# Dynamics of a fault model with two mechanically different regions

- Michele Dragoni
^{1}View ORCID ID profile and - Emanuele Lorenzano
^{1}Email authorView ORCID ID profile

**Received: **21 April 2017

**Accepted: **13 October 2017

**Published: **23 October 2017

## Abstract

## Keywords

## Introduction

It is a common observation that fault slip following an earthquake may continue for some time, at a decreasing rate. This phenomenon is known as afterslip and may last up to several months. It is expected to be the consequence of aseismic slip of a velocity-strengthening region of the fault, which has been loaded by the coseismic slip of a velocity-weakening region (Scholz 1990).

In fact, seismic and geodetic observations have shown that faults can accommodate tectonic motion in different ways. Some fault regions respond with stable, quasi-static motion, with slip rates comparable to tectonic rates; other regions remain locked for decades or centuries and then experience fast slip with emission of seismic waves. Typically, the amount of afterslip is proportional to that of seismic slip. For instance, afterslip can account for a part of the postseismic deformation following the 1906 San Francisco earthquake (Kenner and Segall 2000).

One way to account for such observations is to separate the fault surface into two kinds of regions: stable regions, which mostly creep, and unstable regions, which produce earthquakes (e.g. Johnson 2010). In studying a thrust earthquake at the Japan Trench, Heki et al. (1997) found that the afterslip distribution was relatively even throughout the fault surface, while the coseismic slip was concentrated in a small central part corresponding to an asperity. Yagi and Kikuchi (2002) acknowledged that afterslip following the 1994 Sanriku-Haruka-Oki earthquake took place in a region adjacent to the asperity whose failure generated the mainshock. However, it has been suggested that in some cases the two regions may not be spatially separated (Noda and Lapusta 2013).

There is obviously an interaction between the two fault regions. The interaction between seismic and aseismic fault slip was studied by Dragoni and Tallarico (1992) and Tallarico et al. (2002). Kato (2004, 2014) discussed the interplay between velocity-weakening and velocity-strengthening patches on a fault plane and the conditions for seismic and aseismic slip events. Dublanchet et al. (2013) and Yabe and Ide (2017) considered a fault with a large number of asperities and pointed out that asperity concentration remarkably affects the slip behaviour of heterogeneous faults containing both velocity-weakening and velocity-strengthening regions: specifically, the entire fault may slip seismically if the asperity concentration exceeds a critical threshold. Bulk relaxation of the crust and the effect of pore fluid diffusion may also play a role. Belardinelli and Bonafede (1995) modelled afterslip as driven by viscous flow in the asthenosphere. A study of the effects of viscoelastic relaxation on a complex fault was made by Amendola and Dragoni (2013) and Dragoni and Lorenzano (2015).

Several empirical relationships have been proposed in order to describe aseismic slip. Nason and Weertman (1973) proposed an exponential function approaching a constant value. More recently, observations and theoretical considerations suggested that afterslip can be represented as a logarithmic function of time (Marone et al. 1991). Even though this function represents well the observations in many cases, it yields a slip that increases indefinitely with time and must be truncated ad hoc. A discussion of different time functions was presented by Barbot et al. (2009).

Seismic and aseismic slip in a two-degree-of-freedom spring-block model was studied by Yoshida and Kato (2003). Friction on the blocks was described by means of a rate- and state-dependent friction law. Under the condition of velocity-weakening for one block and velocity-strengthening for the other, afterslip was found as a result of the relaxation of the stress imposed on the latter by the sudden slip of the former. The model was further investigated by Abe and Kato (2013) employing a different expression for the friction law and assuming a velocity-weakening behaviour for both blocks. Numerical simulations were carried out using different initial conditions for the stiffnesses of the coupling springs, and several slip patterns were acknowledged accordingly, including stable sliding, seismic and aseismic slip. In both papers, the evolution equations were solved numerically.

The aim of the present study is to obtain afterslip as a dynamic mode of a fault containing two regions with different mechanical behaviours: a strong, velocity-weakening region (asperity) and a weak, velocity-strengthening region. The two regions may have different areas and seismic wave radiation associated with asperity slip is considered. We base on a model developed by Dragoni and Santini (2015) and Dragoni and Tallarico (2016), where the average values of stress, friction and slip are considered for each region, so that the fault is described as a discrete dynamical system (Ruff 1992; Rice 1993; Turcotte 1997). Such an approach has the advantage of reducing the number of degrees of freedom required to describe the dynamics of the fault; furthermore, it allows the study of the evolution of the system by means of orbits in the phase space, making it possible to better visualize the different aspects of the dynamics. On the whole, models with a finite number of degrees of freedom allow to focus on the main features of the seismic source, while avoiding the more complicated characterization based on continuum mechanics.

We describe the different stages of fault behaviour, including the interaction between the weak region and the asperity responsible for earthquakes. During the interseismic intervals, the asperity accumulates stress that is released by asperity slip in seismic events. When the asperity slips, it transfers stress to the weak region, triggering the afterslip. In the same way, we expect that afterslip transfers stress back to the asperity, thus changing its state and subsequent evolution. An analytical solution for the evolution equations of the system will be obtained.

As an example, we apply the model to the fault of the 2011 Tohoku-Oki earthquake. The moment rate function associated with this event was dominated by one single hump (Wei et al. 2012), making it appropriate to ascribe the seismic slip to a large unstable region on the fault plane; what is more, this earthquake was followed by a prolonged afterslip episode (Ozawa et al. 2011). Observations show that seismic slip concentrated in a compact area at shallow depth, while afterslip occurred on a similar area downslip (Lay et al. 2012; Silverii et al. 2014). A part of postseismic deformation was certainly due to viscoelastic relaxation in the asthenosphere (Sun et al. 2014; Yamagiwa et al. 2015), and an attempt will be made to discriminate between the two mechanisms.

## The model

We consider a plane fault embedded in a shear zone between two tectonic plates moving at constant relative velocity *v*. The shear zone is assumed to be a homogeneous and isotropic Hooke solid with rigidity \(\mu\). As a consequence of plate motion, the fault is subject to a tangential strain rate \({\dot{e}}\).

The two regions have areas \(A_1\) and \(A_2\) , respectively and the distance between their centres is *a* (Fig. 1). They are allowed to slip in the direction of the tangential traction imposed by the motion of tectonic plates. Their slip is controlled by their constitutive equations and by the forces exerted by the surrounding medium. We introduce a discrete fault model where the average values of stress, friction and slip on each region are considered. The state of the fault is described by two variables: the slip deficits *x*(*t*) and *y*(*t*) of the asperity and of the weak region, respectively, as functions of time *t*. At a certain instant *t* in time, the slip deficit of a fault region is defined as the slip that such region should undergo in order to recover the relative plate displacement occurred up to time *t*.

*t*. The force \(f_1\) applied to the asperity includes three terms: the elastic force due to plate motion, the elastic force due to a difference in the slip deficits of the two regions and a rate-dependent force due to radiation damping (e.g. Rice 1993). It can be written as

*s*is the average tangential traction (per unit moment) that the dislocation of one region imposes to the other one. According to (3) and (4), the forces applied to each region depend on

*x*and

*y*, so that a change in

*x*or

*y*entails a change in both forces, hence a change in the stress distribution on the fault. The interaction between the two regions is controlled by the constant \(K_c\).

*x*and

*y*express the relative motion of fault walls. The effect of gravity on the motion of one wall is opposite to the effect on the other one. There is not exact compensation, but this suffices to render gravity a second-order effect that can be ignored in a first approximation. Neglecting gravity has the effect to slightly overestimate or underestimate dynamic friction. As to friction, we use simplified versions of the general rate and state-dependent law. For the asperity, we assume a velocity-weakening law, characterized by a static friction \(f_{\mathrm{s}}\) and an average dynamic friction \(f_{\mathrm{d}}\):

- 1.During interseismic intervals, \(f_1 < f_{\mathrm{s}}\) by definition, so that the asperity is stationary. However, its slip deficit
*x*increases due to tectonic loading. As to the weak region, we allow a steady-state creep at constant stress, so that its slip deficit*y*increases with time, but slower than*x*. Equations (9) and (10) become$$\begin{aligned} {\ddot{x}}= 0 \end{aligned}$$(11)$$\begin{aligned} (K_2 + K_c) y - K_c x= f_0 \end{aligned}$$(12) - 2.During asperity slip, both tectonic loading and steady-state creep can be neglected, because they give negligible contributions in such a short time. Equations (9) and (10) become$$\begin{aligned}&\mu _1 {\ddot{x}} + \varGamma {\dot{x}} + (K_1 + K_c) x - K_c y - f_{\mathrm{d}} = 0 \end{aligned}$$(13)$$\begin{aligned}&{\dot{y}} = 0 \end{aligned}$$(14)
- 3.During afterslip in the weak region, the asperity is stationary. Afterslip has a much shorter duration than typical interseismic intervals; therefore, we neglect tectonic loading also in this case. Equations (9) and (10) become$$\begin{aligned}&{\dot{x}} = 0 \end{aligned}$$(15)$$\begin{aligned}&\mu _2 {\ddot{y}} + \varLambda {\dot{y}} + (K_2 + K_c) y - K_c x - f_0 = 0 \end{aligned}$$(16)

## Solutions

*V*is the nondimensional velocity of tectonic plates. The parameter values have the following ranges: \(\alpha \ge 0, 0< \beta< 1, \gamma \ge 0, \lambda> 0, 0<\epsilon <1, \xi>0, V>0.\) We also assume that the masses associated with the two regions are proportional to their areas: accordingly,

### Interseismic intervals

*T*. With initial conditions

*X*of the asperity increases in time with the velocity

*V*of tectonic plates, so that the asperity is stationary. The slip deficit

*Y*of the weak region also increases in time, but with a lower rate

### Seismic slip

*U*. From (49) and (44), we obtain

### Afterslip

*T*, we can write to a very good approximation

### Subsequent evolution

## Discussion

In order to test the model, it is necessary to assign appropriate values to the parameters. They will be chosen within the allowed ranges, but do not pertain to any real case. A different set of parameters would imply different values of the quantities characterizing the seismic cycle, but would not alter the general conclusions of this section.

As an example, we consider a fault zone with rigidity \(\mu =30\) GPa, in the presence of a relative plate velocity \(v = 3\) cm a\(^{-1}\) and a tangential strain rate \({\dot{e}} = 10^{-14}\, \hbox {s}^{-1}\). We consider a medium-size earthquake produced by a fault with \(A_1 = A_2 = 500 \,\mathrm{km^2}, a = 25\) km and a strike-slip mechanism. We assume that the seismic event has a duration \(t_{\mathrm{s}} = 10\) s and a slip amplitude \(u_{\mathrm{s}} = 1\) m. The seismic moment is then \(m_{\mathrm{s}} = 1.5 \times 10^{19}\) N m.

It is evident that the value of the distance *a* may have a strong influence on \(\alpha\). In fact, if we increase \(a, \alpha\) becomes smaller and smaller. But we cannot decrease *a* below a certain value without decreasing the area \(A_2\); otherwise, the two regions would overlap. It follows that \(\alpha\) is always much smaller than 1 (in the present example, \(\alpha =0.2\)). From the solutions of the governing equations, it can be seen that \(\alpha\) is always summed to unity or to the parameter \(\xi\), which is in the order of unity. Therefore, the results are moderately sensible to the value of \(\alpha\) and the sensitivity decreases with decreasing \(\alpha\). The parameter \(\beta\) is by definition smaller than 1. Since asperity models assume that weak regions may slip at a much lower stress level than asperities, a value \(\beta\) = 0.1 is reasonable, meaning that the force on the weak region is one-tenth of the force on the asperity at the onset of seismic slip. In applications to real cases, the value can be chosen in order that the model gives the best fit with observations. The parameter \(\epsilon\) is by definition smaller than 1. Experiments on rock samples suggest values ranging between 0.5 and 1 (Jaeger and Cook 1976). We take \(\epsilon =0.7\). Small variations do not have important consequences.

*V*can be calculated from the observed plate velocity

*v*, the duration \(t_{\mathrm{s}}\) and the slip amplitude \(u_{\mathrm{s}}\) of the seismic event. From the definitions, we obtain

With these values of the model parameters, Fig. 4 shows the main features of the seismic event. The slip amplitude \(\Delta X\), the moment rate \(\dot{M}_{\mathrm{s}}\) and the forces \(F_1\) and \(F_2\) are shown as functions of time in the interval \(0 \le T \le T_{\mathrm{s}}\). The event duration is \(T_{\mathrm{s}} \simeq 3.2\), with a final slip amplitude \(U_{\mathrm{s}} =0.30\) and a seismic moment \(M_{\mathrm{s}}/M_1=0.60\). In Fig. 4c, we note a decrease (in magnitude) of the force \(F_1\) on the asperity and a corresponding increase of the force \(F_2\) on the weak zone. At the end of the event, the force drop on the asperity is \((1+\alpha )U_{\mathrm{s}} =0.36\), while \(F_2\) has received a contribution \(-\,\alpha U_{\mathrm{s}} = -\,0.06\) from the asperity, reaching the value \(-\,0.16\).

The sequence of three dynamic modes (seismic slip, afterslip, interseismic evolution) makes a cycle that can be represented in the plane *XY* (Fig. 6). The coordinates of points \(P_1, P_2, P_3\) are given in (40), (57) and (72), respectively. During the interseismic interval, the representative point of the system moves on line (39). When it reaches \(P_1\), the seismic event takes place. The point moves by a quantity \(U_{\mathrm{s}}\) given by (50), reducing the value of *X* and reaching \(P_2\). Here, afterslip begins and lowers the value of *Y* by a quantity \(U_{\mathrm{a}}\) given by (71), driving the point to \(P_3\): this is again on line (39) and a new interseismic interval begins. The orbit is independent of \(\lambda\).

Of course, this cycle represents an ideal situation. There are several factors that may change this scheme. Often the seismic source is not made of a single asperity, but of two or more asperities that introduce complexity in the dynamics (Dragoni and Tallarico 2016). Secondly, the fault is not an isolated system, but is subject to stress perturbations from the slipping of neighbouring faults (Dragoni and Piombo 2015). These complexities may be introduced without difficulty in the model and would break the periodicity of the cycle.

*b*is a constant and \(\tau\) is a characteristic time (Scholz 1990; Marone et al. 1991; Heki et al. 1997; Barbot et al. 2009). This function becomes arbitrarily large for \(t\rightarrow \infty\), even though its derivative tends to zero. In many cases, it fits reasonably well the postseismic deformation data in finite time intervals. On the contrary, (99) yields an afterslip approaching a maximum value \(U_{\mathrm{a}}\) and the surface displacement would be

## An application: the 2011 Tohoku-Oki earthquake

Data for the 2011 Tohoku-Oki earthquake

| 40 GPa | Rigidity of the lithosphere |

| \(8\hbox { cm} \hbox { a}^{-1}\) | Relative plate velocity |

\({\dot{e}}\) | \(4 \times 10^{-15}\hbox { s}^{-1}\) | Tangential strain rate on the fault |

\(\delta\) | \(20^\circ\) | Average dip angle of the fault |

\(A_1\) | \(6 \times 10^4\hbox { km}^2\) | Area of the asperity |

\(A_2\) | \(6 \times 10^4\hbox { km}^2\) | Area of the weak region |

| 150 km | Distance between the centres of the two regions |

\(m_{\mathrm{s}}\) | \(3.5 \times 10^{22}\) N m | Seismic moment of the event |

\(u_{\mathrm{s}}\) | 15 m | Average slip of the asperity |

\(t_{\mathrm{s}}\) | 80 s | Duration of seismic slip |

\(t_{\mathrm{a}}\) | 15 days | Assumed duration of afterslip |

The duration of the seismic event was about 160 s, with a moment rate concentrated in a time interval \(t_{\mathrm{s}}=80\) s and a seismic moment \(m_{\mathrm{s}} = 3.5 \times 10^{22}\) N m (Maercklin et al. 2012; Wei et al. 2012; Bletery et al. 2014). Accordingly, the average seismic slip was \(u_{\mathrm{s}} = 15\) m.

*V*. The value of \(\epsilon\) is taken as in the previous section. The value of \(\gamma\), corresponding to a seismic efficiency \(\eta = 0.17\), is chosen in order to obtain the best fit for the moment rate. The value of \(\lambda\) will be evaluated on the basis of the assumed afterslip duration. The values are given in Table 2. Notice that we aim to investigate only the seismic slip and afterslip phases associated with the event, whose evolutions are independent of \(\beta\) (as shown in “Seismic slip” and “Afterslip” sections); therefore, we do not need to assign a value to this parameter.

Values of model parameters for the 2011 Tohoku-Oki fault

\(\alpha\) | 0.3 | Coupling parameter between the two fault regions |

\(\gamma\) | 1.5 | A measure of seismic efficiency |

\(\epsilon\) | 0.7 | Ratio between dynamic and static frictions on the asperity |

\(\lambda\) | \(10^5\) | A measure of intensity of velocity strengthening |

\(\xi\) | 1 | Ratio between the areas of the two regions |

| \(10^{-9}\) | Nondimensional plate velocity |

*u*is the slip that would be observed in the case \(\gamma =0\). Hence,

The ground displacement produced by afterslip has been calculated making use of Okada’s (1985) formulae. The graphs of the horizontal and vertical displacement components are shown in Fig. 10. The direction and magnitude of the calculated displacement are broadly comparable with displacements obtained from GPS data over a time interval comparable with \(t_{\mathrm{a}}\). For instance, Silverii et al. (2014) reported a maximum horizontal displacement of the order of 1 m at the eastern coasts of the Iwate/Miyagi prefectures of Japan and a maximum vertical displacement of about 20 cm in the same area. These figures are in good agreement with the results shown in Fig. 10, where the eastern coasts of the Iwate/Miyagi prefectures approximately correspond with the projection of the lower margin of the weak fault region on the Earth’s surface.

Of course, the present model cannot reproduce the details of the Tohoku-Oki earthquake nor of any other seismic event, but its aim is rather to investigate the basic mechanical processes occurring on the fault surface and to enlighten the relationships between them.

## Conclusions

We presented a model that describes in a unique frame both seismic and aseismic slip on a fault, taking into account the interaction between the two processes. This has been achieved by considering a fault containing two regions with different mechanical behaviours: a strong, velocity-weakening region (asperity) and a weak, velocity-strengthening region.

During the interseismic intervals, the asperity is locked, while the weak region is subject to a very slow creep. Consequently, the slip deficit of the asperity increases in time with the velocity of tectonic plates, while the slip deficit of the weak region increases much slower. The asperity accumulates stress and eventually releases it, producing an earthquake, when a frictional threshold is exceeded. As a consequence of coseismic stress imposed by the asperity failure, the weak region is then subject to aseismic slip (afterslip).

The evolution equations of the system have been solved analytically for the interseismic intervals, the asperity slip and the afterslip in the weak region. The amount of afterslip is found to be proportional to the seismic slip of the asperity, in agreement with observations. The proportionality factor depends on the geometry of the fault and on the velocity of tectonic motion.

The model shows that afterslip is a natural consequence of seismic slip in a fault containing a velocity-strengthening region. Afterslip may have any duration, according to the intensity of velocity strengthening, thus accounting for the wide range of observed durations. According to the model, afterslip approaches an asymptotic value as time increases. We have shown that the higher rate that is often observed in postseismic deformation may be due to the superposition of the effects of afterslip and viscoelastic relaxation in the asthenosphere.

The model has been applied to the fault of the 2011 Tohoku-Oki earthquake. On the basis of data, the fault has been considered as made of a single large asperity, extending from the Japan trench to about 50 km of depth, and a weak region located downdip of the asperity to a depth of about 100 km. With the appropriate values of the model parameters, the dominant part of the seismic moment rate has been reproduced. The stress transfer from the slipping asperity produces a substantial increase of shear stress on the weak region, which is responsible for afterslip. The results suggest that the first four months after the event were dominated by afterslip, while the subsequent postseismic deformation was probably due to viscoelastic relaxation in the asthenosphere. Of course, the model presents a simplified description of real fault dynamics. It necessarily disregards complicated physical processes that may be dealt with via a numerical approach. However, a discrete model as the one we presented offers an alternative insight on the essential dynamics of the seismic source in an analytical framework.

## Declarations

### Authors’ contributions

MD developed the model, solved the equations and wrote a preliminary version of the paper; EL checked the solutions and gave further contributions to the text and to data collection. Both authors discussed extensively the results. Both authors read and approved the final manuscript.

### Acknowlegements

The authors are grateful to the editor Ryosuke Ando, to Suguru Yabe and to anonymous referees for useful comments on the first version of the paper.

### Competing interests

The authors declare that they have no competing interests.

### Availability of data and materials

All data and results supporting this work were gathered from the papers listed in References and are freely available to the public.

### Consent for publication

Not applicable.

### Ethics approval and consent to participate

Not applicable.

### Funding

Not applicable.

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## Authors’ Affiliations

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