Microslips as precursors of large slip events in the stick-slip dynamics of sheared granular layers: A discrete element model analysis
Department of Civil, Environmental and Geomatic Engineering, Swiss Federal Institute of Technology Zürich, Zürich, Switzerland
Swiss Federal Laboratories for Materials Science and Technology, Empa, ETH Domain, Dübendorf, Switzerland
Corresponding author: B. Ferdowsi, Department of Civil, Environmental and Geomatic Engineering, Swiss Federal Institute of Technology Zürich, Zürich, Switzerland. (firstname.lastname@example.org; email@example.com)
 We investigate the stick-slip behavior of a granular system confined and sheared by deformable solid blocks using three-dimensional discrete element method simulations. Our modeling results show that large slip events are preceded by a sequence of small slip events—microslips—whose occurrence accelerates exponentially before the large slip event onset. Microslips exhibit energy release several orders of magnitude smaller than the large slip events. The microslip event rate is proposed as a measure of slip activity in the granular gouge layer. A statistical analysis shows that microslip event rate correlates well with large slip event onset and that variations in it can be used to predict large slip events. The emergence of microslips and their duration are found to be controlled by the value of the slipping contact ratio and are therefore related to the jamming/unjamming transition of frictional granular packings.
 Tectonic fault zones contain granular media known as fault gouge. Fault gouge plays a fundamental role in determining the fault's frictional strength and the earthquake slip dynamics. Fault systems accumulate strain energy during the interseismic period of the seismic cycle, just as a sheared granular layer does during the “stick” phase of the stick-slip cycle [Brace and Byerlee, 1966; Johnson et al., 1973]. Understanding and characterizing the behavior of sheared granular layers in numerical simulations and laboratory studies can provide important insight into the influence of microscopic physics of granular friction on faults' behavior.
 There is increasing evidence that some proportion of large earthquakes are preceded by a period of accelerating slip activity of small- to moderate-sized earthquakes, foreshocks, or slow slip events [Sykes and Jaume, 1990; Kato et al., 2012; Bouchon et al., 2012, 2013]. These observations are in agreement with a scenario where foreshocks are the manifestation of an initiation process leading to the main shock [Ohnaka, 1993; Abercrombie and Mori, 1996]. At the fault gouge scale, stick-slip experiments show that slow displacements occur prior to the onset of large and rapid slip events, such that there is a transition from quasi-static creep to rapid and dynamic slip, e.g., Marone  and Nasuno et al.  for sheared granular gouge layers, and Rubinstein et al.  and Baumberger et al.  for solid-on-solid frictional interaction. The slow creep-like displacements can be explained by the occurrence of microslips and microfailures in the system [Amon et al., 2013; Pica Ciamarra et al., 2010; Papanikolaou et al., 2012]. Microslips drive the granular system to a local energy minimum, while large slip events accompany a large amount of energy release and correspond to a significant reorganization of the granular assembly itself, in terms of contacts and contact forces [Pica Ciamarra et al., 2010].
 In this paper, the stick-slip behavior of a sheared fault gouge is simulated by the 3-D discrete element method (DEM) [Cundall and Strack, 1979; Place and Mora, 1999; Wang et al., 2006]. We report a detailed investigation of microslip occurrence, their grain-scale mechanisms, and their relation to energy release during granular shearing and large slip events.
 Figure 1a illustrates the simulated granular gouge layer. The model consists of three layers of particles: a driving block at the top, a granular gouge layer, and a substrate block at the bottom. The driving and substrate blocks are used to confine the granular gouge by applying a constant normal force in the Y direction. The top driving block moves at constant velocity in the positive X direction and applies a shear force to the granular gouge layer. Each variable/parameter in our 3-D DEM model is expressed in terms of the following basic dimensional units: L0=150μm, t0=1 s, and M0=1 kg, for length, time, and mass, respectively. We run sheared granular layer simulations at a confining pressure of σn=40 MPa and shearing velocity of to achieve stick-slip dynamics. Further details about the model are provided in the supporting information.
 The stick-slip behavior of the granular gouge layer is monitored by friction coefficient time series. The friction coefficient, μ, is defined as the ratio of shear stress developed at the boundary layers to the imposed normal stress. In the following, we focus on a large slip event and associated activity that takes place prior to this event. Figure 1b shows a characteristic time series of the friction coefficient for the time interval 255;280t0. Slip events are identified when the first-order derivative of the friction coefficient becomes negative and lower than a threshold equal to −3×10−2t0−1. This threshold is chosen to be small enough to capture small events, but large enough to avoid capturing the intrinsic fluctuations of the friction coefficient due to the granular dynamics (By “granular dynamics,” we mean slow rearrangement of particles due to constant shear velocity applied to the top boundary of the granular layer. These small rearrangements of particles are always present in the friction coefficient signal as a background fluctuation.) In Figure 1b, a characteristic large slip event occurs at t1≈263t0. Figure 1c shows the time series of the granular layer's total kinetic energy, which is the energy due to particles movement (complete definition is provided in the supporting information). We distinguish three primary categories of slip events: large slips, microslips before a large slip (red symbols, Figure 1b) and afterslips that follow the large slip event (blue symbols, Figure 1b). The distinction between the three types of slip events is based on the change in the granular layer's total kinetic energy during the slip event. At the onset of a slip event, the kinetic energy increases sharply from the background level. The largest energy release, which results from the elastic potential energy stored in particle contacts, occurs at the moment of a large slip event. A detailed look at the friction coefficient and kinetic energy before the large event at t1≈263t0 is given in the insets. We observe that many small slip events manifest by abrupt increases of kinetic energy compared to the background level. These slip events release between 1 and 6 orders of magnitude smaller energy compared to the large slip event. If an event is large enough (like at t1≈263t0) to activate other susceptible jammed regions of the granular layer, we observe afterslips closely clustered in time after the large event. During afterslips, the kinetic energy signal is elevated above the background level preceding the large event, and afterslips involve an energy release comparable to the large slip event. Figure 1d shows the thickness of the granular layer. At the time of the large slip event, the thickness of the granular gouge layer shows a drastic compaction (thickness reduction). Microslips occur in association with dilation of the granular gouge during the stick phase (inset). The grain rearrangements can be characterized by study of the slipping contacts [Aharonov and Sparks, 2004]. The ratio of the number of slipping contacts, i.e., those contacts in which the tangential contact force is at the Coulomb threshold, to the total number of contacts is called Slipping Contact Ratio (SCR) and is presented in Figure 1e. We distinguish three levels of the SCR during the stick phase of a characteristic large event. At the beginning of the stick phase, the SCR is about 0.01 (level 1) and no microslips occur at this stage. As the friction coefficient increases, the shear force between the particles increases; hence, the SCR increases until it reaches a value of about 0.04 (level 2). From this level on, the SCR slightly and gradually increases, and microslips appear with an increasing frequency until the SCR reaches a value of about 0.06 (level 3) where the large slip event occurs. The afterslips cluster after the large event where the SCR is still higher than level 3. The large values for the SCR during afterslips explain why their energy release differs noticeably from microslips.
 We use the Slip Event Rate (SER), defined as the number of slip events per time unit as a measure of slip activity. The SER includes microslips, large slip events, and afterslips without distinguishing between them. Figure 1f shows the slip event rate for the time interval 255;280t0. We observe that the large slip event coincides with a noticeable increase in the slip event rate.
 So far, we have presented one characteristic large event in our model, its properties, and the event activity that takes place before and after its occurrence. We next investigate the microslip activity that precedes large slip events with different slip event size. The size of a slip event is measured in terms of its total energy release, E. The definition of E is explained in the supporting information. In order to compare slip events with different time scales, we use a normalized time (to failure) as described in Figure 2a. tnorm = −1 corresponds to the beginning of the stick phase of a specific slip event, while tnorm = 0 corresponds to when the slip/failure happens. Figure 2b shows the probability of microslips occurrence during the stick phase of slip events with different event size ranges. The plot summarizes the results for a total of 44,000 slip events taking place during a long simulation interval, H=200;8000t0. Figure 2b indicates that microslips occur randomly during the stick phase. However, their occurrence accelerates exponentially close to the event onset (−0.02 <tnormalized < 0.0) for events with size E> 1.0 × 10−6M0·L02·t0−2. The nonlinear acceleration of microslips occurrence exists less significantly for events with 1.0×10−7<E<1.0×10−6M0·L02·t0−2, and it disappears for smaller events. We therefore call those events with E>Ethresh = 1.0×10−6M0·L02·t0−2 as “large” events. The events occurring at t1≈263t0, t2≈265.2t0, and t3≈274.5t0shown with black markers in Figure 1a are examples of large events in our model. We choose to explore the precursory behavior for these “large” events.
 Figure 2c shows the complementary Cumulative Distribution Function (cCDF) of the energy release, E, for a total of 898 large slip events during the interval H. The distribution follows a power law (for E>1.0×10−6), cCDF(E)∝E−βwith β≃1.23.
 Having confirmed the exponential acceleration of microslips occurrence before large slip events, we perform a pairwise sliding window cross-correlation analysis for the time series of the event energy release E(t) and slip event rate, SER(t). This is to statistically evaluate whether or not the increase in SER can be used as a further indicator (precursor) for a large slip event. After calculating the cross-correlation coefficient ρ(t) as a function of time, we determine for each large slip event the maximum cross-correlation coefficient, ρmax, and the time lag between the instant when the cross-correlation is maximal and the onset of the large slip event, τmax. An example of the events energy and SER time series as well as cross-correlation results for those time series are provided in the supporting information. A high cross-correlation coefficient ρmax(>0.8) and negative time lag means that on average, microslips anticipate a large slip event. The distributions of ρmax and τmax considering all large slip events (E>Ethresh) are plotted in Figures 2d and 2e, respectively. The average maximal cross-correlation coefficient is 0.8187, and the average time lag is −0.0525t0. More than 85%of large slip events have a cross-correlation coefficient higher than 0.75. About 75% of large slip events have negative time lags. The remaining 25%of large slip events have zero time lags. The zero time lags are mainly due to those events whose afterslip activity masks the precursory microslips. It is also partially due to the resolution of the cross-correlation analysis and the running average used for smoothing the SER and the energy release time series.
 The observation of microslips in our numerical simulation is in agreement with laboratory and theoretical studies which suggest that earthquakes are preceded by a nucleation process where quasi-static creep develops into dynamically driven motion within a confined zone on a fault [Dieterich, 1978; Marone, 1998; Kawamura et al., 2012]. Precursory activities in the form of creep deformation and aseismic slips have also been reported for large earthquakes worldwide [Ellsworth and Beroza, 1995; Peng and Gomberg, 2010]. The most recent and profound example is the evidence of small repeating earthquakes that led to the 2011 moment magnitude Mw9.0 Tohoku-Oki event [Kato et al., 2012; Bouchon et al., 2012]. There also exist evidences of similarities between microslip phenomena and slow slip events (Episodic Tremor and Slip, ETS) observed on the deeper interface of the northern Cascadia subduction zone [Garry and Dragert, 2003]: ETS events are several orders of magnitude smaller than regular earthquakes in terms of stress drop and energy release [Vidale and Houston, 2012]. In addition, a majority of ETS events occur in the dilatational quadrants of the strain field on both sides of the plate interface [Kao et al., 2006]. Similarly, in our DEM model, microslips energy release are orders of magnitude smaller than large slips, and they occur during dilatant strengthening of the granular gouge layer. It is suggested that pore pressure can be a competing mechanism with dilation for further occurrence of either slow or fast slip in the course of ETS activities [Segall et al., 2010]. Development of a fluid coupled DEM model can provide insight on the pore pressure evolution during microslips.
 Our numerical observations reveal that microslips occurrence accelerates exponentially for events designated “large” based on their energy release. A similar pattern of nonlinear acceleration of precursory activity has been observed in experimental stick slip in sheared granular layers by Nasuno et al.  and in large interplate earthquakes worldwide by Bouchon et al. . The precursory activity becomes less important and significant for smaller event size. Statistical analysis of SER and slip events energy release further confirms the occurrence of precursory activity of microslips before large slip events. SER might be a better measure in this sense since we do not distinguish between different event size in forming its time series.
 The evolution of the SCR (Slipping Contact Ratio) during a characteristic stick phase implies that a background level of slipping contacts always exists in a dense sheared granular layer and plays a major role in slow rearrangement of particles and loss/formation of old/new contacts. Microslips start to appear only when the SCR rises above a certain minimum level. Further increase of the SCR above a certain level results in an increase of SER, which leads to the onset of a large slip event. The critical SCR value controls the lower bound of the isostatic coordination number for frictional packings; therefore, its increase forces the medium from a marginally (shear-) jammed to unjammed state [Shundyak et al., 2007; Song et al., 2008; Bi et al., 2011]. This also hints that the controlling parameters of minimum and critical slipping ratio values (grain-scale friction, particle packing, particle size distribution, shearing velocity, and confining pressure) are among the parameters which influence the duration and intensity of microslips occurrence. The analysis of the microslips, e.g., by investigating the development of affine and nonaffine deformations in the granular gouge layer during microslips [Griffa et al., 2011, 2012] may allow us to further characterize how the microslips signal the approaching of a large slip event.
 The distribution of large slip events in our model follows a power law with β≃1.23. The β value complies with observationally found values for earthquakes [Kanamori and Anderson, 1975; Kagan, 1991; Godano and Pingue, 2001]. The observed scaling is also in agreement with the avalanche experiments of microsized and nanosized crystals [Dimiduk et al., 2006; Friedman et al., 2012; Papanikolaou et al., 2012].
 We have presented the results of 3-D DEM modeling of a sheared granular gouge layer in the stick-slip regime. We show that there is precursory activity due to the occurrence of small slip events, called microslips, that precede the onset of large slip events. Microslips occurrence accelerates exponentially shortly before the onset of large slip events. The slip event rate is examined as a more rigorous measure that shows significant increase when approaching a large slip event. The increase is particularly accentuated before the large slip onset. The onset and duration of microslips emergence are controlled by the slipping contact ratio in the granular layer which connects the precursory microslips to the jamming-unjamming transition of the granular layer. The results of this study allow us to advance our understanding of the earthquake initiation on mature faults and to develop analysis methods to be used in seismology for improving probabilistic hazard assessment.
 We thank D. Weatherley and S. Abe for support for the ESyS-Particle code and D. Passerone and C. Pignedoli for the help with the computational cluster of Empa, Ipazia. Our work has been supported by the Swiss National Science Foundation (projects 206021-128754 and 200021-135492) and by the LDRD Program (Institutional Support) at the Los Alamos National Laboratory, Department of Energy, USA.
 The Editor thanks Karen Mair for her assistance in evaluating this paper.