A model for fast prediction of tunnel settlement

Abstract

The application discloses a model for quickly predicting tunnel settlement, and establishes a control equation under the framework of a wave propagation model, can link the disturbance input of tunnel excavation with the settlement response of strata at different depths, thereby providing a solid mathematical and physical basis for predicting the settlement amount at any position and any depth of the upper part of the tunnel, and providing a theoretical basis for subsequent engineering safety evaluation and protection measure design.

Description

Technical Field

[0001] This invention belongs to the field of tunnel engineering technology, specifically relating to a model for rapidly predicting tunnel settlement. Background Technology

[0002] To delve into the distribution characteristics and evolution of ground settlement under different burial depths and horizontal offsets from the tunnel centerline, this study utilizes the mature analytical framework of stochastic media theory. Its core assumption is that the movement of the soil above the tunnel after excavation is treated as a statistically consistent stochastic settlement process. This theory posits that surface and ground settlement caused by the excavation of countless micro-units can be superimposed using probability integration, thus more accurately reflecting the nonlinear and discontinuous displacement characteristics of the soil. Tunnel excavation inevitably leads to ground loss, which is the fundamental cause of displacement and deformation of the surrounding soil. At the instant the tunnel excavation is completed, the stress release is most significant at the top axis directly above the excavation face due to the direct loss of the original soil support, resulting in the maximum instantaneous settlement at this point. Simultaneously, due to the strong constraint of the soil near the excavation face, the impact of settlement is confined to a relatively narrow area, manifested as the narrowest settlement trough at this depth and a steep settlement curve, exhibiting strong local concentration characteristics. As settlement originates at the tunnel top and propagates upwards in a wave-like manner to the overlying strata, its propagation path and morphological evolution follow certain physical laws. During this process, friction, interlocking, and stress redistribution between soil particles cause the settlement energy to gradually dissipate as it is transmitted upwards. Therefore, when the settlement curve finally reaches the surface, its maximum settlement value is significantly reduced compared to the tunnel top; simultaneously, due to the stress diffusion effect, the horizontal range affected by settlement expands dramatically, and the shape of the settlement trough gradually evolves from narrow and steep at the deep end to wide and gentle at the shallow end. This gradual change in settlement amplitude and morphology from the epicenter (tunnel top) to the free surface (surface) constitutes the core characteristic of ground settlement propagation.

[0003] Accurately revealing and quantifying the nonlinear distribution characteristics and morphological evolution of ground settlement during tunnel excavation, under different burial depths and horizontal offset distances from the tunnel centerline, as it propagates from the top of the tunnel upwards to the overlying strata and finally to the surface, is a key factor in tunnel support and an urgent problem to be solved. Summary of the Invention

[0004] In order to systematically and scientifically describe this settlement diffusion mechanism from deep to shallow, embodiments of the present invention propose a model for rapidly predicting tunnel settlement.

[0005] The present invention provides a rapid prediction model for tunnel settlement. This model employs a wave propagation model to describe the propagation mechanism of settlement in the top strata caused by tunnel excavation. Taking the tunnel top as the settlement initiation point, the governing equations of the model are:

[0006] (1)

[0007] In the formula, x is the horizontal distance from the central axis of the tunnel, and h is the burial depth of the tunnel. This represents the ground settlement curve perpendicular to the tunnel's orientation at any depth above the tunnel. The model parameters are determined using the boundary conditions of actual ground subsidence. This is the boundary function for actual measured ground subsidence;

[0008] The solution to the governing equation is:

[0009] (6)

[0010] In the formula, Let S be the integral variable, and S be the Fourier transform of the initial deformation of the top after tunnel excavation.

[0011] The ground settlement curve perpendicular to the tunnel strike at any depth above the tunnel, as defined by the governing equations, is as follows:

[0012] (8)

[0013] In the formula, β represents the maximum ground settlement, and β is the ground settlement trough width coefficient.

[0014] The ground settlement curve above the tunnel, parallel to the tunnel's strike, for the governing equations is as follows:

[0015] (11)

[0016] In the formula, Let x represent the ground settlement curve above the tunnel, parallel to the tunnel's direction. x is the horizontal distance from the tunnel's central axis, and y is the longitudinal coordinate along the tunnel's direction. This represents the maximum ground subsidence.

[0017] The formula for calculating the settlement of surface and deep soil caused by shield tunnel construction, as defined in the governing equations, is as follows:

[0018] (12)

[0019] In the formula, Formulas for soil settlement at different locations and depths at the top of the tunnel. β represents the maximum ground settlement, and β is the ground settlement trough width coefficient. These are the model parameters, determined by the boundary conditions of actual ground subsidence.

[0020] The beneficial effects of this invention are that the governing equations established within the framework of the wave propagation model can link the disturbance input of tunnel excavation with the settlement response of strata at different depths, thereby providing a solid mathematical and physical basis for predicting the settlement at any location and depth above the tunnel, and providing a theoretical basis for subsequent engineering safety assessment and protection measure design. Attached Figure Description

[0021] Figure 1 It refers to the distribution characteristics of ground settlement curves perpendicular to the tunnel orientation at different depths h (from the ground surface downwards to the top of the tunnel).

[0022] Figure 2 These are the ground settlement curves (along the tunnel excavation direction) of the upper part of the tunnel at different depths h, parallel to the tunnel's orientation.

[0023] Figure 3 This describes the influence of the settlement trough width coefficient β on the settlement curve of the strata above the tunnel, parallel to the tunnel's orientation.

[0024] Figure 4 It describes the influence of model parameter 'a' on the settlement curve of the strata above the tunnel, parallel to the tunnel's orientation. Detailed Implementation

[0025] Embodiments of the present invention are described in detail below, examples of which are illustrated in the accompanying drawings. The embodiments described below with reference to the accompanying drawings are exemplary and intended to explain the present invention, and should not be construed as limiting the present invention.

[0026] This invention introduces a wave propagation model as a tool for theoretical analysis. This model draws upon the concepts of attenuation and diffusion of energy and disturbances in a medium from the physics of wave motion, treating the initial ground disturbance caused by tunnel excavation as a settlement wave source. Within the framework of the wave propagation model, the tunnel top is taken as the zero point of settlement initiation or the wave source point, and the stratum is considered as a continuous, homogeneous, or layered homogeneous propagation medium. The model's governing equations aim to quantitatively describe the attenuation law of the settlement wave's amplitude (i.e., settlement) with propagation distance (depth) during upward propagation, as well as the diffusion effect of its influence range (settlement trough width) with propagation distance. By establishing such governing equations, the disturbance input from tunnel excavation can be linked to the settlement response of strata at different depths, thus providing a solid mathematical and physical foundation for predicting settlement at any location and depth above the tunnel, and providing a theoretical basis for subsequent engineering safety assessments and protective measure design.

[0027] This invention employs a wave propagation model to describe the propagation mechanism of ground settlement above the tunnel after excavation. Taking the tunnel top as the settlement initiation point, the governing equations of the model are:

[0028] (1)

[0029] In the formula, x is the horizontal distance from the central axis of the tunnel, and h is the burial depth of the tunnel. This represents the ground settlement curve perpendicular to the tunnel's orientation at any depth above the tunnel. These are model parameters, which can be determined through the boundary conditions of actual ground subsidence. This is the boundary function for actual measured ground subsidence.

[0030] To solve the partial differential equation (1), we first assume... The Fourier transform exists, and then a Fourier transform with respect to x yields:

[0031] (2)

[0032] In the formula, For settlement The image function, Boundary function The image function, where i is an imaginary number. is the independent variable of the image function.

[0033] Taking a Fourier transform of both sides of equation (1) with respect to x, we can obtain:

[0034] (3)

[0035] Solve the first-order differential equation (3) and substitute the boundary conditions. We can obtain:

[0036] (4)

[0037] function The Fourier inverse transform is:

[0038] (5)

[0039] Using Fourier transform properties From equations (4) and (5), the solution to the partial differential equation (1) can be obtained as follows:

[0040] (6)

[0041] In the formula, Let S be the integral variable, and S be the Fourier transform of the initial deformation of the top after tunnel excavation.

[0042] Taking surface settlement as a boundary condition, the settlement can be described using Peck's formula, i.e.:

[0043] (7)

[0044] In the formula, β represents the maximum ground settlement, and β is the ground settlement trough width coefficient.

[0045] Substituting equation (7) into equation (6) and calculating based on the integral of the complex function, we can obtain:

[0046] (8)

[0047] Equation (8) is the analytical solution for ground settlement at any depth above the tunnel caused during tunnel excavation.

[0048] Equation (8) is only applicable to describing the ground settlement perpendicular to the tunnel direction at any depth at the top of the tunnel after tunnel excavation, when the instantaneous settlement of the soil at the tail of the shield has basically stabilized. It cannot calculate the ground settlement parallel to the tunnel direction caused by this. In terms of coordinate settings, it is assumed that the tunnel axis is parallel to the y-axis and located directly below the origin, the coordinate of the tunnel face is y=0, and the shield excavation direction is consistent with the positive direction of the y-axis.

[0049] Based on Peck's formula, the cumulative probability curve can be used to obtain a formula for calculating the ground settlement of the strata directly above the tunnel axis, parallel to the tunnel's direction. The expression is as follows:

[0050] (9)

[0051] In the formula, y is the longitudinal coordinate along the tunnel direction. This is the starting point of the tunnel. This is the end point of the tunnel. Let be a probability function.

[0052] probability function for:

[0053] (10)

[0054] When establishing the coordinate system, it is assumed that the location coordinates of the tunnel face are point 0, i.e., x=0. =0, then the tunnel starting point Substituting the coordinates of the tunnel's start and end points and the probability function (10) into equation (9), we can obtain the ground settlement on the surface parallel to the tunnel's direction caused by tunnel excavation as follows:

[0055] (11)

[0056] Combining equations (10), (11), and (8), the formula for calculating the settlement of surface and deep soil caused by shield tunnel construction can be obtained as follows:

[0057] (12)

[0058] In the formula, Formulas for the settlement of soil at different locations and depths at the top of the tunnel.

[0059] Model parameter analysis

[0060] To deeply reveal the laws governing ground settlement caused by shield tunnel construction, this invention conducts a systematic parameter analysis based on a typical parallel tunnel project. To ensure the accuracy and comparability of the research results, key parameters in the calculation model were uniformly set: the settlement trough width coefficient β = 11m was selected, reflecting the comprehensive influence of ground conditions and tunnel construction factors on the settlement range; the model calculation parameter a was set to 2.0, which is used in stochastic medium theory to control the morphological characteristics of the settlement curve. Regarding tunnel geometry, the tunnel diameter was set to 6m, and the tunnel axis burial depth was set to h = 20m. This combination of burial depth and diameter can well represent the general working conditions of urban subway tunnels. Simultaneously, the maximum settlement u at the tunnel top was used as the basis for the analysis. max =12mm was used as the baseline input value, and the settlement response under different depth conditions was analyzed based on this.

[0061] Figure 1 The distribution characteristics of ground settlement curves perpendicular to the tunnel orientation at different depths h (from the surface downwards to the tunnel top) are shown. The following patterns can be observed from the figure: as the depth of the observation point increases, the width of the settlement trough tends to increase, while the maximum settlement gradually increases. This phenomenon can be explained from a mechanical perspective: deeper soil layers are closer to the tunnel excavation face and are more directly affected by construction disturbances, resulting in more complete stress release and thus greater settlement. Simultaneously, due to the stronger constraint of the overlying strata on the deeper soil layers, the range of displacement diffusion to both sides is relatively limited, leading to a relatively narrow settlement trough shape and a larger settlement gradient. As the observation point moves upwards closer to the surface, the loose overlying soil layer has a diffusion and homogenization effect on stress, causing the energy of the settlement wave to gradually dissipate during propagation, manifested as a decrease in the maximum settlement, an expansion of the settlement influence range, and a smoother settlement trough.

[0062] Figure 2This displays the ground settlement curves (along the tunnel excavation direction) at different depths *h* above the tunnel, parallel to the tunnel's orientation. The settlement curves show the smallest settlement value at the surface, with a relatively gentle change, indicating a certain lag and homogenization in the surface's response to excavation disturbance. With increasing depth, the shape of the settlement curves changes significantly: on the one hand, the absolute settlement value at each point increases significantly; on the other hand, the slope of the settlement curve gradually increases, meaning the settlement amplitude changes from gentle to steep, and the settlement transition section shortens. This indicates that the settlement of deep soil is more sensitive to the location of the excavation face and can respond more quickly to the spatial effects generated by tunnel excavation. Specifically, in the deep region near the tunnel crown, settlement begins to occur before the excavation face approaches the observation section; however, once the excavation face passes the observation section, the settlement rapidly reaches a large value, and the curve exhibits a steep rise and fall characteristic. This pattern has important guiding significance for determining the timing of synchronous grouting during shield tunneling and controlling differential settlement of surface buildings and structures.

[0063] In summary, by comparing and analyzing the ground settlement curves perpendicular to and parallel to the tunnel's direction at different depths above the tunnel, it is clear that ground settlement caused by tunnel excavation is a spatial process that varies significantly with depth. Settlement in deeper strata exhibits characteristics of concentration, rapidity, and strong interaction, while settlement in shallower strata exhibits characteristics of diffusion, lag, and weak coupling. This understanding provides a theoretical basis for establishing more refined settlement prediction models, optimizing shield tunneling parameters, and developing targeted ground reinforcement measures.

[0064] To delve into the patterns of surface settlement caused by shield tunnel construction, the influence of two key calculation parameters—the settlement trough width coefficient β and the model parameter a—on the settlement curve of the strata above the tunnel parallel to its direction was analyzed. By comparing the calculation results under different parameter values, the controlling role of these parameters in the settlement development process can be revealed, providing a theoretical basis for parameter selection and settlement prediction in practical engineering.

[0065] Figure 3This paper demonstrates the influence of the settlement trough width factor β on the settlement curve of the strata above the tunnel parallel to the tunnel's direction. The settlement trough width factor β is a crucial comprehensive parameter in stochastic medium theory, reflecting to a certain extent the coupled influence of factors such as stratum conditions, tunnel depth, construction technology, and stratum loss rate on the settlement range. The figure clearly shows that as the β value gradually increases, the settlement curve of the strata above the tunnel parallel to the tunnel's direction exhibits two significant characteristics: first, the final settlement value at each measuring point increases significantly; second, the rate of settlement growth as the excavation face advances increases significantly. This phenomenon can be explained from the perspective of soil deformation response: a larger β value indicates a stronger settlement transmission capacity of the strata, or in other words, relatively soft soil, which can more fully propagate the disturbance caused by tunnel excavation to the surrounding area, leading to a wider range of soil deformation. Therefore, under the same excavation progress, the amount of settlement is greater, and the rate of settlement development is faster. Further observation of the curve's characteristics as it stabilizes reveals that as the β value increases, the distance between measuring points corresponding to the stable state of surface settlement (i.e., the longitudinal range from the start of excavation to when settlement essentially stops) gradually decreases. This indicates that strata with larger β values ​​are more sensitive to settlement response and can complete the entire process from initial deformation to final stability within a shorter distance. In other words, the larger the β value, the more easily the surface is disturbed by excavation and settles, and the settlement is more concentrated and converges faster. This pattern has significant guiding implications for monitoring point placement and grouting reinforcement timing during shield tunneling: in soft strata with large β values, greater attention needs to be paid to short-term settlement control before and after the excavation face, and synchronous and secondary grouting measures should be strengthened to prevent excessively rapid settlement development.

[0066] Figure 4This demonstrates the influence of model parameter 'a' on the settlement curve of the strata above the tunnel, parallel to the tunnel's direction. In stochastic medium theory, parameter 'a' is mainly used to control the shape of the settlement curve, reflecting the soil's deformation compatibility and stress transfer characteristics under excavation disturbance. An important critical point phenomenon can be observed in the figure: at the location of the tunnel excavation face (i.e., where the distance between measuring points is zero), the settlement values ​​corresponding to different 'a' values ​​are basically the same, exhibiting a convergent characteristic. This has clear physical significance—the instantaneous settlement at the excavation face is mainly determined by the immediate effect of excavation unloading, and is relatively less affected by subsequent settlement development patterns. With the excavation face as the boundary, the influence of parameter 'a' exhibits obvious zonal characteristics. In the unexcavated area (i.e., in front of the excavation face), the surface settlement decreases as the value of 'a' increases. This is because in the unexcavated area, settlement is mainly caused by the advance disturbance of the soil in front of the excavation face. With a larger 'a' value, the soil has good integrity and relatively high stiffness, and is less sensitive to advance disturbance, thus resulting in smaller advance settlement. In the area already excavated (i.e., behind the excavation face), the situation is quite the opposite: the rate of settlement growth of the strata parallel to the tunnel's direction increases significantly with the increase of the 'a' value, and the final stable settlement value also increases accordingly. This indicates that strata with larger 'a' values, although relatively stable ahead of the excavation face, experience more rapid and complete post-construction settlement once the excavation face has passed. It is also noteworthy that the distance between measuring points corresponding to near-stabilized settlement decreases with increasing 'a' value. This trend is similar to the effect of the 'β' value, but the mechanism is different: a larger 'a' value means a more concentrated settlement curve, with settlement mainly occurring in a shorter section near the excavation face; therefore, settlement converges faster and the stabilization distance is shorter. In summary, the larger the parameter 'a', the stronger the sensitivity of the surface to excavation response, and the more likely it is to experience significant settlement in the short term. This understanding suggests that during shield tunneling, for strata with a large a value, although the advance settlement in front of the excavation face is small, special attention should be paid to the immediate settlement control after the excavation face passes through, and the timeliness and effectiveness of grouting behind the wall should be strengthened to prevent damage to surface buildings and structures due to excessive settlement rate.

[0067] In summary, parameters β and a characterize the settlement response of the strata to tunnel excavation from different perspectives: β mainly affects the overall amplitude and range of settlement, reflecting the settlement transmission capacity of the strata; while a mainly affects the shape and development rate of the settlement curve, reflecting the concentration of stratum deformation. In practical engineering, the values ​​of these two parameters should be reasonably determined based on geological survey data and field monitoring data, and targeted construction control measures should be formulated in combination with their combined influence to achieve effective prediction and precise control of surface settlement.

[0068] Although the above embodiments have been shown and described, it is understood that the above embodiments are exemplary and should not be construed as limiting the present invention. Any changes, modifications, substitutions and variations made to the above embodiments by those skilled in the art are within the protection scope of the present invention.

Claims

1. A model for rapidly predicting tunnel settlement, characterized in that, The model employs a wave propagation model to describe the propagation mechanism of ground settlement at the top of the tunnel after excavation. Taking the tunnel top as the settlement initiation point, the governing equations of the model are: （1） In the formula, x is the horizontal distance from the central axis of the tunnel, and h is the burial depth of the tunnel. This represents the ground settlement curve perpendicular to the tunnel's orientation at any depth above the tunnel. The model parameters are determined using the boundary conditions of actual ground subsidence. This is the boundary function for actual measured ground subsidence; The solution to the governing equation is: （6） In the formula, Let S be the integral variable, and S be the Fourier transform of the initial deformation of the top after tunnel excavation.

2. The model for rapidly predicting tunnel settlement according to claim 1, characterized in that, The ground settlement curve perpendicular to the tunnel strike at any depth above the tunnel, as defined by the governing equations, is as follows: （8） In the formula, β represents the maximum ground settlement, and β is the ground settlement trough width coefficient.

3. The model for rapidly predicting tunnel settlement according to claim 1, characterized in that, The ground settlement curve above the tunnel, parallel to the tunnel's strike, for the governing equations is as follows: （11） In the formula, Let x represent the ground settlement curve above the tunnel, parallel to the tunnel's direction. x is the horizontal distance from the tunnel's central axis, and y is the longitudinal coordinate along the tunnel's direction. This represents the maximum ground subsidence.

4. The model for rapidly predicting tunnel settlement according to claim 1, characterized in that, The formula for calculating the settlement of surface and deep soil caused by shield tunnel construction, as defined in the governing equations, is as follows: （12） In the formula, Formulas for soil settlement at different locations and depths at the top of the tunnel. β represents the maximum ground settlement, and β is the ground settlement trough width coefficient. These are the model parameters, determined by the boundary conditions of actual ground subsidence.

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