A convection-conduction model for analysis of the freeze-thaw
conditions in the surrounding rock wall of a
tunnel in permafrost regions
Abstract
Based on the analyses of fundamental meteorological and hydrogeological conditions at the site of a tunnel in the cold regions, a combined convection-conduction model for air flow in the tunnel and temperature field in the surrounding has been constructed. Using the model, the air temperature distribution in the Xiluoqi No. 2 Tunnel has been simulated numerically. The simulated results are in agreement with the data observed. Then, based on the in situ conditions of sir temperature, atmospheric pressure, wind force, hydrogeology and engineering geology, the air-temperature relationship between the temperature on the surface of the tunnel wall and the air temperature at the entry and exit of the tunnel has been obtained, and the freeze-thaw conditions at the Dabanshan Tunnel which is now under construction is predicted.
Keywords: tunnel in cold regions, convective heat exchange and conduction, freeze-thaw.
A number of highway and railway tunnels have been constructed in the permafrost regions and their neighboring areas in China. Since the hydrological and thermal conditions changed after a tunnel was excavated，the surrounding wall rock materials often froze, the frost heaving caused damage to the liner layers and seeping water froze into ice diamonds，which seriously interfered with the communication and transportation. Similar problems of the freezing damage in the tunnels also appeared in other countries like Russia, Norway and Japan .Hence it is urgent to predict the freeze-thaw conditions in the surrounding rock materials and provide a basis for the design，construction and maintenance of new tunnels in cold regions.
Many tunnels，constructed in cold regions or their neighbouring areas，pass through the part beneath the permafrost base .After a tunnel is excavated，the original thermodynamical conditions in the surroundings are and thaw destroyed and replaced mainly by the air connections without the heat radiation, the conditions determined principally by the temperature and velocity of air flow in the tunnel，the coefficients of convective heat transfer on the tunnel wall，and the geothermal heat. In order to analyze and predict the freeze and thaw conditions of the surrounding wall rock of a tunnel，presuming the axial variations of air flow temperature and the coefficients of convective heat transfer, Lunardini discussed the freeze and thaw conditions by the approximate formulae obtained by Sham-sundar in study of freezing outside a circular tube with axial variations of coolant temperature .We simulated the temperature conditions on the surface of a tunnel wall varying similarly to the periodic changes of the outside air temperature .In fact，the temperatures of the air and the surrounding wall rock material affect each other so we cannot find the temperature variations of the air flow in advance; furthermore，it is difficult to quantify the coefficient of convective heat exchange at the surface of the tunnel wall .Therefore it is not practicable to define the temperature on the surface of the tunnel wall according to the outside air temperature .In this paper, we combine the air flow convective heat ex-change and heat conduction in the surrounding rock material into one model，and simulate the freeze-thaw conditions of the surrounding rock material based on the in situ conditions of air temperature，atmospheric pressure，wind force at the entry and exit of the tunnel，and the conditions of hydrogeology and engineering geology.
Mathematical model
In order to construct an appropriate model, we need the in situ fundamental conditions as a ba-sis .Here we use the conditions at the scene of the Dabanshan Tunnel. The Dabanshan Tunnel is lo-toted on the highway from Xining to Zhangye, south of the Datong River, at an elevation of 3754.78-3 801.23 m, with a length of 1 530 m and an alignment from southwest to northeast. The tunnel runs from the southwest to the northeast.
Since the monthly-average air temperature is beneath 0`}C for eight months at the tunnel site each year and the construction would last for several years，the surrounding rock materials would become cooler during the construction .We conclude that, after excavation, the pattern of air flow would depend mainly on the dominant wind speed at the entry and exit，and the effects of the temperature difference between the inside and outside of the tunnel would be very small .Since the dominant wind direction is northeast at the tunnel site in winter, the air flow in the tunnel would go from the exit to the entry. Even though the dominant wind trend is southeastly in summer, considering the pressure difference, the temperature difference and the topography of the entry and exit，the air flow in the tunnel would also be from the exit to entry .Additionally，since the wind speed at the tunnel site is low，we could consider that the air flow would be principally laminar.
Based on the reasons mentioned，we simplify the tunnel to a round tube，and consider that the
air flow and temperature are symmetrical about the axis of the tunnel，Ignoring the influence of the air temperature on the speed of air flow, we obtain the following equation:
where t，x，r are the time，axial and radial coordinates; U，V are axial and radial wind speeds; T is temperature; p is the effective pressure(that is，air pressure divided by air density); v is the kinematic viscosity of air; a is the thermal conductivity of air; L is the length of the tunnel; R is the equivalent radius of the tunnel section; D is the length of time after the tunnel construction;，
(t), (t) are frozen and thawed parts in the surrounding rock materials respectiv
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A convection-conduction model for analysis of the freeze-thaw
conditions in the surrounding rock wall of a
tunnel in permafrost regions
Abstract
Based on the analyses of fundamental meteorological and hydrogeological conditions at the site of a tunnel in the cold regions, a combined convection-conduction model for air flow in the tunnel and temperature field in the surrounding has been constructed. Using the model, the air temperature distribution in the Xiluoqi No. 2 Tunnel has been simulated numerically. The simulated results are in agreement with the data observed. Then, based on the in situ conditions of sir temperature, atmospheric pressure, wind force, hydrogeology and engineering geology, the air-temperature relationship between the temperature on the surface of the tunnel wall and the air temperature at the entry and exit of the tunnel has been obtained, and the freeze-thaw conditions at the Dabanshan Tunnel which is now under construction is predicted.
Keywords: tunnel in cold regions, convective heat exchange and conduction, freeze-thaw.
A number of highway and railway tunnels have been constructed in the permafrost regions and their neighboring areas in China. Since the hydrological and thermal conditions changed after a tunnel was excavated，the surrounding wall rock materials often froze, the frost heaving caused damage to the liner layers and seeping water froze into ice diamonds，which seriously interfered with the communication and transportation. Similar problems of the freezing damage in the tunnels also appeared in other countries like Russia, Norway and Japan .Hence it is urgent to predict the freeze-thaw conditions in the surrounding rock materials and provide a basis for the design，construction and maintenance of new tunnels in cold regions.
Many tunnels，constructed in cold regions or their neighbouring areas，pass through the part beneath the permafrost base .After a tunnel is excavated，the original thermodynamical conditions in the surroundings are and thaw destroyed and replaced mainly by the air connections without the heat radiation, the conditions determined principally by the temperature and velocity of air flow in the tunnel，the coefficients of convective heat transfer on the tunnel wall，and the geothermal heat. In order to analyze and predict the freeze and thaw conditions of the surrounding wall rock of a tunnel，presuming the axial variations of air flow temperature and the coefficients of convective heat transfer, Lunardini discussed the freeze and thaw conditions by the approximate formulae obtained by Sham-sundar in study of freezing outside a circular tube with axial variations of coolant temperature .We simulated the temperature conditions on the surface of a tunnel wall varying similarly to the periodic changes of the outside air temperature .In fact，the temperatures of the air and the surrounding wall rock material affect each other so we cannot find the temperature variations of the air flow in advance; furthermore，it is difficult to quantify the coefficient of convective heat exchange at the surface of the tunnel wall .Therefore it is not practicable to define the temperature on the surface of the tunnel wall according to the outside air temperature .In this paper, we combine the air flow convective heat ex-change and heat conduction in the surrounding rock material into one model，and simulate the freeze-thaw conditions of the surrounding rock material based on the in situ conditions of air temperature，atmospheric pressure，wind force at the entry and exit of the tunnel，and the conditions of hydrogeology and engineering geology.
Mathematical model
In order to construct an appropriate model, we need the in situ fundamental conditions as a ba-sis .Here we use the conditions at the scene of the Dabanshan Tunnel. The Dabanshan Tunnel is lo-toted on the highway from Xining to Zhangye, south of the Datong River, at an elevation of 3754.78-3 801.23 m, with a length of 1 530 m and an alignment from southwest to northeast. The tunnel runs from the southwest to the northeast.
Since the monthly-average air temperature is beneath 0`}C for eight months at the tunnel site each year and the construction would last for several years，the surrounding rock materials would become cooler during the construction .We conclude that, after excavation, the pattern of air flow would depend mainly on the dominant wind speed at the entry and exit，and the effects of the temperature difference between the inside and outside of the tunnel would be very small .Since the dominant wind direction is northeast at the tunnel site in winter, the air flow in the tunnel would go from the exit to the entry. Even though the dominant wind trend is southeastly in summer, considering the pressure difference, the temperature difference and the topography of the entry and exit，the air flow in the tunnel would also be from the exit to entry .Additionally，since the wind speed at the tunnel site is low，we could consider that the air flow would be principally laminar.
Based on the reasons mentioned，we simplify the tunnel to a round tube，and consider that the
air flow and temperature are symmetrical about the axis of the tunnel，Ignoring the influence of the air temperature on the speed of air flow, we obtain the following equation:
where t，x，r are the time，axial and radial coordinates; U，V are axial and radial wind speeds; T is temperature; p is the effective pressure(that is，air pressure divided by air density); v is the kinematic viscosity of air; a is the thermal conductivity of air; L is the length of the tunnel; R is the equivalent radius of the tunnel section; D is the length of time after the tunnel construction;，
(t), (t) are frozen and thawed parts in the surrounding rock materials respectiv
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