Numerical Modeling of Strain Driven Fractures Around Tunnels In ...

11th Congress of the International Society for Rock Mechanics – Ribeiro e Sousa, Olalla & Grossmann (eds) © 2007 Taylor & Francis Group, London, ISBN 978-0-415-45084-3

Numerical modeling of strain driven fractures around tunnels in layered rock masses S. Stefanizzi & G. Barla Department of Structural and Geotechnical Engineering, Politecnico of Turin, Italy

P.K. Kaiser MIRARCO (Geomechanical Research Center) Laurentian University, Sudbury, Canada

ABSTRACT: The understanding of the patterns, characteristics, and genesis of fractures in a rock mass is essential in coping with the solution of a number of problems in rock mechanics and rock engineering. This paper demonstrates that in multilayered systems, the development of fractures and their spacing vary according to the extension strain localization. The extension strain criterion of Stacey (1981) is indirectly validated and the concept of “strain-driven” fractures is introduced. FEM/DEM modeling has been used as available with the ELFEN code, which allows the transition of a rock mass from continuum to discontinuum to be simulated effectively assuming that, if the fracture criterion within the intact rock (represented by FEM) is met, a crack (represented by DEM) is initiated. Re-meshing allows the fracture process through the FEM mesh to be tracked and visualized, thus contact properties can be assigned to pre-existing fractures and newly generated fractures. 1 INTRODUCTION

being considered. Experimental studies of this phenomenon (Garrett&Bailey, 1977; Narr&Suppe, 1991; Gross, 1993; Wu&Pollard, 1995) show that the spacing initially decreases as extension strain increases in the direction perpendicular to the fractures. At a certain ratio of spacing to layer thickness, no new fractures form and the additional strain is accommodated by further opening of the existing fractures. The spacing then simply scales with layer thickness, a phenomenon which is called “fracture saturation” (Wu&Pollard, 1995; Avenston et al. 1971). This is in marked contrast with existing theories of fracture, such as the stress-transfer theory (Cox, 1952; Hobbs, 1967), which predict that spacing should decrease with increasing strain at infinitum. According with Bai & Pollard (2000) it is possible to say that with increasing applied stress, the normal stress acting between such fractures undergoes a transition from tensile to compressive, suggesting a cause for fracture saturation. In addition, the extension strain profile cannot reach in the space between two existing fractures the critical value over which a fracture opens, so no new fractures can generate. The development of fractures and their spacing varies according to extension strain localization.

In layered rock masses, such as in certain sedimentary rocks, fractures may develop preferentially within, and be relatively confined to, the less ductile layer. Gross (1993) defines a lithology-controlled mechanical layer as “a unit of rock that behaves homogeneously in response to an applied stress and whose boundaries are located where changes in lithology mark contrasts in mechanical properties”. Field observations of layered rocks show that opening-mode fractures often are confined by layer boundaries (Fig. 1). This remarkable behavior can influence the mode of failure and instability around underground excavations in layered rock masses, which is related to the magnitude of the in situ stress relative to the rock mass strength and to rock mass quality, a parameter which depends strongly on the degree of jointing and persistence. In layered rock masses fractures may be strain driven. The spacing of opening-mode fractures in layered media is shown to be proportional to the thickness of the fractured layer and is controlled by the geometry of the problem

2 FRACTURE DEVELOPMENT IN LAYERED MEDIA 2.1 Simulation model

Figure 1. Example of opening-mode fractures in the layers of the Flysh Formation in Algeria (photo courtesy of Geodata, Torino).

To investigate the fracture development in layered media, a three-layer numerical model has been constructed, using the finite/discrete element code ELFEN (Rockfield Software, 2006). ELFEN provides a unique ability to model fracture initiation and growth, as it simulates the transition of a rock mass from continuum to discontinuum. If the fracture criterion within the intact rock (represented by FEM) is met then a crack (represented by DEM) is initiated. Re-meshing allows the fracture process through the FEM mesh to be tracked and visualized; thus contact properties 971

Figure 2. Mesh and geometry of the three-layer numerical model. Table 1.

Time step 0.17s

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Material properties.

Central layer Neighboring layers

E∗ [GPa]

v∗ [–]

σt∗ [MPa]

G∗f [N/mm]

50 25

0.25 0.25

3 5

0.01 0.05

∗ E =Young’s modulus, v = Poisson’s ratio, σt = tensile strength, Gf = fracture energy.

Figure 3. Fracture patterns at different time steps.

can be assigned to pre-existing cracks and newly generated fractures. Geometry and mesh of the numerical model are indicated in Figure 2. The central layer has a thickness of 20 cm. The width of the entire model is 2 m. No constraint conditions are applied at the sides. A constant vertical velocity that acts to the upper and lower boundary of the model is applied to simulate loading. A plane strain condition for the entire model is postulated. The Rankine rotating fracture model, which has two distinct material parameters, tensile strength and fracture energy, is used in the simulation. The initial failure surface is defined by a tension failure surface. Post initial yield, the rotating crack formulation represents the anisotropic damage evolution by degrading the elastic modulus in the direction of the major principal stress. The damage parameter is dependent on the fracture energy. Stresses are computed in the FEM elements and checked against the fracture criterion in each loading step. This explicit dynamic finite/discrete element tool enables the user to track the fracturing process in a unique fashion. Material properties of the three layers are listed in Table 1. The central layer has aYoung’s modulus greater than that of the neighboring layers, so that the central layer is the less ductile layer in which fractures are expected to occur.

Figure 4. Line along which maximum principal stress and extension strain are computed.

to, the less ductile layer. The fracture pattern is related to the geometry of the problem and to the relative stiffness of different layers. It is shown that changes in lithology that mark contrasts in mechanical properties, in terms of stiffness, can drive the process. In the last step the process is stable and this confirms the “fracture saturation” theory as explained by Bai&Pollard (2000). In the space between two existing fractures a stress transition from tension to compression takes place, so that no new tensile fractures can generate. A strain-driven behaviour can be recognized in the fracture patterns development, so fracture initiation, propagation and then saturation are driven by extension strain localization. To demonstrate this, the extension strain distribution along a horizontal line (AA) located in the middle of the central layer (Fig. 4) at different time steps is computed. The maximum principal stress is also computed along the same line, to compare stress and strain distributions. As shown in Figure 5, the maximum principal stress distribution is uniform. Also, it is noted that every time a fracture opens a local increase in the maximum principal stress occurs due to stress concentration in a point corresponding to fracture localization. It is observed that based on the maximum principal stress distribution it is not possible to localize where a tensile fracture will open. The extension strain distribution (Fig. 6, 7, and 8) is not uniform as the maximum principal stress distribution. It is observed that fracture initiation starts when the extension strain, that is generated, overpasses a limit value that for the particular model is equal to 150 µε. Also, at a point where the

2.2 Results The results in terms of fracture pattern are shown in Figure 3 at different time steps. It is noted that with the load increase, a tensile stress acts in the middle of the central layer and a tensile fracture opens exactly in the middle of the model. At different time steps, new fractures are generated and are always confined to the less ductile layer. At time step 0.25 s the new fracture that appears propagates in the space between two existing fractures. The process continues to reach a final configuration in which the final spacing between fractures is proportional to the thickness of the central layer. The crack propagation process can be tracked in detail by the numerical tool, showing that in layered materials fractures may develop preferentially within, and be relatively confined 972

Time interval 0.191s-0.195s

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Figure 8. Extension strain along line AA in the time interval 0.247 s–0.251 s.

Figure 5. Maximum principal stress distribution along line AA.

It is clear that fracture initiation cannot be predicted on the basis of the principal stress, due its uniform distribution. Therefore, the extension strain distribution appears to be a more reliable tool to predict when and where a fracture can generate. Stacey’s empirical strain criterion (1981) states that “fracture of the rock will initiate when the total extension strain in the rock exceeds a critical value which is characteristic of that rock type”. The numerical model and the analysis in terms of extension strain as described above is an indirect validation of this criterion.

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3 UNDERGROUND EXCAVATION IN A LAYERED ROCK MASS

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It is inferred that the phenomena described above can influence the mode of failure and instability around underground excavations in layered rock masses. Therefore, in order to understand in which way the presence of bedding planes and discontinuities in a rock mass can influence the Excavation Disturbed Zone (EDZ) around a tunnel, some preliminary models have been run by the ELFEN code as discussed in the following. The excavation of a circular tunnel in:

Figure 6. Extension strain along line AA in the time interval 0.174 s–0.178 s. Time interval 0.247s-0.251s 300

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exx [10^-6]

150

– a continuum rock mass (model A) – a homogeneous rock mass containing two pre-existing parallel discontinuities (model B) – a multi-layer rock mass, with different material properties (model C)

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Figure 7. Extension strain along line AA in the time interval 0.191 s–0.195 s.

fracture opens, the value of the extension strain drops down, to increase in the surrounding area, giving the opportunity to a new fracture to start. The extension strain distribution between two existing fractures has a parabolic shape. If the extension strain between two fractures reaches the limit value of 150 µε, then a new fracture will open. With the applied stress increasing, a reduction of fracture spacing takes place and fracture saturation occurs, as the spacing between existing fractures is not such as to allow the extension strain to grow until the limit value.

has been considered. The results are interpreted in terms of fracture pattern. A modified Mohr-Coulomb material model with strain softening and discrete fracturing is employed. The relevant material properties are listed in Table 2. A 90 m × 90 m plane strain model has been constructed for all analyses.The excavation (a circular tunnel with 3.6 m diameter) is placed in the centre of the model. The initial stress field is obtained by appropriate loading on the boundaries of the model. The vertical stress is 6.5 MPa and the horizontal stress 4 MPa, with a stress ratio of 0.62. The analysis is conducted in two stages. In the first stage a uniform stress field is obtained with the elements representing the tunnel active. In the second stage the tunnel is excavated. This is achieved by recording the internal loads on the excavation boundary at the time of excavation and ramping these 973

Table 2.

4 CONCLUSION

Material properties.

Material 1 Material 2

E∗ [GPa]

ν∗ [–]

σt∗ [MPa]

G∗f [N/mm]

c∗ [MPa]

φ∗ [◦ ]

2.7 3.7

0.3 0.3

5 5

0.01 0.01

2.6 2.6

25 25

In layered media tensile fractures can be localized in the less ductile layer and can be confined to it, due to the stiffness difference with the neighboring layers. Fracture spacing is proportional to the thickness of the layer where fractures take place. Fracture formation and spacing are controlled by extension strain localization. This is an indirect validation of Stacey’s empirical strain criterion. It has been shown that the phenomenon as described can influence the modes of instability in underground excavations and that in layered rock masses a strain driven process should be considered and analyzed.

∗

E =Young’s modulus, v = Poisson’s ratio, σt = tensile strength, Gf = fracture energy, c = cohesion, φ = friction angle.

ACKNOWLEDGMENT Part of this research was sponsored by the National Research Council Canada (NSERC). The authors wish to thank Giovanni Grasselli for his valuable comments and suggestions during this research. REFERENCES

Figure 9. Fracture distribution around tunnel for model (a), model (b), model (c).

loads down to zero over a user-specified time period or number of steps. The resultant fracture distributions are presented in Figure 9 for the three models A, B, and C. It is shown that the presence of discontinuities or the change in lithology influence the mechanism of failure around the tunnel. In the continuum rock mass (model A) a spalling process takes place at the tunnel wall, which is governed by the strength of the rock mass: the process can be interpreted as stress-driven. Due to the presence of discrete fractures (model B) the EDZ localizes in it: fracture initiation and propagation can be interpreted as a strain-driven process. This type of behaviour takes place also with no changes in stiffness: the strain-driven behaviour in this case is related to the shear at the interfaces. In model C the presence of different materials with different stiffness properties localizes the fracture in the less ductile layer. The simple models considered demonstrate that in layered materials, such as sedimentary rocks, a strain-driven behaviour can be recognized: this behaviour can be due to a change in mechanical properties of different layers or to the sliding at the interfaces of existing discontinuities. A behaviour like this can influence the EDZ around tunnels.

Avenston J., Cooper G.A., Kelly A., 1971. The properties of Fiber Composite 15 (IPC Sci. Technol. Press, London) Bai T., Pollard D.D., 2000. Fracture spacing in layered rocks: a new explanation based on the stress transition. J. Struct. Geol., 22: pp.43–57. Bai T., Pollard D.D., Gao H., 2000. Explanation for fracture spacing in layered materials. Nature, 403: pp.753–755. Cai M., Kaiser P.K.K. 2004, Numerical simulation of the Brazilian Test and the tensile strength of anisotropic rocks and rocks with pre-existing cracks, SINOROCK 2004 Symposium, Int. J. Rock Mech. Min. Sci., 41. Cox H.L., 1952. The elasticity and strength of paper and other fibrous materials. Br. J. Appl. Phys., 3: pp.72–79 ELFEN, Rockfield Software Ltd, 2006. Version 3.7.1 Garret K.W., Bailey J.E., 1997. Multiple transverse fracture in 90◦ cross-ply laminates of a glass fibre-reinforced polyester. J. Mater. Sci. 12: pp.157–168 Gross M.R., 1993. The origin and spacing of cross joints: examples from Monterey formation, Santa Barbara Coastline, California. J. Struct. Geol. 15: pp. 737–751. Hobbs, D.W., 1967. The formation of tension joints in sedimentary rocks: an explanation. Geol. Mag. 104: pp. 550–556 Narr N., Suppe J., 1991. Joint spacing in sedimentary rocks. J .Struct. Geol. 13: pp.1037–1048 Stacey T.R., 1981. A simple extension strain criterion for fracture of brittle rock. Int. J. Rock Mech. Min. Sci. & Geomech. Abstr., 18: pp. 469–474. Wu H., Pollard D.D., 1995. An experimental study of the relationship between joint spacing and layer thickness. J. Struct. Geol. 17: pp. 887–905

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