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A simplified 3D model for tunnel construction using tunnel boring machines
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Tunnelling and Underground Space Technology 23 (2008) 38–45
Tunnelling and Underground Space Technology incorporating Trenchless Technology Research
www.elsevier.com/locate/tust
A simplified 3D model for tunnel construction using tunnel boring machines H. Mroueh *, I. Shahrour Laboratoire de Me´canique de Lille (UMR 8107), Universite´ des Sciences et Technologies de Lille, F-59655 Villeneuve d’Ascq, France Received 5 September 2006; received in revised form 21 November 2006; accepted 28 November 2006 Available online 24 January 2007
Abstract This paper includes a presentation of a simplified three-dimensional numerical model for the prediction of soil movement induced during tunnel construction using tunnel boring machines (TBM). The model is based upon the generalization of the convergence-confinement concept to 3D tunnel construction. It uses two parameters (Ldec and adec) which stand for the length of the unlined zone and the partial stress release, respectively. The value of the parameter Ldec can be taken equal to the tunnel diameter, while the value of adec can be determined by fitting the model to empirical formula, and then adjusted based on settlement registered during tunnel construction. The capacity of the model is illustrated through an application to a shallow tunnel in soft soil. The comparison of the numerical results to those suggested by different authors shows good agreement. Ó 2006 Elsevier Ltd. All rights reserved. Keywords: Tunnel boring machine; TBM design; Finite element method; Three-dimensional; Non-linear; Shield tunnelling; Convergence-confinement
1. Introduction Construction of tunnels in soft soils induces generally soil movement, which could seriously affect the stability and integrity of existing structures (pile foundations, buildings. . .). In order to reduce such movement, in particular in urban areas, contractors use more and more the tunnel boring machines (TBM) for the construction of tunnels. Indeed, thanks to the application of a face pressure and to the temporary support, the TBM allows to reduce the soil disturbance due to tunneling, providing enhanced safety to existing structures (Herrenknecht, 1998; Kurihara, 1998; Kuwahara, 1999). Analysis of the impact of the tunnel construction using TBM on the soil movement requires the solution of large 3D non-linear soil–structure interaction problem. Non-linearity results from the non-linear behavior of geomaterials, the condition at the soil–structure interface (soil – grouting-lining, soil – shield,) and the evolution of *
Corresponding author. E-mail address: [email protected] (H. Mroueh).
0886-7798/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.tust.2006.11.008
the geometry during excavation. The 3D aspect is due to the significant stress disturbance and soil movement induced ahead the excavation front. 3D elastic analyses conducted by Panet and Guenot (1982) showed ground convergence at the tunnel face which was equal to about 27% of the total settlement. Higher values, up to 50%, were observed in field measurements and computational analyses in soft ground (Moraes, 1999). Finite element modeling of the tunnel construction using TBM requires also the consideration of the complex tunnel process which includes the advance of the TBM, the application of the face pressure, the soil excavation, the installation of an immediate support behind the rotating front, the installation of the definitive support (lining ring) and the tail void grouting. A realistic consideration of these issues in the 3D calculation constitutes a high challenge (Dias et al., 2000; Cheng et al., 2002; Galli et al., 2004), because of the large effort for numerical modeling and calculation and the large uncertainties concerning the interaction between the shield and the soil, the behavior of the grouting, and the distribution of the tail void. Consequently, the use of this approach in tunnel design is still limited, because it requires impor-
H. Mroueh, I. Shahrour / Tunnelling and Underground Space Technology 23 (2008) 38–45
Step i Phase i α dec * σ(i- 1)
Ldec
Llin Step Phasei+1 i+1
(1−αdec) * σ(i) α dec * σ(i)
revêtement lining
p
grouting coulis
Fig. 1. Method used for the tunnel construction using TBM.
tant modeling effort and computational time. In order to overcome this difficulty, a simplified method is proposed in this paper to model the TBM tunneling process using a three-dimensional model based on the convergenceconfinement method (Panet and Guenot, 1982) with two release parameters: adec and Ldec, which stand for the partial stress release and the length of the unlined zone, respectively (Fig. 1). This method can be easily implemented and employed using existing programs based on either the finite element or the finite difference method. The paper presents successively, the proposed method, its application to a model tunnel and the sensitivity of the method to the release factors adec and Ldec. 2. Presentation of the numerical model Numerical modelling of the tunnel construction using TBM constitutes a hard task, because it requires consideration of complex aspects such as the soil excavation, the overcut or annular space between the jacking pipe and the excavation, the application of the face pressure, the installation of the definitive support constituted of lining rings and the grouting of the annular space. It also requires the description of the non-linear behavior of both the soil and the lining and the condition at the soil–structure interface. Modelling of the tunnel construction is also three-dimensional, because the TBM induces an important stress disturbance and soil movement ahead the excavation front. Modelling of the annular space between the ground and the lining extrados is still problematic, because of the difficulties to collect effective data on the distribution and grouting of this space. Up to now, it seems very difficult to consider the abovementioned issues in the practical design of tunnels. In order to overcome this difficulty, a simplified method is proposed
39
in this paper to model the TBM process using a threedimensional model based on the convergence-confinement method (Panet and Guenot, 1982). This method uses a step-by-step procedure. Each step corresponds to the progression of the tunnel face by a distance Llin (Fig. 1). At each step of the procedure, the stress release around the tunnel head is modeled using the parameters adec and Ldec, which stand for the ratio of the stress release and the length of the unlined zone, respectively. The calculation procedure at the step (incr) includes: (a) Determination of the incremental force resulting from the soil excavation (DF). This force is equal to the difference between the nodal force vector (F(incr)) due to the external forces (self-weight, surface loads, front pressure, . . .) and the nodal internal forces at the previous step ‘incr-1’; calculation of DF is carried out using the following expression: Z t ðincrÞ DF ¼ F Be rðincr1Þ dV ð1Þ V ðincrÞ
V(incr) represent the volume of the soil mass at the step (incr); Be is the strain interpolation matrix which contains the spatial derivatives of the interpolation functions (e = BeÆu; u nodal displacement); r(incr1) denotes the stress tensor at the previous step (incr1).In order to take into account the partial deconfinement resulting from the tunnel construction process (overcut, injection of the annular void, installation of the definitive tunnel support,. . .), a parameter adec is used for considering the partial release on the unsupported section of the tunnel; the length of this section is assumed to be equal to Ldec. The incremental nodal force vector in this section (DF) is transformed using the following expression: DF 0 ¼ adec DF
ð2Þ
(b) Activation of the lining elements located in the new section and a full release of stresses in this section. (c) Application of the face pressure ‘p’ (Fig. 1); the pressure is assumed to be constant with depth; it corresponds to a ‘compressed-air pressure’ TBM. Note that this pressure can vary with depth to model ‘slurry shield’ machines or ‘earth pressure balance’ (EPB) machines. The soil movement is controlled through the partial release factor adec and the parameter Ldec which enable users to consider the influence of the void space and grouting around the tunnel. The determination of these parameters can be carried out by an adjustment procedure using empirical models and measurements during tunnel construction. The following section presents the application of the proposed method to a model tunnel, which will be followed by a sensitivity analysis of the model to the variation of the partial release parameters adec and Ldec. This analysis allows the elaboration of a methodology for the determination of these factors.
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3. Application to a tunnel model
Table 1 Properties of geomaterials used in the tunnel model
The proposed method has been implemented in the finite element code PECPLAS which provides flexible features for the analysis of three-dimensional and non-linear soil– structure interaction problems (Shahrour, 1992; Mroueh and Shahrour, 1999). This program uses a sparse storage scheme for the stiffness matrix, and the bi-CGSTAB iterative method (Van der Vorst, 1992) coupled to the SSOR preconditioning operator (Successive Symmetrical OverRelaxation) for the solution of the resulting linear systems.
Geomaterial
E (MPa)
m
c 0 (MPa)
u (°)
w (°)
c (kN/m3)
Soil Lining
30 35,000
0.3 0.25
0.005
27
5
20 25
3.1. Geometry and numerical parameters Fig. 2 shows the tunnel model geometry. The tunnel is characterized by its outer diameter D = 7.5 m, depth H = 2.5 D and lining thickness e = 0.5 m. The distance of the tunnel centre to the bottom boundary (rigid substratum) is assumed to be equal to 2.5D. The soil behavior is assumed to be governed by an elastic perfectly–plastic constitutive relation based on the Mohr–Coulomb criterion with a non-associative flow rule. The yield function and the plastic potential are given by rffiffiffiffiffi pffiffiffiffiffi J2 f ¼ p sin u þ J 2 cos h sin u sin h C cos u ð3Þ 3 rffiffiffiffiffi pffiffiffiffiffi J2 sin u sin h ð4Þ g ¼ p sin w þ J 2 cos h 3 where C, u and w designate the soil cohesion, friction angle and dilatancy angle, respectively; p, J2 and h stand for the mean stress, second invariant of the deviatoric stress tensor and Lode angle, respectively. Their expressions are given by p ¼ rii =3 1 J 2 ¼ sij sij where sij ¼ rij pdij 2 ! pffiffiffi 1 3 3 J3 sij sjk ski 1 : 3=2 h ¼ sin where J 3 ¼ 3 2 J2 3
ð5Þ ð6Þ ð7Þ
H = 2.5D
Table 1 summarizes the characteristics of both the soil and the lining. Homogeneous silty sand is considered with the following characteristics: friction angle u = 27°, cohesion C = 5 kPa, dilatancy angle w = 5°, Young’s modulus E = 30 MPa, and Poisson’s ratio m = 0.3. The lining is assumed to be governed by a linear-elastic behavior with a Young’s modulus E = 35,000 MPa and a Poisson’s ratio m = 0.25. The initial stress in the soil media is determined using a coefficient of lateral earth pressure at rest K0 = 0.5 and an effective bulk unit weight of the soil of c 0 = 10 kN/m3. 3.2. Finite element mesh Finite element analysis was carried out using the mesh presented in Fig. 3. This mesh consists of 2214 20-nodes hexahedral elements, which give rise to 10,494 nodes and 28,471 degrees of freedom. The lateral boundaries of the model are located a distance 4D from the tunnel axis in order to minimize their interaction with the tunneling construction. The longitudinal length of the mesh is fixed to 8D, and the tunnel excavation is performed for a final position of 4D. This mesh is used to illustrate the application of the proposed method. In tunnels design, an enhanced mesh must be used in order to well capture the soil deformation and the development of plasticity around the tunnel. An extension of the lateral boundaries of the soil mass should also be considered. 3.3. Calculation process Computation was carried out in 12 steps using the following parameters for the excavation modelling: ratio of stress release adec = 0.5, length of the unlined zone Ldec = 1D, and length of the excavated section at each step Llin = D/3. The face pressure is assumed to be uniform and equal to p ¼ r0h , where r0h stands for the initial axial stress at the tunnel axis. 3.4. Results
D = 7.5m e= 50 cm
Fig. 2. Numerical example used for the illustration of the model performances.
3.4.1. Settlement along longitudinal profiles Fig. 4a shows the evolution of the surface soil settlement along the longitudinal axis (A–AP 0 ) during the tunnel construction. It can be observed that the maximal surface settlement increases with the tunnel progression, and tends to stabilize at a value of wsurf max ¼ 0:07%D (D denotes the tunnel outer diameter. The stabilisation of the surface settlement
H. Mroueh, I. Shahrour / Tunnelling and Underground Space Technology 23 (2008) 38–45
41
u=0 v=0
4.5D v=0 u=0
z
8D
y
x
4D u=v=w=0
Fig. 3. 3D Finite element mesh used in numerical analysis (2214 20-node elements; 10,494 nodes; 28,471 ddl).
Fig. 4. Reference example: vertical displacement along longitudinal section: (a) at the ground surface, along the line (A–A 0 ) and (b) at the tunnel crown, along the line (B–B 0 ).
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H. Mroueh, I. Shahrour / Tunnelling and Underground Space Technology 23 (2008) 38–45
is observed after an excavation length of 3D). It can be noted that the surface settlement (wa) at the tunnel face is constant during the tunnel progression. It is equal to 0.03%D, which corresponds to 46% of the maximal value of settlement at the end of the simulation wsurf max . The amount of settlement induced just before the lining installation (wa + wb) is about 77% of wsurf max (0.05%D), which means that 31% of the total settlement is induced in the unlined zone of the tunnel. This result shows that 23% of the total settlement (wc) is due to the complete release of the confinement (1 adec). Table 2 summarises the proportion of settlement, in comparison with some published results. It can be noted that these values are in good agreement with observed or computed values reported by various authors. The amount of settlement in the front of the tunnel face wa appears higher than reported values, but the cumulative settlement wa + wb is in good agreement. Note that these values highly depend on the factors adec and Ldec.
The spreading of displacements from the tunnel to the surface can be analysed through a ‘softening’ coefficient or factor of diffusion Rdif, according to (AFTES, 1999): cr Rdif ¼ wsurf max =wmax
Vertical displacements at the soil surface and the tunnel crown give Rdif = 0.4, which means that 40% of the vertical displacement at the crown tunnel is transmitted to the surface. This value agrees with empirical approaches and in situ observations. Ward and Pender (1981) reported values for the diffusion coefficient varying from 0.2 (for sands) to 0.74 (for over-consolidated clays), while Sagaseta (1987) reported values between 0.2 (for frictional material) and 0.67 (for low frictional clay). 3.4.3. Settlement in a cross direction Fig. 5 shows the evolution of the settlement in the transverse section along the axis C C 0 . It can be observed that a moderate settlement appears (0.01%D) when the tunnel face is about 1D behind the cross section (C C 0 ), then the settlement increases when the tunnel face crosses the traverse section and tends to stabilize when the distance between the tunnel face and the section (C C 0 ) exceeds +2D. Numerical results illustrated in Fig. 5 were used for the determination of the parameters of Peck formula (Peck, 1969): the location of the point of inflection of the settlement curve ‘i’, the length of the settlement profile ‘Ls’ and the volume loss at the ground surface ‘vs’. Results are summarised in Table 3a. It is noted that the distance ‘‘i’’ and the length ‘Ls’ decrease with the progression of the tunnel face, and tend to stabilise to a value of i = 1.17D when the relative distance between the cross section and the tunnel face exceeds +1D. The volume loss at the surface is estimated to Vs = 0.26%Vexc, where Vexc denotes for the volume of excavated soil. The parameters of Peck formula can be estimated from semi-empirical methods. Table 3b shows a comparison of the numerical results with the empirical expressions proposed by
3.4.2. Crown displacement Fig. 4b shows the variation of vertical displacement along the longitudinal axis (BB 0 ) during the tunnel construction, at the tunnel crown location wcr. It can be observed that the major part of the crown displacement results from the TBM progression. Indeed, 40% (respectively 90%) of the total displacement at the tunnel crown is observed at the TBM passage (respectively at the lining activation). After the lining installation, the displacement shows a rapid stabilization around the value wcr max ¼ 0.16%D. Table 2 Soil settlement along the longitudinal axis (A A 0 )
Numerical model AFTES (1999)a In situ measurements (Pantet, 1991)a a
wa /wsurf max (%)
wb /wsurf max (%)
wc /wsurf max (%)
46 10–20 15–35 (Mean: 27)
31 40–50 20–65 (Mean: 48)
23 30–50 9–55 (mean: 25)
ð8Þ
Observed in case of shield driven excavation, with lining installation.
Surface settlement w (%D)
0 -0.01
(C-C’) -0.02
2D
-0.03
Tunnel face location y = -1D y = -D/3 y = 0 (face) y = +1D y = +2D (D-D')
-0.04 (C-C')
-0.05 -0.06
0
(D-D’)
z y
x
-0.07 0
0.5
1
1.5
2
2.5
3
3.5
4
Distance from tunnel axis (x/D) Fig. 5. Reference example: surface settlement in transversal section during excavation Line (C C 0 ) is located at y = 2D from the line (D D 0 ) (y = 0).
H. Mroueh, I. Shahrour / Tunnelling and Underground Space Technology 23 (2008) 38–45 Table 3a Determination of the parameter of Peck’s formula for the settlement in the cross section (Fig. 5) Position of the tunnel face
wsurf max (%D)
i/D
Ls/D
Vs/Vexc (%)
(C C 0 ): 1D (C C 0 ): 0D (C C 0 ): +1D (C C 0 ): +2D Profile (D D 0 )
0.01 0.03 0.05 0.06 0.07
1.31 1.21 1.18 1.18 1.17
3.28 3.03 2.96 2.96 2.93
0.04 0.12 0.19 0.23 0.26
Table 3b Comparison of numerical results with suggested values for the parameter of Peck’s formula
i/D Vs/Vexc
Model
Attewell (1977)
O’Reilly and New (1982)
Oteo and Sagaseta (1982)
1.17 0.26%
1.25 1–5%
0.68–1.22 0.5–3%
0.77–1.43
Attewell (1977): ‘i’ varies from 0.8D (sands) to 1.25D (clays), while the ratio Vs/Vexc lies in the range 1% and 5%; O’Reilly and New (1982): ‘i’ varies from 0.68D (sands) to 1.22D (cohesive soil), while the ratio Vs/Vexc lies in the range 0.5% and 3%; Oteo and Sagaseta (1982): ‘i’ varies between 0.77D and 1.43D.
a
It can be observed that the model presented in this paper leads to some results in good agreement with the suggested values for the position of the inflection point ‘i’, but underestimates the volume loss at the ground surface ‘vs’. This last result is dependent to the numerical model and may be optimised with enhanced meshes. 3.4.4. Extension of plasticity Fig. 6a–b shows the extension of plasticity in two transverse sections: the tunnel face (Fig. 6a) and the traverse section located at a distance y = 1D behind the tunnel face (Fig. 6b). It can be observed that plasticity at the tunnel face is concentrated around the tunnel with mainly an extension in the horizontal direction; while behind the tunnel face, the extension of plasticity is mainly observed along the directions +/ 30° with regard to the horizontal axe. It should be emphasised that the extension of the plasticity depends on the soil characteristics, the tunnelling process and the depth of the tunnel. 4. Influence of the tunnel modelling parameters Ldec and adec The proposed model includes the partial release factors Ldec and adec which control the soil movement due to the construction of the tunnel. The determination of these parameters can be carried out by comparison of the model to empirical values flowed by an adjustment on
1
Plastic zone
Z (z/D)
0.5
Tunnel face 0 0
0.5
1
1.5
2
-0.5
-1 X (x/D)
b
z y0
1
x
Plastic zone
Z (z/D)
0.5
0 0
0.5
1
1.5
2
43
y = -1D behind the tunnel face
-0.5
-1 X (x/D)
Fig. 6. Spread of plasticity in transverse section: (a) at the tunnel face (y = 0) and (b) behind the tunnel face (y = 1D).
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H. Mroueh, I. Shahrour / Tunnelling and Underground Space Technology 23 (2008) 38–45
Surface settlement w surf (%D)
0 -0.02 -0.04 Lining installation
-0.06
L = 1D dec L = 5D/3
-0.08
dec
-0.1 -4
-3
-2 -1 0 1 2 Distance from tunnel face (y/D)
3
4
Fig. 7. Influence of the parameter Ldec (length of the unlined zone) on the soil settlement.
-0.05 -0.1
5. Conclusion -0.15
αdec = 0.3 αdec = 0.5
-0.2
αdec = 0.7 -0.25 -4
-3
-2 -1 0 1 2 Distance from tunnel face (y/D)
3
4
Fig. 8. Influence of the factor adec (partial stress release) on the soil settlement.
a
1
Z (z/D)
measurements registered during the first stage of the tunnel construction. This adjustment requires a sensitivity analysis in order to investigate the influence of the factors Ldec and adec on the soil settlement profile.
0
A simplified non-linear three-dimensional numerical model is proposed for the determination of the soil movement induced by TBM. The method uses the factors Ldec and adec, which stand for the length of the unlined zone and the partial stress release, respectively. The value of the parameter Ldec can be taken equal to the tunnel diameter, while the value of adec can be determined by fitting the model to empirical formula then by adjustment on settlement registered during tunnel construction. The capacity of the model is illustrated through an application to a shallow tunnel in soft soils. The work is under progress for the construction of charts which provide values of the factor adec for different tunnels configurations (depth, diameter, lining stiffness, etc.) and soil properties.
b1
0
0.5
1
-1
1.5
2
Z (z/D)
Surface settlement w surf (%D)
0
The parameter Ldec corresponds to the length of the unlined area of the tunnel. It can be approximated to the length of the tunnel boring machine (TBM), which is approximately equal to 1D. The influence of Ldec on the soil movement along the longitudinal profile (A A 0 ) is presented in Fig. 7. It may be noted that an increase of Ldec of about 66% leads to a moderate increase in the maximum settlement (about 15%), without significant influence on the ratio between the settlements behind and ahead the tunnel face. This parameter can then be fixed to Ldec = 1D. The factor adec stands for the ratio of the stress release before the lining installation. Fig. 8 illustrates the influence of a variation of this factor on the soil settlement. It shows that the reduction of adec from 0.5 to 0.3 leads to a reduction of the about 70% of the maximum settlement, whereas an increase of adec from 0.5 to 0.7 leads to an increase of about 160% of the settlement. Fig. 9a and b illustrate the extension of plasticity in the transversal section located at a distance of y = 1D from the tunnel face. It clearly shows that the factor adec highly affects the extension of plasticity in the soil mass. This factor is strongly related to the construction procedure, the tunnel depth and the soil properties. It should evaluated and adjusted using in situ measurements.
0 0
0.5
1
1.5
2
-1 X (x/D)
X (x/D)
Fig. 9. Influence of the factor adec on the spread of plasticity: (a) adec = 0.3 and (b) adec = 0.7.
H. Mroueh, I. Shahrour / Tunnelling and Underground Space Technology 23 (2008) 38–45
References AFTES, 1999. Settlements induced by tunnelling. Recommendations of the working group 16, chairman LEBLAIS, Y., with the collaboration of Andre, D.C., Chapeau, P. Dubois, J.P. Gigan, J. Guillaume, E. Leca, A. Pantet, G. Riondy, Special issue of ‘‘Tunnels et Ouvrages souterrains’’ available online www.aftes.asso.fr. 1st issue published in 1995, in French: Tunnels et Ouvrages Souterrains, vol. 132, pp. 373– 395. Attewell, P.B., 1977. Ground movements caused by tunnelling in soil. In: Geddes, Cardiff J.D. (Ed.), . In: 1st Conf. on Large Ground Movements and Structures. Pentech Press, London, pp. 812–948. Cheng , L.T., Dasari, G.R., Leung, C.F. Chow, Y.K., 2002. A novel FE technique to predict tunnelling induced ground movements in clays. In: Queck, S.T., Ho, D.W.S. (Eds.), Proceedings of the 15th KKCNN Symposium on Civil Engineering. Dias, D., Kastner, R., Maghazi, M., 2000. Three-dimensional simulation of slurry shield tunnelling. Proceedings of International Symposium on Geotechnical Aspects of Underground Construction in Soft Ground. Balkema, Rotterdam, pp. 351–356. Galli, G., Grimaldi, A., Leonardi, A., 2004. Three-dimensional modelling of tunnel excavation and lining. Comput. Geotech. 31, 171–183. Herrenknecht, M., 1998. New developments in large-diameter tunnel design manufacture and utilisation for world-wide projects. In: World Tunnel Congress’98 on Tunnels and Metropolises. Balkema, Sao Paulo, pp. 869–875. Kurihara, K., 1998. Current mechanized shield tunnelling methods in Japan. In: World Tunnel Congress’98 on Tunnels and Metropolises. Balkema, Sao Paulo, pp. 615–622. Kuwahara, S., 1999. Mechanized and automated tunnelling in Japan. In: Proceedings of the International Symposium on Ground Challenges and Expectations in Tunnelling Projects, Cairo, Egypt, pp. 81–91.
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Moraes Jr., A.H., 1999. Three-dimensional numerical simulation of tunnels excavated with NATM. Msc thesis, University of Brazilia, Brazil (in Portugese). Mroueh, H., Shahrour, I., 1999. Use of sparse iterative methods for the resolution of three-dimensional soil/structure interaction problems. In: Int. J. Numer. Anal. Methods Geomech., 23. Wiley & sons (Ltd), pp. 1961–1975. O’Reilly, M.P., New, B.M., 1982. Settlements above tunnels in the United Kingdom – their magnitude and prediction. In: Proceedings of the International Conference Tunnelling ’82. Institution of Mining and Metallurgy, London, pp. 55–64. Oteo, C.S., Sagaseta, C., 1982. Prediction of settlements due to underground openings. In: Int. Symp. on Numerical Methods in Geomechanics, Zurich, 13–17 September, pp. 653–659. Panet, M., Guenot, A., 1982. Analysis of convergence behind the face of a tunnel. In: Proceedings of the International Conference Tunnelling ’82. Institution of Mining and Metallurgy, London, 187–204. Pantet, A. 1991. Creusement des galeries a` faible profondeur a` l’aide d’un tunnelier a` pression de boue. Mesures in situ et e´tude the´orique du champ de de´placement, Ph.D. thesis, INSA, Lyon. Peck, R.B., 1969. Deep excavations and tunnelling in soft ground. In: Proc. 7th Int. Conf. on Soil Mechanics and Foundation Engineering, Mexico City, State-of-the-Art volume, pp. 225–290. Sagaseta, C., 1987. Evaluation of surface movements above tunnels: A new approach. Colloque Interactions sols-structures, Paris, ENPC press, 445–452. Shahrour, I., 1992. PECPLAS: A finite element package for the resolution of geotechnical problems. In: Colloque Ge´otechnique et Informatique. ENPC press, Paris, pp. 327–334. Van der Vorst, H.A., 1992. Bi-CGSTAB: a fast and smoothing converging variant of bi-CG for the solution of non-symmetric linear systems. SIAM J. Sci. Statist. Comput. 13 (2), 631–644. Ward, W.H., Pender, M.J., 1981. Tunnelling in soft ground – General report. In: 10th Int. Conf. on Soil Mechanics and Foundation Engineering, vol. 4, Stockholm, pp. 261–275.
Tunnelling and Underground Space Technology 23 (2008) 38–45
Tunnelling and Underground Space Technology incorporating Trenchless Technology Research
www.elsevier.com/locate/tust
A simplified 3D model for tunnel construction using tunnel boring machines H. Mroueh *, I. Shahrour Laboratoire de Me´canique de Lille (UMR 8107), Universite´ des Sciences et Technologies de Lille, F-59655 Villeneuve d’Ascq, France Received 5 September 2006; received in revised form 21 November 2006; accepted 28 November 2006 Available online 24 January 2007
Abstract This paper includes a presentation of a simplified three-dimensional numerical model for the prediction of soil movement induced during tunnel construction using tunnel boring machines (TBM). The model is based upon the generalization of the convergence-confinement concept to 3D tunnel construction. It uses two parameters (Ldec and adec) which stand for the length of the unlined zone and the partial stress release, respectively. The value of the parameter Ldec can be taken equal to the tunnel diameter, while the value of adec can be determined by fitting the model to empirical formula, and then adjusted based on settlement registered during tunnel construction. The capacity of the model is illustrated through an application to a shallow tunnel in soft soil. The comparison of the numerical results to those suggested by different authors shows good agreement. Ó 2006 Elsevier Ltd. All rights reserved. Keywords: Tunnel boring machine; TBM design; Finite element method; Three-dimensional; Non-linear; Shield tunnelling; Convergence-confinement
1. Introduction Construction of tunnels in soft soils induces generally soil movement, which could seriously affect the stability and integrity of existing structures (pile foundations, buildings. . .). In order to reduce such movement, in particular in urban areas, contractors use more and more the tunnel boring machines (TBM) for the construction of tunnels. Indeed, thanks to the application of a face pressure and to the temporary support, the TBM allows to reduce the soil disturbance due to tunneling, providing enhanced safety to existing structures (Herrenknecht, 1998; Kurihara, 1998; Kuwahara, 1999). Analysis of the impact of the tunnel construction using TBM on the soil movement requires the solution of large 3D non-linear soil–structure interaction problem. Non-linearity results from the non-linear behavior of geomaterials, the condition at the soil–structure interface (soil – grouting-lining, soil – shield,) and the evolution of *
Corresponding author. E-mail address: [email protected] (H. Mroueh).
0886-7798/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.tust.2006.11.008
the geometry during excavation. The 3D aspect is due to the significant stress disturbance and soil movement induced ahead the excavation front. 3D elastic analyses conducted by Panet and Guenot (1982) showed ground convergence at the tunnel face which was equal to about 27% of the total settlement. Higher values, up to 50%, were observed in field measurements and computational analyses in soft ground (Moraes, 1999). Finite element modeling of the tunnel construction using TBM requires also the consideration of the complex tunnel process which includes the advance of the TBM, the application of the face pressure, the soil excavation, the installation of an immediate support behind the rotating front, the installation of the definitive support (lining ring) and the tail void grouting. A realistic consideration of these issues in the 3D calculation constitutes a high challenge (Dias et al., 2000; Cheng et al., 2002; Galli et al., 2004), because of the large effort for numerical modeling and calculation and the large uncertainties concerning the interaction between the shield and the soil, the behavior of the grouting, and the distribution of the tail void. Consequently, the use of this approach in tunnel design is still limited, because it requires impor-
H. Mroueh, I. Shahrour / Tunnelling and Underground Space Technology 23 (2008) 38–45
Step i Phase i α dec * σ(i- 1)
Ldec
Llin Step Phasei+1 i+1
(1−αdec) * σ(i) α dec * σ(i)
revêtement lining
p
grouting coulis
Fig. 1. Method used for the tunnel construction using TBM.
tant modeling effort and computational time. In order to overcome this difficulty, a simplified method is proposed in this paper to model the TBM tunneling process using a three-dimensional model based on the convergenceconfinement method (Panet and Guenot, 1982) with two release parameters: adec and Ldec, which stand for the partial stress release and the length of the unlined zone, respectively (Fig. 1). This method can be easily implemented and employed using existing programs based on either the finite element or the finite difference method. The paper presents successively, the proposed method, its application to a model tunnel and the sensitivity of the method to the release factors adec and Ldec. 2. Presentation of the numerical model Numerical modelling of the tunnel construction using TBM constitutes a hard task, because it requires consideration of complex aspects such as the soil excavation, the overcut or annular space between the jacking pipe and the excavation, the application of the face pressure, the installation of the definitive support constituted of lining rings and the grouting of the annular space. It also requires the description of the non-linear behavior of both the soil and the lining and the condition at the soil–structure interface. Modelling of the tunnel construction is also three-dimensional, because the TBM induces an important stress disturbance and soil movement ahead the excavation front. Modelling of the annular space between the ground and the lining extrados is still problematic, because of the difficulties to collect effective data on the distribution and grouting of this space. Up to now, it seems very difficult to consider the abovementioned issues in the practical design of tunnels. In order to overcome this difficulty, a simplified method is proposed
39
in this paper to model the TBM process using a threedimensional model based on the convergence-confinement method (Panet and Guenot, 1982). This method uses a step-by-step procedure. Each step corresponds to the progression of the tunnel face by a distance Llin (Fig. 1). At each step of the procedure, the stress release around the tunnel head is modeled using the parameters adec and Ldec, which stand for the ratio of the stress release and the length of the unlined zone, respectively. The calculation procedure at the step (incr) includes: (a) Determination of the incremental force resulting from the soil excavation (DF). This force is equal to the difference between the nodal force vector (F(incr)) due to the external forces (self-weight, surface loads, front pressure, . . .) and the nodal internal forces at the previous step ‘incr-1’; calculation of DF is carried out using the following expression: Z t ðincrÞ DF ¼ F Be rðincr1Þ dV ð1Þ V ðincrÞ
V(incr) represent the volume of the soil mass at the step (incr); Be is the strain interpolation matrix which contains the spatial derivatives of the interpolation functions (e = BeÆu; u nodal displacement); r(incr1) denotes the stress tensor at the previous step (incr1).In order to take into account the partial deconfinement resulting from the tunnel construction process (overcut, injection of the annular void, installation of the definitive tunnel support,. . .), a parameter adec is used for considering the partial release on the unsupported section of the tunnel; the length of this section is assumed to be equal to Ldec. The incremental nodal force vector in this section (DF) is transformed using the following expression: DF 0 ¼ adec DF
ð2Þ
(b) Activation of the lining elements located in the new section and a full release of stresses in this section. (c) Application of the face pressure ‘p’ (Fig. 1); the pressure is assumed to be constant with depth; it corresponds to a ‘compressed-air pressure’ TBM. Note that this pressure can vary with depth to model ‘slurry shield’ machines or ‘earth pressure balance’ (EPB) machines. The soil movement is controlled through the partial release factor adec and the parameter Ldec which enable users to consider the influence of the void space and grouting around the tunnel. The determination of these parameters can be carried out by an adjustment procedure using empirical models and measurements during tunnel construction. The following section presents the application of the proposed method to a model tunnel, which will be followed by a sensitivity analysis of the model to the variation of the partial release parameters adec and Ldec. This analysis allows the elaboration of a methodology for the determination of these factors.
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3. Application to a tunnel model
Table 1 Properties of geomaterials used in the tunnel model
The proposed method has been implemented in the finite element code PECPLAS which provides flexible features for the analysis of three-dimensional and non-linear soil– structure interaction problems (Shahrour, 1992; Mroueh and Shahrour, 1999). This program uses a sparse storage scheme for the stiffness matrix, and the bi-CGSTAB iterative method (Van der Vorst, 1992) coupled to the SSOR preconditioning operator (Successive Symmetrical OverRelaxation) for the solution of the resulting linear systems.
Geomaterial
E (MPa)
m
c 0 (MPa)
u (°)
w (°)
c (kN/m3)
Soil Lining
30 35,000
0.3 0.25
0.005
27
5
20 25
3.1. Geometry and numerical parameters Fig. 2 shows the tunnel model geometry. The tunnel is characterized by its outer diameter D = 7.5 m, depth H = 2.5 D and lining thickness e = 0.5 m. The distance of the tunnel centre to the bottom boundary (rigid substratum) is assumed to be equal to 2.5D. The soil behavior is assumed to be governed by an elastic perfectly–plastic constitutive relation based on the Mohr–Coulomb criterion with a non-associative flow rule. The yield function and the plastic potential are given by rffiffiffiffiffi pffiffiffiffiffi J2 f ¼ p sin u þ J 2 cos h sin u sin h C cos u ð3Þ 3 rffiffiffiffiffi pffiffiffiffiffi J2 sin u sin h ð4Þ g ¼ p sin w þ J 2 cos h 3 where C, u and w designate the soil cohesion, friction angle and dilatancy angle, respectively; p, J2 and h stand for the mean stress, second invariant of the deviatoric stress tensor and Lode angle, respectively. Their expressions are given by p ¼ rii =3 1 J 2 ¼ sij sij where sij ¼ rij pdij 2 ! pffiffiffi 1 3 3 J3 sij sjk ski 1 : 3=2 h ¼ sin where J 3 ¼ 3 2 J2 3
ð5Þ ð6Þ ð7Þ
H = 2.5D
Table 1 summarizes the characteristics of both the soil and the lining. Homogeneous silty sand is considered with the following characteristics: friction angle u = 27°, cohesion C = 5 kPa, dilatancy angle w = 5°, Young’s modulus E = 30 MPa, and Poisson’s ratio m = 0.3. The lining is assumed to be governed by a linear-elastic behavior with a Young’s modulus E = 35,000 MPa and a Poisson’s ratio m = 0.25. The initial stress in the soil media is determined using a coefficient of lateral earth pressure at rest K0 = 0.5 and an effective bulk unit weight of the soil of c 0 = 10 kN/m3. 3.2. Finite element mesh Finite element analysis was carried out using the mesh presented in Fig. 3. This mesh consists of 2214 20-nodes hexahedral elements, which give rise to 10,494 nodes and 28,471 degrees of freedom. The lateral boundaries of the model are located a distance 4D from the tunnel axis in order to minimize their interaction with the tunneling construction. The longitudinal length of the mesh is fixed to 8D, and the tunnel excavation is performed for a final position of 4D. This mesh is used to illustrate the application of the proposed method. In tunnels design, an enhanced mesh must be used in order to well capture the soil deformation and the development of plasticity around the tunnel. An extension of the lateral boundaries of the soil mass should also be considered. 3.3. Calculation process Computation was carried out in 12 steps using the following parameters for the excavation modelling: ratio of stress release adec = 0.5, length of the unlined zone Ldec = 1D, and length of the excavated section at each step Llin = D/3. The face pressure is assumed to be uniform and equal to p ¼ r0h , where r0h stands for the initial axial stress at the tunnel axis. 3.4. Results
D = 7.5m e= 50 cm
Fig. 2. Numerical example used for the illustration of the model performances.
3.4.1. Settlement along longitudinal profiles Fig. 4a shows the evolution of the surface soil settlement along the longitudinal axis (A–AP 0 ) during the tunnel construction. It can be observed that the maximal surface settlement increases with the tunnel progression, and tends to stabilize at a value of wsurf max ¼ 0:07%D (D denotes the tunnel outer diameter. The stabilisation of the surface settlement
H. Mroueh, I. Shahrour / Tunnelling and Underground Space Technology 23 (2008) 38–45
41
u=0 v=0
4.5D v=0 u=0
z
8D
y
x
4D u=v=w=0
Fig. 3. 3D Finite element mesh used in numerical analysis (2214 20-node elements; 10,494 nodes; 28,471 ddl).
Fig. 4. Reference example: vertical displacement along longitudinal section: (a) at the ground surface, along the line (A–A 0 ) and (b) at the tunnel crown, along the line (B–B 0 ).
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H. Mroueh, I. Shahrour / Tunnelling and Underground Space Technology 23 (2008) 38–45
is observed after an excavation length of 3D). It can be noted that the surface settlement (wa) at the tunnel face is constant during the tunnel progression. It is equal to 0.03%D, which corresponds to 46% of the maximal value of settlement at the end of the simulation wsurf max . The amount of settlement induced just before the lining installation (wa + wb) is about 77% of wsurf max (0.05%D), which means that 31% of the total settlement is induced in the unlined zone of the tunnel. This result shows that 23% of the total settlement (wc) is due to the complete release of the confinement (1 adec). Table 2 summarises the proportion of settlement, in comparison with some published results. It can be noted that these values are in good agreement with observed or computed values reported by various authors. The amount of settlement in the front of the tunnel face wa appears higher than reported values, but the cumulative settlement wa + wb is in good agreement. Note that these values highly depend on the factors adec and Ldec.
The spreading of displacements from the tunnel to the surface can be analysed through a ‘softening’ coefficient or factor of diffusion Rdif, according to (AFTES, 1999): cr Rdif ¼ wsurf max =wmax
Vertical displacements at the soil surface and the tunnel crown give Rdif = 0.4, which means that 40% of the vertical displacement at the crown tunnel is transmitted to the surface. This value agrees with empirical approaches and in situ observations. Ward and Pender (1981) reported values for the diffusion coefficient varying from 0.2 (for sands) to 0.74 (for over-consolidated clays), while Sagaseta (1987) reported values between 0.2 (for frictional material) and 0.67 (for low frictional clay). 3.4.3. Settlement in a cross direction Fig. 5 shows the evolution of the settlement in the transverse section along the axis C C 0 . It can be observed that a moderate settlement appears (0.01%D) when the tunnel face is about 1D behind the cross section (C C 0 ), then the settlement increases when the tunnel face crosses the traverse section and tends to stabilize when the distance between the tunnel face and the section (C C 0 ) exceeds +2D. Numerical results illustrated in Fig. 5 were used for the determination of the parameters of Peck formula (Peck, 1969): the location of the point of inflection of the settlement curve ‘i’, the length of the settlement profile ‘Ls’ and the volume loss at the ground surface ‘vs’. Results are summarised in Table 3a. It is noted that the distance ‘‘i’’ and the length ‘Ls’ decrease with the progression of the tunnel face, and tend to stabilise to a value of i = 1.17D when the relative distance between the cross section and the tunnel face exceeds +1D. The volume loss at the surface is estimated to Vs = 0.26%Vexc, where Vexc denotes for the volume of excavated soil. The parameters of Peck formula can be estimated from semi-empirical methods. Table 3b shows a comparison of the numerical results with the empirical expressions proposed by
3.4.2. Crown displacement Fig. 4b shows the variation of vertical displacement along the longitudinal axis (BB 0 ) during the tunnel construction, at the tunnel crown location wcr. It can be observed that the major part of the crown displacement results from the TBM progression. Indeed, 40% (respectively 90%) of the total displacement at the tunnel crown is observed at the TBM passage (respectively at the lining activation). After the lining installation, the displacement shows a rapid stabilization around the value wcr max ¼ 0.16%D. Table 2 Soil settlement along the longitudinal axis (A A 0 )
Numerical model AFTES (1999)a In situ measurements (Pantet, 1991)a a
wa /wsurf max (%)
wb /wsurf max (%)
wc /wsurf max (%)
46 10–20 15–35 (Mean: 27)
31 40–50 20–65 (Mean: 48)
23 30–50 9–55 (mean: 25)
ð8Þ
Observed in case of shield driven excavation, with lining installation.
Surface settlement w (%D)
0 -0.01
(C-C’) -0.02
2D
-0.03
Tunnel face location y = -1D y = -D/3 y = 0 (face) y = +1D y = +2D (D-D')
-0.04 (C-C')
-0.05 -0.06
0
(D-D’)
z y
x
-0.07 0
0.5
1
1.5
2
2.5
3
3.5
4
Distance from tunnel axis (x/D) Fig. 5. Reference example: surface settlement in transversal section during excavation Line (C C 0 ) is located at y = 2D from the line (D D 0 ) (y = 0).
H. Mroueh, I. Shahrour / Tunnelling and Underground Space Technology 23 (2008) 38–45 Table 3a Determination of the parameter of Peck’s formula for the settlement in the cross section (Fig. 5) Position of the tunnel face
wsurf max (%D)
i/D
Ls/D
Vs/Vexc (%)
(C C 0 ): 1D (C C 0 ): 0D (C C 0 ): +1D (C C 0 ): +2D Profile (D D 0 )
0.01 0.03 0.05 0.06 0.07
1.31 1.21 1.18 1.18 1.17
3.28 3.03 2.96 2.96 2.93
0.04 0.12 0.19 0.23 0.26
Table 3b Comparison of numerical results with suggested values for the parameter of Peck’s formula
i/D Vs/Vexc
Model
Attewell (1977)
O’Reilly and New (1982)
Oteo and Sagaseta (1982)
1.17 0.26%
1.25 1–5%
0.68–1.22 0.5–3%
0.77–1.43
Attewell (1977): ‘i’ varies from 0.8D (sands) to 1.25D (clays), while the ratio Vs/Vexc lies in the range 1% and 5%; O’Reilly and New (1982): ‘i’ varies from 0.68D (sands) to 1.22D (cohesive soil), while the ratio Vs/Vexc lies in the range 0.5% and 3%; Oteo and Sagaseta (1982): ‘i’ varies between 0.77D and 1.43D.
a
It can be observed that the model presented in this paper leads to some results in good agreement with the suggested values for the position of the inflection point ‘i’, but underestimates the volume loss at the ground surface ‘vs’. This last result is dependent to the numerical model and may be optimised with enhanced meshes. 3.4.4. Extension of plasticity Fig. 6a–b shows the extension of plasticity in two transverse sections: the tunnel face (Fig. 6a) and the traverse section located at a distance y = 1D behind the tunnel face (Fig. 6b). It can be observed that plasticity at the tunnel face is concentrated around the tunnel with mainly an extension in the horizontal direction; while behind the tunnel face, the extension of plasticity is mainly observed along the directions +/ 30° with regard to the horizontal axe. It should be emphasised that the extension of the plasticity depends on the soil characteristics, the tunnelling process and the depth of the tunnel. 4. Influence of the tunnel modelling parameters Ldec and adec The proposed model includes the partial release factors Ldec and adec which control the soil movement due to the construction of the tunnel. The determination of these parameters can be carried out by comparison of the model to empirical values flowed by an adjustment on
1
Plastic zone
Z (z/D)
0.5
Tunnel face 0 0
0.5
1
1.5
2
-0.5
-1 X (x/D)
b
z y0
1
x
Plastic zone
Z (z/D)
0.5
0 0
0.5
1
1.5
2
43
y = -1D behind the tunnel face
-0.5
-1 X (x/D)
Fig. 6. Spread of plasticity in transverse section: (a) at the tunnel face (y = 0) and (b) behind the tunnel face (y = 1D).
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H. Mroueh, I. Shahrour / Tunnelling and Underground Space Technology 23 (2008) 38–45
Surface settlement w surf (%D)
0 -0.02 -0.04 Lining installation
-0.06
L = 1D dec L = 5D/3
-0.08
dec
-0.1 -4
-3
-2 -1 0 1 2 Distance from tunnel face (y/D)
3
4
Fig. 7. Influence of the parameter Ldec (length of the unlined zone) on the soil settlement.
-0.05 -0.1
5. Conclusion -0.15
αdec = 0.3 αdec = 0.5
-0.2
αdec = 0.7 -0.25 -4
-3
-2 -1 0 1 2 Distance from tunnel face (y/D)
3
4
Fig. 8. Influence of the factor adec (partial stress release) on the soil settlement.
a
1
Z (z/D)
measurements registered during the first stage of the tunnel construction. This adjustment requires a sensitivity analysis in order to investigate the influence of the factors Ldec and adec on the soil settlement profile.
0
A simplified non-linear three-dimensional numerical model is proposed for the determination of the soil movement induced by TBM. The method uses the factors Ldec and adec, which stand for the length of the unlined zone and the partial stress release, respectively. The value of the parameter Ldec can be taken equal to the tunnel diameter, while the value of adec can be determined by fitting the model to empirical formula then by adjustment on settlement registered during tunnel construction. The capacity of the model is illustrated through an application to a shallow tunnel in soft soils. The work is under progress for the construction of charts which provide values of the factor adec for different tunnels configurations (depth, diameter, lining stiffness, etc.) and soil properties.
b1
0
0.5
1
-1
1.5
2
Z (z/D)
Surface settlement w surf (%D)
0
The parameter Ldec corresponds to the length of the unlined area of the tunnel. It can be approximated to the length of the tunnel boring machine (TBM), which is approximately equal to 1D. The influence of Ldec on the soil movement along the longitudinal profile (A A 0 ) is presented in Fig. 7. It may be noted that an increase of Ldec of about 66% leads to a moderate increase in the maximum settlement (about 15%), without significant influence on the ratio between the settlements behind and ahead the tunnel face. This parameter can then be fixed to Ldec = 1D. The factor adec stands for the ratio of the stress release before the lining installation. Fig. 8 illustrates the influence of a variation of this factor on the soil settlement. It shows that the reduction of adec from 0.5 to 0.3 leads to a reduction of the about 70% of the maximum settlement, whereas an increase of adec from 0.5 to 0.7 leads to an increase of about 160% of the settlement. Fig. 9a and b illustrate the extension of plasticity in the transversal section located at a distance of y = 1D from the tunnel face. It clearly shows that the factor adec highly affects the extension of plasticity in the soil mass. This factor is strongly related to the construction procedure, the tunnel depth and the soil properties. It should evaluated and adjusted using in situ measurements.
0 0
0.5
1
1.5
2
-1 X (x/D)
X (x/D)
Fig. 9. Influence of the factor adec on the spread of plasticity: (a) adec = 0.3 and (b) adec = 0.7.
H. Mroueh, I. Shahrour / Tunnelling and Underground Space Technology 23 (2008) 38–45
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Moraes Jr., A.H., 1999. Three-dimensional numerical simulation of tunnels excavated with NATM. Msc thesis, University of Brazilia, Brazil (in Portugese). Mroueh, H., Shahrour, I., 1999. Use of sparse iterative methods for the resolution of three-dimensional soil/structure interaction problems. In: Int. J. Numer. Anal. Methods Geomech., 23. Wiley & sons (Ltd), pp. 1961–1975. O’Reilly, M.P., New, B.M., 1982. Settlements above tunnels in the United Kingdom – their magnitude and prediction. In: Proceedings of the International Conference Tunnelling ’82. Institution of Mining and Metallurgy, London, pp. 55–64. Oteo, C.S., Sagaseta, C., 1982. Prediction of settlements due to underground openings. In: Int. Symp. on Numerical Methods in Geomechanics, Zurich, 13–17 September, pp. 653–659. Panet, M., Guenot, A., 1982. Analysis of convergence behind the face of a tunnel. In: Proceedings of the International Conference Tunnelling ’82. Institution of Mining and Metallurgy, London, 187–204. Pantet, A. 1991. Creusement des galeries a` faible profondeur a` l’aide d’un tunnelier a` pression de boue. Mesures in situ et e´tude the´orique du champ de de´placement, Ph.D. thesis, INSA, Lyon. Peck, R.B., 1969. Deep excavations and tunnelling in soft ground. In: Proc. 7th Int. Conf. on Soil Mechanics and Foundation Engineering, Mexico City, State-of-the-Art volume, pp. 225–290. Sagaseta, C., 1987. Evaluation of surface movements above tunnels: A new approach. Colloque Interactions sols-structures, Paris, ENPC press, 445–452. Shahrour, I., 1992. PECPLAS: A finite element package for the resolution of geotechnical problems. In: Colloque Ge´otechnique et Informatique. ENPC press, Paris, pp. 327–334. Van der Vorst, H.A., 1992. Bi-CGSTAB: a fast and smoothing converging variant of bi-CG for the solution of non-symmetric linear systems. SIAM J. Sci. Statist. Comput. 13 (2), 631–644. Ward, W.H., Pender, M.J., 1981. Tunnelling in soft ground – General report. In: 10th Int. Conf. on Soil Mechanics and Foundation Engineering, vol. 4, Stockholm, pp. 261–275.