TY - JOUR
AU - Ji,, Yuntao
AB - SUMMARY The effects of fault waviness on the fault slip modes are unclear. Laboratory study on the effects of the centimetre-scale fault contact distribution, which is mainly controlled by the fault waviness, on granodiorite stick-slip instabilities may help to unveil some aspects of the problem. The fast and slow stick-slip motions were separately generated in two granodiorite samples of the same roughness but different fault contact distributions in the centimetre scale in the laboratory. The experimental results show the following: (1) the fault with the small contact area and heterogeneous contact distribution generates fast stick-slip instabilities, while the fault with the large contact area and homogeneous contact distribution produces slow stick-slip events; (2) the nucleation processes of the fast stick-slip events are characterized by abrupt changes once the nucleation zones expand to the critical nucleation length that is observed to be shorter than the fault length, while the slow stick-slip events appear as a gradual evolution of the nucleation zones leading to total fault sliding. These indicate that, unlike the micron-scale fault contact distribution controlled by roughness, which depends mainly on the grain size of the abrasives used for lapping the fault surface, the centimetre-scale fault contact distribution, which depends mainly on the waviness of the fault surface profile, also plays an important role in the fault slip modes. In addition, the effects of the fault waviness on the fault friction properties are preliminarily analysed based on the rate- and state-dependent friction law. Image processing, Instability analysis, Earthquake dynamics, Rheology and friction of fault zones, Dynamics and mechanics of faulting, Mechanics, theory, and modelling 1 INTRODUCTION Earthquakes usually appear as sudden and rapid fault sliding events, which are characterized by a rapid release of energy in a short time. However, some sliding events are found to develop slowly, with durations lasting up to several months; these are referred to as slow slip events (Dragert et al. 2001; Rogers & Dragert 2003; Bürgmann 2018). The seismic moment of slow events is thought to be proportional to the characteristic duration, and the moment rate function of slow events is thought to be constant, with a spectral high-frequency decay of f−1 (Ide et al. 2007). Furthermore, the duration-energy scale relationship of different slip events around the world indicates a continuous transition between regular earthquakes and slow slip events (Peng & Gomberg 2010). However, regular earthquakes are quite different from slow slip events in terms of both their occurrence and preparation. A slow process known as nucleation is found to precede the occurrence of regular earthquakes (Ellsworth & Beroza 1995, 1998; Kato et al. 2012; Kato & Nakagawa 2014; Ellsworth & Bulut 2018); however, this process is rarely reported before the occurrence of slow slip events, which is probably due to the low signal-to-noise ratio of the slow slip event observations. Thus, an experimental study and comparison of the nucleation processes of regular earthquakes and slow slip events may further the understanding of the factors controlling their differences. In laboratory tests, the earthquake mechanism is usually simulated by a stick-slip motion (Brace & Byerlee 1966). Specifically, fast and slow stick-slip motions are the mechanisms of regular earthquakes and slow slip events, respectively. Experiments were carried out to study the factors controlling the fault slip modes and the underlying physics. For instance, the increasing roughness of a fault surface may lead to a slower stick-slip event under lower normal stress in granite, gabbro or poly methyl methacrylate samples (Okubo & Dieterich 1984; Marone & Cox 1994; Ohnaka & Shen 1999; Harbord et al. 2017; Zhou et al. 2018). Experiments conducted on acrylic poly blocks showed that the rupture propagation speed increases with an increasing local friction coefficient (Ben-David et al. 2010). The slip changes from creeping to slow and fast instability in carbonate samples as the confining pressure and bulk temperature increase (Passelègue et al. 2019). The fault gouges of powdered granular quartz or powdered silica transition from slow slip to dynamic ruptures as the normal stress increases (Leeman et al. 2016; Tinti et al. 2016). The stick-slip events in granite or polycarbonate samples become faster as the loading rate increases (McLaskey & Yamashita 2017; Guérin-Marthe et al. 2019). Long-term sliding of NaCl crystal leads to the transition from fast to slow stick-slip events (Voisin et al. 2007, 2008). Models have been proposed to explain how these factors control the fault slip modes. One of the successful models proposed involved the sliding stability analysis of the spring-slider system based on the rate- and state-dependent friction law, which revealed that the fault sliding stability depends on the comparison between the system stiffness (k) and the critical stiffness (kc) (Dieterich 1981; Ruina 1983). Namely, the stick-slip motion occurs when k is smaller than kc. Furthermore, the stick-slip motion of the fault changes from fast to slow as k/kc increases from being much less than 1 to close to 1 (Tinti et al. 2016). k can be directly measured during the experiment. For the case of constant normal stress, kc can be expressed by eq. (1) (Ruina 1983), as follows: $$\begin{eqnarray*} {k_{\rm c}} = \frac{{{\sigma _n}\left( {b - a} \right)}}{{{D_{\rm c}}}}, \end{eqnarray*}$$ (1) where a and b are the constitutive parameters, σn is the normal stress and Dc is the characteristic sliding distance. For the case of a variable normal stress condition, the sliding stability of an inclined spring-slider system was studied (Dieterich & Linker 1992; He et al. 1998), where kc can be expressed by a more complicated form, but the main parameters are still a, b and Dc, and their relationship is still the same as shown in eq. (1). Thus, different stick-slip behaviours can be produced via modulating the parameters shown on the right side of eq. (1) (e.g. Leeman et al. 2016; Tinti et al. 2016). The value of (b–a) is related to the fault slip velocity, normal stress, temperature and gouge composition, etc. (Kilgore et al. 1993; He et al. 2007; Voisin et al. 2007; He et al. 2013; Kaproth & Marone 2013). It was proposed that the larger the average size of the fault contact zones is, the larger the Dc is (Dieterich & Kilgore 1994; Voisin et al. 2007) or the greater the thickness of the localized shear zone of the gouge is, the larger the Dc is (Marone & Kilgore 1993). In addition, (b–a) and Dc also depend on the shear strain and fabric of the fault gouges (Scuderi et al. 2017). In laboratory studies, the nucleation process was observed prior to slick-slip events (Dieterich 1978; Ohnaka & Kuwahara 1990; Ohnaka 1992; Ohnaka & Shen 1999; Ma et al. 2002, 2003; Nielsen et al. 2010; Latour et al. 2013; McLaskey & Kilgore 2013; Zhuo et al. 2018b; Passelègue et al. 2019). The experimental results revealed that the shear failure nucleation process of a fault is composed of the following two phases: a slow (quasi-static) phase followed by an accelerated propagation of the nucleation zone, the transition between which is the nucleation zone expanding to a critical length (Lc) (Ohnaka & Kuwahara 1990; Ohnaka 1992; Ohnaka & Shen 1999; Nielsen et al. 2010; Latour et al. 2013; Guérin-Marthe et al. 2019). According to these studies, the expansion speed of the nucleation zone increases abruptly when the nucleation length reaches Lc. The relationship between Lc and kc was obtained via theoretical analysis and can be expressed by eq. (2) (Dieterich 1992), as follows: $$\begin{eqnarray*} {k_{\rm c}} = \frac{{2G}}{{{L_{\rm c}}}}, \end{eqnarray*}$$ (2) where G is the shear modulus. For faults that do not contain gouges, one of the key factors affecting the friction properties of the faults is the fault contact distribution, which is mainly controlled by the topography of the fault surfaces. Roughness and waviness are two main parameters that describe the topography of a macroscopic planar fault surface. The roughness determines the small scale (e.g. the micron-scale) fault contact distribution (Dieterich & Kilgore 1994; Dieterich & Kilgore 1996), while the waviness controls a larger scale (e.g. centimetre-scale) fault contact distribution. Previous studies have pointed out that Dc increases with increasing fault roughness (Dieterich 1979; Ohnaka 1996; Ohnaka & Shen 1999), which depends mainly on the grain size of the abrasives used for lapping the fault surface (Ohnaka 1996; Ohnaka & Shen 1999). As a result, the fracture energy, which is a function of Dc, also relates to the fault roughness. Furthermore, it was shown that the fracture energy increases with increasing normal stress (Bayart et al. 2016; Passelègue et al. 2017), which tends to enlarge the fault contact areas (Dieterich & Kilgore 1994). However, little is known about the effects of the fault waviness on the fault friction properties. One of the main factors affecting the waviness is the grinding process used on the fault surface. Because it is difficult to maintain the consistency of the sample grinding process, the waviness may differ for different samples of the same roughness. Such a phenomenon will be more obvious when the samples are relatively large, because the grinding process of large samples is more complicated. The centimetre-scale fault contact distribution is related to the fault roughness, but it is mainly controlled by the fault waviness (Fig. 1). Studying the influences of the centimetre-scale fault contact distribution on stick-slip instability can help to explore the roles of the fault waviness plays in the fault friction properties. Therefore, we used two granodiorite faults, which have the same roughness but different centimetre-scale contact distribution, to compare their stick-slip instability characteristics and nucleation processes. Figure 1. Open in new tabDownload slide The centimetre-scale fault contact distributions of the samples. The fault contact distributions of Experiment EX1 and EX2 are shown in (a)–(c) and (e)–(g), respectively. Panels (a) and (e) show the contact area changes along the fault parallel to the slip direction. (b) and (f) show the distributions of the contact zones in white on the fault surfaces. Panels (c) and (g) show the contact pressure (CP) distributions on the fault surfaces. Panels (d) and (h) schematically show the fault contact, roughness and waviness of the fault surfaces in Experiments EX1 and EX2, respectively. Figure 1. Open in new tabDownload slide The centimetre-scale fault contact distributions of the samples. The fault contact distributions of Experiment EX1 and EX2 are shown in (a)–(c) and (e)–(g), respectively. Panels (a) and (e) show the contact area changes along the fault parallel to the slip direction. (b) and (f) show the distributions of the contact zones in white on the fault surfaces. Panels (c) and (g) show the contact pressure (CP) distributions on the fault surfaces. Panels (d) and (h) schematically show the fault contact, roughness and waviness of the fault surfaces in Experiments EX1 and EX2, respectively. The nucleation processes of the stick-slip instability have been densely studied in the laboratory setting (e.g. Dieterich 1978; Ohnaka & Kuwahara 1990; Ohnaka 1992; Ohnaka & Shen 1999; Ma et al. 2002, 2003; Nielsen et al. 2010; Latour et al. 2013; McLaskey & Kilgore 2013; McLaskey & Lockner 2014; Zhuo et al. 2018b; Acosta et al. 2019; Guérin-Marthe et al. 2019; Passelègue et al. 2019). In addition to the use of strain gauges and displacement metres, the optical interference method was also used in the transparent samples to observe the spatiotemporal evolution of the nucleation process in some of these studies. Although the point contact type observation methods (e.g. strain gauges, displacement metres) used in the previous studies have high precision, it is difficult to observe the accurate nucleation length with these methods due to the limited number of sensors. Thus, a high-speed camera and a digital image correlation method were used in our study to observe the detailed spatiotemporal evolution of the nucleation process in opaque granodiorite samples. 2 EXPERIMENTS AND METHODS 2.1 Samples Two granodiorite samples of the same size with areas of 300 mm by 300 mm and thicknesses of 50 mm (Figs 1 and 2) were used in the experiments. The Young's modulus, Poisson's ratio and shear modulus values of the samples are 60 GPa, 0.27 and 23.6 GPa, respectively. The samples were cut diagonally to form macroscopic planar faults. The fault surfaces of both samples were ground by a 150-mesh (with abrasive grain size of ∼110 μm) RVD (Resinoid and Vitrified bonded Diamond grains) abrasive before the experiments were conducted. According to the method used to describe the fault roughness in Ohnaka & Shen (1999), the topographical lengths of a 20-mm-long local profile along each fault were measured by using different ruler lengths. The topographical length of the local profile showed segmented power functions of the ruler length in the two experiments. The corner ruler lengths reflecting the abrasive grain size of the wheel are ∼130 and ∼110 μm in Experiments EX1 and EX2, respectively. This indicates that the roughness degrees of the two fault surfaces before the experiments were equivalent when considering statistical error. However, the centimetre-scale fault contact distributions of the two samples are different (Fig. 1). The total fault contact area is small and the fault contact distribution is heterogeneous in Experiment EX1, which is due to the higher relative translational motion speed between the grinding wheel and the fault surface during the grinding process. On the other hand, the fault surface has a larger total contact area and a more homogeneous contact distribution in Experiment EX2 under the slower relative translational motion speed between the grinding wheel and the sample during the grinding process. The differences in the centimetre-scale fault contact distributions between the two experiments can be measured by pressure sensitive films (Fujifilm, Japan, with a measurement range from 2.5 to 10 MPa). The pressure sensitive films were placed between the two slider blocks to measure the distribution of the fault contact pressure (CP) under the condition of σ1= σ2 = 5 MPa before the experiments (Figs 1c and g). The CP measure is derived from comparing the pressure-induced colour on the pressure sensitive films with standard colour samples. Although the standard measurement range is from 2.5 to 10 MPa, the actual measurement results show a wider range due to weak contact or stress concentration due to strong contact. The distribution and sizes of the contact zones on the fault surfaces can be estimated from the distribution of CP (Selvadurai & Glaser 2015). According to the iterative method described in Selvadurai & Glaser (2015), the thresholds of 5.3 and 6.5 MPa are obtained for Experiments EX1 and EX2, respectively. Namely, only the zones with CP values equal to or greater than the thresholds can be treated as contact zones (denoted by the white zones in Figs 1b and f). By doing this, the total sizes of the contact zones are 72.65 and 93.17 cm2 in Experiments EX1 and EX2, respectively. In addition, the distribution of the centimetre-scale contact zone is more homogeneous in Experiment EX2 than that in Experiment EX1, which indicates that the waviness of the fault surface profile is flatter in Experiment EX2 than that in Experiment EX1 (Figs 1d and h). Accordingly, fast and slow stick-slip events were generated in Experiments EX1 and EX2, respectively. Figure 2. Open in new tabDownload slide The loading and observation systems. (a) Schematic diagram of the horizontal biaxial hydraulic servo control loading apparatus at the State Key Laboratory of Earthquake Dynamics, Institute of Geology, China Earthquake Administration. 1-Hydraulic actuator, 2-Piston, 3-Ball seat, 4-linear variable differential transformer (LVDT), 5-Steel pad, 6-Load cell, 7-Steel roller, 8-Steel framework, 9-Base, 10-Sample. (b) Schematic diagram of the loading manner and observation system. The maximum principal stress (σ1) and the minimum principal stress (σ2) are parallel to the Y- and X-axes, respectively. The fault and D-axis overlap and the centre of the fault is the common origin of axes X, Y and D. The high-speed camera is mounted over the upper sample surface. The red lines denoted by l1 and l2 are each composed of thousands of pixels, which form an array of digital fault slip sensors to measure the fault slip via the DIC method. Figure 2. Open in new tabDownload slide The loading and observation systems. (a) Schematic diagram of the horizontal biaxial hydraulic servo control loading apparatus at the State Key Laboratory of Earthquake Dynamics, Institute of Geology, China Earthquake Administration. 1-Hydraulic actuator, 2-Piston, 3-Ball seat, 4-linear variable differential transformer (LVDT), 5-Steel pad, 6-Load cell, 7-Steel roller, 8-Steel framework, 9-Base, 10-Sample. (b) Schematic diagram of the loading manner and observation system. The maximum principal stress (σ1) and the minimum principal stress (σ2) are parallel to the Y- and X-axes, respectively. The fault and D-axis overlap and the centre of the fault is the common origin of axes X, Y and D. The high-speed camera is mounted over the upper sample surface. The red lines denoted by l1 and l2 are each composed of thousands of pixels, which form an array of digital fault slip sensors to measure the fault slip via the DIC method. 2.2 Loading processes During the experiments, the samples were placed in a horizontal biaxial hydraulic servo control loading apparatus, as shown in Fig. 2 (a) (Ren et al. 2017; Chen et al. 2018; Zhuo et al. 2018b). The loads along the X- and Y-directions were synchronously increased from 0 to 5 MPa at a rate of 0.13 MPa s–1; then, the load along the X-direction was maintained at 5 MPa, while the Y-direction was shifted to the displacement rate control mode at a rate of 0.5 μm s–1 until regular stick-slip events occur. Then, the displacement rate along the Y-direction was shifted to 1 μm s–1 and successively followed by 0.1 μm s–1 in Experiment EX1, while the displacement rate along the Y-direction was maintained at 0.1 μm s–1 in Experiment EX2 because the 1 μm s–1 loading process was unintentionally omitted. The average friction coefficient (μ) and average normal stress (σn) obtained by the load cells of the loading system and the displacement along the Y-direction of piston (dy) with time are shown in Fig. 3. Each drop of μ corresponds to a fault stick-slip event, and the seven stick-slip events denoted by E1–E7 were captured by a high-speed camera for analysis. Figure 3. Open in new tabDownload slide Loading processes of the two experiments. (a) Evolution of the average friction coefficient (μ) and displacement of the piston along the Y-axis (dy) with time during the two experiments. (b) shows the detailed evolution of μ and dy, which is a magnified view of the black dashed rectangular area in (a). E1 through E7 denote the seven stick-slip events observed by the high-speed camera. (c) Evolution of the average normal stress (σn) with time during the two experiments. Figure 3. Open in new tabDownload slide Loading processes of the two experiments. (a) Evolution of the average friction coefficient (μ) and displacement of the piston along the Y-axis (dy) with time during the two experiments. (b) shows the detailed evolution of μ and dy, which is a magnified view of the black dashed rectangular area in (a). E1 through E7 denote the seven stick-slip events observed by the high-speed camera. (c) Evolution of the average normal stress (σn) with time during the two experiments. Fig. 3 shows the loading curves of the two experiments. The stick-slip events in Experiment EX1 occur at lower friction strengths than those in Experiment EX2. In addition, the stress drop, the stress drop rate and the stick-slip event interval values are large in Experiment EX1 compared with those in Experiment EX2 under the same loading rates. 2.3 The digital image correlation (DIC) method A high-speed camera (Photron Fastcam SA2, Japan) was used to record images of the upper sample surface during the experiments (Fig. 2b). Three and four stick-slip events in Experiments EX1 and EX2, respectively, were recorded. The recording duration of each event was 5.455 and 5.822 s in Experiments EX1 and EX2, respectively. The sampling rate was 1000 frames per second in the two experiments. The resolution of each image of Experiment EX1 is 2048 × 2048 pixels, which covers the whole upper sample surface. The actual size of each pixel corresponding to the surface of the sample is 150 × 150 μm2. In Experiment EX2, the high-speed camera was rotated to make the array of pixels parallel to the fault to simplify the fault slip calculation. Furthermore, the distance between the high-speed camera and the upper sample surface in Experiment EX2 was appropriately lengthened to ensure that the spacing between the adjacent digital fault slip sensors (they will be described below) are the same as that in Experiment EX1. Accordingly, the resolution of each image of Experiment EX2 is set to 2048 × 200 pixels, which is a strip-shaped area covering the entire fault. The actual size of each pixel corresponding to the sample surface is 197 × 197 μm2 in Experiment EX2. Since it will take approximately half an hour to export the data from the camera buffer to the computer after each recording, during which time the camera cannot record new images, it is impossible to record all the stick-slip events with the high-speed camera. As a result, only seven stick-slip events (Events E1–E7) were recorded, of which Events E1–E3 belong to Experiment EX1, while Events E4–E7 are from Experiment EX2. The DIC method, which is an object recognition method based on pattern matching via a correlation algorithm in computer graphics (Yamaguchi 1981; Peters & Ranson 1982; Sutton et al. 1983; Zhuo et al. 2013; Ji et al. 2015; Rubino et al. 2015), was used to obtain the fault slip in our study. The chosen region of interest (ROI) was a rectangle-shaped zone covering the entire faults in the two experiments, which was used to calculate the displacement field. Based on the principle of selecting the minimum side length (R) of the square subregion for pattern matching under the condition that the correlation coefficient (CC) corresponding to the subregion size was sufficiently high via comprehensive consideration of the mean and standard deviation values of CC (Zhuo et al. 2015, 2018a, 2019), the subregion size of 39 by 39 pixels was determined in the two experiments. The subregion was moved pixel by pixel in the calculated image when the calculation was performed. Calculations were performed for all of the images with respect to the first image in each stick-slip event and, accordingly, the cumulative displacement field of the ROI was obtained. The threshold for determining whether a calculation result was reliable depended on the CC (Zhuo et al. 2019). Namely, the calculated displacement at a location was reliable when the CC at the location is larger than the value that is twice the standard deviation lower than the average value of the CC at R = 39. Two lines on both sides of the fault (the red symmetric lines on both sides of the fault, denoted by l1 and l2 in Fig. 2b) were selected to calculate the fault slip. Each line was offset a certain distance from the fault, which just kept the fault from intersecting the subregion, with the centre point located at the lines, and ensured the accuracy of the fault slip measurement (Zhuo et al. 2015, 2018a, 2019). Each line contains 1770 pixels and is offset by approximately 4.2 mm from the fault in Experiment EX1. In Experiment EX2, the two lines, each of which contains 1920 pixels, are offset by approximately 4.0 mm from the fault. Every two symmetrical pixels from l1 and l2 form a digital fault slip sensor. The spacing between the digital fault slip sensors in the two experiments is approximately 0.2 mm due to the pixel by pixel moving step of the subregion during calculation, which helps to determine the exact length of the nucleation zone. The noise level of the displacement measurement of the DIC method was ±5 µm, which was obtained under the condition that the sample was static without loading. Segmented smoothing of the time series of the fault slip was performed to improve the measurement precision. Then, smoothing was performed in the spatial series of the fault slip to suppress noise. By performing these steps, the noise can be reduced to a level less than ±1 µm. 3 RESULTS 3.1 Average fault slip and slip velocity of the fast and slow stick-slip events Fig. 4 shows the evolution of the average fault slip (⁠|$\bar{d}$|⁠) (over all the digital fault slip sensors) and the average slip velocity (⁠|$\bar{v}$|⁠) (over all the digital fault slip sensors) with time for all the seven stick-slip events. |$\bar{d}$| and |$\bar{v}$| increase abruptly in Events E1–E3 but change gradually in Events E4–E7. Therefore, the zero time definitions in the diagrams of the two types of events are different. The zero time represents the onset of the sudden rising of |$\bar{d}$| and |$\bar{v}$| for Events E1–E3 but represents the peak |$\bar{v}$| for Events E4–E7. Figure 4. Open in new tabDownload slide The evolution of the average fault slip (⁠|$\bar{d}$|⁠) and the average fault slip velocity (⁠|$\bar{v}$|⁠) with time (t) during Events E1–E7, as shown in Fig. 3. Panels (a) and (b) show the evolution of |$\bar{d}$| and |$\bar{v}$| with time, respectively. Panels (c) and (d) are the magnified views of the black dashed rectangular areas in (b). Figure 4. Open in new tabDownload slide The evolution of the average fault slip (⁠|$\bar{d}$|⁠) and the average fault slip velocity (⁠|$\bar{v}$|⁠) with time (t) during Events E1–E7, as shown in Fig. 3. Panels (a) and (b) show the evolution of |$\bar{d}$| and |$\bar{v}$| with time, respectively. Panels (c) and (d) are the magnified views of the black dashed rectangular areas in (b). The values of |$\bar{d}$| and |$\bar{v}$| during the pre-slip prior to the instability increase with the increasing loading rate. In addition, the peak |$\bar{v}$| (⁠|${\bar{v}_p}$|⁠) is affected by the cumulative fault slip under the same loading rate. The larger the cumulative fault slip is, the larger |${\bar{v}_p}$| is. The |${\bar{v}_p}$| values of Events E1–E3 are approximately three orders of magnitude higher than those of Events E4–E7. Therefore, we refer to Events E1–E3 as fast stick-slip events and Events E4–E7 as slow stick-slip events. Next, we analyse the differences in the nucleation processes of the two types of events via the spatiotemporal distribution of the fault slip (d) of each event. 3.2 The nucleation processes of the fast stick-slip events The slow nucleation processes of Events E1–E3 are shown in Fig. 5. Since |$\bar{d}$| and |$\bar{v}$| abruptly increase after the onset of the instability, the distribution of d after the onset of the abrupt change of the fault slip is not added to Fig. 5 to clearly show the slow nucleation process. Event E1 shows a single nucleation zone, which initiates near D = 0 mm, while coalescing double nucleation zones can be identified in Events E2 and E3. Figure 5. Open in new tabDownload slide The spatiotemporal evolution of the fault slip (d) during the slow nucleation processes of Events E1–E3. Zero time denotes the onset of the abrupt changes in |$\bar{d}$| and |$\bar{v}$|⁠. The red dashed arrow and its corresponding number represent a scale for the expansion direction and expansion speed (vr) of the nucleation zone; the red double arrows denoted by Lc represent the critical nucleation length. The noise levels are ±0.5, ±0.4 and ±0.3 μm in events E1–E3, respectively, after smoothing is performed. A slip within these levels is illustrated in white in these diagrams. Figure 5. Open in new tabDownload slide The spatiotemporal evolution of the fault slip (d) during the slow nucleation processes of Events E1–E3. Zero time denotes the onset of the abrupt changes in |$\bar{d}$| and |$\bar{v}$|⁠. The red dashed arrow and its corresponding number represent a scale for the expansion direction and expansion speed (vr) of the nucleation zone; the red double arrows denoted by Lc represent the critical nucleation length. The noise levels are ±0.5, ±0.4 and ±0.3 μm in events E1–E3, respectively, after smoothing is performed. A slip within these levels is illustrated in white in these diagrams. The high-speed camera used in this study can observe the expansion speed of the nucleation zone at least on the order of 10 m s–1 (Zhuo et al. 2018b) under the same conditions. The expansion speeds of the nucleation zones prior to the instability of Events E1–E3 in Experiment EX1 are on the order of 100 mm s–1 (Fig. 5), which are values far below the observation capability of the camera, indicating that the nucleation process was completely observed until the last 1 ms prior to the occurrence of instability. On the other hand, the nucleation zones accelerate sharply once the nucleation length reaches a critical value (Lc) less than the fault length (39.4 cm) within the last 1 ms before the instability occurs. Although the nucleation process within the last 1 ms prior to the instability occurrence was not recorded due to the sampling rate limit of the camera, this abrupt increase in the expansion of the nucleation length can be estimated as follows. Before the nucleation length reaches Lc, the nucleation zones of Events E1–E3 expand at an accelerated speed from the static state up to the order of 100 mm s–1 in durations of 1–2 s, namely, the expansion acceleration is only on the order of 0.1 m s–2. After the nucleation length reaches Lc, the length of the nucleation zone expands from Lc to the entire fault length during the last 1 ms (actually less than 1 ms) before the instability occurs, which indicates that the expansion of the nucleation zone accelerates from the order of 100 mm s–1 to at least several tens of m s–1 in Events E1–E3. Namely, the acceleration of the nucleation zone expansion is at least on the order of 10 km s–2. Thus, for Events E1–E3, the expansion speed and acceleration of the nucleation zones increase by at least two and five orders of magnitude, respectively, before and after the length of the nucleation zone reaches Lc. The abrupt increase in the expansion of the nucleation zones as the nucleation length reaches Lc indicates that Lc divides the nucleation process into a slow phase, followed by an accelerated phase. This indicates that Lc is the critical nucleation length, as defined by the previous studies (Ohnaka & Kuwahara 1990; Ohnaka 1992; Ohnaka & Shen 1999; Nielsen et al. 2010; Latour et al. 2013). 3.3 The nucleation processes of the slow stick-slip events Compared with the fast stick-slip events in Experiment EX1, the four slow stick-slip events observed by the high-speed camera in Experiment EX2 show different features in their nucleation processes (Fig. 6). The slips nucleate at multiple locations in Events E4–E7 and coalesce to form larger nucleation zones quickly, which causes the slip of the overall fault to occur in a short time after the nucleation initiates. There is not an abrupt change in the expansion speed of the nucleation zone before and after it expands to the entire fault. Accordingly, the nucleation length is gradually increasing up to the total fault length without an observable critical nucleation length. The measured parameters of the two experiments are shown in Table 1. Figure 6. Open in new tabDownload slide The spatiotemporal evolution of fault slip (d) during the nucleation and occurrence of Events E4–E7. Zero time denotes the peak average slip velocity. The red dashed arrow and its corresponding number represent the scale for the expansion direction and expansion speed (vr) of the nucleation zone. Please note that logarithmic values of d are used in the figures to clearly show the nucleation processes. The noise levels are ±1 μm after smoothing is performed in all of the four events. A slip within this level is set to 1 μm (the corresponding logarithmic value is zero), which is illustrated in white in the diagrams. Figure 6. Open in new tabDownload slide The spatiotemporal evolution of fault slip (d) during the nucleation and occurrence of Events E4–E7. Zero time denotes the peak average slip velocity. The red dashed arrow and its corresponding number represent the scale for the expansion direction and expansion speed (vr) of the nucleation zone. Please note that logarithmic values of d are used in the figures to clearly show the nucleation processes. The noise levels are ±1 μm after smoothing is performed in all of the four events. A slip within this level is set to 1 μm (the corresponding logarithmic value is zero), which is illustrated in white in the diagrams. Table 1. The measured parameters of the experiments. Event no. . Loading rate (μm s–1) . Lc (cm) . K* (MPa mm−1) . |${\bar{v}_p}$| (mm s–1) . E1 1.0 36.2 18.03 24.3 E2 0.1 30.4 18.03 37.4 E3 0.1 32.9 18.50 46.0 E4 0.5 — 13.25 0.00771 E5 0.1 — 17.63 0.0533 E6 0.1 — 18.46 0.0582 E7 0.1 — 18.64 0.0584 Event no. . Loading rate (μm s–1) . Lc (cm) . K* (MPa mm−1) . |${\bar{v}_p}$| (mm s–1) . E1 1.0 36.2 18.03 24.3 E2 0.1 30.4 18.03 37.4 E3 0.1 32.9 18.50 46.0 E4 0.5 — 13.25 0.00771 E5 0.1 — 17.63 0.0533 E6 0.1 — 18.46 0.0582 E7 0.1 — 18.64 0.0584 *k is derived from the linear fit of the linear segment of the average shear stress versus the dlp (fault displacement measured from the movement of the loading pistons) curves in each stick-slip cycle. Open in new tab Table 1. The measured parameters of the experiments. Event no. . Loading rate (μm s–1) . Lc (cm) . K* (MPa mm−1) . |${\bar{v}_p}$| (mm s–1) . E1 1.0 36.2 18.03 24.3 E2 0.1 30.4 18.03 37.4 E3 0.1 32.9 18.50 46.0 E4 0.5 — 13.25 0.00771 E5 0.1 — 17.63 0.0533 E6 0.1 — 18.46 0.0582 E7 0.1 — 18.64 0.0584 Event no. . Loading rate (μm s–1) . Lc (cm) . K* (MPa mm−1) . |${\bar{v}_p}$| (mm s–1) . E1 1.0 36.2 18.03 24.3 E2 0.1 30.4 18.03 37.4 E3 0.1 32.9 18.50 46.0 E4 0.5 — 13.25 0.00771 E5 0.1 — 17.63 0.0533 E6 0.1 — 18.46 0.0582 E7 0.1 — 18.64 0.0584 *k is derived from the linear fit of the linear segment of the average shear stress versus the dlp (fault displacement measured from the movement of the loading pistons) curves in each stick-slip cycle. Open in new tab 4 DISCUSSION 4.1 Effects of the centimetre-scale fault contact distribution on the nucleation process The previous studies focused on the effects of loading rates on the nucleation process (McLaskey & Yamashita 2017; Guérin-Marthe et al. 2019), the evolution of pre-slip and expansion of the nucleation zone (e.g. Latour et al. 2013), microfracture events (acoustic emissions) generated during the nucleation process (McLaskey & Lockner 2014; Passelègue et al. 2017), types of nucleation (McLaskey & Lockner 2014; McLaskey 2019) and so on. Furthermore, the role of the micron-scale fault contact derived from the roughness in the nucleation process has also been studied (Harbord et al. 2017; Selvadurai et al. 2017). However, the effects of the centimetre-scale fault contact distribution, which is mainly controlled by fault waviness, on the nucleation process, has not been reported before. The differences in the nucleation processes between the slow and fast stick-slip events in the two experiments are as follows. (1) Before the coalescence of the nucleation zones, there are one or two nucleation zones, which initiate at the locations concentrating between D = 0 mm and D = 100 mm, in the fast stick-slip events (Fig. 5). However, there are a greater number of nucleation zones, which have scattered distributions along the fault, in the slow stick-slip events before they coalesce to expand to the whole fault (Fig. 6). This difference in the spatial distributions of nucleation processes of the stick-slip events is correlated with the difference in the centimetre-scale fault contact distributions of the faults. (2) The durations of the nucleation processes are longer in the fast stick-slip events than those in the slow stick-slip events. Moreover, the pre-slip amount and velocity are smaller in the fast stick-slip events than those in the slow stick-slip events (Fig. 3). These findings indicate that the duration of the nucleation process increases with the increasing heterogeneity of the fault contact distribution, which is very similar to previous numerical simulation studies that found that the duration of the nucleation process becomes longer as the fault heterogeneity increases (Ionescu & Campillo 1999; Campillo et al. 2001). Therefore, the increase in the fault heterogeneity may lead to a slower initial weakening rate and may weaken the interactions between the nucleation zones, which delays the overall instability of the fault, as proposed by the numerical simulation studies (Ionescu & Campillo 1999; Campillo et al. 2001). On the other hand, the increasing duration of the nucleation process may provide time for stress to concentrate on the tips of the nucleation zones, which prepares for the fast instability of the entire fault. 4.2 Implications for the origin of slow slip events The argument of k/kc that is used to represent the stick-slip behaviours is based on the single-degree-of-freedom spring-slider model (Ruina 1983). The complexity of the faults in our experiments is much higher than that of the single spring-slider model. However, based on the assumptions that there is a single nucleation zone in the elastic medium and that the properties and conditions in the centre of the nucleation zone can represent the entire nucleation zone, it is proposed that the stiffness of the nucleation zone at its maximum length (the critical nucleation length) before its accelerated expansion is equal to kc (Dieterich 1986, 1992). Therefore, the k/kc argument can be used to approximately explain the stick-slip behaviour of the fault with a single nucleation zone. Figs 5 and 6, as well as previous studies (Fukuyama et al. 2018; Zhuo et al. 2018a), show that a bigger nucleation zone can be formed from the coalescence among the double or multiple small nucleation zones in the nucleation process. Therefore, the k/kc argument may be used to approximately explain the occurrences of the fast and slow stick-slip events in experiments EX1 and EX2, respectively (although the actual mechanism is probably more complicated). The direct observation of Lc in Experiment EX1 can help us to explore the origin of the slow stick-slip events that occurred in Experiment EX2. Although the values of k in Experiments EX1 and EX2 are approximately equivalent to 18.5 MPa mm−1 under the loading rate of 0.1 μm s–1 (Table 1), k/kc is larger in Experiment EX2 than in Experiment EX1 since the stick-slip events in Experiment EX2 are obviously slower than those in Experiment EX1. As a result, kc is smaller in Experiment EX2 than in Experiment EX1, which indicates that Lc is larger in Experiment EX2 than in Experiment EX1 according to eq. (2). As the Lc in Experiment EX1 is nearly equal to the fault length, it is reasonable to propose that the Lc in Experiment EX2 is larger than the fault length, which is why we cannot observe Lc in the four slow stick-slip events (Fig. 6). This is similar to the condition that slow events occur when instability cannot fully nucleate before reaching the sample ends, as proposed in McLaskey & Yamashita (2017). Since the initial conditions of the samples are identical except for the centimetre-scale fault contact distribution, the different stick-slip behaviours between the two experiments may be caused by the centimetre-scale fault contact distribution. In this sense, slow slip events may tend to occur in homogeneous fault contact scenarios under the condition that Lc is larger than the fault length. 4.3 Effects of the roughness and waviness on the stick-slip behaviour The roughness of a fault depends mainly on the grain size of the abrasives used to lap the fault surface (Dieterich 1979; Ohnaka 1996; Ohnaka & Shen 1999). The topography of a fault surface usually exhibits band-limited self-similarity, where the topographical length of the fault surface profile can be expressed as the power function of the ruler length in certain ranges of the ruler length (Ohnaka & Shen 1999). One of the corner ruler lengths (⁠|${\lambda _{\rm c}}$|⁠) is approximately equal to the grain size of the abrasives used to lap the fault surface (Ohnaka & Shen 1999); as a result, |${\lambda _c}$| is a measurement of the fault roughness and increases with the increasing roughness of the fault surface. The previous studies have shown that as the roughness of the fault increases, the stick-slip speed under lower normal stress decreases (Okubo & Dieterich 1984; Marone & Cox 1994; Ohnaka & Shen 1999; Harbord et al. 2017; Zhou et al. 2018). Dieterich (1979) proposed that the Dc increases with increasing roughness based on the rate- and state-dependent friction constitutive law. On the other hand, it was also revealed that Dc (the critical slip displacement) is proportional to the roughness represented by |${\lambda _{\rm c}}$| (Ohnaka 1996; Ohnaka & Shen 1999) based on the slip weakening of the shear rupture model. The terms of Dc that they used differ in their names but have similar meaning (Marone 1998). This indicates that faults with the same roughness will have the same Dc values. Thus, Dc is the same in both of our experiments. This suggests that the main differences in the stick-slip behaviours between Experiments EX1 and EX2 may be caused by the differences in the fault waviness of the two samples. Let η = b–a (the meanings of a and b here as well as Dc, kc and σn below are the same as those shown in eq. 1) and let subscript letters 1 and 2 represent Experiments EX1 and EX2, respectively. From the discussion in the last paragraph, we can obtain the following: $$\begin{eqnarray*} {D_{c1}} = {D_{c2}} \end{eqnarray*}$$ (3) According to Section 4.2, kc1 is larger than kc2. Then, together with eqs (1) and (3), we can obtain the following: $$\begin{eqnarray*} {\eta _1}{\sigma _{n1}} > {\eta _2}{\sigma _{n2}}. \end{eqnarray*}$$ (4) Since σn1 is smaller than σn2, as shown in Fig. 3(c), we can obtain the following: $$\begin{eqnarray*} {\eta _1} > {\eta _2}. \end{eqnarray*}$$ (5) That is, η may increase with the increasing heterogeneity of the centimetre-scale fault contact distribution. Therefore, we propose that the waviness probably affects the stick-slip behaviours of a fault via controlling the value of b–a. The heterogeneous fault contact distribution of Experiment EX1 was prefabricated in our experiment. In fact, the loading system may also cause the heterogeneous fault contact distribution by inducing the stress concentration in the fault ends (Langer et al. 2013). Such a loading system induced heterogeneous fault contact distribution usually controls the initial nucleation locations, as well as dynamic instability locations (Dieterich 1978; McLaskey et al. 2015; Zhuo et al. 2018b). However, the factors controlling the fault contact distribution, as well as the influences of the fault contact distribution on the stick-slip behaviours, require further study. 5 CONCLUSIONS Two granodiorite sample faults, which have the same roughness but vary greatly in terms of their centimetre-scale contact distributions, show different stick-slip behaviours in the simple shear experiments loaded by a biaxial apparatus. The fault with the small contact area and heterogeneous contact distribution in the centimetre scale exhibits fast stick-slip motions with sudden increases in the fault slip velocity and expansion of the nucleation zones, while the fault with the large contact area and homogeneous contact distribution in the centimetre scale shows slow stick-slip motions with gradual changes in the fault slip velocity and expansion of the nucleation zones. The nucleation zones are concentrated and present either a single nucleation zone or a coalescing double expanding nucleation zones prior to the fast stick-slip events, which expand to a critical nucleation length smaller than the fault length and then abruptly accelerate to instability. The slow stick-slip events are characterized by the gradual expansion of multiple nucleation zones, which eventually coalesce and cover the entire fault without an abrupt acceleration or a distinguishing critical nucleation length. The experimental results show that, unlike the roughness, which depends mainly on the grain size of the abrasives used for lapping the fault surface, the centimetre-scale fault contact distribution, which depends mainly on the waviness of the fault surface profile, also plays an important role in the fault slip modes. ACKNOWLEDGEMENTS Wenbo Qi participated in the experiments. Liqiang Liu and Haijian Hao provided assistance in measuring roughness. Peixun Liu, Changrong He, Kaiying Wang, Lei Zhang and Lu Yao provided useful suggestions and discussions. Many thanks to G.C. McLaskey for his helpful reviews on the previous and current versions of this paper. We thank the other two anonymous reviewers for their constructive comments and corrections. This research was funded by the National Natural Science Foundation of China (grant no. 41702226, 41981240686 and 41572181) and the Basic Research Funds from the Institute of Geology, China Earthquake Administration (grant no. IGCEA1605). 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TI - Laboratory study on the effects of fault waviness on granodiorite stick-slip instabilities
JF - Geophysical Journal International
DO - 10.1093/gji/ggaa088
DA - 2020-05-01
UR - https://www.deepdyve.com/lp/oxford-university-press/laboratory-study-on-the-effects-of-fault-waviness-on-granodiorite-CCFotrx2G1
SP - 1281
VL - 221
IS - 2
DP - DeepDyve
ER -