Analytical method for leakage problem of three-dimensional tunnel segment joint

Abstract

The invention relates to an analytical method for a three-dimensional tunnel segment joint leakage problem, and the method comprises the steps: converting a complex three-dimensional seepage problem into a two-dimensional annular region problem through conformal transformation, solving the two-dimensional annular region problem, converting the solution into an original three-dimensional plane through inverse transformation, obtaining an explicit analytical solution of a total water head of a seepage field, and effectively reducing the calculation difficulty and cost; through the combined application of conformal transformation, variable substitution and a variable separation method, the accuracy and reliability of an analytical solution are improved, the pore water pressure and the leakage amount at the tunnel joint are accurately calculated, an accurate seepage field analytical solution is obtained instead of an approximate solution or an improved analytical solution on the basis of other people, and the method has the advantages of being high in practicability and easy to popularize. And a scientific basis is provided for leakage prevention of tunnel engineering.

Description

Technical Field

[0001] The invention belongs to the technical field of tunnel engineering, and in particular relates to an analytical method for the leakage problem of joints of three-dimensional tunnel segments. Background Art

[0002] In tunnel engineering, leakage at lining joints has always been an important factor affecting the safety and stability of tunnel structures. In the prior art, there are generally approximate solutions or improved solutions based on others for the analytical solution analysis of seepage field for tunnel leakage. Although these solutions can reflect the leakage phenomenon to a certain extent, they have the problems of complex calculation, high cost, and difficulty in obtaining accurate analytical solutions. Therefore, the present invention provides an efficient and accurate analytical method to analyze the leakage problem at tunnel joints. Summary of the invention

[0003] The purpose of the present invention is to overcome the deficiencies in the prior art and provide an analytical method for the problem of leakage in the joints of three-dimensional tunnel segments.

[0004] The analytical method for the leakage problem of the joints of the three-dimensional tunnel segments comprises the following steps:

[0005] Step 1: Establish a 3D tunnel segment joint leakage model and determine the basic assumptions and boundary conditions of the model;

[0006] Step 2: Decompose the total head function using the separation of variables method, and solve the equation for a specific solution in combination with boundary conditions;

[0007] Step 3: Using the principle of conformal transformation, the three-dimensional seepage problem is transformed into a two-dimensional annular region problem for solution, and then the solution is transformed back to the original three-dimensional plane through the inverse transformation of the conformal transformation to obtain the explicit analytical solution of the total water head of the seepage field;

[0008] Step 4: Based on the explicit analytical solution of the total water head of the seepage field and the Bernoulli equation, the analytical solution of the pore water pressure at the tunnel joint is obtained;

[0009] Step 5: Based on the explicit analytical solution of the total water head of the seepage field, the analytical solution of the normal hydraulic gradient at the leakage point of the tunnel lining joint is obtained, and then the leakage amount of the total model is obtained.

[0010] Preferably, in step three, based on the conformal transformation of the complex function, each vertical plane with x=const in the seepage area in the original three-dimensional plane is mapped to the circular area in the ω plane, and the three-dimensional seepage problem is converted into a two-dimensional circular area problem, and the non-steady-state equation after the decomposition of the total head function in step two is solved.

[0011] Preferably, in step 2, the total head function is decomposed into two independent equations using the separation of variables method:

[0012]

[0013] Among them, U and are spatial functions about coordinates (y, z) and x respectively; λ is an undetermined parameter, and A is a calculation constant;

[0014] In step three, the conformal transformation formula is substituted into the first formula in the above formula, and the general solution is obtained after variable substitution. Then, according to the boundary conditions and velocity continuity conditions determined in step one, the explicit analytical solution of the total water head of the seepage field is obtained.

[0015] Preferably, in step 3, an explicit analytical solution of the total water head of the seepage field is obtained:

[0016]

[0017] in, where i=1 represents the characteristic function of the lining leakage interval, and i=2 represents the characteristic function of the lining non-leakage interval; U in In (), i=1 represents the characteristic function of the lining leakage interval, i=2 represents the characteristic function of the lining non-leakage interval, and l is the longitudinal length of the leakage at the lining joint.

[0018] Preferably, in step one, the tunnel with a non-circular cross-section is simplified to a circular cross-section; the parts at infinity away from the tunnel and where no water leakage occurs in the tunnel lining are regarded as impermeable boundaries, and the cross-section at the tunnel cutoff corresponding to the boundary of the calculation domain is regarded as a normal impermeable vertical boundary.

[0019] As a preferred method, in step 2, the total head function is decomposed into:

[0020]

[0021] Combining Darcy's law, we get:

[0022]

[0023] Among them, U and are spatial functions about coordinates (y,z) and x respectively.

[0024] The beneficial effects of the present invention are:

[0025] 1) The present invention transforms the complex three-dimensional seepage problem into a two-dimensional circular area problem through conformal transformation, and then transforms the solution back to the original three-dimensional plane through inverse transformation to obtain an explicit analytical solution of the total water head of the seepage field, which effectively reduces the calculation difficulty and cost.

[0026] 2) The analytical method proposed in the present invention improves the accuracy and reliability of the analytical solution by combining conformal transformation, variable substitution and separation of variables method, accurately calculates the pore water pressure and leakage at the tunnel joints, and obtains an accurate analytical solution of the seepage field, rather than an approximate solution or an improved analytical solution based on others, providing a scientific basis for leakage prevention and control in tunnel engineering.

[0027] 3) The method proposed in the present invention is suitable for analyzing leakage conditions of different lining joints, and can consider the leakage capacity of the vault leakage position separately, and has strong versatility and practicality. BRIEF DESCRIPTION OF THE DRAWINGS

[0028] Figure 1 The overall framework flow chart provided for the present invention;

[0029] Figure 2 A schematic diagram of a three-dimensional tunnel annular water leakage model provided by the present invention;

[0030] Figure 3 A schematic diagram of a tunnel cross section provided by the present invention;

[0031] Figure 4 This is a schematic diagram of the conformal transformation provided by the present invention. DETAILED DESCRIPTION

[0032] The present invention is further described below in conjunction with embodiments. The description of the following embodiments is only used to help understand the present invention. It should be noted that for ordinary persons in the art, without departing from the principle of the present invention, the present invention can also be modified in some ways, and these improvements and modifications also fall within the scope of protection of the claims of the present invention.

[0033] Embodiment 1

[0034] As an embodiment, this embodiment 1 proposes a method for analyzing the leakage problem of a three-dimensional tunnel segment joint, comprising the following steps:

[0035] Step 1: Establish a 3D tunnel segment joint leakage model, simplify the non-circular cross-section of the tunnel into a circular cross-section, and determine the basic assumptions and boundary conditions of the model;

[0036] The basic assumptions of the model in step 1 are:

[0037] The part at infinity away from the tunnel can be approximately regarded as an impermeable boundary. The part of the tunnel lining where no water seepage occurs can be regarded as an impermeable boundary because its permeability coefficient is much smaller than that of the soil. And after analyzing the right half of the center of the tunnel water seepage, the cross section at the tunnel cutoff can be regarded as a normal impermeable vertical boundary according to the symmetry.

[0038] Since the longitudinal dimension of the leakage water is generally small, it can be assumed that the head distribution of the leakage part is consistent along the longitudinal direction, and the total head at the joint of the tunnel segment where the leakage water occurs is constant as its position head;

[0039] The soil is homogeneous and isotropic, and the permeability coefficient remains constant;

[0040] The tunnel is in a stable seepage state and the water flow obeys Darcy's law.

[0041]

[0042] In this embodiment, Figure 2 As shown, the boundary conditions of the model in step 1 are:

[0043]

[0044] Where BC1, BC2, BC3, BC4, and BC5 are the boundary water head of the leakage at the tunnel lining joint, the boundary water head at the non-leakage of the tunnel lining, the boundary water head on the surface, the impermeable boundary condition at infinity, and the impermeable boundary condition at the vertical symmetry plane of the model.

[0045] Step 2: Use the separation of variables method to decompose the total head function H(x,y,z) into:

[0046]

[0047] In the formula, U and are spatial functions about coordinates (y,z) and x respectively.

[0048] Substituting formula (3) into formula (1), we can obtain:

[0049]

[0050] The total head function is decomposed into two independent equations by using the separation of variables method, which is specifically expressed as formula (5), and the special solution of the equation is solved by combining the boundary conditions, which is specifically expressed as formula (6) and formula (7):

[0051]

[0052] In the formula, λ is an undetermined parameter, is a calculation constant.

[0053] From BC4 and BC5 in equation (2), we can get the special solution of the second equation in equation (5) that satisfies the boundary conditions:

[0054]

[0055] In the formula, λ n is the nth eigenvalue of the characteristic function in formula (6).

[0056] Step 3: Using the principle of conformal transformation, the three-dimensional seepage problem is transformed into a two-dimensional annular region problem for solution, and then the solution is transformed back to the original three-dimensional plane through the inverse transformation of the conformal transformation, thereby obtaining an explicit analytical solution of the total water head of the seepage field;

[0057] Step 4: Combine the explicit analytical solution of the total water head in the seepage field in step 3 with the Bernoulli equation to obtain the analytical solution of the pore water pressure at the tunnel joint;

[0058] Step 5: Based on the explicit analytical solution of the total water head of the seepage field in step 3, the normal hydraulic gradient at the leakage point of the tunnel lining joint is obtained, and then the leakage of the total model is obtained based on the analytical solution of the hydraulic gradient.

[0059] Embodiment 2

[0060] As another embodiment, this embodiment 2 proposes a more specific analytical method for the leakage problem of three-dimensional tunnel segment joints based on the embodiment 1:

[0061] In step 2, the first equation in equation (5) satisfies:

[0062]

[0063] Where: U n is the nth characteristic function.

[0064] In step 3, the explicit analytical solution of the total water head of the seepage field when the tunnel vault leaks is obtained by using the principle of conformal transformation. The specific solution is as follows:

[0065] According to Darcy's law and the law of conservation of mass, the two-dimensional seepage flow around the tunnel can be described by the Laplace equation, but Figure 2 It is very difficult to solve it directly in the three-dimensional plane. Therefore, based on the conformal transformation of complex variable functions, the more complex seepage area in the original three-dimensional plane can be transformed into each vertical plane with x = const, such as Figure 3 As shown, it is then mapped to the annular region in the ω plane, as Figure 4 As shown, the solution is obtained, and finally the solution is transformed back to the original three-dimensional plane through the inverse transformation of the conformal transformation to obtain the explicit analytical solution of the total water head of the seepage field when the tunnel vault leaks.

[0066] As a further solution of this solution, the angle-preserving transformation in step 3 is specifically expressed as formula (8):

[0067]

[0068] In the formula, z0=y+iz, ω=u+iv are respectively Figure 3 and Figure 4The complex number representation of the midpoint, (y,z) and (u,v) are the coordinate parameters before and after the conformal transformation.

[0069] As a further solution of this scheme, the first decomposition and derivation process of the total water head of the seepage field in step 3 is as follows:

[0070] Substituting the first equation of equation (8) into equation (7), we can obtain:

[0071]

[0072] Right now:

[0073]

[0074] Where (ρ, θ) is the angle-conformal transformation. Figure 3 Polar coordinate representation of a space point. nρρ , U nρθ , U nθθ Denote the characteristic function U n The two partial derivatives with respect to ρ, the two partial derivatives first with respect to ρ and then with respect to θ, and the two partial derivatives with respect to θ.

[0075] Substituting variables in formula (10) yields:

[0076]

[0077] Formula (10) becomes:

[0078]

[0079] Where χ is the variable generated in variable substitution. nχ and U nχχ Denote the characteristic function U n The first and second partial derivatives of χ with respect to .

[0080] Formula (12) is a 0th-order modified Bessel function, and its general solution is:

[0081]

[0082] Where U in In (χ), i=1 represents the characteristic function of the lining leakage interval, i=2 represents the characteristic function of the lining non-leakage interval, and a in , a in , b in , n=1,2..., i=1,2 are all unknown constants, I0(x) and K0(x) are the 0th-order first-kind modified Bessel function and the second-kind modified Bessel function of the variable x, respectively.

[0083] According to BC1~BC3 in formula (2), we can get:

[0084]

[0085] Formula (14) holds for every coordinate z, where To calculate the constants, H0 and H1 are the water head at the surface leakage of the lining joint and the constant water head on the ground surface, respectively, and l is the longitudinal length of the leakage at the lining joint. I1(x) and K1(x) are the first-order derivatives of I0(x) and K0(x), respectively. χ(ρ,θ) is the variable generated in the variable substitution.

[0086] The head and velocity continuity condition of the model at the vertical plane x=l / 2 is:

[0087]

[0088] According to equations (14) to (15), combined with the orthogonality of trigonometric functions and the orthogonality of Bessel functions, MATLAB programming can be used to calculate and solve a in , b in It is worth noting that when solving the above parameters, the series in the formula are infinite series. In order to solve the unknown coefficients, the series must be truncated at the Nth term.

[0089] Then the total water head solution of the steady-state seepage field when the tunnel vault leaks in the z plane is:

[0090]

[0091] It should be noted that the parts in this embodiment that are the same or similar to those in the first embodiment can be referenced to each other and will not be described in detail in this application.

[0092] Embodiment 3

[0093] As another embodiment, this embodiment 3 proposes a more specific analytical method for the leakage problem of three-dimensional tunnel segment joints based on embodiments 1 and 2. The second decomposition and derivation process of the total water head of the seepage field in step 3 is as follows:

[0094]

[0095] From BC4 and BC5 in equation (2), the special solution of the second equation in equation (17) that satisfies the boundary conditions is obtained as follows:

[0096]

[0097] Accordingly, the first equation of formula (17) can be written as:

[0098]

[0099] Substituting the first conformal transformation of equation (8) into equation (19) yields:

[0100]

[0101] Right now:

[0102]

[0103] Where (ρ, θ) is the angle-conformal transformation Figure 3 Polar coordinate representation of a space point. inρρ 、u inρθ , U inθθ Denote the characteristic function U in The two partial derivatives of ρ, the two partial derivatives of ρ and then of θ, and the two partial derivatives of θ. i=1 represents the characteristic function of the lining leakage interval, and i=2 represents the characteristic function of the lining non-leakage interval.

[0104] Substituting the variables in equation (11) into equation (21), we can obtain:

[0105]

[0106] Where χ is the variable generated in variable substitution. inχ and U inχχ Denote the characteristic function U in The first and second partial derivatives of χ, and i=1 represents the characteristic function of the lining leakage interval, and i=2 represents the characteristic function of the lining non-leakage interval.

[0107] Formula (22) is the standard Bessel function of order 0, and its general solution is:

[0108]

[0109] In the formula, C in d in , n=1,2..., i=1,2 are all unknown constants, J0(x) and Y0(x) are the 0th-order Bessel function of the first kind and the second kind of Bessel function of the variable x respectively.

[0110] According to BC1~BC3 in formula (2), we can get:

[0111]

[0112] The head and velocity continuity condition of the model at the vertical plane x=l / 2 is:

[0113]

[0114] According to equations (24) to (25), combined with the orthogonality of trigonometric functions and the orthogonality of Bessel functions, MATLAB programming can be used to calculate and solve C ind in It is worth noting that when solving the above parameters, the series in the formula are infinite series. In order to solve the unknown coefficients, the series must be truncated at the Nth term.

[0115] Then the total water head solution of the steady-state seepage field when the tunnel vault leaks in the z plane is:

[0116]

[0117] As a further solution of this scheme, in step 4: combining the explicit analytical solution of the total water head of the seepage field in step 3 with the Bernoulli equation, the analytical solution of the pore water pressure at the tunnel joint is obtained as follows:

[0118] According to the Bernoulli equation, the total water head consists of position head, pressure head, and kinetic head. In the case of three-dimensional steady-state seepage, the kinetic head is approximately 0. Taking the surface y = 0 as the starting position for position head calculation, the pore water pressure p can be expressed as the water head H as follows:

[0119] p=γ w (H(x,y,z)-y),y<0 (27)

[0120] In the formula, γ w The weight of water.

[0121] It should be noted that the parts in this embodiment that are the same or similar to those in Embodiments 1 to 2 can be referenced to each other and will not be described in detail in this application.

[0122] Embodiment 4

[0123] As another embodiment, this embodiment 4 proposes a more specific analytical method for the leakage problem of three-dimensional tunnel segment joints based on embodiments 1 to 3. In step 5, according to the explicit analytical solution of the total water head of the seepage field in step 3, the normal hydraulic gradient at the leakage point of the tunnel lining joint is obtained as follows:

[0124] According to formula (16) in embodiment 2 or formula (26) in embodiment 3, we can get Figure 2 The normal hydraulic gradient at the leakage point of the lining joint of the middle tunnel is specifically expressed as formula (28):

[0125]

[0126] In the formula, Here, i=1 represents the characteristic function of the lining leakage interval, and i=2 represents the characteristic function of the lining non-leakage interval. is the hydraulic head gradient at every point on the exterior of the tunnel lining.

[0127] As a further solution of this scheme, the leakage of the total model obtained by the analytical solution of the hydraulic gradient in step 5 is specifically expressed as formula (29):

[0128]

[0129] Where Q is the total leakage at the tunnel lining joints. Figure 2 The symmetrical right side of the model is taken for analysis, so the total leakage amount Q needs to be multiplied by 2 after summing.

[0130] It should be noted that the parts in this embodiment that are the same or similar to those in Embodiments 1 to 3 can be referenced to each other and will not be described in detail in this application.

[0131] The various embodiments in this specification are described in a progressive manner, and each embodiment focuses on the differences from other embodiments. The same or similar parts between the various embodiments can be referenced to each other.

Claims

1. A method for analyzing the leakage problem of three-dimensional tunnel segment joints, characterized in that: The following steps are involved: Step 1: Establish a 3D tunnel segment joint leakage model and determine the basic assumptions and boundary conditions of the model; Step 2: Decompose the total head function using the separation of variables method, and solve the equation for a specific solution in combination with boundary conditions; Step 3: Using the principle of conformal transformation, the three-dimensional seepage problem is transformed into a two-dimensional annular region problem for solution, and then the solution is transformed back to the original three-dimensional plane through the inverse transformation of the conformal transformation to obtain the explicit analytical solution of the total water head of the seepage field; Step 4: Based on the explicit analytical solution of the total water head of the seepage field and the Bernoulli equation, the analytical solution of the pore water pressure at the tunnel joint is obtained; Step 5: Based on the explicit analytical solution of the total water head of the seepage field, the analytical solution of the normal hydraulic gradient at the leakage point of the tunnel lining joint is obtained, and then the leakage amount of the total model is obtained.

2. The analytical method for the three-dimensional tunnel segment joint leakage problem according to claim 1 is characterized in that: In step three, based on the conformal transformation of the complex function, each vertical plane with x=const in the seepage area in the original three-dimensional plane is mapped to the circular area in the ω plane, and the three-dimensional seepage problem is transformed into a two-dimensional circular area problem, and the non-steady-state equation after the decomposition of the total head function in step two is solved.

3. The analytical method for the three-dimensional tunnel segment joint leakage problem according to claim 1 is characterized in that: In step 2, the total head function is decomposed into two independent equations using the separation of variables method: Among them, U and are spatial functions about coordinates (y, z) and x respectively; λ is an undetermined parameter, and A is a calculation constant; In step three, the conformal transformation formula is substituted into the first formula in the above formula, and the general solution is obtained after variable substitution. Then, according to the boundary conditions and velocity continuity conditions determined in step one, the explicit analytical solution of the total water head of the seepage field is obtained.

4. The analytical method for the three-dimensional tunnel segment joint leakage problem according to claim 1 is characterized in that: In step 3, the explicit analytical solution of the total hydraulic head in the seepage field is obtained: in, where i=1 represents the characteristic function of the lining leakage interval, and i=2 represents the characteristic function of the lining non-leakage interval; U in In (), i=1 represents the characteristic function of the lining leakage interval, i=2 represents the characteristic function of the lining non-leakage interval, and l is the longitudinal length of the leakage at the lining joint.

5. The analytical method for the three-dimensional tunnel segment joint leakage problem according to claim 1 is characterized in that: In step 1, the non-circular cross-section of the tunnel is simplified to a circular cross-section; the parts at infinity away from the tunnel and where no water seepage occurs in the tunnel lining are regarded as impermeable boundaries, and the cross-section at the tunnel cutoff corresponding to the boundary of the computational domain is regarded as a normal impermeable vertical boundary.

6. The analytical method for the three-dimensional tunnel segment joint leakage problem according to claim 1 is characterized in that: In step 2, the total head function is decomposed into: Combining Darcy's law, we get: Among them, U and are spatial functions about coordinates (y,z) and x respectively.

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