Numerical Studies for the Application of the Methodology for Volume Loss of Cohesionless (Loose) Soils (V_L,LSR) and the Additional Settlement (S_max) During Shield Tunneling

Abstract

This paper presents results of numerical modeling of tunneling using mechanized tunnel boring machines (TBMs) based on a methodology for determining the volume loss cohesionless (loose) soils, denoted as V_L,LSR, for shallow tunnels in dispersive soils to estimate surface and foundation on settlement natural ground. Existing methods for estimating ground surface and structural settlements have significant drawbacks, caused by several factors, including the complexity of determining volume loss using the proposed methodologies, a limited number of empirical parameters describing the technological features of TBM operations, the absence of methods in Russian regulatory documentation for determining volume loss in tunnels with diameters of 6 m or more, among other issues. The study aims to validate a previously developed method for estimating V_L,LSR and an empirical equation for predicting surface settlements, S_max, to assess additional settlements induced by tunneling. The proposed volume loss methodology and the modified S_max expression from Peck R.B. (1969), derived from monitoring data, are used in empirical calculations and numerical modeling of surface and building settlements during TBM tunneling. Validation results include back-analysis of geotechnical “tunnel–ground–structure” interaction models, comparisons of additional settlements from design calculations and field monitoring data, as well as comparisons with existing empirical relationships and relevant regulatory documents, followed by recommendations for their integrated application. The validated methods demonstrate good agreement with observed monitoring data, while providing sufficient engineering safety margins, confirming the applicability of the V_L,LSR and the modified S_max expression by Peck R.B. (1969) for predicting settlements of tunneling and identifying directions for further research.

1. Introduction

Tunnel boring operations are highly complex construction activities inherently associated with the risk of emergency situations, which can sometimes lead to serious or even catastrophic consequences. Numerous cases of such emergencies are documented in the work of Sousa R.L. and Einstein H.H. (2021) [1], as well as in studies [2,3,4]. These works collectively emphasize the critical importance of carefully selecting, monitoring, and managing the technological processes involving tunnel boring machines (TBMs) to mitigate such risks.

Modern tunnel boring machines (TBMs) have been extensively used for many years in urban tunnel construction. As complex technological systems, TBMs incorporate numerous automated processes for underground excavation, each requiring temporary support of the soil mass and the tunnel face. This construction method employs temporary movable supports, enabling simultaneous soil excavation and precise installation of permanent tunnel lining segments. TBMs differ in operating principles and dimensions, as well as in their methods of excavation and face support.

TBMs may be categorized into mechanized, semi-mechanized, and manual types, depending on the method of excavated soil transport. They are automatically guided along the designed tunnel alignment using system guidance and computer control, but may also be manually adjusted through mine surveying techniques. After the tunnel lining segments are installed, and as the TBM shield slides off the lining behind the brushes, a high-pressure grout (plugging solution) is automatically injected. During this stage, additional soil deformation occurs around the tunnel, as described by Mazein S.V. (2011) [5], often resulting in further surface settlement. The TBM is propelled forward by hydraulic jacks acting against the installed segments.

When constructing tunnels in Moscow, several types of tunnel boring machines (TBMs) are predominantly used: earth pressure balance, slurry pressure balance, and clay–earth pressure balance TBMs. Regardless of the technological advancement of the TBM equipment, soil mass excavation induces changes in the stress–strain state (SSS), leading to additional deformations caused by the technological process itself. Furthermore, there are several factors that can adversely affect ground surface settlement and the additional settlement of structures:

• Excavation of excess soil volume beyond the capacity of one ring, as indicated in the research by Mazein S.V. (2009) [6];
• Dispersion and mixing of the injected mortar, as shown in [7], and the remaining bentonite solution, according to Nagel F. and Meschke G. (2011) [8];
• Shrinkage during grouting mortar hardening, as noted in [9,10,11], and rheological processes in soil, described in [12].

In his study, ref. [7] identified three possible flow scenarios of bentonite and cement mortar during tunnel boring along the TBM shield:

• Bentonite flows from the face to the tail of the TBM, pushing the injected cement mortar in the area where the last segment of the lining is installed;
• The injected cement mortar moves from the tail to the face, displacing the bentonite;
• Bentonite flows from the face to the tail of the TBM, while the injected cement mortar moves in the opposite direction.

These processes were also simulated in [8]. For shallow tunnels, the pressure of the injected cement mortar and bentonite is sometimes limited due to the risk of breakthroughs to the surface. Bezuijen A. (2007) [7] suggests that both fluids behave as Bingham fluids, meaning their flow is governed by a yield stress, and the pressure exerted by the fluids decreases along the TBM shield.

Attewell P.B. et al. (1986) [13] noted that the radial displacement of the TBM shield additionally influences settlement, together with possible subsidence of the TBM weight itself, due to the formation of a rock pressure column in the absence of vaulting.

Many researchers have studied, and continue to study, the influence of technological factors during tunnel boring operations that lead to additional soil deformations, resulting in surface subsidence and further settlement of structures within the affected area. Numerical methods are now regarded as the most advanced approaches for determining an additional settlement during tunneling. These methods are summarized in [14], most of which focus on geotechnical modeling of the additional settlement induced by tunnel excavation through the application of the volume loss, V_L, i.e., the relative compression (shrinkage) of the simulated tunnel cavity. The V_L parameter, is empirical and, according to the authors, is most comprehensively examined in [15,16,17,18,19,20]; and in an early study [21].

However, many well-known and widely cited tunneling researchers have continued their studies primarily in the field of assessing the influence of structural dimensions and other parameters of protective walls on reducing building deformations induced by tunneling [22,23,24,25,26,27], or in applied research focused on foundation settlement analysis [28,29,30,31,32,33,34]. Others have explored different aspects of tunneling processes, but not the volume loss itself [35].

The new methodology proposed in [21] for determining the volume loss of cohesionless (non-cohesive, loose) soils, denoted as V_L,LSR, was tested against known monitoring data and volume loss calculations performed by the sequential iteration reverse calculation method [36] for a project located at Moscow, Ryazansky ave., 4A, bldg 2. To facilitate practical application of this methodology for volume loss in cohesionless soils V_L,LSR during tunnel design and construction using TBM methods, the authors highlight the necessity of further validation at several sites with available field monitoring data of subsidence.

The necessity for developing improved methods to determine settlements induced by tunneling arose from several factors, namely the complexity of existing analytical methods; insufficient validation of empirical parameters related to the technological features of TBMs in current approaches; the existence of methodologies that, according to the authors, are more region-specific; the need to develop methods considering a broader range of soil characteristics identified during site investigations; the availability and detailed analysis of extensive tunneling monitoring data in Moscow in recent years; and the absence of provisions in the regulatory documentation of the Russian Federation for determining the soil volume loss for tunneling for tunnels with diameters of 6 m and larger. The effectiveness of the V_L,LSR methodology and the modified S_max expression based on Peck R.B. (1969) [37] was confirmed through analysis of monitoring results for buildings and structures on natural foundations.

In study [38], a methodology for determining the V_L was developed primarily for dense clays. The paper indicates that its application to sandy soils, highly plastic clays, or strongly water-saturated soils requires further validation. In addition, the compression and face pressure (in cutterhead) coefficients are often taken as averaged values, which may not adequately represent the complex tunneling conditions, particularly in heterogeneous or stratified soils. For the proper application of the formula, monitoring data (such as face pressure, deformations, and grouting volumes) are required; without these, the calculations remain preliminary.

In another study [39], a method for determining the V_L was developed to provide a more accurate analytical solution for volume loss. However, similar to the approach presented in [38], it requires operational shield parameters. These data are obtained from tunneling monitoring and include excavated volume per cycle V_excavated (m³/cycle), grouting volume V_grout (m³/m of advance), face pressure P_face (kPa), grouting pressure V_g (kPa), pressure gradient i (dimensionless), as well as the soil compressibility coefficient C (m²/kN), defined as C = (1 + ν)(1 − 2ν)/E, and the misalignment factor (0.8–1.2).

This paper aims to validate a previously developed methodology for calculating volume loss in cohesionless (loose) soils V_L,LSR through geotechnical modeling using numerical methods to predict the maximum surface and structural settlements induced by tunneling in dispersive soils. This is achieved by comparison of modeled additional settlements with field monitoring data obtained from various projects.

2. Materials and Methods

This study describes the basic technological operation of the TBM, its structural features, and additional factors that may adversely affect ground surface settlements and structural settlements. The research presents design values of the soil volume loss V_L and monitoring data for structures in different sections during tunneling works of the Moscow Metro and related engineering networks. The tunneling operations considered were performed by TBMs with external diameters of ∅4.1 m, ∅6.2 m, and ∅10.5 m to ensure the maximum objectivity of the results. The works were carried out in sands and loams from Quaternary deposits of the Moscow region. Geotechnical modeling of the SSS and assessment of tunneling conditions were performed using both two-dimensional (2D) and three-dimensional (3D) finite element analyses (FEM) implemented in the PLAXIS software (version 23).

This study builds upon a previously developed methodology for calculating volume loss in cohesionless (loose) soils, denoted as V_L,LSR [21], where the definition of the technological volume loss for the TBM (Figure 1) [39] was first proposed in [40] as defined in Annex J of SP 249.1325800.2016 [41]:

$$V_{L-T}=K_L\left(1-{\displaystyle \frac{A_1}{A_2}}\right)\%$$

(1)

where K_L is the degree of soil filling of the annular gap between the constructed tunnel and the contour of the formed excavation, taken as the ratio of the difference between the shield skirt area (outer diameter) and the tunnel lining area to the difference between the TBM cutting area and the tunnel lining area during automatic and continuous filling of the annular space;

A₁ is the cross-sectional area of the constructed tunnel, m²;

A₂ is the cross-sectional area of the closed excavation, m².

$$A_1={\displaystyle \frac{\pi D_1^2}{4}}$$

(2)

$$A_2={\displaystyle \frac{\pi D_2^2}{4}}$$

(3)

where D₁ is the outer diameter of the lining, m;

D₂ is the diameter of the TBM cutting tool (rotor—cutterhead), m.

$$K_L=1-\left({\displaystyle \frac{A_3-A_1}{A_2-A_1}}\right)$$

(4)

where A₃ is the cross-sectional area of the TBM shell (skirt), m.

$$A_3={\displaystyle \frac{\pi D_3^2}{4}}$$

(5)

where D₃ is the diameter of the TBM shell (skirt), m.

Considering the provisions of the hypothesis presented in the study [40], where the soil component was characterized by the rock strength coefficient (f) according to Protodyakonov, M.M. [42], based on the degree of labor intensity in rock excavation (rocky soils), as well as the generality and limitations of the coefficient (f) values for a number of dispersed soils, the authors proposed to replace the coefficient (f) with a new parameterized approach for describing the geomechanical conditions of soil mass excavation. This new approach is designed to incorporate conventional physical and mechanical characteristics of ground conditions by tunneling.

Based on an analysis of the method for determining the vault formation mechanism [42], the authors focused on the geomechanical classifications: the Rock Mass Rating (RMR) proposed by Bieniawski Z.T. [43], which aggregates the quality indicators of rock masses, and the Q-classification (or NGI) developed by Barton N. et al. [44] at the Norwegian Geotechnical Institute. As a result, a calculation methodology was developed for determining the LSR (Loose Soil Ratio) coefficient for cohesionless (loose) soils as a function of their physical and mechanical parameters. This approach was based on identifying and analyzing the most influential factors affecting ground settlement during tunneling operations, where the principal strength criterion is the Mohr–Coulomb theory [45,46], expressed by the following expression:

$$LSR=R_C+R_{\phi }+R_v+R_{E_0}+R_e+R_{I_L}$$

(6)

where C is the cohesion of the soil, kPa; φ is the angle of internal friction of the soil, degrees (°); υ is Poisson’s ratio; E₀ is Young’s modulus, MPa; e is the coefficient of porosity of the soil; I_L is the number (indicator) of fluidity of the soil.

R_C, R_φ, R_υ, R_E0, R_e, R_IL are coefficients (ratios) of the physical and mechanical characteristics of the soil, determined in accordance with the physical and mechanical parameters of the soil by linear interpolation of nearby values between the columns. If the values of the physical and mechanical characteristics of the soil exceed or fall below the limit values of the table, the maximum or minimum specified value is taken. Data for the calculation coefficients (ratios) are presented in

Table 1. Loose soil ratio (LSR) coefficient calculation table.

Ratio	Parameter of Soil	Intervals of Values of the Characteristics of Cohesionless (Loose) Soils
RC	C, kPa	≥60	50	40	30	20	10	0
		0.32	0.25	0.20	0.15	0.10	0.05	0
Rφ	φ, (°)	≥35	30	25	20	15	10	0
		0.15	0.10	0.08	0.05	0.03	0.02	0
Re	υ	≤0.25	0.27	0.3	0.35	0.4	≥0.45
		0.15	0.10	0.07	0.04	0.02	0
RE0	E0, MPa	≥50.0	35.0	25.0	20.0	15.0	≤10.0
		0.08	0.06	0.04	0.02	0.01	0
Re	e	≤0.4	0.5	0.55	0.6	0.7	0.75	≥0.8
		0.15	0.12	0.09	0.07	0.04	0.03	0
RIL	IL	≤0	0.25	0.5	0.75	≥1
		0.15	0.10	0.05	0.01	0

Note: The top line is the physical and mechanical characteristics of the soil; the bottom line is the value of the R parameter corresponding to this soil characteristic.

.

As a result, the volume loss for cohesionless (loose) soils (V_L,LSR) is determined by the following expression:

$$V_{L,LSR}={\displaystyle \frac{V_{L-T}}{2LSR}}={\displaystyle \frac{\left(1-\frac{\frac{\pi D_3^2}{4}-\frac{\pi D_1^2}{4}}{\frac{\pi D_2^2}{4}-\frac{\pi D_1^2}{4}}\right)\left(1-\frac{\frac{\pi D_1^2}{4}}{\frac{\pi D_2^2}{4}}\right)\%}{2LSR}}$$

(7)

To validate the methodology for calculating volume loss in cohesionless (loose) soils, V_L,LSR, several tunnel boring sites were analyzed. The soil at the tunnel face predominantly comprising water-saturated sands and loams from the Quaternary deposits of the Moscow region was modeled using the Mohr–Coulomb failure criterion model, as concluded in [47], as this model is the most suitable for tunneling simulation.

The lower boundary of the calculation models was established following the recommendations of previous researchers [21,36,47,48,49,50] and Annex E of SP 249.1325800.2016 [41], and approximately set at 0.5 times the tunnel diameter (0.5 Ds). Since the adopted Mohr–Coulomb model does not capture the secondary phase of soil deformation, the predicted settlement values may exhibit considerable deviation from the actual behavior.

The horizontal boundaries of the models were set based on the following conditions: >three times the tunnel depth for sands, and >six times the tunnel depth for clay (loam) soils from the surface to the tunnel invert level according to Annex E of SP 249.1325800.2016 [41].

The model boundary conditions were set as follows:

• Lateral boundaries are fixed against lateral displacements but free in the vertical direction;
• The bottom boundary is fully fixed in all directions.

The calculation stages in the geotechnical software sequentially include the following:

• First step: Creation of geometry and assignment of necessary loads, influences, stiffnesses, and boundary conditions;
• Second step: Specification of the initial SSS of the soil massive with generation of pore water pressure;
• Third step: Resetting the deformations of the previous step while retaining all stresses and continuing activation of existing structures under load conditions;
• Fourth step: Resetting the deformations of the previous step while retaining all stresses and continuing activation of the tunnel excavation with lining stiffness and at a given soil volume loss, which is modeled by structure shrinkage (axial or volumetric deformation).

After each calculation phase (second to fourth steps), deformation resetting of the system is performed while retaining stresses.

The stiffness of the reinforced concrete tunnel lining structures was determined based on the corresponding concrete strength class and on the number of segments per ring according to Muir Wood A.M. (1975) [51].

Mesh sensitivity was assessed by analyzing iterative results to ensure that the final results were independent of the finite element size.

The study includes a direct validation method that involves checking the validity and efficacy of the methodology for calculating the volume loss cohesionless (loose) soils, V_L,LSR, and a modified maximum settlement equation, S_max, based on Peck R.B. (1969) [37], at surface and subsurface (foundation) levels. This validation is performed by comparing the calculated additional settlement of structures with design values and the monitoring results presented later in this work. Additionally, a comparative analysis with selected regulatory recommendations is carried out. Finally, the paper presents a discussion and conclusions concerning the validation of the proposed methodologies against the monitoring of data.

3. Results

Technological parameters, excavation conditions, and geotechnical (geodetic) monitoring data results were supplied by Mosinzhproekt JSC.

3.1. Validation Example 1

Tunneling was carried out as part of constructing the connecting branch tunnels at the Nizhegorodskoe depot on the Kozhukhovskaya line (KZL) of the Moscow Metro.

Building and foundation structures:

• Smirnovskaya str.; 10 bldg; seven prefabricated reinforced concrete frame with columns and beams on column foundations;
• Smirnovskaya str.; two bldg; 13 brick walls on rubble-stone strip foundations.

The tunneling soil conditions consist primarily of sands, as shown in Figure 2, whose characteristics are detailed in

Table 2. Physical and mechanical characteristics of soils for validation example 1.

Soil	Cohesion c, kPa	Internal Friction Angle φ, Deg (°)	Poisson’s Ratio ν	Young’s Modulus E0, MPa	Porosity Coefficient e	Number of Fluidity IL
EGE-7 Sand	0	31	0.3	25.0	0.67	-
EGE-7a Sand	0	27	0.3	19.0	0.72	-

. The validation results for additional settlements obtained from the design solution (a, c) and from the V_L,LSR methodology (b, d) are shown in Figure 3.

3.2. Validation Example 2

Tunnel boring was performed as part of the Big Circle Line (BCL) of the Moscow Metro beneath the open sections of the Arbatsko–Pokrovskaya (APL) and Filevskaya (FL) lines.

Foundation of column structures:

• Lobby—slab foundation;
• Platform—column foundation;
• Tracks—rail trestle lattice on a rigid concrete base.

The tunneling soil conditions consist primarily of sands, as shown in Figure 4, with their characteristics detailed in

Table 3. Physical and mechanical characteristics of soils for validation example 2.

Soil	Cohesion c, kPa	Internal Friction Angle φ, Deg (°)	Poisson’s Ratio ν	Young’s Modulus E0, MPa	Porosity Coefficient e	Number of Fluidity IL
EGE-2/16 Sand	4	34	0.3	29.0	0.55	-
EGE-5/16 Sand	3	35	0.3	37.0	0.54	-
EGE-8/16 Sand	2	35	0.3	45.0	0.52	-

. The layout of geodetic deformation markers in the cross-section is shown in Figure 5. The validation results of additional settlements obtained from the design solution (a, c, e) and using the V_L,LSR methodology (b, d, f) are shown in Figure 6.

3.3. Validation Example 3

Tunneling was performed as part of the removal and reorganization of the heating system beneath the Akademicheskaya station of the Big Circle Line (BCL) and the Akademicheskaya station of the Kaluzhsko–Rizhskaya Line (KRL).

The tunnel structure consists of prefabricated reinforced concrete wall panels, a monolithic reinforced concrete channel, and ribbed reinforced concrete roof panels.

The tunneling soil conditions consist of loam and sand, as shown in Figure 7, with their characteristics summarized in

Table 4. Physical and mechanical characteristics of soils for validation example 3.

Soil	Cohesion c, kPa	Internal Friction Angle φ, Deg (°)	Poisson’s Ratio ν	Young’s Modulus E0, MPa	Porosity Coefficient e	Number of Fluidity IL
EGE-112 Loam	36	22	0.352	29.0	0.47	0.1
EGE-134 Sand	2	35	0.3	45.0	0.52	-

. The layout of geodetic deformation markers in the cross-section is shown in Figure 8. The validation results of additional settlements obtained from the design solution (a) and using the V_L,LSR methodology (b) are shown in Figure 9.

3.4. Summary Validation Results

The data for the analyzed structures, tunneling conditions (technological and excavation), soil volume loss coefficients, additional settlements, and monitoring results for various structures have been compared and are presented below (

Table 5. Summary table of validation results for different objects.

Ex. No.	Address of the Structure/Name of the Object	Distance to Foundations or Structure, m	Tunnel Diameter, m, and Type of TBM	Ultimate Additional Settlement Smax, mm	Design Volume Loss VL, %	Validation Settlement Smax, mm	Methodology VL,LSR, %	Validation Settlement Smax, mm	Monitoring Data
1	Tunneling under buildings at str. Smirnovskaya
1.1	Smirnovskaya str., 10 bldg. 7	~15.8	∅6.2 EBP	30	1.75	28.8	1.19 2 1.51	21.0	9.1 4
1.2	Smirnovskaya str., two bldg. 13	~10.0		30		45.2 13.3 1	1.51	29.3	10.8 4
2	Tunneling under buildings at the Kuntsevskaya station
2.1	Western lobby Kuntsevskaya St. (APL/FL)	~5.0	∅10.5 EBP	31	1.3	28.0	0.75	15.3	9.3 / 12.7
2.2	Platform Kuntsevskaya St.	~6.8		31		8.0		4.1	3.2
2.3	Tracks 1 and 2 APL and track 1 FL Kuntsevskaya St.	~8.5		11		41.0		22.8	16.4
3	Tunneling under buildings at the Akademicheskaya station
3.1	Akademicheskaya St.	~4.0	∅4.1 EBP	30	1.5	8.7	1.16% 2.21% 3	3.8	3.6

Note: ¹ settlement when using horizontal compensatory injection; ² combined coefficient of soil overburden according to the V_L,LSR method for different soils (numerator—sand; denominator—loam); ³ combined coefficient of soil overburden according to the V_L,LSR method for different soils (numerator—loam; denominator—sand); ⁴ maximum additional settlements are given taking into account the construction of two tunnels.

).

3.5. Validation of Modified Empirical Settlement Expression

For the comparative numerical modeling results presented in this article, as well as other related results, calculations were performed using the new modification of the maximum settlement expression S_max proposed by Peck R.B. (1969) [37] and the proposed expression for determining the distance to the inflection point of the settlement curve i_x, introduced earlier in [21]:

$$i_x=\left(1-\sin \phi \right)\left(1-v^2\right)\left(z_0-z\right)+R$$

(8)

$${\sigma }_P=\gamma \left(z_0-z\right)+{\sigma }_S·e^{-\left({\displaystyle \frac{x^2}{2i_x^2}}\right)}$$

(9)

$$S_{max}=e^{-\left({\displaystyle \frac{\tau }{{\sigma }_V}}\right)}{e}^{\left({\displaystyle \frac{{\sigma }_S^2}{{\sigma }_P^2}}\right)}\sqrt{{\displaystyle \frac{\pi }{2}}}{\displaystyle \frac{V_L{D}^2}{4i_x}}{e}^{-\left({\displaystyle \frac{x^2}{2i_x^2}}\right)}=e^{-\left(\tan \phi +{\displaystyle \frac{C}{\gamma z_0}}\right)}{e}^{\left({\displaystyle \frac{{\sigma }_S^2}{{\sigma }_P^2}}\right)}\sqrt{{\displaystyle \frac{\pi }{2}}}{\displaystyle \frac{V_L{D}^2}{4i_x}}{e}^{-\left({\displaystyle \frac{x^2}{2i_x^2}}\right)}$$

(10)

where z₀ is the depth of the tunnel axis (closed excavation), m; z is the distance from the surface to the “subsurface” (foundation); γ is the weighted average value of the specific gravity of the soil above the tunnel, kN/m³; τ is maximum tangential stresses in the ground at the tunnel axis level, kPa; σ_V is the natural vertical stress at the tunnel axis level, kPa; σ_P is the total stress at the tunnel axis level, taking into account the additional load, kPa; σ_S is the additional stress (load) on the surface or below, kPa.

The basis of the modified S_max precipitation formula is the integrated Formula (4) in [21] from [37] for undrained clays with the introduction of two additional factors:

• $e^{-\left(\tau /{\sigma }_V\right)}=e^{-\left(\tan \phi +C/\gamma z_0\right)}$ is the Mohr–Coulomb strength criterion [45,46] expressed through a negative exponential, which is consistent with the general theory of arching in non-rocky ground;

• $e^{\left({\sigma }_S^2/{\sigma }_P^2\right)}$ is an expression that accounts for relative additional stresses at the ground surface level or at the foundation depth, considering the that the applied load may be located at some distance from the tunnel axis.

This formulation incorporates soil strength characteristics and load distribution effects into the prediction of maximum settlement during tunneling.

Additionally, soil volume loss, V_L, calculated using the method proposed by the Loganathan N. (2011) [19], is presented for comparison. The results of calculating the volume loss, V_L, coefficients for the above-mentioned objects, and several other cases are listed in

Table 6. Summary table of volume loss V_L for different objects, taking into account the characteristics of soil tunneling.

No.	Name or Address of the Facility	Tunnel Diameter, m	VL Calculation Method, %
			Loganathan N. (2011) [19]	VL,LSR [21]
1	Western lobby Kuntsevskaya St. (APL/FL)	∅10.5	1.15	0.75
2	Platform Kuntsevskaya St.
3	Tracks 1 and 2 APL and track 1 FL Kuntsevskaya St.
4	Akademicheskaya St.	∅4.3	0.44	1.16
5	ZML 1 tunnels from PC 115 + 53.140 to PC 117 + 57.600	∅6.0	0.63	1.22
6	Smirnovskaya str., 10 bldg. 24		1.36	1.51
7	Smirnovskaya str., 10 bldg. 7		1.38
8	Smirnovskaya str., two bldg. 13		1.18
9	1-ya Novokuzminskaya str., 1	∅10.5	1.03	0.78
10	1-ya Novokuzminskaya str., 3		1.15	0.82
11	Andropova ave., 39 bldg. 46		1.31	2.17
12	Andropova ave., 39 bldg. 63		1.04	2.08
13	Andropova ave., 39 bldg. 82		1.23	2.03
14	Shtatnaya str., 19			2.17
15	Shtatnaya str., 19A
16	Bolshaya str., 128		1.07	1.58
17	2nd Silikatny proezd, 7 bldg. 2, str. 4	∅6.0	0.89	1.88
18	2nd Silikatny prospekt, 9 bldg. 11		1.14	1.39
19	Shenogina str., two bldg. 33		1.07	2.53
20	Ryazansky prospekt, 22	∅10.5	1.15	1.20
21	Ryazansky prospekt, two bldg. 27 [36]		1.13
22	Ryazansky prospekt, 4A bldg. 2 [36]		1.13
23	MCAR 2 8 km, vl. 3 bldg. 2. BC “Drive” [49]		1.36	0.8
24	Ferganskaya str., 25 bldg. 2 [49]		1.41
25	Samarkand boulevard, 134 bldg. 5 [49]		1.39
26	Ferganskaya str., 23 [49]		1.32

Note: ¹ Zamoskvoreckaya Line Moscow metro; ² Moscow Circle Automobile Road.

and in Figure 10.

In the absence of field data on undrained shear strength (c_u) for calculating V_L according to [19], which is not a mandatory characteristic in geological surveys in the Russian Federation, this parameter was determined using the expression by Skempton A.W. and Henkel D.J. [52]:

$${\displaystyle \frac{c_u}{{\sigma }_1^{\prime }}}=0.11+0.37{\displaystyle \frac{I_P}{100}}$$

(11)

where ${\sigma }_1^{\prime }$ is the effective vertical stress, i.e., the total vertical stress minus the pore pressure, ${\sigma }_1^{\prime }=({\sigma }_1-u)$.

The main parameters for calculating additional settlements S_max for various structures are presented in

Table 7. Summary table of calculation parameters additional settlements S_max.

No.	Name or Address of the Facility	Depth from the Surface to the Center of the Tunnel	Depth from the Surface to the Bottom of the Foundation	Diameter of the Tunnel Along the Outer Edge of the Lining	Distance from the Edge of the Foundation to the Tunnel Axis	Pressure Under the Base of the Building Foundation	Cohesion	Internal Friction Angle	Poisson’s Ratio	Young’s Modulus	Porosity Coefficient	Number of Fluidity
		z0 [m]	z [m]	D [m]	L [m]	P [kPa]	c [kPa]	˚ [deg]	υ [–]	E0 [MPa]	e [–]	IL [–]
1	Table 6 and Table 8. Position 1	13.9	3.7	10.5	0	66	4	34	0.3	29	0.55	0
2	Table 6 and Table 8. Position 2	13.9	1.7	10.5	18	130	4	34	0.3	29	0.55	0
3	Table 6 and Table 8. Position 3	13.9	0	10.5	0	101.7	4	34	0.3	29	0.55	0
4	Table 6 and Table 8. Position 4	14.1	7.9	4.3	0	130.8	36	22	0.352	29	0.47	0.10
5	Table 6 and Table 8. Position 5	16.8	8.3	6.0	0	135.3	50	25	0.35	28.7	0.47	0.17
6	Table 6 and Table 8. Position 6	19.9	3.5	6.0	0	390	0	27	0.3	19	0.72	0
7	Table 6 and Table 8. Position 7	21.1	2	6.0	0	250	0	27	0.3	19	0.72	0
8	Table 6 and Table 8. Position 8	14.5	1.3	6.0	0	155	0	27	0.3	19	0.72	0
9	Table 6 and Table 8. Position 9	19.2	1.6	10.5	0	60	3	38	0.29	32	0.65	0
10	Table 6 and Table 8. Position 10	24.2	1.6	10.5	3.4	72	3	35	0.31	29	0.63	0
11	Table 6 and Table 8. Position 11	27.2	1.5	10.5	1.5	27	5	31	0.29	26.2	0.71	0
12	Table 6 and Table 8. Position 12	21.0	2.3	10.5	7	36	1	33	0.31	31.8	0.65	0
13	Table 6 and Table 8. Position 13	22.6	1.8	10.5	0.5	55	2	34	0.31	25.6	0.63	0
14	Table 6 and Table 8. Position 14	23.2	2.4	10.5	0	36	5	31	0.29	26.2	0.71	0
15	Table 6 and Table 8. Position 15	23.2	1.8	10.5	30.9	27	5	31	0.29	26.2	0.71	0
16	Table 6 and Table 8. Position 16	18.5	1.7	10.5	22.2	27	30	20	0.35	20.5	0.63	0.33
17	Table 6 and Table 8. Position 17	23.4	1.9	6.0	6.6	281.3	30	23	0.36	21	0.60	0.35
18	Table 6 and Table 8. Position 18	25.9	1.9	6.0	23.4	281.3	54	20	0.32	22	0.73	0.09
19	Table 6 and Table 8. Position 19	24.9	1.8	6.0	17.4	187.5	2	32	0.31	28	0.58	0
20	Table 6 and Table 8. Position 20	30.2	2.0	10.5	9.6	24	4	32	0.31	26	0.62	0
21	Table 6 and Table 8. Position 21	27.3	1.9	10.5	0	24	4	32	0.31	26	0.62	0
22	Table 6 and Table 8. Position 22	26.7	2.2	10.5	0	236.3	4	32	0.31	26	0.62	0
23	Table 6 and Table 8. Position 23	25.3	3.4	10.5	0	75	4	35	0.29	23.8	0.63	0
24	Table 6 and Table 8. Position 24	27.3	2.35	10.5	34.8	225	4	35	0.29	23.8	0.63	0
25	Table 6 and Table 8. Position 25	26.4	3.3	10.5	25.4	300	4	35	0.29	23.8	0.63	0
26	Table 6 and Table 8. Position 26	22.2	2.5	10.5	0	45	4	35	0.29	23.8	0.63	0

Note: physical and mechanical characteristics of soils are indicated in the in top (vault) of the TBM.

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The results of the calculations of additional settlements S_max using various methods and monitoring data are presented in

Table 8. Summary table of additional settlements S_max for different objects.

No.	Name or Address of the Facility	Peck R.B. (1969) [37]	Peck R.B. (1969) [37] with VL по [19]	O’Reilly M.P. and New B.N. (1982) [53]	Lee C.J. et al. (1999) [54]	Tupikov M.M. (2010) [55]	Chakeri H. and Ünver B. (2014) [56]	Wang F. et al. (2016) [57]	Smax by VL,LSR (2025) [21]	Monitoring Data
1	Western lobby Kuntsevskaya St. (APL/FL)	26.0	39.9	36.7	26.6	21.3	19.4	37.3	14.8	9.3/ 12.7
2	Platform Kuntsevskaya St.	5.1	7.8	36.7 1	26.6 1	7.3	22.9 1	5.9	3.4	3.2
3	Tracks 1 and 2 APL and track 1 FL Kuntsevskaya St.	26.0	39.9	36.7	26.6	21.3	21.4	24.5	13.0	16.4
4	Akademicheskaya St.	12.8	4.8	17.5	10.0	13.6	10.7	90.5	9.3	3.6
5	ZML tunnels from PC 115 + 53.140 to PC 117 + 57.600	17.0	8.8	29.9	16.0	12.8	12.9	71.6	12.4	5.6
6	Smirnovskaya str., 10 bldg. 24	16.4	14.8	17.6	18.0	18.0	28.7	30.2	12.2	10.8 2
7	Smirnovskaya str., 10 bldg. 7	16.4	15.1	16.8	17.4	21.0	23.2	37.7	9.5	9.1 2
8	Smirnovskaya str., two bldg. 13	20.2	15.9	23.5	21.7	12.5	24.0	55.3	12.4	1.8 2
9	1-ya Novokuzminskaya str., 1	23.3	30.9	28.8	22.9	14.5	14.4	21.0	10.9	14.7
10	1-ya Novokuzminskaya str., 3	19.0	26.8	24.5 1	21.0 1	11.7	15.7 1	16.7	10.0	12.3
11	Andropova ave., 39 bldg. 46	47.5	28.5	58.8 1	52.3 1	31.9	17.8 1	39.5	24.4	17.2
12	Andropova ave., 39 bldg. 63	41.4	20.6	71.2 1	58.2 1	26.3	15.0 1	48.2	25.2	9.4
13	Andropova ave., 39 bldg. 82	44.5	27.0	65.0 1	54.4 1	26.4	17.8 1	46.3	26.6	6.4
14	Shtatnaya str., 19	47.7	27.1	68.0 1	57.4 1	28.4	18.7 1	50.2	28.3	12.5
15	Shtatnaya str., 19A	7.0	4.0	66.7 1	56.7 1	14.5	18.3 1	10.0	3.3	3.3
16	Bolshaya str., 128	1.3	0.9	107.4 1	47.4 1	6.9	25.4 1	11.8	7.8	5.8
17	2nd Silikatny proezd, 7 bldg. 2, str. 4	18.8	8.9	33.0 1	20.4 1	35.2	19.5 1	38.7	9.7	7.4
18	2nd Silikatny prospekt, 9 bldg. 11	1.6	1.3	20.2 1	13.5 1	2.4	19.1 1	13.4	2.9	1.2
19	Shenogina str., two bldg. 33	8.8	3.7	20.3 1	23.7 1	17.5	11.1 1	31.1	5.7	1.7
20	Ryazansky prospekt, 22	21.4	20.6	31.0 1	28.0 1	17.1	16.3 1	19.3	11.2	13.4
21	Ryazansky prospekt, two bldg. 27 [22]	26.7	25.3	32.3	28.7	18.0	16.5	22.2	13.7	3.0
22	Ryazansky prospekt, 4A bldg. 2 [22]	27.0	24.5	33.6	29.1	17.5	19.0	23.2	14.6	13.0
23	MCAR 8 km, vl. 3 bldg. 2. BC “Drive” [49]	19.0	32.3	23.1	20.0	11.8	18.8	17.7	10.1	6.7
24	Ferganskaya str., 25 bldg. 2 [49]	1.4	2.5	21.5 1	19.2 1	5.1	23.0 1	3.4	0.8	1.8
25	Samarkand boulevard, 134 bldg. 5 [49]	4.1	7.1	22.2 1	19.5 1	7.3	25.5 1	6.8	2.8	1.8
26	Ferganskaya str., 23 [49]	22.3	36.9	26.0	21.6	13.3	18.6	19.5	10.6	3.5

Note: ¹ the method does not take into account the displacement of the structure in plan from the axis of the tunnel; ² the maximum additional settlements are given taking into account the sinking of two tunnels (right then left tunnel); all compared methods do not account for additional stress (load) on or below the surface (σ_s), except for [56], but only on the surface itself; the indication is built in a five-color ratio between min and max value: min, ~mid-min, ~mid, ~mid-max, ~max; indication when the min settlement calculated is less than the actual one during monitoring.

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3.6. Additional Validation for Modified Empirical Settlement Expression

Furthermore, the validation of the V_L,LSR method was considered alongside research by Tupikov M.M. (2010) [55], which formed the basis for the Recommendations “Assessment of the Impact of Collector Drilling on the Settlement of Surrounding Buildings and Underground Structures,” ANO ANTTS RAASN, Moscow, 2007 [58]. This research primarily used empirical methods to estimate settlements resulting from shield tunneling of utility tunnels and numerical simulations in PLAXIS were also used to verify the empirical equations, producing graphs of surface settlement differential, settlement curvature, and horizontal displacement across seven engineering–geological sections typical of Moscow.

To compare the volume loss in cohesionless (loose) soils, V_L,LSR, the most representative engineering–geological sections were selected: a surface layer of anthropogenic soil, followed by clays and loams of hard or semi-hard consistency (Type I) (Figure 11a), and various sands ranging from coarse to fine, dense, and medium density (Type III) (Figure 11b), based on extensive physical and mechanical data for sands, clays, and loams in Moscow, considering a tunnel of ∅4 m.

4. Discussion

In this study, the previously developed methodology was validated using data from cases with available monitoring results. The volume loss methodology for cohesionless (loose) soils, V_L,LSR Equation (7), which links important geomechanical soil parameters and the geometric (technological) of TBMs operation to calculate additional settlements, has previously shown good agreement with the measured data on additional structure settlement in the tunnel-affected zone, as demonstrated in [21]. This provides the required engineering safety margin to ensure reliability.

According to the validation results of the V_L,LSR calculation models, a noticeable reduction in additional structural settlement was occasionally observed as compared with calculations performed in accordance with Annex J of SP 249.1325800.2016 [41] and the recommendations of ANO ANTTS RAASN, Moscow, 2007 [58], based on the research by Tupikov M.M. (2010) [55].

According to the calculated volume loss, V_L [19] and V_L,LSR [21] (see Table 6), it can be concluded that the solution proposed is not optimal for the considered objects, primarily because the V_L values for TBMs ~∅6 m and ~∅10 m are almost the same or equal (see Table 6, §3–8), and sometimes differ significantly for TBMs around ~∅10 m (see Table 6, §13–16). However, the method volume loss, V_L [19] and V_L,LSR [21], still yields results of the same order of magnitude. Although methods V_L [19] and V_L,LSR [21] produce similar magnitudes, they sometimes exhibit significant discrepancies in values, which, according to the authors, may depend on the use of the parameter shear strength (c_u), as mentioned in [14].

According to the study [38], for the site located at Ryazansky prospekt, 4A bldg. 2, the calculated value of V_L = 0,85%. Despite the high computational complexity involved in determining volume loss, the result is ~30% lower than the values obtained of V_L = 1.13% [19] и V_L,LSR = 1.20% [21] (see Position 22, Table 6 and Table 8). Consequently, the settlement predicted by FEM analysis is expected to be lower than the monitored data. This may indicate that the method [38] is most likely unsuitable for determining the volume loss in sandy soils.

The authors also note that the volume loss, V_L, for sandy soils is ~25–45% higher than for silt-clay soils, excluding liquefied or fluid soils. This observation is consistent with data [50], as well as [59,60], and the findings of several other researchers.

The newly proposed modified expression for calculating surface settlement S_max [21] based on Peck R.B. (1969) [37] and Equation (10) demonstrates very good agreement, ensuring an adequate design safety margin when compared with numerical calculations, field geodetic data, and well-known empirical expressions for the determining S_max during the construction for various tunnels and related structures.

As indicated by the results of empirical calculations (see Table 8), it is particularly important to note that all methods described in [61] inherently assume homogeneous soil conditions, and therefore yield inaccurate results for strongly layered, heterogeneous soil masses, particularly when tunneling occurs in sandy soils overlain by thick clay layers.

It should be noted that geodetic monitoring data may include errors caused by various external factors, including instrument calibration, weather conditions (temperature, wind, precipitation, visibility, and illumination), refraction, vibrations, electromagnetic interference, software-related errors, such as those associated with network adjustment algorithms, and human factors, both in-office and in-field, which significantly influence the permissible geodetic measurement error in monitoring data, typically not less than 1 mm for elevation measurements [62].

The derivation of the new V_L,LSR methodology is based on a linear relationship between strain and stress, and is applicable to TBMs operating with earth pressure balance (EPB), slurry (SPB), and mixed face support systems (MPB). This approach does not consider for the applied face support pressure, since it serves as a design parameter for tunneling planning rather than a direct characteristic of soil rheological behavior. Static analysis introduces certain limitations, and the methodology does not consider dynamic tunneling parameters such as cutterhead rotation speed or TBM advance rate. Furthermore, the new model is not applicable to collapsible soils of Types 2.

The modified expression for calculating the surface settlements S_max was not considered for structures with pile foundations due to the absence of corresponding monitoring data. This aspect will constitute one of the objectives of future research.

5. Conclusions

• The previously presented methods have been confirmed as effective, particularly when applied jointly in empirical calculations. This leads to a significant reduction in additional structural settlements under the soil conditions of the Moscow region compared with calculations based on previously established methods.
• The calculated volume loss in cohesionless (loose) soils, V_L,LSR, generally shows good agreement with field monitoring data, indicating an acceptable engineering margin of safety.
• The newly modified Peck’s expression for calculating the maximum surface settlement S_max shows good agreement with field monitoring data, and its relative simplicity allows for a qualitative justification of design decisions for tunnel design in dense urban environments.
• Given that the application of the newly developed methodologies enables a reduction in final surface settlements for particularly critical tunneling sections and high-risk structures, this can significantly decrease the required scope of monitoring and the associated techniques. Since the frequency of monitoring during tunnel construction depends on the deformation rate, the methods applied may be limited to geodetic observations with a reduced number of deformation markers. It may also be possible to exclude the use of compensation and protective measures, as the expected structural settlements could remain well below critical thresholds. The proposed methodologies can assist in optimizing monitoring strategies and types, as well as compensation and protective actions; however, their use does not necessarily eliminate the need for such measures—the final decision always remains the responsibility of the designer.
• By analyzing a series of numerical simulation results, the authors observed that deformation patterns obtained from various software packages with different solvers, and those carried out by different engineers, exhibited similar patterns. This finding underscores that the accurate determination of volume loss (V_L), a fundamental geomechanical parameter for predicting tunneling-induced settlements, is crucial.
• The authors intend to continue their research by validating the V_L,LSR methodology and determining additional settlements using the modified Peck’s expression for calculating surface settlement S_max for a wider range of structures with available monitoring data. This effort aims to enhance the quality of tunnel design and construction, thereby reducing the likelihood of accidents.