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

- Hideo Aochi
^{1, 2, 3}Email authorView ORCID ID profile and - Satoshi Ide
^{4}

**Received: **10 January 2017

**Accepted: **27 June 2017

**Published: **11 July 2017

## 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.

## Keywords

## 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

### Fractal patch model with a slip-strengthening process

### Numerical method and parameters

*i*th element in the discrete form

*n*th to

*i*th element. Note that \(\tau_{i}\) is influenced by not only the

*i*th 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.

Model parameters

Parameter | Quantity |
---|---|

Medium rigidity \(\mu\) | 50 GPa |

Grid size \(\Delta s\) | 4 km, 2 km, 1.5 km |

Strength drop \(\Delta \tau\) | 0.5 MPa |

Increment of tectonic loading \(w_{pl}\) | 0.01 m |

Loading rate \(v_{pl}\) /Time step \(\Delta t\) | 0.01 m/\(\Delta t\) |

First patch level (size \(R\), characteristic distance \(w_{c}\)) | 40 km, 80 cm |

## 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 \(w_{c}\))

Variables for a single patch case

Parameter | Quantity |
---|---|

Patch size \(R\) | 40, 20, 10, and 5 km |

Characteristic distance \(w_{c}\) | Proportional to \(R\) |

Shape parameter \(\gamma\) | 1, 1.1, 1.5, 2, and 3 |

Phase factor \(w_{0}\) | 0 (fixed) |

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 \(w_{0}\))

Variables for the two-patch case

Parameter | Quantity |
---|---|

Patch size \(R\) of large and small patches | 40 km (L) and 10 km (S) |

Characteristic distance \(w_{c}\) | Proportional to \(R\) |

Shape parameter \(\gamma\) | 2 (fixed) |

Phase factor \(w_{0}\) | 0 m (fixed) for L 0.2, 0.6, and 1.0 m for S |

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

## Discussion

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 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.

## Declarations

### Authors’ contributions

HA developed the numerical codes. Both authors worked on the model conception, parameterization, simulation execution, and interpretation, and contributed to discussions and writing of the manuscript. Both authors read and approved the final manuscript.

### Acknowledgements

This work started in the framework of the French national project S4 (Subduction Slow and Standard Seismology; ANR-11-BS56-0017), with funding from Agence Nationale des Recherches, and then was partially supported by MEXT, Japan, under its Earthquake and Volcano Hazards Observation and Research Program, and KAKENHI (16H02219 and 16H06477).

### Availability of data and materials

The simulation datasets supporting the conclusions of this article are included within the article.

### Competing interests

The authors declare that they have no competing interests.

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## Authors’ Affiliations

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