Nonlinear seepage field analysis method for submarine tunnel based on mirror image method in finite plane

Abstract

The invention discloses a method for analyzing the nonlinear seepage field of an undersea tunnel based on a mirror image method in a finite plane, comprising determining the geological conditions of an undersea tunnel excavation section, wherein the geological conditions are that the undersea tunnel excavation section is located in a homogeneous stratum, the homogeneous stratum is regarded as being distributed circumferentially around the undersea tunnel section, and the inner / outer diameters from each homogeneous stratum to the center of the tunnel section and the permeability coefficients corresponding to each homogeneous stratum are determined; the actual undersea tunnel and the virtual undersea tunnel are used to form an undersea tunnel seepage field, and surrounding rock seepage field parameters, grouting circle seepage field parameters and lining seepage field parameters are calculated for the undersea tunnel seepage field; the undersea tunnel seepage field is analyzed and predicted according to the surrounding rock seepage field parameters, grouting circle seepage field parameters and lining seepage field parameters; the problem of predicting the nonlinear seepage field of the undersea tunnel under the condition that the undersea tunnel seepage field cannot consider the condition that the lower boundary is impermeable and the nonlinear seepage of the grouting circle and the lining is considered at the same time is solved.

Description

Technical Field

[0001] The present invention relates to the technical field of civil engineering, and particularly relates to a method for analyzing the non-linear seepage field of a submarine tunnel based on the mirror image method within a finite plane. Background Art

[0002] In recent years, with the in-depth implementation of the national strategy of building a transportation power in China, the rapid construction of coastal city tunnels has led to a large number of them. However, most tunnels are designed as drainage tunnels, or there are seepage channels due to the process itself, or a large number of deformation cracks occur during long-term service. These factors will change the distribution of water and soil pressure around the tunnel. Moreover, the excavation of the tunnel often affects the surrounding soil mass and its permeability coefficient.

[0003] Currently, the surrounding rock of submarine tunnels is mostly considered as a semi-infinite space, that is, the influence of the impermeable boundary of the lower boundary on the seepage field is not considered, which does not conform to the actual situation. And it is impossible to consider the impermeable boundary of the lower boundary and accurately predict the non-linear seepage field of the submarine tunnel under the condition that both the grouting circle and the lining are non-linear seepage, which reduces its safety and applicability before and after tunnel construction. Summary of the Invention

[0004] The present invention provides a method for analyzing the non-linear seepage field of a submarine tunnel based on the mirror image method within a finite plane to overcome the problem that currently, the surrounding rock of submarine tunnels is mostly considered as a semi-infinite space, that is, the influence of the impermeable boundary of the lower boundary on the seepage field is not considered, which does not conform to the actual situation. And it is impossible to consider the impermeable boundary of the lower boundary and accurately predict the non-linear seepage field of the submarine tunnel under the condition that both the grouting circle and the lining are non-linear seepage, which reduces its safety and applicability before and after tunnel construction.

[0005] To achieve the above object, the technical solution of the present invention is as follows:

[0006] A method for analyzing the non-linear seepage field of a submarine tunnel based on the mirror image method within a finite plane, comprising

[0007] Step S1: Determine the geological conditions of the excavation section of the submarine tunnel. The geological conditions are that the excavation section of the submarine tunnel is located in a homogeneous stratum area, and the plane where the top of the homogeneous stratum area is located is defined as a permeable boundary, and the plane where the bottom of the homogeneous stratum area is located is defined as an impermeable boundary;

[0008] The homogeneous stratum includes a surrounding rock layer, a grouting layer, and a lining layer;

[0009] Step S2: Consider the homogeneous stratum area as circumferentially distributed around the cross-section of the submarine tunnel, and determine the inner / outer diameter of each homogeneous stratum to the center of the tunnel cross-section and the corresponding permeability coefficient of each homogeneous stratum;

[0010] The seepage coefficient includes the seepage coefficient of the surrounding rock, the seepage coefficient of the grouting circle, and the seepage coefficient of the lining;

[0011] Step S3: Set the horizontal plane at the center of the actual undersea tunnel as the zero potential surface, use the water level line as the x-axis reference baseline, and use the straight line perpendicular to the axis of the actual undersea tunnel as the y-axis reference baseline to establish a rectangular coordinate system; Based on the mirror image superposition principle, map the actual undersea tunnel with the permeable boundary and the impermeable boundary as the mirror surfaces respectively to obtain a virtual undersea tunnel, and superimpose the potential of the actual undersea tunnel and the virtual undersea tunnel to form an undersea tunnel seepage field model considering the influence of the permeable boundary and the impermeable boundary of the surrounding rock. The undersea tunnel seepage field model includes the surrounding rock seepage field, the grouting circle seepage field, and the lining seepage field;

[0012] Step S4: Calculate the surrounding rock seepage field based on Darcy's seepage law to obtain surrounding rock seepage field parameters, where the surrounding rock seepage field parameters include the water inflow of the undersea tunnel section, the total head of the surrounding rock area, and the seepage flow within the surrounding rock of the undersea tunnel;

[0013] Step S5: Based on the Hansbo nonlinear seepage model, calculate the grouting circle seepage field and the lining seepage field to obtain grouting circle seepage parameters and lining seepage parameters;

[0014] The grouting circle seepage parameters include the seepage flow of the grouting circle, the total head of the grouting circle, and the pore pressure of the grouting circle;

[0015] The lining seepage parameters include the seepage flow of the lining, the total head of the lining, and the pore pressure of the lining;

[0016] Step S6: Analyze and predict the undersea tunnel seepage field according to the surrounding rock seepage parameters, the grouting circle seepage field parameters, and the lining seepage field parameters.

[0017] Further, in step S4, calculating the surrounding rock seepage field parameters based on Darcy's seepage law specifically includes:

[0018] Step S4.1: Reflect the actual undersea tunnel with the permeable boundary as the mirror surface to obtain a virtual undersea tunnel with equal flow and opposite signs and define it as a source, and the permeable boundary remains an equipotential line after reflection;

[0019] Reflect the actual undersea tunnel with the impermeable boundary as the mirror surface to obtain a virtual undersea tunnel with equal flow and the same sign and define it as a sink, and the impermeable boundary remains a streamline after reflection;

[0020] Step S4.2: When the seepage mode of the soil conforms to Darcy's seepage:

[0021]

[0022] Where, i represents the hydraulic gradient of the surrounding rock; k represents the permeability coefficient; v represents the seepage velocity of the surrounding rock;

[0023] Then, when the undersea tunnel is a single - hole circular tunnel, the flow rate through each cross - section of the tunnel is:

[0024] Q s = 2πrv (2)

[0025] In the formula: Q s represents the water inflow of the calculated cross - section of the surrounding rock layer; r represents the distance from any point of the calculated cross - section to the center of the undersea tunnel. Substituting formula (1) into formula (2), we get: It can be obtained that:

[0026]

[0027] In the formula: represents the total head at any point within the surrounding rock layer; Q s represents the water inflow of the calculated cross - section of the surrounding rock layer; k s represents the permeability coefficient of the surrounding rock; n s represents the soil disturbance coefficient; r represents the distance from any point of the calculated cross - section to the center of the undersea tunnel;

[0028] Integrating formula (3) gives:

[0029]

[0030] In the formula: represents the total head at any point within the surrounding rock layer; Q s represents the water inflow of the calculated cross - section of the surrounding rock layer; k s represents the permeability coefficient of the surrounding rock; r represents the distance from any point of the calculated cross - section to the center of the undersea tunnel; n s represents the soil disturbance coefficient; C represents the integration constant;

[0031] Step S4.3: According to the principle of potential superposition, the water head at any point in the plane of the mirror image method can be determined. Therefore:

[0032]

[0033] In the formula: Q si represents the seepage flow rate of the i - th tunnel, taking a positive value when flowing in and a negative value when flowing out; r i represents the distance from the calculation point to the center point of the i - th tunnel; C 1 represents a undetermined constant value;

[0034] The central coordinates of various tunnels after mirror reflection include sink coordinates and source coordinates; the sink coordinates are (0, 4nh + b), (0, 2h + 4nh - b); the source coordinates are (0, 4nh - b), (0, 2h + 4nh + b); where h represents the distance between the water supply boundary and the impermeable boundary; b represents the distance from the center of the actual tunnel to the water supply boundary; n = 0, ±1, ±2, ···, ±∞; therefore,

[0035]

[0036] In the formula: r 1 represents the distance from any point M(x, y) in the finite plane to the center of the first type of tunnel; the first type of tunnel is a virtual submarine tunnel with the center coordinate (0, 4nh + b) obtained by the mirror method; r 2 represents the distance from any point M(x, y) in the finite plane to the center of the second type of tunnel; the second type of tunnel is a virtual submarine tunnel with the center coordinate (0, 2h + 4nh - b) obtained by the mirror method; r 3 represents the distance from any point M(x, y) in the finite plane to the center of the third type of tunnel; the third type of tunnel is a virtual submarine tunnel with the center coordinate (0, 4nh - b) obtained by the mirror method; r 4 represents the distance from any point M(x, y) in the finite plane to the center of the fourth type of tunnel; the fourth type of tunnel is a virtual submarine tunnel with the center coordinate (0, 2h + 4nh + b) obtained by the mirror method;

[0037] Step S4.4: Substitute Equation (6) into Equation (5) to obtain the water head value of any point M in the soil mass at the time of seepage stability as:

[0038]

[0039] In the formula: represents the total water head of any point in the surrounding rock layer; Q s represents the water inflow of the calculation section of the surrounding rock layer; k s represents the permeability coefficient of the surrounding rock; n s represents the soil disturbance coefficient; C 2 represents the undetermined constant;

[0040] Step S4.5: According to the Bessette formula:

[0041]

[0042] The formula (7) can be simplified to:

[0043]

[0044] At the water supply boundary condition y = 0, The constant C can be obtained 2 = b; Then, substitute the outer boundary condition of the grouting circle x = 0, y = b - r g When, r g represents the outer radius of the grouting circle; h g represents the total head at the outer boundary of the grouting circle; Substitute into Equation (9), the seepage flow rate in the surrounding rock of the submarine tunnel can be obtained:

[0045]

[0046] In the formula: Q represents the seepage flow rate in the surrounding rock of the submarine tunnel: h g represents the total head at the outer boundary of the grouting circle; h represents the distance between the water supply boundary and the impermeable boundary; b represents the distance from the center of the actual tunnel to the water supply boundary; r 0 represents the inner radius of the lining.

[0047] Furthermore, in step S5, the seepage parameters of the grouting circle and the lining are calculated based on the Hansbo non - linear seepage model. Specifically, step S5.1: The Hansbo non - linear seepage model is

[0048]

[0049]

[0050] In the formula: k 0s represents the permeability coefficient of the curve segment of the non - linear seepage model; k 0 represents the permeability coefficient of the straight - line segment; m represents the non - linear parameter; i 0 、i l represents the critical hydraulic gradient;

[0051] Step S5.2: Define that the fluid is incompressible and the seepage direction is perpendicular to the y - axis. The continuity equation of the fluid seepage in the grouting circle is expressed as:

[0052]

[0053] Decompose the velocity vector in the seepage field:

[0054] u = v i cosθ, v = v i sinθ, v i = k 0s i m (14)

[0055] In the formula: u represents the seepage velocity in the x - axis direction; v represents the seepage velocity in the y - axis direction; v i represents the seepage velocity in any direction; im represents the m-th power of the hydraulic gradient; k os represents the permeability coefficient of the non-linear seepage model curve segment; θ represents the angle between the seepage velocity in any direction and the x-axis direction;

[0056] Step S5.3: Based on the fact that the direction of the hydraulic gradient of seepage is the same as the streamline direction, the hydraulic gradient can be expressed as:

[0057]

[0058] According to the radial seepage conditions of the circular section of the submarine tunnel, it can be obtained that:

[0059]

[0060] In the formula: represents the total head; i represents the hydraulic gradient of the surrounding rock; r represents the distance from any point on the calculation section to the center of the circle;

[0061] Substituting formulas (11), (12), (14), (15), and (16) into formula (13) and simplifying, the fluid non-linear continuous equation in polar coordinate form of the grouting circle and the lining under the radial seepage conditions of the submarine tunnel can be obtained:

[0062] The fluid non-linear continuous equation in polar coordinate form of the grouting circle is:

[0063]

[0064] In the formula: represents the total head of the grouting circle area; m g 、i lg represent the non-linear parameters of the grouting area; r 1g represents the distance from any point in the grouting area to the center of the actual tunnel;

[0065] The fluid non-linear continuous equation in polar coordinate form of the lining is:

[0066]

[0067] In the formula: represents the total head of the lining area; m l 、i ll represent the non-linear parameters of the lining area; r 2l represents the distance from any point in the lining area to the center of the actual tunnel;

[0068] Step S5.4: Solve formulas (17) and (18) by the method of separating variables, and substitute the boundary condition (19) to obtain the total head of the grouting circle and the total head of the lining:

[0069]

[0070] In the formula: h g represents the total head at the outer boundary of the grouting circle; h l represents the total head at the outer boundary of the lining area; h 0 represents the total head at the inner boundary of the lining area; r 0 represents the inner radius of the lining; r l represents the outer radius of the submarine tunnel excavated inside the rock mass;

[0071] The total head of the grouting circle is:

[0072]

[0073] The total head of the lining area is:

[0074]

[0075] Step S5.5: Integrate Formula (20) and Formula (21) to obtain the seepage flow rate of the grouting circle and the seepage flow rate of the lining:

[0076] The seepage flow rate Q of the grouting circle g is:

[0077]

[0078] The seepage flow rate Q of the lining l is:

[0079]

[0080] Step S5.6: According to Formula (22) and Formula (23), the pore pressure of the grouting circle and the pore pressure of the lining are:

[0081] The calculation formula for the pore pressure of the grouting circle is:

[0082]

[0083] The calculation formula for the pore pressure of the lining is:

[0084]

[0085] In the formula: In the formula: p g represents the pore pressure of the grouting circle; p l represents the pore pressure of the lining; y represents the position head of the grouting area or the lining area; k g represents the permeability coefficient of the grouting circle; k l represents the permeability coefficient of the lining; r l represents the outer radius of the submarine tunnel; h 0 represents the total head at the inner boundary of the lining area; r 0 represents the inner radius of the lining; mg , i lg , k gs , i 0g represent the nonlinear parameters of the grouting area; m l , i ll , k ls , k 0l represent the nonlinear parameters of the lining area; h l represents the total water head at the outer boundary of the lining area; γ w represents the unit weight of water.

[0086] Furthermore, in step S6, based on the seepage parameters of the surrounding rock, the seepage field parameters of the grouting circle, and the seepage field parameters of the lining, the seepage field of the submarine tunnel is analyzed and predicted as follows:

[0087] Step S6.1: Analyze the seepage mode of the grouting circle and the lining based on the seepage continuity condition to obtain the water head h g outside the grouting area and the water head h l outside the lining area;

[0088] The seepage continuity condition is that the seepage flow rate of the surrounding rock is equal to the seepage flow rate of the grouting circle and equal to the seepage flow rate of the lining;

[0089] Step S6.2: When both the grouting circle and the lining are in the Hansbo nonlinear seepage curve section, that is, when m g > 1, i ≤ i lg ; m l > 1, i ≤ i ll under the condition; then from formulas (10), (22), and (23), we can obtain

[0090]

[0091] When the lining is in the Hansbo nonlinear seepage curve section; and the grouting circle is in the Hansbo nonlinear seepage straight line section, that is, when m g > 1, i ≤ i lg ; m l > 1, i > i ll under the condition; from formulas (10), (22), and (23), we can obtain

[0092]

[0093] When both the grouting circle and the lining are in the Hansbo nonlinear seepage straight line section, that is, when m g = 1; m l = 1 under the condition, from formulas (10), (22), and (23), we can obtain

[0094]

[0095] Step S6.3: Calculate the water head h outside the grouting area according to Formulas (26), (27), and (28). gt and the water head h outside the lining area lt ; and h gt is h g ; h lt is h l ; h 0t is h 0 ;

[0096] Substitute h gt and h lt back into the calculation formulas of the seepage field parameters of the surrounding rock, the grouting circle, and the lining to obtain the seepage field parameters of the surrounding rock, the grouting circle, and the lining under the current seepage conditions;

[0097] Predict the seepage field of the submarine tunnel according to the seepage field parameters of the surrounding rock, the grouting circle, and the lining under the current seepage conditions.

[0098] Beneficial effects: The present invention provides a method for analyzing the non-linear seepage field of a submarine tunnel based on the mirror image method in a finite plane. Considering the condition of an impermeable lower boundary, based on the mirror image method, by considering the cases where both the grouting circle and the lining of the submarine tunnel are non-linear seepage, the Hansbo non-linear seepage model is introduced. It can consider the situation where when the water pressure is too high, the seepage mode enters the linear seepage situation. Thus, the seepage modes of the grouting circle and the lining under different water pressure conditions can be discussed separately, and then the actual situation of the seepage field of the submarine tunnel can be described more completely and accurately. The tunnel water inflow and the pore pressure borne by the grouting circle and the lining can be solved, making the simulation of the situation more real and reliable. In actual situations, the grouting and lining materials can be better selected, improving the accuracy of predicting the non-linear seepage field of the submarine tunnel and reducing the safety and applicability problems before and after tunnel construction. Description of the Drawings

[0099] In order to more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the following will briefly introduce the drawings required for the description of the embodiments or the prior art. Obviously, the following drawings are some embodiments of the present invention. For those of ordinary skill in the art, other drawings can be obtained based on these drawings without creative efforts.

[0100] Figure 1 is the step block diagram of the method for analyzing the non-linear seepage field of a submarine tunnel based on the mirror image method in a finite plane of the present invention;

[0101] Figure 2 is the flow block diagram of the method for analyzing the non-linear seepage field of a submarine tunnel based on the mirror image method in a finite plane of the present invention;

[0102] Figure 3 This is a simplified calculation model of the seepage field of a submarine tunnel for the analysis method of the non - linear seepage field of a submarine tunnel based on the mirror method in a finite plane;

[0103] Figure 4 This is a schematic diagram of the mirror method for the analysis method of the non - linear seepage field of a submarine tunnel based on the mirror method in a finite plane;

[0104] Figure 5 This is the Hansbo non - linear seepage model for the analysis method of the non - linear seepage field of a submarine tunnel based on the mirror method in a finite plane. Specific implementation mode

[0105] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the following will clearly and completely describe the technical solutions in the embodiments of the present invention with reference to the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are some, but not all, of the embodiments of the present invention. All other embodiments obtained by those of ordinary skill in the art based on the embodiments of the present invention without creative efforts shall fall within the protection scope of the present invention.

[0106] This embodiment provides an analysis method for the non - linear seepage field of a submarine tunnel based on the mirror method in a finite plane, as Figure 1 shown, including

[0107] Step S1: Determine the geological conditions of the excavation section of the submarine tunnel. The geological conditions are that the excavation section of the submarine tunnel is located in a homogeneous stratum area, and the plane where the top of the homogeneous stratum area is located is defined as a permeable boundary, and the plane where the bottom of the homogeneous stratum area is located is defined as an impermeable boundary;

[0108] The homogeneous stratum includes a surrounding rock layer, a grouting layer, and a lining layer;

[0109] Step S2: Consider the homogeneous stratum area as circumferentially distributed around the cross - section of the submarine tunnel, and determine the inner / outer diameters of each homogeneous stratum to the center of the tunnel cross - section and the corresponding permeability coefficients of each homogeneous stratum;

[0110] The permeability coefficients include the surrounding rock permeability coefficient, the grouting ring permeability coefficient, and the lining permeability coefficient;

[0111] Step S3: Set the water level at the center of the actual undersea tunnel as the zero potential surface. Take the water level line as the x-axis reference baseline and the line perpendicular to the axis of the actual undersea tunnel as the y-axis reference baseline to establish a rectangular coordinate system. Based on the mirror superposition principle, map the actual undersea tunnel with the permeable boundary and the impermeable boundary as mirrors respectively to obtain virtual undersea tunnels, and superimpose the potentials of the actual undersea tunnel and the virtual undersea tunnels to form an undersea tunnel seepage field model considering the influence of the permeable boundary and the impermeable boundary of the surrounding rock. The undersea tunnel seepage field model includes the surrounding rock seepage field, the grouting ring seepage field, and the lining seepage field.

[0112] Step S4: Calculate the surrounding rock seepage field based on Darcy's seepage law to obtain surrounding rock seepage field parameters, where the surrounding rock seepage field parameters include the water inflow of the undersea tunnel section, the total head of the surrounding rock area, and the seepage flow within the surrounding rock of the undersea tunnel.

[0113] Step S5: Calculate the grouting ring seepage field and the lining seepage field based on the Hansbo non-linear seepage model to obtain grouting ring seepage parameters and lining seepage parameters.

[0114] The grouting ring seepage parameters include the seepage flow of the grouting ring, the total head of the grouting ring, and the pore pressure of the grouting ring.

[0115] The lining seepage parameters include the seepage flow of the lining, the total head of the lining, and the pore pressure of the lining.

[0116] Step S6: Analyze and predict the undersea tunnel seepage field according to the surrounding rock seepage parameters, the grouting ring seepage field parameters, and the lining seepage field parameters.

[0117] In a specific embodiment, calculating the surrounding rock seepage field parameters based on Darcy's seepage law in Step S4 specifically includes:

[0118] Taking a single-hole circular tunnel in the impermeable horizontal plane of the lower boundary as the research object, as shown in Figure 3 ; the research object is a saturated, homogeneous, continuous, and isotropic rock mass medium. An undersea tunnel with an outer radius of r l is excavated inside the rock mass. The distance from the center of the undersea tunnel to the water supply boundary is b, the distance from the water supply boundary to the impermeable boundary is h, and the permeability coefficient of the surrounding rock is k s ; the inner radius of the lining is r 0 ; the permeability coefficient is k l ; the radius of the grouting ring is r g ; the permeability coefficient is k g ; assume that the ratio of the permeability coefficient of the surrounding rock after excavation disturbance to the original surrounding rock is n s ; set the water level at the center of the tunnel as the zero potential surface and establish a rectangular coordinate system with the water level line as the reference baseline.

[0119] Step S4.1: Reflect the actual submarine tunnel with the permeable boundary as the mirror surface to obtain a virtual submarine tunnel with equal flow rates and opposite signs, and define it as the source. After reflection, the permeable boundary remains an equipotential line;

[0120] Consider the surrounding rock as a finite plane, and transform the seepage field in the finite plane into the superposition of an infinite number of source-sink systems, as Figure 4 shown. The principle of mirror image method reflection is that the influence of the boundary on the seepage field can be regarded as mapping the actual tunnel with the boundary as the mirror surface, imaging a virtual tunnel at the position symmetric to the actual tunnel with respect to the boundary, and superimposing the potentials of the actual tunnel and the virtual tunnel to form a seepage field equal to the seepage field formed considering the influence of the boundary; for the permeable (water supply) boundary, perform an equal amount (flow rate Q) and opposite sign reflection, and the mirrored virtual tunnel is called the "source". After reflection, the permeable boundary remains an equipotential line; for the impermeable boundary, perform an equal amount (flow rate Q) and same sign reflection, and the mirrored virtual tunnel is called the "sink". After reflection, the impermeable boundary remains a streamline; before and after mirror image reflection, the true seepage field remains unchanged.

[0121] Reflect the actual submarine tunnel with the impermeable boundary as the mirror surface to obtain a virtual submarine tunnel with equal flow rates and the same sign, and define it as the sink. After reflection, the impermeable boundary remains a streamline;

[0122] Step S4.2: When the seepage mode of the soil conforms to Darcy seepage, we have:

[0123]

[0124] where i represents the hydraulic gradient of the surrounding rock; k represents the permeability coefficient; v represents the seepage velocity of the surrounding rock;

[0125] Then when the submarine tunnel is a single-hole circular tunnel, the flow rate through each cross-section of the tunnel is:

[0126] Q s = 2πρv (2)

[0127] In the formula: Q s represents the water inflow of the calculation cross-section of the surrounding rock layer; ρ represents the distance from any point of the calculation cross-section to the center of the circle of the submarine tunnel. Substitute Equation (1) into Equation (2), and from we can obtain:

[0128]

[0129] In the formula: represents the total head of any point in the surrounding rock layer; Q s represents the water inflow of the calculation cross-section of the surrounding rock layer; k s represents the permeability coefficient of the surrounding rock; n s represents the soil disturbance coefficient; ρ represents the distance from any point of the calculation cross-section to the center of the circle of the submarine tunnel;

[0130] Integrating formula (3) gives:

[0131]

[0132] In the formula: represents the total head at any point within the surrounding rock layer; Q s represents the water inflow of the calculated cross-section of the surrounding rock layer; k s represents the permeability coefficient of the surrounding rock; ρ represents the distance from any point of the calculated cross-section to the center of the circular tunnel of the seabed tunnel; n s represents the soil disturbance coefficient; C represents the integration constant;

[0133] Step S4.3: The water head at any point in the plane of the mirror image method can be determined according to the superposition principle of potential, so:

[0134]

[0135] In the formula: Q si represents the seepage flow of the i-th tunnel, taking a positive value when flowing in and a negative value when flowing out; ρ i represents the distance from the calculation point to the center point of the i-th tunnel; C 1 represents a constant value to be determined;

[0136] As Figure 4 shown, the center coordinates of various tunnels after mirror reflection include the point sink coordinates and the point source coordinates; the point sink coordinates are (0, 4nh + b), (0, 2h + 4nh - b); the point source coordinates are (0, 4nh - b), (0, 2h + 4nh + b); where h represents the distance between the water supply boundary and the impermeable boundary; b represents the distance from the center of the actual tunnel to the water supply boundary; n = 0, ±1, ±2, ···, ±∞; therefore,

[0137]

[0138] In the formula: ρ 1 represents the distance from any point M(x, y) within the finite plane to the center of the first type of tunnel; the first type of tunnel is a virtual seabed tunnel with the center coordinate (0, 4nh + b) obtained by the mirror image method; ρ 2 represents the distance from any point M(x, y) within the finite plane to the center of the second type of tunnel; the second type of tunnel is a virtual seabed tunnel with the center coordinate (0, 2h + 4nh - b) obtained by the mirror image method; ρ 3 represents the distance from any point M(x, y) within the finite plane to the center of the third type of tunnel; the third type of tunnel is a virtual seabed tunnel with the center coordinate (0, 4nh - b) obtained by the mirror image method; ρ 4denotes the distance from any point M(x, y) in the finite plane to the center of the fourth type of tunnel; the fourth type of tunnel is a virtual submarine tunnel with the center coordinates of (0, 2h + 4nh + b) obtained by the mirror method;

[0139] Step S4.4: Substitute Equation (6) into Equation (5) to obtain the water head value of any point M in the soil body at the time of seepage stability as:

[0140]

[0141] In the formula: denotes the total water head of any point in the surrounding rock layer; Q s denotes the water inflow of the calculated cross-section of the surrounding rock layer; k s denotes the permeability coefficient of the surrounding rock; n s denotes the soil disturbance coefficient; C 2 denotes the undetermined constant;

[0142] Step S4.5: According to the Bessette formula:

[0143]

[0144] The formula (7) can be simplified to:

[0145]

[0146] From the water supply boundary condition when y = 0, the constant C can be obtained 2 = b; then substitute the outer boundary condition of the grouting circle x = 0, y = b - r g when, r g denotes the outer radius of the grouting circle; h g denotes the total water head at the outer boundary of the grouting circle; Substitute it into Equation (9) to obtain the seepage flow rate in the surrounding rock of the submarine tunnel:

[0147]

[0148] In the formula: Q denotes the seepage flow rate in the surrounding rock of the submarine tunnel: h g denotes the total water head at the outer boundary of the grouting circle; h denotes the distance between the water supply boundary and the impermeable boundary; b denotes the distance from the center of the actual tunnel to the water supply boundary; r 0 denotes the inner radius of the lining.

[0149] In a specific embodiment, as Figure 5 shown, in Step S5, the seepage parameters of the grouting circle and the lining are calculated based on the Hansbo nonlinear seepage model, specifically

[0150] Step S5.1: The Hansbo non-linear seepage model is

[0151]

[0152]

[0153] In the formula: k 0s represents the permeability coefficient of the non-linear seepage model curve segment; k 0 represents the permeability coefficient of the straight line segment; m represents the non-linear parameter; i 0 , i l represents the critical hydraulic gradient;

[0154] Step S5.2: Define that the fluid is incompressible, and the seepage direction is perpendicular to the y-axis. The continuity equation of fluid seepage in the grouting circle is expressed as:

[0155]

[0156] Decompose the velocity vector in the seepage field:

[0157] u = v i cosθ, v = v i sinθ, v i = k 0s i m (14)

[0158] In the formula: u represents the seepage velocity in the x-axis direction; v represents the seepage velocity in the y-axis direction; v i represents the seepage velocity in any direction; i m represents the m-th power of the hydraulic gradient; k os represents the permeability coefficient of the non-linear seepage model curve segment; θ represents the angle between the seepage velocity in any direction and the x-axis direction;

[0159] Step S5.3: Based on the fact that the direction of the hydraulic gradient of seepage is the same as the direction of the streamline, the hydraulic gradient can be expressed as:

[0160]

[0161] According to the radial seepage conditions of the circular section of the submarine tunnel, we can get:

[0162]

[0163] In the formula: represents the total head; i represents the hydraulic gradient of the surrounding rock; r represents the distance from any point on the calculated section to the center of the actual tunnel;

[0164] Substituting Eqs. (11), (12), (14), (15), and (16) into Eq. (13) and simplifying, the nonlinear fluid continuity equation in polar coordinates for the grouting circle and the lining under the radial seepage condition of the submarine tunnel can be obtained:

[0165] The nonlinear fluid continuity equation in polar coordinates for the grouting circle is:

[0166]

[0167] In the formula: represents the total head in the grouting circle area; m g , i lg represent the nonlinear parameters of the grouting area; r 1g represents the distance from any point in the grouting area to the center of the actual tunnel;

[0168] The nonlinear fluid continuity equation in polar coordinates for the lining is:

[0169]

[0170] In the formula: represents the total head in the lining area; m l , i ll represent the nonlinear parameters of the lining area; r 2l represents the distance from any point in the lining area to the center of the actual tunnel;

[0171] Step S5.4: Solve Eqs. (17) and (18) by the method of separation of variables and substitute the boundary conditions (19) to obtain the total head of the grouting circle and the total head of the lining:

[0172]

[0173] In the formula: h g represents the total head at the outer boundary of the grouting circle; h l represents the total head at the outer boundary of the lining area; h 0 represents the total head at the inner boundary of the lining area; r 0 represents the inner radius of the lining; r l represents the outer radius of the submarine tunnel excavated in the rock mass;

[0174] The total head of the grouting circle is:

[0175]

[0176] The total head of the lining area is:

[0177]

[0178] Step S5.5: Integrate Equation (20) and Equation (21) to obtain the seepage flow rate of the grouting circle and the seepage flow rate of the lining:

[0179] The seepage flow rate Q of the grouting circle g :

[0180]

[0181] The seepage flow rate Q of the lining l is:

[0182]

[0183] wherein, r in Equation (20) and Equation (22) is actually r 1 , r 1 represents the distance from any point in the grouting area to the center of the actual tunnel; r in Equation (21) and Equation (23) is actually r 2 , r 2 represents the distance from any point in the lining area to the center of the actual tunnel;

[0184] Step S5.6: According to Equation (22) and Equation (23), the pore pressure of the grouting circle and the pore pressure of the lining can be obtained as:

[0185] The calculation formula for the pore pressure of the grouting circle is:

[0186]

[0187] The calculation formula for the pore pressure of the lining is:

[0188]

[0189] In the formula: p g represents the pore pressure of the grouting circle; p l represents the pore pressure of the lining; y represents the position head of the grouting area or the lining area; k g represents the permeability coefficient of the grouting circle; k l represents the permeability coefficient of the lining; r l represents the outer radius of the subsea tunnel; h 0 represents the total head at the inner boundary of the lining area; r 0 represents the inner radius of the lining; m g , i lg , k gs , i 0g represent the nonlinear parameters of the grouting area; m l , i ll , k ls , k 0l represent the nonlinear parameters of the lining area; h l represents the total head at the outer boundary of the lining area; γ w represents the unit weight of water.

[0190] In a specific embodiment, in step S6, based on the surrounding rock seepage parameters, the seepage field parameters of the grouting circle, and the seepage field parameters of the lining, the seepage field of the submarine tunnel is analyzed and predicted as follows:

[0191] Step S6.1: Analyze the seepage mode of the grouting circle and the lining based on the seepage continuity condition to obtain the water head h outside the grouting area g and the water head h outside the lining area l ; the seepage continuity condition is that the seepage flow of the surrounding rock is equal to the seepage flow of the grouting circle and equal to the seepage flow of the lining;

[0192] According to the seepage continuity condition, that is, the seepage flow of the surrounding rock is equal to the seepage flow of the grouting circle and equal to the seepage flow of the lining, and the seepage modes of the grouting circle and the lining are not always the same. When the water level is low, both the grouting circle and the lining will stay in the Hansbo non - linear seepage curve section. Therefore, it can be divided into three different actual situations for discussion; the basis for the division of the three situations is: when the water head is low, both the grouting circle and the lining are in the Hansbo non - linear seepage curve section. According to formula (2) and formulas (12), (13), it can be known that the threshold between the Hansbo non - linear seepage curve section and the straight line section of the grouting circle is 2πr l k g i lg mg , since the lining is relatively dense, it will not enter the straight line section first. Therefore, the judgment threshold for case one is 2πr l k g i lg mg ; as the water head continues to increase, the tunnel water inflow increases continuously. The grouting circle enters the Hansbo non - linear seepage straight line section, while the lining stays in the curve section. The threshold between the Hansbo non - linear seepage curve section and the straight line section of the lining area is 2πr 0 k l i ll ml , so when the tunnel water inflow is greater than 2πr l k g i lg mg , less than 2πr 0 k l i ll ml it is case two; as the water head continues to increase, when the tunnel water inflow exceeds the threshold between the Hansbo non - linear seepage curve section and the straight line section of the lining area, the seepage modes of both the grouting circle and the lining are the Hansbo non - linear seepage straight line section;

[0193] Step S6.2: If both the grouting circle and the lining are in the Hansbo non - linear seepage curve section, that is, m g >1, i ≤ ilg ; m l > 1, i ≤ i ll Under the condition; then from formula (10), formula (22) and formula (23), it can be obtained that

[0194]

[0195] If the lining is in the Hansbo non - linear seepage curve section; and the grouting ring is in the Hansbo non - linear seepage straight - line section, that is, m g > 1, i ≤ i lg ; m l > 1, i > i ll Under the condition; from formula (10), formula (22) and formula (23), it can be obtained that

[0196]

[0197] If both the grouting ring and the lining are in the Hansbo non - linear seepage straight - line section, that is, m g = 1; m l Under the condition of = 1, from formula (10), formula (22) and formula (23), it can be obtained that

[0198]

[0199] Step S6.3: Calculate the water head h gt outside the grouting area and the water head h lt outside the lining area according to formula (26), (27) and (28); and h gt is h g ; h lt is h l ; h 0t is h 0 ; Substitute h gt and h lt back into the calculation formulas of the seepage field parameters of the surrounding rock, the grouting ring and the lining, as shown in Figure 2 , to obtain the seepage field parameters of the surrounding rock, the grouting ring and the lining under the current seepage condition; predict the seepage field of the submarine tunnel according to the seepage field parameters of the surrounding rock, the grouting ring and the lining under the current seepage condition.

[0200] The present invention solves the problem that the existing calculation methods cannot consider the impermeable lower boundary while taking into account the non-linear seepage of both the grouting circle and the lining. Moreover, the Hansbo non-linear seepage model is introduced, which can consider the situation where the seepage mode enters linear seepage when the water pressure is too high. Thus, different seepage modes of the grouting circle and the lining under different water pressure conditions can be discussed case by case, and then the actual situation of the seepage field can be described more completely and accurately. The tunnel water inflow and the pore pressure borne by the grouting circle and the lining can be solved, making the simulation of the situation more realistic and reliable, and enabling better selection of grouting and lining materials in actual situations.

[0201] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and are not intended to limit them. Although the present invention has been described in detail with reference to the foregoing embodiments, those of ordinary skill in the art should understand that they can still modify the technical solutions described in the foregoing embodiments, or perform equivalent replacements on some or all of the technical features; and these modifications or replacements do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention.

Claims

1. A nonlinear seepage field analysis method for submarine tunnels based on the mirror method in a finite plane, characterized in that: include Step S1: determining the geological conditions of the subsea tunnel excavation section, wherein the geological conditions are that the subsea tunnel excavation section is located in a homogeneous stratum region, and the plane boundary where the top of the homogeneous stratum region is located is defined as a permeable boundary, and the plane where the bottom of the homogeneous stratum region is located is defined as an impermeable boundary; The homogeneous stratum includes a surrounding rock layer, a grouting ring layer and a lining layer; Step S2: The homogeneous stratum area is regarded as being distributed annularly around the periphery of the submarine tunnel section, and the inner / outer diameter of each homogeneous stratum to the center of the tunnel section and the permeability coefficient corresponding to each homogeneous stratum are determined; The permeability coefficient includes the permeability coefficient of surrounding rock, the permeability coefficient of grouting ring and the permeability coefficient of lining; Step S3: setting the horizontal plane at the center of the actual submarine tunnel as the potential zero plane, taking the water level as the x-axis reference datum line, and taking the straight line perpendicular to the axis of the actual submarine tunnel as the y-axis reference datum line, to establish a rectangular coordinate system; Based on the mirror superposition principle, the actual submarine tunnel is mapped by mirroring the permeable boundary and the impermeable boundary to obtain a virtual submarine tunnel, and the actual submarine tunnel and the virtual submarine tunnel are superimposed with each other to form a submarine tunnel seepage field model that takes into account the influence of the permeable boundary and the impermeable boundary of the surrounding rock. The submarine tunnel seepage field model includes the surrounding rock seepage field, the grouting circle seepage field and the lining seepage field; Step S4: Calculate the surrounding rock seepage field based on Darcy's seepage law to obtain surrounding rock seepage field parameters, wherein the surrounding rock seepage field parameters include water inflow at the cross section of the submarine tunnel, total water head in the surrounding rock area, and seepage volume in the surrounding rock of the submarine tunnel; Step S5: Based on the Hansbo nonlinear seepage model, the seepage field of the grouting circle and the seepage field of the lining are calculated to obtain the seepage parameters of the grouting circle and the seepage parameters of the lining; The grouting circle seepage parameters include the grouting circle seepage volume, the grouting circle total water head and the grouting circle pore pressure; The lining seepage parameters include lining seepage, lining total water head and lining pore pressure; Step S6: Analyze and predict the seepage field of the submarine tunnel according to the surrounding rock seepage parameters, the grouting circle seepage field parameters and the lining seepage field parameters.

2. According to the method for analyzing nonlinear seepage field of submarine tunnel based on mirror method in a finite plane in claim 1, it is characterized in that: In step S4, the surrounding rock seepage field parameters are calculated based on Darcy's seepage law, specifically: Step S4.1: The actual submarine tunnel is subjected to equal flow and different sign reflection with the permeable boundary as a mirror surface to obtain a virtual submarine tunnel and define it as a source, and after the reflection, the permeable boundary remains as an equipotential line; The actual submarine tunnel is subjected to equal flow and same sign reflection with the impermeable boundary as a mirror surface to obtain a virtual submarine tunnel and define it as a sink, and after the reflection, the impermeable boundary is kept as a flow diversion line; Step S4.2: The seepage pattern of the soil conforms to Darcy seepage: Where i represents the hydraulic gradient of the surrounding rock; k represents the permeability coefficient; v represents the seepage velocity of the surrounding rock; When the submarine tunnel is a single-hole circular tunnel, the flow rate flowing through each section of the tunnel is: Q s =2πrv (2) Where: Q s represents the water inflow of the calculated section of the surrounding rock layer; r represents the distance from any point of the calculated section to the center of the submarine tunnel. Substituting formula (1) into formula (2), we get We can get: Where: It represents the total water head at any point in the surrounding rock layer; Q s Indicates the water inflow of the calculated section of the surrounding rock layer; k s represents the permeability coefficient of surrounding rock; n s represents the soil disturbance coefficient; Integrating formula (3) yields: Where: It represents the total water head at any point in the surrounding rock layer; Q s Indicates the water inflow of the calculated section of the surrounding rock layer; k s represents the permeability coefficient of the surrounding rock; r represents the distance from any point on the calculation section to the center of the submarine tunnel; n s represents the soil disturbance coefficient; C represents the integral constant; Step S4.3: The water potential at any point in the mirror normal plane can be determined according to the principle of superposition of potentials, thus: Where: Q si represents the seepage volume of the ith tunnel, which takes a positive value when it flows in and a negative value when it flows out; r i represents the distance from the calculation point to the center point of the i-th tunnel; C1 represents the value of a constant to be determined; The center coordinates of various tunnels after mirroring include point sink coordinates and point source coordinates; the point sink coordinates are (0, 4nh+b), (0, 2h+4nh-b); the point source coordinates are (0, 4nh-b), (0, 2h+4nh+b); where h represents the distance between the water supply boundary and the impermeable boundary; b represents the distance from the center of the actual tunnel to the water supply boundary; n = 0, ±1, ±2, ···, ±∞; therefore, Wherein: r1 represents the distance from any point M(x,y) in the finite plane to the center of the first type of tunnel; the first type of tunnel is a virtual submarine tunnel with the center coordinates (0,4nh+b) obtained by the mirror method; r2 represents the distance from any point M(x,y) in the finite plane to the center of the second type of tunnel; the second type of tunnel is a virtual submarine tunnel with the center coordinates (0,2h+4nh-b) obtained by the mirror method; r3 represents the distance from any point M(x,y) in the finite plane to the center of the third type of tunnel; the third type of tunnel is a virtual submarine tunnel with the center coordinates (0,4nh-b) obtained by the mirror method; r4 represents the distance from any point M(x,y) in the finite plane to the center of the fourth type of tunnel; the fourth type of tunnel is a virtual submarine tunnel with the center coordinates (0,2h+4nh+b) obtained by the mirror method; Step S4.4: Substitute equation (6) into equation (5) to obtain the water head value at any point M in the soil when the seepage is stable: Where: It represents the total water head at any point in the surrounding rock layer; Q s Indicates the water inflow of the calculated section of the surrounding rock layer; k s represents the permeability coefficient of surrounding rock; n s represents the soil disturbance coefficient; C2 represents the undetermined constant; Step S4.5: According to Bessette's formula: Formula (7) can be simplified as: According to the water supply boundary condition y=0, The constant C2 = b can be obtained; then the outer boundary conditions of the grouting circle are x = 0, y = br g When g Indicates the outer radius of the grouting circle; h g It represents the total water head at the outer boundary of the grouting circle; Substituting into formula (9), we can get the seepage flow rate in the surrounding rock of the submarine tunnel: Where: Q represents the seepage flow rate in the surrounding rock of the submarine tunnel: h g It is expressed as the total water head at the outer boundary of the grouting circle; h is the distance between the water supply boundary and the impermeable boundary; b is the distance from the center of the actual tunnel to the water supply boundary; r0 is the inner radius of the lining.

3. According to the method for analyzing nonlinear seepage field of submarine tunnel based on mirror method in a finite plane in claim 1, it is characterized in that: In step S5, the seepage field of the grouting circle and the seepage field of the lining are calculated based on the Hansbo nonlinear seepage model to obtain the seepage parameters of the grouting circle and the seepage parameters of the lining, which is specifically step S5.1: the Hansbo nonlinear seepage model is Where: k 0s represents the permeability coefficient of the curve segment of the nonlinear seepage model; k0 represents the permeability coefficient of the straight line segment; m represents the nonlinear parameter; i0, i l represents the critical hydraulic gradient; Step S5.2: Define the fluid as incompressible and the seepage direction is perpendicular to the y-axis. The continuity equation of the fluid seepage in the grouting circle is expressed as: Decompose the velocity vector in the seepage field: u=v i cosθ,v=v i sinθ,v i =k 0s i m (14) Where: u represents the seepage velocity in the x-axis direction; v represents the seepage velocity in the y-axis direction; v i represents the seepage velocity in any direction; i m represents the hydraulic gradient to the power of m; k os represents the permeability coefficient of the curve segment of the nonlinear seepage model; θ represents the angle between the seepage velocity in any direction and the x-axis direction; Step S5.3: Based on the fact that the hydraulic gradient direction of the seepage is the same as the streamline direction, the hydraulic gradient can be expressed as: According to the radial seepage condition of circular cross section of submarine tunnel, it can be obtained that: Where: It is expressed as the total water head; i represents the hydraulic gradient of the surrounding rock; r represents the distance from any point of the calculation section to the center of the circle; Substituting formulas (11), (12), (14), (15), and (16) into formula (13) for simplification, we can obtain the nonlinear continuity equation of the fluid in the polar coordinate form of the grouting ring and the lining under the radial seepage condition of the submarine tunnel: The nonlinear continuity equation of the fluid in the grouting circle in polar coordinate form is: Where: Indicates the total water head in the grouting circle; m g 、i lg Represents the nonlinear parameter of the grouting area; r 1g Indicates the distance from any point in the grouting area to the actual tunnel center; The nonlinear continuity equation of the fluid in the lining polar coordinate form is: Where: Indicates the total water head in the lining area; m l 、i ll Represents the nonlinear parameter of the lining area; r 2l It indicates the distance from any point in the lining area to the actual center of the tunnel; Step S5.4: Solve formula (17) and formula (18) by separation of variables method, and substitute boundary condition (19) to obtain the total water head of the grouting circle and the total water head of the lining: Where: h g represents the total water head at the outer boundary of the grouting circle; h l It is expressed as the total water head at the outer boundary of the lining area; h0 is the total water head at the inner boundary of the lining area; r0 is the inner radius of the lining; r l It represents the outer radius of the submarine tunnel excavated inside the rock mass; Total water head of grouting ring for: Total head in lining area for: Step S5.5: Integrate formula (20) and formula (21) to obtain the seepage of the grouting ring and the seepage of the lining: Seepage volume of grouting ring Q g : Lining seepage Q l for: Step S5.6: According to formula (22) and formula (23), the pore pressure of the grouting ring and the pore pressure of the lining can be obtained as follows: The calculation formula of pore pressure in grouting ring is: The calculation formula of lining pore pressure is: Where: p g Indicates the pore pressure of the grouting circle; p l represents the lining pore pressure; y represents the water head at different positions in the grouting area or lining area; k g Indicates the permeability coefficient of the grouting circle; k l represents the lining permeability coefficient; r l represents the outer radius of the submarine tunnel; h0 represents the total water head at the inner boundary of the lining area; r0 represents the inner radius of the lining; m g 、i lg , k gs 、i 0g Represents the nonlinear parameter of the grouting area; m l 、i ll , k ls , k 0l represents the nonlinear parameter of the lining area; h l represents the total water head at the outer boundary of the lining area; γ w Indicates the weight of water.

4. According to the method for analyzing nonlinear seepage field of submarine tunnel based on mirror method in a finite plane in claim 1, it is characterized in that: In step S6, the seepage field of the submarine tunnel is analyzed and predicted according to the surrounding rock seepage parameters, the grouting ring seepage field parameters and the lining seepage field parameters, specifically: Step S6.1: Analyze the seepage mode of the grouting ring and the lining based on the seepage continuity condition and obtain the water head h outside the grouting area g The water head outside the lining area h l ; The seepage continuity condition is that the surrounding rock seepage is equal to the grouting ring seepage and equal to the lining seepage; Step S6.2: If both the grouting ring and the lining are Hansbo nonlinear seepage curve segments, that is, m g >1,i≤i lg ;m l >1,i≤i ll Under the condition, then from formula (10), formula (22) and formula (23), we can get If the lining is a Hansbo nonlinear seepage curve segment and the grouting circle is a Hansbo nonlinear seepage straight line segment, that is, m g >1,i≤i lg ;m l >1,i>i ll Under the condition; from formula (10), formula (22) and formula (23), we can get If the grouting ring and the lining are both Hansbo nonlinear seepage straight line segments, that is, m g =1;m l =1, from formula (10), formula (22) and formula (23) we can get Step S6.3: Calculate the water head h outside the grouting area according to formulas (26), (27) and (28): gt The water head outside the lining area h lt ; and h gt That is h g ;h lt That is h l ;h 0t That is h0; h gt With h lt Substituting back into the calculation formulas of surrounding rock seepage field parameters, grouting circle seepage field parameters and lining seepage field parameters, the surrounding rock seepage field parameters, grouting circle seepage field parameters and lining seepage field parameters under current seepage conditions are obtained; The seepage field of the submarine tunnel is predicted according to the surrounding rock seepage field parameters, the grouting circle seepage field parameters and the lining seepage field parameters under the current seepage conditions.

Images