A prediction method for the maximum freezing depth of tunnels in seasonally frozen soil areas

Abstract

The present invention discloses a method for predicting the maximum frozen depth of a tunnel in a seasonal frozen soil area. Since latent heat is much greater than sensible heat, a functional relationship between freezing time and freezing front radius is obtained based on a quasi-steady-state assumption and a heat flow equation of a phase change moving interface. By integrating the functional relationship, a display expression of freezing time and freezing front radius can be obtained. After that, the maximum frozen depth of the tunnel in the seasonal frozen soil area can be obtained by substituting the equivalent longest freezing time into the display expression. Alternatively, the predicted value of the freezing front radius is continuously substituted into the display expression. When the predicted value of the freezing front radius is accurate to a predetermined value, the predicted value of the freezing front radius at this time is the predicted value of the maximum frozen depth of the tunnel in the seasonal frozen soil area. The prediction result of the present invention is close to the measured value and has high reliability.

Description

Technical Field

[0001] The present invention relates to the technical field of tunnel engineering in cold regions, and in particular to a method for predicting the maximum freezing depth of tunnels in seasonally frozen soil regions, providing a reference for cold-proof design and later maintenance of tunnel engineering in cold regions. Background Art

[0002] Affected by the severe cold climate, tunnel projects in high altitude or high latitude areas often suffer from frost damage such as lining ice, surrounding rock fragmentation, and concrete frost heave cracking in cold seasons, which has a very adverse effect on the normal function of the tunnel. On the one hand, the freezing depth of cold-zone tunnels is the reflection of the development depth of the sub-zero temperature in the radial structure, which directly affects the buried depth of the tunnel drainage ditch and the thickness of the insulation layer; on the other hand, the freezing depth is related to the size of the frost heave force, which in turn affects the degree of destruction of the surrounding rock and the degree of lining damage. Therefore, accurately and quickly calculating the maximum freezing depth of cold-zone tunnels is of great significance for the adoption of insulation defense measures, the determination of the buried depth of drainage ditches, and the selection of concrete antifreeze grades.

[0003] Many scholars have conducted in-depth research on the analytical calculation of frost depth in tunnels in cold regions. Zhang Yao published "A Practical Calculation Method for Design Parameters of Thermal Insulation Layer in Cold Region Tunnels" in "China Railway Science" in 2009 and ZHAO published "Temperature field characteristics and influencing factors on frost depth of a highway tunnel in a cold region" and "An analytical solution for frost heave force by the multifactor of coupled heat and moisture transfer in cold-region" in "Cold Regions Science and Technology" in 2020. In 2001, Lai Yuanming published an article entitled "Analytical Solution of Temperature Field of Circular Section Tunnels in Cold Regions" in "Glacier and Cryogenics", which adopted the idea of ​​asymptotic method and applied the perturbation method to obtain an approximate solution of the temperature field of tunnels in cold regions considering the phase change effect, so that the moving position of the phase change interface can be obtained, that is, the freezing depth of the tunnel surrounding rock can be obtained. However, it can only be calculated under the action of only one layer of surrounding rock material, and the influence of the insulation layer and lining cannot be considered. Moreover, it can only be calculated under specific initial ground temperature conditions of the surrounding rock, and the complex conditions of the current tunnel structure in cold regions cannot be effectively considered. In 2008, Xia Caichu published "Semi-analytical Method for Calculating the Maximum Freezing Depth of Surrounding Rock of Tunnels in Cold Regions" in "Rock and Soil Mechanics" and in 2022, LIANG published "Cold Regions Science" in "Cold Regions Science". In the paper "Temperature field spatio-temporal law and frozen-depth calculation of a tunnel in a seasonally frozen region" published in "Energy and Technology", a quasi-steady-state assumption and energy conservation method were used to propose an analytical calculation method for the maximum freezing depth of the surrounding rock of tunnels in cold regions. However, on the one hand, the calculation process is too complicated, and on the other hand, the solution obtained is implicit, which is difficult to master and use.

[0004] In view of the problems existing in the current methods, this application proposes a method for predicting the maximum frozen depth of tunnels in seasonal frozen soil areas. In the solution process of this method, since the latent heat is much greater than the sensible heat, based on the quasi-steady-state assumption and the heat flow equation of the freezing front, the functional relationship between the freezing time and the freezing front radius is obtained, and the explicit expression between the two can be obtained by integrating this. After that, the length of time that the inner surface temperature of the lining is below zero in a year is substituted into the formula to obtain the position of the freezing front radius, and the maximum frozen depth of the tunnel in the cold region can be obtained. Summary of the invention

[0005] The purpose of the present invention is to provide a method for predicting the maximum freezing depth of a tunnel in a seasonally frozen soil area in view of the technical defects in the prior art in predicting the maximum freezing depth of a tunnel in a cold region.

[0006] The technical solution adopted to achieve the purpose of the present invention is:

[0007] A method for predicting the maximum freezing depth of a tunnel in a seasonally frozen soil area comprises the following steps:

[0008] Step 1: Calculate the equivalent temperature of the inner surface of the lining and the duration of the lining inner surface temperature below zero in a year :

[0009] ;

[0010] in: is the equivalent temperature of the inner surface of the lining, in °C; is the annual average temperature of the inner surface of the lining, in °C; is the annual amplitude of the temperature on the inner surface of the lining, in °C; and They are the time for the inner surface temperature of the lining to change from positive temperature to negative temperature and from negative temperature to positive temperature in one year, in seconds; The duration of time that the inner surface temperature of the lining is below zero in one year, in seconds;

[0011] Step 2: Establish a freeze time Freeze front radius The explicit expression for is:

[0012] ;

[0013] in: They are the thermal conductivity of the lining, the thermal conductivity of the frozen surrounding rock, and the thermal conductivity of the unfrozen surrounding rock, in W·m -1 ℃ -1 ; is the volume phase change latent heat, in J·m -3 ; is the freezing time, in seconds; is the exponential integral function; and They are the inner surface radius of the lining, the interface radius between the lining and the surrounding rock, the freezing front radius and the outer boundary radius of the surrounding rock, in m;

[0014] Equivalent temperature of inner surface of lining Calculated by step 1; is the freezing front temperature, in °C; is the outer boundary temperature of the surrounding rock (the radial depth of the tunnel is The temperature at the location), in °C;

[0015] Step 3: Calculate the maximum freezing depth of tunnels in seasonally frozen soil areas:

[0016] In the explicit expression, take the freezing time The length of time that the inner surface temperature of the lining is below zero in a year , substituting this formula into the calculation, we can get the corresponding freezing front radius at this time Value , that is, the maximum value that the freezing front radius can reach within a year;

[0017] Alternatively, keep changing the Freeze Front Radius Substitute the predicted value into the explicit expression and solve the freezing time multiple times If you get Greater than Reduce the freezing front radius The predicted value is then brought back into the explicit expression for calculation. Conversely, when we obtain Less than Increase the freezing front radius The predicted value is substituted into the explicit expression to solve. When the front radius is frozen When the predicted value is accurate to the predetermined value, the front radius is frozen at this time The predicted value is the maximum value that the freezing front radius can reach within a year. ;

[0018] The predicted value of the maximum frozen depth of a tunnel in a seasonally frozen soil area is defined as the difference between the maximum value of the freezing front radius that can be reached within a year and the radius of the interface between the lining and the surrounding rock. The predicted value of the maximum frozen depth of a tunnel in a seasonally frozen soil area can be calculated according to the following formula: :

[0019] ;

[0020] in: It is the predicted value of the maximum freezing depth of the tunnel in the seasonal frozen soil area, in meters.

[0021] In the above technical solution, in step 2, ,or Taken from experience or literature records.

[0022] In the above technical solution, in step 2, , The temperature of the surrounding rock of the isothermal layer, in degrees Celsius. The isothermal layer refers to the area where the temperature no longer changes with time and remains unchanged all year round when reaching a certain depth from the surface. is the tunnel burial depth, in m; is the burial depth of the isothermal layer, in m; is the influence depth of the tunnel surrounding rock temperature field, in m; is the geothermal gradient, in units of .

[0023] In the above technical solution, the predetermined value in step 3 is 0.01 m.

[0024] Compared with the prior art, the present invention has the following beneficial effects:

[0025] 1. In the present invention, the temperature distribution of each layer of the tunnel material under the quasi-steady state is brought into the heat flow equation on the phase change moving interface for solution, so the calculation mechanism of the method is clearer and easier to understand and master.

[0026] 2. The present invention can obtain a display solution for the maximum freezing depth of tunnels in cold regions. When the corresponding parameters are known, the corresponding explicit expressions can be directly brought in for calculation, thereby making the prediction process more convenient and quick. BRIEF DESCRIPTION OF THE DRAWINGS

[0027] Figure 1 This is a schematic diagram of the equivalent negative accumulated temperature in the seasonal frozen soil area of ​​the present invention.

[0028] Figure 2 It is a schematic diagram of the radial structure of the cold region tunnel in the present invention.

[0029] Figure 3 The freezing time of the RK6+645 section of the cold region tunnel selected in the present invention is Freeze front radius relationship diagram. DETAILED DESCRIPTION

[0030] The present invention is further described in detail below in conjunction with specific embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not used to limit the present invention.

[0031] A method for predicting the maximum freezing depth of a tunnel in a seasonally frozen soil area comprises the following steps:

[0032] Step S1, model assumptions: In actual engineering, the tunnel structure is complex and the following assumptions need to be made: (1) The tunnel cross section is circular; (2) The contact between the structural layers is good, and the contact thermal resistance between the layers is ignored; (3) The radius value at the constant temperature of the outer boundary of the tunnel surrounding rock remains unchanged; (4) The tunnel structure layers are continuous, homogeneous, isotropic materials.

[0033] Affected by the external environment, the temperature in the tunnel changes in a sinusoidal manner with an annual cycle. The annual average temperature in tunnels in seasonally frozen areas is generally higher than 0°C. In addition, there is a period of time throughout the year when the temperature in tunnels is If the indoor temperature is below 0°C, Figure 1 As shown. The equivalent temperature of the inner surface of the lining is calculated using formula (1): ;

[0034] ;

[0035] Formula (1) Figure 1 middle, is the equivalent temperature of the inner surface of the lining, in °C; is the annual average temperature of the inner surface of the lining, in °C; is the annual amplitude of the temperature on the inner surface of the lining, in °C;

[0036] The duration of time that the inner surface temperature of the lining is below zero in one year, in seconds; , and They are respectively the time when the inner surface temperature of the lining changes from positive temperature to negative temperature and the time when the negative temperature changes to positive temperature in one year, in seconds.

[0037] S2, solve the temperature distribution of the tunnel structure under quasi-steady state conditions:

[0038] The radial structure of the tunnel is as follows Figure 2 As shown. Since the latent heat of phase change is much greater than the sensible heat, the temperature distribution under quasi-steady state conditions can be used for calculation. At this time, the heat transfer differential equation of each layer is:

[0039] (2);

[0040] The boundary conditions are:

[0041] (3);

[0042] (4);

[0043] (5);

[0044] (6);

[0045] (7);

[0046] in: is the radius, in m; For the The thermal diffusivity of the layer is When it is 1, 2, and 3, it represents lining, frozen surrounding rock, and unfrozen surrounding rock, respectively, and the unit is ; For the The temperature of the layer, in °C; and are the freezing front temperature and the surrounding rock outer boundary temperature, both in °C; and They are the inner surface radius of the lining, the interface radius between the lining and the surrounding rock, the freezing front radius and the outer boundary radius of the surrounding rock, in m; and are the thermal conductivity of the lining and the thermal conductivity of the frozen surrounding rock, both in units of .

[0047] By solving equations (2) to (7), we can obtain a temperature distribution of the tunnel structure as follows:

[0048] (8);

[0049] (9);

[0050] (10);

[0051] in:

[0052] (11);

[0053] In formula (10), , is the temperature of the surrounding rock of the isothermal layer, in degrees Celsius; the isothermal layer refers to the area where the temperature no longer changes with time and remains unchanged all year round when reaching a certain depth from the surface; is the tunnel burial depth, in m; is the burial depth of the isothermal layer, in m; is the influence depth of the tunnel surrounding rock temperature field, in m; is the geothermal gradient, in units of .

[0054] Step S3, solving the heat flow equation of the freezing front:

[0055] For the freezing front, since it will produce phase change during the movement, its heat flow equation is:

[0056] (12);

[0057] in: is the thermal conductivity of the unfrozen surrounding rock, in ; is the volume phase change latent heat, in units of ; is the freezing time, in seconds.

[0058] Substituting equations (9), (10) and (11) into equation (12), we can obtain:

[0059] (13);

[0060] make: (14);

[0061] (15)

[0062] Then, from formula (13), we can get:

[0063] (16);

[0064] Integrating both sides, we get:

[0065] (17);

[0066] In the formula, Is a constant.

[0067] Since the initial conditions are: (18);

[0068] Therefore, the values ​​that can be obtained are:

[0069] (19);

[0070] Therefore, the relationship between time and freezing front radius can be obtained as follows:

[0071] (20);

[0072] Substituting equations (14) and (15) into equation (20) for calculation, we can obtain:

[0073] ;

[0074] in: is the exponential integral function.

[0075] This results in freezing time Freeze front radius The display relationship of . Take the freezing time The length of time that the inner surface temperature of the lining is below zero in a year , and the equivalent temperature of the inner surface of the lining in formula (1) and other corresponding parameters are directly substituted into equation (21) to solve, then the freezing front radius is The value is the maximum value that the freezing front radius can reach within a year .

[0076] You can also continuously adjust the Freeze Front Radius Substitute the predicted value into formula (21) and solve the freezing time multiple times If you get Greater than Reduce the freezing front radius The predicted value of is re-substituted into formula (21) for calculation. On the contrary, when we get Less than Increase the freezing front radius Substitute the predicted value of into equation (21) to solve. When the radius of the freezing front is accurate to 0.01 m, it can be considered The predicted value is the maximum value that the freezing front radius can reach within a year. .

[0077] The predicted value of the maximum frozen depth of tunnels in seasonal frozen soil areas is the difference between the maximum value of the freezing front radius that can be reached within one year and the radius of the interface between the lining and the surrounding rock. The predicted value of the maximum frozen depth of tunnels in seasonal frozen soil areas can be calculated according to the following formula:

[0078] ;

[0079] in: It is the predicted value of the maximum freezing depth of the tunnel in the seasonal frozen soil area, in meters.

[0080] Example 2

[0081] The Alatan Tunnel is located in Zhalute Banner, Tongliao City, Inner Mongolia Autonomous Region. The left and right lines are 775 m and 730 m long respectively. It is a controlling project on the Tongliao to Huolin River National Highway and a tunnel in the seasonal frozen soil zone.

[0082] The effectiveness of the analytical method proposed in this paper is verified by using the temperature data of the Alatan Tunnel on-site monitoring. The tunnel is equivalent to the inner surface radius of the lining. 5.5 m, the radius of the interface between lining and surrounding rock is 6.2 m (i.e. the lining thickness is 0.7 m) and the outer boundary radius of the surrounding rock The volume phase change latent heat is 12 m. Pick , thermal conductivity of lining, thermal conductivity of frozen surrounding rock and thermal conductivity of unfrozen surrounding rock , , Take separately , and .

[0083] The maximum freezing depths of sections RK6+645, RK6+665, RK6+690, and RK6+740 were predicted using the method of Example 1 (R refers to the right line of the highway, and K6+645, K6+665, K6+690, and K6+740 refer to the distances of these points from the starting point of the route, which are 6.645 km, 6.665 km, 6.690 km, and 6.740 km, respectively), and compared with the calculation results of the Stefan formula and the field measurement results. Among them, the annual average temperature of the inner surface of the lining at the four sections is The annual amplitude of the temperature on the inner surface of the lining is 0.70 ℃, 2.83 ℃, 2.70 ℃, and 2.46 ℃ respectively. The freezing front temperatures are 13.20 ℃, 11.02 ℃, 10.65 ℃, and 9.90 ℃ respectively. The outer boundary temperature of the surrounding rock is 0 °C. The temperature is 7 ℃.

[0084] The calculation results are as follows:

[0085] The duration of the inner surface temperature of the lining at section RK6+645 being below zero in one year is 1.5235×10 7 s, equivalent temperature of inner surface of lining The maximum freezing depth of tunnels in seasonal frozen soil areas is -7.98 ℃ and the measured freezing depth is 1.50 m. The predicted value of the maximum freezing depth of tunnels in seasonal frozen soil areas obtained by the method of the present invention is is 1.93 m, while the frozen depth calculated according to Stefan’s formula is 3.76 m;

[0086] The duration of the inner surface temperature of the lining at section RK6+665 being below zero in one year is 1.3161×10 7 s, equivalent temperature of inner surface of lining The maximum freezing depth of tunnels in seasonal frozen soil areas is -5.29 ℃ and the measured freezing depth is 1.50 m. The predicted value of the maximum freezing depth of tunnels in seasonal frozen soil areas obtained by the method of the present invention is is 1.36 m, while the frozen depth calculated according to Stefan’s formula is 2.85 m;

[0087] The duration of the inner surface temperature of the lining at section RK6+690 being below zero in one year is 1.3195×10 7 s, equivalent temperature of inner surface of lining The maximum freezing depth of tunnels in seasonal frozen soil areas is -5.14 ℃ and the measured freezing depth is 1.50 m. The predicted value of the maximum freezing depth of tunnels in seasonal frozen soil areas obtained by the method of the present invention is is 1.35 m, while the frozen depth calculated according to Stefan’s formula is 2.81 m;

[0088] The duration of the inner surface temperature of the lining at section RK6+740 being below zero in one year is 1.3247×10 7 s, equivalent temperature of inner surface of lining The maximum freezing depth of tunnels in seasonal frozen soil areas is -4.81 ℃ and the measured freezing depth is 0.85 m. The frozen depth calculated by Stefan formula is 1.26 m, while the frozen depth calculated by Stefan formula is 2.72 m.

[0089] It can be seen that the predicted value of the maximum frozen depth of tunnels in seasonal frozen soil areas obtained by the method of the present invention is generally close to the measured value, and most of the predicted values ​​are within 0.5 m of the measured value. However, the Stefan formula cannot consider the influence of multi-layer material media in the tunnel structure, and the solution result is too large.

[0090] Taking the RK6+645 section as an example, the calculated freezing time is Freeze front radius The curve diagram is as follows Figure 3 When the freezing front radius When the freezing time is greater than 7.8 m, the corresponding curve has no actual physical meaning and should be discarded. Growth, Freeze Front Radius It grows larger and freezes the front radius over time The movement speed gradually decreases; when the freezing time As it approaches infinity, the radius of the freezing front It is about 7.8 m.

[0091] The above is only a preferred embodiment of the present invention. It should be pointed out that, for ordinary technicians in this technical field, several improvements and modifications can be made without departing from the principle of the present invention. These improvements and modifications should also be regarded as the scope of protection of the present invention.

Claims

1. A method for predicting the maximum freezing depth of tunnels in seasonally frozen soil areas, characterized in that: The following steps are involved: Step 1: Calculate the equivalent temperature of the inner surface of the lining and the duration of the lining inner surface temperature below zero in a year : ； in: is the equivalent temperature of the inner surface of the lining, in °C; is the annual average temperature of the inner surface of the lining, in °C; is the annual amplitude of the temperature on the inner surface of the lining, in °C; and They are the time for the inner surface temperature of the lining to change from positive temperature to negative temperature and from negative temperature to positive temperature in one year, in seconds; The duration of time that the inner surface temperature of the lining is below zero in one year, in seconds; Step 2: Establish the freeze time Freeze front radius The explicit expression for is: ； in: They are the thermal conductivity of the lining, the thermal conductivity of the frozen surrounding rock, and the thermal conductivity of the unfrozen surrounding rock, in W·m -1 ℃ -1 ; is the volume phase change latent heat, in J·m -3 ; is the freezing time, in seconds; is the exponential integral function; and They are the inner surface radius of the lining, the interface radius between the lining and the surrounding rock, the freezing front radius and the outer boundary radius of the surrounding rock, in m; Equivalent temperature of inner surface of lining Calculated by step 1; is the freezing front temperature, in °C; is the outer boundary temperature of the surrounding rock, in °C; Step 3: Calculate the maximum freezing depth of tunnels in seasonally frozen soil areas: In the explicit expression, take the freezing time The length of time in a year when the inner surface temperature of the lining is below zero , substituting this formula into the calculation, we can get the corresponding freezing front radius at this time Value , that is, the maximum value that the freezing front radius can reach within a year; Alternatively, keep changing the Freeze Front Radius Substitute the predicted value into the explicit expression and solve the freezing time multiple times If you get Greater than Reduce the freezing front radius The predicted value is then brought back into the explicit expression for calculation. Conversely, when we obtain Less than Increase the freezing front radius The predicted value is substituted into the explicit expression to solve. When the front radius is frozen When the predicted value is accurate to the predetermined value, the front radius is frozen at this time The predicted value is the maximum value that the freezing front radius can reach within a year. ; The predicted value of the maximum frozen depth of a tunnel in a seasonally frozen soil area is defined as the difference between the maximum value of the freezing front radius that can be reached within a year and the radius of the interface between the lining and the surrounding rock. The predicted value of the maximum frozen depth of a tunnel in a seasonally frozen soil area can be calculated according to the following formula: : ； in: It is the predicted value of the maximum freezing depth of the tunnel in the seasonal frozen soil area, in meters.

2. The method for predicting the maximum frozen depth of a tunnel in a seasonally frozen soil area according to claim 1, characterized in that: In the step 2, ,or Taken from experience or literature records.

3. The method for predicting the maximum frozen depth of a tunnel in a seasonally frozen soil area according to claim 1, characterized in that: In the step 2, , The temperature of the surrounding rock of the isothermal layer, in degrees Celsius. The isothermal layer refers to the area where the temperature no longer changes with time and remains unchanged all year round when reaching a certain depth from the surface. is the tunnel burial depth, in m; is the burial depth of the isothermal layer, in m; is the influence depth of the tunnel surrounding rock temperature field, in m; is the geothermal gradient, in °C·m -1 .

4. The method for predicting the maximum frozen depth of a tunnel in a seasonally frozen soil area according to claim 1, characterized in that: The predetermined value in step 3 is 0.01 m.

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