Although squeezing ground may undergo rapid convergences following tunnel excavation, its behaviour is often markedly time dependent due to creep or consolidation. The effects of creep (a purely mechanical rheological process) and consolidation (a coupled hydromechanical process) on shield tunnelling are discussed, with the aim of demonstrating their qualitative similarities and distinctive features.
Part A: Basic Time Effects
The first paper investigates the basic time effects, looking at the time development of ground deformations and the complex interaction between ground, the TBM and tunnel support during both excavation and construction standstills.
Numerical simulations indicate several qualitative similarities between the two mechanisms of time dependency, such as: time development of rock deformation and shield loading during advance and increased shield loading with increasing advance rate under certain conditions in creep and consolidation.
However, the investigation herein also underscores two prominent differences and these result from the fundamentally different nature of creep and consolidation:
- First, consistently more extensive plastic yielding in consolidating ground, which is partially associated with the seepage forces exerted by the pore water on the solid rock constituents;
- Second, the role of seepage forces as a potential destabilising agent, particularly for the tunnel face, which does not happen in the case of creep and may be critical for shield and cutterhead jamming.
Part B: Transferability of Experience
The second paper builds upon the comparison of creep and consolidation on shield tunnelling, considering the practical transferability of experiences from existing tunnels as a reference for the required thrust force at tunnels of different diameters or to adjacent tunnels.
First, the effect of the tunnel diameter on the risk of shield jamming is examined. The paper demonstrates that larger-diameter tunnels are more favourable in poor-quality ground, while the opposite holds in better-quality ground, as well as where there is pronounced time-dependent ground behaviour.
Second, the effect of a tunnel on the required thrust force in a neighbouring tunnel built later is examined. The paper shows that this interaction effect is particularly important in water-bearing ground of low permeability: the drainage action of the first tunnel induces pore pressure relief and ground consolidation in an extensive area, leading to a substantial reduction of the thrust force in the second tunnel.
Conversely, in the case of creep the interaction is negligible even under extremely squeezing conditions, even for extreme conditions and large overcuts, due to the fundamentally different nature of the purely mechanical rheological processes.
The investigations into transferability are valuable for tunnelling practice, in cases of twin tunnels as well as where a smaller-diameter tunnel is built first (e.g., a pilot tunnel, advance drainage or ground improvement), or also the opposite (e.g., upgrade of a road tunnel by adding a safety tunnel with a smaller diameter).
The summarised papers, Parts A and B, are presented below.
PART A: BASIC TIME EFFECTS
1 INTRODUCTION
Squeezing, the phenomenon of rock deformations or pressures in tunnelling, may occur rapidly or develop slowly, depending on the rheological properties of the ground (creep), by stress redistributions associated with transient seepage flow in the case of low-permeability, saturated ground (consolidation), or both.
Rheological processes are of a purely mechanical nature and in general independent of the presence of pore water. It has been shown to be more pronounced when the ground is overstressed, particularly approaching failure state; hence, it is especially evident under squeezing conditions.
Consolidation is relevant in water-bearing ground of low permeability and is associated with the transient seepage flow triggered by the tunnel excavation. The flow enables progressive dissipation of the excess pore pressures, inducing variations in the effective stresses and ground deformations. Pore pressure has been known to intensify squeezing phenomena.
In tunnelling practice, it is not always directly distinguishable whether the source of time dependency of the ground behaviour is creep, consolidation, or the superposition of both. This is partially due to the different perception concerning the presence and influence of the pore water depending on the rock.
The questions arise: what are the fundamental differences in the phenomenological ground behaviour? Can these help to distinguish creep from consolidation in practical situations? Although several existing works have separately examined the effects of creep, this question has not been addressed in the literature. This paper evaluates comparatively some fundamental aspects of creep and consolidation in tunnelling.
Emphasis is placed on the time development of ground deformations and the problems of shield or cutterhead jamming in mechanised tunnelling.
2 COMPUTATIONAL ASSUMPTIONS
Numerical models were formulated in Abaqus® (Dassault Systèmes 2018) for a 12m-diameter cylindrical tunnel (R0 = 6m; tunnel radius, undeformed config) (see Figure 1).
2.1 Assumptions Common to Both Models
In the transient mechanical analyses considering creep, a traction equal to the in-situ stress σ0 prevailing at the depth of the tunnel is prescribed at the far-field boundary of the computational domain, 400m (ca. 67 R0) away from the tunnel axis.
A linear elastic-viscous perfectly plastic constitutive model is adopted for the rock. Also employed are Mohr–Coulomb yield condition and a non-associated visco-plastic flow rule. The model uses five mechanical parameters – Young’s Modulus E, Poisson’s ratio v, uniaxial compressive strength fc, angle of internal friction ϕ, and angle of dilation ψ, as well as a single rheological parameter – viscosity η, which determines the rate of visco-plastic deformation development due to creep.
In the coupled hydromechanical consolidation analyses, the uniform traction σ0 is also prescribed at the far-field boundary, and a uniform pore pressure equal to the in-situ value po prevailing at the depth of the tunnel.
On account of the potential development of negative pore pressures during ground excavation under undrained conditions, a mixed hydraulic boundary condition is prescribed. The rock is modelled as a two-phase porous medium, according to Terzaghi’s principle of effective stresses. Seepage flow is modelled on Darcy’s law. The solid grains and pore water are assumed incompressible in relation to the rock.
2.2 Model of Cross Section Far Behind the Advancing Face
Far behind the face, plane strain conditions can be assumed. The plane strain model (Figure 1a) simulates the ground response to excavation via an instantaneous unloading of the tunnel boundary. The tunnel boundary subsequently remains unsupported and evolution of its radial displacement is monitored.
Due to rotational symmetry, the problem can be analysed as 1D – a single strip, discretised with a finite element (FE) mesh of four-noded, linear, quadrilateral, axisymmetric elements.
2.3 Model of Advancing Tunnel Heading
The model simulates the transient processes during ongoing mechanised excavation and lining installation, as well as during a TBM standstill. It presupposes negligible TBM weight, and thus uniform tunnel support and overcut, as well as backfilling around the lining.
The construction process for the model is simulated as continuous with an average advance rate v, as shown to be a sufficiently accurate simplification (Leone et al. 2023). The step-by-step method is adopted with steps of length s = 1m; (Figure 1b), equal to lining is installed immediately behind the shield.
The tunnel face is considered unsupported, which is reasonable considering that open shield TBMs are employed in most practical cases of mechanised tunnelling through squeezing rocks. The shield of length L is modelled with non-linear radial springs, which consider no loading (zero stiffness) for convergences below the radial overcut ΔR, and a linear elastic stiffness Ks for the portion of convergences that exceeds that angular gap ΔR.
The lining is modelled with elastic radial springs of stiffness Kl, assuming direct contact with the ground immediately upon installation due to backfilling. The ground unloading behind the shield tail and its reloading over the lining are considered via distinct installation points.
The computational domain was a structured FE mesh of 11,532 four-noded, linear, quadrilateral, axisymmetric element, their sizes increasing in the radial direction, and along the tunnel axis remaining constant, equal to step length – which is sufficiently small to ensure enhanced prediction accuracy. An excavation length of 10R0 (60 excavation steps) was simulated.
With the model enabling a determination of the longitudinal profile of rock pressure, the average value was evaluated over the shield length, the simulated excavation length being long enough for the average value to become constant.
3 Time-Dependent Contraction of a Tunnel Cross Section Far Behind the Face
The radial displacement of the unsupported tunnel boundary depends, in general, on all independent problem parameters, i.e., in-situ stress, tunnel radius, material constants, time, and, in the case of creep, viscosity.
In the case of consolidation, the permeability, in-situ pore pressure, unit weight of the pore water, and the size (far-field radius) of the seepage flow domain are considered in the numerical model.
Figure 2 shows the normalised radial displacement as a function of the normalised time for creep (black line) and consolidation (dashed and solid red lines, for two values of the in-situ pore pressure, 1MPa and 4MPa, respectively).
Very low values of normalised time correspond to rapid excavation, where instantaneous ground response can be considered; very high values correspond to steady-state conditions, where all visco-plastic deformations have developed in the case of creep and all excavation-induced excess pore pressures have dissipated in the case of consolidation.
Time-development of displacements is qualitatively similar in both cases. It can be directly inferred that the curves nearly overlap with appropriate scaling between the viscosity and the permeability, and selection of the material parameters. This demonstrates the difficulty of distinguishing creep from consolidation, as well as of back-analysing ground parameters from—or even simply interpreting—the observed time-dependent deformations.
All strength and stiffness parameters being equal, the instantaneous and steady-state displacements are consistently higher in the case of consolidation, even more so in the case of the higher pore pressure due to the more extensive ground plastification.
The seepage forces do not depend on ground characteristics and can be arbitrarily high depending on the in-situ pore pressure. It may then happen that equilibrium near the tunnel boundary is impossible and thus excessive cavity contraction or even complete cavity closure occurs. In this sense, seepage forces constitute a potential destabilising agent for the tunnel cross section and face. This is a distinguishing feature between creep and consolidation.
4 SHIELD LOADING DURING TBM ADVANCE AND STANDSTILLS
The average rock pressure acting upon the shield depends, in general, on all independent problem parameters, i.e., in-situ vertical stress, tunnel radius, material constants, TBM parameters (Ks, Kl , L, ΔR), the advance rate, and the standstill time. In the case of creep it also depends on viscosity; in the case of consolidation on the permeability, in-situ pore pressure, unit weight of pore water, and size of the seepage flow domain.
4.1 Shield Loading During TBM Advance
For the advancing tunnel face, plastic strain contours, displacements, convergences, and radial stresses could be graphed and evaluated for the cases of creep and consolidation, respectively, and for two values of in-situ pore pressures (1MPa and 4MPa), for two limit cases of rapid and slow excavation.
The limit cases are considered via normalised advance rates—how fast one tunnel radius is excavated in relation to the development rate of time-dependent deformations. During rapid excavation the ground response is purely elastic in the case of creep, and undrained elastoplastic in the case of consolidation; during slow excavation there is sufficient time for steady-state conditions to develop.
Plastic strains and displacements are higher in the case of consolidation, both during rapid and slow excavation.
However, this is not necessarily reflected as higher rock pressure on the shield as the mechanisms of the rock–shield interaction are complex.
The pressure that ultimately develops on the shield comes from the interplay of two counteracting effects of rock plastification: more plastic yielding leads to increased ground convergences, increased contact area and tentatively increased shield loading; and, more stress relief ahead of the face though tentatively lessens load transfer to the shield. Based on this interaction, the differences in rock pressure between creep and consolidation for the cases of rapid and slow excavation can be interpreted.
4.2 Counter-Intuitive Effect of Advance Rate
The interplay of the two competing effects also produces a counter-intuitive result on the effect of advance rate. Figure 3 shows the average shield pressure as a function of the normalised advance rate for creep (black lines) and consolidation (red). This seemingly paradoxical behaviour is observed for both creep and consolidation, with the two distributions being qualitatively identical.
Leone et al. (2023) showed that it is attributed to delayed plastic deformation development with increasing viscosity, leading to smaller contact area between shield and ground, but limits stress relief ahead of the face thereby having a stiffening effect.
Ramoni and Anagnostou (2011a) originally reported and examined the same paradox for the case of consolidation, explaining it on the basis of the slower development of plastic deformations in the case of lower permeability. However, there is the additional effect of the negative pore pressures have a stabilising effect by increasing effective stresses and so reducing deformations and plastification.
4.3 Shield Loading During a Standstill
Within the range of normalised advance rate, the conditions are also more unfavourable during a TBM standstill (see dashed lines in Figure 3). The rock pressure variation is qualitatively similar for both creep and consolidation, but its increase is in general more pronounced for the latter, since seepage flow progressively starts during standstill and seepage forces induce more extensive ground plastification.
The above becomes clearer for creep and consolidation, respectively. Standstills are unfavourable in both cases. Therefore, models that disregard time dependency and assume that plastic deformations develop instantaneously are not conservative: they may overestimate shield loading during excavation but underestimate it during even short standstills.
5 ON THE DESTABILISING EFFECT OF THE SEEPAGE FORCES
Permeability governs the rate of squeezing for consolidation, but an additional parameter to consider is in-situ pore pressure or its gradient, the seepage force. In homogeneous ground the steady-state pore pressure field, and thus the magnitude of the seepage forces, depend solely on the hydraulic boundary conditions.
To fulfil equilibrium, the seepage forces must be resisted by stresses in the ground; however, the maximum resistance that the ground can provide depends on its mechanical characteristics. Under certain conditions the seepage forces may be sufficiently high to prevent equilibrium near the tunnel, resulting in excessive convergences and instability.
As these effects, are induced by an external agent they must be distinguished from familiar instability phenomena, e.g., those in rocks exhibiting strain softening that are relevant both in creep and consolidation; therefore, they may also occur in the absence of softening, in the perfectly plastic rocks examined in the present work.
5.1 Tunnel Cross Section Far Behind the Face
The rotationally symmetric, plain strain problem of an unsupported tunnel cross section was considered. Where effective radial stress cannot increase with the radius, and therefore cannot reach a state of equilibrium with the far-field stress, this according to Egger et al. (1982) means that an opening would be unstable. This points to a qualitative difference for ground behaviour in the case of creep, where a stable stress field always exists in the absence of softening. The difference is, however, only apparent; the instability would manifest itself by increasing convergences, which would result in an increasing curvature of the tunnel boundary, and equilibrium can always be reached if the curvature becomes sufficiently big.
The instability as postulated cannot actually occur; there is no difference between creep and consolidation in respect of equilibrium or lack thereof in the case of a circular opening. The reason is of a geometric nature: at the onset of instability the curvature increases and stabilises the system, even if this may happen at big convergences which, in practical terms, may fail to satisfy serviceability criteria.
However, this finding points to a fundamental difference between the behaviours of a tunnel’s cross section and face: the geometry of the former becomes increasingly favourable during cavity contraction, whereas the geometry of the latter becomes more unfavourable during extrusion, as it becomes convex (Figure 4).
5.2 Tunnel Face
The extrusion of the face was numerically investigated over the course of a sufficiently long TBM stand-still to reach steady state conditions. The model assumed no face support (open shield TBM), but in practice the ground establishes contact with the cutterhead after a certain amount of deformation. The face–cutterhead interaction was considered by introducing nonlinear springs in the model.
From a practical viewpoint, the combined effect of the extrusion and instability of the tunnel face, and the contraction of the tunnel cross section under seepage forces, may be particularly critical in respect of the thrust force and the torque of the cutterhead required for the TBM to restart its advance after a standstill. The studies also showed that, at restart, the skin friction to be overcome increases monotonically with standstill duration.
Conclusively, these results underscore the practical significance of the seepage forces, which may result in significant loading of the cutterhead and have severe implications in respect of the design of the TBM, or even assessment of a TBM drive in consolidating ground— which does not exist for creep.
6 CONCLUSIONS
The investigations highlighted several qualitative similarities between the mechanisms of time dependency of the ground behaviour, and underscored two prominent differences—resulting from the fundamentally different nature of the purely mechanical rheological creep processes and the coupled hydromechanical consolidation processes.
The first difference is the consistently more pronounced plastification in consolidating compared to creeping ground.
The second difference can be traced to the seepage forces, their magnitude unrelated to mechanical ground characteristics and depending only on hydraulic boundary conditions.
The paper demonstrated that these phenomena are particularly critical for shield jamming and cutterhead blocking in consolidating ground.
Despite the above fundamental differences, these do not allow for a distinction of creep from consolidation based upon the observed or observable ground behaviour in practice, particularly with uncertainty on strength and stiffness parameters.
Concerning instability, there is no way of distinguishing this based on the observed behaviour. In this sense, the only viable way of distinction is advance exploration of the hydrogeological conditions of a project site and also appropriate laboratory tests to determine the rheological properties as well as the degree of saturation and permeability of the ground.
PART B: TRANSFERABILITY OF EXPERIENCE
1 INTRODUCTION
In this paper, the comparison between creep and consolidation investigates whether experiences gained from previously constructed tunnels, about required thrust force, can be transferred to tunnels of different diameter or to adjacent tunnels of the same diameter built under the same geotechnical conditions.
The investigations into transferability of experience are valuable in practical situations where, for example, a smaller-diameter pilot tunnel is constructed prior to the main tunnel for exploration, advance drainage or ground improvement; or, the opposite (e.g., upgrading of a road tunnel by later construction of a safety tunnel with a smaller diameter); and, cases of sequentially excavated twin tunnels.
The paper addresses: (i), the effect of the tunnel diameter on the risk of shield jamming (‘scale effect’); and, (ii), the effect of a tunnel on the required thrust force in an adjacent, later excavated tunnel (‘interaction’).
While there is a significant volume of literature on creep and consolidation in tunnelling in general, and on shield jamming in mechanised tunnelling through squeezing ground, however the ‘interaction’ effect has not been studied so far. The ‘scale effect’ has only been investigated by Ramoni and Anagnostou (2011) for the case of consolidation.
- Scale Effect
Besides the question of transferability, investigating scale effect is also motivated by a theoretical consideration— that consolidation time increases with the second power of the drainage path length, e.g., the thickness of a low permeability layer.
Based on this, squeezing would develop slower for a larger-diameter tunnel, as the drainage paths are longer in relation to a smaller diameter tube. In turn, this should also affect the rock pressure acting on the advancing shield and thus the force required to overcome shield skin friction and the risk of shield jamming.
There may be significant differences in scale effects between consolidation and creep.
The effect of the tunnel diameter on the risk of shield jamming is more complex to assess, for three reasons:
- Depending on diameter, there are technical limitations for certain TBM parameters.
- Diameter poses limitations on certain operational parameters.
- A larger cross section has a higher potential of encountering adverse conditions.
Considering the limitations, Ramoni and Anagnostou (2011) showed that a smaller diameter is consistently less prone to shield jamming in the case of consolidation; however, the differences between them decrease in weaker ground. Two questions arise: Is the scale effect the same in the case of creep? Is a larger diameter more favourable in weaker ground?
1.2 Interaction
Tunnelling through heavily squeezing ground can cause stress redistribution and deformations in an extended area, in turn affecting other existing or planned underground structures. Interaction effects may be more or less pronounced by the method of construction, moreso with conventional tunnelling with yielding support, and less so where ground deformations are limited by stiff support close to the face. Shield tunnelling is closer to the second case.
A special situation arises in the case of saturated, water-bearing ground, as excavation-induced drainage results in pore pressure relief over an extended area. In the case of a twin tunnel, the pore pressure relief due to construction of the first tube results in higher effective stresses and helps the second tube, such as experienced at Simplon rail tunnel and the Gotthard road tunnel in Switzerland.
It is expected that time-dependent interaction effects will be more pronounced in the case of consolidation than in the case of creep.
2 Scale Effect
Starting with a simple plane-strain problem, the study looked at the time-development of tunnel convergences in a cross section far behind the face, and then the scale effect with respect to the risk of shield jamming.
2.1 Time-Dependent Contraction of a Deep Tunnel
2.1.1 Homogeneous Ground
The problem is analysed based on the classic, rotationally symmetric, plane-strain model of a cylindrical and uniformly supported tunnel located deep below the ground surface and the water table. The rock is assumed to obey a linear elastic and perfectly plastic constitutive model with a Mohr– Coulomb yield condition and a non-associated plastic flow rule. The rheological ground behaviour due to creep is modelled as purely viscoplastic. Seepage flow in water-bearing ground is modelled according to Darcy’s law.
It is inferred that the time required to attain a given percentage of displacement during consolidation is proportional to R2 or where given ground parameters, initial stress and pore pressure. Such a dependency does not exist in the case of creep. Practically, this means that given displacements would develop slower in the case of consolidation. The quadratic dependency holds as long as the tunnel lies deep under the water table.
2.1.2 Confined Aquitard
In the case of an aquitard (low permeability layer confined between permeable layers), the in-situ pore pressure practically acts at the layer interfaces. The hydraulic head difference between the aquifer and tunnel is dissipated only within the aquitard, along a drainage path whose length corresponds to half the aquitard thickness (d/2 in Figure 5).
This constraint may give rise to a secondary, nontrivial scale effect, which becomes relevant when the ratio of the aquitard thickness to the tunnel diameter (d/D) is small, and generates two competing effects: the drainage path becomes shorter, accelerating consolidation, but the hydraulic gradients and seepage forces are higher, which induces more extensive ground plastification and thereby decelerates consolidation.
The key takeaway is that in confined aquitards the consolidation rate may deviate dramatically from the quadratic rule of consolidation theory—even moreso for lower aquitard thickness and weaker ground, indicating that deformations may continue for a long time after excavation. This may prove critical in respect of serviceability requirements or structural safety of the final lining.
2.2 Risk of Shield Jamming
The effect of the tunnel diameter on the risk of shield jamming was investigated only for homogeneous ground.
Due to technical limitations, the TBM parameters (over-cut, shield length, and shield and lining stiffnesses) in practice vary within a specific range, independently of the tunnel diameter, while the advance rate decreases with increasing diameter. Consequently, the nondimensional parameters in general cannot take the same values for different tunnel diameter, which gives rise to a scale effect with respect to the risk of shield jamming (Ramoni and Anagnostou 2011).
The strength and stiffness of the ground may also tentatively decrease with increasing representative volume, and thus tunnel diameter.
Assessing the scale effect qualitatively is cumbersome. Therefore, numerical simulations were necessary to scale effect.
2.2.1 Simplified Theoretical Analysis
The start of the analysis considered conditions during TBM restart after a standstill. As rock pressure may already have developed, its value depends not only on the standstill duration but also on the advance rate. To eliminate the influence of the latter, rapid excavation was assumed.
Conditions during the advance phase were examined. In the case of creep, for a given average advance rate, the advance would be slower in a larger-diameter tunnel (equivalent effect to lower viscosity), and hence a higher rock pressure will develop; alternatively, to achieve a given rock pressure the advance rate must be higher in the larger-diameter tunnel.
In the case of consolidation, the effect is opposite. To compare rock pressure in different diameter tunnels the average rate alone is not the most suitable measure of the rate of advance. At any given tunnel cross section, a shield remains exposed to pressure locally while it passes. Shield pressure, ultimately, depends on the time for the TBM to advance by one shield length, which becomes a function of v/R only, as L/R, the normalised shield length, is considered fixed.
For a given v/R: in the case of creep, the normalised advance rate [(v/R) (η/Ε)] being constant means the pressure develops at the same rate regardless of R; in the case of consolidation, its normalised advance rate [(v/R) (γw R2)/(kE)] is proportional to R2, and the pressure develops at a rate inversely proportional to R2, and thus slower in a larger-diameter tunnel.
The effect of the tunnel diameter, therefore, becomes the same here as in the case of the standstill examined previously. Both during advance and restart, in the case of creep there will be no scale effect for tunnels excavated at the same rate, v/R; in the case of consolidation, though, the dependency on the square of the characteristic length—here R—holds.
2.2.2 Theoretical Analysis Under Realistic Assumptions
The assumption adopted in the preceding section that v/R can be the same for the two tunnels is theoretical, for as is known, in practice, the larger the boring diameter the slower the TBM advance. In fact, v can be assumed to be inversely proportional to R, which then makes v/R inversely proportional to R2.
Then: in the case of creep, its normalised advance rate becomes inversely proportional to R2, which means that the pressure develops faster in a larger tunnel (effect equivalent to lower viscosity); in the case of consolidation, its normalised advance rate becomes independent of R, and hence the shield pressure develops at the same rate in both tunnels.
The assumption concerning the advance rate does not influence the conditions during a standstill preceded by a rapid advance.
2.2.3 Time-Independent Ground Behaviour
The theoretical analysis of the previous sections assumed that the dimensionless parameters do not depend on the tunnel radius, when in fact the dimensionless TBM parameters in general cannot take the same values for a large and a small tunnel. This generates a scale effect, even without time-dependent ground behaviour.
The scale effect related to the TBM parameters was assessed by comparing two tunnels with diameters of 12m and 4m, respectively, excavated under identical ground conditions, disregarding creep and consolidation.
With respect to the risk of shield jamming, it is not sufficient to consider only the shield loading or the required thrust force, Fr, but also the installed or installable thrust force, Fi, the latter increasing with tunnel diameter, which also introduces a scale effect.
The comparison between the two tunnels is based on ratio Fr/Fi, known as the Thrust Utilisation Factor (TUF). The installed TBM thrust force is assumed to increase proportionally to the cross-sectional area, and thus to R2.
The intuitive perception that the smaller-diameter tunnel is less vulnerable than the larger-diameter one is unconditionally true for other potential hazards (e.g., an instability of the tunnel face) but not for the risk of shield jamming, where it is true only in better-quality ground. The opposite holds in weaker ground.
Interestingly, this has been shown to be generally true for any two tunnels with different radii, by means of a more extensive parametric study considering a wide range of practically relevant in-situ stresses, boring radii, ground, and TBM parameters.
For the parameters adopted herein, the TUF in the larger-diameter tunnel is less sensitive to variations of rock quality than in the smaller-diameter tunnel, indicating that its excavation is feasible irrespective of the ground conditions. The small tunnel excavation, on the other hand, is only feasible in better-quality ground.
2.2.4 Time-Independent Ground Behaviour: TBM Advance
The scale effect during TBM advance in the case of creep and consolidation was examined under the same assumptions, setting fc = 3.2MPa to eliminate the influence of the TBM parameters, and allowing advance rates of 90m/day and 30m/day for the 4m- and 12m-diameter tunnels, respectively.
A scale effect clearly exists with respect to the risk of shield jamming (expressed by the TUF) both in creep and consolidation. This is consistent with the results of the last section: time dependency delays squeezing, the rock thus responds to tunnel excavation as if it were of a higher quality, and an increase in rock quality renders a smaller diameter more favourable.
The vertical distance between the two lines reflects the scale effect.
Standstill
The same conclusions can essentially be drawn for the thrust force needed for restart after a standstill. There are the same parameters as well as a high viscosity and low permeability, which ensure that the behaviour during advance is elastic (creep) or undrained (consolidation), and hence relevant shield loading develops only during the standstill.
In graphing, the distances between lines indicate the existence of a scale effect. The widths of those bands reflects the effect of tunnel radius.
3 Interaction
The hypothesis that interaction is significant for consolidation but negligible for creep was tested, considering the problem of shield jamming in a twin tunnel (Figure 6). Tunnel 2 is assumed to have been built long after Tunnel 1, such that steady state conditions have been re-established. The limit cases examined were minimum interaction in consolidation versus maximum interaction in creep.
Consolidation: the effect on Tunnel 2 is, first, due to stress redistribution, and, second, pore pressure relief and the increase in undrained shear strength. This effect is minimum when the excavation-induced deformations are as small as possible, which is the case when the ground permeability is very low and its response to excavation practically undrained.
Creep: the effect on Tunnel 2 is due to stress redistribution. This effect is maximum if the deformations ahead of the face and around the shield of Tunnel 1 are as large as possible, as when the advance is slow and all viscoplastic deformations, due to creep, occur practically simultaneously with excavation progress.
The problem was analysed in a simplified manner using, sequentially, two models: (i) an axisymmetric model of the advancing tunnel heading; and, (ii), a plane-strain, twin-tunnel model of a cross section.
3.1 Consolidation Case
The short-term shield–lining–ground interaction in Tunnel 1 was analysed using the axisymmetric model (Figure 1b) to establish rock pressure distribution along the shield and, so, the thrust force required to overcome shield skin friction. The model was used to calculate the required thrust force in Tunnel 2, with different initial conditions that reflect stress redistribution, pore pressure relief and ground consolidation induced by Tunnel 1.
The alteration of the initial stress and pore pressure fields were quantified using the plane-strain model (Figure 7).
The comparison of the rock pressure in the two tunnels reflects the very favourable effect of the drainage-induced consolidation in the vicinity of Tunnel 2. It is remarkable that the Tunnel 2 thrust force is about 65% less.
3.2 Creep Case
The creep case essentially follows the same approach. Computations start with the axisymmetric analysis of Tunnel 1, which provides the longitudinal stress distributions on the shield and lining, as well as the equilibrium point of the rock.
The analysis showed that construction of Tunnel 1 does not have a relevant effect on the Tunnel 2 thrust force for the considered spacing and this lack of relevant interaction can be observed also in the extreme squeezing and large overcut case. The difference to the consolidation case is remarkable—at a spacing of 2.5D or even greater, the interaction in the consolidation case leads to a reduction of the Tunnel 2 thrust force by 65%–90%.
Conclusions
The investigations into the scale effect showed time development of the convergences of an unsupported opening gave the findings of: independence of the tunnel diameter (creep); and, being inversely proportional to the square of the diameter (consolidation), which means that convergences develop slower in larger tunnels.
The latter is directly analogous to the well-known quadratic dependency between consolidation time and length of the drainage path, but holds only if there is one significant geometric parameter: the tunnel diameter.
In the more complex problem of shield jamming risk, the investigations demonstrated that a larger diameter is more favourable than a smaller one in poor-quality ground with time-independent behaviour; the opposite is true for better-quality ground, creep or consolidation.
Funding: Open access funding provided by Swiss Federal Institute of Technology Zurich. No funding was received for conducting this study.
Declarations (Conflict of interest): The authors have no relevant financial or non-financial interests to disclose. These papers were originally published online in Rock Mechanics and Rock Engineering (RMRE) Journal, from Springer, in 2024, as Parts A and B of ‘Creep Versus Consolidation in Tunnelling through Squeezing Ground,’ under a Creative Commons Attribution 4.0 International Licence, http://creativecommons.org/licenses/by/4.0/.
As permitted under the particular open access facility, this version of the original papers is combination of two full papers, abridged and edited for space, and images adapted to house-style. The original papers are available in full, online: Part A – Rock Mechanics and Rock Engineering (2024) 57:5537–5555; https://doi.org/10.1007/ s00603-024-03968-6; and, Part B – Rock Mechanics and Rock Engineering (2024) 57:5519–5536; https://doi. org/10.1007/s00603-023-03720-6.
