Role of multiscale heterogeneity in fault slip from quasi-static numerical simulations

Abstract

Quasi-static numerical simulations of slip along a fault interface characterized by multiscale heterogeneity (fractal patch model) are carried out under the assumption that the characteristic distance in the slip-dependent frictional law is scale-dependent. We also consider slip-dependent stress accumulation on patches prior to the weakening process. When two patches of different size are superposed, the slip rate of the smaller patch is reduced when the stress is increased on the surrounding large patch. In the case of many patches over a range of scales, the slip rate on the smaller patches becomes significant in terms of both its amplitude and frequency. Peaks in slip rate are controlled by the surrounding larger patches, which may also be responsible for the segmentation of slip sequences. The use of an explicit slip-strengthening-then-weakening frictional behavior highlights that the strengthening process behind small patches weakens their interaction and reduces the peaks in slip rate, while the slip deficit continues to accumulate in the background. Therefore, it may be possible to image the progress of slip deficit at larger scales if the changes in slip activity on small patches are detectable.

Introduction

Slow slip events (SSEs) at various scales have been observed and studied in many regions worldwide over the past two decades (e.g., Linde et al. 1996; Dragert et al. 2001; Obara 2002; Kostoglodov et al. 2003; Ide et al. 2007a, 2008; Ide 2012). It is considered that SSEs occur along the same interface as ordinary tectonic earthquakes (Ide et al. 2007b), generally at the deeper extent of the shallow locked region, below the brittle–ductile transition. Elucidating the mechanics of SSEs is thus important, in terms of both understanding their complex behaviors, such as their spatiotemporal migration and regular recurrence intervals, and also recognizing their potential relationship to large earthquakes, particularly when considering the evolution of frictional strength with sliding (e.g., Shibazaki and Iio 2003; Liu and Rice 2005; Colella et al. 2012).

The dynamic process of earthquake rupture is governed by the energy balance between the released elastic energy and the fracture energy along the interface, with its transient frictional behavior being characterized by a slip-weakening process after the onset of rupture (Ida 1972; Palmer and Rice 1973). This has been proved for many earthquakes through kinematic inversions (Ide and Takeo 1997; Mikumo et al. 2003), forward modeling (Olsen et al. 1997; Aochi and Fukuyama 2002), and dynamic inversions (Peyrat et al. 2004; Di Carli et al. 2010; Ruiz and Madariaga 2011). The most important feature is the scaling relation, which states that fracture energy is proportional to earthquake size (e.g., Ohnaka 2003), regardless of the complexity of the source process. Ide and Aochi (2005) proposed a multiscale heterogeneity concept for earthquakes to uniformly describe the complexity of the earthquake rupture process at any scale. This concept requires that: (1) fault heterogeneity is expressed as the superposition of circular patches of different sizes following a fractal distribution and (2) each patch is attributed fracture energy that is proportional to the patch size. Since stress drop does not change significantly over several magnitudes (e.g., Ide and Beroza 2001), condition (2) is represented in terms of the characteristic slip distance of slip-weakening friction with a scale-invariant stress condition (Ide and Aochi 2005). Such frictional scaling can be a universal feature of the interfacial frictional behavior or the fracture medium and can be applied to both single coseismic processes and sequences of the rupture process along an interface (e.g., Ito et al. 2007; Aochi and Ide 2009; Rohmer and Aochi 2015).

This multiscale heterogeneity concept has been successfully applied to the 2011 Mw9.0 Tohoku earthquake to explain its complex rupture behavior (Aochi and Ide 2011; Ide and Aochi 2013, 2014), particularly in identifying how this mega-earthquake had grown from a small initial process. The vast rupture area of this event covered the rupture areas of several past large earthquakes (e.g., a M7.3 foreshock two days before, and the 1978 M7.5 and 2005 M7.0 earthquakes offshore from Miyagi prefecture), as well as a shallower zone that had previously been considered a stable aseismic zone. This new and expansive rupture process indicates that fault heterogeneity is more complicated that what was thought in interpreting the mechanics of mega-earthquakes. Instead of characterizing a mono-scale feature, various asperities of different scales and behaviors are likely to be involved in the fault processes of such mega-earthquakes (Lay et al. 2012; Ide 2014).

Recent numerical simulations of the fault interface have considered this intricate frictional behavior (Noda and Lapusta 2013; Galvez et al. 2014). Both SSEs and standard earthquakes can be mechanically solved in the same framework, where continuum mechanics are coupled with a given friction law and the fault parameters (slip, slip velocity, and/or state of the fault rheology) evolve during seismic cycles. Essentially, the shear stress decreases during an earthquake (weakening process) and increases during the interseismic period (strengthening) of the seismic cycle. The simplified multiscale concept, consisting of a small patch and a large patch, has been applied to the discussion of full seismic cycles under the rate- and state-dependent friction law, with a particular focus on how small heterogeneities can affect the initiation process of a large earthquake (Noda et al. 2013). Some large earthquakes can nucleate from either a dynamic perturbation on a small heterogeneity or a static nucleation process (Ellsworth and Beroza 1995), thus highlighting the importance of the shear stress loading mechanism on the system (Ando et al. 2010). It is also proposed that the many aspects of SSEs and their associated tremors can be modeled by a critical (near-) zero weakening relationship during slip (Ben-Zion 2012; Zigone et al. 2015) or some combination of velocity weakening and strengthening (Ando et al. 2012). One of the aims of this study is to explicitly model the fault healing process after coseismic slip by applying a slip-strengthening/slip-weakening frictional behavior to the system.

Ide (2014) proposed a hierarchical structure over a wide range of heterogeneities, including a characteristic large asperity and small SSE patches, which implies that the upper and lower limits of the hierarchical structure (i.e., the size of the largest and smallest patches) might control the overall behavior of the fault slip. We therefore address, in a general sense, the question of what may happen to the system if the frictional behavior of strengthening and weakening is changed during seismic cycles and how this multiscale concept is applicable, yet different, to both a stable loading system and a stick–slip system (Aochi and Ide 2009). We start with a single patch to explore the scaling relationship in fault slip, and then discuss the interaction between two patches. Finally, we investigate many patches possessing multiscale heterogeneity. Extending from the suggested framework of Ide (2014), this study employs numerical simulations to demonstrate how the occurrence of small events (i.e., the behavior of small patches) changes due to the surrounding large patch or background behavior.

Model and method

Model configuration

Figure 1A shows the model configuration of this study. We prepare a finite model domain for an infinite planar fault $\varSigma_{\infty }$ in a 3D infinite elastic medium and assume that the rest of the fault plane is dragged at a constant tectonic speed $V_{pl}$ (i.e., the imposed slip evolves linearly with time: $w_{pl} = V_{pl} t$, where $w_{pl}$ is the steady slip progress of the rest of the fault plane). In the model domain, slip $w$ is expressed as

$$w = w_{pl} + dw,$$

(1)

where ($- dw$) is a differential slip with respect to the steady slip progress of the surrounding part $w_{pl}$ and is often cited as “slip deficit.” The shear stress at any point on the fault is written as the spatial convolution over the fault plane $\varSigma_{\infty }$:

$$\tau \left( {\vec{x}} \right) = \int\limits_{{\varSigma_{\infty } }} {G\left( {\vec{x} - \vec{\xi }} \right) \cdot \left( {w_{pl} + dw\left( {\vec{\xi }} \right)} \right){\text{d}}\varSigma_{\infty } } ,$$

(2)

where $G$ is a response function of the medium (Green’s function). In this study, we assume the fault plane is embedded in a homogeneous, infinite elastic medium, such that the discrete forms of $G$ are analytically defined (Tada et al. 2000). The contribution of $w_{pl}$ over the whole infinite fault plane is zero, indicating a rigid body movement. Thus, Eq. (2) can be rewritten with only the slip-deficit ($- dw$) component on the modeled area $\varSigma$:

$$\tau \left( {\vec{x}} \right) = \int\limits_{\varSigma } {G\left( {\vec{x} - \vec{\xi }} \right) \cdot {\text{d}}w\left( {\vec{\xi }} \right){\text{d}}\varSigma ,}$$

(3)

which simplifies the analysis. This stress can then be equilibrated with any given friction relation for $\tau$ (as demonstrated in the following section).

Fig. 1

A Schematic illustration of the model configuration for a single patch surrounded by aseismic slip. The model area is described by: a a patch with radius R, governed by a friction law that is dependent on the parameters of the patch; b the background, which may possess a zero-friction condition or simply be the largest patch existing behind a; and c a constant loading outside of the model area. The infinite domain $\varSigma_{\infty }$ from Eq. (2) consists of a to c, while the modeled area is limited to a and b. B Examples of frictional behavior, from Eq. (4), with $\gamma = 1,2$ and $w_{c} = 0.2,0.4,\;{\text{and}}\;0.8\;{\text{m}}$. In this case, $\Delta \tau = 0.5\;{\text{MPa}}$ and $w_{0} = 0$ m

Fractal patch model with a slip-strengthening process

We extend the fractal patch model previously developed for explaining the coseismic phase of earthquake dynamics (Aochi and Ide 2004, 2009; Ide and Aochi 2005) by allowing aseismic slip prior to the slip-weakening process. If we take the 2011 M9 Tohoku earthquake, Japan, for which the average coseismic slip was ~20 m (maximum of ~60 m), the accumulated slip deficit is ~100 m (assuming 0.1 m/yr and a recurrence time of 1000 years). Some limited areas of the interface are recognized as slip-deficit zones (strong seismic coupling), such as the Miyagi-oki ~M7.5 earthquakes (recurring every ~40 years; e.g., Hashimoto et al. 2009). However, it is considered that most part of the fault surface was aseismically sliding, thus contributing to the stress release associated with the 2011 earthquake. To capture these heterogeneities, we introduce a slip-dependent friction law to our model that varies gradually from strengthening to weakening behavior. This transition is observed in laboratory experiments shortly before slip-weakening (e.g., Ohnaka 2003; Brantut and Viesca 2015) and has been implemented in previous numerical modeling studies (Ohnaka and Yamashita 1989; Shibazaki and Matsu’ura 1995). In most numerical modeling of the coseismic earthquake process, this strengthening process is neglected, because the contribution from the strengthening phase to fracture energy release is considered minor once the instability develops. However, we aim to model much longer strengthening processes that correspond to the long-term healing process after the previous weakening process during a seismic cycle (e.g., Nakatani, 2001; Aochi and Matsu’ura 2002). By extending the slip-dependent law used by Shibazaki and Matsu’ura (1995), we describe friction as an evolution of the shear stress $\tau$:

$$\tau = f\left( w \right) = \left\{ {\begin{array}{*{20}l} {\Delta \tau \left( {\frac{{w - w_{0} }}{{w_{c} }}} \right)^{\gamma } \exp \left( {1 - \left( {\frac{{w - w_{0} }}{{w_{c} }}} \right)^{\gamma } } \right)} \hfill & {{\text{for}}\;w \ge w_{0} } \hfill \\ 0 \hfill & {{\text{for}}\;w < w_{0} } \hfill \\ \end{array} } \right. ,$$

(4)

where $\Delta \tau$ is the strength drop (peak strength, assuming zero residual stress), $\gamma > 0$, and $w_{c}$ and $w_{0}$ are constants. As illustrated in Fig. 1B, $w_{c}$ characterizes the amount of slip (or distance) until the peak strength is achieved (as well as the following weakening process), and the shape parameter $\gamma$ defines the curve of frictional behavior. Ohnaka (2003) found a linear scaling relationship between the distances of the strengthening and weakening processes, demonstrating that this one-parameter description is a good approximation. We thus define $w_{c}$ as the characteristic distance and $w_{0}$ as the phase shift. Equation (4) has a peak at $w = w_{0} + w_{c}$. Shibazaki and Matsu’ura (1995) explored the nucleation process of earthquakes for $\gamma = 1$ and $w_{0} = 0$; we extend their study and also consider the cases where $\gamma \ne 1$ and $w_{0} \ne 0$. When $\gamma > 1$, $\tau$ remains low for longer durations and then increases quickly [as the derivative of Eq. (4) is zero at $w = w_{0}$], corresponding to a fast healing process arriving just prior to weakening process.

Fault heterogeneity is generally described by patches of different sizes (Ide and Aochi 2005; Aochi and Ide 2009). For each patch, we assign a particular frictional parameter ($w_{c}$) based on the size of the patch, but we keep the same value of $\Delta \tau$ on all patches to sustain the broad scaling relationship for stress drop (e.g., Kanamori and Anderson 1975; Ide and Beroza 2001). According to Ide and Aochi (2005), $w_{c}$ is proportional to the patch size $R$, such that

$$w_{c} \propto R.$$

(5)

This assumption highlights the fact that earthquakes are generally scale-independent processes from the viewpoint of earthquake seismology. Ohnaka (2003) summarized the scaling relationship from laboratory-scale studies to large earthquakes. Uchide and Ide (2010) analyzed earthquakes over a wide range of magnitudes to statistically show that rupture growth is self-similar. We then assume that the same scaling relationship is applicable both to the dynamic, coseismic process of earthquakes (slip-weakening friction) and to the preparation process (slip-strengthening friction).

Numerical method and parameters

Equations (3) and (4) are solved simultaneously for the two unknowns $\tau$ and $dw$ over the model area $\varSigma$, such that the loaded shear stress due to the slip distribution is balanced by the frictional behavior of the system. We discretize the fault plane as a series of small square elements of size $\Delta s$ that experience uniform slip ($dw$) at a given time step. We can then write the shear stress $\tau_{i}$ for the ith element in the discrete form

$$\tau_{i} = \mathop \sum \limits_{n} G_{\text{in}} \cdot dw_{n} = f\left( {w_{pl} + dw_{i} } \right) ,$$

(6)

where $G_{\text{in}}$ is a response function from the nth to ith element. Note that $\tau_{i}$ is influenced by not only the ith element but also all other elements ($n \ne i$). We adapted the Levenberg–Marquardt method to solve Eq. (6) iteratively (e.g., Press et al. 1992). Note that the solution $dw_{i} = 0$ is only valid when, independent of our slip calculation, the friction is modeled as constant. The physical quantities of the key parameters that define our model space are summarized in Table 1. Large patches are defined as a few tens of kilometers in size, based on significant aseismic phenomena observed in some subduction zones (e.g., Ide et al. 2007a, b). We note that our fixed $\Delta \tau$ may be smaller than that inferred for standard earthquakes. For example, Gao et al. (2012) infer a range between 0.01 and 1 MPa from SSEs in Cascadia, and Maury et al. (2014) infer a value of ~0.1 MPa from the large 2010 SSE in Guerrero, Mexico. We fix a value of $\Delta \tau = 0. 5\,{\text{MPa}}$ for our study. We further note that the stress accumulation is not led by the stress increment but by the slip deficit. While our study focuses on a fault interface, the following results and discussions can be downscaled to local microseismicity or laboratory experiments. At time $t = 0$, we set $w_{pl} = 0$, $dw = 0$, and $\tau = 0$ everywhere, and $w_{pl}$ increases at a constant rate (via the loading rate $v_{pl}$) for each time step. As there is no time dependency in Eqs. (3) and (4), the time step can be viewed as a slip step imposed outside the model area. Therefore, the system described here is relative in its stress level and in time, and can be extrapolated to other scales.

Table 1 Model parameters

Parameter studies

In this section, we study the characteristics of the introduced slip-dependent law, with an emphasis on the role of the slip-strengthening process. We focus on the scaling issue to address how the stress (slip deficit) is accumulated prior to the slip-weakening process, according to the size of a single patch (Scale dependency in the accumulation process (scaling in w _c ) section), and then explore the behavior of a small patch surrounded by a large patch (Incoherent onset of stress accumulation (variation in w ₀) section). An understanding of both features is important in explaining the consequent weakening process.

Scale dependency in the accumulation process (scaling in $w_{c}$)

We first explore the behavior of a single patch of varying size (different $w_{c}$). According to Eq. (4), the peak strength $\tau = \Delta \tau$ at the turning point from the strengthening to the weakening process does not vary with patch size. However, the stress accumulation is more rapid on a smaller patch than on a larger patch under the same background loading rate. As shown in Fig. 1A, we set a single patch in the model domain and employ the model parameters summarized in Tables 1 and 2. We then vary the patch radius and the scaling relation of $w_{c}$ in Eq. (4). Figure 1B illustrates the frictional relationship, assuming $w_{c}$ scales with respect to $R$. Here, $w_{c} = 0.2,0.4,{\text{and}}\;0.8\;{\text{m}}$ correspond to $R = 10,20,{\text{and}}\;40\;{\text{km}}$, respectively.

Table 2 Variables for a single patch case

Figure 2a and b provides a snapshot of a single patch with radius $R = 20\;{\text{km}}$, $w_{0} = 0$, and $\gamma = 2$. The slip deficit initially increases with time, leading to stress accumulation. The patch then slips to catch up with the surrounding region, releasing the accumulated stress after the peak friction is reached in Eq. (4). We study the effect of different values of $\gamma$ and $R$ on slip deficit, listed in Table 2, where $w_{c}$ is scale-dependent. The stress and slip evolutions are qualitatively similar in all cases. Here we are interested in how the slip-deficit phenomena are scaled. As shown in Fig. 2c, the time necessary to achieve the peak slip deficit is proportionally scaled with $R$ and independent of $\gamma$; a similar trend is observed in the amount of peak slip deficit (Fig. 2d). Note that if there is no scale dependency on $w_{c}$, the time to the peak slip deficit scales differently for different values of $\gamma$ and $R$ (pale gray small circles and line in Fig. 2c, where $w_{c} = 0.4\;{\text{m}}$). Therefore, one can conclude that the time to the peak slip deficit is only scale-invariant according to the scale dependence of $w_{c}$. On the other hand, the amount of peak slip deficit is dependent on the geometrical size of the patch and is independent of its frictional parameters.

Fig. 2

Snapshot of the quasi-static evolution of fault slip for a single patch of $R = 20\;{\text{km}}$, $w_{c} = 0.4\,{\text{m}}$, $w_{0} = 0\,{\text{m}}$, and $\gamma = 2$, in terms of a slip deficit and b shear stress. At approximately time step 70, the peak in slip deficit is observed. c Time step necessary to reach the peak in slip deficit, with variable R and $\gamma$ (Table 2). All the cases (γ = 1, 1.1, 1.5, and 2) are superposed, represented by solid large symbols (crosses: γ = 2; triangles: γ = 1, 1.1, and 1.5) and lines. The pale gray small circles and lines represent the cases where $w_{c}$ is not scaled with patch size R ($w_{c}$ = 0.4 m fixed). The small pale gray circles and lines represent the cases where $w_{c}$ is not scaled with patch size R ($w_{c} = 0.4\,{\text{m}}$ fixed). d Peak slip deficit at the center of the patch for each case. Markers are the same as in the previous panel

Figure 3 shows the evolution of the slip deficit and the average slip rate for each patch to quantify the scaling relationship for different values of $\gamma$ ($\gamma = 1\;{\text{and}}\;2$), assuming that $w_{c}$ is scale-dependent (see Fig. 1B). The weakening process after the peak stress is reached accelerates the fault slip (i.e., reduces the slip deficit). Figure 3a shows that in the case where $\gamma = 1$, the increasing rate of the slip deficit with time is common for all cases, such that the initial slip rate at $t = 0$ is invariant from $R$. However, the amplitude of the slip rate is inversely related to $R$ if $\gamma = 2$ (Fig. 3a). The stress (slip deficit) is accumulated and released faster on a smaller patch than on a larger patch. This is a generality for cases where $\gamma > 1$.

Fig. 3

Evolution of slip behavior in terms of slip deficit and slip rate, averaged for each patch (center and right panels), for a $\gamma = 1$ and b $\gamma = 2$. The three lines correspond to $R = 10$ (green), 20 (blue), and 40 (red), respectively. In the right-hand panels, gray lines represent the behaviors of different parts of a patch, and the solid colored lines give the average behavior for an entire patch

From these preliminary tests, we find that the behavior of this system for the case where $\gamma = 1$ is characterized by universal functions, which can be normalized over time according to the scale. Such scaling is known for the nucleation and the dynamic process of earthquakes (Aochi and Ide 2004 and references therein), which is important for understanding both the loading and weakening processes of aseismic slip. This is based on the fact that the elastic equation for a fault (Eq. 6) is governed by a non-dimensional rigidity, $L_{c} = \left( {\mu /\Delta \tau } \right) \cdot \left( {w_{c} /R} \right)$ (Matsu’ura et al. 1992), with medium rigidity $\mu$. In the case that the frictional evolution is scale-dependent (as in Eq. 5), it is expected that slip behaviors appear as scale-independent. Since the slip rate is controlled by a factor of $R \times ({\text{maximum}}\;{\text{slope}}\;{\text{of}}\;{\text{friction}}\;{\text{law}})$, one expects that the slip rate is scale-independent if $\gamma = 1$ in Eq. (4). However, for cases where $\gamma > 1$, this scale invariance is not applicable, as a smaller patch has a sharper and higher slip rate function than a large patch. This scale dependency might be consistent with slow earthquake observations where only small slow earthquakes (tremors) are detected. The particularity of Eq. (4) is that $\left. {\frac{{{\text{d}}\tau }}{{{\text{d}}w}}} \right|_{w = 0} \ne 0$ if $\gamma = 1$, while $\left. {\frac{{{\text{d}}\tau }}{{{\text{d}}w}}} \right|_{w = 0} = 0$ for $\gamma \ge 1$. This means that the evolution of the slip deficit at $w = w_{0}$ is discontinuous for $\gamma = 1$; the slope of the slip-deficit function ($dw \ne 0$ on the patch) shows a sudden change at $t = 0$, while it changes gradually ($dw\sim 0$ on the patch) in the latter case.

Incoherent onset of stress accumulation (variation in $w_{0}$)

We now consider two patches of different sizes superposed along the fault surface. Their frictional parameter distributions are shown in Fig. 4. A large patch (L) of radius $R = 40\;{\text{km}}$ and a small patch (S) of radius $R = 10\;{\text{km}}$ are modeled, with the model parameters summarized in Table 3. The frictional characteristics of the smaller patches appear locally if different patches are superposed. We aim to show how the behavior of a small patch is affected by the friction of the surrounding large patch. The small patch has minimal influence on the large patch behavior in this given situation, while smaller patch(es) may impact the occurrence of the large patch quasi-statically (e.g., Noda et al. 2013) or dynamically (Aochi and Ide 2009). As $\Delta \tau$ is the same for both patches, the different attributes of $w_{c}$ and $w_{0}$ vary the behavior of each patch. Here we focus on the role of $w_{0}$, setting $\gamma = 2$ to avoid the discontinuous evolution of slip deficit at $w = w_{0}$, such that $dw = 0$. The initial condition is uniform everywhere ($w = 0$ and $\tau = 0$).

Fig. 4

Model with two patches (L: large; S: small). a Distribution of two patches in terms of peak strength $\Delta \tau$ and characteristic distance $w_{c}$. The cross section is taken at $x = 5\,{\text{km}}$, running through the center of the small patch. b Frictional behavior, from Eq. (4), for both L and S, with $w_{0} = 0.2,0.6,{\text{and}}\;1.0\,{\text{m}}$

Table 3 Variables for the two-patch case

The simulation results shown in Fig. 5 represent different $w_{0}$ values for the small patch, such that there are different positions of the frictional peak ($w_{0} + w_{c}$) on S with respect to just one ($w_{c}$) on L. The two time steps ($T_{1}$ and $T_{2}$) for each sequence are selected when the shear stress has a peak accumulation on either of the two patches. For the first two cases, where $w_{0} = 0.2\;{\text{m}}$ (Fig. 5a) and $w_{0} = 0.6\;{\text{m}}$ (Fig. 5b), S is the first to reach the peak stress and slide, while L does not begin to slide before S until $w_{0} = 1.0\;{\text{m}}$ (Fig. 5c). In all cases, S does not have a significant influence on the aseismic process of L, as the time step of the maximum shear stress accumulation does not change.

Fig. 5

Evolution of shear stress and slip behaviors for L (red) and S (blue), where a $w_{0} = 0.2\,{\text{m}}$, b $w_{0} = 0.6\,{\text{m}}$, and c $w_{0} = 1.0\,{\text{m}}$. Plots of temporal changes in shear stress, total slip, slip deficit, and slip rate are shown for each case. Stress values are extracted from the center of each patch, while the slip quantities are averaged across each patch. The times $T_{1}$ and $T_{2}$ are indicated by broken lines when the shear stress has a peak at either of two patches (L or S) just before the weakening process, and the corresponding spatial distributions of slip deficit and shear stress are illustrated in the following figures

Figure 5 also illustrates the detailed evolution of stress at the center of the two patches (S and L), as well as the average slip behavior on each patch. The time taken to reach the peak slip deficit on S changes with $w_{0}$. In the first case where $w_{0} = 0.2\;{\text{m}}$ (Fig. 5a), the slip deficit on S is charged and released prior to the peak on L. S also experiences simultaneous and passive slip again when L releases its stress. In the latter cases, where $w_{0} = 0.6\,{\text{m}}$ (Fig. 5b) and $w_{0} = 1.0\;{\text{m}}$ (Fig. 5c), S reaches its peak shear stress when L is already charged or releasing its stress. The slip rate on S becomes much smaller when $w_{0} = 0.2\;{\text{m}}$ (Fig. 5a) compared with larger $w_{0}$ values (Fig. 5b, c). This means that the slip behavior of S is overprinted by L, even though the shape of the frictional behavior is the same. It is thus interpreted that the difference arises from the deformation state surrounding S, namely the practical stiffness of the system at each moment. Comparing these results with the coseismic process, it is expected that small patches may keep a high slip rate if rupture onset on them is delayed. Note that the small patches are mostly hidden when a surrounding patch is sliding, if the phase of the peak friction is simultaneous among the patches (e.g., Aochi and Ide 2014).

Multiscale patch models applied to seismic cycles

Here we numerically study the aseismic slip behavior in a multiscale patch model by changing the geometry of patch distributions and the limit of the hierarchical structure. It is expected that the system behavior depends on the spatial distribution of patches even if the same fractal size–number distribution is applied. We focus our discussion by simplifying the model setting and carrying out 15 simulations. We consider three ranks ($k = 2 , 3 , 4$) of different patch sizes from $R = 20\;{\text{to}}\,5\,{\text{km}}$ along a $60\,{\text{km}} \times 60\;{\text{km}}$ fault surface (one-quarter the size of the previous simulation), such that the background corresponds to the characteristics of the larger patch ($k = 1$). According to Ide and Aochi (2005) and Aochi and Ide (2009), the number of patches at each rank follows a fractal relationship:

$$N_{k} = 2^{{2\left( {k - 1} \right)}} ,$$

(7)

and $w_{c}$ is proportional to $R$.

Figure 6 illustrates our 15 examples of the randomly generated patch distribution, where an average of 1, 4, and 16 patches are present for ranks $k = 2,3,{\text{and}}\;4$, respectively. A few patches of the largest rank are sometimes found, but their total area is statistically equivalent to that of one patch. For $k = 2,3,{\text{and}}\;4$, $w_{c} = 0.4,0.2,\;{\text{and}}\;0.1\;{\text{m}}$, respectively, and it is assumed that the frictional behavior of each patch has a cycle of $3 \times w_{c}$ to represent the fault healing process and then the seismic cycle. This cyclic period is long enough that the shear stress is reduced to approximately zero across each patch. Extending this cyclic period leads to a quasi-zero-stress state that lasts longer, resulting in a longer seismic cycle with weaker coupling on average. This suggests that the fault is always sliding and that the slip amount from a large event (seismic or aseismic) may only form a portion of the total slip. The condition where $w_{c} = 0.2\;{\text{m}}$ for a patch of $R = 10\;{\text{km}}$ (cycle of 0.6 m) leads to a slip event of ~M6, with a potential recurrence interval of several tens of years if the tectonic convergence speed is fast enough. We assume $\gamma = 2$ in Eq. (4) (see Fig. 1B) so that the frictional behavior is quasi-zero during most of interseismic period. The onset distance $w_{0}$ is randomly defined. For the background fault interface, we define $w_{c} = 0.8\;{\text{m}}$ and $w_{0} = 2.4\;{\text{m}}$, allowing us to analyze the slip behavior for different background states, namely the zero-stress condition, stress accumulation, and stress releasing phases. As the slip rate is set at 0.01 m per time step, each phase briefly corresponds to the time until step 240, 320, and 400, respectively.

Fig. 6

Spatial distribution of randomly generated multiscale patches for 15 cases, randomly generated under the same fractal size–number distribution. Each model includes three different ranks of patches ($k = 2,3,\;{\text{and}}\;4$), denoted by light blue, dark blue, and purple, respectively. The background region is yellow, with $w_{c} = 0. 8 {\text{m}}$ and $w_{0} = 2. 4 {\text{m}}$, and is assumed to represent a larger patch of rank $k = 1$

Figure 7 shows the temporal evolution of stress, slip deficit, and slip rate for two cases (4 and 6) from Fig. 6, representing the minimum (4) and maximum (6) background areas among the 15 examples. Shear stress is plotted at the center of each patch, noting that a large patch is sometimes occupied by another small patch. Slip deficit and slip rate are averaged across each patch. After stabilization during the initial 50 time steps, the system behavior becomes quite active, with increases in both the frequency and maximum value of the slip rate. Pink curves represent the seismic moment release $M_{0}$ (Fig. 7b) and seismic moment release rate along the entire fault plane $\dot{M}_{0}$ (Fig. 7c), while red points show the local maximum (estimated from the sign change of the derivative) in slip rate for each time step (Fig. 7c). Although the two extreme cases are different in terms of small-scale interactions between patches, closer inspection of the peaks in slip rate indicates that fault rupture due to these patches may be seismologically detectable and that $\dot{M}_{0}$ may be geodetically detectable in both instances (Fig. 7c). The peaks in slip rate become less frequent in case (4) between time steps 250 and 350 (Fig. 7c), corresponding to the stress accumulation phase of the slip-deficit curve (Fig. 7b). All 15 simulations are summarized in Fig. 8, in terms of both the locally observed maximum slip rate and $\dot{M}_{0}$ as a function of time. Less frequent peaks in slip rate between time steps 250 and 350 are observed across all simulations (Fig. 8). A distinct and consistent relationship between the locally observed maximum slip rate and $\dot{M}_{0}$ is also observed, with some variations in amplitude arising due to the specifics regarding patch distribution and interaction.

Fig. 7

Simulated temporal evolution for cases (4) and (6), whose patch distributions are shown in Fig. 6. a Shear stress evolution at the center of each patch. The colors correspond to the order of the patches, with blue, green, and gray used for $k = 2 , 3 , {\text{and 4}}$, respectively. b Average slip deficit across each patch. $M_{0}$ is shown by a pink curve. c Average slip rate across each patch. $\dot{M}_{0}$ is shown with a pink curve. The local peaks in slip rate are identified by blue dots, whereas the red circles correspond to the peaks in $\dot{M}_{0}$ for each time step

Fig. 8

Temporal evolution of all 15 simulations. The maximum slip rate at each time step is stated in blue (left axis), and $\dot{M}_{0}$ from the whole area is in red (right axis)

Moving averages of the number of peaks in slip rate and the maximum slip rate are determined for all 15 simulations to further quantify the temporal variation in slip rate (Figs. 9, 10). A clear reduction in activity between 250 and 350 time steps is evident in both the overall behavior and the local patch slip (Fig. 9). This reduction is also observed in both $\dot{M}_{0}$ (Fig. 10a) and the observed minimum peak slip rate at about time step 300 (Fig. 10b). Therefore, slip on small patches occurs more frequently and at a faster rate when the background stress is decreasing. However, when the background stress is increasing, the short-term activity evident as peaks in slip rate becomes even weaker than at the other period of the seismic cycle. This observation might be unexpected because we often expect enhanced seismic activity in the case of high stress or rapid stress buildup in the system, but it is in agreement with the simulation using only two patches (Fig. 5), where the slip rate is enhanced during stress release in the background region. This observation is therefore important when interpreting SSEs in nature.

Fig. 9

Number of peaks per time step, averaged over 50 time steps, from the 15 simulations. In a and b, peaks in $\dot{M}_{0}$ (corresponding to the red points in Fig. 7) and in the local slip rate (blue points in Fig. 7, selected for a peak rate higher than 0.02 m/$\Delta t$) are shown, respectively, as a function of time step. The bold colored curves are the average values for all the simulations

Fig. 10

Peak values of slip rate, averaged over 20 time steps, from the 15 simulations. The overall behavior and local movement are shown in the left and right panels, respectively. The averages of all the simulations are shown by bold curves

Discussion

We assume that the characteristic distance $w_{c}$ is proportional to patch size $R$ and randomly define the onset distance $w_{0}$ to demonstrate the influence of patches on slip along the fault interface. Here we use cases (4) and (6), and employ the same parameters as shown in Fig. 7, except for defining $w_{0} = 0$ (Fig. 11). As the smallest patches follow a seismic cycle of $3 \times w_{c} = 0.3\;{\text{m}}$ and the loading rate is 0.01 m/$\Delta t$, most of smallest patches slip almost simultaneously along the fault every 30 time steps, with a visible periodicity in the total slip deficit (Fig. 11b) and seismic moment release over the fault (Fig. 11c). Since case (4) is governed by a large background area, the peaks in slip rate for the smallest patches become sparse and the slip deficit (variation in $M_{0}$) for the total system increases due to the broader background strengthening process between time steps 240 and 360. This result likely arises from the fact that the distance between two slip-weakening patches affects the interaction between them (Kaneko et al. 2010). It also implies that the background strengthening makes the interaction between patches difficult. This mechanism is unclear for case (6), because the background area is minimized for this simulation.

Fig. 11

Temporal evolution of the simulations for cases (4) and (6), with the same parameters settings as shown in Fig. 7, except that $w_{0} = 0$. a Shear stress evolution at the center of each patch. The colors correspond to the order of patches, with blue, green, and gray for $k = 2 , 3 , {\text{and 4}}$, respectively. b Average slip deficit for each patch. $M_{0}$ is shown with a pink curve. c Average slip rate for each patch. $\dot{M}_{0}$ is shown with a pink curve. The local peaks of each curve are identified by red points according to the change in sign of the derivative

Finally, we change the lower limit of the patch ranks, following Ide (2014), and observe the resultant perturbations to the system behavior (Fig. 12). We simulate our 15 cases, with the same initial patch distribution as that shown in Fig. 6, but exclude the smallest patches of rank $k = 4$. The system behavior becomes simpler as the smallest patches disappear, in both the frequency and amplitude of the perturbations. Figure 13 summarizes these behaviors in terms of the local maximum and overall slip rates. As all the parameters are kept the same except for the existence of the small patches, a direct comparison of Fig. 13a with Fig. 8 reveals that local peaks in slip rate are mostly associated with the smallest patches in the system. The same periodic slip rate bursts exist in both cases (Figs. 8, 13), because this is governed by the larger patches that commonly exist in the system. According to the frictional state of the larger patches, the peaks in slip rate for the smallest patches may vary, as explored in the two-patch system. The relation between the peaks in slip rate is shown in Fig. 13b. Again, the overall behavior of the system (horizontal axis) does not change because it is governed by the larger patches; however, the peak value of the local slip rate is reduced because of the disappearance of the smallest patches.

Fig. 12

Spatial distribution of multiscale patches for all 15 simulations, originating from those shown in Fig. 6. The lower limit of the structure is changed so that the smallest patches of rank $k = 4$ are removed

Fig. 13

Temporal evolution of all 15 simulations from Fig. 11. a The maximum slip rate at each time step is in blue (left axis) and the seismic moment release rate from the whole area is in red (right axis). b Relationship between the above two values from all 15 simulations, denoted by red points. Gray points represent the relationship for the case that includes the smallest patches $k = 4$, from Fig. 8

In this study, we have adopted a slip-dependent friction law to characterize the variations along the fault interface that are necessary for sliding (fracture energy). As demonstrated by Noda et al. (2013), and which is also well known in the coseismic process (Bizzarri and Cocco 2003), this could be translated to a rate and state friction law (Ruina 1983). In particular, we have a priori fixed a slip cycle, but the healing process should be more complex due to the fault rheology. Nakatani (2001) argued that the state variable represents the fault strength such that its evolution obeys both slip-weakening and log-time healing. The scale of the characteristic distance in the rate- and state-dependent law also produces stable and unstable behaviors within the system (Hori and Miyazaki 2011). The advantage of the introduction of an explicit form of the slip-dependent relation, Eq. (4), is the clear definition of when the turning point in frictional behavior from strengthening to weakening occurs, such that any point on the fault interface experiences both states repeatedly. This slip dependency can provide a simpler way to describe the shallower part of the fault interface, which is largely stable during seismic cycles but sometimes becomes unstable and can potentially generate a tsunami earthquake or a mega-earthquake (e.g., Sun et al. 2017). Such a change from strengthening to weakening, or extra weakening during a mega-earthquake, can be regarded as the result of either a hydrothermal interaction during sliding (e.g., Kanamori and Heaton 2000; Noda and Lapusta 2013), the presence of weak clay minerals (Ujiie et al. 2013), or fault-related melting (Hirose and Shimamoto 2005). In our model, slip on small patches can be repeated very quickly once the surrounding slip is large enough. This supports the initial point of view of the multiscale concept in fracture energy (Ohnaka 2003; Ide and Aochi 2005), where the fracture energy reflects an irregular surface topography (Matsu’ura et al. 1992). Nevertheless, the rate effect or healing is oversimplified in our study, since we assume a strictly periodic behavior over slip and the same peak strength, even though the rate effect was weak during SSEs in Guerrero, Mexico (Maury et al. 2014). It is unknown whether a small heterogeneity remains in its original state after a large slip event. A more complex time-dependent process is probably required, which is also dependent on patch size (Aochi and Matsu’ura 2002).

The simulation results are informative with respect to the conceptual earthquake model. We have increased the complexity from a single patch to multiple patches and observed that the system behaviors become increasingly complex. In our numerical simulations, the peaks in slip rate from the smallest patches are the most visible, with the existence of large patches hidden in the background. Nevertheless, the frictional status of large-scale patches changes the slip rate on the small patches, such that the background strengthening process weakens the peaks in slip rate peaks on the small patches. In the case of multiscale patches, the system complexity is controlled by the lower and upper fractal limits (Ide 2014). Slip rates on the small patches become visible as tremor-like burst activity, with the amplitude and frequency of the bursts controlled by the status of the surrounding larger patches (Fig. 14). Since it is considered that a large patch evolves gradually with time (Fig. 13), the upper limit of the fractal structure evolves with time during a long seismic cycle. If a variation in slip along the fault interface (which arises from small patches) is detected, this would help us to better understand changes in the surrounding, or background, frictional status at a larger scale. Conversely, a change in this lower limit (small patches) has an immediate impact on the system behavior, as the lack of small patches leads to a repeating behavior with moderate slip rates.

Fig. 14

Schematic illustration of fault hierarchy heterogeneity and different appearances of seismic behavior. a When the smallest scale is missed, the fault behavior appears as repeating slow slip events. b When the largest scale is missing (or there is a zero-stress background), burst seismicity controlled by small patches becomes dominant. c When the largest scale patch plays a strengthening role, the burst behavior is reduced

The wide variety of SSEs worldwide might be explained by their differences in hierarchical structure. For example, in Guerrero, Mexico, where very large SSEs occur, it is possible that intermediate-scale patches are poorly developed. The segmentation structures of tremors observed at Nankai (Obara 2010) and Cascadia (Brudzinski and Allen 2007) may correspond to a hierarchical structure consisting of a nearly continuous distribution of many small patches along the trench, with a few very large patches controlling the recurrence intervals of tremor burst, together with some intermediate-scale patches. Conversely, in some spotty tremor regions in southwestern Japan (Ide and Tanaka 2014) and Taiwan (Ide et al. 2015), where tremors occur quasi-regularly, the interface heterogeneity is represented principally by isolated small patches.

While this study applies hierarchical structures to model stable slip along the plate interface, similar models might also be applicable to regions with small-scale unstable patches on larger-scale stable structures. One example would be eastern Japan, where many small to moderate repeating earthquakes are observed (e.g., Uchida et al. 2003). These events are usually attributed to the repeated rupture of small patches on a homogeneous and stably sliding background. There may exist a multiscale structure of stable patches behind them, but it would be difficult to observe this complex structure within the limited period of the seismic cycle (Uchida et al. 2016). Nevertheless, the long-term observation of slip-deficit evolution and differences in amplitude for repeating events may provide further insight into the complex structure of the plate interface.

Conclusions

We performed quasi-static numerical simulations of fault slip on an interface characterized by multiscale heterogeneity (patches), assuming that the characteristic distance in frictional behavior is scale-dependent. Stress accumulation on patches (e.g., strengthening process) prior to the weakening process depends on the patch size. For an isolated single patch, the evolution of slip rate is self-similar under the same loading if friction is scale-dependent, as is known for standard earthquakes. In the interaction between two patches, the behavior of a small patch can be changed by the stress condition of the surrounding region. In the case of multiscale patches, slip on small patches becomes visible, while the role of a large patch remains hidden behind. However, the frequency and maximum slip rate of slow events decrease when the surrounding region is in the strengthening regime. It is thus important to know the stress state of the large patch behind the small patches in order to describe the behavior of the whole system.

The results of this systematic study are important in improving our understanding of the heterogeneity along a fault interface for both ordinary (fast) and slow earthquakes. Since fault heterogeneity must be associated with the multiscale geometrical irregularity of the plate interface, such structures may be long-lived and remain unchanged during a few seismic cycles. These potential changes in stability may also depend on the scale of the concerned heterogeneity of the events, suggesting that there are various types of mixed behaviors associated with fast and slow slip. The slip behavior modeled in the present paper should be considered as one end-member of various slip behaviors (slow earthquakes); the other end-member (fast earthquakes) has previously been demonstrated by Aochi and Ide (2009) and related works.