FEA Based Level 3 Assessment of Deformed Tanks with Fluid Induced Loads
Cundall 2008 rock mass workshop
1. An approach to rock mass modelling
Peter A Cundall
Itasca Consulting Group, Inc
Minneapolis, Minnesota, USA
SHIRMS Workshop:
FROM ROCK MASS TO ROCK MODEL WORKSHOP
15 September 2008
2. 1. What is a rock mass? – intact failure + joint failure.
2. Synthetic Rock Mass (SRM) uses bonded-particle model to
represent both components.
3. Summary of the bonded-particle model and smooth joint model.
4. Assembly of the SRM, using pre-compacted bricks + DFN
(Discrete Fracture Network).
5. SRM can be used in two ways (examples given) –
A. to perform element tests, to derive behaviour that can be
imported into other methods (e.g., FE or LE);
B. to perform complete simulations of boundary-value problems
(time consuming!).
6. Conclusions
Contents
3. 1. What is a “rock mass”?
The term “rock mass” refers to a large volume of rock, containing
discontinuities such as fractures, joints, bedding planes and
faults. Typically, such discontinuities are not continuous, so that
the failure of a large system (e.g., an open pit or a block cave)
will involve fracture of intact rock as well as movement on the
discontinuities (slip and opening).
Thus, a complete numerical model of a rock mass allows both
mechanisms to take place, with full interaction between them.
We describe the Synthetic Rock Mass (SRM) approach which
is implemented as bonded particle assembly, using the distinct
element method (DEM). Such a model reproduces well the
combined effects of intact fracture and discontinuity movement.
4. Current design methods make an assessment of the rock
mass using a classification scheme, such as GSI.
Then, the GSI may be used to derive parameters for the
Hoek-Brown strength criterion (which is then used in a
continuum model of the designed object – slope, cave-mine)
Both of these steps are empirical.
The “Synthetic Rock Mass” approach is a direct, quantitative
way to go from measurable properties (intact strength and
fracture statistics) to large-scale rock-mass properties
Rock mass properties (such as strength) cannot be tested,
because of the large sizes involved.
5. Fracture
representation –
3D DFN (Discrete
Fracture Network)
Intact rock
representation
(including brittle
fracture)
2. Synthetic Rock Mass
Bonded-particle assembly
intersected with fractures
(Smooth Joint Model – SRM)
6. Thus, we have two components:
The bonded-particle model, consisting of spheres
attached together with brittle material, and
The smooth-joint model, which allows slip and
opening on joint planes, without the “bumpy-road”
effect of contact between spheres.
We describe the components separately -
7. 3A. Introduction to the Bonded Particle Model
The bonded particle model uses the code PFC …
8. Particle Flow Code (PFC)
• Simulates the static & dynamic behavior of a
system of circular/spherical rigid particles that
may be bonded together.
– All deformation lumped at contacts
• Released as a commercial code by Itasca in
1995, funded by groups interested in modeling
block caving and powder-forming processes.
9. PFC is a distinct element code
– simulates movement & interaction of discrete bodies
• PFC2D: circular, PFC3D: spherical
– allows complete detachment & recognizes new
contacts automatically during calculation cycle.
– Newton’s law of motion solved via explicit finite-
difference procedure (allows dynamic instabilities to
be followed without numerical problems).
– damping formulation allows quasi-static solution.
13. Material Calibration
given micro-properties macro-response
perform simulation
to predict
grains & bonds grain shape grain packing lab scale
simulations
field scale
(bulk behavior &
relevant mechanisms)represent microstructure
anisotropic properties
biaxial tests
mining : excavation damage
& stability
rock cutting : chip formation
liner anchor : pullout strength
Weaker and less stiff biotite bands
within rock matrix
Slot in floor of URL Mine-by Tunnel
Brazilian tests
circular balls &
parallel bonds
• deformability
• strength
• clusters
• clumps
• porosity
14. Typical simulated lab test:
Typically, we calibrate the synthetic material to measured values of
modulus,
Poisson’s ratio,
UCS,
tensile strength and
fracture toughness.
Optionally, we can match crack-initiation stress, post-peak softening
and dilation rate.
15. 3B. Smooth-Joint Model (SJM)
Illustration of smooth joint in 3D:
Example with several sliding joints:
A “joint” in a PFC bonded
assembly consists of
modified properties of
contacts whose 2 host-
particle centroids span
the desired joint plane.
Note that the SJM also represents normal joint opening
To avoid the “bumpy-road” effect, a
new smooth joint model is employed
that allows continuous slip
16. We have performed an extensive series of validation comparisons with
laboratory experiments, in 2D (Wong et al, 2001) and 3D (Germanovich
and Dyskin, 2000)
R.H.C. Wong, K.T. Chau, C.A. Tang, P. Lin. Analysis of crack coalescence in rock-like
materials containing three flaws: Part I: experimental approach. Int. J. Rock Mech. &
Min. Sci. 38 (2001).
Laboratory Numerical
Initial cracks
17. Summary of the Synthetic Rock Mass (SRM)
We model the intact rock combined with in situ discontinuities.
This allows fracturing across rock bridges. The overall response
represents the rock mass strength, and its post-peak behaviour.
The strength envelope (e.g., Hoek-Brown) of the rock mass is
not prescribed – it develops as emergent behavior, given the
known properties and geometry of the constituents.
It is very difficult to estimate the strength envelope and brittleness
of a large rock mass. However, the length-scales of the
constituents are much smaller, and individually testable.
Typically, we have data for calibration of the constituents.
18. Example of model creation process
CALIBRATION STAGE
30 particles
10m x 5m
PERIODIC STAGE
3,000 particles
100m x 50m
PRE-MINE STRESS INITIATION STAGE
300,000 particles
SMOOTH JOINT ADDITION (1000m x 500m)
.
.
.
CONSTRUCT FULL MODEL OUT OF BRICKS
. .
.
.
19. Rapid sample Construction
xy
z
(1) (2) (3)
1. Small “pbrick” built rapidly, using periodic space,
and brought to mechanical equilibrium
2. Pbricks combined to form intact assembly (they fit
together perfectly!)
3. DFN inserted
Pbrick derived
from a sample
made in periodic
space:
21. Discrete Fracture Network (DFN)
• assume fractures are
3D circular disks
• faults and joints are
generated
stochastically
fracture
length
(x0,y0)
W
d/dir
fracture
center
22. Stochastic Fracture Disk in DFN
#
LLmin
Disk Diameter
(Power Law)
N
k1
k2
k1 < k2
E
Pole Orientation
(Fisher Distribution)
Fracture Disk Position
(Poisson Distribution)
Scan Line Validation
Is average spacing correct?
NO
YES
DFN
#
LLmin
#
LLmin
Disk Diameter
(Power Law)
N
k1
k2
k1 < k2
E
N
k1
k2
k1 < k2
E
Pole Orientation
(Fisher Distribution)
Fracture Disk Position
(Poisson Distribution)
Scan Line Validation
Is average spacing correct?
NO
YES
DFN
23. Samples of various sizes may be built
5m10m
20m
40m
80m
This sample is composed
of a million bonded
particles.
27. Joint Size Distribution
• The joint traces mapped are almost always censored on
both ends by the mapping window (open pit and
underground).
• As a result, it is usually necessary to make an assumption
about the fracture size distribution.
• A power-law model with an exponent equal to –4 and a
minimum fracture size of 10 m may employed for the
Carbonatite unit.
• Cases like this would clearly benefit from digital mapping
of pit walls on a larger scale, e.g. several benches. Trace
maps developed from such campaigns would be very
helpful in reducing uncertainty in the joint size distribution
and resulting DFN.
28. Typical joint density statistics
(Fracture Frequency)
Lithology Approximate
traverse
orientation
Mean
fracture
frequency
(P10)
Standard
dev
Min Max Count
(number
of
traverses
used)
DOL West 1.88 1.19 0.02 4.80 33
DOL South 2.54 1.95 0.16 10.00 80
BCB West 0.83 0.58 0.16 3.33 53
BCB South 0.53 0.39 0.04 3.64 200
FOS West 0.34 0.27 0.03 1.15 20
FOS South 0.33 0.24 0.06 0.74 7
MPY West 0.39 0.32 0.04 0.94 7
MPY South - - - - 0
29. E-W sectionS-N section Horizontal section
Example of DFN realization – cross-sections
(typically, 50,000 3D joint segments may be generated, and applied
to the simulated block of rock)
33. Joint Properties
0%
10%
20%
30%
40%
50%
60%
70%
1 2 3 4 5 More
Roughness
Dolerite
BCB
FOS
MPY
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
1 2 3 4 5 More
Hardness
Dolerite
BCB
FOS
MPY
Roughness Hardness
Estimated Shear Strength and Stiffness
(based on comparisons with tests conducted by SKB, 2005; SKB, 2006a and 2006b, Olofsson et al. 2007)
34. Intact Strength
Carbonatite Foskorite Micaceous
Pyroxenite
Dolerite
Mean Measured UCS (MPa) 139 63 90 320
Estimated Rock Block Strength:
80% of UCS
111.2 50.4 72 256
Young’s Modulus, E (GPa) 58 78 72 90
Poisson’s Ratio, v 0.33 0.27 0.35 0.30
35. The SRM approach can be used in two main ways –
A. to perform element tests, deriving behaviour that can be
imported into other methods (e.g., FE, FD or LE);
B. to perform complete simulations of boundary-value
problems.
5. SRM applications
36. 5A. Element tests
Such tests can be used (among other things) to determine:
1. The equivalent material properties of the rock mass –
strength, modulus & anisotropy;
2. The size effect (how the properties depend on sample
size);
3. The apparent brittleness of the rock mass;
4. The variability (e.g., standard deviation) of measured
quantities.
5. Fragmentation – blocks sizes resulting from straining
37. 3D Carbonatite 40m x 80m sample:
typical triaxial stress-strain response
0.0E+00
1.0E+07
2.0E+07
3.0E+07
4.0E+07
5.0E+07
6.0E+07
7.0E+07
8.0E+07
0.0E+00 1.0E-02 2.0E-02 3.0E-02 4.0E-02 5.0E-02 6.0E-02
Axial Strain
AxialStress(Pa)
X
Y
Z
(5 MPa confinement) Note anisotropy
39. Carbonatite SRM samples for size-effect tests
80m x 40m x 40m
20m x 10m x 10m
from mid-sized U2
U1
U4
U2
U3
L1
L4
L2
L340m x 20m x 20m
(a) (d)
(b)
(c)
N
W
E
S
Y
Z
X
L2
L3
U1
U4
U2
U3
L1
L4
40. 2D SRM biaxial results for
different scales: same DFN
0.0E+00
1.0E+06
2.0E+06
3.0E+06
4.0E+06
5.0E+06
6.0E+06
7.0E+06
8.0E+06
9.0E+06
1.0E+07
0.0E+00 5.0E-04 1.0E-03 1.5E-03 2.0E-03 2.5E-03 3.0E-03 3.5E-03 4.0E-03 4.5E-03 5.0E-03
Strain
AxialStress(Pa)
20m_dia_SJ @ 1 MPa
50m_dia_SJ @ 1 MPa
100m_dia_SJ @ 1 MPa
Increasing Specimen Scale
psr = 1e-4 s-1
stress
strain
20 m sample
100 m sample
50 m sample
42. We know brittleness is important but it is very difficult to
estimate it for a large rock mass.
For example, in a cave mine the cavability is a strong function
of apparent brittleness.
Brittle
Less
Brittle
ORCaves
Readily
More Difficult
to Cave
44. 10x20m
0
10
20
30
40
50
60
70
80
L1 L2 L3 L4 U1 U2 U3 U4 U2-
20x40m
UCS(MPa)
X
Y
Z
U2-20x40m-X
U2-20x40m-Y
U2-20x40m-Z
10x20m
0
5
10
15
20
25
30
L1 L2 L3 L4 U1 U2 U3 U4 U2-
20x40m
E(GPa)
X
Y
Z
U2-20x40m-X
U2-20x40m-Y
U2-20x40m-Z
The SRM approach may also
be used to quantify variability
by testing many realizations,
each conforming to the same
DFN and particle properties.
Eight 3D samples:
UCS
Young’s modulus
(note correlation)
45. Why does variability matter? Jefferies et al (2008) performed
1000 slope stability simulations with strengths normally distributed
(in space) about the mean value …
Contours of cohesion
46. Result of 1000 realizations
Histogram
0
20
40
60
80
100
120
140
0.56
0.6
0.64
0.68
0.72
0.76
0.8
0.84
0.88
0.92
0.96
1
1.04
1.08
1.12
1.16
1.2
1.24
1.28
FS
Frequency
FS (using uniform
average strength)
= 1.23
Factor of safety
FS = 1.0
Most cases are less stable
than the uniform case with
average strength
47. Finally, we can quantify how fragmentation evolves
during a loading test
in situ, from a cave mine
3D synthetic
rock mass
after testing
increasing strain
Fragment size
Cumulative % passing
48. 5B. Full simulations using the SRM
Full 2D example simulation have been performed, and
there are various 3D possibilities.
49. Property Value
Bulk Density (kg/m3
) 2600.0
Young’s Modulus (GPa) 4.6
Poisson’s Ratio 0.25
Bulk Modulus (GPa) 3.07
Shear Modulus (GPa) 1.84
Cohesion (MPa) 0.22
Friction (degrees) 38.0
Tensile Strength (MPa) -0.09
Dilation (degrees) 0.0
40°
39°
34°
366 m
297 m
168 m
40 m
40 m
40 m
400 m N
Profile 65.6°
Orientation of
the profile
350 m
40°
39°
34°
366 m
297 m
168 m
40 m
40 m
40 m
400 m N
Profile 65.6°
Orientation of
the profile
350 m
CSIRO 2D Slope Problem
This is a simplified model of West-Wall Slope at Chuquicamata
50. Blocky system realized by DFN
• 2,890 faults and 37,335 joints for 40,225 discontinuities
• ~330,000 particles
• 38,656 blocks (clusters*)
1 Km X 500 m
51. The slope is excavated in stages, bench by bench
bonded particles
(intact rock)
SJM
(discontinuity)
ROCK MASS
52. PFC2D Results – major movement
(red vectors)
Depth of movement-region ~ 130 m
- similar to that observed in the field
Most of the slip shows toppling
failure (motion on these joints has
been favoured automatically, out
of many possibilities)
53. 0.0 – 0.1 m
0.1 – 0.2 m
< 0.0 m
0.3 – 0.4 m
0.4 – 0.5 m
0.2 – 0.3 m
0.6 – 0.7 m
0.7 – 0.8 m
0.5 – 0.6 m
0.9 – 1.0 m
1.0 – 1.1 m
0.8 – 0.9 m
1.1 – 1.2 m
> 1.2 m
Accumulated
horizontal
displacement
Toppling is the main mechanism
Ramp 1
54. Cross-section of 3D slope to be modelled
z
y
x
190m
IRA = 50º
3m
BA = 80º
30.5m
225 m
17.215m
(0,30.4294,95.0)
x
x = {4m, 8m,
16m, 24m}
z
(after Stewart et al, 2000)
55. 155,000 balls
4m 8m 24m16m
967,000 balls656,000 balls345,000 balls
Realizations of 4 slices of rock mass, given a particular DFN
56. Thus, we can reasonably model a significant portion of a large rock
slope, but with a limited-thickness slice – of 24 m, using PFC3D.
However, we now have a specialized code, similar to PFC3D, using
a lattice of nodes and springs, rather than spheres. This allows
model sizes of about 10 times those of PFC3D. We may therefore
expect 240 m slices to be possible, which will permit full 3D
mechanisms to occur in the example slope. The new code has
been written, and is currently being tested.
57. SRM applied to continuum simulations
If a full 3D model cannot be made with the SRM (at the same
resolution as the 2D example presented earlier), we may use 3D tests
on SRM samples (containing the field-derived DFN), and subject it to
strain paths determined from a continuum (with faults) simulation. The
resulting mechanical behavior (notably brittleness, which depends on
rock-bridge fracture) is imported into the continuum model as a
constitutive relation.
Test SRM samples,
using PFC3D
Get strain paths with
(e.g.) FLAC3D
Element
behavior
Actual
strain
paths
58. This approach has been used successfully to model
the progressive damage around a block cave
Sections through full 3D
mine-scale model
59. Conclusions
We expect that the SRM may replace empirical methods. Thus, we
can jump directly from small-scale properties and the DFN to large-
scale rock mass behaviour (strength, size effect, brittleness,
variability etc), bypassing the GSI/Hoek-Brown methodology (for
example).
The SRM approach takes account of all available field data.
With limited field data, the SRM approach can inform the
designer of the consequences of spending more money on field
tests – e.g. how would the standard deviation of predictions
reduce as a function of extra cost?
The SRM approach has already been validated in several ways,
but the process is on-going, and there must always be feedback
from field observations in any large-scale application.
