Numerical Investigation Of The Mechanical Behavior Of Rock

Paper 2A 04 — SINOROCK2004 Symposium Int. J. Rock Mech. Min. Sci. Vol. 41, No. 3, CD-ROM, © 2004 Elsevier Ltd.

NUMERICAL INVESTIGATION OF THE MECHANICAL BEHAVIOR OF ROCK UNDER CONFINING PRESSURE AND PORE PRESSURE C.A. Tang 1, 2, T. Xu 1, 2 , T.H. Yang 1 , Z.Z. Liang 1 ¹) Centre for Rock Instability and Seiscimity Research, Northeastern University, Shenyang, 110004, China [email protected] ²) Research Centre for Numerical Tests on Material Failure, Dalian University, Dalian, 116622, China [email protected]

Abstract: Many of the important problems of rock engineering are concerned with mechanical behaviours of rock where the internal rock structure sustains pore pressure and confining pressure from the surrounding rocks. A basic understanding of rock mechanical properties under confining pressure and pore pressure conditions is of great importance in rock mechanics and rock engineering. In this paper, the newly-developed pore-pressure incorporated Rock Failure Process Analysis model (RFPA) is briefly outlined at first. Then a series of numerical tests on rock under different constant confining pressure and pore pressure are conducted to illustrate how the overall macroscopic responses and mechanical properties of brittle heterogeneous rocks under different confining pressure and pore pressure were revealed by RFPA code. In addition, through the modelling of acoustic emission sequences in rock progressive failure, the AE characteristics and the correlation between AE events and stress-strain curves under different confining pressure and pore pressure were also investigated. From the numerically simulated results, it can be possible to analyze large-scale practical rock engineering problems such as mining induced seismicit ies and rock bursts. Keywords: Numerical simulation, confining pressure, pore pressure, mechanical behaviours, acoustic emission, RFPA.

1. INTRODUCTION A basic understanding of rock mechanical properties under different stress conditions is of great importance in rock mechanics and rock engineering since many of the important problems of rock engineering are concerned with mechanical behaviours of rock where the internal rock structure sustains pore pressure and confining pressure. Pore pressure greatly affects the probability of rock failure. Meanwhile, rock in deep ground also suffers from confining pressure from the surrounding rocks. Considerable attention has been given to the mechanical behaviours of rocks under different confining pressure and pore pressure by laboratory tests and in-situ measurements, but the general precise theoretical formulations and laws of rock mechanical properties are still not impossible due to its extreme complexity. When rock is subjected to stress, multiple cracks can nucleate, propagate, interact and coalesce, which induces the change of the pore pressure in the rock. This complex multiplicity of

fracturing interaction events also causes the complexity of mechanical breakdown of heterogeneous rocks. Numerical models that simulate the detailed fracturing sequence are thus useful for understanding rock failure mechanisms under pore pressure and confining pressure. In this paper, the further improved Rock Failure Process Analysis (RFPA) code by integration of porepressure was employed to investigate the mechanical properties of brittle heterogeneous rocks. A series of numerical tests on rock under different confining pressure and pore pressure were conducted illustrating how the overall macroscopic response of brittle heterogeneous rocks was presented by pore-pressure incorporated RFPA. Moreover, through the modelling of acoustic emission sequences, the AE characteristics of rock in failure process were also investigated to gain some possible insight into some large-scale practical problems such as mining induced seismicit ies and rock bursts.

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Paper 2A 04 — SINOROCK2004 Symposium Int. J. Rock Mech. Min. Sci. Vol. 41, No. 3, CD-ROM, © 2004 Elsevier Ltd.

2. PHILOSOPHY OF RFPA Numerical simulation is currently the most popular method used for modelling deformation behaviour of rock-like materials. RFPA code, as a numerical tool for simulating the geomaterials, has been widely used in investigating and contrasting the simulated results and laboratory findings (Wong & Lin 2001; Tang & Lin 2001; Xu and Tang, 2003). On the basis of the RFPA code, besides considering the deformation of an elastic material containing an initial random distribution of micro-features, the interactions between the individual elements, pore pressure and stress are also taken into account in the code. The governing equations for rock deformation in further developed RFPA code mainly consist of the equilibrium, the continuity, and the constitutive equations. According to elastic theory, equilibrium equation is given

σij , j + f j = 0

The constitutive equation of deformation fields can be expressed for elastic isotropic materials.

σ' ij = λδij εv + 2Gεij where G is shear modulus and is Lame’s constant. On the basis of the above the equilibrium, the continuity, and the constitutive equations, the governing equations for rock deformation considering the gas pressure in rock can be represented as:

( λ + G ) ⋅ u j , ji + Gui , jj + f j + (α ⋅ p) , i = 0 For heterogeneity, the material mechanical parameters (failure strength 0 and elastic modulus E0 ) for elements are randomly distributed throughout the specimen by following a Weibull distribution, a detailed information can be referred to published literature (Tang & Tham, 2000).

3. NUMERICAL MODEL

(1)

1

where σij is the stress tensor,( i , j = 1,2,3 ), MPa.

f j is the body forces per unit volume, MPa. The generalized effective stress principle based on Terzaghi’s law was invoked in the stress equilibrium equations:

σij = σ 'ij + α ⋅ p ⋅ δij '

(3)

Thus, the equilibrium equation is expressed according to the effective stress principle. According to the continuous conditions, the geometrical equation can be expressed

εij =

1 ( ui , j + u j ,i ) 2

3

(2)

where σij is the solid total stress, σij is the solid effective stress, P is the pore pressure, α is a positive constant equal to 1 when individual grains are much more incompressible than the grain skeleton, and δij is the Kronecker delta function. Substitution of equation (2) into equation (1) leads to:

σ ' ij , j + f j + (α ⋅ p ⋅ δij ) , j = 0

P 2

(4)

where εij is strain tensor,( i , j = 1,2,3 ). εv is the volumetric strain and u is the displacement of element.

Figure 1. Numerical model The mesh for the plane strain numerical sample consists of 200×100 elements with geometry of 100mm×50mm in size (as shown in Fig.1), and all the elements have the same size in scale (square in shape). The pore pressure in rock specimen is denoted as P, confining pressure and axial pressure acted on numerical rock specimen are respectively denoted as σ0 and σ1 . The elements are characterized by their failure strength, σ0 , Young’s modulus, E0 , and Poisson’s ratio, ν. The elements provide resistance against compressive or tensile deformations that are governed by constitutive equations described above. In order to consider the heterogeneity of rock specimen, a widely used Weibull distribution (Weibull, 1951) was introduced to describe the material properties of elements such as failure strength, Young’s modulus, and Poisson’s ratio at mesoscopic level. The input material mechanical properties parameters used to simulate numerical model rock

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Paper 2A 04 — SINOROCK2004 Symposium Int. J. Rock Mech. Min. Sci. Vol. 41, No. 3, CD-ROM, © 2004 Elsevier Ltd.

specimen are listed in the Table 1 below. An external displacement at a constant rate of 0.002mm/step in the axial direction was applied to the rock specimen and the stress acted on rock specimen as well as induced deformation in each element were computed in the numerical tests. Table 1. Mechanical parameters of numerical model. Mechanical parameters Value m 1.5 E0 / GPa 30 200 σ0 / MPa 30 φ/ ° 0.25 µ 0.2 ξ P/MPa 0 1 3 0, 2, 4, 8, 16 σ3 / MPa

deformation up to the peak strength. At low confining pressures, the curves show defined peak strength and a gradual strength decrease in the post failure region until final deformation occurs at about constant axial stress, i.e., residual strength. At higher confining pressures, the rock exhibits work-hardening and the Young’s modulus of rock is higher than that of rock at lower pressure. Meanwhile, transit ion from brittle to ductile deformation in rock with an increase in confining pressure was also clearly demonstrated in Fig.3. 70

0 60

2 4

50

8 16

¦ Ò 1 / MPa

40

30

20

10

4. EFFECT OF CONFINING PRESSURE

0 0

0.05

0.1

0.15

4.1 Deformation and strength behaviors 0MPa

4MPa

0.2

0.25

0.3

¦ 1Å/%

Figure 3. Complete stress-strain curves of rock specimens

8MPa

70 60

Figure 2. Macroscopic specimens

failure

patterns

of

Figure 2 is given the numerically simulated macroscopic failure patterns of model specimens under different confining pressures and the correspondingly numerical complete axial stress versus axial strain curves of rock at constant confining pressure up to 16 MPa with no pore pressure are presented in Fig.3. As shown in Fig.2, the angle between the failure plane and the maximum principal stress direction in uniaxial compression is about 30 degrees, and the angle between macroscopic failure plane and the maximum principal stress direction gradually increases with the increase of confining pressure acted on the rock specimens, which agrees well with theoretical predictions. It can be seen from the stress-strain curves in Fig. 3, the rock deforms linearly and elastically at axial stresses below the yield strength which is dependent on the confining pressure. Further compression leads to inelastic

UCS/ MPa

50 40 30 20 10 0 -5

Tension

0

5

10

Compression

15

20

Confining pressure/ MPa

Figure 4. Curve between compressive strength of rock specimens and confining pressure Figure 4 gives the relationship curve between peak strength of rock specimens and confining pressure at failure and Figure 5 is numerically obtained failure envelopes of rock specimens. As can be seen from Figure 4 and 5, the ultimate compressive failure strength, i.e., peak strength of numerical rock specimens gradually increases with confining pressure. Even though the linear MohrCoulomb failure criterion with tension cut-off is adopted in the model, the macroscopic failure envelope is concave towards the σ axis. The numerical results indicate that the macroscopic

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Paper 2A 04 — SINOROCK2004 Symposium Int. J. Rock Mech. Min. Sci. Vol. 41, No. 3, CD-ROM, © 2004 Elsevier Ltd.

400

40 300 30 200

20

50

0 0 .0 5

0 .1

0 .1 5

0 .2

0 .2 5

60

30 200

20

0

0.15

0.2

0.25

0.2

0.25

0.3

1/%

600 500

σ3 =16MPa

400

40 300 30 200 100

10

0 0.15

0.1

20

100

10

¦Å

0.05

50 ¦ Ò 1 /MPa

300

AE accounts(N)

70

500

0

0.3

0 0

0.05

0.1

0.15 ¦Å

1/%

0.2

0.25

0.3

1/%

Figure 6. Complete stress-strain curves and AE characteristic curves of model specimens 1

600

500

0.9 0.8

500

ó3=2MPa

0.7 0.6

300

0.5

200

0.4 0.3 0.2

100

0.9 0.8 ó3=4MPa

400

0.7 0.6

300

0.5

200

0.4 0.3 0.2

100

0.1 0

0.1

0 0

9

0

18 2 7 3 6 4 5 5 4 6 3 7 2 8 1 9 0 9 9 1 0 8 1 1 7 1 2 6

0 0

9

Loading step

Loading step 600

18 2 7 3 6 4 5 5 4 6 3 7 2 8 1 9 0 9 9 1 0 8 1 1 7 1 2 6

1

600

1

0.9 0.7 0.6

0.9

300

0.5 0.4

200

0.3 0.2

100

AE accounts(N)

ó3=8MPa

400

500

0.8 Normalized AEE

Figure 5. Simulated failure envelope of model rock specimens

AE accounts(N)

500

0.8 0.7 0.6

ó3=16MPa

400 300

0.5 0.4

200

0.3 0.2

100

0.1 0

0 0

9

1 8 2 7 3 6 4 5 5 4 6 3 7 2 8 1 9 0 9 9 1 0 81 1 7 1 2 6

Loading step

4.2 AE characteristics As we know, the monitoring of acoustic emission (AE) or seismic events has proven to be one of the powerful tools available in analyzing damage or brittle fracture during rock deformation. There is generally a good correlation between AE rate and inelastic strain rate so that the AE rate can be used to quantify damage accumulation occurring in the rock sample. Locker (1991), Cox and Meredith (1993) have analyzed catalogues of AE events recorded during compression tests in rock in terms of the information they give about the accumulated state of damage in a material. And combine this measured damage state with a model for the weakening behaviour of cracked solids, which shows that reasonable predictions of the mechanical behaviour are possible. Based on this background knowledge, by recording the counts of failed elements, the seismicities associated with the progressive failure can be simulated in RFPA that allows elements to fail when overstressed. In RFPA code, a single AE event represents a micro-crack forming event to indirectly assess the damage evolution (Tang 1997, 1998).

0.1 0

0 0

9

1 8 2 7 3 6 4 5 5 4 6 3 7 2 8 1 9 0 9 9 1 0 8 1 1 7 1 26

Loading step

Figure 7. AE curves and normalized AE energy curves of model specimens under different confining pressure Fig.6 shows the complete stress-strain curves and corresponding AE characteristic curves of model specimens at different constant confining pressure. Fig.7 shows the AE and corresponding normalized AE energy curves of model specimens under different confining pressure. A comparison among the curves in Fig.6 shows a good relationship between the simulated stress-strain curves and the modelled curves of event rate. It can be seen from Fig. 6 and 7, in general, a sharp increase of AE event rate in AE characteristic curves corresponds to an abrupt stress drop in complete stress-strain curve and the maximum rate of AE events appears in the post-peak range. It indicates that the initiation and propagation of mesoscopic main-fractur ing which precedes the final stage macroscopic fracture development has occurred in the rock. The results show that the maximum AE event or main shock emitting from rock can be regarded as the precursor of macro-

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Normalized AEE

400

1

Normalized AEE

600

AE accounts(N)

AE accounts(N)

Mohr failure envelope

Normalized AEE

¦ Ò 1 /MPa

600

400

0.1

0 0

40

0.05

100

¦Å

3 =8MPa

0

200

0

0 .3

70

50

300

30

Å ¦ 1/ %

60

400

40

10

0 0

500

20

100

10

600

σ3 =4MPa

AE accounts(N)

60

AE accounts(N)

70

500

1/MPa

¦ Ò 1/MPa

50

600

¦ Ò

σ3 =2 MPa

60

AE accounts(N)

70

non-linear phenomena such as rock failure in nature can be described and revealed through some simple linear rules at mesoscopic level. In addition, it is noticing that the residual strength (or frictional) of rock, also dependent on the confining pressure, increases with confining pressure. For rock materials, fracture and friction are macroscopic manifestations of the same processes: e.g., grain crushing, crack growth, healing, and plastic yielding. When viewed in this way, it is not surprising that the difference between intact strength and residual (or frictional) strength should vanish with increasing confining pressure. This is to say, the rock will ideally exhibit a state of plastic flow at extremely high confining pressure.

Paper 2A 04 — SINOROCK2004 Symposium Int. J. Rock Mech. Min. Sci. Vol. 41, No. 3, CD-ROM, © 2004 Elsevier Ltd.

fracture of rock, which is essential for the location of earthquake source, the search for earthquake precursors and earthquake prediction research. In addition, the main-fracturing will occur later when rock specimens are subjected to the same strain value with the increase of the constant confining pressure. The confining pressure enhances the ultimate compressive strength and defers the occurrence of main-fracturing at failure, which is termed as the typical confining pressure effect in rock failure.

P=8MPa

450 400 350 300 250 200 150 100 50 0

10 0

P=16MPa

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

80 70

¦ Ò 1/MPa

=2MPa

3

40 30 20 10 0

0.4

0

0.1

0.2

¦ 1Å/%

450 400 350 300 250 200 150 100 50 0

=8MPa

40 30 20 10

Figure 8. Macroscopic failure patterns under given pore pressure 1 MPa

0 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

80

450

70

3

400

=16MPa

350

60

300

50

250

40

200

30

150

20

100

10

50

0

0.4

0 0

0.05

0.1

0.15

¦ 1Å/%

0.2

0.25

0.3

0.35

0.4

¦ Å /% 1

Figure 10.Complete stress-strain curves and AE characteristic curves of model specimens with pore pressure 1 MPa

70 60

30

P=1MPa

20

P=3MPa

10 0 0

5

10

15

20

350 300 250

Several series of numerical tests were carried out to investigate the effect of pore effect on the mechanical properties of rock. The numerically simulated macroscopic failure patterns of rock with 1MPa pore pressure at different constant confining pressures are presented in Figure 8. Compared with the macroscopic failure modes of rock in Fig. 2, much splitting occurred in the rocks with pore pressure which exhibit remarkably brittle. Fig.9 gives the relationship curves between peak strength of rock and applied confining pressure for given

0.35 0.3 0.25

200 150 100 50

0.2 0.15 0.1 0.05

0

450 400 350

0 0

100 50 0 0

0.04 0.08 0.12 0.16 0 . 2 0.24 0.28 0.32 0.36 0 . 4 1(%)

0.45 1 0.9 0.4 0.8 0.35 0.7 0.3 0.6 0.25 0.5 0.2 0.4 0.15 0.3 0.1 0.2 0.05 0.1 0

3 =8MPa

Normalized AEE Normalized AEE AE AE accounts accounts

AE AE accounts accounts

1(%)

450 450 400 400 350 350 300 300 250 250 200 200 150 150 100 100 50 0

0 0.040.08 0.080.11 0.12 0.16 . 2 0.24 0.320.35 0.360.38 0.4 0 0.04 0.15 0.18 00.21 0.25 0.28 0.28 0.32 1(%)

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

3 =4MPa

300 250 200 150

0.04 0.08 0.12 0.16 0 . 2 0.24 0.28 0.32 0.36 0 . 4

3 /MPa

Figure 9. Relationship between compressive strength and confining pressure for given poro pressure

0.45 0.4

3 =2MPa

450 400 350 300 250

0.45 1 0.9 0.4 0.8 0.35 0.7 0.3 0.6 0.25 0.5 0.2 0.4 0.15 0.3 0.1 0.2 0.05 0.1 0

3 =16MPa

200 150 100 50 50 00

00 0.09 0.11 0.14 0.170.16 0 . 20.23 0.28 0.310.32 0.34 0.36 0.37 0.39 0.04 0.08 0.12 0 . 20.25 0.24 0.28 0.4

(%) 1(%) 1

Figure 11.AE and normalized AE energy curves of model specimens with given pore pressure (1MPa) Fig.10 shows the complete stress-strain curves and corresponding AE characteristic curves of model specimens with 1MPa pore pressure at different constant confining pressure and Fig.11 gives the AE events and corresponding. As stated

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Normalized AEE Normalized AEE

450 400

P=0MPa

Normalized AEE AE accounts

40

Normalized AEE

50

AE accounts

1 /MPa

3

60 50

¦ 1Ò/MPa

¦ Ò 1/MPa

70

0.3

¦ Å1/%

AE accounts

80

450 400 350 300 250 200 150 100 50 0 0.4

=4MPa

60 50

AE accounts

P=4MPa

¦ Ò 1/MPa

5.1 Deformation and strength behaviors

3

60 50 40 30 20

AE accounts

80 70

AE accounts

5.2 AE characteristics

5. EFFECT OF PORE PRESSURE

P=2MPa

pore pressures. As can be seen from Fig. 9, at the same constant confining pressure, the rock specimens with pore pressure have lower peak strength than those with no pore pressure. Generally, the pore pressure decreases the peak strength of rock at failure and increases the brittleness of rock. Moreover, the nonlinear relationship curves between the maximum compressive strength and confining pressure at different constant pore pressure are basically parallel.

Paper 2A 04 — SINOROCK2004 Symposium Int. J. Rock Mech. Min. Sci. Vol. 41, No. 3, CD-ROM, © 2004 Elsevier Ltd.

above, the simulated stress-strain curves, the AE event rate curves and the normalized AE energy curves have a good correlation. In contrast to Fig.6 and 7, it can be seen that the main-fracturing will occur earlier when rock specimens are subjected to the same confining pressure with increasing pore pressure. The pore pressure reduces the ultimate compressive strength and accelerates the occurrence of main-fracturing at failure, which can be termed as the pore pressure effect in rock failure. The confining pressure effect and the pore pressure effect are greatly indicative for the earthquake prediction and hydraulic fracturing research.

8. REFERENCES

6. CONCLUSSIONS

Tang, C.A., Kaiser, P.K. 1998. Numerical Simulation of Cumulative Damage and Seismic Energy Release in Unstable Failure of Brittle Rock-Part I. Fundamentals. Int. J. Rock Mech. & Min. Sci. 35(2): pp.113-121.

The problems of deformation and failure in heterogeneous rock at confining pressure and pore pressure are of great importance in associated rock mechanics and engineering areas. It is important to identify the main failure mechanisms associated with AE characteristics in compression. This identification is crucial for a better understanding and interpretation of the experimental results, and consequently, improves our concepts on rock mechanical properties or analysis of rock engineering structures. It is noted that the model predications of mechanical behaviour of rocks using pore pressure incorporated RFPA in this paper capture most of the experimental observed phenomena, especially the confining pressure effect and pore pressure effect of rock specimens in failure process. Although the simulations are not a quantitative approach and many conclusions presented here may have already been obtained by laboratory tests, the significance of mimicking these phenomena by numerical simulation is obvious. At least, and the most important, the successful reproducing of the experimentally observed failure phenomena with a numerical method implies that our understanding to the mechanisms of rock failure has reached a more reasonable level, which in turn will help us to make further progresses in the field of rock mechanics and rock engineering.

Cox, S.J.D., Meredith, P.G. 1993. Microcrack formation and material softening in rock measured by monitoring acoustic emissions. Int J Rock Mech Min Sci Geomech Abstr 30(1): pp.11-24. Lockner, D.A., Byerlee, J.D., Kuksenko, V., et al. 1991. Quasi-static fault growth and shear fracture energy in granite. Nature 350(7) : pp. 39-42. Tang, C.A. 1997. Numerical simulation of progressive rock failure and associated seismicity. Int. J. Rock Mech. Min. Sci. 34: pp.249-262.

Tang, C.A., Tham, L.G., Liu, H.Y., et al. 2000. Numerical Tests on Micro-Macro Relationship of Rock Failure under Uniaxial Compression, Part I: Effect of heterogeneity. Int. J. Rock Mech. Min. Sci. 37: pp.555-569. Tang, C.A., Lin , P., Wong, R.H.C., et al. 2001. Analysis of crack coalescence in rock-like materials containing three flaws—Part II: Numerical approach. Int. J. Rock Mech. Min. Sci. 38: pp.925-939. XU, T., Tang, C.A., Zhang, Z., et al. 2003. Theoretical, experimental and numerical studies on deformation and failure of brittle rock in uniaxial compression. Journal of Northeastern University 24(1): pp.87-90. Weibull, W. 1951. A statistic al distribution function of wide applicability. Journal of applied mechanics, pp. 293-297. Wong, R.H.C., Lin , P., Tang C.A., Analysis of crack coalescence materials containing three I experimental approach Int. J. Min. Sci 38: pp.909-924.

et al. 2001. in rock-like flaws—Part Rock Mech.

7. ACKNOWLEDGEMENTS The present research in this paper was carried out with the jointly financial support of the National Natural Science Foundation of China (No. 50134040, 50204003 and 50174013).

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