1、A convection-conduction model for analysis of the freeze-thaw conditions in the surrounding rock wall of a tunnel in permafrost regionsAbstractBased on the analyses of fundamental meteorological and hydrological conditions at the site of a tunnel in the cold regions, a combined convection-conduction
2、 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 con
3、ditions 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
4、 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 hydrolo
5、gical 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 t
6、he 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，con
7、structed in cold regions or their neighboring areas，pass through the part beneath the permafrost base .After a tunnel is excavated，the original thermodynamically conditions in the surroundings are and thaw destroyed and replaced mainly by the air connections without the heat radiation, the condition
8、s 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 vari
9、ations 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 temperat
10、ure 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，i
11、t 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-ch
12、ange 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
13、and engineering geology.1Mathematical modelIn 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
14、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 0C for eight months at the tunnel site each year and the construction would la
15、st 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 outsi
16、de 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 southeasterly in summer, considering the pressure difference, the temperature diff
17、erence 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 ro
18、und 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
19、 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
20、;(t), (t) are frozen and thawed parts in the surrounding rock materials respectively; ,and , are thermal conductivities and volumetric thermal capacities in frozen and thawed parts respectively; X= (x , r)，(t) is phase change front; Lh is heat latent of freezing water; and To is critical freezing te
21、mperature of rock ( here we assume To= -0.1).2.Used for solving the modelEquation(1)shows flow. We first solve those concerning temperature at that the temperature of the surrounding rock does not affect the speed of air equations concerning the speed of air flow, and then solve those equations ever
22、y time elapse.2. 1 Procedure used for solving the continuity and momentum equations Since the first three equations in(1) are not independent we derive the second equation by x and the third equation by r. After preliminary calculation we obtain the following elliptic equation concerning the effecti
23、ve pressure p:Then we solve equations in(1) using the following procedures: ( i ) Assume the values for U0，V0; ( ii ) substituting U0，V0 into eq. (2)，and solving (2)，we obtain p0; (iii) solving the first and second equations of(1)，we obtain U0，V1; (iv) solving the first and third equations of(1)，we
24、obtain U2，V2; (v) calculating the momentum-average of U1，v1 and U2，v2，we obtain the new U0，V0，then return to (ii);(vi) iterating as above until the disparity of those solutions in two consecutive iterations is sufficiently small or is satisfied，we then take those values of p0，U0 and V0 as the initia
25、l values for the next elapse and solve those equations concerning the temperature.2 .2 Entire method used for solving the energy equationsAs mentioned previously，the temperature field of the surrounding rock and the air flow affect each other. Thus the surface of the tunnel wall is both the boundary
26、 of the temperature field in the surrounding rock and the boundary of the temperature field in air flow .Therefore， it is difficult to separately identify the temperature on the tunnel wall surface，and we cannot independently solve those equations concerning the temperature of air flow and those equ
27、ations concerning the temperature of the surrounding rock .In order to cope with this problem，we simultaneously solve the two groups of equations based on the fact that at the tunnel wall surface both temperatures are equal .We should bear in mind the phase change while solving those equations conce
28、rning the temperature of the surrounding rock，and the convection while solving those equations concerning the temperature of the air flow, and we only need to smooth those relative parameters at the tunnel wall surface .The solving methods for the equations with the phase change are the same as in r
29、eference 3.2.3 Determination of thermal parameters and initial and boundary conditions2.3.1 Determination of the thermal parameters. Using p= 1013.25-0.1088 H，we calculate air pressure p at elevation H and calculate the air density using formula , where T is the yearly-average absolute air temperatu
30、re，and G is the humidity constant of air. Letting be the thermal capacity with fixed pressure, the thermal conductivity，and the dynamic viscosity of air flow, we calculate the thermal conductivity and kinematic viscosity using the formulas and. The thermal parameters of the surrounding rock are dete
31、rmined from the tunnel site.2.3.2 Determination of the initial and boundary conditions .Choose the observed monthly average wind speed at the entry and exit as boundary conditions of wind speed，and choose the relative effective pressure p=0 at the exit ( that is，the entry of the dominant wind trend)
32、 and on the section of entry ( that is，the exit of the dominant wind trend )，where k is the coefficient of resistance along the tunnel wall, d = 2R，and v is the axial average speed. We approximate T varying by the sine law according to the data observed at the scene and provide a suitable boundary v
33、alue based on the position of the permafrost base and the geothermal gradient of the thaw rock materials beneath the permafrost base.3 A simulated example Using the model and the solving method mentioned above，we simulate the varying law of the air temperature in the tunnel along with the temperatur
34、e at the entry and exit of the Xiluoqi No.2 Tunnel .We observe that the simulated results are close to the data observed6. The Xiluoqi No .2 Tunnel is located on the Nongling railway in northeastern China and passes through the part beneath the permafrost base .It has a length of 1 160 m running fro
35、m the northwest to the southeast, with the entry of the tunnel in the northwest，and the elevation is about 700 m. The dominant wind direction in the tunnel is from northwest to southeast, with a maximum monthly-average speed of 3 m/s and a minimum monthly-average speed of 1 .7 m/s . Based on the dat
36、a observed，we approximate the varying sine law of air temperature at the entry and exit with yearly averages of -5，-6.4 and amplitudes of 18.9 and 17.6 respectively. The equivalent diameter is 5 .8m，and the resistant coefficient along the tunnel wall is 0.025.Since the effect of the thermal paramete
37、r of the surrounding rock on the air flow is much smaller than that of wind speed，pressure and temperature at the entry and exit，we refer to the data observed in the Dabanshan Tunnel for the thermal parameters. Figure 1 shows the simulated yearly-average air temperature inside and at the entry and e
38、xit of the tunnel compared with the data observed .We observe that the difference is less than 0 .2 C from the entry to exit.Figure 2 shows a comparison of the simulated and observed monthly-average air temperature in-side (distance greater than 100 m from the entry and exit) the tunnel. We observe
39、that the principal law is almost the same，and the main reason for the difference is the errors that came from approximating the varying sine law at the entry and exit; especially , the maximum monthly-average air temperature of 1979 was not for July but for August.Fig.1. Comparison of simulated and
40、observed air temperature in XiluoqiNo.2 Tunnel in 1979.1,simulated values;2,observed valuesFig.2.The comparison of simulated and observed air temperature inside The Xiluoqi No.2 Tunnel in 1979.1,simulated values;2,observed values4 Prediction of the freeze-thaw conditions for the Dabanshan Tunnel4 .1
41、 Thermal parameter and initial and boundary conditionsUsing the elevation of 3 800 m and the yearly-average air temperature of -3, we calculate the air density p=0 .774 kg/m.Since steam exists In the air, we choose the thermal capacity with a fixed pressure of air heat conductivity and the dynamic v
42、iscosity After calculation we obtain the thermal diffusivity a= 1 .3788 and the kinematic viscosity， .Considering that the section of automobiles is much smaller than that of the tunnel and the auto-mobiles pass through the tunnel at a low speed，we ignore the piston effects，coming from the movement
43、of automobiles，in the diffusion of the air. We consider the rock as a whole component and choose the dry volumetric cavity ,content of water and unfrozen water W=3% and W=1%, and the thermal conductivity ,heat capacity and ,According to the data observed at the tunnel site，the maximum monthly-averag
44、e wind speed is about 3 .5 m/s，and the minimum monthly-average wind speed is about 2.5 m/s .We approximate the wind speed at the entry and exit as , where t is in month. The initial wind speed in the tunnel is set to be The initial and boundary values of temperature T are set to bewhere f(x) is the
45、distance from the vault to the permafrost base，and R0=25 m is the radius of do-main of solution T. We assume that the geothermal gradient is 3%，the yearly-average air temperature outside tunnel the is A=-3，and the amplitude is B=12. As for the boundary of R=Ro，we first solve the equations considerin
46、g R=Ro as the first type of boundary; that is we assume that T=f(x)3%on R=Ro. We find that, after one year, the heat flow trend will have changed in the range of radius between 5 and 25m in the surrounding rock. Considering that the rock will be cooler hereafter and it will be affected yet by geothe
47、rmal heat, we approximately assume that the boundary R=Ro is the second type of boundary; that is，we assume that the gradient value，obtained from the calculation up to the end of the first year after excavation under the first type of boundary value, is the gradient on R=Ro of T. Considering the sur
48、rounding rock to be cooler during the period of construction，we calculate from January and iterate some elapses of time under the same boundary. Then we let the boundary values vary and solve the equations step by step(it can be proved that the solution will not depend on the choice of initial value
49、s after many time elapses ).4 .2 Calculated resultsFigures 3 and 4 show the variations of the monthly-average temperatures on the surface of the tunnel wall along with the variations at the entry and exit .Figs .5 and 6 show the year when permafrost begins to form and the maximum thawed depth after permafrost formed in different surrounding sections. Fig.3.The
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