Numerical anisotropic fracture mechanics modelling

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predict the propagation of a crack system in both mode I and mode II, which is crucial .... Chevron Bend test results and for KIIC the Punch-Through. Shear with ...
Numerical anisotropic fracture mechanics modelling in crystalline rock T. Siren1, B. Shen2, M. Rinne3, K. Kemppainen1 1 Posiva Oy, Eurajoki, Finland 2 CSIRO Earth Science and Resource Engineering, Queensland, Australia 3 Fracom Ltd, Kirkkonummi, Finland

ABSTRACT: Fracture propagation code (FRACOD) is a two-dimensional Displacement Discontinuity Method (DDM) computer code that was designed to simulate fracture initiation and propagation. The latest development introduced in the code allows the possibility to simulate anisotropy of rock medium using strength anisotropy instead of just explicit joints and bedding planes. Anisotropy related to new fracture initiation is described by direction dependent Mohr-Coulomb and direct tensile strength criteria. The fracture propagation function is converted to anisotropic by formulating F-criterion to be direction dependent. A case example is presented where the code is used in fracture mechanics prediction of Posiva’s Olkiluoto Spalling Experiment (POSE). The POSE experiment will be described in more detail in an accompanying paper by Kemppainen et al. elsewhere in this proceeding. In the case example laboratory results of the anisotropic behaviour of the rock are used in the simulations. Special attention in modelling is paid to analysing the loading conditions under which spalling will occur. The results show that the fracture propagation is very sensitive to changes in the anisotropy direction, friction angle and cohesion. However, the fracture toughness is observed not to be a very sensitive parameter. Modelling results suggest minor spalling on the pillar surface while observations from the field shows slight fracture slipping of an existing fracture. SUBJECT: Modelling and numerical methods KEYWORDS: Numerical modelling, rock stress, nuclear repository, case studies 1 INTRODUCTION 1.1 Fracture propagation code (FRACOD) Most numerical methods for rock engineering are based on methods that cannot genuinely describe the fracture initiation and propagation. Fracture propagation code (FRACOD) is a two-dimensional Displacement Discontinuity Method (DDM) computer code that was designed to simulate fracture initiation and propagation. The code development was started in early nineties by Shen & Stephansson (1993). The FRACOD code is based on the principles of the Boundary Element Method (BEM). Since the BEM doesn’t cope with problems with a fracture that is, two surfaces coinciding with each other - Crouch (1976) developed a method called the Displacement Discontinuity Method (DDM). The advantage of the DDM in simulating fracture propagation, compared with other boundary element techniques, is its direct presentation of a fracture as fracture elements instead of as separate fracture surfaces. FRACOD uses a modified G-criterion, which is also known as the F-criterion, introduced by Shen & Stephansson (1993). The problem with the original Gcriterion is that, in some cases, it predicts shear failure when the failure, in fact, is tensile. However, the F-criterion can predict the propagation of a crack system in both mode I and mode II, which is crucial for such material as rock in which shear and tensile failures are common.

The latest development, described in later chapters, introduced in the code allows the possibility to simulate anisotropy of rock medium using strength anisotropy instead of just explicit joints and bedding planes. Anisotropy related to new fracture initiation is described by direction dependent Mohr-Coulomb and direct tensile strength criteria. The fracture propagation function is converted to anisotropic by formulating F-criterion to be direction dependent. A case example, reported in detail by Siren (2011), is presented where the fracture propagation code is used in fracture mechanics prediction of Posiva’s Olkiluoto Spalling Experiment (POSE). 1.2 Posiva’s Olkiluoto Spalling Experiment (POSE) Currently in Olkiluoto, the construction of the underground rock characterisation facility for the final disposal of spent nuclear fuel named ONKALO is on the way. The site has been under thorough research for many years, but there are still some uncertainties, related to the in situ stress and to the rock spalling strength. (Siren et al. 2011) To answer these questions, an in situ experiment called Posiva’s Olkiluoto Spalling Experiment (POSE) was started (Aalto et al. 2009). The objective of POSE is to establish the in situ spalling strength of the rock in Olkiluoto and also to establish the state of in situ stress at the -345 metre depth level.

F = GI/GIC +GII/GIIC =1.0

2 DEVEPLOMENT OF ANISOTROPY OF ROCK MEDIUM In FRACOD, the fracture initiation occurs when the combination of two principal stresses reaches a critical value. More closely, the tensile and shear stresses and strengths are used to determine the initiation of a new fracture. The fracture propagation is, however, determined by using fracture toughness parameters. To take anisotropy of the rock into account, the parameters (st, c, f, KIC, KIIC) have an elliptical variation from q to q+90°. Calculations are done in all anisotropy directions, and the fracture will initiate in the direction of maximum tension or shearing, and will propagate in the direction where the maximum F value is reached. 2.1 Fracture initiation For the shear failure, the critical strength is presented with the friction angle (φ) and cohesion (c) of the intact rock. The critical normal and shear stresses in anisotropic medium in an arbitrary plane are calculated by: σn(β) = (σ1 + σ3)/2+cos(2β)(σ1 - σ3)/2

(1)

σs(β) = sin(2β)(σ1 - σ3)/2

(2)

where σn is the normal stress, σs is the shear stress, β is the angle to the minor principal stress direction, and σ1 and σ3 are the major and minor concentrated principal stresses. In an anisotropic case the shear strength is calculated by S(β) = σn(β)tan(β) + c(β)

(3)

where S is the shear strength. When the shear stress exceeds the shear strength, a shear failure will occur. However, in an anisotropic case, the entire angle range from 0° to 360° must always be considered. The failure will initiate only in the direction in which the ratio of shear stress and strength is the highest. The ratio can be calculated by: Fs(β) = σs(β)/S(β)

(4)

The initiation of a tensile failure can be determined similarly with a ratio calculated by: Fs(β) = σs(β)/T(β)

(5)

where T(β)=2σt(β) and σt is the tensile strength to the direction β . 2.2 Fracture propagation FRACOD uses the F-criterion to determine the fracture propagation. The F-criterion doesn’t consider microcrack formation, but the macrofracture growth only. In macroscale, the fracture growth can be a combination of microcracking in mode I and mode II. In the F-criterion the resultant strain energy release rate (G) is divided into tension (GI) and shear (GII) components. Mode I and II crack propagations are normalized and summed to produce a factor which expresses whether the crack is propagating and in which direction. The F-criterion is expressed by:

(6)

where the GI and GII are strain energy release rates in modes I and II, and GIC and GIC are the critical strain energy release rate. GIC and GIIC are material constant values that express a stress state where the crack starts to propagate. The equation can also be written in terms of fracture intensity and anisotropy as: F(θ) = (KI/KIC(θ))2+(KII/KIIC(θ))2=1.0

(7)

where θ is the arbitrary direction, KI and KII are stress intensity factors in modes I and II, and KIC and KIC are the corresponding fracture toughness values. The direction in which the fracture starts to propagate is where F(θ) reaches its maximum. 3 CASE EXAMPLE OF POSE The POSE experiment is based on boring three large holes at the bottom of a tunnel, of which two are modelled in this work. The pillar of rock between the two holes is expected to have crack growth due to the high stresses induced by the two holes modelled. There are several different versions of the predictions with the two holes: for isotropic and anisotropic mediums with different sensitivity studies. The layout of the POSE tunnel with the cross-section planes modelled is illustrated in Figure 1. 3.1 Anisotropy direction The parameters for the anisotropy direction for migmatitic gneiss are determined by using Posiva’s geological mapping of the POSE tunnel and projected to plane surfaces of the models. Stereoplot of geological mapping of the foliation direction in the POSE niche is shown in Figure 2, the mean value dipping 52 degrees to direction of 175 degrees. A variation of 30° in all directions with steps of 15° is used in the models to investigate the effect of a change in the anisotropy direction. 3.2 The input parameters The input parameters for the models are determined by using existing test results for pegmatitic rock (PGR), which is assumed to be isotropic, and for migmatitic gneiss (MIGN.GN), which is assumed to be anisotropic. Both types are assumed to be homogeneous and linearly elastic. For migmatitic gneiss, individual rock strength and fracture strength parameters are determined for two perpendicular foliation directions. The input parameters, as listed in Table 1, are used. 3.2.1 Rock mass strength

The “single plane of weakness theory” states that rock sample with a discontinuity should have its weakest direction of 45°+(φ/2) where φ is the friction angle (Hudson & Harrison 1997). "Assuming the single plane of weakness theory is applicable to the foliated rock", the foliation has an effect on the crack damage (CD) and peak strength values. However, "the crack initiation is dominated by the favourably oriented weakest mineral contacts" and is not affected by the foliation. (Hakala et al. 2005)

Figure 1. The layout of the POSE tunnel with the crosssection planes modelled.

For different anisotropy directions, the KIC values are determined by using corrected CB results. The results are separated in two regions corresponding to anisotropy directions shown in Figure 3. For the foliation direction, there are five samples within the range of 1.50 -2.25 MPam1/2 with the average of 1.87 MPam1/2. For the direction perpendicular to the foliation direction, there are five samples within the range of 2.50 -3.75 MPam1/2 with the average of 3.05 MPam1/2. Similarly, the KIIc values are determined by using PTS/CP test results shown in Figure 4. For the foliation direction, there are five samples within the range of 1.50 -3.40 MPam1/2 with the average of 3.00 MPam1/2. For the direction perpendicular to the foliation direction, there are five samples within the range of 3.40 -4.20 MPam1/2 with the average of 3.86 MPam1/2. Table 1. Values of the mechanical parameters of the intact rock and fractures used as input in FRACOD.

Figure 2. Stereoplot of the variation of anisotropy direction. Hakala et al. (2005) reported that the CI strength mean value is 41% - 49 % of peak strength. FRACOD doesn´t support the separate behavior of crack initiation, damage and peak strength. The effect of foliation to the rock strength is taken into account in the rock mass strength values, although the rock mass strength is close to the CI strength where foliation has no effect. The effect of foliation however gradually increases between the CI and CD strengths, which makes the assumption reasonable since the rock mass strength (spalling strength) is estimated to be 57% of the UCS, as reported by Andersson et al. (2009). It should be noted that rock mass strength is estimated after the results in Underground Research Laboratory (URL) in Canada and Äspö Hard Rock Laboratory (HRL) in Sweden. The rock mass strength of the rocks in Olkiluoto is an object of research in the POSE in situ experiment. For pegmatitic granite the 65.6 MPa (57% of UCS) is used. For the migmatitic gneiss, the values of 60 MPa and 70 MPa were estimated as the rock mass strength after results by Hakala et al. (2005). From these values, cohesion and friction angle were determined using the equation cpeak=σci (1-sin∅m)/(2cos∅m)

Rock type Anisotropy

PGR M.GN. M.GN. isotrop. paral. perpendic.

Young’s modulus E (GPa) 55 55 Poisson’s ratio n 0.20 0.20 Anisotropy dip 52° Anisotropy direction 175° Cohesion c (MPa) 12.9 12.4 Friction angle φ (MPa) 47° 45° Tensile strength σT,I 12 10 Fracture toughness I(MPa√m) 1.96 1.87 Fracture toughness II(MPa√m)3.30 3.00 Fracture cohesion c (MPa) 10 10 F. normal stiffness kn (GPa/m) 20000 20000 F. shear stiffness ks (GPa/m) 2000 2000

55 0.20 52° 175° 13.8 47° 14 3.05 3.86 10 20000 2000

(8)

where σci = rock mass strength; and ∅m = friction angle. The friction angle is estimated to be 45° for the anisotropy direction and 47° for the perpendicular direction and for the isotropic rock mass. The cohesion calculated with Equation (8) is correspondingly 12.9 MPa for pegmatitic granite, 12.4 MPa for the anisotropy direction and 13.8 MPa for the perpendicular direction to anisotropy.

Figure 3. Chevron Bend test results separated in regions.

3.2.2 Fracture toughness

The fracture toughness properties are determined by using the results of the Olkiluoto laboratory tests by Geomecon GmbH in 2009. Values for KIC are calculated by using the Chevron Bend test results and for KIIC the Punch-Through Shear with Confining Pressure (PTS/CP) test results. For pegmatitic granite, the values are calculated as the mean of three test values.

Figure 4. The PTS/CP test results.

Table 2. In situ stress components at POSE niche. In situ stress. Value (Mpa)Direction ° Major principal stress σ1 25.1 166 (horizontal) Intermediate principal stress σ2 17.1 256 (horizontal) Minor principal stress σ3 12.3 (vertical)

3.2.3 In situ stress

Several stress measurements has been carried out in the ONKALO in the past few years and it is possible to determine different stress domains to describe the stress distribution at different depths. However new LVDT-cell measurements have been applied in the POSE niche with promising results which are used in this study. 3.2.4 Secondary stresses around POSE niche

The model with the tunnel geometry was calculated in order to achieve the stress state below the tunnel (Fig. 5). Although three anisotropy directions were tested in tunnel model, no change in the stress state was noticed in the numerical results. This is because FRACOD only takes into account the strength anisotropy but not the deformation anisotropy. However, anisotropy direction had influence on models with holes and concentrated stresses below the tunnel. The plot of the horizontal (σxx) and vertical (σyy) stresses along a line between the experiment holes heading downwards from the tunnel floor to a depth of 6 metres are presented in the Figure 5.

Figure 5. Plot of the horizontal (σxx) and vertical (σyy) stresses along a line between the experiment holes heading downwards from the tunnel floor to a depth of 6 metres. The stresses at 1 m and 3 m under the floor are presented with boxes.

3.3 Results 3.3.1 Results of the assumed anisotropy direction

Migmatitic gneissic rock with interpreted anisotropy in the direction of 99° results in some fracture propagation at both sides of pillar, as shown in Figure 6, at the depth of both 1 metre and 3 metres. The results indicate only minor differences between the depths and that spalling will occur in both simulated depths. Notches that form are deeper in anisotropic models than in isotropic models. The fracture propagation in stages with the mean anisotropy direction (99°) is presented in Figure 7. The figures are close-ups of the pillar between the holes. From the figures it can be observed that small notches, which are about the size of the grid density, form at the right side at 2nd cycle and the large wedge on left forms last. 3.3.2 Stresses at the pillar

The maximum compressive stresses at the pillar between the holes are at the highest 73 MPa, which is over the assumed spalling limit of 65.6 MPa. This leads to crack growth that starts with a single crack growing in the anisotropy direction or perpendicular to it. The cracks form crevices, which join and form spalling when the crack growth turns its direction parallel to the tangential stress direction. The forming of cracks stops when a balanced state is reached. At this state, the maximum tangential stress in the pillar surface is 64 MPa. The maximum tensile stress stays much below the tensile strength, at the maximum at 0.5 MPa. The maximum shear stress is 34 MPa.

Figure 6. Anisotropic model showing the spalling and crevice at the depth of 3 metres in the model with anisotropy direction 99°. The direction of anisotropy is shown with thick black line and variation of the direction with thin lines in the bottom right corner.

1

2

3

7

8

17

Figure 7. The fracture propagation in numbered stages from left to right with the mean anisotropy direction (99°). Fractures marked with grey are slipping or open. 3.3.3 Spalling

Fracture growth happens in all of the models with the POSE holes. In the models with the anisotropy of 84 degrees, the cracks form only crevices, but in the rest of the models the fractures form notches—or in other words, spalling. The results are assembled in Table 3. In the table the number of crevices and spalling is stated as observed from the results.

Table 3. Results of the models. Model 1st hole 2nd hole direction width depth type width depth type - depth mm mm mm mm Peg - 1m 44 409 3Nm,Cs 148 593 Ns,2Cm,Cx Peg - 3m 40 295 3Nm 241 421 Nm+Cx 84° - 1m 71 295 Cl,2Cm 48 296 3Cm 84° - 3m 55 294 3Cm 48 295 3Cm 99° - 1m 98 468 2Nm,2Cl 85 468 Nm+Cl,2Cl 99° - 3m 94 352 2Ns,Cl 109 352 Nm+Cl,Cl 114° - 1m 82 409 Nm,2Cl,Cm 78 410 Nm+Cl,2Cl 114° - 3m 97 410 Ns,3Cl 75 410 Nm, 2Cl Abbreviations: N=Notch, C=Crevice Scale: s=Small (depth 100 mm), x=Extra (400...500 mm) In the model with anisotropy direction 114° at depth of 1 m a medium sized crevice at the opposite side of the pillar in the 1st hole. This is possibly due to the fact that, compared to the isotropic model, the rock mass parameters are higher perpendicular to the anisotropy direction, which in this case is the major principal stress direction. However all formed fractures are slipping. It can be observed in the results that the most vulnerable modelled anisotropy direction with regards to spalling is 99 degrees. Least vulnerable is the anisotropy direction across the pillar (84°), no spalling is formed. 3.3.4 Sensitivity studies

The sensitivity studies showed that a 2MPa change in the tensile stress does not have any noted effects. This is due to the significantly lower tensile stresses in the models. It can be also noticed that the direction of anisotropy is very sensitive for spalling. A 15-degree change can determine whether spalling or a crevice occurs. It was noted that changes in the fracture toughness values do not have significant effects on the results. Higher fracture toughness values only changed the form of the cracks with minor direction changes. This is due to the fact that the fracture initiation is controlled by the rock mass strength and the fracture growth by the fracture toughness parameters. The rock mass spalling strength with the current parameters was narrowed down to be between 62.5 MPa and 63.75 MPa for the anisotropy direction. 4 DISCUSSION

The case example shows results that seem to be realistic. The local geological features that were neglected in this study, however can affect to the failure plane so that the failure will proceed through existing weaker surfaces that penetrate the holes. The latest observations from the in situ experiment showed that shear fractures rather than tensile fractures occur in the holes. The shear fracture proceeded in mica contact in both holes. The simulations in this study showed that the maximum tensile stress is well below the tensile strength, but the maximum shear stress is probably enough to displace mica contact. Due to the observations the shear

strength of the sheared mica contact is the next subject of study in this line of research as a part of POSE study. 5 CONCLUSIONS The results show that the fracture propagation is very sensitive to changes in the anisotropy direction, friction angle and cohesion. However, the fracture toughness is observed not to be a very sensitive parameter. Modelling results suggest minor spalling on the pillar surface while observations from the field shows slight fracture slipping of an existing fracture. ACKNLOWDGEMENTS The latest development of the possibility to simulate anisotropy of rock medium using strength anisotropy instead of just explicit joints and bedding planes in FRACOD were developed with by FRACOM (Finland) and CSIRO (Australia). In the POSE experiment major contributions from M. Hakala (KMS Hakala Oy), E. Johansson (SROY), J. Hudson (REC) and T. Backers (Geomecon GmbH) are gratefully acknowledged. REFERENCES Aalto, P., Aaltonen, I., Ahokas, H., Andersson, J., Hakala, M., Hellä, P., Hudson, J., Johansson, E., Kemppainen, K., Koskinen, L., Laaksoharju, M., Lahti, M., Lindgren, S., Mustonen, A., Pedersen, K., Pitkänen, P., Poteri, A., Snellman, M. & Ylä-Mella, M. 2009. Programme for Repository Host Rock Characterisation in ONKALO (ReRoc). Posiva Oy, Working Report 2009-31 Andersson, J., Front, K.; Löfman, J., Poteri, A., Pitkänen, P., Partamies, S., et al. Olkiluoto Site Description 2008. POSIVA 2009-01. Posiva Oy , Eurajoki. 714 p. Crouch S.L. 1976. Solution of plane elasticity problems by the displacement discontinuity method. Int. J. Num. Methods Engng. 10, 301-343. Hakala, M., Kuula, H., & Hudson, J. 2005. Strength and Strain Anisotropy of Olkiluoto Mica Gneiss. Working report 2005-61, Olkiluoto, Finland, Posiva Oy, 2005. pp. 57–58. Hudson, J. A. & Harrison, J. P. 1997. Engineering Rock Mechanics – An Introductio to the Principles. Pergamon. p. 144 Kemppainen, K., Hakala, M., Johansson, E., Kuula, H., Hudson, J. 2011. In situ rock stress-strength comparison: Posiva’s Olkiluoto Spalling Experiment (POSE). 12th ISRM International Congress on Rock Mechanics. Shen, B. & Stephansson O. 1993. Numerical analysis of Mode I and Mode II propagation of rock fractures. Int. J. Rock Mech. Min. Sci. & Geomech. Abst. 30(7), 861-867. Siren, T. 2011. Fracture Mechanical Prediction of Posiva’s ONKALO’s Spalling Experiment. Working Report, Posiva, in prep., expected 2011, pp. 0–27. Siren, T., Martinelli, D., & Uotinen, L. 2011. Assessment of the Potential for Rock Spalling in the Technical Rooms of ONKALO. Working Report Posiva, in prep., expected 2011, pp. 0–37.