V O ~ .42 supp.
SCIENCE IN CHINA (Series D)
A U ~ U S1999
~
A convection-conduction model for analysis of the freeze-thaw
conditions in the surrounding rock wall of a
tunnel in permafrost regions *
HE Chunxiong (
8@ ) ,
(State Key Laboratory of Frozen Soil Engineering, Lanzhou Institute of Glaciology and Geocryology,
Chinese Academy of Sciences, Lanzhou 730000, China; Department of Applied Mathematics,
South China University of Technology, Guangzhou 510640, China)
WU Ziwang ( 3? fXP EE ) and ZHU Linnan (
M)
(State key Laboratory of Frozen Soil Engineering, Lanzhou Institute of Glaciology and Geocryology,
Chinese Academy of Sciences, Lanzhou 730000, China)
Received February 8 , 1999
Based on the analyses of fundamental meteorological and hydrogeological conditions at the site of a tunnel
Abstract
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 o h w e d . Then, based on the in situ conditions of air 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
l
is now under construction is
tunnel has been obtained, and the freezethaw conditions at the Dabanahan T u ~ e which
predicted.
Keywords:
tunnel in cold regions, convective heat exchange and heat conduction, freezethaw.
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 destroyed and replaced mainly by the air convections without the heat radiation, the
freeze and thaw conditions determined principally by the temperature and velocity of air flow
* Project
supported by the National Natural Science Foundation of China (Grant No. 4%71020).
2
SCIENCE IN CHINA
(Series
D)
Vol. 42
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,
~unardini"] discussed the freeze and thaw conditions by the approximate formulae obtained by Shamsundarr2' in study of freezing outside a circular tube with axial variations of coolant temperature. We
~imulated'~'
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 exchange and heat conduction in the surrounding rock material into one model, and simulate the freezethaw conditions of the surrounding rock material based on the in situ conditions of air temperature, atmospheric pressure, wind force at the entty and exit of the tunnel, and the conditions of hydrogeology
and engineering geology.
1 Mathematical model
In order to construct an appropriate model, we need the in situ fundamental conditions as a basis. Here we use the conditions at the scene of the Dabanshan Tunnel. The Dabanshan Tunnel is located on the highway from Xining to Zhangye, south of the Datong River, at an elevation of
3 754.78-3 801 .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 OS: 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 northeastl~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:
SUPP.
A MODEL ANALYZING THE FREEZE-THAW CONDITION
3
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 ; Sf( t ) ,
S u( t ) are frozen and thawed parts in the surrounding rock materials respectively ; A f , A, and Cf , C,
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 Tois critical freezing temperature of rock (here we assume To = - 0.1 'T ) .
2
Procedure used for solving the model
Equation ( 1 ) shows that the temperature of the surrounding rock does not affect the speed of air
flow. We first solve those equations concerning the speed of air flow, and then solve those equations
concerning temperature at every 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 effective pressure p :
4
SCIENCE IN CHINA (Series D)
Vol. 42
Then we solve equations in (1) using the following procedures :
( i ) Assume the values for UO, p;
(ii) substituting UO , into eq. ( 2 ) , and solving ( 2 ) , we obtain
(iii) solving the first and second equations of ( 1 ) , we obtain
v';
(iv) solving the first and third equations of ( 1 ) , we obtain u',
(v) calculating the momentum-averageE4]of U' , V' and u2, V2, we obtain the new UO, VO;
then return to (ii) ;
(vi) iterating as above until the disparity of those solutions in two consecutive iterations is suffias the initial values for the
ciently small or is satisfied, we then take those values of
U0 and
next elapse and solve those equations concerning the temperature.
u',
IF;
2.2
Entire method used for solving the energy equations
As 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 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 equations 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 concerning 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 reference [ 3 ] .
2.3
2.3.1
Determination of thermal parameters and initial m d boundary conditions
Determination of the thermal parameters.
- 0.108 8 H, we calculate
P
using formula p = - , where T is
GT
Using p = 1 013.25
the air pressure p at elevatian H and calculate the air density p
the yearly-average absolute air temperature, and G is the humidity constant of air. Letting Cp be the
thermal capacity with fixed pressure, h the thermal conductivity, and p the dynamic viscosity of air
A
flow, we calculate the thermal conductivity and kinematic viscosity using the formulas a = - and v
CPP
= &. The thermal parameters of the surrounding rock are determined &om the tunnel site.
P
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) and p = ( 1 t k L / d )
x v2/2['] 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 = 2 R , 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
value based on the position of the ~ermafrostbase and the geothermal gradient of the thaw rock materials beneath the permafrost base.
Supp .
A MODEI. ANAI.YZISG
THE FREEZE-THAF CONDITION
5
A simulated example
3
Lsing the model and the solving method mentioned above, we simulate the varying law of the air
temperature in the tunnel along with the temperature at the entry and exit of the Xiluoqi No. 2 Tunr ~ e l .We observe that the simulated results are close to the data observedi6' .
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 1 6 0 m running from the northwest to
the southeast, with the entry of the tunnel in the northwest, and the elevation is about 7 0 0 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 data observed, we
approximate the varying sine law of air temperature at the entry and exit with yearly averages of
- 5 "C ,
- 6 . 4 "C and amplitudes of 18 . 9 "C and 17 . 6 "C respectively. The equivalent diameter is
5 . 8 m , and the resistant coefficient along the tunnel wall is 0.025. Since the effect of the thermal
parameter 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 I shows the simulated yearly-average air temperature inside and at the entry and exit 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 inside (distance greater than 100 m from the entry and exit) the tunnel. W e observe 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.
F
,
0
5
0
s
0
0
\O
0
0
M
O
E:
I>istancc from rhc cntrylm
Fig. I .
-
0
'0
Comparison of simulated and observed air temper-
Fig. 2 .
The comparison of simulated and observed air tem-
ature in Xiluoqi No. 2 Tunnel in 1979. 1 , Simulated val-
perature inside the Xiluoqi No. 2 Tunnel in
ues; 2 , observed sduc.s.
lated values ; 2 , observed values.
4
1979. 1 , Simu-
Prediction of the freeze-thaw conditions for the Dabanshan Tunnel
4.1
Thermal parameter and initial and boundary conditions
Using the elevation of 3 800 m and the yearly-average air temperature of - 3"C, we calculate
the air density p = 0 . 7 7 4 k g / m 3 . Since steam exists in the a i r , we choose['] the thermal capacity
with a fixed pressure of air Cp = I . 8 7 4 4 k ~ / ( k g -OC ) , heat conductivity A = 2.0 x 1 0 - ~ ~ / ( m . )
and the dynamic viscosity p = 9 . 2 1 8 x 1 0 - 6 k g / ( m . ~ ) . After calculation we obtain the thermal diffusivity cx = 1 . 3 7 8 8 x 1 0 - S m 2 / s and the kinematic viscosity v = 1 . 1 9 x 1 0 - 5 m 2 / s .
6
Vol. 42
SCIENCE IN CHINA (Series D)
Considering that the section of automobiles is much smaller than that of the tunnel and the automobiles pass through the tunnel at a low speed, we ignore the piston effects, coming from the movement of automobiles, in the diffusion of the air.
We consider the rock as a whole component and choose the dry volumetric gravity Yd = 2 400
kg/m3, content of water and unfrozen water W = 3 % and W, = 1% , and the thermal conductivity A ,
= 1 . 9 W/m . T , A f = 2 . 0 W/m * " C , heat capacity C , = 0. 8 kJ/kg *OC, and Cf =
(0.8 + 4.128 W,)
( 0 . 8 + 4.128 W) y d .
X Yd, C u =
1+ W
1+ W
According to the data observed at the tunnel site, the maximum monthly-average wind speed is
about 3.5 m/s , and the minimum monthly-average wind speed is about 2 . 5 m/s We approximate
the windspeed at the entry and exit as V ( t ) = [ 0 . 0 2 8 ~( t - 7 ) 2 + 2 . 5 ] ( m / s ) , 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 be
where f ( x ) is the distance from the vault to the permafrost base, and Ro = 25 m is the radius of domain of solution T. We assume that the geothermal gradient is 3 % , the yearly-average air temperature outside the tunnel is A = - 3°C , and the amplitude is B = 120C .
As for the boundary of R = Ro, we first solve the equations considering R = Ro as the first type
of boundary; that is we assume that T = f ( x ) x 3 % OC on R = Ro We find that, after one year, the
heat flow trend will have changed in the range of radius between 5 and 25 m in the surrounding rock.
Considering that the rock will be cooler hereafter and it will be affected yet by geothermal heat, we
appoximately 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, ig the gradient on R = Ro of T.
Considering the surrounding 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 values after many time elapses).
.
4 . 2 Calculated results
Figures 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.
SUPP.
A MODEL ANALYZING THE FREEZE-THAW CONDITION
7
Distance from entrylm
Fig. 3 .
The monthly-average temperatures on the surface of
Dabanshan Tunnel. I , the Ith month, I = 1 , 2 ,
,12.
Fig. 4 .
Comparison of the monthly-average temperature
on the surface of tunnel with that outside the tunnel. 1 , Inner (distance
> 100 m fmm entry and exit) temperature on
the surface ; 2 , outside air temperature.
Distance from entrytm
Fig. 5 .
The year when pennafmst begins to form in dif-
ferent sections of the surrounding mck .
4.3
Distance from entry/m
Fig. 6.
The maximum thawed depth after permafmst
formed in different years.
Preliminary conclusion
Based on the initial-boundary conditions and thermal parameters mentioned above, we obtain the
following preliminary conclusions :
1) The yearly-average temperature on the surface wall of the tunnel is approximately equal to the
air temperature at the entry and exit. It is warmer during the cold season and cooler during the warm
season in the internal part (more than 100 m from the entry and exit) of the tunnel than at the entry
and exit. Fig. 1 shows that the internal monthly-average temperature on the surface of the tunnel wall
is 1 . 2 T higher in January, February and December, 1 OC higher in March and October, and 1 .6 OC
lower in June and August, and 20C lower in July than the air temperature at the entry and exit. In
other months the internal temperature on the surface of the tunnel wall approximately equals the air
temperature at the entry and exit.
2) Since it is affected by the geothermal heat in the internal surrounding section, especially in
the central part, the internal amplitude of the yearly-average temperature on the surface of the tunnel
wall decreases and is 1.6T lower than that at the entry and exit.
3) Under the conditions that the surrounding rock is compact, without a great amount of underground water, and using a thermal insulating layer (as designed PU with depth of 0.05 m and heat
conductivity A = 0.021 6 W/mo OC , FBT with depth of 0.085 m and heat conductivity A = 0.051 7
W/m. OC ) , in the third year after tunnel construction, the surrounding rock will begin to form permafrost in the range of 200 m from the entry and exit. In the first and the second year after construc-
SCIENCE IN CHINA (Series D)
8
Vol. 42
tion, the surrounding rock will begin to form permafrost in the range of 40 and 100 m from the entry
and exit respectively. In the central part, more than 200 m from the entry and exit, permahst will
begin to form in the eighth year. Near the center of the tunnel, permafrost will appear in the 1415th years. During the first and second years after permafrost formed, the maximum of annual thawed
depth is large (especially in the central part of the surrounding rock section) and thereafter it decreases every year. The maximum of annual thawed depth will be stable until the 19-20th years and will
remain in a range of 2-3 m.
4) If permafrost forms entirely in the surrounding rock, the permafrost will provide a water-isolating layer and be favourable for communication and transportation. However, in the process of construction, we found a lot of underground water in some sections of the surrounding rock. It will permanently exist in those sections, seeping out water and resulting in freezing damage to the liner layer.
Further work will be reported elsewhere.
References
1
2
3
4
5
6
7
Lunardini , V . J . , Heat Transfer with Freezing and Thawing, Amsterdam: Elsevier Science Publishers, 1991, 75-86.
Shamsundar, N. , Formulae for freezing outside a circular tube with axial variation of coolant temperature, Znt . J. Heat M a s
Transfer, 1982, 25(10) : 1614.
He Chunxiong, Wu Ziwang, Preliminary prediction of freeze-thaw situations in surrounding rock of the Dabanshan Tunnel, in
Proceedings of the F$h Chinese Conference on Glaciology and Geocryology ( i n Chinese), Vol. 1 , Lanzhou: Gansu Cultural
Press, 19%, 419-425.
Chen , J . C . , Fluid Dynamics and Heat Transfer ( i n Chinese) , Beijing : Press of National Defence Industry, 1984, 245-369.
Nie , F .M . , Dynanuc state of air temperature in railway tunnel in cold regions, Jownal of Glaciology Md Geocryology (in Chinese), 1988, 10(4) : 450.
Jin , X .E . , Chen , W .Y . , Ventilation of Twlnel and Air Dynamics in Twlnel ( in Chinese) , Beijing : Press of Chinese Railway, 1983, 1-49.
Chen Shixiong, Chen Chuangmai, Meteorology (in Chinese) , Beijing : Press of Agriculture, 1982, 211-231.