Keywords

1 Introduction

Deep tunnel drainage systems, as a form of grey sponge technology, have been applied in practice for many years [1,2,3]. The underground tunnels are typically buried within 30–60 m to provide a large storage capacity and drainage capability. The system does not conflict with other underground structures in the shallow layer and provides an effective way to utilize urban land resources.

Generally, storage facility operations have continuous loading and unloading processes, e.g., long-term filling and discharging of water loads, which can be simplified as long load cycles with large fluctuations. Previous research reports that the long-term cyclic loading [4,5,6] could induce a softening effect in soils surrounding the tunnels due to high compressibility and large porosity of deposits. As a result, the storage tunnel structures have high risks of deforming and even cracking, undermining the normal supply, drainage functions, and water environmental management within the city. To ensure the safety and maintenance of the deep tunnel, a proper estimation of the settlement of surrounding soil during the operational period, which also represents the settlement of the tunnels, is beneficial and worth discussing.

Research on deep tunnel drainage systems is still in its early stages. Since relevant studies mainly concentrate on case analyses, structural construction, and planning, there is a lack of research concerning the long-term operation and maintenance aspects. Presently, empirical methods have been successfully applied to predict long-term settlement in subway tunnels and soft soil roadbeds [7,8,9]. However, these methods have not yet been applied to predict the long-term operational settlement of deep drainage tunnels in soft soil areas. In this paper, a combined approach of quasi-static finite element analysis and layered summation method were testified based on a deep drainage system in soft soils. Considering water filling-discharge cycles and the cycle-induced loads, the long-term settlement of deep tunnels was predicted and analyzed.

2 Cumulative Deformation of Saturated Soft Clay Under Cyclic Loading

The long-term settlement of soil under undrained cyclic loading can be divided into two parts: cumulative plastic deformation and end-of-shake consolidation settlement resulting from pore pressure dissipation. Empirical prediction formulas generally require the calculation of the first-cycle plastic strain and pore pressure, and then extensively and uniquely determine the relationship among the number of loading cycles, cumulative strain, and pore pressure. Factors like loading intensity and stress history are simply considered. The earliest wide-used model was an exponential model proposed by Monishmith [10] \({\varepsilon }_{p}=A{N}^{b}\), which directly established the relationship between cumulative plastic strain \({\varepsilon }_{p}\) and the number of cycles \(N\). . Here, parameter A represents the first-cycle strain (\(N=1\)), and parameter \(b\) is a fitted parameter. Based on Monishmith's exponential model, optimized empirical fitting calculation models and formulas were further proposed, and were combined with the layered summation method to calculate the long-term settlement of soft soil foundation [11,12,13,14]. Combining previous research results, the calculation formulas have been widely recognized and applied, as shown in:

$$Q=a{(\frac{{q}_{d}}{{q}_{ult}})}^{n}{{(\frac{{p}_{0}}{{p}_{a}})}^{c}N}^{b}$$
(1)

where \(Q\) is cumulative plastic strain or pore pressure; \({q}_{d}\) is the dynamic deviator stress generated in the foundation after the first loading; \({q}_{ult}\) is the undrained shear strength, \({p}_{0}\) is the consolidation mean stress; \(N\) is the number of load cycles; parameters \(a,n,c,b\) are obtained by experimental fitting, in which \(a\) and \(n\) reflect the influence of dynamic deviator stress on cumulative settlement, and \(c\) and \(b\) respectively reflect the effects of confining pressure and number of repetitions on settlement during operation. In default, \({p}_{a}=101.3kPa\).

3 Prediction of the Long-Term Settlement of Deep Drainage Pipelines

The total length of the deep drainage system project in Shanghai is approximately 15 km, with a tunnel diameter of 10 m and a burial depth of about 50 m. A typical cross-section of the tunnel and shaft is shown in Fig. 1, where the tunnel is called a ‘pipeline’ since its length is much greater than its dimension.

Fig. 1
Two illustrations a and b display the plan view and profile view of the two shafts, which are connected by a pipeline with the dimensions marked.

Schematic diagram of deep drainage pipeline and shafts

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3.1 Stress Distribution in the Foundation Under Water Filling and Discharging Conditions

The settlement prediction method based on empirical fitting formulas only requires the calculation of plastic strain and pore pressure generated in the first cycle, enabling the establishment of a relationship between the number of cycles and cumulative strain as well as pore pressure. The finite element method or elastic theory solution is a proper tools to analyze the static and dynamic shear stresses in the foundation after the first cycle loading as described in Eq. (1). In this case, the finite element method was used for calculation due to its superior capability to simulate engineering environments and its higher computational accuracy. Since the effects of water flow in deep pipelines were approximately neglected, it was transformed into an analysis of statically distributed forces repeated multiple times.

The finite model was built on plane strain condition as shown in Fig. 2, due to the significantly larger axial length in comparison with the radial dimension of the tunnel. The width of the model is 110 m, determined by a circle range that has 5 times the diameter of the tunnel. The height of the models is also 110 m to adequately account for settlement within a substantial range beneath the tunnel. The depth of the tunnel bottom is 50 m and the tunnel diameter is 10 m. Consequently, settlement of the foundation soil within a 60 m range below the tunnel can be considered. The boundary conditions on lateral sides are horizontally constrained, and horizontally and vertically fixed at the bottom. The whole model is saturated and submerged; the soil is evaluated using specific gravity. Figure 3 illustrates the representative cross-sectional distribution of soil layers and the position of the deep drainage tunnel. The specific values of various physical parameters for each soil layer were obtained from the engineering geological survey report as listed in Table 1.

Fig. 2
An illustration depicts a finite element model of the soil layers surrounding the shaft, with the stress distribution indicated by various shades.

FE model for stress distribution analysis

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Fig. 3
An illustration depicts the sectional view of the soil profile surrounding the shaft with the thickness of different soil layers and the elevations indicated.

The distribution of soil in a typical section

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Table 1 Physical parameters of soil layers
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The dynamic-load-induced stress change in underlaying soil was evaluated in this way: (1) extracting the initial static deviation stress \({q}_{s}\) based on stress distribution at different depths along the central axis below the pipeline; (2) then, applying water load on the underlaying soil below the pipeline, by assuming a full flow inside the pipeline and simplifying the load as a uniform distribution with a resultant water pressure of 76.969 \(kPa\); (3) recording the magnitude of the deviatoric stress at different depths along the central axis below the tunnel after applying the water load, where the increment value represents the dynamic-load-induced deviatoric stress \({q}_{d}\) during water filling and draining in the flood season.

The accuracy of the evaluation of stress change is crucial for predicting foundation deformation under cyclic loading. In the determination of the mean effective stress \({p}_{0}\) of the soil after consolidation, it was assumed that the soil layer is under a consolidated state in a lateral earth pressure coefficient \({K}_{0}\). The vertical stress was determined by the self-weight stress of stratified soil, and the horizontal stress was decided upon the lateral earth pressure coefficient \({K}_{0}\). The specific calculation method is shown as follows.

$${p}_{0}=\frac{(1+2{K}_{0})}{3}\sum_{i=1}^{k}{\gamma }_{i}{\prime}{h}_{i}$$
(2)

where \(k\) is the number of upper soil layers at the stress calculation point; \({\gamma }_{i}^{\mathrm{^{\prime}}}\) is the effective unit weight of the \(i\)-th upper layer of soil; \({h}_{i}\) is the thickness of the \(i\)-th upper layer of soil.

The undrained shear strength \({q}_{ult}\) of saturated soft clay consolidation can be calculated using the undrained shear strength formula under the \({K}_{0}\) consolidation state.

$${q}_{ult}=M{p}_{c}{[\frac{M+\alpha }{2M}]}^{\frac{\lambda -\kappa }{\lambda }}$$
(3)
$$\alpha =3(1-{K}_{0})/(1+2{K}_{0})$$
(4)
$$M=6{\text{sin}}{\varphi }{\prime}/(3-{\text{sin}}{\varphi }{\prime})$$
(5)

where \(\lambda\) and \(\kappa\) represent the slopes of the normal consolidation line and rebound line in the \(e-lnp\) space, respectively. Referring to previous test results [15], \(\lambda\) was taken as 0.182 and \(\kappa\) as 0.0347, while for the subsequent layers,\(\lambda\) and \(\kappa\) were both taken as 0.0991 and 0.0129, respectively.

In the selected project, Suzhou River section of Shanghai, the tunnel is buried more than 40 m below the ground surface, and the underlying stratum is composed of silty clay. The soil samples taken in reference [13] are consistent with those used in this study. The parameters obtained through dynamic hollow cylinder torsional shear tests in reference [13] can be employed for analysis. Here, \(a,n,c,b\) represent parameters of the cumulative plastic strain calculation model, while \({a}_{u},{n}_{u},c,{b}_{u}\) denote parameters of the cumulative pore pressure calculation model. The values of \(a,n,c,b\), \({a}_{u},{n}_{u},c,{b}_{u}\) were set to 0.076, 1.408, 0.5, 0.408, 0.0385, 1.37, 0.5, and 0.32, respectively.

3.2 Calculation of Long-Term Settlement in the Foundation of Deep Tunnels

The settlement of the foundation induced by the cumulative deformation of undrained soft clay can be calculated by Eq. (6).

$${s}_{s}={\sum }_{i=1}^{n}{\varepsilon }_{i}^{p}{h}_{i}$$
(6)

where \({\varepsilon }_{i}^{p}\) is the cumulative plastic strain of the \(i\)-th layer, and \(n\) is the total number of compressible layers.

By conducting finite element simulations to obtain stress components and using Eq. (1), it is possible to calculate the undrained cumulative deformation of each soil layer, which can then be superimposed to obtain the settlement of the foundation soil. As the model calculates the settlement of the foundation within a depth range of 60 m below the storage pipeline, the soil layers were divided into more than fifty layers with unequal thicknesses based on the finite element mesh size. Each layer has an approximate thickness of 1 m. The total settlement was calculated using the layered summation method for these fifty-plus layers of soil.

The settlement induced by cumulative pore pressure dissipation can be calculated by Eq. (7).

$${s}_{v}={\sum }_{i=1}^{n}{m}_{\nu i}{h}_{i}{u}_{i}{U}_{i}$$
(7)

where \({s}_{v}\) is the consolidation settlement caused by the dissipation of cumulative pore pressure, \({u}_{i}\) is the undrained cyclic cumulative pore pressure of the \(i\)-th layer, \({m}_{\nu i}\) is the volume compression coefficient of the \(i\)-th layer, \({U}_{i}\) is the degree of consolidation of the \(i\)-th layer, \({h}_{i}\) is the thickness of the \(i\)-th layer, and \(n\) is the number of sub-layers. From a long-term perspective, it can be assumed that the cumulative pore pressure is completely dissipated, indicating a consolidation degree of 100%.

Finally, the settlement \({S}_{s}\) obtained from plastic cumulative deformation was superimposed with the settlement \({S}_{v}\) obtained from cumulative pore pressure dissipation to obtain the total settlement \(S.\)

3.3 Analysis of Long-Term Settlement

For the deep tunnel, considering 50 cycles of water filling and draining each year, the settlements induced by cumulative plastic strain after 1 year, 2 years, 6 years, and 12 years are estimated as 0.084, 0.111, 0.175, and 0.232 m respectively. The consolidation settlements caused by pore pressure dissipation after 1 year, 2 years, 6 years, and 12 years are 0.0006, 0.0008, 0.0011, and 0.0014 m respectively. The total settlements after 1 year, 2 years, 6 years, and 12 years are 0.085, 0.112, 0.176, and 0.234 m. The variation of settlements over time is shown in Fig. 4.

Fig. 4
A line graph plots settlement versus number of cycles. The lines are plotted for settlement induced by cyclic cumulative plastic deformation, settlement induced by cumulative pore pressure dissipation, and total settlement. The lines for plastic deformation and total settlement depict an increasing trend, while the line for pore pressure dissipation remains flat.

Relationship between settlement of foundation and cyclic number

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It is obvious from the graph that the total settlement increases exponentially with operational time. The settlement induced by cyclic cumulative pore pressure dissipation is much smaller than the settlement caused by cyclic cumulative plastic strain, which the former is negligible. The depth distribution of settlements referring to the tunnel bottom is shown in Fig. 5, where N is the number of cycles and depth is the relative depth to the tunnel bottom. According to the requirements for calculating depth in the layered summation method, when the compression of a certain layer of soil is less than 0.025 times the current total compression, the compression of that layer and all layers below it is considered neglected. The curves of 0.025 times the total settlement of the soil under N = 50 cycles (1 year) of cyclic loading intersect with the curves of compressibility of each soil layer under N = 50 cycles at a depth of 20 m, indicating that the compressibility of a single soil layer at 20 m depth is already 0.025 times the total settlement. Two curves also intersect at 20 m depth when subjected to N = 300 cycles (6 years), suggesting that the settlement below 20 m can be considered negligible compared to the total settlement. Hence, compression below 20 m is no longer considered.

Fig. 5
A line graph plots settlement versus depth. The lines are plotted for different N values and 0.025 times the total settlement at N = 50 and 300. The lines plotted for N values depict a downward trend after the initial spike, while the lines for total settlement depict an increasing trend.

Settlement of each compression subjected to different loading cycles

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4 Conclusion

By combining the formulas for cumulative plastic strain and cumulative pore pressure with the layering summation method, a simplified calculation method for the long-term settlement of saturated soft clay under cyclic loading conditions was established. The long-term settlement of urban deep storage tunnels in soft soil areas was studied. The conclusions obtained are as follows:

Considering the low-frequency heavy load characteristics of cyclic water loads, the cumulative deformation calculation model for saturated soft clay under cyclic loading conditions previously proposed was proved to reasonably predict the settlement during the operation period of urban deep storage tunnels. The results show that the consolidation settlement caused by pore pressure dissipation is extremely small compared to the cumulative deformation caused by cyclic loading and it can be almost neglected. Furthermore, cumulative deformation mainly occurred within a certain depth below the deep storage pipeline due to its large diameter and the high water level. In summary, the cyclic water filling and discharging loads tend to cause significant settlement during the operation of the deep tunnel pipelines, so it should be given attention in their planning and construction.

The prediction settlement method combining empirical formulas and layered summation method proven to be a good tool for settlement estimation of deep drainage pipelines in an elastic soil deposited horizontally. However, it cannot directly reflect the nonlinear characteristics of stress and deformation, and further research should be conducted.