Dynamic evolution characteristics and laws of ground settlement due to shield tunneling in water-rich sandy pebble strata

1 Introduction

During construction of water diversion tunnels, the shield method is widely adopted because of its advantages such as fast excavation speed, high safety, and minimal impact on the surrounding environment (Cui et al., 2025; Wang et al., 2023a). In implementation of this process, it is often necessary to pass through sandy pebble strata, which have notable looseness, relatively weak self-stabilization ability, low cohesion, and large interparticle gaps. From a mechanical perspective, in an anhydrous state, the particles of sandy pebble strata mainly transmit forces through point-to-point contact. Once disturbed, the force transmission mode and the stress distribution between particles undergo substantial changes, exhibiting sensitive mechanical response characteristics (Zhang et al., 2024; Yao et al., 2023). Owing to the high particle mobility and the ease of disruption of stress transmission paths in sandy pebble strata, coupled with the influence of construction disturbances, shield tunneling is highly likely to cause instability of excavation surfaces and loss of fine particles. Furthermore, it might lead to geological collapse and surface subsidence, which not only damage the ecological environment but also cause secondary environmental disasters such as water resource depletion. Accordingly, considering the unique characteristics of sandy pebble strata, in-depth investigation into the ground subsidence mechanism induced by shield tunneling in water diversion tunnels within such strata is of considerable theoretical importance and engineering value.

Many earlier studies have conducted theoretical investigation on the stability and surface subsidence of strata during shield tunneling. For example, Horn (1961) initially introduced the limit equilibrium failure mechanism, which has been applied extensively to stability analysis of the excavation face during shield tunneling. Anagnostou (2012) proposed a method for evaluating tunnel face stability in frictional or cohesive-frictional soils under horizontal stress, and derived a design formula for ground collapse. Later studies further improved the model proposed by Anagnostou, thereby enhancing its computational accuracy (Geng et al., 2023; Yu et al., 2020). Klar et al. (2007) proposed a new mechanism with a continuous velocity field based on elasticity theory, which is proportional to the elastic displacement field but only involves local damage to the tunnel face. Yang et al. (2015) studied the effects of soil particle strength anisotropy and heterogeneity strength on the boundary support pressure faced by the tunnel face under two-dimensional conditions, which simultaneously considered the anisotropy of the internal friction angle and cohesion. Huang et al. (2023) predicted shield-induced ground settlement using CART, RF, and GBDT machine learning methods, proving that ensemble trees outperform the single CART approach, with tunnel geometric parameters as the dominant factor. Wang et al. (2024) derived an analytical solution for predicting ground surface settlement induced by shield tunnels based on stochastic medium theory, which accounts for the discrepancy in volume loss between the surrounding soil and the tunnel. They analyzed the effects of tunnel geometry, influence angle, and volume loss on surface settlement characteristics. However, the theoretical model tends to either overpredict or underpredict the actual measured settlement. Most theoretical methods are established under idealized soil assumptions, which cannot fully reflect the complex particle interactions and seepage characteristics of water-rich sandy pebble strata, resulting in certain deviations between calculated results and field monitoring data.

Numerical simulation represents an efficient and cost-effective method for investigating surface settlement during shield tunneling in sandy pebble strata, and it has been widely adopted in the field (Chen et al., 2011; Wang et al., 2023b; Du et al., 2025). Wang et al. (2024) used numerical simulation methods to quantitatively predict the ground settlement caused by shield tunnels in sandy pebble strata, effectively revealing three volume deformation modes of the sand cobble soil above the tunnel crown. Using combined numerical simulations and model tests, Di et al. (2022) studied the progressive settlement characteristics of sandy pebble strata during unstable deformation of shield tunnels at various burial depth ratios. Zhao et al. (2022) performed three-dimensional numerical simulations to investigate the deformation and stress characteristics of an airport runway pavement structure in sandy pebble strata, and analyzed the strata displacement, surface subsidence, and pavement stress under different tunnel depths. In addition to the above finite element method, the discrete element method is used widely to study both the deformation above the soil and the surface subsidence when shield tunnels pass through sandy pebble strata (He et al., 2012; Jiang and Yin, 2012; Yan et al., 2010). Gu et al. (2024) proposed a finite element-based numerical simulation method to dynamically reproduce shield tunneling and capture the interaction between the shield machine and the ground. The discrete element method considers the heterogeneity and anisotropy of sand and pebble media, which is helpful for studying the microscopic movement behavior and pressure transfer of particles during shield tunneling Guo et al. (2023); Zhang et al., 2025). Additionally, Shao et al. (2022) and Lin et al. (2026) used the FDM-DEM coupling method, which is particularly suitable for simulating large-scale ground conditions, to study particle motion and the pressure transfer mechanism in sandy pebble strata. Numerical simulations, which use simplified boundaries and materials, produce less intuitive results than model tests and have obvious practical flaws. Moreover, they oversimplify the particle, pebble, and pore features of sandy pebble strata, and fail to replicate particle contact and force chain evolution. Shield construction disturbances are hard to model accurately, simulations lack intuitive collapse evolution visualization, and the predicted settlement and failure ranges differ from field data, meaning that such models are unable to fully reveal shield-induced surface subsidence mechanisms in such strata.

Model testing, which is a convenient and effective method for in-depth study of the collapse mechanism of shield tunnels, can quantitatively observe the deformation and settlement of different areas above tunnels. Kirsch (2010) conducted a small-scale model test to study the collapse of the tunnel face, exploring the development of failure mechanisms in dry sand and the support force of the face. Liao et al. (2024) conducted large-scale physical model tests using both contact and noncontact measurement methods to study the progressive failure mechanism of soft rock during tunnel excavation. Shin et al. (2008) conducted full-scale model tests to reveal the effects of reinforcement methods and pipe length on the strata deformation and face stability of tunnels in granular soil. Using model tests and field measurements, Hu et al. (2020) investigated the mechanical properties of pebble-rich soils during shield tunnel construction. Their results showed that the width of the underground deformation zone broadly decreases linearly with increasing depth. Shi et al. (2024) developed a self-designed test system for twin-tunnel shield tunneling crossing existing pipelines, and investigated ground settlement and pipeline deformation under minor settlement and leak-free conditions during shield construction. Nevertheless, most existing similar model tests focused on the settlement evolution of clay and pure sand strata, while few dedicated disturbance test platforms have been developed for sandy pebble strata. Furthermore, most tests merely monitor surface settlement without multiparameter synchronous measurements of layered deep soil deformation above the tunnel crown, earth pressure, and segment internal force. Accordingly, it is difficult to quantitatively differentiate the collapse zone, stable zone, and subsidence zone, and to reveal their spatiotemporal coupled evolution mechanism.

Current theoretical and experimental systems for shield tunneling construction through sandy pebble strata still have notable deficiencies. Specialized research targeting sandy pebble strata remains scarce, and a comprehensive theoretical framework for zoning excavation-induced instability in such strata has yet to be established. Both a core theoretical formula capable of quantitatively calculating the thickness of overlying stable strata and a penetration collapse criterion tailored to sandy pebble formations remain lacking. From the perspective of experimental investigation, critical components exclusive to physical simulation tests for construction characteristics of water conveyance tunnels in sandy pebble strata are still missing, including proportional preparation of analogous simulation materials and synchronous multiparameter monitoring technologies. This hinders implementation of high-precision and reliable laboratory simulation tests. Accordingly, based on a deep-buried water diversion shield tunnel project in Mengjin (Luoyang, China), this study established a three-zone collapse mechanical model of the upper soil mass in sandy pebble strata based on ellipsoid instability theory, and derived the calculation formula for the minimum thickness of the stable stratum and the stratum collapse criterion. A self-developed physical simulation test system was adopted to conduct shield excavation model tests with reasonable similarity ratio design and sensor arrangement. Multidimensional monitoring data, including segment strain, surrounding rock earth pressure, deep soil displacement, and ground settlement, were collected synchronously. On this basis, the spatiotemporal evolution characteristics of the collapse zone, stable zone, and subsidence zone above the tunnel were explored systematically, and the influence of stratum cavity size on the settlement development rate was analyzed quantitatively. The research findings provide theoretical and experimental support for collapse disaster prevention, and enable parameter optimization of water diversion shield tunnels in sandy pebble strata.

2 Theoretical model for excavation zones of shield tunnels in sandy pebble strata

2.1 Basic assumptions of the theoretical model

The sandy pebble stratum is a typical uncemented granular medium, in which stress is transmitted through particle point contact with nearly zero cohesion. With consideration of the instability and failure characteristics of the shield tunnel face, this study developed an ellipsoidal cavity collapse zoning model, with the following basic assumptions adopted for theoretical simplification.

• The sandy pebble stratum is regarded as a homogeneous and isotropic granular medium. The inhomogeneity of local interlayers and the short-term weakening effect of groundwater seepage on stratum mechanical properties are neglected.
• The slip failure surface of the tunnel face induced by shield excavation unloading conforms to the logarithmic spiral distribution, and the stratum cavity caused by excavation is simplified as a standard ellipsoid.
• The overlying strata above the tunnel are divided into three independent stress transfer zones from the top to the bottom, and the vertical thickness of each zone satisfies the geometric conservation relationship.
• Stratum settlement deformation is dominated by vertical transmission, and the horizontal disturbance range is controlled by the Peck settlement trough width coefficient.
• The soil arching effect formed in the stable zone can block the stress transfer between the collapse zone and the subsidence zone. Penetrating ground collapse will occur when the thickness of the stable zone decreases to zero.

2.2 Method of division of the three zones and its mechanical action mechanism

Sandy pebble strata are composed of pebbles and sand particles with varying sizes and shapes. Owing to their relatively loose structure and large interparticle voids, the interparticle cohesion is almost zero. When the tunnel experiences water and sand influx due to an unstable excavation surface, it is easy for a collapse zone to form. At this time, sand and pebbles transmit force chains through point contact, forming a key arch-controlled zone. Substantial loss of sand particles further gives rise to ground surface subsidence. Based on this physical mechanism, a three-zone model for ground settlement above the tunnel crown during shield excavation in sandy pebble strata is established comprising the collapse zone, critical control zone, and subsidence zone from the bottom to the top, as shown in Figure 1. The collapse zone refers to the unstable region immediately above the tunnel where soil mass fails and caves inward because of insufficient support force. The critical control zone is the intermediate bearing layer that maintains the arch structure and force chains, the stability of which dominates the overall deformation behavior. The subsidence zone is the uppermost strata extending to the ground surface, where visible deformation and settlement occur under the influence of lower strata movement. Therefore, the thickness of the critical control zone plays a crucial role in ground subsidence. The total thickness H of the overlying rock at the top of the tunnel is expressed as follows:where H represents the total height of the overlying strata at the top of the tunnel, H₁ is the height of the collapse zone, H₂ is the height of the critical control zone, and H₃ is the height of the subsidence zone.

H 1 + H 2 + H 3 = H (1)

2.2.1 Collapse zone (H₁)

Adjacent to the shield arch crown, this zone is formed directly by excavation unloading and soil shear sliding. The soil inside this zone is fully loosened with complete loss of integrity, presenting an inverted funnel collapse shape. The self-weight stress of the cavity propagates upward continuously, acting as the primary stress source triggering stratum deformation. Its thickness is jointly governed by the geometric dimension of the ellipsoidal cavity and the internal friction angle of the surrounding strata.

2.2.2 Critical control zone (H₂)

Located above the collapse zone, a soil arch bearing structure is formed via interlocking of sandy pebble particles. It serves as the core barrier preventing upward transmission of the collapse stress from the lower stratum to the ground surface. Insufficient shield supporting pressure will break the mechanical equilibrium of the soil arch. The soil in this zone slides downward into the collapse zone and migrates upward into the subsidence zone, simultaneously leading to continuous reduction of H₂ and synchronous growth of both H₁ and H₃. The thickness of the critical control zone is the decisive parameter for ground settlement control and overall stratum collapse prevention.

2.2.3 Subsidence zone (H₃)

This zone extends from the top of the critical control zone to the ground surface without obvious shear failure planes. Uniform bulk subsidence occurs merely under the combined effect of stratum soil loss and upward stress transfer from lower layers. The ground surface exhibits a typical Peck settlement trough, with the maximum settlement value directly above the central axis of the tunnel.

As tunnel excavation proceeds, the collapse zone ahead of the tunnel face assumes an approximately elliptical shape under the action of the ultimate supporting force. During this process, voids within the collapse zone migrate upward, and the failure zone at the tunnel face propagates from the bottom to the top, resulting in expansion of the collapse zone. Meanwhile, the surface subsidence zone develops from the top to the bottom, accompanied by increase in the value of H₃. When the subsidence zone and the collapse zone become connected, the value of H₂ decreases rapidly. As the thickness of the critical control zone approaches zero, a geological collapse disaster will be triggered, as illustrated in Figure 2. Therefore, based on Equation 1, the criterion for ground collapse induced by shield tunneling can be derived as follows:

H 1 + H 3 ≥ H (2)

The empirical Equations 3–5 for ground subsidence, derived by Peck (1969), indicates that surface subsidence during shield construction projects can be expressed as follows: where H₃ represents the ground settlement value at distance X from the center axis of the tunnel, m; i is the ground settlement trench width coefficient, m; S is the maximum ground settlement at the centerline of the tunnel, m; and V _*i* is the loss of the ground layer per unit length of the tunnel caused by construction, m³/m.

H 3 = s max exp ‐ X 2 2 i 2 (3)

s max = V i i 2 π ≈ V i 2.5 i (4)

i = H 2 π tan 45 ° ‐ φ 2 (5)

By combining Equations 1–3, the amount of ground subsidence can be calculated as Equation 6:

H 3 = V i 2 π tan 45 ° ‐ φ 2 2.5 H exp ‐ X 2 2 i 2 (6)

From the simplified model shown in Figure 2, it can be seen that center O must exist because of the sliding failure mode of the tunnel face. In Figure 2, D is the diameter of the tunnel, l _n is the horizontal distance from origin O to the excavation surface, l _*j* represents the horizontal distance from the tunnel face to the ellipsoid center in the collapse zone, l _m is the vertical distance from origin O to the midpoint of the excavation face, σ _t is the maximum support force required to maintain the stability of the palm face, h _s is the height from the lowest point of the collapsed ellipsoid to the tunnel arch top, r ₀ represents the initial radius of the logarithmic spiral, r _*a* is the logarithmic spiral radius corresponding to the bottom of the slip surface, a is the short half axis of the ellipsoid in the collapse zone, and b is the major axis of the ellipsoid in the collapse zone. The model can effectively simulate the situation of soil collapse in front of the tunnel face caused by excavation of shield tunnels in sandy pebble strata. Based on the geometric relationship, the following Equations 7–12 can be derived: where L = 2l _*i* is the width of the sliding soil in front of the excavation face, and ε is eccentricity. Assuming the axial ratio of an ellipsoid is equal to the lateral pressure coefficient, then 1 − ε ² = a ²/b ² and K ₀ = 1 − sinφ = a ²/b ². Based on the geometric relationship illustrated in Figure 2, the geometric parameters of the ellipsoid is further obtained as Equation 13, and H₂ can be then expressed as Equation 14:

r 0 = D sin π 4 + φ 2 · ⁡ exp π 4 + φ 2 tan ⁡ φ ‐ ⁡ sin ⁡ φ (7)

r a = r 0 ⁡ exp π 4 + φ 2 tan ⁡ φ (8)

l n = r a ⁡ cos π 4 + φ 2 + φ (9)

l i = r 0 ⁡ cos ⁡ φ ‐ l n / 2 (10)

h s = r 0 l i l n sin ⁡ φ (11)

H 1 = 1 1 ‐ ε 2 · L 2 4 h s (12)

a = b 1 ‐ ε 2 b = l n + 1 h s l i H 1 + h S = 2 b (13)

H 2 = H ‐ H 1 ‐ H 3 = H − 1 1 ‐ ε 2 · L 2 4 h s − V i 2 π tan 45 ° ‐ φ 2 2.5 H exp ‐ X 2 2 i 2 (14)

The evolution law of the three zones is governed by the coupled mechanism of “excavation disturbance–stress redistribution–seepage erosion,” exhibiting progressive development from the bottom to the top as well as dynamic spatiotemporal correlation. After tunnel excavation, the stress around the tunnel wall is released. The rock mass surrounding the tunnel face is prone to instability owing to excavation unloading and particle fragmentation, forming the collapse zone. The collapse zone continuously expands upward, and its self-weight pressure and unloading effect induce stress concentration in the overlying strata, activating the critical zone (plastic flow zone). Obvious settlement begins to occur in the surface subsidence zone. The settlement rate is positively correlated with the deformation rate of the plastic flow zone, forming a “lower flow–upper subsidence” linkage effect. When the collapse zone expands upward and the surface subsidence zone extends downward, the critical zone is gradually reduced to zero. Consequently, the equilibrium between particle loss and seepage erosion in the critical zone is broken, ultimately leading to instability failure of the tunnel face.

2.3 Comprehensive demonstration of the mechanical mechanism of the model

The model is driven by the coupling effect of “excavation unloading–stress redistribution–particle loss,” and the evolution of the three zones presents a progressive linkage law.

• The disturbance induced by shield tunneling causes unloading of the arch crown soil, and soil particle sliding forms the collapse zone H₁, with the self-weight stress transferring upward.
• The load of the collapse zone leads to stress concentration in the soil of the stable zone, and the soil arch serves as the only stability barrier of the stratum. When the supporting force is insufficient, the soil in the stable zone slides toward the upper and lower zones, resulting in continuous reduction of H₂.
• During the attenuation of the stable zone thickness, the subsidence zone develops synchronously from the top to the bottom and H₃ increases continuously. When H₂ ≤ 0, the collapse zone is connected with the subsidence zone, leading to surface collapse.
• The ellipsoid model can fully describe the sliding and collapse range of the soil in front of the tunnel face. In conjunction with the peck settlement model, it realizes the unified quantitative calculation of underground cavity morphology and surface settlement, which is applicable to the stability analysis of shield excavation in uncemented sand and pebble granular stratum.

3 Engineering background and materials

This research is based on an actual engineering project site in Luoyang City in China. According to the actual geological structure of the water-rich sandy pebble strata that the shield tunnel passes through, appropriate similar simulation materials were configured.

3.1 Engineering background

This study was conducted against the engineering background of a shield tunnel in the Luoyang area, where the tunnel passes through water-rich sandy pebble strata. The tunnel, with length of 1683.2 m, diameter of 4.56 m, and buried depth ranging from 33 to 60 m, was constructed using the in situ pressure balance shield method. It is overlain by Quaternary soil deposits along its alignment, and the surrounding rock mass consists mainly of Upper Tertiary ⑤3 conglomerate, and ⑤2 argillaceous fine sandstone interbedded with sandy claystone and conglomerate. The outlet section is partially covered by the Quaternary soil layer. Most pebbles have a particle size in the range of 2–20 cm, with some exceeding 60 cm in diameter. The pebbles generally have a rounded shape. There is almost no cohesion between the pebbles and the soil particles, resulting in a loose state. Occasionally, there are slightly cemented blocks and layered structures that belong to an extremely soft rock. The content of gravel and pebbles in the strata is uneven and undergoes marked phase changes, with interbedded sandstone or claystone, exhibiting medium to strong permeability characteristics. The local strata contain a large amount of clayey sandstone, which is in an unconsolidated or weakly consolidated state. In some other structurally layered areas, there are clay–calcite weakly consolidated substances that are prone to water permeability, softening, and disintegration when exposed to water. In summary, the strata are relatively loose, with low cohesion, high porosity, and strong water permeability. Disturbance during excavation is prone to cause instability at the working face, posing a threat to the safety of on-site construction personnel. The physical parameters of the strata are listed in Table 1.

TABLE 1. Formation physical parameters.

3.2 Similarity theory and similarity ratio determination

Dimensional analysis is used to determine five core similarity criteria that comprise geometry, unit weight, cohesion, internal friction angle, and hydraulic conductivity, which meet the multifield coupling similarity of elasticity, strength, and seepage. The prototype is the sandy pebble stratum of the Mengjin water diversion tunnel in Luoyang, and the geometric similarity ratio of the model is C _*L* = 1:40:

Geometric similarity : C L = L m / L p = 1 / 40

Unit weight similarity : C γ = γ m / γ p = 1

Internal friction angle similarity : C φ = φ m / φ p = 1

Hydraulic conductivity similarity: C _*k* = C L = 0.158, which is used to match the seepage characteristics of the stratum.

The specific model similarity parameters are listed in Table 2.

TABLE 2. Material consumption for layered model pavement.

The pebble particle size of the stratum is in the range of 2–60 cm. According to the geometric similarity ratio, the equivalent pebble particle size of the model is calculated to be 0.05–1.5 cm. The model filler is prepared by compounding multigrade natural fine pebbles and quartz sand to reproduce the heterogeneous and high-porosity bulk structural characteristics of the actual stratum. Sieving tests were performed to plot the particle grading curve of the model to ensure that the particle force transfer law and pore structure of the model are consistent with those of the actual stratum.

3.3 Similar simulation materials

The model soil layer thicknesses were converted from the actual stratum thicknesses based on the similarity criterion. The soil was filled and compacted layer by layer. Each compacted layer was allowed to stand for 30 min before the next layer was laid. All filling materials were uniformly mixed with water with a controlled moisture content of 12%. After completion of the layered filling, the model was cured for 24 h to stabilize the soil mass before the excavation test was undertaken. The mix proportion, material dosage, and layer thickness of each stratum are listed in Table 3.

TABLE 3. Material composition for the layered model pavement.

The main composition of the similar simulation material is shown in Figure 3.

The simulated sandy pebble stratum was prepared by mixing small-sized pebbles, natural sand, lime, and silty clay in a designed proportion. The fundamental mechanical parameters of the soil layers were determined via direct shear tests using the testing procedure illustrated in Figure 4.

The basic physical and mechanical parameters of the similar materials were obtained in accordance with the methods and procedures shown in Figure 4. The test results of the similar materials and the actual strata are compared in Table 4.

TABLE 4. Comparison of the physical and mechanical parameters of the actual strata and the model similar materials.

4 Model preparation and test procedure

According to similarity theory, the model dimensions for indoor physical model tests are determined after the similarity constants are specified. By adopting similar material proportioning methods, the heterogeneity and low-cohesion characteristics of the sandy pebble strata of the study site were reproduced, thereby ensuring a high degree of similarity in terms of the geomechanical behavior between the physical model and the prototype. The permeability and structural differences of the actual strata were simulated by controlling particle size gradation and porosity. A shield tunneling physical simulation test was then performed to monitor variations in earth pressure, ground surface movement, and overlying strata settlement during excavation, as well as the stress–strain responses around the tunnel. The objective was to reveal the settlement evolution law within the three zones above the tunnel during shield construction in water-rich sandy pebble strata.

4.1 Physical simulation model test system

The test bench was independently developed with overall dimensions of 1.5 m (length) × 1.0 m (width) × 1.5 m (height), and the height of the support was 400 mm. Jacks and diagonal braces were installed at the bottom of the test bench to improve the overall structural stiffness and eliminate the interference of boundary deformation on the test results, as shown in Figure 5. The entire system consisted of six submodules, and the detailed structural forms and technical parameters of each module were as follows.

• Model box module: The front and rear panels of the model box are made of 15-mm-thick transparent high-strength endurance plates, which allow intuitive observation of stratum fracture development and soil collapse during the test. The left and right sides of the box are assembled with channel steel and bolts to ensure structural stability. All assembly joints are sealed with glass glue to achieve effective waterproofing and anti-seepage performance.
• Shield tunneling simulation system: A double-layer PVC sleeve structure is adopted to simulate the shield tunneling system. The outer PVC pipe with an outer diameter of 12.5 cm is used to simulate the shield cutterhead, and the inner pipe with an inner diameter of 11 cm is applied to simulate the tunnel segment lining. BX120-6AA strain gauges are pasted circumferentially on the pipe wall at intervals of 45° to monitor the strain variation of the lining structure. The tunneling parameters are set as follows: the single tunneling drawing distance is 5 cm, a 1-h stabilization period is reserved after each excavation for soil stress stabilization, and a total of 20 tunneling operations are carried out in the entire test.
• Earth pressure monitoring system: Miniature earth pressure cells, used for stratum earth pressure monitoring with a measuring range of 0–5 kPa and accuracy of 0.01 kPa, can accurately capture the subtle dynamic changes of stratum earth pressure caused by shield tunneling disturbance.
• Deep settlement monitoring system: Rigid stainless steel settlement rods are adopted for deep stratum settlement monitoring with measurement accuracy of 0.01 mm. The rods have high rigidity and low deformation, ensuring the accuracy of deep soil settlement data.
• Ground displacement monitoring system: Dial gauge displacement meters, employed to monitor the ground displacement of the model with a measuring range of 0–10 mm, stably and continuously collect ground displacement data throughout the tunneling process.
• Data acquisition system: The Donghua DH3816N static strain test system is adopted as the core acquisition equipment with a sampling frequency of 1 Hz. It synchronously collects multichannel test data that include strain, earth pressure, and displacement, thereby ensuring the synchronism and integrity of the test data.

Via detection sensors, the monitoring system of the test bench collects the strain of the inner sleeve of segments under simulated shield construction, the earth pressure at different depths above the inner sleeve of segments, as well as the stratum settlement and surface settlement induced by construction disturbance of the simulated shield, as shown in Figure 6.

The detection sensors were used mainly to monitor the variations in surface displacement, deep soil displacement, segment stress, and formation earth pressure during shield tunneling in sandy pebble strata. Strain gauges of type BX120-6AA were adopted in the test, and three monitoring sections were arranged at the middle of the model with the same sensor layout for all sections. The surface displacement sensors were arranged in direct contact with the ground surface. The settlement rods for monitoring stratum displacement were installed at depths of 15 and 40 cm below the miscellaneous fill layer. A single earth pressure cell was arranged at a horizontal distance of 15 cm on both sides of the tunnel. In the argillaceous calcareous fine sandstone interbedded with sandstone layer, three earth pressure cells and two settlement rods were deployed, with an interval of 30 cm between adjacent devices. An additional earth pressure cell was separately arranged 15 cm from the tunnel waist. Meanwhile, strain gauges were pasted at intervals of 45° along the circumferential direction of the shield tunnel. The layout of all sensors is shown in Figure 7.

The shield excavation system consists of outer and inner PVC plastic pipes with diameter of 12.5 and 11 cm, respectively. The 11-cm-diameter PVC pipe was used to simulate the pipe support structure of the shield machine, while the other pipe was used to simulate the cutterhead of the shield machine. By fitting these two pipes together and pulling out the outer PVC pipe, the movement of the cutterhead is simulated, thus realizing the excavation process of the shield tunnel. By attaching strain gauges to the pipe walls of the simulated segmental support structure, the deformation patterns of the segments during the excavation process were collected, thereby enabling study of the spatiotemporal evolution laws of the three zones of collapse at the top of the sandy pebble stratum tunnel.

A seepage simulation system was added for the research object of water-rich sandy pebble stratum. Water-permeable holes were pre-embedded on both sides of the model box, and a continuous water supply was realized via a constant-pressure water tank. Before the test, the water level was raised stepwise to the preset buried depth and kept static for 12 h to stabilize the pore water pressure of the stratum. Pore water pressure sensors were arranged inside the stratum to synchronously monitor the seepage evolution during excavation, which reproduces the characteristics of particle loss, softening, and disintegration of the water-rich strata.

4.2 Preparation process of the similar physical test models

The preparation process of the similar physical test models consisted mainly of the following steps.

• Assembly and waterproof sealing of the simulation test chamber. First, place the base plate horizontally on a stable support and use a level gauge to adjust the level of the base plate. Then, install the side plates, fix them to the base plate, and connect them with bolts or other fasteners to ensure a tight and seamless connection. Finally, seal the connection between the transparent panel and other components of the chamber to prevent leakage or deformation during the experiment that might affect observation.
• Carry out the mixing of similar materials. Sand, gypsum, clay, graded sand, and gravel of various particle sizes and water are selected and mixed uniformly.
• Place the prefabricated model tunnel in the designated position.
• Reasonably arrange sensors for measuring earth pressure, soil displacement, pore water pressure, and ground surface displacement.
• Connect the soil pressure sensors, pore water pressure sensors, surface displacement sensors, and strain gauges to the DH3816N static strain testing and analysis system, and then conduct a preliminary test to verify the stability of the entire test system.

Upon completion of the model, it was allowed to stand for 1 day prior to testing. The final physical test model is illustrated in Figure 8.

During shield tunneling, the sensors mainly monitor variations in surface displacement, deep soil displacement, pipe stress, and soil pressure in the sandy pebble stratum, as shown in Figures 8c–f. The BX120-6AA strain gauges were adopted in the model test, and three monitoring sections arranged at the middle of the similar model. The surface displacement sensor was placed in contact with the ground surface, while settlement rods for monitoring soil displacement were installed at depths of 20 and 36 cm below the mixed soil layer. Two soil pressure gauges were positioned at 15 cm on each side of the tunnel. In the interbedded stratum composed of mudstone and calcareous fine sandstone, three soil pressure gauges and two seepage settlement rods were arranged, with spacing of 15 cm between adjacent soil pressure gauges and settlement rods. Additionally, strain gauges were installed circumferentially around the shield tunnel at intervals of 45°.

5 Results and discussion

5.1 Analysis of the similar simulation test process

Using the shield excavation device shown in Figure 9, the simulated cutterhead and pipe assembly were extracted 20 times during the similar simulation test. Each extraction length was set to 5 cm, and the outer cylinder sleeve was extracted once per hour. After each extraction step, a period of stabilization was allowed to collect monitoring data and ensure soil stability.

When the tunneling distance reached 15 cm, a few cracks appeared around the edge of the tunnel opening. The frictional shear and compression between the support system and the soil mass induced movement in a certain thickness of surrounding soil, forming a shear disturbance zone. This further caused ground loss, accompanied by the formation of small cracks and loosened zones around the tunnel, as shown in Figure 10.

Owing to the weak or absent cementation between the pebbles and soil particles, the stratum exhibits a loose structure with occasional microaggregates and layered fabrics, and can be classified as extremely soft rock. The content of pebbles and gravel in the stratum is uneven with obvious lithological changes, and the stratum is interbedded with sandstone or mudstone. When the shield machine is shut down for a long time for maintenance or tool replacement, it will cause collapse of the soil mass above the cutterhead, resulting in large voids and stratum loss. Meanwhile, the water-rich sandy pebble stratum forms a self-stabilized semi-spherical arch, as shown in Figure 11.

When excavating through a geological cavity, the arching effect of the soil layer causes the overlying soil inside the cavity to collapse downward, leading to upward propagation of the cavity. Influenced by factors such as rising groundwater levels and surface dynamic loads, the self-stabilized structure of the sandy pebble stratum gradually peels away from the dome surface and develops toward the ground surface, as shown in Figure 12. Once the shield tunneling distance exceeds a certain threshold, the soil around the cavity collapses downward, resulting in ground settlement that is referred to as “delayed settlement.” Furthermore, the duration of such delayed settlement is related to the volume of the soil cavity.

As the shield tunnel continues to advance, the original equilibrium state of the cavity disrupts, causing the cavity to propagate throughout the stratum, resulting in subsidence of the strata above the tunnel, as shown in Figure 13.

As illustrated in Figure 14, ground settlement patterns induced by shield tunneling in sandy pebble strata show pronounced spatiotemporal heterogeneity, leading to a distinct three-zone distribution in the surrounding ground: the subsidence zone, critical control zone, and collapse zone.

The subsidence zone exhibits the largest settlement magnitude and widest distribution, forming a distinct settlement trough with its maximum point directly above the tunnel axis. Within the critical control zone, when the thickness of the collapsed soil layer immediately above the tunnel exceeds that of the stable zone, a settlement transfer pathway forms and propagates upward through the ground, enabling ground settlement to extend continuously to the surface. In contrast, when the thickness of the collapsed soil layer immediately above the tunnel is less than that of the stable zone, the stable zone acts as a natural barrier that impedes the upward propagation of ground settlement. Thus, the soil thickness in the critical control zone plays a pivotal role in controlling ground subsidence. Within the collapse zone, the collapse range adjacent to the tunnel crown displays an obvious inverted funnel shape, and the corresponding collapse height is governed closely by the parameters adopted during shield tunneling.

With progressive excavation of the shield tunnel, ground settlement induced in water-rich sandy pebble strata follows an evolutionary sequence: initial ground disturbance → collapse of the soil layer at the tunnel crown → cavity development → formation of a loosened zone → coupled evolution of surface subsidence and downward propagation of cavities. Strata loss induced by the initial stage of shield tunneling gives rise to approximately symmetrical triangular loosened zone fractures on both sides of the tunnel, accompanied by stress redistribution in the surrounding strata. The loose collapse zone on both sides of the tunnel within the strata will thus reach a new stable state. As shield tunneling proceeds, construction-induced disturbance further exacerbates the loss of the underlying strata. Once the gravitational load of the excavated soil surpasses the inherent bearing capacity of the strata, the initial stable state is disrupted. This unstable condition persists until the soil mass above the tunnel crown collapses onto the shield segment lining, thereby triggering stress release and redistribution within the surrounding rock mass. During subsequent shield excavation, the segment lining carries most of the strata forces. Following repeated stress redistributions and restabilization processes within the ground, the earth pressure acting on the segments increases gradually. The loading evolution of the segments includes a period of rapid growth, which is controlled primarily by the volume of the collapsed soil immediately above the tunnel.

5.2 Analysis on the strain law of pipe segments

Full-cycle data including pipe segment strain, stratum pressure, vertical displacement of deep soil, and ground surface settlement were synchronously acquired using the DH3816N static data acquisition system. With consideration of macroscopic phenomena such as stratum crack propagation, cavity development, and formation of the three characteristic zones during the step-by-step shield tunneling process, the coupling relationship among tunneling disturbance, stratum cavity expansion, dynamic evolution of the three zones, and multi-index mechanical responses was established. Quantitative mechanism analysis was carried out for each indicator, and specific investigation conducted on the evolution mechanism of delayed settlement in water-rich sandy pebble strata.

The stress variation curve of the segment lining during the shield tunneling process is presented in Figure 15.

Analysis of Figure 15 reveals that the monitoring curves of segment strain indicate that the lining force throughout the entire shield tunneling process can be divided into three stages: uniform growth, rapid sharp increase, and slow stabilization. Each stage is highly consistent with the expansion of the vault collapse zone.

• Tunneling distance of 0–30 cm (before cavity formation): No obvious underground cavities exist, the stratum is uniformly disturbed, and soils in all zones remain intact. The soil arch effect of the stable zone remains steady, and the strain at all monitoring points of the segments increases uniformly and only slightly. The height of the collapse zone H₁ stays at a low level, and stratum stress transfers uniformly to the lining.
• Tunneling distance of 30–60 cm (cavity development stage): An ellipsoidal stratum cavity forms behind the tunnel face. The lower collapse zone H₁ continuously expands upward, and the soil in the stable zone slides and loses support toward the cavity under the self-weight load of the overlying strata, leading to continuous reduction in the thickness of the stable layer H₂. After the soil above the cavity loses support, loads concentrate on the vault lining of the tunnel. The strain growth rate of the vault measuring point Y-1 is remarkably higher than that of the other circumferential measuring points, presenting a steep rising trend of the strain curve. This phenomenon directly verifies that the development of stratum cavities breaks the original equilibrium of the soil arch, and the upward expansion of the collapse zone induces local stress concentration on the lining.
• Tunneling distance >60 cm (stable stage with the three zones fully formed): The soil above the cavity collapses completely to form a new arch, the range of the subsidence zone stops continuous expansion, and a new bearing system of a soil arch reconstructs in the stable zone accompanied by redistribution of stratum stress. The growth rate of segment strain slows and gradually becomes constant.

Through consideration of both strain data and macroscopic stratum failure modes, it can be concluded that the peak strain of the vault lining corresponds to the critical state where the collapse zone expands upward to the bottom of the stable zone, which acts as a critical node for early warning of stratum settlement risks.

5.3 Variation law of soil pressure in geological strata

Owing to the unique characteristics of water-rich sandy pebble strata, they have high internal porosity and contain various-sized stratum cavities. As the stratum cavities move upward, the continuity of the stratum is altered, which in turn causes changes in the thickness and density of a specific soil layer and thus produces variations in the data collected by the soil pressure gauge, as shown in Figure 16.

The sandy pebble stratum features high porosity, well-developed cavities, and a loose soil mass. The loosening degree of the soil directly alters the effective self-weight stress of the soil layer and, intuitively, the variation of earth pressure reflects the evolution of soil compactness in both the collapse zone and the stable zone.

• Excavation distance of 0–50 cm: No large-scale cavities exist in the stratum, the three soil layers remain intact with distinct stratification, and the soil in the stable zone maintains high compactness. The soil pressure remains steady at approximately 0.1 kPa, the range of the collapse zone is extremely small, and no obvious fluctuation is observed in soil pressure.
• Cavity penetration stage (excavation distance of 50–60 cm): The soil above the cavity continuously collapses to fill the gap between the segment lining and the surrounding stratum. The soil in the collapse zone becomes loose with drastically increased porosity, which reduces the effective unit weight of the soil layer. The monitoring value of soil pressure cell T-8 decreases continuously to 0 kPa. During this stage, the soil in the stable zone slides downward constantly, the thickness of the stable layer keeps decreasing, and the transfer channels for ground settlement are gradually formed.
• Excavation distance of > 60 cm: The collapsed soil from the upper part of the cavity completes gap filling. The soil in the upper subsidence zone undergoes slow consolidation and densification under self-weight, the effective unit weight of the soil layer recovers, and the monitored soil pressure values rise gradually.

Sustained reduction of earth pressure indicates expansion of the lower collapse zone and loss of soil mass in the stable zone. The rebound of soil pressure marks the termination of the upward expansion of the collapse zone, the reformation of a stable soil arch inside the stratum, and the evolution of the three characteristic zones enters an equilibrium state.

5.4 Analysis on variation law of surface displacement

During the shield tunnel excavation process, the movement of the cutterhead will cause disturbance to the geological strata, resulting in over excavation of the tunnel face and causing settlement of the geological strata and ground surface. The main analysis focuses on the displacement change law of the ground surface using three sensors. Among them, the area directly below the displacement sensor W-4 is a simulated formation cavity, the area directly below W-5 has no cavity, and the area directly below W-6 is a stratum cavity, but the tunnel interior and the surface are jointly grouted. The settlement law of the ground surface with excavation in different states is shown in Figure 17.

Analysis of the surface monitoring data of measuring points W-4 (directly above the cavity), W-5 (cavity-free control group), and W-6 (grouting-reinforced zone) during tunnel excavation, shown in Figure 17, clearly reveals the blocking and controlling effect of the stable zone on ground surface settlement.

• Excavation distance of 0–30 cm: The shield machine has not reached the cavity cross section, and only slight stratum loss is induced by the excavation process. The thickness H₃ of the subsidence zone increases slowly, and the settlement values of all three measuring points are < 0.5 mm. The intact stable zone blocks the upward transmission of underlying disturbances, resulting in trivial ground surface deformation.
• Excavation distance > 30 cm: The cavity beneath measuring point W-4 expands upward continuously, and the collapse zone erodes the stable zone persistently, which drastically reduces the thickness of the stable layer and connects the settlement transfer channels. Consequently, the ground surface settlement rate rises sharply, with a maximum settlement of 4.5 mm. In contrast, the stable zone remains intact at W-5 with no underground cavities, and the soil mass at the grouting-reinforced zone W-6 possesses strong integrity without damage to the soil arch structure. The peak settlement value at both measuring points is <1 mm.

The above data directly verify the aforementioned three-zone division theory: the thickness of the stable zone acts as the core barrier governing ground surface settlement. When underground cavities trigger a dramatic reduction in stable zone thickness, the subsidence zone simultaneously develops rapidly and obvious ground surface settlement occurs. If the stable zone stays intact or the soil arch is preserved via grouting reinforcement, ground surface deformation can be substantially restrained.

5.5 Analysis of stratum settlement laws

Stratum displacement and subsidence refers to the positional changes and subsidence phenomena that occur in the stratum under the influence of external forces or natural factors. The main factor causing stratum subsidence during shield tunnel construction is stratum loss. To study the evolution law of the top three zones of a shield tunnel, settlement rods were arranged in different strata. Three settlement rods were arranged 30 cm from the top of the tunnel, and monitoring data from measuring point C-4 were selected for analysis. Additionally, six settlement rods were arranged 60 cm from the top of the tunnel, and monitoring data from measuring point C-1 were selected for analysis. The analysis results are shown in Figure 18.

The data from two layers of settlement rods installed 30 and 60 cm above the tunnel crown can realize cross-verification of the spatial distribution of the three characteristic zones.

• Measuring point C-4 at depth of 30 cm exhibits large settlement and a fast deformation rate. This area is fully covered by the influence range of the collapse zone. The soil mass is directly disturbed by cavity collapse, and vertical displacement is dominated by progressive upward collapse induced by the underlying cavity.
• Measuring point C-1 at depth of 60 cm shows negligible settlement, with only slight compressive deformation caused by soil consolidation. It is free from shear failure triggered by the collapse zone because this stratum lies within the critical stable zone where the soil arch effectively blocks upward transmission of collapse stress from the lower part.

The layered deep settlement data corroborate the three-zone layered model from the internal spatial perspective of the stratum: the area adjacent to the tunnel crown is the collapse zone, the thick middle soil layer serves as the stable control zone, and the region from the ground surface to the upper part of the stable zone corresponds to the subsidence zone. Clear vertical boundaries exist among the three zones, which are highly consistent with the theoretically calculated thicknesses H₁, H₂, and H₃.

5.6 Special analysis on formation mechanism of delayed settlement in water-rich sandy pebble strata

In the test investigating water-rich sandy pebble strata, the ground surface settlement fails to stabilize immediately after excavation, presenting an obvious delayed settlement phenomenon. In conjunction with the evolution characteristics of the three zones, the formation mechanism of delayed settlement is divided into three stages.

• Instantaneous settlement (synchronous with excavation): Unloading induced by shield excavation and instantaneous stratum loss rapidly form the collapse zone. This type of deformation is immediate and triggered directly by excavation disturbance.
• Short-term delayed settlement (0–2 h after excavation): The sandy pebble particles above the cavity are cementation-free. Under the combined action of self-weight and groundwater seepage, soil particles continuously spall off. The collapse zone expands upward slowly, the soil mass in the stable zone gradually loses support, and the subsidence zone expands synchronously.
• Long-term consolidation delayed settlement (2 h after excavation): Sustained groundwater seepage causes the loss of fine particles in the stratum and the soil porosity decreases continuously. Consolidation compression occurs in the soil of the subsidence zone, leading to persistent slight settlement on the ground surface. Larger underground cavity volume corresponds to more developed seepage channels and greater loss of fine particles, prolonging the duration of delayed settlement. Synchronous grouting for stratum cavity sealing in advance can block seepage paths and substantially reduce delayed settlement in the middle and later stages.

5.7 Summary of coupling and linkage law of multiple monitoring indicators

Combining the measured data of segment strain, earth pressure, deep layered settlement, and ground surface settlement, a unified coupling response logic exists for the dynamic evolution of the three characteristic zones in a stratum.

Underground cavities are generated by shield excavation → the collapse zone with thickness H₁ expands upward → the thickness of the critical control zone H₂ decreases and the soil loosens → stratum earth pressure declines and stress concentration occurs on the crown lining → settlement transfer channels are connected → the subsidence zone H₃ develops continuously and ground surface settlement rises rapidly.

After the internal strata collapse to form a new stable soil arch, all monitoring indicators gradually converge and stabilize. This multi-indicator linkage law fully interprets the synergistic evolution mechanism of “lower collapse and upper settlement” during shield tunneling in sandy pebble strata. It compensates for the deficiency of previous studies that analyzed ground surface settlement separately while ignoring the linked deformation of the three internal stratum zones.

6 Engineering application analysis

Based on the above analysis, the maximum stratum settlement value can be obtained through similar simulation tests, while the theoretical stratum settlement value can be calculated using the collapse zone thickness model proposed for sandy pebble strata. In this section, measured values from similar simulation tests are compared with theoretical settlement values to verify the applicability of the theoretical model describing the stratum subsidence mechanism. This comparison is intended to provide a reference for the analysis and control of ground settlement in sandy pebble strata under similar engineering conditions in the future. First, the surface settlement of the cavity-containing strata is calculated at the position directly above the tunnel axis, i.e., at a distance of 0 cm from the tunnel axis. Based on the mass of the cavity-containing soil measured in the model test box and its density determined in the laboratory, the ground loss is obtained as approximately 260 cm³. The internal friction angle of the surface soil layer in the test box is then substituted into the surface settlement formula derived earlier, which gives the following:

H 3 = V i 2 π tan 45 ° ‐ φ 2 2.5 H ⁢ exp ⁡ ‐ X 2 2 i 2 = 260 × 2 π × tan 45 ° ‐ 18 2 2.5 × 150 × exp ⁡ ‐ 0 2 2 i 2 = 1.26 cm

In the similar simulation test, the surface settlement value of the first cavity-containing section recorded by the data acquisition system is 1.26 cm. This value shows a small discrepancy compared with the result calculated by the theoretical model, which effectively validates the rationality of the proposed theoretical model for surface settlement.

Substituting the model dimensions and soil sample parameters of the indoor simulation test into the theoretical formulas presented earlier produces the following calculation results:

r 0 = D sin π 4 + φ 2 · e θ ⁡ tan ⁡ φ ‐ ⁡ sin ⁡ φ = 11 sin π 4 + 13 × π 2 × 180 × e π 4 + 13 × π 2 × 180 tan 13 × π 180 ‐ ⁡ sin 13 × π 180 = 26.76 cm

r a = r 0 e θ ⁡ tan ⁡ φ = 13.05 × e π 4 + 13 × π 2 × 180 tan 13 × π 180 = 32.86 cm

r a = r 0 e θ ⁡ tan ⁡ φ = 14.30 × e π 4 + 13 × π 180 × 2 tan 13 × π 180 = 32.86 cm

l n = r a ⁡ cos θ + φ = 17.6 × cos π 4 + 13 × π 180 × 2 + 13 × π 180 = 14.15 cm

l i = r 0 ⁡ cos ⁡ φ ‐ l n / 2 = 26.76 × cos ⁡ 13 ° ‐ 14.15 / 2 = 5.96 cm

h s = r 0 l i l n sin ⁡ φ = 26.76 × 5.96 14.15 × sin ⁡ 13 ° = 2.53 cm

Elliptical major axis length : a = l n + 1 h s l i = 14.15 + 1 × 2.53 5.96 = 6.43 cm

Elliptical short axis length : b = a 1 ‐ ε 2 = 6.43 × 1 ‐ ⁡ sin ⁡ 13 ° = 5.66 cm

H 1 = 1 1 ‐ ε 2 · L 2 4 h s = 1 1 ‐ ⁡ sin ⁡ 13 ° × 2 × 5.96 2 4 h s = 9.06 cm

From the above analysis, the theoretical ground settlement at a position 30 cm directly above the tunnel is 15.6 mm, whereas the maximum collapse height above the tunnel face in the similar simulation test is approximately 11 mm. The difference between the two values is relatively small, demonstrating that the theoretical formula is in good agreement with the experimental results.

The height of the critical control zone can be calculated as follows:

H 2 = H ‐ H 1 ‐ H 3 = 110 ‐ 9.06 ‐ 1.26 = 99.68 cm

Given that H₂ > H₁ + H₃, it can be inferred that during the test, the collapse zone induced by the cavity above the tunnel and the surface subsidence zone can hardly penetrate the boundary of the stable zone, i.e., the strata remain stable at the initial stage. Following cavity formation and with its further development, the formation gradually transitions from its initial stable state to a new stable state that subsequently persists. Therefore, it is vitally important to reinforce the critical control zone during construction and to interrupt the upward propagation path of ground deformation, thereby effectively mitigating ground deformation at corresponding spatial locations. The comparison results between the theoretically calculated values and the measured values from the model tests are listed in Table 5.

TABLE 5. Comparison of theoretical calculation values and measured values from the model tests.

6.1 Classification analysis of error sources in model calculation

Slight deviations exist between the theoretical results and the measured data, which are jointly induced by multiple factors including theoretical simplification, test conditions, and measurement operations. The error sources can be divided into the following four categories.

Inherent errors caused by theoretical model simplification. The theoretical model assumes a homogeneous granular stratum, neglects the weakening effect of groundwater seepage, and simplifies irregular collapse cavities into standard ellipsoids. However, the actual sandy pebble strata contain interlayers of coarse and fine particles with partial weak cementation. Groundwater seepage accelerates the loss of fine particles, resulting in a slightly larger actual collapse range than that of the theoretical calculation, which leads to higher measured values of H₁.

Influence of boundary effects in model tests. The limited size of the model box restricts the lateral expansion of soil mass via sidewall constraints, and the rigid boundary of the box alters the stress transfer path of the strata. Additionally, only one buried depth condition is adopted in the test, which cannot fully reproduce the in situ infinite strata stress field, thereby introducing minor systematic errors.

Deviations of mechanical parameters between similar materials and prototype soil. The mixing ratio of similar materials only approximately matches the unit weight, internal friction angle, and cohesion of the prototype, failing to completely replicate the particle gradation and pore structure of the in situ strata. Moreover, artificial compaction control during layered filling brings tiny fluctuations that alter the overall mechanical response of the soil mass.

Operational errors from measurement and excavation simulation. Small amounts of fine particles are lost during the collection of the soil mass caused by strata loss. There are discrepancies between the staged pulling excavation simulation in the test and the continuous shield tunneling on site. Moreover, soil pressure and displacement sensors carry inherent precision errors of measuring instruments.

6.2 Applicability and limitations of the theoretical model

6.2.1 Applicable conditions

This model applies to earth pressure balance shield water conveyance tunnels buried at depths of 30–60 m in weakly cemented or uncemented sandy pebble strata with pebble particle sizes of 2–60 cm.

It is suitable for tunnel projects where strata collapse is dominated by self-weight, strong tectonic stress is absent, and the groundwater level is moderate.

The model can predict strata cavity development and calculate the minimum thickness of the safe stable layer, thereby guiding parameter design of advance support and synchronous grouting.

6.2.2 Model limitations

The high hydraulic pressure seepage field is not coupled in the theoretical derivation. For water-rich strata with a large water head, fine particle loss will be aggravated, leading to underestimated calculation results; thus, an extra seepage correction coefficient needs to be introduced.

The superimposed effects of surface dynamic loads and additional loads from ground buildings on the evolution of the three zones are not considered.

This is merely a two-dimensional ellipsoid instability model for a single cross section, without establishment of a three-dimensional calculation system for spatial collapse evolution.

Working conditions with extra-large boulders (particle size > 60 cm) and complex alternating soft–hard strata are not covered.

6.3 Correction suggestions for engineering application

In view of the above limitations of the model, the following corrections can be implemented in field engineering:

For highly water-rich sandy pebble strata, multiply the collapse zone height H₁ by a correction coefficient of 1.1–1.3 according to field permeability test results.

If heavy-load structures exist on the ground surface, supplementary reduction calculation of the stable zone thickness caused by additional stress should be carried out.

Conduct multiple cross-section monitoring of surface and deep subsidence on site, invert the actual thickness of the stable layer using the formulas presented in this paper, and dynamically adjust the shield chamber pressure and grouting volume.

7 Conclusion

This study adopted a self-developed similarity simulation test system for tunnels through sandy pebble strata to conduct model tests on stratum settlement induced by shield tunneling. The entire evolutionary process of soil displacement and stress under construction disturbance was systematically investigated. The strata above the tunnel crown were divided into three zones, and the minimum thickness of the overlying stable stratum in sandy pebble ground was deduced. The main conclusions are as follows.

• On the basis of ellipsoid theory and the instability characteristics of sandy pebble strata, a three-zone settlement model from the bottom to the top during tunnel excavation in sandy pebble ground was established, consisting of a collapse zone, a critical control zone, and a subsidence zone. In conjunction with engineering data, the thicknesses of the collapse, critical control, and subsidence zones were calculated as 9.06 cm, 99.68 cm, and 1.26 cm, respectively, which agree well with measured test data. The spatiotemporal evolution law of the three zones above the tunnel crown in sandy pebble strata was revealed. Targeted advanced precise support should be implemented in the collapse zone both to prevent the formation of underground cavities and to block upward propagation of stratum deformation. It was clarified that the stable stratum forms an earth arch barrier relying on interparticle interlocking, and its thickness acts as the core control parameter for preventing ground surface collapse.
• Taking the stratum settlement of the water conveyance tunnel in Luoyang as the engineering background, model tests reproduced the full process of stratum settlement triggered by shield tunnel construction, which can be divided into four stages: crack initiation at the arch crown, crack expansion, collapse and arch formation, and stratum restabilization. The volume of the underground cavities directly governs the soil collapse rate; a larger cavity accelerates the expansion of the collapse zone and the degradation of the stable zone. The test results verified that the peak settlement of sections with large cavities is remarkably higher than that of cavity-free sections, highlighting the dominant effect of cavities on stratum deformation. Monitoring data indicated that cavity development leads to sharp surges in segment strain and attenuation of earth pressure. Shallow soils at the arch crown suffer severe disturbance from collapse, while deep soils experience only slight consolidation settlement owing to the isolation effect of the earth arch.
• Differentiated prevention and control strategies were proposed according to the deformation characteristics of the three zones above the tunnel. For the collapse zone, dynamic earth pressure regulation and synchronous grouting filling should be adopted to mitigate ground loss. The safety thickness of the stable stratum should be strictly controlled through auxiliary advanced grouting reinforcement to block deformation transmission. A comprehensive technical framework was formed, including cavity elimination at the source, active reinforcement of the stable stratum, staged grouting to restrain delayed settlement, and early warning based on dual indicators of strain and settlement rate. Optimized shield tunneling parameters offer technical support for safe construction of water conveyance tunnels in sandy pebble strata.

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Footnotes

Edited by: Nibir Mandal, Jadavpur University, India

Reviewed by: Yaxiong Peng, Hunan University of Science and Technology, China

Aizhong Luo, Guizhou University of Engineering Science, China