)=13.27exp(-x/3.63)
R2=1.00数值模拟 tM(r)=-6.77+6.30ln(r)
R2=1.00tV(x)=12.77exp(-x/2.99)
R2=1.00
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表 6 工况14~17的理论解析和数值模拟计算结果统计
Table 6. Statistics of analytical and numerical simulation calculation results of cases 14-17
工况 年平均温度 年温度振幅 拟合函数 最大差值/℃ 拟合函数 最大差值/℃ 14 衬砌结构 理论解析 tM(r)=-4.24+4.31ln(r)
R2=1.000.04 tV(x)=15.77exp(-x/2.10)
R2=1.001.13 数值模拟 tM(r)=-4.31+4.35ln(r)
R2=0.99tV(x)=15.82exp(-x/1.72)
R2=1.00围岩 理论解析 tM(r)=-1.25+2.67ln(r)
R2=1.00tV(x)=13.90exp(-x/3.36)
R2=1.00数值模拟 tM(r)=-1.25+2.67ln(r)
R2=1.00tV(x)=13.42exp(-x/2.80)
R2=1.0015 衬砌结构 理论解析 tM(r)=-3.16+3.66ln(r)
R2=1.000.03 tV(x)=15.85exp(-x/2.45)
R2=1.001.19 数值模拟 tM(r)=-3.23+3.71ln(r)
R2=1.00tV(x)=15.92exp(-x/2.00)
R2=1.00围岩 理论解析 tM(r)=-1.52+2.67ln(r)
R2=1.00tV(x)=14.76exp(-x/3.32)
R2=1.00数值模拟 tM(r)=-1.48+2.74ln(r)
R2=1.00tV(x)=14.42exp(-x/2.77)
R2=1.0016 衬砌结构 理论解析 tM(r)=-1.84+2.88ln(r)
R2=1.000.03 tV(x)=16.00exp(-x/3.33)
R2=0.031.23 数值模拟 tM(r)=-1.84+2.88ln(r)
R2=1.00tV(x)=16.10exp(-x/2.60)
R2=1.00围岩 理论解析 tM(r)=-1.42+2.64ln(r)
R2=1.00tV(x)=15.85exp(-x/3.63)
R2=1.00数值模拟 tM(r)=-1.44+2.65ln(r)
R2=1.00tV(x)=15.53exp(-x/3.03)
R2=1.0017 衬砌结构 理论解析 tM(r)=-0.68+2.19ln(r)
R2=1.000.02 tV(x)=16.11exp(-x/4.78)
R2=0.991.27 数值模拟 tM(r)=-0.70+2.20ln(r)
R2=1.00tV(x)=16.27exp(-x/3.34)
R2=1.00围岩 理论解析 tM(r)=-1.60+2.69ln(r)
R2=1.00tV(x)=16.91exp(-x/3.64)
R2=1.00数值模拟 tM(r)=-1.61+2.69ln(r)
R2=1.00tV(x)=16.78exp(-x/3.01)
R2=1.00
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表 7 工况18~21的理论解析和数值模拟计算结果统计
Table 7. Statistics of analytical and numerical simulation calculation results of cases 18-21
工况 年平均温度 年温度振幅 拟合函数 最大差值/℃ 拟合函数 最大差值/℃ 18 衬砌结构 理论解析 tM(r)=-4.75+4.61ln(r)
R2=1.000.05 tV(x)=15.77exp(-x/2.08)
R2=0.041.05 数值模拟 tM(r)=-4.80+4.65ln(r)
R2=1.00tV(x)=15.80exp(-x/1.67)
R2=1.00围岩 理论解析 tM(r)=3.47+1.09ln(r)
R2=0.94tV(x)=13.43exp(-x/3.88)
R2=1.00数值模拟 tM(r)=3.49+1.09ln(r)
R2=0.95tV(x)=12.84exp(-x/3.14)
R2=1.0019 衬砌结构 理论解析 tM(r)=-5.35+4.97ln(r)
R2=1.000.04 tV(x)=15.69exp(-x/1.81)
R2=0.990.91 数值模拟 tM(r)=-5.40+5.00ln(r)
R2=1.00tV(x)=15.72exp(-x/1.50)
R2=1.00围岩 理论解析 tM(r)=-1.08+2.63ln(r)
R2=1.00tV(x)=13.07exp(-x/3.32)
R2=1.00数值模拟 tM(r)=-1.03+2.61ln(r)
R2=1.00tV(x)=12.63exp(-x/2.76)
R2=1.0020 衬砌结构 理论解析 tM(r)=-5.76+5.22ln(r)
R2=1.000.05 tV(x)=15.63exp(-x/1.67)
R2=0.990.77 数值模拟 tM(r)=-5.89+5.29ln(r)
R2=1.00tV(x)=15.67exp(-x/1.40)
R2=1.00围岩 理论解析 tM(r)=3.57+1.22ln(r)
R2=0.95tV(x)=14.87exp(-x/2.51)
R2=1.00数值模拟 tM(r)=3.61+1.20ln(r)
R2=0.95tV(x)=12.44exp(-x/2.52)
R2=1.0021 衬砌结构 理论解析 tM(r)=-6.50+5.65ln(r)
R2=1.000.03 tV(x)=15.57exp(-x/1.51)
R2=0.990.63 数值模拟 tM(r)=-6.43+5.62ln(r)
R2=1.00tV(x)=15.60exp(-x/1.29)
R2=1.00围岩 理论解析 tM(r)=-1.63+2.99ln(r)
R2=1.00tV(x)=12.59exp(-x/2.65)
R2=1.00数值模拟 tM(r)=3.74+1.16ln(r)
R2=0.94tV(x)=12.20exp(-x/2.26)
R2=1.00
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