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Forecast analysis of the refreezing of Kunlun mountain permafrost tunnel on Qing–Tibet railway in China
Cold Regions Science and Technology 39 (2004) 19 – 31 www.elsevier.com/locate/coldregions
Forecast analysis of the refreezing of Kunlun mountain permafrost tunnel on Qing–Tibet railway in China Zhang Xuefu, Lai Yuanming *, Yu Wenbing, Zhang Shujuan, Xiao Jianzhang State Key Laboratory of Frozen Soil Engineering, Cold and Arid Regions Environmental and Engineering Research Institute, Chinese Academy of Sciences, Gansu Lanzhou 730000, China Received 12 March 2003; accepted 14 December 2003
Abstract According to the in situ measured data of the air temperature and ground temperature inside of Kunlun mountain tunnel on Qing – Tibet railway, China, the thawed extent of the permafrost surrounding the tunnel is very large, caused by the latent heat of the cast in situ concrete lining insulation during excavation in winter and other artificial activities. In this paper, taking the coupled effects of moisture transfer and heat conduction into account, a finite element formula of this problem is derived from the governing differential equations with phase change and moisture transfer equations using Galerkin’s solution. Using the computer program, predictions for the refreezing of Kunlun mountain permafrost tunnel are made. The analysis results show that the vault of Kunlun mountain tunnel would have been refrozen by 31 March 2004, the arch bottom by 30 September 2004 and the total tunnel by 30 September 2005, without insulation material. On the other hand, if the tunnel is fitted with insulation material, whose thermal conductivity is equal to 0.03 W/m.K and thickness of 0.05 m, the vault of Kunlun mountain tunnel would have been refrozen by 31 March 2004, the arch bottom by 31 December 2004, the left wall by 31 March 2006, and the right wall by 30 September 2006. Thus, insulation material will delay the refreezing of Kunlun mountain permafrost tunnel. The thawed extent of the permafrost surrounding the tunnel in cold regions caused by construction must be considered, and the time of the field observation should be extended, or observational results, such as their temperature and stress field, will greatly be different from results in stability. D 2004 Elsevier B.V. All rights reserved. Keywords: Moisture transfer; Construction effect; Insulation material; Numerical analysis; Heat conduction; Phase change; Tunnel; China
1. Introduction Up to now, the effects of construction on the freezing– thawing situations of cold region engineering have only been studied for cold region embankments and dams, but there has rarely been studies on these effects for cold region tunnels. Xuefu et al. (2002a,b) studied in detail the effects on the freez* Corresponding author. Fax: +86-931-8273894. E-mail address: [email protected] (L. Yuanming). 0165-232X/$ - see front matter D 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.coldregions.2003.12.002
ing –thawing situations of cold region tunnels caused by construction time using finite element method. The study of Xuefu et al. (2002a,b) limited only different excavation time to the effects on the freezing –thawing situations of cold region tunnel, but the effects of the latent heat of the cast in situ concrete lining insulation during excavation in winter and other artificial activities are not taken into consideration. However, according to the in situ measured data of the air temperature and ground temperature inside of Kunlun mountain tunnel on Qing –Tibet railway, the thawed
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extent of the permafrost surrounding the tunnel is very large and the air temperature in the tunnel rises during excavation, caused by the latent heat of the cast in situ concrete lining insulation during excavation in winter and other artificial activities. Up to now, the thawed extents of the permafrost surrounding the tunnel and the changes of the air temperature in the tunnel during excavation are not considered during the design and research. But nobody studies these problems of how many thawed extents of the permafrost surrounding the tunnel during excavation are, when the thawed extent will be refrozen and whether the thawed extent should be taken into consideration during the design and research on the cold region tunnel engineering. On the other hand, in general, to keep the initial thermal situations of the rock surrounding tunnels from being broken since the construction of tunnels and to prevent their linings from being destroyed by the frost heave or thaw settlement, the insulation material is usually installed in cold region tunnels. In Russia, the pipe-electric heaters heat the tunnels, and the new-type and effective insulation materials preserve the heats of the cold region tunnel. In Norway, some of the drainages in the tunnels are heated with the heated cables, some are fitted double thermal insulation doors, and others are installed insulation materials. In Japan, the insulation materials are fitted on the lining surfaces of Shang Yu Huang tunnel at Hokkaido to prevent the local geothermal from giving out of the rock surrounding the tunnel and to keep the temperatures on the lining surface from falling below the frozen temperature (Lai, 1999). Lai et al. (1998) considered the coupled effects of temperature, seepage and stress fields, studied the temperature fields of the Da Ban Shan tunnel with insulation material in Qing Hai province, China. Lai et al. (1999) considered the coupled effects of temperature and seepage fields,
studied that the insulation material of the Da Ban Shan tunnel are valid and essential. Xuefu et al. (2002a,b) researched on how to select and install the insulation material of the cold region tunnel by finite element method, etc. However, nobody studies the problems of what the functions of the insulation material on the refreezing of permafrost tunnel are, whether the functions are positive or negative and how large the functions are. In allusion to these problems, in this paper, taking the coupled effects of moisture transfer and heat conduction into account, a finite element formula of this problem are derived from the governing differential equations with phase change and moisture transfer equations using Galerkin’s solution. According to the in situ measured data of the air temperature inside of Kunlun mountain tunnel on Qing –Tibet railway and the ground temperature of the rock surrounding the tunnel, predictions for the refreezing of Kunlun mountain permafrost tunnel are made by the computer program. Some significant results are presented, which can provide reference for actual tunnel engineering design and research in cold regions.
2. General situation of Kunlun mountain tunnel Kunlun mountain tunnel region belongs to the arid – cold and periglacial type climate regions of the Qing – Tibet plateau. Its air temperature and air pressure are both low and its annual average air temperature is about 5.2 jC. Its mean annual ground temperature is 2.5 jC. Kunlun mountain tunnel is located in the mileage from DK976 + 250 to DK977 + 936 of Qing– Tibet railway in China. Its total length, depth maximum, permafrost thickness and altitude are about 1686, 106, 125 and 4600– 4750 m, respectively.
Fig. 1. The vertical section of the Kunlun mountain tunnel.
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Kunlun mountain tunnel lies in mainly schist with some slate. Its plateau tunnel totally buried in permafrost. There is ice-poor permafrost around Kunlun mountain tunnel except for its inlet and outlet where ice-rich permafrost exist. Fig. 1 shows the vertical section of the Kunlun mountain tunnel.
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where a is a coefficient of heat convection, and Ta is the equilibrium temperature for which no convection occurs. Initial conditions are. In Xf, Tf jt¼0 ¼ T0
ð6Þ
where T0 is negative. Or in the nonfrozen zone Xu,
3. Governing differential equations and their finite element formula
Tu jt¼0 ¼ T0
According to heat transfer and moisture transfer theory, not considering the thermal dissipation during the evaporation of the moisture in soil and taking the coupled effects of moisture transfer and heat conduction into account, the differential equations of the problem with phase change are given as follows. In Xf, B B Bhi 1 BTf 1 BTf 1 BTf kf kf Cf ¼ þ þ Lqi Bx By Bt Bx By Bt
where T0 is positive. Assuming that the effects of air and vapor on pure moisture transfer can be ignored, in general, the mass transfer equations of the two-dimensional stable or unstable fluid in the process of saturated or unsaturated freezing and thawing are given by (Carrol and Holt, 1972). B Bw B Bw Bhu q Bhi Kx Ky þ i ð8Þ þ ¼ Bx Bx By By Bt qw Bt
ð1Þ In Xu, BTu B BTu B BTu ku ku ¼ Cu þ Bx By Bt Bx By
ð2Þ
Where subscripts f and u represent the frozen and the nonfrozen states, respectively. Tf, C1f and k1f are the temperature, volumetric heat capacity and thermal conductivity of surrounding rock in the frozen area Xf, respectively. L, hi, qi and t are the latent heat per unit volume, volumetric ice content, density of the ice and time, respectively. At the frost front position s(t), the continuous condition and the conservation of energy should be met, i.e., Tf ðsðtÞ; tÞ ¼ Tu ðsðtÞ; tÞ ¼ Tm
ð3Þ
BTf BTu dsðtÞ ku ¼L dt Bn Bn
ð4Þ
k1f
where Tm is the temperature at the frost front position. The first and the second type boundary conditions are T ¼ Ta ;
k
BT ¼ aðTa T Þ Bn
ð5Þ
ð7Þ
Where w = U + Z; w, U and Z represent the total potential energy, volumetric energy and gravity energy of the moisture in soil, respectively. Kx and Ky are the hydraulic conductivity of the moisture in the x and y directions, respectively. hu is volumetric content of the nonfrozen moisture. qw is the density of the water. Other parameters are the same as forenamed. When the gravity energy Z is usually ignored, w = U. Substituting Eq. (8) into Eq. (1), we can obtain B B 1 BTf 1 BTf k k þ Bx f Bx By f By BTf Bhu þ Lqw Bt Bt B Bw B Bw Kx Ky Lqw þ Bx Bx By By
¼ Cf1
ð9Þ
Tailor and Luthin (1978) thought that ignoring the function of soil – ice – moisture system, volumetric energy becomes equal to the chemical energy in the salt moisture under the condition of the same temperature. That is to say, the characteristic curve of the moisture in the nonfrozen soil adapts to the one in the frozen soil. Thus, according to the relationship of the content of the nonfrozen moisture and the temperature
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in the frozen soil, the volumetric energy of the nonfrozen moisture in the frozen soil can be obtained BU BU Bhu ¼ BT Bhu BT
ð10Þ
Also according to the formula c = Bhu/BU, the formula of the transmissibility coefficients of the moisture D = K/c can be gained. Thus, Eq. (9) can be simplified to Bhu BT 1 Cf þ Lqw BT Bt B Bhu BT 1 kf þ Lqw D ¼ Bx BT Bx B Bhu BT 1 kf þ Lqw D þ By BT By
ð11Þ
Thus, in Xf Cf
BTf B BTf B BTf kf kf ¼ þ Bx By Bt Bx By
ð12Þ
where: Cf ¼ Cf1 þ Lqw
kf ¼
k1f
Bhu BT
8 kf ; T < Tm DT > > > > < k k k* ¼ kf þ u f ½T ðTm DT Þ; Tm DTVTVTm þDT > 2DT > > > : T > Tm þ DT ku ;
ð14Þ
Using Eqs. (13) and (14), Eqs. (2) –4) and (12) can be written as simply BT B BT B BT ¼ k* k* C* þ ð15Þ Bt Bx Bx By By Because the specific heat and thermal conductivity of the rock are functions of temperature and the frost front position is variable, the condition of energy conservation at the frost front position is nonlinear. Its analytical solution can’t be obtained. We obtained its solution by using a numerical analytical method. Using Galerkin’s method, the following finite element formula can be obtained.
BT ½M þ ½KfT g ¼ fFg ð16Þ Bt where, Mij ¼
Bhu þ Lqw Dx BT
XZ Xe
C*Ni Nj dX
BNi BNj BNi BNj þ Kij ¼ k* dX Bx Bx By By Xe XZ aNi Nj dC þ Ce2 Z X Fi ¼ aTa Ni dC XZ
Other parameters are the same as forenamed. It is assumed that the phase change occurs in a range of temperatures (Tm F DT). When the equivalent heat capacity is constructed, the effect of the temperature interval DT should be included. Suppose that Cf, Cu, kf and ku do not depend on temperature T. Then in the interval TmDT V T V Tm + DT, the following definitions may be assumed (Lai et al., 1998). 8 Cf ; > > > > < L Cf þ Cu C* ¼ þ ; > 2DT 2 > > > : Cu ;
and
ð17Þ
ð18Þ ð19Þ
Ce2
Ni ; Nj the shape functions
T < Tm DT
The relationship of the content of the nonfrozen moisture and the temperature in the frozen soil will be obtained from experiments. First solve Eq. (15), and then solve the volumetric ice content hi from Eq. (8).
Tm DT VT VTm þ DT
4. The refreezing analysis of Kunlun Shan tunnel
T > Tm þ DT ð13Þ
There is ice-poor permafrost around Kunlun mountain tunnel except for its inlet and outlet
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Fig. 2. The structure of tunnel.
where ice-rich permafrost exist. Thus, we select the cross-section at DK796 + 410, which is a representative section in the inlet of Kunlun mountain tunnel, to be the analytical model. The structure of Kunlun mountain tunnel in this section is shown in Fig. 2. In this paper, four-node quadrilateral elements are used, and each node has 1 degree of freedom. The rock surrounding tunnel is divided into 2750 elements and 2911 nodes, and the outer lining of the tunnel has 330 elements and 448 nodes, the insulation material with the thickness of 0.05 m is divided into 110 elements and 224 nodes, and the inner lining of the tunnel is divided into 220 elements and 336 nodes, respectively. Thus, the whole model holds 3410 elements and 3583 nodes. The mesh of elements of the model is shown in Fig. 3. The distance EF from the vault of the tunnel to the ground surface is 50 m; GJ = 30 m; AE = EB = DJ = JC = 30 m. Because the computational domain is large enough and the heat fluxes on the boundaries AD and BC affect little on the rock nearby the tunnel, the heat fluxes of the boundaries AD and BC are 0. The
Fig. 3. The mesh of elements.
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geothermal heat flux 0.06 W/m2 flows in the computational domain from the boundary DC. Boundary AB is in contact with air. The second type boundary condition is imposed on the boundary AB, and the convective heat transfer coefficient a between the air and the ground surface is 15.0 W/m2.k (Zhang, 1989). The air temperature curves may be obtained from the meteorological data in this region. Ta ¼ 5:2 þ 12sin
2p 11 th þ p 8760 12
ð20Þ
where the parameter th denotes the step time. The inner surface FIGH of the tunnel is in contact with air too. And the convective heat transfer coefficient a between the air and the surface of the outer lining or inner lining of the tunnel is 15.0 W/m2 k too. However caused by the latent heat of the cast in situ concrete lining insulation during excavation in winter and other artificial activities, the air temperatures in the tunnel are different, and the air temperatures at different sites of the same section of the tunnel are different too. The inner linings of the DK976 + 410 section were lined on 30 September 2001. On 30 September 2002, the insulation materials and the outer linings of the DK976 + 410 section had been constructed. The total tunnel will be finished in 2 years since the inner linings of the DK976 + 410 section were lined. The air temperatures at the different sites of the cross-section at DK796 + 410 can be obtained from the in situ measured data during constructing the tunnel.At the vault of the tunnel
8 2p 11 > > < 14:9sin 8760 th þ 12 p Ta ¼ > 2p 11 > : 3sin th þ p 8760 12
2p 11 sin th þ p >0 8760 12 2p 11 sin th þ p V0 8760 12
ð21Þ
At the arch bottom of the tunnel 8 2p 11 2p 11 > > 14:2sin þ þ t p sin t p >0 h h < 8760 12 8760 12 Ta ¼ > 2p 11 2p 11 > :5sin th þ p sin th þ p V0 8760 12 8760 12
ð22Þ
At the left wall of the tunnel 8 2p 11 2p 11 > > sin th þ p >0 <14:1sin 8760 th þ 12 p 8760 12 Ta ¼ > 2p 11 2p 11 > :4sin th þ p sin th þ p V0 8760 12 8760 12
ð23Þ
Because the vent pipe is fitted in the right wall of the tunnel, the temperatures of the left wall and right wall of the tunnel are different. At the right wall of the tunnel 8 2p 11 2p 11 > > sin th þ p >0 <13:6sin 8760 th þ 12 p 8760 12 Ta ¼ > 2p 11 2p 11 > : 4:5sin th þ p sin th þ p V0 8760 12 8760 12
ð24Þ
After the total tunnel was finished on 30 September 2003, the airflow in the style of laminar flow will happen in the tunnel. Thus, the air temperatures inside of the tunnel are assumed to be the air temperatures in Eq. (20). The thermal conductivity of the insulation material with the thickness of 0.05 m is 0.03 W/(m.K). The initial temperatures of the rock surrounding the tunnel can be gained from the in situ measured data. The volumetric heat capacity and thermal conductivity of the rock are derived from the following formulae (Xu et al., 2001). Cu ¼
Cus þ xCw 1þx
ð25Þ
Cf ¼
Cfs þ xu Cw þ ðx xu ÞCi 1þx
ð26Þ
ku ¼ khs s khww
ð27Þ
kf ¼ khs s khwuu khi i
ð28Þ
Where hs, hw, hi represent the volumetric contents of the soil grain, moisture and ice in the soil, respectively. hu is the volumetric content of the nonfrozen moisture in the frozen soil. x stands for the mass content of the moisture in the soil. xu represents the mass content of the nonfrozen moisture in the frozen soil. Other parameters are the same as the forenamed. According to the character of the rock surrounding the tunnel, we can know the thermal parameters of the
Z. Xuefu et al. / Cold Regions Science and Technology 39 (2004) 19–31 Table 1 The thermal parameters of water, ice, frozen soil grain and thawed grain The thermal parameters
Value
Heat capacity of the ice Ci [kJ/(kg.K)] Heat capacity of the water Cw [kJ/(kg.K)] Heat capacity of the thawed soil grain Cus [kJ/(kg.K)] Heat capacity of the frozen soil grain Cfs [kJ/(kg.K)] Thermal conductivity of the ice ki [W/(m.K)] Thermal conductivity of the water kw [W/(m.K)] Thermal conductivity of the soil grain ks [W/(m.K)]
2.09 4.18 0.84 0.75 2.22 0.55 3.00
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from 2.24 to 3.25 m around the left wall would have had residual thawed layer with the thickness of 1.01 m, and that the thickness range from 1.87 to 3.66 m around the right wall would have had residual thawed layer with the thickness of 1.79 m by 31 March 2002. Fig. 4(b) displays that the thawed depth around the vault would have been 2.39 m, that the thawed depth around the left wall would have been 3.52 m, that the thawed depth around the right wall had been 3.75 m, and that the thawed depth around the arch bottom would have been 2.36 m by 30 September 2002.
water, the ice, the frozen soil grain and the thawed grain. These thermal parameters are listed in Table 1. From the in situ measured data, we know that the cross-section at DK976 + 410 of Kunlun mountain tunnel lies in ice-rich soil whose volumetric content of the moisture is 55% and whose transmissibility coefficients D of the moisture is equal to 2.2 10 9 m2/s. According to the experimental data, the relationship of the content of the nonfrozen moisture and the temperature can be shown by the following Eq. (29). 8 < 6:989AT A0:26 T VTm hu ¼ ð29Þ : hw T zTm Substituting the abovementioned parameters into the finite element formula, the calculated results are obtained by computers, as follows. 4.1. Predicting the refreezing of Kunlun mountain tunnel with insulation material whose thermal conductivity is equal to 0.03 W/m.k and whose thickness is equal to 0.05 m In light of the actual design and construction of Kunlun mountain tunnel, after the inner lining is fitted, the waterproof layer is installed first, the insulation material, whose thermal conductivity is equal to 0.03 W/m.K and whose thickness is equal to 0.05 m, is constructed second, then the waterproof layer is fixed again, the outer lining is fixed finally. Because the thickness the waterproof layer is very thin, the effects of the two waterproof layers can be ignored. The detail analysis is made, as follows. From Fig. 4(a), it is found that the rock surrounding the vault and the arch bottom of Kunlun mountain tunnel would have been frozen, that the thickness range
Fig. 4. The temperature distributions of the rock surrounding tunnel with insulation material in the first freezing – thawing cycle when the inner linings are constructed.
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From Fig. 6(a), it is seen that the thickness range from 1.76 to 2.49 m around the vault would have had residual thawed layer with the thickness of 0.73 m, that the thickness range from 1.15 to 4.70 m around the left wall would have had residual thawed layer with the thickness of 3.55 m, that the thickness range from 1.12 to 5.36 m around the right wall would have had residual thawed layer with the thickness of 4.24 m, and that the thickness range from 1.00 to 3.34 m around the arch bottom would have had residual thawed layer with the thickness of 2.34 m by 31 March 2004. From Fig. 6(b),
Fig. 5. The temperature distributions of the rock surrounding tunnel with insulation material in the second freezing – thawing cycle when the inner and outer linings are constructed.
Fig. 5(a) indicates that the thickness range from 0.39 to 2.31 m around the vault would have had residual thawed layer with the thickness of 1.92 m, and that the thawed depths around the left wall, the right wall and the arch bottom would have been 4.04, 4.60 and 2.73 m by 31 March 2003, respectively. From Fig. 5(b), it is shown that the thawed depths around the vault, the left wall, the right wall and the arch bottom would have been 2.68, 4.25, 4.88 and 2.80 m by 30 September 2003, respectively.
Fig. 6. The temperature distributions of the rock surrounding tunnel with insulation material in the third freezing – thawing cycle when the inner and outer linings are constructed.
Z. Xuefu et al. / Cold Regions Science and Technology 39 (2004) 19–31
it is shown that the thickness range from 3.75 to 4.04 m around the arch bottom would have had residual thawed layer with the thickness of 0.29 m, and that the thawed depths around the vault, the left wall and the right wall would have been 0.76, 4.56 and 5.34 m by 30 September 2004, respectively. It is obvious that, after three freezing – thawing cycles, the thawed extent caused by construction around the vault of Kunlun mountain tunnel will basically be refrozen, and that the thawed extent around the left wall, the right wall and the arch bottom will not yet be refrozen. Fig. 7(a) indicates that the rock surrounding the vault and the arch bottom would have been frozen, that the thickness range from 2.28 to 4.45 m around the left wall would have had residual thawed layer with the thickness of 2.17 m, and that the thickness range from 2.23 to 5.30 m around the right wall would have had residual thawed layer with the thickness of 3.07 m by 31 March 2005. From Fig. 7(b), it is seen that the thickness range from 3.44 to 3.91 m around the arch bottom would have had residual thawed layer with the thickness of 0.47 m, that the thickness range from 2.97 to 5.11 m around the arch bottom would have had residual thawed layer with the thickness of 2.74 m, and that the seasonal thawed depths around the vault and the arch bottom would have been 0.50 and 0.27 m by 30 September 2005, respectively. It is obvious that, by 31 March 2004, the thawed extent caused by construction around the vault and the arch bottom of Kunlun mountain tunnel will basically have been refrozen, that the thawed extent around the left wall and the right wall would not have been refrozen. However, by 30 September 2005, the thawed extent around the left wall and the right wall would not yet have been refrozen. Fig. 8(a) indicates that the rock around the vault, the left wall and the arch bottom would have been frozen, that the thickness range from 3.96 to 4.42 m around the right wall would have had residual thawed layer with the thickness of 0.46 m by 31 March 2006. From Fig. 8(b), it is seen that the thawed extent around the total tunnel would basically have been refrozen, and that there would only have been the seasonal thawing around the total tunnel by 30 September 2006. It is obvious that by 31 March 2006, the thawed extent around the right wall would not yet have been
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Fig. 7. The temperature distributions of the rock surrounding tunnel with insulation material in the fourth freezing – thawing cycle when the inner and outer linings are constructed.
refrozen. However, by 30 September 2006, the thawed extent around the total tunnel would basically have been refrozen. 4.2. Predicting for the refreezing of Kunlun mountain tunnel without insulation material In order to research the function of the insulation material on the refreezing of permafrost tunnel, predictions for the refreezing of Kunlun mountain tunnel without insulation material are made, as follows.
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31 March 2003. From Fig. 9(b), it is shown that the thawed depths around the vault, the left wall, the right wall and the arch bottom would have been 2.73, 4.26, 4.86 and 2.93 m by 30 September 2003, respectively. From Fig. 10(a), it is seen that the rock around the vault will have been frozen, that the thickness range from 2.26 to 4.63 m around the left wall would have had residual thawed layer with the thickness of 2.73 m, that the thickness range from 2.20 to 5.22 m around the right wall would have had residual thawed layer with the thickness of 2.92 m, and that the
Fig. 8. The temperature distributions of the rock surrounding tunnel with insulation material in the fifth freezing – thawing cycle when the inner and outer linings are constructed.
Fig. 9(a) indicates that the thickness range from 0.87 to 2.30 m around the vault of Kunlun mountain tunnel would have had residual thawed layer with the thickness of 1.33 m, that the thickness range from 0.61 to 4.08 m around the left wall would have had residual thawed layer with the thickness of 3.47 m, that the thickness range from 0.60 to 4.59 m around the right wall would have had residual thawed layer with the thickness of 3.99 m, and that the thickness range from 0.09 to 2.78 m around the arch bottom would have had residual thawed layer with the thickness of 2.69 m by
Fig. 9. The temperature distributions of the rock surrounding tunnel without insulation material in the second freezing – thawing cycle when the inner and outer linings are constructed.
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It is obvious that, after three freezing – thawing cycles, the thawed extent caused by construction around the vault and the arch bottom of Kunlun mountain tunnel will basically be refrozen, and that the thawed extent around the left wall and the right wall will not yet be refrozen. From Fig. 11(a), we can know that the thawed depth around the vault and the arch bottom would have been frozen, that the thickness range from 3.53 to 4.33 m around the left wall would have had residual thawed layer with the thickness of 0.80 m,
Fig. 10. The temperature distributions of the rock surrounding tunnel without insulation material in the third freezing – thawing cycle when the inner and outer linings are constructed.
thickness range from 1.85 to 3.32 m around the arch bottom would have had residual thawed layer with the thickness of 1.47 m by 31 March 2004. Fig. 10(b) indicates that the seasonal thawed depth around the vault would have been 0.91 m, that the seasonal thawed depth around the arch bottom would have been 0.34 m, that the thickness range from 3.32 to 4.49 m around the left wall would have had residual thawed layer with the thickness of 1.17 m, and that the thickness range from 3.33 to 5.09 m around the left wall would have had residual thawed layer with the thickness of 1.76 m by 30 September 2004.
Fig. 11. The temperature distributions of the rock surrounding tunnel without insulation material in the fourth freezing – thawing cycle when the inner and outer linings are constructed.
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and that the thickness range from 3.59 to 4.52 m around the left wall would have had residual thawed layer with the thickness of 0.93 m by 31 March 2005. Fig. 11(b), it is seen that the thawed extent around the total tunnel would basically have been refrozen, and that there would only have been the seasonal thawed around the total tunnel by 30 September 2005. It is obvious that, by 31 March 2005, the thawed extent caused by construction around the vault and the arch bottom of Kunlun mountain tunnel would basically have been refrozen, and that the thawed extent around the left wall and the right wall would not yet have been refrozen. After four freezing – thawing cycles, the thawed extent around the total tunnel would basically have been refrozen. Comparing Fig. 5 with Fig. 9, Fig. 6 with Fig. 10 and Fig. 7 with Fig. 11, the temperature distributions of the rock surrounding Kunlun mountain tunnel with insulation material are greatly different from the temperature distributions without insulation material at the same time. For example, from Figs. 7(b) and 10(b), by 30 September 2004, the thawed extent around the vault and the arch bottom of Kunlun mountain tunnel with insulation material would have been refrozen; the thawed extent around the left wall and the right wall would not yet have been refrozen. However, at the same time, the rock surrounding the Kunlun mountain tunnel without insulation material would have been refrozen, expect that there would be a little residual thawed layer around the vault. Thus, the insulation material will delay the refreezing of the permafrost tunnel.
5. Conclusions From the computational analyses mentioned above, we can make the following conclusions: (a) Fitting with insulation material whose thermal conductivity is equal to 0.03 W/(m.K) and thickness is 0.05 m, the vault of Kunlun mountain tunnel would have been refrozen by 31 March 2004; the left wall would have been refrozen by 31 March 2006; and the right wall would have been refrozen by 30 September 2006.
(b) The vault of Kunlun mountain tunnel without insulation material would have been refrozen by 31 March 2004; the arch bottom would have been refrozen by 30 September 2004; and the total tunnel would basically have been refrozen by 30 September 2005. (c) The insulation material will delay the refreezing of the thawed extent of the permafrost tunnel caused by the latent heat of the cast in situ concrete lining, insulating during excavating in winter and other artificial activities. (d) The thawed extent of the permafrost surrounding the tunnel in cold regions caused by construction must be considered, and the time of observation should be extended when their temperature and stress are observed in situ, or observation results will largely be different from the results in stability.
Acknowledgements This study was supported in part by the National Science Fund for Distinguished Young Scholars of China (40225001), in part by the National Natural Science Foundation of China (40171019), in part by the Foundation of ‘‘Hundred People Plan’’ of Chinese Academy of Sciences (to Dr. Y.M. Lai) and by the grant of the Knowledge Innovation Program of the Chinese Academy of Sciences (KZCX1-SW-04).
References Carrol, M.M., Holt, A.C., 1972. Static and dynamic pore collapse relation for ductile porous material. Journal of Applied Physics 43, 1626 – 1635. Lai, Y.M., 1999. Nonlinear analysis for the coupled problem of temperature, seepage and stress fields in cold-region tunnels (PhD dissertation). LanZhou Institute of Glaciology and Geocryology, China Academy of Science, pp. 2 – 3. Lai, Y.M., et al., 1998. Nonlinear analysis for the coupled problem of temperature, seepage and stress fields in cold-region tunnels[J]. Tunneling and Underground Space Technology 13 (4), 435 – 436. Lai, Y.M., et al., 1999. Nonlinear analysis for the coupled problem of temperature and seepage fields in cold-region tunnels. Cold Regions Science and Technology 29 (1), 89 – 96. Tailor, G.S., Luthin, J.N., 1978. A model for coupled heat and moisture transfer during soil freezing. Canadian Geotechnical Journal 15, 545 – 555.
Z. Xuefu et al. / Cold Regions Science and Technology 39 (2004) 19–31 Xu, X., Wang, J., Zhang, L., 2001. Physics of Frozen Ground [M]. Science Press, Beijing, pp. 75 – 90. Xuefu, Z., Yuanming, L., Fengcai, Y., Wenbing, Y., 2002a. Numerical analysis for the freezing – thawing situation of the rock surrounding tunnel in cold regions[J]. Journey of the China Railway Society 24 (4), 92 – 96. Xuefu, Z., Yuanming, L., Wenbing, Y., Shujuan, Z., 2002b. Non-
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linear analysis for the freezing – thawing situation of the rock surrounding the tunnel in cold regions under the conditions of different construction seasons, initial temperatures and insulations. Tunneling and Underground Space Technology 17 (3), 315 – 325. Zhang, Z.R., 1989. Heat Transfer. High Educational Press, Beijing, pp. 133 – 134.
Forecast analysis of the refreezing of Kunlun mountain permafrost tunnel on Qing–Tibet railway in China Zhang Xuefu, Lai Yuanming *, Yu Wenbing, Zhang Shujuan, Xiao Jianzhang State Key Laboratory of Frozen Soil Engineering, Cold and Arid Regions Environmental and Engineering Research Institute, Chinese Academy of Sciences, Gansu Lanzhou 730000, China Received 12 March 2003; accepted 14 December 2003
Abstract According to the in situ measured data of the air temperature and ground temperature inside of Kunlun mountain tunnel on Qing – Tibet railway, China, the thawed extent of the permafrost surrounding the tunnel is very large, caused by the latent heat of the cast in situ concrete lining insulation during excavation in winter and other artificial activities. In this paper, taking the coupled effects of moisture transfer and heat conduction into account, a finite element formula of this problem is derived from the governing differential equations with phase change and moisture transfer equations using Galerkin’s solution. Using the computer program, predictions for the refreezing of Kunlun mountain permafrost tunnel are made. The analysis results show that the vault of Kunlun mountain tunnel would have been refrozen by 31 March 2004, the arch bottom by 30 September 2004 and the total tunnel by 30 September 2005, without insulation material. On the other hand, if the tunnel is fitted with insulation material, whose thermal conductivity is equal to 0.03 W/m.K and thickness of 0.05 m, the vault of Kunlun mountain tunnel would have been refrozen by 31 March 2004, the arch bottom by 31 December 2004, the left wall by 31 March 2006, and the right wall by 30 September 2006. Thus, insulation material will delay the refreezing of Kunlun mountain permafrost tunnel. The thawed extent of the permafrost surrounding the tunnel in cold regions caused by construction must be considered, and the time of the field observation should be extended, or observational results, such as their temperature and stress field, will greatly be different from results in stability. D 2004 Elsevier B.V. All rights reserved. Keywords: Moisture transfer; Construction effect; Insulation material; Numerical analysis; Heat conduction; Phase change; Tunnel; China
1. Introduction Up to now, the effects of construction on the freezing– thawing situations of cold region engineering have only been studied for cold region embankments and dams, but there has rarely been studies on these effects for cold region tunnels. Xuefu et al. (2002a,b) studied in detail the effects on the freez* Corresponding author. Fax: +86-931-8273894. E-mail address: [email protected] (L. Yuanming). 0165-232X/$ - see front matter D 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.coldregions.2003.12.002
ing –thawing situations of cold region tunnels caused by construction time using finite element method. The study of Xuefu et al. (2002a,b) limited only different excavation time to the effects on the freezing –thawing situations of cold region tunnel, but the effects of the latent heat of the cast in situ concrete lining insulation during excavation in winter and other artificial activities are not taken into consideration. However, according to the in situ measured data of the air temperature and ground temperature inside of Kunlun mountain tunnel on Qing –Tibet railway, the thawed
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extent of the permafrost surrounding the tunnel is very large and the air temperature in the tunnel rises during excavation, caused by the latent heat of the cast in situ concrete lining insulation during excavation in winter and other artificial activities. Up to now, the thawed extents of the permafrost surrounding the tunnel and the changes of the air temperature in the tunnel during excavation are not considered during the design and research. But nobody studies these problems of how many thawed extents of the permafrost surrounding the tunnel during excavation are, when the thawed extent will be refrozen and whether the thawed extent should be taken into consideration during the design and research on the cold region tunnel engineering. On the other hand, in general, to keep the initial thermal situations of the rock surrounding tunnels from being broken since the construction of tunnels and to prevent their linings from being destroyed by the frost heave or thaw settlement, the insulation material is usually installed in cold region tunnels. In Russia, the pipe-electric heaters heat the tunnels, and the new-type and effective insulation materials preserve the heats of the cold region tunnel. In Norway, some of the drainages in the tunnels are heated with the heated cables, some are fitted double thermal insulation doors, and others are installed insulation materials. In Japan, the insulation materials are fitted on the lining surfaces of Shang Yu Huang tunnel at Hokkaido to prevent the local geothermal from giving out of the rock surrounding the tunnel and to keep the temperatures on the lining surface from falling below the frozen temperature (Lai, 1999). Lai et al. (1998) considered the coupled effects of temperature, seepage and stress fields, studied the temperature fields of the Da Ban Shan tunnel with insulation material in Qing Hai province, China. Lai et al. (1999) considered the coupled effects of temperature and seepage fields,
studied that the insulation material of the Da Ban Shan tunnel are valid and essential. Xuefu et al. (2002a,b) researched on how to select and install the insulation material of the cold region tunnel by finite element method, etc. However, nobody studies the problems of what the functions of the insulation material on the refreezing of permafrost tunnel are, whether the functions are positive or negative and how large the functions are. In allusion to these problems, in this paper, taking the coupled effects of moisture transfer and heat conduction into account, a finite element formula of this problem are derived from the governing differential equations with phase change and moisture transfer equations using Galerkin’s solution. According to the in situ measured data of the air temperature inside of Kunlun mountain tunnel on Qing –Tibet railway and the ground temperature of the rock surrounding the tunnel, predictions for the refreezing of Kunlun mountain permafrost tunnel are made by the computer program. Some significant results are presented, which can provide reference for actual tunnel engineering design and research in cold regions.
2. General situation of Kunlun mountain tunnel Kunlun mountain tunnel region belongs to the arid – cold and periglacial type climate regions of the Qing – Tibet plateau. Its air temperature and air pressure are both low and its annual average air temperature is about 5.2 jC. Its mean annual ground temperature is 2.5 jC. Kunlun mountain tunnel is located in the mileage from DK976 + 250 to DK977 + 936 of Qing– Tibet railway in China. Its total length, depth maximum, permafrost thickness and altitude are about 1686, 106, 125 and 4600– 4750 m, respectively.
Fig. 1. The vertical section of the Kunlun mountain tunnel.
Z. Xuefu et al. / Cold Regions Science and Technology 39 (2004) 19–31
Kunlun mountain tunnel lies in mainly schist with some slate. Its plateau tunnel totally buried in permafrost. There is ice-poor permafrost around Kunlun mountain tunnel except for its inlet and outlet where ice-rich permafrost exist. Fig. 1 shows the vertical section of the Kunlun mountain tunnel.
21
where a is a coefficient of heat convection, and Ta is the equilibrium temperature for which no convection occurs. Initial conditions are. In Xf, Tf jt¼0 ¼ T0
ð6Þ
where T0 is negative. Or in the nonfrozen zone Xu,
3. Governing differential equations and their finite element formula
Tu jt¼0 ¼ T0
According to heat transfer and moisture transfer theory, not considering the thermal dissipation during the evaporation of the moisture in soil and taking the coupled effects of moisture transfer and heat conduction into account, the differential equations of the problem with phase change are given as follows. In Xf, B B Bhi 1 BTf 1 BTf 1 BTf kf kf Cf ¼ þ þ Lqi Bx By Bt Bx By Bt
where T0 is positive. Assuming that the effects of air and vapor on pure moisture transfer can be ignored, in general, the mass transfer equations of the two-dimensional stable or unstable fluid in the process of saturated or unsaturated freezing and thawing are given by (Carrol and Holt, 1972). B Bw B Bw Bhu q Bhi Kx Ky þ i ð8Þ þ ¼ Bx Bx By By Bt qw Bt
ð1Þ In Xu, BTu B BTu B BTu ku ku ¼ Cu þ Bx By Bt Bx By
ð2Þ
Where subscripts f and u represent the frozen and the nonfrozen states, respectively. Tf, C1f and k1f are the temperature, volumetric heat capacity and thermal conductivity of surrounding rock in the frozen area Xf, respectively. L, hi, qi and t are the latent heat per unit volume, volumetric ice content, density of the ice and time, respectively. At the frost front position s(t), the continuous condition and the conservation of energy should be met, i.e., Tf ðsðtÞ; tÞ ¼ Tu ðsðtÞ; tÞ ¼ Tm
ð3Þ
BTf BTu dsðtÞ ku ¼L dt Bn Bn
ð4Þ
k1f
where Tm is the temperature at the frost front position. The first and the second type boundary conditions are T ¼ Ta ;
k
BT ¼ aðTa T Þ Bn
ð5Þ
ð7Þ
Where w = U + Z; w, U and Z represent the total potential energy, volumetric energy and gravity energy of the moisture in soil, respectively. Kx and Ky are the hydraulic conductivity of the moisture in the x and y directions, respectively. hu is volumetric content of the nonfrozen moisture. qw is the density of the water. Other parameters are the same as forenamed. When the gravity energy Z is usually ignored, w = U. Substituting Eq. (8) into Eq. (1), we can obtain B B 1 BTf 1 BTf k k þ Bx f Bx By f By BTf Bhu þ Lqw Bt Bt B Bw B Bw Kx Ky Lqw þ Bx Bx By By
¼ Cf1
ð9Þ
Tailor and Luthin (1978) thought that ignoring the function of soil – ice – moisture system, volumetric energy becomes equal to the chemical energy in the salt moisture under the condition of the same temperature. That is to say, the characteristic curve of the moisture in the nonfrozen soil adapts to the one in the frozen soil. Thus, according to the relationship of the content of the nonfrozen moisture and the temperature
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in the frozen soil, the volumetric energy of the nonfrozen moisture in the frozen soil can be obtained BU BU Bhu ¼ BT Bhu BT
ð10Þ
Also according to the formula c = Bhu/BU, the formula of the transmissibility coefficients of the moisture D = K/c can be gained. Thus, Eq. (9) can be simplified to Bhu BT 1 Cf þ Lqw BT Bt B Bhu BT 1 kf þ Lqw D ¼ Bx BT Bx B Bhu BT 1 kf þ Lqw D þ By BT By
ð11Þ
Thus, in Xf Cf
BTf B BTf B BTf kf kf ¼ þ Bx By Bt Bx By
ð12Þ
where: Cf ¼ Cf1 þ Lqw
kf ¼
k1f
Bhu BT
8 kf ; T < Tm DT > > > > < k k k* ¼ kf þ u f ½T ðTm DT Þ; Tm DTVTVTm þDT > 2DT > > > : T > Tm þ DT ku ;
ð14Þ
Using Eqs. (13) and (14), Eqs. (2) –4) and (12) can be written as simply BT B BT B BT ¼ k* k* C* þ ð15Þ Bt Bx Bx By By Because the specific heat and thermal conductivity of the rock are functions of temperature and the frost front position is variable, the condition of energy conservation at the frost front position is nonlinear. Its analytical solution can’t be obtained. We obtained its solution by using a numerical analytical method. Using Galerkin’s method, the following finite element formula can be obtained.
BT ½M þ ½KfT g ¼ fFg ð16Þ Bt where, Mij ¼
Bhu þ Lqw Dx BT
XZ Xe
C*Ni Nj dX
BNi BNj BNi BNj þ Kij ¼ k* dX Bx Bx By By Xe XZ aNi Nj dC þ Ce2 Z X Fi ¼ aTa Ni dC XZ
Other parameters are the same as forenamed. It is assumed that the phase change occurs in a range of temperatures (Tm F DT). When the equivalent heat capacity is constructed, the effect of the temperature interval DT should be included. Suppose that Cf, Cu, kf and ku do not depend on temperature T. Then in the interval TmDT V T V Tm + DT, the following definitions may be assumed (Lai et al., 1998). 8 Cf ; > > > > < L Cf þ Cu C* ¼ þ ; > 2DT 2 > > > : Cu ;
and
ð17Þ
ð18Þ ð19Þ
Ce2
Ni ; Nj the shape functions
T < Tm DT
The relationship of the content of the nonfrozen moisture and the temperature in the frozen soil will be obtained from experiments. First solve Eq. (15), and then solve the volumetric ice content hi from Eq. (8).
Tm DT VT VTm þ DT
4. The refreezing analysis of Kunlun Shan tunnel
T > Tm þ DT ð13Þ
There is ice-poor permafrost around Kunlun mountain tunnel except for its inlet and outlet
Z. Xuefu et al. / Cold Regions Science and Technology 39 (2004) 19–31
Fig. 2. The structure of tunnel.
where ice-rich permafrost exist. Thus, we select the cross-section at DK796 + 410, which is a representative section in the inlet of Kunlun mountain tunnel, to be the analytical model. The structure of Kunlun mountain tunnel in this section is shown in Fig. 2. In this paper, four-node quadrilateral elements are used, and each node has 1 degree of freedom. The rock surrounding tunnel is divided into 2750 elements and 2911 nodes, and the outer lining of the tunnel has 330 elements and 448 nodes, the insulation material with the thickness of 0.05 m is divided into 110 elements and 224 nodes, and the inner lining of the tunnel is divided into 220 elements and 336 nodes, respectively. Thus, the whole model holds 3410 elements and 3583 nodes. The mesh of elements of the model is shown in Fig. 3. The distance EF from the vault of the tunnel to the ground surface is 50 m; GJ = 30 m; AE = EB = DJ = JC = 30 m. Because the computational domain is large enough and the heat fluxes on the boundaries AD and BC affect little on the rock nearby the tunnel, the heat fluxes of the boundaries AD and BC are 0. The
Fig. 3. The mesh of elements.
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geothermal heat flux 0.06 W/m2 flows in the computational domain from the boundary DC. Boundary AB is in contact with air. The second type boundary condition is imposed on the boundary AB, and the convective heat transfer coefficient a between the air and the ground surface is 15.0 W/m2.k (Zhang, 1989). The air temperature curves may be obtained from the meteorological data in this region. Ta ¼ 5:2 þ 12sin
2p 11 th þ p 8760 12
ð20Þ
where the parameter th denotes the step time. The inner surface FIGH of the tunnel is in contact with air too. And the convective heat transfer coefficient a between the air and the surface of the outer lining or inner lining of the tunnel is 15.0 W/m2 k too. However caused by the latent heat of the cast in situ concrete lining insulation during excavation in winter and other artificial activities, the air temperatures in the tunnel are different, and the air temperatures at different sites of the same section of the tunnel are different too. The inner linings of the DK976 + 410 section were lined on 30 September 2001. On 30 September 2002, the insulation materials and the outer linings of the DK976 + 410 section had been constructed. The total tunnel will be finished in 2 years since the inner linings of the DK976 + 410 section were lined. The air temperatures at the different sites of the cross-section at DK796 + 410 can be obtained from the in situ measured data during constructing the tunnel.At the vault of the tunnel
8 2p 11 > > < 14:9sin 8760 th þ 12 p Ta ¼ > 2p 11 > : 3sin th þ p 8760 12
2p 11 sin th þ p >0 8760 12 2p 11 sin th þ p V0 8760 12
ð21Þ
At the arch bottom of the tunnel 8 2p 11 2p 11 > > 14:2sin þ þ t p sin t p >0 h h < 8760 12 8760 12 Ta ¼ > 2p 11 2p 11 > :5sin th þ p sin th þ p V0 8760 12 8760 12
ð22Þ
At the left wall of the tunnel 8 2p 11 2p 11 > > sin th þ p >0 <14:1sin 8760 th þ 12 p 8760 12 Ta ¼ > 2p 11 2p 11 > :4sin th þ p sin th þ p V0 8760 12 8760 12
ð23Þ
Because the vent pipe is fitted in the right wall of the tunnel, the temperatures of the left wall and right wall of the tunnel are different. At the right wall of the tunnel 8 2p 11 2p 11 > > sin th þ p >0 <13:6sin 8760 th þ 12 p 8760 12 Ta ¼ > 2p 11 2p 11 > : 4:5sin th þ p sin th þ p V0 8760 12 8760 12
ð24Þ
After the total tunnel was finished on 30 September 2003, the airflow in the style of laminar flow will happen in the tunnel. Thus, the air temperatures inside of the tunnel are assumed to be the air temperatures in Eq. (20). The thermal conductivity of the insulation material with the thickness of 0.05 m is 0.03 W/(m.K). The initial temperatures of the rock surrounding the tunnel can be gained from the in situ measured data. The volumetric heat capacity and thermal conductivity of the rock are derived from the following formulae (Xu et al., 2001). Cu ¼
Cus þ xCw 1þx
ð25Þ
Cf ¼
Cfs þ xu Cw þ ðx xu ÞCi 1þx
ð26Þ
ku ¼ khs s khww
ð27Þ
kf ¼ khs s khwuu khi i
ð28Þ
Where hs, hw, hi represent the volumetric contents of the soil grain, moisture and ice in the soil, respectively. hu is the volumetric content of the nonfrozen moisture in the frozen soil. x stands for the mass content of the moisture in the soil. xu represents the mass content of the nonfrozen moisture in the frozen soil. Other parameters are the same as the forenamed. According to the character of the rock surrounding the tunnel, we can know the thermal parameters of the
Z. Xuefu et al. / Cold Regions Science and Technology 39 (2004) 19–31 Table 1 The thermal parameters of water, ice, frozen soil grain and thawed grain The thermal parameters
Value
Heat capacity of the ice Ci [kJ/(kg.K)] Heat capacity of the water Cw [kJ/(kg.K)] Heat capacity of the thawed soil grain Cus [kJ/(kg.K)] Heat capacity of the frozen soil grain Cfs [kJ/(kg.K)] Thermal conductivity of the ice ki [W/(m.K)] Thermal conductivity of the water kw [W/(m.K)] Thermal conductivity of the soil grain ks [W/(m.K)]
2.09 4.18 0.84 0.75 2.22 0.55 3.00
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from 2.24 to 3.25 m around the left wall would have had residual thawed layer with the thickness of 1.01 m, and that the thickness range from 1.87 to 3.66 m around the right wall would have had residual thawed layer with the thickness of 1.79 m by 31 March 2002. Fig. 4(b) displays that the thawed depth around the vault would have been 2.39 m, that the thawed depth around the left wall would have been 3.52 m, that the thawed depth around the right wall had been 3.75 m, and that the thawed depth around the arch bottom would have been 2.36 m by 30 September 2002.
water, the ice, the frozen soil grain and the thawed grain. These thermal parameters are listed in Table 1. From the in situ measured data, we know that the cross-section at DK976 + 410 of Kunlun mountain tunnel lies in ice-rich soil whose volumetric content of the moisture is 55% and whose transmissibility coefficients D of the moisture is equal to 2.2 10 9 m2/s. According to the experimental data, the relationship of the content of the nonfrozen moisture and the temperature can be shown by the following Eq. (29). 8 < 6:989AT A0:26 T VTm hu ¼ ð29Þ : hw T zTm Substituting the abovementioned parameters into the finite element formula, the calculated results are obtained by computers, as follows. 4.1. Predicting the refreezing of Kunlun mountain tunnel with insulation material whose thermal conductivity is equal to 0.03 W/m.k and whose thickness is equal to 0.05 m In light of the actual design and construction of Kunlun mountain tunnel, after the inner lining is fitted, the waterproof layer is installed first, the insulation material, whose thermal conductivity is equal to 0.03 W/m.K and whose thickness is equal to 0.05 m, is constructed second, then the waterproof layer is fixed again, the outer lining is fixed finally. Because the thickness the waterproof layer is very thin, the effects of the two waterproof layers can be ignored. The detail analysis is made, as follows. From Fig. 4(a), it is found that the rock surrounding the vault and the arch bottom of Kunlun mountain tunnel would have been frozen, that the thickness range
Fig. 4. The temperature distributions of the rock surrounding tunnel with insulation material in the first freezing – thawing cycle when the inner linings are constructed.
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From Fig. 6(a), it is seen that the thickness range from 1.76 to 2.49 m around the vault would have had residual thawed layer with the thickness of 0.73 m, that the thickness range from 1.15 to 4.70 m around the left wall would have had residual thawed layer with the thickness of 3.55 m, that the thickness range from 1.12 to 5.36 m around the right wall would have had residual thawed layer with the thickness of 4.24 m, and that the thickness range from 1.00 to 3.34 m around the arch bottom would have had residual thawed layer with the thickness of 2.34 m by 31 March 2004. From Fig. 6(b),
Fig. 5. The temperature distributions of the rock surrounding tunnel with insulation material in the second freezing – thawing cycle when the inner and outer linings are constructed.
Fig. 5(a) indicates that the thickness range from 0.39 to 2.31 m around the vault would have had residual thawed layer with the thickness of 1.92 m, and that the thawed depths around the left wall, the right wall and the arch bottom would have been 4.04, 4.60 and 2.73 m by 31 March 2003, respectively. From Fig. 5(b), it is shown that the thawed depths around the vault, the left wall, the right wall and the arch bottom would have been 2.68, 4.25, 4.88 and 2.80 m by 30 September 2003, respectively.
Fig. 6. The temperature distributions of the rock surrounding tunnel with insulation material in the third freezing – thawing cycle when the inner and outer linings are constructed.
Z. Xuefu et al. / Cold Regions Science and Technology 39 (2004) 19–31
it is shown that the thickness range from 3.75 to 4.04 m around the arch bottom would have had residual thawed layer with the thickness of 0.29 m, and that the thawed depths around the vault, the left wall and the right wall would have been 0.76, 4.56 and 5.34 m by 30 September 2004, respectively. It is obvious that, after three freezing – thawing cycles, the thawed extent caused by construction around the vault of Kunlun mountain tunnel will basically be refrozen, and that the thawed extent around the left wall, the right wall and the arch bottom will not yet be refrozen. Fig. 7(a) indicates that the rock surrounding the vault and the arch bottom would have been frozen, that the thickness range from 2.28 to 4.45 m around the left wall would have had residual thawed layer with the thickness of 2.17 m, and that the thickness range from 2.23 to 5.30 m around the right wall would have had residual thawed layer with the thickness of 3.07 m by 31 March 2005. From Fig. 7(b), it is seen that the thickness range from 3.44 to 3.91 m around the arch bottom would have had residual thawed layer with the thickness of 0.47 m, that the thickness range from 2.97 to 5.11 m around the arch bottom would have had residual thawed layer with the thickness of 2.74 m, and that the seasonal thawed depths around the vault and the arch bottom would have been 0.50 and 0.27 m by 30 September 2005, respectively. It is obvious that, by 31 March 2004, the thawed extent caused by construction around the vault and the arch bottom of Kunlun mountain tunnel will basically have been refrozen, that the thawed extent around the left wall and the right wall would not have been refrozen. However, by 30 September 2005, the thawed extent around the left wall and the right wall would not yet have been refrozen. Fig. 8(a) indicates that the rock around the vault, the left wall and the arch bottom would have been frozen, that the thickness range from 3.96 to 4.42 m around the right wall would have had residual thawed layer with the thickness of 0.46 m by 31 March 2006. From Fig. 8(b), it is seen that the thawed extent around the total tunnel would basically have been refrozen, and that there would only have been the seasonal thawing around the total tunnel by 30 September 2006. It is obvious that by 31 March 2006, the thawed extent around the right wall would not yet have been
27
Fig. 7. The temperature distributions of the rock surrounding tunnel with insulation material in the fourth freezing – thawing cycle when the inner and outer linings are constructed.
refrozen. However, by 30 September 2006, the thawed extent around the total tunnel would basically have been refrozen. 4.2. Predicting for the refreezing of Kunlun mountain tunnel without insulation material In order to research the function of the insulation material on the refreezing of permafrost tunnel, predictions for the refreezing of Kunlun mountain tunnel without insulation material are made, as follows.
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31 March 2003. From Fig. 9(b), it is shown that the thawed depths around the vault, the left wall, the right wall and the arch bottom would have been 2.73, 4.26, 4.86 and 2.93 m by 30 September 2003, respectively. From Fig. 10(a), it is seen that the rock around the vault will have been frozen, that the thickness range from 2.26 to 4.63 m around the left wall would have had residual thawed layer with the thickness of 2.73 m, that the thickness range from 2.20 to 5.22 m around the right wall would have had residual thawed layer with the thickness of 2.92 m, and that the
Fig. 8. The temperature distributions of the rock surrounding tunnel with insulation material in the fifth freezing – thawing cycle when the inner and outer linings are constructed.
Fig. 9(a) indicates that the thickness range from 0.87 to 2.30 m around the vault of Kunlun mountain tunnel would have had residual thawed layer with the thickness of 1.33 m, that the thickness range from 0.61 to 4.08 m around the left wall would have had residual thawed layer with the thickness of 3.47 m, that the thickness range from 0.60 to 4.59 m around the right wall would have had residual thawed layer with the thickness of 3.99 m, and that the thickness range from 0.09 to 2.78 m around the arch bottom would have had residual thawed layer with the thickness of 2.69 m by
Fig. 9. The temperature distributions of the rock surrounding tunnel without insulation material in the second freezing – thawing cycle when the inner and outer linings are constructed.
Z. Xuefu et al. / Cold Regions Science and Technology 39 (2004) 19–31
29
It is obvious that, after three freezing – thawing cycles, the thawed extent caused by construction around the vault and the arch bottom of Kunlun mountain tunnel will basically be refrozen, and that the thawed extent around the left wall and the right wall will not yet be refrozen. From Fig. 11(a), we can know that the thawed depth around the vault and the arch bottom would have been frozen, that the thickness range from 3.53 to 4.33 m around the left wall would have had residual thawed layer with the thickness of 0.80 m,
Fig. 10. The temperature distributions of the rock surrounding tunnel without insulation material in the third freezing – thawing cycle when the inner and outer linings are constructed.
thickness range from 1.85 to 3.32 m around the arch bottom would have had residual thawed layer with the thickness of 1.47 m by 31 March 2004. Fig. 10(b) indicates that the seasonal thawed depth around the vault would have been 0.91 m, that the seasonal thawed depth around the arch bottom would have been 0.34 m, that the thickness range from 3.32 to 4.49 m around the left wall would have had residual thawed layer with the thickness of 1.17 m, and that the thickness range from 3.33 to 5.09 m around the left wall would have had residual thawed layer with the thickness of 1.76 m by 30 September 2004.
Fig. 11. The temperature distributions of the rock surrounding tunnel without insulation material in the fourth freezing – thawing cycle when the inner and outer linings are constructed.
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and that the thickness range from 3.59 to 4.52 m around the left wall would have had residual thawed layer with the thickness of 0.93 m by 31 March 2005. Fig. 11(b), it is seen that the thawed extent around the total tunnel would basically have been refrozen, and that there would only have been the seasonal thawed around the total tunnel by 30 September 2005. It is obvious that, by 31 March 2005, the thawed extent caused by construction around the vault and the arch bottom of Kunlun mountain tunnel would basically have been refrozen, and that the thawed extent around the left wall and the right wall would not yet have been refrozen. After four freezing – thawing cycles, the thawed extent around the total tunnel would basically have been refrozen. Comparing Fig. 5 with Fig. 9, Fig. 6 with Fig. 10 and Fig. 7 with Fig. 11, the temperature distributions of the rock surrounding Kunlun mountain tunnel with insulation material are greatly different from the temperature distributions without insulation material at the same time. For example, from Figs. 7(b) and 10(b), by 30 September 2004, the thawed extent around the vault and the arch bottom of Kunlun mountain tunnel with insulation material would have been refrozen; the thawed extent around the left wall and the right wall would not yet have been refrozen. However, at the same time, the rock surrounding the Kunlun mountain tunnel without insulation material would have been refrozen, expect that there would be a little residual thawed layer around the vault. Thus, the insulation material will delay the refreezing of the permafrost tunnel.
5. Conclusions From the computational analyses mentioned above, we can make the following conclusions: (a) Fitting with insulation material whose thermal conductivity is equal to 0.03 W/(m.K) and thickness is 0.05 m, the vault of Kunlun mountain tunnel would have been refrozen by 31 March 2004; the left wall would have been refrozen by 31 March 2006; and the right wall would have been refrozen by 30 September 2006.
(b) The vault of Kunlun mountain tunnel without insulation material would have been refrozen by 31 March 2004; the arch bottom would have been refrozen by 30 September 2004; and the total tunnel would basically have been refrozen by 30 September 2005. (c) The insulation material will delay the refreezing of the thawed extent of the permafrost tunnel caused by the latent heat of the cast in situ concrete lining, insulating during excavating in winter and other artificial activities. (d) The thawed extent of the permafrost surrounding the tunnel in cold regions caused by construction must be considered, and the time of observation should be extended when their temperature and stress are observed in situ, or observation results will largely be different from the results in stability.
Acknowledgements This study was supported in part by the National Science Fund for Distinguished Young Scholars of China (40225001), in part by the National Natural Science Foundation of China (40171019), in part by the Foundation of ‘‘Hundred People Plan’’ of Chinese Academy of Sciences (to Dr. Y.M. Lai) and by the grant of the Knowledge Innovation Program of the Chinese Academy of Sciences (KZCX1-SW-04).
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