Discrete Element Models of the Micromechanics of Sedimentary Rock: The Role of Organization vs. Friction
David F. Boutt, B.J.O.L. McPherson, Department of Earth and Environmental Sciences, Hydrology Program, New Mexico Tech, Socorro, NM, 87801, dboutt@nmt.edu.
ABSTRACT:
The micromechanics of sedimentary rock deformation are a fundamental aspect of many research fields, ranging from geotechnical engineering to petroleum recovery and hazardous waste disposal.  Laboratory triaxial tests yield information concerning macroscopic behaviors but are not capable of quantifying micromechanical processes such as microcracking and localization.  Thus, to quantify micromechanical processes we employed the discrete element method (DEM) of rock deformation, calibrated with triaxial test results.  This DEM simulates rock using rigid disc shaped particles bonded at contacts between particles.  Previous studies demonstrated that this type of DEM can qualitatively and quantitatively mimic macroscopic behaviors of triaxial tests.  An important conclusion of these studies is that a number of particles must be bonded together with higher bond strengths than the surrounding particles to achieve a steeper strength envelope of rocks.  This process, termed clustering, is the focus of this study.
We hypothesize that since clusters posses a more complicated geometry, they may increase failure strength at elevated confining pressures by interlocking and creating a higher apparent friction.  An alternative hypothesis is that the clusters change force chain development by allowing chains to persist longer in specimens.  This ultimately causes failure to occur at higher strengths compared to unclustered material.  A systematic study comparing effects of cluster shape, particle friction, and force chain development was undertaken.  Several model simulations with various cluster shapes and sizes were compared with each other as well as single particle models with high friction coefficients (>1).  Preliminary results suggest that the organization of the particle clusters play a key role in increasing the strength envelope.  Particle friction coefficients needed to increase slopes of the strength envelopes are well beyond those of geological materials measured in the laboratory.  We conclude that models of rock using the DEM with single spherical and disc shaped elements do not fully capture the appropriate physics of most sedimentary rocks.
PREVIOUS OBSERVATIONS:
NUMERICAL MODEL DESCRIPTION:
Contact Stiffness Model Slip Model
if
where:
slip will occur
Parallel-Bond Model
where:
Contact-Bond Model
•inactivates slip model
•allows forces to
 build until bond
 strength is exceeded
Model Equations - Force-Displacement Law and Contact Constitutive Model
Law of Motion
(applied to each particle)
•resultant force + moment
Force-Displacement
(applied to each contact)
•relative motion
•constitutive law
Contact Forces
Update particle + contacts
Calculation Cycle
A
B
Contact Force
k[B]
k[A]
B
Parallel Bond
k,FP
A
B
Friction
m
A
A
Contact Bond
Fc
B
0
2
4
6
x 10
-3
0
20
40
60
80
100
120
140
Strain
Stress (MPa)
2
1
4
5
6
3
7
5 6    7
1       2 3   4
0  
0.01
0.02
0.03
0.04
0.05
PROBLEM STATEMENT:
•Why do clusters increase the internal coefficient of friction (slope of the failure envelope) of these discrete assemblies?
PARTICLE ASSEMBLY GENERATION (CONT’D):
INTRODUCTION:
 
Discrete Element Models (DEM) of rock are popular tools for extending laboratory observations to make inferences on the micromechanics of granular materials and consolidated rock (Boutt and McPherson, In Press).  Despite this, little is known about the sensitivity of fundamental parameters controlling macroscopic model behavior.  This study addresses the particle shape issue.
Clusters in DEM models are groups of particles bonded together with higher strength bonds in the cluster than between clusters.  Figure 1b to the right shows groups of three particles in a cluster.  Ideally, clusters are envisioned to reproduce the micromechanics of a breakable grain. Two properties of clusters are significant.  First are the shapes of the resulting group of particles (focus of this poster).  The second important property is the finite strength of the cluster.  In traditional DEM’s grains are not deformable but the contacts are.  This is one technique to allow grains or groups of them deform differently.
Figure 1: Thin Section Micrograph (a) and
particle clusters (b).
Figure 5: The spatial gradient of the displacement field and the stress-strain curve for the simulation of sample 5U-4.  Spatial gradients of the displacement field were calculated to amplify areas where strong changes in displacement magnitudes occurred.  Brighter areas are interpreted as zones of differential movement or fracture surfaces.  The model is consistent with laboratory observations of fracture orientation and location.
Figure 5:  Spatial Gradient of Displacement Field
Post-Failure
Pre-Failure
Failure
Particle Rotations
Figure 6:  Particle Rotations During Compression Test
Figure 2: Simulated and observed failure envelopes show a good match at all confining pressures.  Clusters of both 3 and 4 particles were used in these simulations.  All models were executed with a particle friction coefficient of 0.5.
Figure 3:  The sensitivity of particle friction coefficients is shown here for a clustered model.  Low confining pressures do not exhibit a large change in strength for different friction coefficients.  The differences in higher confining pressures is significant.
Figure 4: Failure envelopes for unclustered and clustered models of 2, 3, and 4.  Clustering a model changes not only the cohesion (due to a larger % of higher strength bonds) but also changes the internal coefficient of friction for the model. 
A new algorithm to build assemblies with equal amounts of clusters was necessary (See previous box).  This consisted of first generating clusters dropping them into a box under gravity and then compacting the assembly.  This is shown in figure 7.  This results in particle assemblies that have clusters of all the same size but the particles can very in size between different clusters. The particle are sizes then adjusted to remove anisotropies in stresses that build-up during the sample generation process in attempt to remove large locked-in stresses.
Figure 7:  Sequence of images showing the particle generation process. A new method was necessary to isolate the effects of particle shapes and to reduce the the effects of a distribution of particle cluster sizes
PARTICLE CLUSTERING AND FRICTION SENSITIVITY:
Method:
•4 different cluster sizes
•Simulations at 0, 5, 10, 20, 40 MPa confining pressures obtained for 5 different friction coefficients
•Other parameters held constant, Intracluster bond strength of 650 MPa
0
MPa
5
MPa
40
MPa
10
MPa
20
MPa
Frequency
Cluster Size
The previous algorithm to generate clusters consisted of searching through all particles and attempting to group them in groups of a similar number.  This resulted in a distribution of cluster sizes as shown to the right.  Clusters of 3 are black, clusters of 2 are red, and single particles are blue.  The new algorithm described below avoids this complication.
PARTICLE ASSEMBLY GENERATION:
CONCLUSIONS:
•Cluster shape has a large influence on the internal friction coefficient of discrete assemblies
•Single particle models need an unreasonably high particle friction coefficient (>>1) to match the same value of the internal friction coefficient of a clustered model.
•The Savage et al. (1996) model of internal friction coefficients is consistent with these results.
REFERENCES:
Boutt, D.F., and McPherson, B.J.O.L., In Press, Simulation of Sedimentary Rock Deformation:  Lab-Scale Model Calibration and Parameterization: Geophysical Research Letters.
Mogi, K., 1974, On the pressure dependence of strength of rocks and the Coulomb fracture criterion: Tectonophysics, v. 21, p. 273-285.
Savage, J.C., Byerlee, J.D., and Lockner, D.A., 1996, Is Internal Friction Friction?: Geophysical Research Letters, v. 23, p. 487-490.
Figure 9:  Strength envelopes for 3 different shapes of clusters.  Triangle shaped clusters are much stronger than other shapes.
Figure 8:  Failure Strength surfaces as a function of particle cluster size and particle friction coefficient for 5 different confining pressures.  Notice that as confining pressure is increased that the assemblies reponse to stress is very different. 
A model first proposed by Mogi (1974) and later modified by Savage et al. (1996) concluded that the internal coefficient of friction can be related to the sliding coefficient of friction of the material by:
Here, mi is the internal coefficient of friction, mi is the sliding friction, A1 is the sliding area, and A2 the intact areas. In order to achieve the same mi for a unclustered material compared to a clustered material m must be at least 5 times greater.  Thus if this model applies the area of sliding versus intact rock must be different in the unclustered vs. clustered models.  We argue that in clustered materials the area of intact rock is smaller due to the larger mechanical units and effectively increases the ratio of slipping surfaces to intact surfaces.  This ratio for unclustered material is much smaller due to the small size the mechanical units and greater number of contacts allowed to slip
Paper No. T32E-0913
Failure Envelope Comparison