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Applied Numerical Modeling in Geomechanics – 2020 – Billaux, Hazzard, Nelson & Schöpfer (eds.) Paper: 03-01
©2020 Itasca International, Inc., Minneapolis, ISBN 978-0-9767577-5-7
1 INTRODUCTION
The Swedish Nuclear Fuel and Waste Management Company (SKB) is developing a methodology for
long-term storage of nuclear spent fuel in an underground repository in compact crystalline rock (SKB
2011). The repository design calls for the placement of the spent nuclear fuel in copper canisters in verti-
cal 1.8 meter diameter boreholes of 8 meter length. It is likely that the stress redistribution around the ex-
cavations, along with the thermal load from the decaying spent nuclear fuel, will induce stress at the
borehole walls that is greater than the rock mass strength, and thereby produce an excavation damaged
zone (EDZ) in which hydromechanical and geochemical modifications may induce significant changes in
flow and transport properties. The mechanical characteristics of this EDZ must be understood to evaluate
its impact on repository performance.
Our understanding of the damage process can be enhanced by performing in-situ testing. The Äspö Pillar
Stability Experiment (Andersson & Martin 2009, Andersson et al. 2009) is one such in-situ test. The ex-
periment was performed in Äspö diorite and consisted of drilling two vertical 1.75 meter diameter bore-
holes of approximately 6 meter length separated by a one meter pillar. The pillar was loaded by a combi-
nation of excavation-induced stresses and heating of the surrounding rock. In this way, the tangential
stress on the pillar boundary could be controlled in a very precise manner.
In competent crystalline rock, the EDZ takes the form of spalling on the excavation boundaries, whereby
a localized damage region forms at the location of the maximum tangential stress. Spalling begins with
small rock chips forming on the boundaries of the excavation. In the APSE experiment, the chips ranged
in size from a finger nail to approximately 0.1 m2. Larger chips formed beneath the small chips. The
chips are tangential to the hole wall and are believed to have formed in tension. The spalling region
evolves into a v-shaped notch as the tangential stress increases with crushing and/or shearing at the notch
tip, suggesting that the chip-forming process is suppressed at the notch tip. In the APSE experiment, rock
mass yielding occurred when the tangential stress on the hole boundary reached 59% of the unconfined-
compressive strength of the intact rock. Changes in the tangential stress of approximately one MPa be-
yond this threshold expanded the extent of rock mass yielding, indicating that a distinct stress level
threshold needs to be reached before the rock yields.
Our understanding of the damage process can also be enhanced by numerical modeling. This paper sum-
marizes the results of a study to evaluate the ability of the PFC (Itasca 2018) flat-jointed material model
to simulate spalling (Potyondy 2019). Both 2D and 3D flat-jointed Äspö diorite materials were created,
and their response during direct-tension and compression tests was studied. The material behavior is de-
scribed in Section 2. Borehole models of a cylindrical hole in an infinite medium were created and used to
approximate the conditions in the APSE experiment. The behavior of the borehole models is described in
Section 3. Conclusions of this study are provided in Section 4.
Simulating spalling with a flat-jointed material
David O. Potyondy1 & Diego Mas Ivars2,3
1Itasca Consulting Group, Inc., Minneapolis, MN, USA
2Swedish Nuclear Fuel and Waste Management Co (SKB), Solna, Sweden
3Division of Soil and Rock Mechanics, Royal Institute of Technology (KTH), Stockholm, Sweden
2 FLAT-JOINTED MATERIALS
The flat-joint contact model provides the macroscopic behavior of a finite-size, linear elastic, and either
bonded or frictional interface that may sustain partial damage. A flat-jointed material mimics the micro-
structure of angular, interlocked grains. We refer to the balls of a flat-jointed material as faced grains,
each of which is depicted as a spherical core and a set of skirted faces (see Fig. 1a). The flat joint model
formulation is given in Potyondy (2018).
Both 2D and 3D flat-jointed materials to represent Äspö diorite were created and subjected to direct ten-
sion and compression tests. These materials match the Young’s modulus, direct-tension strength, uncon-
fined-compressive strength, and peak strength at 7-MPa confinement. The crack-initiation stress is too
low, the crack-damage stress is too high, and the Poisson’s ratio is too low. No attempt was made to
match the crack-initiation and crack-damage stresses. Attempts to increase the Poisson’s ratio were un-
successful. The material behavior during these tests is like the brittle behavior of compact rock (Potyondy
2015), with the exception that transgranular cracking occurring within and across grains during the com-
pression tests is absent. In compact rock under near-zero confinement, axial splitting occurs, and with in-
creased confinement, shear fractures form. These behaviors are exhibited by the 2D flat-jointed material
but appear to be absent in the 3D flat-jointed material. Although there is extensive dilation after the crack-
initiation point in the 3D flat-jointed material, localizations associated with axial splitting and shear-
fracture formation are absent; instead, the deformation field remains homogeneous.
Microstructural validity describes whether the grain faces overlap one another; a flat-jointed material has
a valid microstructure if and only if the faces of each grain can be connected to the grain center with no
overlap. A valid microstructure is physically realizable — i.e., a physical replica of such a material could
be constructed. An invalid microstructure may produce useful behavior, and as such, its use can be justi-
fied. For the present modeling effort, microstructurally valid and invalid instances of the flat-jointed ma-
terial that match the Äspö diorite properties listed above were created (see Fig. 1b). The microstructurally
valid 2D material was selected for the borehole models because it is physically realizable. The micro-
structurally invalid 3D material was selected for the borehole models because the microstructurally valid
3D material was deemed to be unrealistic, having a very small amount of cement with excessively long
cement bridges that overlap across bonded regions and microproperties approximately 3.5 times larger
than their corresponding macroproperties to compensate for the fact that the interfaces do not fully cover
the ball surfaces. This suggests that the 3D flat-jointed material may not be well-suited to model a com-
pact rock like Äspö diorite and may be better suited to model a more porous rock like a sandstone. The
PFC3D bonded-block model that represents the rock as a bonded collection of rigid polyhedral blocks
with an initial porosity of zero may be better suited to model compact rock.
The flat-jointed material exhibits a size effect with modulus and peak compressive strengths increasing
with increasing specimen resolution (number of grains across the specimen width) until a representative
volume is reached. There is no size effect for the tensile strength and Poisson’s ratio. The representative
volumes for the 2D and 3D materials correspond with specimen resolutions of approximately 25 and 50,
respectively. For the 3D material, the values reach approximately 92% of their asymptotic values at a res-
olution of 20. The specimen resolution is the controlling parameter — i.e., the same response is obtained
by either varying the specimen size while keeping the average grain diameter constant or varying the av-
erage grain diameter while keeping the specimen size constant.
3 BOREHOLE MODELS
Both two- and three-dimensional borehole models of a cylindrical hole in an infinite medium, have been
developed. The two-dimensional models approximate the conditions for the Kirsch solution (long excava-
tion with a circular cross-section in a medium subject to biaxial stress), whereas the three-dimensional
models are quasi 3D, meaning that plane-strain conditions are enforced on a plane normal to the hole axis
and the model thickness in this direction is small relative to the hole diameter (with approximately four
grains through the thickness). These borehole models include PFC models (both 2D and 3D) and a cou-
pled FLAC3D-PFC3D model. The coupled model allows greater hole resolution (number of grains across
the hole diameter) to be obtained in the near-borehole region.
Figure 1. (a) Flat-jointed material; and (b) microstructural characteristics of the four materials (left: microstructural-
ly valid, right: microstructurally invalid) that match the properties of Äspö diorite.
The borehole models approximate the conditions in the Äspö Pillar Stability Experiment. After excava-
tion of the hole, the maximum tangential stress reaches 165 MPa according to the Kirsch solution, but on-
ly reaches 128 MPa according to a stress analysis accounting for the drift and both holes. A set of coupled
borehole models is constructed for which the material grain size is varied to produce models with a range
of hole resolutions. The displacement field is compatible across the coupling interface (see Fig. 2a), and
when the models are run elastically, the excavation-induced displacements provide a good match to the
Kirsch solution for all hole resolutions. The damage increases with increasing hole resolution, with grad-
ual delineation of a spalling zone and two damage lobes (see Fig. 2b). These damage features become
more clearly delineated as grain size is reduced from 55 to 9 mm. This is approaching the grain size of
Äspö diorite (which ranges from 0.1 to 5 mm). It is expected that the model response will best match the
rock behavior when the average grain size of the model is equal to that of the rock; therefore, the general
damage characteristics are studied for the model with the 9-mm grain size. This model has 718,000 balls
with a total run time of 48 hours on a PC with an Intel Xeon CPU at 3.4 GHz.
Figure 2. (a) Excavation-induced displacement of the coupled model with 9-mm grain size showing displacement-
magnitude contours of the balls and zones. (b) Damage summary overlaid on damage after excavation. The crack
magnification is chosen to draw each crack as a disk with a 36-mm diameter. When the cracks are magnified in this
way, the crack density cannot be visually assessed, appearing larger than it is because the neighboring cracks over-
lap one another.
Damage consists of a spalling zone and two damage lobes. The spalling depth is 8 grains over a 100-
degree sector and reduces to sparse single grains over the remaining perimeter. The damage lobes consist
of a crack swarm located at 40 degrees from the horizontal and extending by the distance of the hole radi-
us into the rock. The crack density and dilation are greater in the spalling zone than in the damage lobes,
and the swarm cracks are aligned with the compressive force chains. The spalling zone corresponds with
an early stage of rock mass yielding before a well-defined v-shaped notch has formed. The cracks in the
spalling zone are like the rock chips that formed in the APSE experiment: formed in tension, aligned par-
allel to the surface, and dilated. Discrete chips have not formed in the model; such chips might form if the
model grain size was reduced to equal the grain size of Äspö diorite. It is likely that the damage lobes are
formed where the stress is near to or greater than the crack-initiation stress. The crack-initiation stress of
the model material is less than that of the rock (60 versus 90 MPa, respectively); thus, these lobes may be
present only in the model and not in the APSE experiment. They were not detected by the AE monitoring
of the experiment; however, damage in the lobe regions may be occurring as individual (not clustered)
grain-scale cracks that have not been detected by conventional acoustic emission monitoring. The damage
lobes are also present in a 2D borehole model of the APSE experiment; however, the damage lobes are
not well formed until the APSE load has been scaled up by a factor of two.
Two variants of the base material that match the Äspö diorite properties listed above were created and
used in the borehole modeling. In the first material, 25% of the fully bonded interfaces were replaced with
slits to increase the Poisson’s ratio. In the second material, a distribution of microstrengths was used. The
overall damage characteristics (spalling zone and two damage lobes) were only minimally affected by
these material modifications.
4 CONCLUSIONS
The PFC3D flat-jointed material can be used to model spalling. By using a coupled FLAC3D-PFC3D
model, a 9-mm grain size for a model that approximates the 1.75 meter diameter APSE experiment could
be run in a reasonable amount of time (48 hours). This model produced a spalling zone and two damage
lobes. The spalling zone corresponds with an early stage of rock mass yielding before a notch has formed
with tensile cracks aligned parallel to the hole surface and dilated. Work is planned to model the URL
Mine-by Experiment (Martin 1997) using PFC3D flat-jointed, soft-bonded and bonded-block models to
provide a quantitative assessment of PFC3D spall-modeling capability.
REFERENCES
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