Abstract
An improved method tailored for anisotropic soft soils is presented, integrating theoretical models and field data to calculate the grouting quantity required for tunnel foundations. Given the complexities of soil interactions, particularly under variable geological conditions, this approach incorporates nonlinear behaviors and empirical field data to improve accuracy. Our findings reveal that integrating these theoretical frameworks significantly enhances the understanding of stress–strain behavior during grouting, enabling precise calculations of both axial and vertical expansion. Validation against numerical simulations demonstrates the model’s reliability, highlighting the influence of soil types and grouting depths on expansion dynamics. This method not only helps mitigate risks in tunnel construction but also enhances foundation reinforcement strategies, driving progress in geotechnical engineering. It is particularly valuable for urban tunnel projects in complex geological conditions, where ensuring ground stability and safety is crucial.
Keywords: Grouting, Anisotropic soil, Circular hole expansion, Mindlin solution, Tunnel foundations
Introduction
Grouting is a crucial technique for enhancing the stability and integrity of underground structures1,2. The interaction between the grouting material and the surrounding soil plays a significant role in determining the effectiveness of grouting operations, particularly in anisotropic soft soil formations where soil properties exhibit directional variability3–5. Understanding the diffusion behavior of grouting bodies in these contexts is essential for optimizing construction methods and ensuring long-term performance. Among the various factors influencing this behavior, the volume of grout injected is a critical parameter that directly affects the expansion dynamics of the grouting material. This is especially pertinent in soft soil tunnel construction, where accurately predicting the required amount of grout is vital for optimizing processes, reducing costs, and ensuring effective foundation reinforcement.
Over the years, methods and models for estimating the grouting volume for tunnel foundations in soft soil conditions have continuously evolved. Notably, the empirical formula method, grouting diffusion model, and volume balance model have made significant contributions6–9. Slenders et al. introduced a calculation model that incorporates the properties of oil and grout to analyze their effects on structural safety, surface settings, and oil deformations10. Yang et al. developed longitudinal and transverse calculation models to analyze tunnel uplift in relation to increased grout strength, thereby verifying safety criteria for the structural lining11. Qi et al. proposed a method for comparing and rectifying the horizontal disposition of existing tunnels by integrating a mechanical model for the volume expansion of grouped pipe systems12. Zhou et al. presented a pressure distribution model for simultaneous grouping in shield tunneling, accounting for variations in slurry properties during construction13. Tao et al. proposed a Pasternak model-based upgrade method for tunnel segments in subway shield tunnels, incorporating the hardening process of grouted bodies over time14. Zhong et al. developed a theoretical approach to predict the diffusion radius of fractures in soil-rock mixtures, deriving a Bingham fluid fracture diffusion formula and validating it through field and laboratory tests15. With advancements in computational mechanics and geotechnical engineering, the analysis of grouting processes has become more complex. Finite element modeling, combined with laboratory and field tests, provides deeper insights into the behavior of soft soil under grouting pressure. Qi et al. investigated the impact of grouping reinforcement on the settlement of existing tunnels, presenting theoretical calculation methods consistent with finite element model data16. Peng and Chen studied the surface reinforcement effects of small clearance tunnels in soil-rock composite strata, establishing a numerical model to assess tunnel stability17. However, these models often assume idealized conditions, which may not fully capture the complex interactions in soft soil, particularly regarding anisotropy and time-dependent properties of soft soil materials. Consequently, existing research has limited application in engineering practice and necessitates more accurate and practical models, especially in urban tunnel construction where geological conditions can change significantly over short distances.
Based on this, a new method utilizing circular hole expansion theory and the Mindlin solution is proposed for calculating the grouting quantity of tunnel bases, incorporating the nonlinear and anisotropic behavior of soft soil to provide a more accurate estimate. Additionally, the model integrates empirical field measurement data, further enhancing its applicability in practical engineering scenarios. The results demonstrate that both the circular hole expansion theory and the Mindlin solution for elastic bodies in semi-infinite space are crucial for understanding the stress–strain behavior of soil during grouting. The combination of these methods allows for the calculation of both axial and vertical expansion of the grouting body. The reliability of the calculations is confirmed by comparing numerical simulation results of segment lifting with measured values. Furthermore, the impact of different soil types and grouting depths on the expansion of the grouting body is analyzed. This method offers a more reliable approach to estimating grouting quantity, effectively reducing risks associated with tunnel construction under complex geological conditions, and is expected to enhance the design and implementation of foundation reinforcement strategies in soft soil tunnel engineering, thereby advancing the practice of geotechnical engineering.
Methods and theories
Solution of circular hole expansion problem
The circular hole expansion theory primarily addresses the problem of a circular hole expanding in an infinite medium under uniformly distributed internal pressure p, solved using a combination of equilibrium differential equations, geometric equations, and soil yield criteria. The theory assumes the soil initially behaves elastically, transitioning to plasticity governed by the Mohr–Coulomb yield criterion. The geometric description is illustrated in Fig. 1. In the figure, p0 denotes the initial isotropic in-situ stress. σr and σθ represent the radial and tangential stresses, respectively, within the soil elements surrounding the circular hole. a0 indicates the initial radius of the hole, while a refers to the radius of the hole after expansion. rp represents the radius of the plastic deformation zone around the hole, and up corresponds to the displacement at the interface between the elastic and plastic regions.
Fig. 1.
Illustration of circular hole expansion.
Elastic solution
The problem of circular hole expansion is typically treated as a plane strain axisymmetric problem and is primarily described using polar coordinates. For this analysis, consider the infinitesimal element illustrated in Fig. 2.
Fig. 2.
Stress state of the element under axisymmetric condition.
The equilibrium differential Eq. (1) is derived from the force analysis of the element and the axisymmetric conditions of the problem.
 |
1 |
where 
denotes radial stress, 
denotes tangential stress, and r represents the distance from the point to the center of the circular hole.
Strain in the elastic region follows the small deformations theory, as follows:
 |
2 |
where εr denotes radial strain, εθ denotes tangential strain, and u denotes radial displacement.
For the plane stress problem, the elastic constitutive equation, accounting for the initial stress p0, can be expressed by Hooke’s law as shown:
 |
3 |
For circular hole expansion, the boundary conditions are as follows:
 |
4 |
where 
represents the Poisson’s ratio, 
represents the stress at the interface between the plastic and elastic regions, rp denotes the position of this boundary, p0 is the initial stress in the soil, and a is the distance from the edge of the hole to its center.
The circular hole expansion process consists of two stages. The first stage is the initial expansion phase, during which the expansion of the hole is purely elastic, and the volumetric strain is zero under undrained elastic conditions, expressed as follows:
 |
5 |
As expansion continues, the area surrounding the circular hole transitions from the elastic region to the plastic region, and with increasing pressure, the plastic region progressively expands. Solving the elastoplastic problem of circular hole expansion requires consideration of the soil yield criterion and flow rule18. For Mohr–Coulomb materials, the yield condition is expressed in polar coordinates by Eq. (6).
 |
6 |
The above equation can be transformed to yield Eq. (7), as shown below:
 |
7 |
where φ denotes the internal friction angle of the soil, and c denotes the cohesion.
Elastic–plastic solution
Firstly, the stress and strain in the elastic zone are derived. From Eqs. (3) and (5), the following can be obtained:
 |
8 |
The following equation can be derived by solving the above equation in conjunction with Eq. (1).
 |
9 |
where D1 is the integration constant.
Based on boundary condition (4) and considering the initial stress p0, the stress at a point located at a distance r from the expansion center in the elastic region can be expressed as:
 |
10 |
The strain in the elastic region can be derived from Eq. (3), as follows:
 |
11 |
For the elastic region, by combining the strain Eq. (2) with the volumetric strain condition (5), the displacement u of the soil can be expressed as follows:
 |
12 |
The displacement at the boundary of the plastic zone is as follows:
 |
13 |
At the boundary between the elastic and plastic regions, both the yield criterion (7) and the condition for the elastic region (8) must be satisfied. Let 
, 
and 
. The stress at the boundary 
can be determined as follows:
 |
14 |
At the boundary between the plastic and elastic zones, where both the equilibrium Eq. (1) and the yield criterion (7) are simultaneously satisfied, the following results:
 |
15 |
The differential equation is also solved to determine the relationship between the plastic zone radius rp, the reaming radius a, and the reaming pressure p based on the boundary conditions at r = rp.
 |
16 |
By combining Eqs. (14) and (16), the plastic zone radius rp can be expressed in terms of the reaming radius a and the reaming pressure p as follows:
 |
17 |
Using Eq. (16) as the boundary condition, solving differential Eq. (1) yields the stress expression in the plastic region as follows:
 |
18 |
From Eqs. (3) and (18), the elastic strain in the plastic zone can be determined as follows:
 |
19 |
In the plastic region, the plastic volumetric strain does not satisfy Eq. (5). Unlike metals, geotechnical materials typically follow a non-associated flow rule in the plastic region19. Therefore, this paper adopts a non-associated flow rule. Under plane strain conditions, the relationship between the radial plastic strain 
and the tangential plastic strain 
in the plastic region can be expressed as follows:
 |
20 |
where M is the soil shear dilatancy parameter, defined as 
, and 
is the soil shear dilatancy angle.
The volumetric strain of soil in the plastic zone is expressed as follows:
 |
21 |
Substituting the small strain condition Eq. (2) and the plastic elastic strain Eq. (20) into Eq. (21) yields:
 |
22 |
For Eq. (22), it can be seen from Eq. (19) that the right side of the equation depends solely on r. Therefore, the particular solution to Eq. (22) can be derived as follows:
 |
23 |
where D2 is the integration constant. Similarly, by substituting Eq. (23) as a particular solution into Eq. (22) and applying the method of variation of constants to introduce a strain term, the general solution for the displacement in the plastic region is obtained, as shown in Eq. (24).
 |
24 |
From the boundary condition Eqs. (4) and (13), D2 can be determined.
 |
25 |
From this, it can be concluded that at any point located at a distance r from the expansion center, the displacement resulting from the grouting pressure p is given by:
 |
26 |
The parameter values in the equation are determined by soil parameters, which are fully described in this section.
Mindlin solution of elastic body in semi-infinite space
Mindlin provided a solution for concentrated forces acting within a half-infinite elastic medium, extending the Boussinesq solution for concentrated forces on the surface of such a medium20. This extension offers greater theoretical consistency in calculations and results21. Although research on the Mindlin solution has advanced significantly, its practical application remains limited due to the complexity and cumbersome nature of the formulas. While it has been utilized in theoretical methods, its practical use is infrequent, primarily found in areas like foundations and piles22–25. In this study, the grouting body is simplified to a cylindrical form, and the grouting expansion process is assumed to occur in two stages. When the axial expansion of the grouting body concludes and vertical expansion begins, it is assumed that this vertical expansion is solely driven by stress. Therefore, employing the Mindlin solution to calculate the vertical expansion of the grouting body near the segment is of significant practical value.
As illustrated in Fig. 3, within a three-dimensional elastic half-space, a vertical concentrated force q is applied at the point (0, 0, c). The horizontal displacement U and vertical displacement W at any point (x, y, z) within this space are given by:
 |
27 |
 |
28 |
where ν is the Poisson’s ratio, G is the shear modulus, 
and 
.
Fig. 3.
Mindlin solution of elastic body in semi-infinite space.
Hypothesis of tunnel foundation grouting expansion model
In the analysis of grouting body expansion, the process is treated as nonuniform, primarily due to the influence of grouting depth. As the grouting depth increases, the initial stress in the surrounding soil rises correspondingly, leading to an expansion pattern characterized by a broader exterior and a narrower interior. This depth-dependent variation in initial stress results in a distinctive expansion profile. In the tunnel foundation lifting grouting case considered in this study, the grouting pressure is typically controlled to remain below 0.4 MPa. When the grouting pressure is less than 0.5 MPa, the pressure diffusion in soft clay primarily results in compaction, preventing the formation of a continuous grout vein and inhibiting further splitting26. This is attributed to the relatively low injection pressure during the grouting process, where grout expansion generally stays within the compressive strength limits of the surrounding soil. Consequently, the dominant effect is soil compaction, which restricts the development of significant fractures, thereby limiting the effectiveness of fracture and permeability grouting. Tunnel segments are affected by the expansion of the grouting slurry during the foundation grouting uplift process, often resulting in certain deformations that impact the outcome. Although adding support can mitigate some deformation, its effect is limited27. Additionally, since the tunnel base grouting in this study is performed around the segment range, the grouting angle may also influence the final lifting effects. Therefore, the following assumptions are made for tunnel base grouting:
Compaction grouting is considered as the entire grout expansion; splitting grouting and seepage grouting, which may be included, are not considered separately.
The soil conforms to the Mohr–Coulomb criterion, satisfies the small deformation condition, and is undrained.
The grouting body is approximated as a cylinder with the same volume, as shown in Fig. 4.
Fig. 4.
Simplified calculation of grouting body.
The assumption of approximating the grouting body as a cylindrical shape is a simplification aimed at facilitating calculations and analyses. Its validity lies in its applicability under conditions where the grouting material is relatively homogeneous and the surrounding soil exhibits straightforward properties. In such scenarios, the cylindrical model effectively captures the expansion process and provides preliminary estimations of displacement and stress distribution. Furthermore, as a standard geometric shape, the cylinder is extensively utilized in elasticity theory, allowing for streamlined mathematical modeling and numerical analysis28–30. Consequently, some studies have demonstrated the use of equivalent cylindrical models to represent diffusion processes in soil matrices and to calculate the expansion of grouting materials behind tunnel linings31,32. However, this assumption has certain limitations, primarily in neglecting the interaction between the grouting material and the surrounding soil as well as the influence of complex boundary conditions. In reality, the expansion of the grouting body is constrained by the soil, particularly due to factors such as soil stiffness, porosity, and cohesion, which can alter the expansion behavior. Therefore, while the cylindrical model is suitable for preliminary analyses, more precise calculations should incorporate actual engineering conditions and advanced numerical simulations.
Calculation of grouting body expansion
Axial expansion calculation based on the circular hole expansion theory
Given the limited range and volume of supplementary grouting, the significant depth of the tunnel, and the relatively uniform initial stress of the soil within the grouting range, the issue of base supplementary grouting is simplified to a circular hole expansion problem. Circular hole expansion encompasses cylindrical and spherical holes, which correspond to two-dimensional and three-dimensional cases, respectively. The theory of circular hole expansion is widely utilized in stress analysis for various problems, including tunnels, shafts, and pile foundations. In this study, a cylindrical hole model is employed for analysis based on the Mohr–Coulomb constitutive model for circular hole expansion. The axial expansion of the grouting body is calculated through vertical integration along the body.
In the selected lining segments, a reserved grouting hole is placed every 22.5°, leading to a total of 16 holes. For compensation grouting in shield tunnels, the soil volume Vsoil in the grouting area per unit length is determined using Eq. (29) of the project.
 |
29 |
where Rout represents the outer diameter of the segment, and L denotes the length of the grouting pipe.
For grouting design, a cement grout fill factor ζ is specified, typically ranging from 10 to 30%. In practice, the grouting volume is usually 10% to 30% of the soil volume within the grouting range. In this project, the maximum fill factor is set to 15%. Consequently, the total grouting volume per ring is calculated by multiplying the soil volume by this fill factor. This total volume is then divided by the number of grouting holes N to determine the grouting volume per hole Vj as follows:
 |
30 |
When applying the circular hole expansion method, it is assumed that the grouting body is cylindrical, which satisfies the circular hole expansion conditions in any plane and expands outward along its axis. The range of expansion induced by grouting pressure at various depths can be determined using Eq. (26). Given a grouting body with length L and radius a, for a differential element dy located at a distance y from the top of the grouting body and a point at a distance x from its center, the displacement in the x-direction under grouting pressure p can be derived from Eq. (26). This study primarily examines the case where the grout expands outward along the grouting hole, particularly when r = a. Therefore, the solution of Eq. (26) at r = a, denoted as ur=a, is utilized to calculate the radial strain 
for any grouting depth.
 |
31 |
Therefore, the expansion distance for any differential element can be determined using Eq. (31). For a differential element dy, its initial volume is 
, and after expansion, its volume becomes 
. Consequently, the volume increment 
of this differential element is given by:
 |
32 |
Integrating with respect to y yields the grouting volume Vjx in the x-direction as follows:
 |
33 |
Due to the nature of compaction grouting, which does not cause splitting, and because the grouting volume exceeds the expansion volume in the y-direction, further injection of grout will result in volume expansion Vjx in the y-direction. Given that the total grout volume Vj = Vjx + Vjy, and assuming uniform expansion of the grout along the y-direction, where each differential element initially has a volume of 
, the volume expansion Vjy in the y-direction, where each differential element experiences the same strain 
, can be expressed as follows:
 |
34 |
Thus, the strain 
in the y-direction can be determined, and the deformation and uplift of the grout body after injection can be derived from the strains 
and 
. In practice, however, due to the control of grouting pressure and in the absence of special geological conditions, the actual grouting volume is often less than the design volume calculated using Eq. (29). Therefore, the grouting volume needs to be controlled further.
Mindlin solution for vertical expansion
Vertical grouting body
Due to the angle of grouting and the small distance between grouting holes, interaction occurs between each pair of grouting holes. Therefore, the vertical and horizontal grouting bodies are calculated separately. In this study, the design grouting volume for the tunnel base is 0.45 m3, the radius of the grouting body’s influence is 0.6 m, and the grouting tube length L is 2 m. At different depths, the strain of the grouting body in different directions, calculated using Eq. (34), should be the designed grouting volume minus the axial expansion volume. However, as the designed grouting volume increases, vertical extrusion becomes more pronounced, making vertical expansion progressively more difficult. This clearly deviates from actual conditions under constant grouting pressure.
Therefore, calculating the vertical expansion of the grouting body is necessary. It is assumed that the grouting process of the tunnel foundation involves two stages: axial expansion of the soil under grouting pressure, followed by vertical expansion when axial expansion reaches the strain limit and can no longer continue. As the soil expands vertically, due to uniform extrusion and expansion between each grouting hole during the foundation grouting process, a single grouting body experiences maximum lateral expansion, followed by vertical compaction. This process continues until the pressure at the lower contact with the soil equals the grouting pressure. Given that the overall soil displacement is small, the Mindlin solution for a semi-infinite body is employed to calculate the vertical displacement.
As discussed in Section "Mindlin solution of elastic body in semi-infinite space", in a semi-infinite elastic space, a vertical concentrated force q applied at the point (0, 0, c) generates horizontal displacement U and vertical displacement W at any point (x, y, z) in that space. For the grouting body, a uniformly distributed pressure p acts at the circular base, as shown in Fig. 5. An arbitrary infinitesimal element is selected from this circular surface, and its position differs from the primary coordinate system. Therefore, cylindrical coordinate transformation is required for parameter evaluation. The area of the infinitesimal element is ρdθdρ, and the load acting on it is − pρdθdρ.
Fig. 5.
Circular area load acting on a semi-infinite body.
By combining Eqs. (27) and (28), the relevant parameters from the original equation are transformed as follows:
 |
35 |
where, ρ represents the radial distance, θ is the azimuthal angle, and z denotes the height. The displacement generated by this infinitesimal element is transformed as follows:
 |
36 |
 |
37 |
From the stress–strain transformation in polar coordinates, the following expression can be derived:
 |
38 |
where, 
. By integrating Eq. (38), the displacement at any point within the soil body due to the uniformly distributed pressure p at a depth c in the semi-infinite body can be obtained as:
 |
39 |
From Eq. (39), the vertical displacement of the soil body caused by the grouting pressure p can be determined, enabling the calculation of the vertical expansion volume Vjy. Assuming that during vertical expansion, the grouting pressure acts on the grouting port, the problem is modeled as a semi-infinite elastic body with a depth H, subjected to a uniform circular pressure p. This uniform pressure induces displacement at the location (x, y, L + H) of the grouting pipe, which has a length L. By applying Eq. (39), the vertical deformation distribution along the r direction can be obtained. Integrating this distribution along the r direction allows the determination of the resulting vertical expansion volume.
Horizontal grouting body
Given that the grouting body is distributed around the circumference of the segment, it is essential to consider the horizontal expansion of the grouting body. In a semi-infinite elastic space subjected to a horizontal concentrated force Ph applied at (0, 0, c), the displacement U along the xxx direction at any point (x, y, z) is described by:
 |
40 |
The application of the Mindlin basic solution requires that the point of action of the concentrated force be located at the z-axis coordinate (0, 0, c), which poses challenges for practical application. Hence, by employing coordinate transformation, the general form of the Mindlin solution at any point in spatial coordinates is derived, as illustrated in Fig. 6.
Fig. 6.
Calculation of vertical strain of horizontal grouting body based on Mindlin solution integral.
Let xyz be the global coordinate system and x′y′z′ be the local coordinate system, with the two systems being parallel. The local coordinate system is offset from the origin of the global coordinate system by distances m and n. The relationship between these two coordinate systems is given as follows:
 |
41 |
In the calculations of this section, since the local coordinate system x′y′z′ only undergoes changes along the y-axis, the parameters are specified as m = 0, n = rcosθ, and c = H − rsinθ. Therefore, in Eq. (40), R1 and R2 are defined as:
 |
42 |
Integrating over the infinitesimal element, the displacement U along the x-direction at point (x, y, z) caused by the horizontal circular stress is given by:
 |
43 |
Segment lift verification
Modeling and material parameters
Building on the earlier analysis of grouting and expansion effects, a two-dimensional model of tunnel foundation grouting has been developed to simulate the lifting displacement of the tunnel and calculate the expansion coefficient. This model integrates practical aspects such as the impact of grouting volume and pressure on tunnel segments, as discussed previously.
The geometric model is based on the Fuzhou Rail Transit Line 4 Project, which serves as a representative case study for this research. As described earlier, the soil primarily consists of soft clay, with sections of muddy soil. The model, detailed in Fig. 7, represents soil dimensions of 60 m × 40 m. Detailed geometric parameters are provided in Table 1, while the material properties of the soil and tunnel segments are listed in Table 2.
Fig. 7.
Two-dimensional model of tunnel foundation lifting grouting.
Table 1.
Geometric parameters of the model.
| Segment outer diameter (m) | Segment thickness (m) | Soft soil thickness (m) | Buried depth of tunnel (m) | Grouting angle (°) |
|---|---|---|---|---|
| 6.2 | 0.35 | 13 | 15 | 22.5 |
Table 2.
Material properties of soil and tunnel segments.
| Material | Density (kg/m3) | Elastic modulus (MPa) | Poisson’s ratio | Cohesion (kPa) | Friction angle (°) |
|---|---|---|---|---|---|
| Muddy soil | 1650 | 5 | 0.42 | 16 | 5 |
| Silty sandy soil | 2000 | 20 | 0.35 | 13 | 29 |
| Tunnel segment | 2500 | 35e3 | 0.2 | - | - |
| Grouting body | 1900 | 20 | 0.32 | - | - |
In practical engineering, the volume of cement injected into each grouting hole is typically less than 0.8 m3, which corresponds to the scenarios analyzed in our grouting expansion models. For consistency with real-world applications, the grouting body in the model is assigned a radius of 0.6 m and an expansion volume of 5%. By comparing simulation results with field data, we validate the theoretical models discussed earlier.
For the model boundaries, roller supports are applied without restricting surface displacement. A triangular mesh with better adaptability to local refinement is utilized, consisting of 2.774 × 104 elements, which provides a reasonable balance between computational speed and accuracy.
It is important to note that COMSOL cannot automatically balance the in-situ stress to achieve initial stress conditions. Poor contact settings can easily lead to non-convergence in calculations. In this two-dimensional model, steady-state calculations were performed in 2 to 4 steps to achieve stress equilibrium. The initial displacement of the model was controlled to be within 10−6, indicating equilibrium. Special attention is required for soft soils (where the elastic modulus E ≤ 10 MPa). Achieving equilibrium with only two steps may be insufficient. Due to the plastic nature of soft soils, non-equilibrium is likely to occur in the first step of the steady-state calculation. Therefore, it is assumed that the soil behaves elastically in the first step, while the calculation continues in the second step. By increasing the number of steps in the equilibrium calculation, the in-situ stress balance can be adequately achieved.
Analysis of verification results
In this study, based on the Fuzhou Rail Transit Line 4 project, vertical displacement data of tunnel segments before and after grouting were collected for an engineering section. The monitoring results for rings 800 to 850 in this section were compared with the calculated results, as shown in Fig. 8.
Fig. 8.
Comparison between monitoring value and simulation value.
Figure 8 shows that in the two-dimensional simulation model, the simulation values obtained by considering the vertical expansion of the grouting body along the segment direction are closer to the maximum vertical displacement of the segment. In contrast, simulation values obtained without considering vertical expansion are closer to the minimum vertical displacement of the tunnel. This discrepancy arises because different sections of the segment experience varying impacts due to different soil distributions. The 95% confidence band represents the uncertainty in the vertical displacement of tunnel segments during grouting. It indicates the range within which 95% of observed values are expected to fall, reflecting the variability in the displacement data and the inherent uncertainty in real-world conditions.
In actual construction, environmental factors may prevent all designed grouting holes from being filled. In this section, approximately 80% of the designed grouting holes were filled. The more grouting holes there are, the more the soil becomes compacted due to the applied pressure, making it more challenging to grout under the same pressure. This effect is reflected in the observed fluctuations in vertical displacement monitoring results. Sections with a higher number of grouting holes exhibit more pronounced effects along the tunnel’s longitudinal direction, leading to greater vertical uplift extremes of the segment. Conversely, sections with fewer grouting holes show smaller uplifts.
Therefore, to better assess the extreme conditions during the grouting process, subsequent chapters will incorporate vertical expansion in the simulation analysis. However, in practical engineering, it is recommended to use the average values of both scenarios for design reference.
Anisotropic expansion coefficient
To facilitate subsequent simulation calculations using the volume strain method, an anisotropic expansion coefficient is proposed based on the aforementioned calculation method of grout body expansion. This coefficient reflects the difference between axial and vertical expansion during the grouting process. Specifically, the axial expansion coefficient represents the total volume strain along the axial direction, while the vertical expansion coefficient corresponds to the total volume strain in the vertical direction.
Using the calculated axial volume strain as a reference, a comparison is made with the isotropic expansion coefficient. When the grouting volume is set at 5%, the resulting displacement effects of basement grouting on the tunnel segments are illustrated in Fig. 9.
Fig. 9.
Segment displacement under different expansion coefficients.
As shown in Fig. 9, the segment displacement obtained using the isotropic expansion coefficient deviates significantly from the actual monitoring results depicted in Fig. 8, with its calculated displacement being much greater than the observed values in real-world grouting. Thus, the simulation of tunnel basement grouting utilizing the anisotropic expansion coefficient provides a more accurate representation of the actual conditions.
Parameter analysis based on numerical simulation
Working condition design
Grouting volumes for tunnel bases at various depths were calculated. For this, grouting pipe outlets were designed at depths of 5 m, 10 m, 15 m, 20 m, and 25 m according to the project’s tunnel depths. Different burial depths correspond to varying confining pressures at the grouting outlet. To validate the results, three representative soil types requiring reinforcement through grouting were selected. The parameters for these soil types are detailed in Table 3. For consistency in comparison, the same confining pressure was applied across all soil types, and vertical stress 
was assumed to be equal to radial stress 
.
Table 3.
Calculation parameters of soil mass.
| Material | Elastic modulus (MPa) | Poisson’s ratio | Cohesion (kPa) | Friction angle (°) |
|---|---|---|---|---|
| Muddy soil | 5 | 0.42 | 16 | 5 |
| Silty clay soil | 12 | 0.33 | 35 | 7 |
| Silty sand soil | 20 | 0.3 | 13 | 29 |
Similarly, the radius of the grouting body is assumed to be 0.6 m, with its depth defined as the distance between the grouting hole location and the top of the soil body.
Expansion calculation in different directions
Following the control method described in Section "Axial expansion calculation based on the circular hole expansion theory", the plastic strain range and the transverse expansion volume Vjx of the soil for the three different soil types were calculated. By integrating these values, the corresponding transverse volumetric strain 
was obtained. The calculation results are presented in Fig. 10.
Fig. 10.
Calculation results of lateral expansion of different soils.
Based on the calculations from Sections "Axial expansion calculation based on the circular hole expansion theory" and "Mindlin solution for vertical expansion", the vertical displacement W at the boundary of the vertical grouting body was determined, with the results presented in Fig. 11. For a horizontal grouting body, considering the confining pressure 
, the axial expansion result should match that shown in Fig. 9. The vertical expansion is now calculated by integrating according to Eq. (39), and the result is displayed in Fig. 12.
Fig. 11.
Vertical displacement of vertical grouting body.
Fig. 12.
Vertical displacement of horizontal grouting body.
Analysis of grouting quantity calculation results
By integrating the results from Figs. 11 and 12, the vertical grouting volume Vjy and the vertical strain 
of the grouting body can be obtained, as shown in Fig. 13.
Fig. 13.
Vertical strain of the grouting body in different directions.
As shown in Fig. 13, under the same confining pressure, the vertical strain of the grouting body exhibits only a minor difference between vertical and horizontal orientations. Specifically, the vertical expansion of the horizontal grouting body is approximately 15% greater than that of the vertical grouting body. Additionally, the vertical strain of the soil, as derived from the Mindlin solution, varies linearly due to the assumption of elasticity in the half-space model. By fitting these data, they can be integrated into the simulation model to control both vertical and lateral expansion of the grouting body. By superimposing the vertical and axial expansions, the overall expansion of the grouting body at various burial depths under a 0.5 MPa grouting pressure can be determined, as illustrated in Fig. 14.
Fig. 14.
Total volumetric strain of soil.
From Fig. 14, it can be seen that when the soil confining pressure is equal (i.e., 
), both vertical and horizontal grouting bodies exhibit larger vertical strains at shallower depths, with horizontal grouting bodies showing greater strains than vertical ones. As the burial depth increases, the difference between the two types of grouting bodies gradually diminishes. Regarding the total volumetric strain of the soil, horizontal grouting bodies display greater overall strain compared to vertical grouting bodies. However, since the total volumetric strain of the soil is less influenced by vertical strain, the difference in strain is less than 1%. In practical engineering, with sufficient burial depth, vertical stress in the soil often exceeds axial stress. Therefore, the effect of vertical strain is minimal, allowing for simplification in the calculation process by substituting horizontal grouting body calculations with vertical grouting body calculations.
From past engineering experience, it is known that under the same grouting pressure, the grouting volume should be limited and decreases as the depth increases. When the grouting pressure approaches the confining pressure, the amount of grout that can be injected becomes smaller. Additionally, the grouting volume varies significantly depending on soil properties. These factors lead to a mismatch between the designed grouting volume and the actual grouting volume. To further validate the reliability of the grouting volume calculation method proposed in this study, a comparison was made between the designed grouting volume, calculated grouting volume, and actual grouting volume, as shown in Fig. 15. The 95% confidence band represents the potential range of grouting volumes, emphasizing the uncertainty between the designed and actual quantities. Derived from field data, it offers a statistical measure that underscores the reliability of the grouting volume estimation within the framework of this study.
Fig. 15.
Comparison of grouting quantity.
Figure 15 highlights a significant discrepancy between the designed grouting volume, calculated grouting volume, and actual grouting volume for a specific section of the tunnel construction. In practical engineering, while the designed grouting volume addresses complex geological conditions, adhering strictly to this volume can lead to substantial material waste. The grouting results obtained using the calculation method proposed in this chapter are more consistent with the actual observations. This suggests that the calculation method used in this study is relatively reliable and offers valuable insights for practical construction.
Conclusion
This study emphasizes the critical importance of accurate grouting volume estimation for tunnel foundations in anisotropic soft soils. By integrating circular hole expansion theory and the Mindlin solution, it presents a framework that addresses the complexities of soil behavior during grouting. Key contributions include:
The development of a simplified approach for calculating grouting quantities, addressing the gap between designed and actual grouting amounts. This method assumes each grouting hole corresponds to a fixed cylindrical reinforcement area, expanding during grouting, providing a theoretical basis for controlling grouting volumes.
The use of circular hole expansion theory and the Mohr–Coulomb small deformation model to calculate axial strain and expansion, along with Mindlin’s solution for vertical strain, which is approximated for efficient calculation.
An analysis of swelling quantities under different confining pressures, revealing significant variations based on soil properties, particularly with higher swelling in muddy soils. The study also finds that vertical expansion for horizontal and vertical grouting is similar at greater depths, simplifying vertical strain calculations.
The introduction of an anisotropic expansion coefficient and the validation of a two-dimensional axisymmetric model, demonstrating that the proposed method provides more accurate estimates of expansion under fixed grouting pressure. This method is more practical for engineering applications, offering reduced material waste and better control over construction costs.
Future research should focus on extending the proposed methodology to dynamic grouting scenarios, incorporating machine learning algorithms to predict grouting behavior based on comprehensive soil parameter databases. This approach could significantly enhance the precision and adaptability of grouting operations in real-time construction settings. A key challenge in implementing this method is the accurate determination of in-situ soil parameters. To address this, future studies could explore the integration of real-time monitoring technologies, such as ground-penetrating radar and borehole sensors, enabling dynamic adjustments to grouting parameters and improving the overall effectiveness of the technique in variable soil conditions.
Acknowledgements
The financial support from the National Natural Science Foundation of China (51308552) and 2022 degrees Guangzhou Metro Design and Research Institute Co. (KY-2022-014) are acknowledged and appreciated. The authors also thank the reviewers and editors who helped to improve the quality of this paper.
Author contributions
Conceptualization, B. D. and P. L.; methodology, C. H.; validation, W. L; formal analysis, B. D. and Y. W.; writing—original draft preparation, B. D.; writing—review and editing, Y. W. All authors have read and agreed to the published version of the manuscript.
Data availability
The datasets generated during and/or analyzed during the current study are available from the corresponding author upon reasonable request.
Declarations
Competing interests
The authors declare no competing financial interests or personal relationships that could have influenced the work.
Footnotes
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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Associated Data
This section collects any data citations, data availability statements, or supplementary materials included in this article.
Data Availability Statement
The datasets generated during and/or analyzed during the current study are available from the corresponding author upon reasonable request.