1 Introduction

Metro tunnels are increasingly common in major cities around the world as urbanisation progresses. As older metro tunnels start aging while new tunnels see adoption of innovative technologies aiming at optimisation of the construction process, it is becoming more important to monitor and evaluate the performance and enhance the resilience of these structures. Past experiences have shown that many tunnels experience structural problems during their service life, such as excessive deformations, severe cracking and dislocation of joints. These problems can cause significant issues to the performance, durability, and safety of the tunnels.

Tunnel deformations and damage can be traced primarily to changes in the ground stress, which could result from various sources. For example, uneven consolidation of the underlying soil stratum can lead to long-term ground deformation and pronounced tunnel settlement [1]. Lack of lateral resistance due to sensitive soft soil conditions can also lead to excessive expansion of the tunnel cross-sections. Furthermore, nearby engineering activities such as deep excavation can create additional disturbance to the tunnel structure, exacerbating existing conditions and causing effects such as surcharge [2] and offloading [3]. Severe structural problems have been reported for tunnels built in soft ground [4, 5].

Tunnels are buried structures, and it is difficult to predict accurately the effects of variation of loads and disturbances arising from the surrounding soil, with further complication due to tunnel-soil interaction. The disturbances can also occur at different stages in the life cycle of a tunnel. Therefore, assessment of the performance of tunnel structures in practice has relied heavily on field monitoring to provide vital information [6, 7]. Especially, if a tunnel is found to experience severe deformation, appropriate measures need to be taken to mitigate such effects [8, 9] and ideally to correct the over-deformation so as to ensure the tunnel’s structural integrity and performance.

Despite technological developments in the various aspects related to tunnel deformation monitoring and rectification, there has however been a general lack of research studies in the literature that provide a comprehensive treatment of the tunnel deformation problems in real life scenarios, with a systematic involvement of advanced measurement and computational techniques. This paper presents a comprehensive overview of some recent studies on the monitoring, evaluation and remediation of over-deformation in a metro tunnel, with the goal of promoting a holistic approach towards enhancing the resilience of tunnel structures in real application environment. The paper firstly examines the considered case where large deformation occurred due to deep excavation in adjacent areas. Techniques for monitoring the tunnel global deformations, both within a cross section and in the longitudinal direction, are summarised. An in-situ investigation into the effect of retrofitting an over-deformed tunnel structure in the cross-section, i.e. reducing the convergence, is then discussed. The development of a numerical modelling framework utilising the Material Point Method (MPM), which enables a direct simulation of the progressive grouting process, is reviewed, along with some parametric observations based on the numerical simulation. Further research needs along the lines of retrofitting an over-deformed tunnel with the grouting treatment and the associated numerical modelling are discussed.

2 Metro tunnel structural integrity: a typical case of over-deformation

Over-deformation is a major problem to the structural integrity of tunnels. It can bring about various forms of damage to the tunnel structure locally, such as spalling and cracks, damage to the joints, segmental dislocation, leakage, and can also cause performance issues to the metro lines that the tunnel carries [10, 11]. Over-deformation can be caused by general disturbance to the surrounding soil medium, and in particular, it can be caused by construction activities in the adjacent areas especially if large and deep excavations are involved.

For the cross-sectional deformation, the increase in a tunnel’s horizontal diameter, or horizontal convergence, ΔD, is commonly adopted as a key performance indicator [12, 13]. In some soft ground cases, a tunnel convergence after construction could reach the level that corresponds to the ultimate bearing capacity of the segmental linings [14].

This section gives an overview of an example case of a shield metro tunnel that was affected by large-scale excavations in the soft soil area during the redevelopment of a city area in China, which was reported in detail in Han et al. [15]. The twin tunnels were put into operation in 2010, and two years later, large excavations took place near the tunnels as part of the construction of 10 high-rise towers (total basement area around 90,000 m2). The excavation site was divided into four zones, and Zones I and III, which were adjacent to the tunnels, were identified as the primary cause of the over-deformation.

Zone I was a triangular excavation zone with a total area of about 10,300 m2. Zone III was a fan-shaped excavation zone with a total area of about three times that of Zone I, or 30,500 m2. The two excavation zones reached a final depth of 19.25 to 21.25 m, respectively. The nearest clearance from the diaphragm wall to the centre line of the tunnels was 15.0 m.

The excavation in Zone I took place first and lasted for 5.7 months. The main excavation in Zone III then took place, lasting 12 months. As a result, the twin tunnels were subjected to offloading on both sides. Coupled with the soft soil condition, this caused excessive deformation of the tunnels. Because of the heavy impact, grouting treatment was later used to retrofit the lining structures.

The geology of this area is characterized by a typical layer-structured soft deposit in the Yangtze River delta. The strata are primarily comprised of made-ground, muddy silty clay, and fine sand. The upper stratum is dominated by a thick layer of muddy silty clay. Engineering experience shows that these soils tend to lose strength significantly once disturbed, resulting in substantial deformation.

Under the muddy silty clay, there is another major thick layer of fine sand with much higher permeability. Beneath the fine sand layer are layers of coarse sand and weathered mudstone. Figure 1 shows the spatial relationship between the two tunnels and Zone I and Zone III of the excavation. Figure 1b shows the representative geological profile at the construction site corresponding to cross section A-A. At the grouting region, which will be discussed later, the soil layer thicknesses of the made-ground and the muddy silty clay were estimated to be around 4.7–5.7 m and 24–26 m, respectively.

Fig. 1
figure 1

Excavation zones and existing tunnels: (a) Plan view, (b) Elevation (after [16])

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The tunnel lining is constructed by segmental shields and made up of six staggered-jointed prefabricated RC segments, each 1.2 m wide and 0.35 m thick. The outer diameter of the lining ring is 6.2 m. The neighbouring segments are joined using curved steel bolts in both the longitudinal and circumferential directions.

Figure 2 shows examples of the deformation recorded by the field monitoring system. Figure 2a depicts representative horizontal convergence of the left-line tunnel, including i) recording before any excavation took place, ii) recording before any grouting treatment, and iii) incremental convergence, which may be attributed to the effect caused by the excavation. The absolute pre-excavation convergence ranges from 8 to 48 mm, or 0.13% to 0.77% in terms of the convergence ratio, ΔD/D, where D is the initial outer diameter of the tunnel. It can be seen that due to the sensitive soft soil condition, the maximum pre-excavation convergence within a few tunnel rings already reached relevant design serviceability limit of 0.3%-0.5% for tunnels in normal operating conditions [17, 18].

Fig. 2
figure 2

Examples of circumferential and longitudinal deformation of the left-line tunnel before grouting operations:(a) Absolute and relative horizontal convergence; (b) Differential settlement and horizontal displacement

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As the excavation took place, a steep increase in the tunnel convergence occurred. In fact, the excavation-induced convergence (i.e. the increment) became comparable to the magnitude of the pre-excavation convergence in the affected region, which appeared to occur around the region of Zone I. Thus, the combined effect of tunnelling construction and the subsequent adjacent excavation introduced a rather severe over-deformation in the worst affected tunnel section, with a maximum total convergence reaching 86.9 mm (convergence ratio of 1.40%), which exceeded significantly the design serviceability limit. As a consequence, site inspections revealed leakages in the longitudinal joint seams and segment cracking in the lining rings.

In terms of the global deformation, Fig. 2b shows the vertical (settlement) and horizontal displacement profiles of the monitored tunnel right before grouting mitigations. As these longitudinal displacements were relatively stable until the construction activities took place, the increments were clearly attributable to the effect of the excavations. From the spatial distribution of the displacement profiles, it can be seen that the most affected region was also associated with the location of Zone I, which is consistent with the observation from the cross-section convergence, and this confirms that the excavation in Zone I was the primary contributing factor to the over-deformation of the tunnel. The settlement recording also revealed obvious formation of a settlement trough around rings 580 to 640, with a maximum magnitude of 41.3 mm. Site inspection also spotted lining leakages, as well as ballast-bed detachment in affected region of the tunnel, showing the unfavourable consequences of the longitudinal over deformation.

3 Methods of monitoring tunnel deformations

Monitoring of tunnel deformations plays an important role in enabling continual assessment of the structural performance of tunnels. Tunnels are linear structures, and their deformational behaviours can be broadly classified into two categories, namely global transversal displacements introducing curvature and shear deformations with respect to the longitudinal axis, and cross-sectional displacements introducing convergences. From a member and material perspectives, these displacements result in deformations (or equivalent strains) in the longitudinal and circumferential directions. These two types of deformations are not mutually exclusive. In fact, they often occur together.

Shield tunnels are complex assembly structures, and their deformation indicators may be categorized into three major scales: systematic (global), member, and material. Figure 3 and Table 1 illustrate the key deformational indicators for each scale in both the longitudinal and circumferential directions.

Fig. 3
figure 3

Schematic diagram of the shield tunnel multi-scale deformational information set (after [19])

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Table 1 Multi-scale deformational indicators of shield tunnel structure [19]
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Conventionally, the circumferential performance of shield metro tunnels is assessed using global (systematic) deformation indicators such as convergence. Similarly, the longitudinal performance is indicated by global indicators such as differential settlement and horizontal displacement. These deformation indicators can be measured using a variety of methods, including total stations, precise levelling, and laser scanners. They can provide a general picture of the tunnel's performance, reflecting global external influences such as soil conditions, environmental changes, or nearby constructions. Measurements are typically taken at regular spatial intervals, such as every 5–10 rings for convergence and every 10 or 20 m for settlement. Figure 4 showcases a total station in use inside a shield tunnel in the aforementioned monitoring project. This total station was used to monitor the deformation of 20 lining rings.

Fig. 4
figure 4

Total station mounted on the lining of a shield tunnel [15]

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In shield tunnels, the joints between the segments are a major source of complexity as they can deform in multiple modes involving both rotation and dislocation. Different deformation modes can induce different stress states in the bolts and segments, leading to different types of structural damage and degradation such as bolt yielding at critical joints and plastic hinges forming at adjacent ring segments [20, 21]. Therefore, in addition to the global deformations, monitoring the behaviours of joints and concentrated deformation in segments is critical for a realistic condition assessment of shield tunnels. However, because there are numerous links and joints even within a given potential critical structural zone in a shield tunnel, it is difficult to establish a comprehensive picture of these local behaviours with a limited number of measurement spots.

Recent advances in distributed inspection and monitoring technologies have opened up the possibility to obtain a more refined picture of tunnel deformations. Laser scanning can serve as a promising tool for large-scale tunnel deformation inspection, and when aided with suitable software, the retrieval of deformation information could be done in real time [22]. For example, a systematic approach was proposed for tunnel deformation monitoring using terrestrial laser scanning (TLS) [23], in which the point cloud data was registered using the iterative closest point (ICP) algorithm, and the centreline of the tunnel was extracted using a moving least squares (MLS) algorithm. The deformation of the tunnel was analysed by comparing the cross-sections of the tunnel at different time intervals, thus reflecting the deformation of the tunnel in both the longitudinal and circumferential directions.

Another methodology was proposed for detecting the convergence diameter of shield tunnels using self-driven mobile laser scanning (SDMLS) [22]. A least trimmed squares method was adopted in removing the noise points in the point cloud. After identifying the tunnel transverse seams and endpoints, the convergence diameter and radial dislocation can be detected by the endpoints of the segments. The method was shown to be capable of achieving a repeated detection accuracy of better than 2 mm. Cui, et al. [24] also investigated the use of mobile laser scanning to detect shield tunnel deformation. Their system incorporated a stepwise ellipse fitting method and was shown to be capable of measuring the convergence diameter and radial dislocation of the tunnel with accuracy of 1.5 mm.

Generally speaking, laser scanning provides an efficient tool for large-scale deformation measurement in tunnel structures. However, research has so far mainly focused on retrieving deformational information at a global level, namely convergence, ovality, and longitudinal displacement, whereas more detailed information regarding local structural responses at the joint and material levels is still lacking.

In contrast to displacement-based methodologies such as laser scanning mentioned above, strain-based monitoring especially using the Optical fibre sensing technology offers detailed deformation information at local scale, such as tunnel joint openings and material damages [25]. Optical fibre sensing technology uses optical fibres to sense and transmit information. Brillouin-based distributed fibre optic sensing technologies could serve as promising deformation sensing technology suitable for tunnel deformation monitoring. Brillouin Optical Time-Domain Analysis (BOTDA) is a distributed fibre optic sensing technology that uses the Brillouin scattering effect to measure strain and temperature along an optical fibre (BOTDR, BOFDA, etc. offers a similar capability). The Brillouin scattering effect is a phenomenon in which light waves are scattered by acoustic waves in a fibre optic. The frequency of the scattered light is shifted by an amount that is proportional to the strain and temperature of the fibre. The technology can be used to measure strain and temperature over a length scale of kilometres and more (Fig. 5).

Fig. 5
figure 5

Illustration of Brillouin optical time-domain analysis fibre optic sensing technology

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BOTDA technology has been used in a variety of applications to monitor the tunnel structural performance both in the longitudinal and circumferential directions [26, 27]. The long-term stability of BOTDA allows for the comparison of deformational behaviours of the tunnel over a relatively long period of time using the distributed strain data recorded [28, 29]. A field trial of distributed fibre optic sensing and wireless sensor network in a cross-rail project in London [30] suggested that two deformational modes, namely inter-ring joint rotation and dislocation illustrated in Fig. 6, were both presented in the tunnel under monitoring when a new tunnel was bored in its vicinity. However, decoupling the rotational and shear (sliding) dislocation remains to be a challenge.

Fig. 6
figure 6

Longitudinal movement modes of a shield tunnel: (a) Bending mode; (b) Shearing mode [19]

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A novel optical sensor layout has recently been proposed to address this issue [19]. The measurement array combined sensors arranged in a straight line and in a zigzag manner to form a spatially distributed sensor network. The technology has been deployed in the measurement of the deformations in the tunnels under consideration in this paper. Figure 7 shows the sensor layout and typical measurement results.

Fig. 7
figure 7

Decoupling the inter-ring movement into dislocation and rotational components using a novel fiber-optic sensor network layout

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It should be noted that although Brillouin-based distributed fibre optic sensing is well-suited for measuring global and member scale tunnel deformations, the technology has inherent limitation when applied to measure material-scale deformation such as cracks while maintaining a large scale coverage. This is because the achievable spatial resolution of standard BOTDA analysers is typically close to 1 m to guarantee a sufficient signal-to-noise ratio, which is too coarse for detection and measurement of cracks. Researchers have attempted to enhance the spatial resolution of BOTDA analysers using e.g. pre-excitation of the acoustic field [31,32,33]. However, such an approach tends to increase the complexity and costs of the measurement systems, and the achievable spatial resolutions are still not sufficient for accurate crack measurement. Besides, the enhancement of spatial resolution is often accompanied by a decrease in the coverage length of the sensing system.

Recently, a novel sensor named short-gauged Brillouin fibre optic sensors has been developed for crack monitoring [34]. The sensor adopts a point-fixation installation scheme, with a gauged length smaller than a half of the spatial resolution length of its corresponding analyser system. The sensor works by exploiting the fact that the Brillouin gain spectrum of a BOTDA sensor is sensitive to crack-induced strain. When a crack is present, the strain in the sensor will generate a pronounced rectangular peak from the background of the ambient strain, which will cause a series of predicted nonlinear evolvement in the Brillouin gain spectrum as the crack propagates, such as peak lowing, peak shifting, bump and double peak formation. These changes in the Brillouin gain spectrum can be leveraged to detect, monitor, and quantify the crack, as schematically illustrated in Fig. 8. According to the research, the proposed sensor could achieve early crack detection and accurate crack measurement (repeatability of ± 0.015 mm) with long-distance coverage capability.

Fig. 8
figure 8

Short-gauged Brillouin fiber optic sensor for crack monitoring (after [34]

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4 Typical remedial measures against over-deformed tunnel structures and effects of retrofitting with grouting

There are a number of different methods that can be used to control tunnel deformation, such as ground improvement, active support, and monitoring. The most commonly used methods from a structural strengthening point of view include using aramid fiber-reinforced polymer (AFRP) reinforcement [2, 35] and bonded steel plates [36,37,38,39]. However, these measures do not usually have an effect of correcting the deformation that has already occurred, and the installation work requires closure of the tunnel and hence usually needs to be carried out in the mid-night hours.

On the other hand, grouting based treatment methods aim at active control and restoration of excessively deformed tunnel lining rings to an acceptable level. Among other grouting techniques, compensation grouting has been used worldwide for deformational control of geo-structures and the ground [40,41,42]. However, the grouting operations involved may generate significant disturbance to the surrounding soil and this can be particularly problematic in the case of sensitive soft deposits.

A number of studies have been devoted to developing grouting methods with reduced disturbance to the surrounding soil. Most of these methods involve a process in which fast-setting grout is injected into the soil during the grouting treatment, creating a soil-concrete mixture with improved ground resistance to the deformational movements of tunnel linings [43]. At the same time, the treatment also creates counter-pressure against the imbalanced pressure induced by surrounding construction activities. The grouting technique proved to be effective in mitigating excessive tunnel deformation, and has been implemented successfully in a number of grouting projects, especially in soft soil areas in Eastern China [9, 35, 44,45,46,47,48]. It should be noted that most of the reported case histories tend to illustrate the effectiveness of the grouting technique by the total effects of the deformational control, whereas detailed observation of the temporal and spatial evolution of the tunnel structural response is generally lacking. Without such detail information, it is difficult to scrutinise the mechanisms underlying the overall effect of the grouting treatment.

A recent study [15] reported a full-scale field experiment of grouting treatment in Eastern China soft ground in connection to a retrofitting operation to reduce the convergence of the tunnels under consideration. The effects of different routing arrangements and grouting history on the grouting efficiency, the time-varying response of the tunnel linings, and the characteristics of the tunnel lining rebounding after grouting, were examined.

As introduced in Sect. 2, the twin tunnels under consideration incurred significant over-deformation with a convergence reaching the order of twice the design serviceability limit in the worst affected region, primarily due to the deep excavation taking place in the close vicinity on both sides of the tunnels. In order to rectify the problem and maintain the safety of the metro lines carried by the tunnels, the metro company decided to resort to grouting treatment with an aim to correct the over deformation of the tunnels back to an acceptable level and to strengthen the resistance of the ground. However, there was limited engineering knowledge regarding the effect of grouting treatment in reducing the excessive tunnel deformation under the type of geological condition of the site. A field grouting test was therefore conducted before the full-scale implementation of the technique. The experiment was conducted over an 11-days window in late 2013 when the main affecting excavations had already been completed, therefore the observations of the changes in the tunnel deformations could be attributed simply to the grouting treatment. The tunnel convergence response was continually measured over an additional 26-day period after the completion of the grouting operations to monitor rebounding and the associated tunnel responses.

The particular grouting treatment involved injecting a mixture of cement slurry and sodium silicate deep into a target region adjacent to the portion of the tunnel being treated, using a system as schematically shown in Fig. 9. By adjusting the mixture ratio of the sodium silicate and Portland cement, a desirable initial setting time of the grouting slurry can be achieved, avoiding plugging of the injection pipe and ensuring a target diffusion radius of grout in the soil. Further details of the mix ratios, the flow rates and the corresponding initial setting times can be found from Han et al. [15].

Fig. 9
figure 9

Schematics of the grouting system and tunnel structural response incurred

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In the field experiment, 10 grouting holes were arranged on each side of the tunnel subjected to the testing, stretching over a length of 8.4 m, which is about 7 shield tunnel rings, and at a horizontal distance between 6.1 m to 6.7 m from the centre (or 3 m to 3.6 m from the outer clearance) of the shield tunnel (Fig. 10). The vertical position (elevation) of the grouting as measured between the bottom of the grouting region and the bottom of the tunnel varied between 0.0 and 2.7 m, and the net height of the grouting regions varied between 4.2 and 5.2 m. The volume of the grouts varied in a range of 1350 to 1765 L.

Fig. 10
figure 10

Grouting locations of the field test wrt. the tunnel

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The tunnel response to each grouting operation typically experienced a convergence recovering stage and a subsequent rebounding stage. This is because the grouting injections cause the expansion movement of the soil body around the grouting nozzles during a grouting operation, this introduces a thrust pressure onto the nearby tunnel linings, pushing the rings to deform inwards and thus recovers a certain amount of the existing outward convergence. After completion of the grouting process, however, the positive excess pore pressure undergoes dissipation, and together with solidification of the grout, a rebounding of the tunnel convergence inevitably occurs [49]. Figures 11 and 12 shows the convergence responses of a set of targeted shield tunnel rings during and after the grouting operations. It should be noted that the convergence time histories are the accumulated convergence results over the whole set of grouting operations, with each data point corresponding to a particular stage of staggered grouting scheme (for details see [15]. Several observations may be made from the graphs, as summarised below:

  1. i)

    the maximum recovery convergence at the time of completion of all 10 grouting operations ranged from 15.6 to 20.9 mm, with an average at about 19.1 mm. This was about 25% of the average absolute existing convergence measured before the grouting treatment.

  2. ii)

    the rebounding convergence at 17 days after ranged from 5.3 to 7.5 mm. On average, the rebounding convergence was about 35% of the maximum recovery convergence.

  3. iii)

    the rebounding convergence tended to stabilise towards the end of the test monitoring period, i.e. 17 days after the completion of the grouting operations.

  4. iv)

    with reference to the rebounding convergence at the end of the monitoring period, the net convergence recovery ranged from 5.9 mm to 13.8 mm. With respect to the initial existing convergence before the grouting, an average net recovery of 12.6 mm. As a result, the absolute convergence among the shield tunnel rings under investigation was reduced to around 1%D.

Fig. 11
figure 11

Tunnel convergence response during and after the grouting treatment: (a) Relative convergence; (b) Absolute convergence [15, 16]

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Fig. 12
figure 12

Correlation between convergence recovery and convergence rebound [16]

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The detailed grouting operation and convergence monitoring data from the field experiment also allowed for a closer-look at the tunnel deformational behaviours during the grouting injection period, particularly with regard to the efficiency of the injected volume of the grout. Representative convergence response curves of a tunnel ring to the varying grouting injection volume, under a single grouting operation on one side of the ring and a double grouting operation on both sides of the tunnel ring, respectively, are illustrated in Fig. 13. Note that the grouting volume injected and the corresponding convergence reaction of the tunnel ring are normalized values with respect to the total injected volume and the maximum convergence recovery value in the specific grouting operation, respectively. It can be seen that a relatively high grouting efficiency is achieved in the lower range of the grout injection volume, as evidenced by a steep slope of the convergence response curve. The slope tends to reduce as the grout volume increases beyond a certain limit, and eventually becomes virtually flat meaning further increase of the grout injection volume beyond a certain critical volume would have little effect in the convergence recovery. Understandably, the critical grouting volume depends upon the various grouting parameters and geotechnical properties. Based on the results from this particular field test, for the single-sided grouting operations, the critical grouting volume ranges from 300 to 600 L, and for the double-sided process, this value raises to about 1000 L.

Fig. 13
figure 13

Example of normalised convergence recovery vs. injected grout volume curves [15]

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It should be pointed out that the above efficiency observation is valid with regard to a single grouting position next to the particular ring. In fact, as can be seen from the cumulative convergence response to the whole series of grout positions earlier in Fig. 10, continued accumulation of the convergence recovery can be achieved from grouting at other positions even though these positions are located next to other rings meaning farther away from the ring under consideration. This phenomenon reflects the complexity of the mechanisms underlying the grouting effect, resulting in a complex interaction between the grout and the immediate surrounding region of the soil, and the transfer of the pressure towards the tunnel rings. Apart from primary soil response and interaction between the grout and the soil, there exist uncertainties with regard to the effect of the spatial arrangement of the group of grouting positions, including for example the actual profile of the grout body which is influenced by the non-homogeneity of the properties of the surrounding soil, and the interference from the already formed grout regions to the current grout position.

All of these make a quantitative assessment of the effect of the spatial arrangement of the grout positions challenging. Nonetheless, the present field experimental data suggests that concerning a particular ring, the grout positions within a range of the adjacent 7 shield rings (or 8.4 m) are generally effective for the convergence recovery, and the effect tends to diminish as the grout position moves farther away in the longitudinal direction.

5 Numerical simulation of grouting and parametric analysis

5.1 General modelling considerations

Field experiment and monitoring can play an indispensable role in the understanding and acquisition of real-life data with regard to the effects of grouting on the convergence recovery of tunnels [50, 51]. Considering that data from field tests are generally dependent upon the specific conditions of a particular project, mock-up experiments could be conducted in a controlled environment to replicate targeted conditions [52,53,54]. However, such experiments can be costly and time-consuming.

Several theoretical methods have been developed to describe the evolution of grouting behaviours, especially concerning the grouting pressure (e.g. [55]). To enable a theoretical solution, simplified assumptions are necessary and these include constant grouting and flow parameters, namely stress, velocity, and volume, as such the complexities that can often arise in real engineering conditions cannot be represented.

On the numerical modelling front, traditional finite element analysis (FEA) has been widely used in the modelling of grouting. In the FEA framework, grouting is commonly simulated by applying internal pressure to the FE elements in the grouted region, through one of the two approaches as follows. a) applying the internal pressure to the embedded interface elements [56, 57], and b) applying pressure to all the grout layer elements [58, 59]. An apparent limitation with the traditional FEA framework is that it cannot model the progressive material injection process of grouting and the associated complex interaction between the grouting operation and the soil. It should be noted at this juncture that the grout injection usually involves large localized ground deformation around the grouting nozzles, which is difficult for mesh-based methods like FEM to model adequately due to elemental-distortion related issues.

To enable the simulation of grout injection while overcoming the mesh-distortion related numerical problems, a numerical modelling approach using material point method (MPM) has been developed by a team involving the authors to simulate the complex interaction between the grout and the surrounding soil during the grout injection process [16]. MPM is one of the latest developments in Particle in Cell (PIC) methods, and it is free from mesh related limitations. In essence, the MPM discretises an object into a system of material particles, which carry with them all state variables including mass, velocity, strain and stress. Comparing to other mesh free methods such as Smooth Particle Hydrodynamics (SPH), the formulation of MPM has similarity to the traditional FEM in that spatial derivatives are calculated based on a regular computational grid in MPM, and in this way the time consuming neighbour searching, which is required in most mesh-free methods, is avoided. General boundary conditions can be applied in MPM conveniently as in FEM, and similarly the simulation of contact. Moreover, all time steps can use the same computational grid, and this enables the use of a constant time step in an MPM simulation. For these reasons, MPM is suitable to simulate problems involving large deformations in geotechnical engineering, such as excavations, footing strips and slope failures [60,61,62].

5.2 MPM based numerical model setup and analysis scheme

For simplicity and considering that the convergence deformation of the tunnel structure is a measure from a cross-section perspective, a 2D plane-strain MPM model was adopted in the numerical simulation of the grouting treatment. Besides, for a focus on the short-term behaviour and considering a ground condition with low permeability, a total stress analysis was considered.

For the particular tunnel case herein, a computational domain of 100 m (width) \(\times\) 50 m (depth) was created as the overall model space, which contains a layer of fine sand below the tunnel elevation, a 25 m deep silty clay, and a 5 m thick top filling layer. The tunnels are embedded in the soil at the depth of 25 m (centre to ground surface). The bottom side of the model is fully fixed and the horizontal movement on the two sides of the model are restrained. Through trial analysis, a grid size of 0.5 m was chosen for the background mesh, and each element consists of 4 particles.

It should be noted that, as generally recognised, the interaction between the grouting and the surrounding soil involves a compensation (or compaction) mechanism and a fracturing mechanism [63], and their respective significance depends upon the injection volume, property of soil and viscosity of the grout. Considering that the main transfer mechanisms of the pressure from grouting to act on the tunnel is through compensation, the fracture phenomenon was not simulated in the present model.

The entire simulation includes three key processes, namely i) stress field initialization, ii) tunnelling excavation, and iii) grouting treatment. The initial stresses are calculated by K0 method, \({K}_{0}=1-\mathrm{sin}\phi\), where ϕ is the friction angle of the soil, allowing some stabilisation steps to reach equilibrium. For the creation of the tunnel(s), firstly the particles within the tunnel area are removed, and then particles at the position of linings are assigned the tunnel lining material to model the tunnel structure, again allowing stabilization steps. After the creation of the tunnel structures, the model then undergoes the grouting procedure.

With MPM, the injection of grouting can be simulated in a straightforward manner by adding new particles into the system at specified positions and in a continuous manner. Figure 14 depicts the addition of new particles. To simulate the grouting process that starts from the bottom and moves up with a constant velocity, a certain number of control points, in the case herein ten control points, are arranged along the grouting zone in the MPM model, such that the ‘grouting’ is implemented at these positions one after another. A group of 50 particles are added to each point continuously to keep a uniform grouting work. By ensuring that the total mass of all new grout particles is equal to the actual grouting mass and that the initial velocities of the added particles resemble the velocities in the actual grout injection process, the conservation of the mass and momentum of the system can be maintained.

Fig. 14
figure 14

Schematic of simulation of the grout injection process in the numerical model

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In the present simulation, the tunnel structure (linings) is modelled as linear elastic with a Young’s modulus of 20 GPa. The grout is considered as in a virtually incompressible flow state, with a small elastic modulus. The three soil layers are all assumed to be isotropic and homogeneous within each layer, and the soil properties are defined using Tresca model (TR), with cohesion set equal to the shear strength of the corresponding soil layer. The Poisson's ratio of the soil layers is set equal to 0.45 to represent an undrained condition. Details of other parameters can be found in Zhao et al. [16].

5.3 Performance of the MPM based model and some parametric observations

Four cases selected from the field experiment, representing different positions and height of the grouting as well as single- and double-sided arrangement, are firstly simulated to verify the numerical modelling scheme and provide detailed information about the responses in the soil and in the tunnel cross-section. Figure 15 shows two example contour plots of the horizontal displacement field, with a single-side grouting of equal grout height but different elevations. It can be observed that the horizontal displacement is almost symmetrical about the grout line, despite that the grout position is markedly closer to the left tunnel in the model. It can also be seen that moving the grout to a lower position reduces the effect overall, and this is deemed to be attributable to the fact that below the tunnels is a stiffer sand layer, which tends to impede the expansion of the soil and hence reduces the effect in the case of grouting at a lower elevation. This indicates that the grouting will be more effective if it is carried out in a soft layer. These results agree favourably with the observations from the field experiment.

Fig. 15
figure 15

Examples of horizontal displacement contours induced by grouting at different positions

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Other observations from the detailed results indicate that an increase of the grouting volume leads to a larger excess deformation, and when the compensation grouting is made on both sides of the tunnel, the result is similar to a superposition of two one-side grouting operations. It is worth noting that a symmetrical grouting arrangement on both sides of the tunnel ensures recovery of the convergence deformation of the tunnel without inducing a whole-body movement (shift) of the tunnel cross-section.

From the numerical simulation, the reduction in convergence is about 4 ~ 5 mm for the one-sided grouting and increases to about 10 mm for the two-sided grouting case. The recovery in the case with a lower grouting volume and elevation is only about 3 ~ 4 mm. These quantitative results also show good agreement with the field data.

Using the validated basic model, parametric analysis has been carried out to examine the effects of varying the grout volume, relative position in the horizontal direction and in elevation, and the net grout height.

Figure 16 illustrates a representative convergence recovery vs. grout volume curve under a one-side grouting condition shown in Fig. 15a. It can be seen that, similar to what was observed from the experiment in Fig. 13, the curve is comprised of an initial steeper increase part where increase of the grout volume is most effective, then a transitional part, followed by a rather flat part where further increase of the grout volume has virtually no incremental effect on the convergence. The critical volume is about 600 L, and this value also echoes favourably with the experimental result.

Fig. 16
figure 16

Example of grouting-induced convergence recovery in the MPM model vs. experiment

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Figure 17 illustrates the (horizontal) convergence recovery vs. vertical position of the grout curves under a fixed grout height of 5.2 m (volume 1593 L), where the elevation is measured as the vertical distance from the bottom of the grout to the bottom of the tunnel ring. It can be observed that the grouting starting from the bottom of the tunnel ring (elevation = 0) appears to give the best convergence recovery result. In general, a lift of the grout position leads to a decrease of the convergence recovery.

Fig. 17
figure 17

Recovery of convergence with different vertical positions

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Figure 18 shows the convergence recovery vs. horizontal position curves when the elevation of the grout is fixed at zero position (i.e. the bottom of the grout is aligned with the bottom of the tunnel ring), with the horizontal distance D measured from the centre of the tunnel ring to the grout (noting the outer radius of the tunnel ring being 3.1 m). It can be observed that there appears to be an optimum distance which in this case is about 6.7 m from the tunnel centre (or 3.6 m from the nearer tunnel edge). As the position of the grout away from the tunnel beyond this point, the effect on the convergence reduces. On the other hand, moving the grout position towards the tunnel from this point also reduces the grout effect. This indicates that there requires a sufficient soil layer in-between the grout and the tunnel body to enable the grout to take effect, an observation that also echoes well the experimental results.

Fig. 18
figure 18

Recovery of convergence with different horizontal positions

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5.4 Discussion on further research needs in association with numerical modelling of grouting treatment

The development of a MPM based numerical model, which allows an explicit simulation of the progressive grouting operation, has opened up scope for investigating into the various parameters influencing the effects of grouting treatment in the correction of over-deformation of the tunnel structure. Such a numerical model can also facilitate examination of the underlying mechanisms of the complex interaction between the grout and the surrounding soil, and the load-pressure transfer onto the tunnel structure. To achieve these in a more reliable manner, several aspects need to be considered in the further development of the numerical modelling framework, and these are elaborated as follows.

  1. i)

    A suitable 3D modelling scheme will need to be considered to not only allow simulation of grouting at multiple positions in a spatial arrangement, as is usually the case in an actual project, but also represent the tunnel behaviour in the longitudinal direction in a more realistic manner. A 3D model will also allow a more direct and reliable representation of the volumetric effect of grouting, without requiring a ‘conversion’ as is implied in a 2D model.

  2. ii)

    An advanced constitutive description of the soil capable of representing the pressure dependent soil behaviour and also the nonlinear loading and unloading response of soil which could occur under grouting at multiple positions in consecutive operation. The incorporation of such a constitutive description of the soil will enable a realistic preservation of the history of the soil response from different grouting positions, and thus allowing the simulation of an important mechanism in influencing the effectiveness of the grouting from multiple positions on the convergence recovery of a particular tunnel ring.

  3. iii)

    More comprehensive calibration of the progressive grouting operation via the scheme of addition of particles. Validation and calibration may be carried out on representative soil samples and under dedicated controlled conditions, for which the soil response may be assessed by analytical means or via relatively simple laboratory experiment, which can then employed as benchmark results to be compared with the simulation using the particle addition scheme.

  4. iv)

    With a holistic numerical modelling framework comprehensive parametric studies can be conducted to establish in more quantitative terms the effects of various pertinent parameters. These could include the properties of the grout, grout parameters such as positions both in the longitudinal and horizontal directions relative to the position of specific tunnel rings, volume of the grout, and spatial arrangement of multiple grout positions. It will also be of interest to examine the margin of variation in such effects under possible variation of the geological condition and properties of the soil layers.

6 Concluding remarks

An overview of recent developments in field monitoring and assessment of shield tunnels hosting urban metro lines in soft soil has been presented in this paper. In particular, the over-deformation of tunnel rings due to nearby construction activities, especially deep excavation, and its rectification using grout treatment is discussed on the basis of relevant field experiment and numerical modelling reported in detail in recent publications. Key considerations in the development of the Material Point Method (MPM) based modelling framework for simulating explicitly the grouting operations are highlighted, and further research needs in this direction are discussed.

As far as global deformation is concerned, a system of instrumentation is deemed to be necessary to acquire pertinent deformation data not only over the cross section but also in the longitudinal direction, such that both the cross-sectional convergence and the (bending and shear) composition of longitudinal deformation can be determined. Such information will allow for a more accurate assessment of the structural state and the characteristics of the tunnel-soil interaction mechanisms.

Grouting is considered to be an effective approach to mitigating and rectifying over-deformed tunnel rings especially concerning the convergence. Field experiment has proved that the effect of grouting on convergence recovery is affected by a range of factors, and from the grouting design and control side these include the properties of the grout, the relative positions of grouting in both horizontal direction and elevation, the height of the grout, as well as the spatial arrangement of multiple grout positions.

As the effects of grouting essentially depend upon a complex system of grout–soil-tunnel structure interaction in a three-dimensional domain, and given the various uncertainties with almost all key constituent entities involved, a comprehensive guidance on the design of a grouting treatment would require a range of sensitivity analysis. To this end, a robust numerical simulation model capable of describing the grouting process is desirable.

The MPM based numerical modelling approach provides a versatile framework for a direct simulation of the grouting treatment. The model has been verified to be capable of simulating single- and double-sided grouting operations with favourable agreement with the relevant field experimental results. Further developments in the numerical simulation front will need to look into more robust constitutive description of the soil media in a realistic geological environment and incorporation of the three-dimensional effects, among other enhancement considerations.