3D Numerical Simulation of Heading Face Support in ...

The 12th International Conference of International Association for Computer Methods and Advances in Geomechanics (IACMAG) 1-6 October, 2008 Goa, India

3D Numerical Simulation of Heading Face Support in Partially Saturated Soils for Shield Tunnelling Felix Nagel, Janosch Stascheit, Günther Meschke Ruhr University Bochum, Germany Keywords: Finite Element Method, Shield Tunnelling, Theory of Porous Media, Partially Saturated Soil ABSTRACT: This paper is concerned with the modelling of the cutting face support within a 3D-simulation model for shield tunnelling. While two-phase formulations for partially saturated soils are, in general, sufficient for numerical analysis in tunnelling, a three phase formulation is required in cases, where compressed air is used to prevent water inflow. This technique is frequently used in compressed air shields and in case of maintenance interventions, where the repair of the cutting wheel is performed under compressed air conditions. In the presentation, a three phase model for partially saturated soil is formulated within the framework of the Theory of Porous Media and the effective stress concept considering large deformations. The soil skeleton, the pore water and the pore air are considered as separate phases. The constitutive relations are described via the soil characteristic curve, relative permeabilities and the stress strain relation for the soil skeleton. The paper also gives an overview on different possibilities of modelling of heading face support with such a model. Selected results from simulations of heading face support by means of compressed air in shield tunnelling are presented. These analyses demonstrate the capability of the presented model to describe the main time variant effects of two phase flow in partially saturated soils during mechanized tunnelling.

1 Introduction In mechanized tunnelling surface settlements ahead, along and behind the TBM, resulting from interactions between the face support, the conical TBM, the tail void grouting and the lining with the surrounding partially or fully saturated soil, may occur and have to be controlled by the appropriate choice of support measures. Since, in particular in urban areas with sensitive existing infrastructure severe restrictions concerning the tolerable tunnelling-induced surface settlements are set by regulations, reliable prognoses of these settlements are an indispensable prerequisite for the design and a valuable tool for decisions to be made during the construction of mechanized tunnelling. The representation of these interactions within a numerical model requires, besides the realistic representation of all relevant components involved in shield tunnelling (the lining, the tail void grouting, the hydraulic jacks and the different types of face support) a sufficiently realistic model for the fully or partially saturated soil. Whereas for the support by means of a support liquid or earth slurry a two phase soil model is generally sufficient even in the case of partially saturated soils, a three phase model considering compressed air as a separate phase has to be used if the face support by means of compressed air should be simulated numerically. Only a relatively few number of simulation models for mechanized tunnelling have been proposed that allow for a detailed consideration of the shield supported tunnel construction as a time dependent problem (Komiya et al., 1999, Abu-Krisha, 1998. Kasper and Meschke, 2004). A relatively comprehensive and automated three-dimensional FEmodel for simulations of shield-driven tunnels in soft, water saturated soil proposed by Kasper and Meschke, 2004 has been successfully employed for the investigation of various design and process parameters (Kasper and Meschke, 2005, 2006). Coupled numerical models for the simultaneous flow of air and water in partially saturated soil within the Theory of Porous Media (Bluhm and de Boer, 1997; Ehlers and Bluhm, 2002; Ehlers, 1996; Lewis and Schrefler, 1999) is a topic of pertinent research in computational geomechanics (see, e.g. Ehlers and Graf, 2002, Sanavia et al., 2002). Applications of three phase models for the analysis of heading face support by means of compressed air with multi-phase soil models have been performed by (Öttl, 2003). The influence of interactions between supporting liquid and the pore water on the stability of the soil in front of the heading face was examined by measurements and analytical models (Bezuijen et al., 2001, Broere 2002). However, no numerical model seems to exist that combines both the realistic simulation of the construction process with an advanced, fully coupled triphasic soil model capable to take into account the space and time variant 3835

interactions between the heading face support and the pore fluids on the soil deformations during tunnel advance. In the framework of the European Integrated Project TUNCONSTRUCT (URL: http//:www.tunconstruct.org) a finite element model (ekate) based on the object-oriented FE-code KRATOS (Dadvand et al., 2002) is being developed for the simulation of shield driven tunnels as a part of an Integrated Design Support (Meschke et al., 2007). This model is characterized by a realistic consideration of the construction process in mechanized tunnelling involving all relevant components and their complex interactions (Nagel, Stascheit and Meschke, 2008). It has been supplemented with an automatic model generator allowing for a user-friendly generation of the simulation model (Stascheit, Nagel and Meschke, 2007). Consideration of large deformations, which may be relevant for analyses of TBM tunnelling in squeezing ground conditions, are taken into account in the model. In this paper, the three-phase soil model, implemented within the above described simulation model, and the capabilities of this model to simulate heading face support in hydro and EPB shield tunnelling are presented. The paper is organized as follows: Section 2 describes the underlying theory of partially saturated soils. Section 3 addresses consideration of different types of heading face support within this model. The applicability of the model for numerical simulation of heading face support is demonstrated in Section 4.

2

Three-phase soil model for partially saturated soils

2.1 Partially saturated soil as a triphasic material Partially saturated soil consists of three phases: the solid soil skeleton and the fluid phases water and air moving through the connected pore structure of the soil. In the context of engineering problems the exact geometry of the pore volume and the interactions of the phases within this pore volume are unknown and of minor interest. An up-scaling procedure is required to describe the microstructural processes in terms of averaged quantities on a macroscopic scale. In the proposed model the Theory of Porous Media (TPM) (Bluhm and de Boer, 1997, Schrefler and Simoni, 1988) is used. Within the TPM each phase has its own state of motion (see Figure 1) and is represented via its volume fraction n

α

β

and for the fluid phases (β =w[ater], a[ir]) via the degree of saturation S of the pore volume:

nβ =

dv β = nSβ dv

For the fluid phases their motion within the pore volume is expressed in terms of the velocity skeleton, which leads in an integral form to the well known DARCY velocity

(1)

v βs

~ v βs = nSβ v βs

relative to the soil

(2)

Figure 1: Independent motion of the soil constituents soil skeleton, water and air By averaging the states and interactions of the phases and their mixture using the TPM the problem can be expressed 3836

on a macroscopic scale by its governing balance equations. Considering geometrically nonlinear continuum mechanics the pore volume can be derived from the mass balance of the soil as a function of the soil skeleton displacements

us as

( )

n u s = 1 − (1 − n 0 )e − div u

s

(3)

The three phase model is characterized by balance equations for each of the phases expressed in terms of the actual (deformed) configuration: the overall momentum balance of the mixture

div σ + ρg = 0

(4)

where σ denotes the overall CAUCHY stress tensor of the mixture and ρ the averaged density of the mixture and the balance equations of the fluid phases, assuming isothermal conditions and neglecting phase interchange:

0=

(

)

D s nSβ ρ β + nSβ ρ β div v β . Dt

(5)

In the model air is treated as a compressible and water as an incompressible fluid. The primary variables of the model are chosen to be the two fluid pressures and the soil skeleton displacements. The stress-strain relation of the soil ˆ s according to BISHOP’s formulation (Bishop, 1956) for three skeleton is formulated in terms of effective stresses σ phase continua

(

) ( )(

)

σˆ s = σ − σ a + χ S w σ a − σ β ,

(6)

considering the stress states σ of the fluids. In general, the BISHOP parameter χ is a material function of the soil depending on the water saturation. Within the presented model the BISHOP parameter is assumed to be equal to the water saturation S w . This is a common assumption and holds for a wide range of soils. The air phase is treated as an ideal gas, using a linear relation between pressure and density after BOYLE-MARRIOT’s law. The water content of the pore volume S w is described by the soil characteristic curve after VAN GENUCHTEN (van Genuchten and Nielson, 1985) (see Figure 2) as β

S =S w

w min

+ (S

w max

−S

w min

⎡ ⎛ pc ) ⎢1 + ⎜⎜ b ⎢⎣ ⎝ p r

⎞ ⎟⎟ ⎠

n

⎤ ⎥ ⎥⎦

−m

,

(7)

w w ( S min ) are upper (lower) limits of the water saturation and p br , m and n are model parameters. p c where S max

indicates the pressure difference between water and air, also denoted as capillary pressure. Due to capillary effects water is able to rise within the pore tubes against an air pressure. For a negative or low capillary pressure the water can stay in all pores; with rising capillary pressure, however, the water can only exist in the smaller capillaries. Hence, the saturation decreases.

Figure 2. Pressure dependent saturation of the pore volume after VAN GENUCHTEN (van Genuchten and Nielson, 1985) (left), comparison of calculated saturation-dependent relative permeabilities and measurements (Mualem, 1976) (right)

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The fluid flow ~ v βs is described in terms of DARCY’s law β

(

)

~v βs = − k grad p β − ρ β g , γβ where the permeability β rel

k . k

β rel

kβ

(8)

of the soil is given by the product of its intrinsic permeability k 0 and a relative permeability

as a function of the saturation, which is equal to one if only the fluid β fills the whole pore volume and

decreases for partially saturated case. This function is derived directly from the soil characteristic curve. For the flow of w is obtained as (Mualem, 1976) the water phase, k rel

⎡ w k rel = S e ⎢1 − ⎛⎜1 − S e ⎣ ⎝

( )

Se =

with

w S w − S min w w S max − S min

2

1

m

m ⎞ ⎤ , ⎟ ⎥ ⎠ ⎦

(9)

(10)

see Figure 2. 2.2 Computational Aspects To allow for the modular implementation of different material models for soil in a large deformation context the spectral decomposition of the deformation tensor was used (Simo, 1992; Simo and Meschke 1993). In the context of the finite element formulation of the model, the balance equations (4,5) are transformed to their corresponding weak forms and discretized in space and time. For the spatial discretization quadratic LAGRANGEan shape functions are used for the displacement field and linear approximations for the gaseous and liquid pressure while for the temporal integration the midpoint rule is adopted. The solution of the highly nonlinear discretized three-phase problem is based on NEWTON’s method together with a consistently linearized tangent matrix. For further details of the triphasic formulation for soils and its implementation see (Nagel, Stascheit and Meschke, 2007). The proposed model has been implemented into the Finite Element software package ekate. This software, which is specifically designed for numerical simulations of mechanized tunnelling is based upon the finite element kernel KRATOS. The triphasic model has passed validation by means of the simulation of THERZAGHI’s consolidation problem and the back-analysis of dewatering of a sand column tested in laboratory (Liakopolous, 1965).

3 Numerical modelling of heading face support For shield supported tunnel advance in a closed mode the heading face beneath the groundwater level may be supported by one of the following support measures; (i) support by an earth slurry, (ii) a bentonite suspension or (iii) by means of compressed air. While in earth slurry shields the pressure of the slurry is directly transmitted onto the soil grains, hydro or compressed air shields are characterized by the application of flow forces of the infiltrating support fluid. The lower the permeability of the soil in front of the heading face the higher is the flow force, or in other words the pressure gradient of the infiltrating support liquid, and the more effectively is the support pressure transmitted onto the soil grains (see Figure 3). This effect is, to a large extent, attributed to the existence of a filter cake consisting of a material with a very low permeability sealing the heading face. In case of tunnelling with bentonite support this filter cake automatically evolves due to the infiltration of the bentonite suspension into the soil pores. In case of compressed air support during repair interventions such a filter cake may persist from an earlier bentonite support.

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Figure 3. Heading face support by means of a supporting fluid (air or bentonite suspension): pressure gradient and pressures on the soil for a support without filter cake (left) and with filter cake (right) Due to the macroscopic nature of the three-phase formulation of the soil model, the micromechanical support mechanism can not be taken directly into account. Within FE-simulations for the prediction of surface settlements due to tunnel advance the heading face support therefore has to be modelled via adequate boundary conditions at the heading face. While for mechanical support prescribed displacements and for earth pressure support a prescribed area loading at the heading face is the adequate representation of the support mechanism, the question of adequate boundary condition is more difficult to answer for liquid or compressed air support, since these conditions depend on the existence of a filter cake. If a perfect filter cake seals the heading face, the supporting liquid acts on an impermeable membrane and the prescription of a corresponding distributed load covers this situation (Kasper and Meschke, 2004, Kasper and Meschke 2006). However, the assumption that a perfect filter cake seals the heading face is not valid in general. For a slurry shield it could be shown that a filter cake builds up only for down time of the machine, whereas for the advancement phase of the machine the evolving filter cake is excavated faster than the bentonite infiltrates into the soil pores (Bezuijen et al., 2001). This has been corroborated by numerical analyses (Kasper and Meschke, 2004). In this case fluid pressures at the heading face should be prescribed. The supporting fluid flows into the pore volume and affects the effective stresses and pore water pressures of the surrounding soil. The disturbance of the hydrostatic ground water pressure by the slurry face support has been measured during the construction of the 2nd Heinenoordtunnel (Bakker et al., 1999). It was observed that the fluid flow due to heading face support resulted in excess pore pressures within a distance of up to 3 times of the tunnel diameter in front of the heading face (Bezuijen et al., 2001). These excess pore pressures have a profound influence on the face stability (Broere, 2002) and the minimum required support may be considerably larger than for the standard case assuming a hydrostatic pore pressure. This holds, in particular, also for compressed air interventions, where the filter cake may become ineffective due to the air flow drying the filter cake. A sufficiently realistic 3D-FEM model should be capable to describe these effects of heading face support and to consider these influences of the flow of the supporting medium on the state of the groundwater in the vicinity of the tunnel face. The soil model presented above allows for the consideration of these effects by the application of support liquid pressures at the heading face. In the following benchmark analyses the capabilities of the model to account for a change in the pore liquid pressure distributions will be demonstrated for the case of a compressed air support without filter cake.

4 Numerical Examples A compressed air intervention of a tunnel with 10m diameter and 15m overburden has been simulated by means of the proposed simulation model. Such an intervention may be conducted for a hydro shield if the excavation chamber has to be entered by the working staff for a repair of the cutting tools. In such a situation the support fluid is being replaced temporarily by compressed air. Computations have been performed for 8 hours of compressed air support for two different types of soils. The following results show the influence of the support on the pore fluids for a soft soil with high permeability and one with low permeability with respect to water and air flow. Within the simulation the surrounding soil has been modelled as an elastic material permeable for both water and air. The ground water level has been assumed at the ground surface. Besides the permeabilities against water and air flow the material parameters of the soil have been chosen as: density ρ = 2000 kg/m3, YOUNG’s modulus E= 5250 2 kN/m , POISSON’s ratio ν = 0.45. The porosity of the soil is assumed as 20%. The soil characteristic curve is given by an air entry pressure of p br = 3kN/m2, n= 2.5 and m= 0.4, which is corresponds to a sharp transition from the fully saturated to the unsaturated case. The simulation started with an initial support liquid pressure equal to the 3839

undisturbed pore water pressure. During compressed air intervention, this prescribed water pressure at the heading face has been replaced by a constant air pressure of 253.4kN/m2.

4.1

Heading face support by means of compressed air in a soil with high permeability

Figure 4: Compressed air support in a soil with high permeability. Saturation of the pore volume with water after 8h (left), isolines for a pore air pressure of 220 kN/m2 (centre), isolines for a pore water pressure of 210kN/m2 (right) The numerical analysis has been performed for a soil with an initial permeability for air flow of 14.4 cm/h and for water flow of 144.0 cm/h. Due to the applied air pressure at the heading face air flows into the pore volume by replacing the pore water. A partially saturated zone establishes in front of the heading face which extends during the compressed air intervention up to a size of approximately 1 D in front of the tunnel face (see Figure 4). It can be observed that an unsaturated or nearly unsaturated zone evolves the longer the intervention lasts. Due to air inflow an air pressure distribution within the soil volume establishes which affects the soil deformations within the partially saturated zone and the water in the vicinity of the tunnel face. The zone of excess pore pressures extents up to 1.5 D in front of the tunnel face. Due to the high permeability of the soil this excess water pressure reached its maximum shortly after the beginning of the intervention and dissipates again fast afterwards (see Figure 4).

4.2

Heading face support by means of compressed air in a soil with low permeability

Figure 5: Compressed air support in a soil with low permeability. Saturation of the pore volume with water after 8h (left), isolines for a pore air pressure of 220 kN/m2 (centre), isolines for a pore water pressure of 210kN/m2 (right) For the second example the initial permeability of the soil has been assumed as ~100 times larger compared to the previous example (0.12 cm/h for air flow of and 1.58 cm/h for water flow). The inflow of air and the low permeability of the soil against an air flow results in a comparably smaller unsaturated zone in the vicinity of the tunnel. A zone of excess pore water pressure extending ~ 1 D in front of the heading face is observed. In contrast to the previous example, however, this excess pore water pressure does not dissipate fast but remains during the complete duration 3840

of the compressed air intervention (see Figure 5).

5 Conclusions A numerical simulation model for partially saturated soils has been presented in the context of a 3D simulation model (ekate) for numerical simulations of shield tunnelling. The proposed formulation of the soil within the framework of the Theory of Porous Media in conjunction with the simulation model allows consideration of different types of heading face support by taking into account the interaction of the support liquid and the pore water of the surrounding soil. In particular, the triphasic coupled formulation of partially saturated soils offers the possibility to simulate the application of compressed air as a supporting medium at the tunnel face. This may be relevant in case of repair interventions during mechanized tunnelling in fully or partially saturated soils. Numerical benchmark analyses of a compressed air intervention in tunnel have demonstrated that the proposed model covers the main effects connected with the interactions between the face support and the groundwater. Further extensions of the model will include the effect of the changing filter cake during the construction phases with consideration of driving and still-stand phases. The model is designed as a part of an Integrated Design Support System currently being developed in the framework of the European Research project TUNCONSTRUCT. Results from numerical analyses employing the presented simulation model may be used within the design and construction process for the assessment of the soil stability in front of the heading face, for the prediction of settlements due to the construction process and for the determination of design acting on the tunnelling machine and the lining and the amount of compressed air needed to conduct the intervention in case of compressed air interventions.

6 Acknowledgements This work has been supported by the European Commission within the Integrated Project TUNCONSTRUCT (IP011817-2). Co-funding to the first two authors was also provided by the Ruhr University Research-School funded by the DFG in the framework of the Excellence Initiative. This support is gratefully acknowledged.

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nd

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