the Creative Commons Attribution 4.0 License.

the Creative Commons Attribution 4.0 License.

# A study of earthquake recurrence based on a one-body spring-slider model in the presence of thermal-pressurized slip-weakening friction and viscosity

### Jeen-Hwa Wang

Earthquake recurrence is studied from the temporal variation in slip through
numerical simulations based on the normalized form of equation of motion of a
one-body spring-slider model with thermal-pressurized slip-weakening friction
and viscosity. The wear process, whose effect is included in the friction
law, is also taken into account in this study. The main parameters are the
normalized characteristic displacement, *U*_{c}, of the friction law
and the normalized damping coefficient (to represent viscosity), *η*.
*T*_{R}, *D*, and *τ*_{D} are the recurrence time
of events, the final slip of an event, and the duration time of an event,
respectively. Simulation results show that *T*_{R} increases when
*U*_{c} decreases or *η* increases, *D* and *τ*_{D}
decrease with increasing *η*, and *τ*_{D} increases with
*U*_{c}. The time- and slip-predictable model can describe the
temporal variation in cumulative slip. When the wear process is considered,
the thickness of slip zone, *h*, which depends on the cumulated slip,
*S*(*t*) = ∑*D*(*t*), i.e., *h*(*t*) = *C**S*(*t*) (*C* is a
dimensionless increasing rate of *h* with *S*), is an important parameter
influencing *T*_{R} and *D*. *U*_{c} is a function of *h* and
thus depends on cumulated normalized slip, ∑*U*, with an increasing rate of *C*. In
the computational time period, the wear process influences the recurrence of
events and such an effect increases with *C* when *C* > 0.0001. When
viscosity is present, the effect due to wear process becomes stronger. Both
*T*_{R} and *D* decrease when the fault becomes more mature, thus
suggesting that it is more difficult to produce large earthquakes along a
fault when it becomes more mature. Neither the time-predictable nor the
slip-predictable model can describe the temporal variation in cumulative slip
of earthquakes under the wear process with large *C*.

Earthquake recurrence that is relevant to the physics of faulting is an
important factor in seismic hazard assessment. It is related to the seismic
cycle, which represents the occurrence of several earthquakes in the same
segment of a fault during a time period. Figure 1 exhibits the general pattern
of time variation in slip and particle velocity during a seismic cycle. In
the figure, *T*_{R} is the recurrence (also denoted by repeat or inter-event)
time of two events in a seismic cycle, *τ*_{D} is the duration time of
slip of an event, *D* is the final slip of an event, and *V*_{m} is the peak
value of particle velocity of an event. The four parameters could
be constants in the time history when all model parameters do not vary with
time and could also vary with time, represented by *T*_{R}(*t*),
*τ*_{D}(*t*), *D*(*t*), and *V*_{m}(*t*), when one of model parameters does vary with time.
Sykes and Quittmeyer (1981) pointed out that the major factors in
controlling *T*_{R} are the plate moving speed and the geometry of the
rupture zone. Based on Reid's elastic rebound theory (Reid, 1910), Schwartz
and Coppersmith (1984) assumed that an earthquake occurs when the tectonic
shear stress on a fault is higher than a critical level, which is dependent
on the physical conditions of the fault and the loading by regional
tectonics. Since in their work a fault has a homogeneous distribution of
physical properties under constant tectonic loading, earthquakes could happen regularly.

Some observations exhibit periodicity for different size earthquakes. Bakun
and McEvilly (1979) obtained *T*_{R} ≈ 23 ± 9 years for
*M* ≈ 6 earthquakes at the Parkfield segment of the San Andreas fault, USA
since 1857. Sykes and Menke (2006) estimated *T*_{R} ≈ 100 years for
*M* ≥ 8 earthquakes in the Nankaido segments of the Nankai Trough, Japan. Okada et
al. (2003) gained *T*_{R} = 5.5 ± 0.7 years for earthquakes with
*M* = 4.8 ± 0.1 off Kamaishi, Japan, since 1957. Nadeau and Johnson (1998)
inferred an empirical relation between *T*_{R} and seismic moment, *M*_{o}:
*T*_{R} ∝ ${M}_{\mathrm{o}}^{\mathrm{1}/\mathrm{6}}$. To make this relation valid, the stress
drop, Δ*σ*, or the long-term slip velocity of a fault,
*v*_{l}, must be in terms of *M*_{o}. Based on three data set from eastern
Taiwan, Parkfield, USA, and northeastern Japan, Chen et al. (2007) inferred
*T*_{R} ∼ ${M}_{\mathrm{o}}^{\mathrm{0.61}}$. Beeler et al. (2001) proposed a theoretical relation:
*T*_{R} = $\mathrm{\Delta}{\mathit{\sigma}}^{\mathrm{2}/\mathrm{3}}{M}_{\mathrm{o}}^{\mathrm{1}/\mathrm{3}}/\mathrm{1.81}\mathit{\mu}{v}_{\mathrm{l}}$, where
*μ* is the rigidity of the fault-zone materials.

However, the main factors in influencing earthquake occurrences commonly are
spatially heterogeneous and also vary with time. Thus, the recurrence times
of earthquakes, especially large events, are not constant inferred either
from observations (Ando, 1975; Sieh, 1981; Kanamori and Allen, 1986; Wang
and Kuo, 1998; Wang, 2005; Sieh et al., 2008) or from modeling (Wang, 1995,
1996; Ward, 1996, 2000; Wang and Hwang, 2001). Kanamori and Allen (1986)
observed that faults with longer *T*_{R} are stronger than those with shorter
*T*_{R}. Davies et al. (1989) proposed that the longer it has been since the
last earthquake, the longer the expected time until the next. Wang and Kuo (1998)
observed that for *M* ≥ 7 earthquakes in Taiwan *T*_{R} strongly
follows the Poissonian processes. Enescu et al. (2008) found that the
probability density distribution of *T*_{R} can be described by an
exponential function. From the estimated values of *T*_{R} of earthquakes
happened on the Chelungpu fault in central Taiwan from trenching data, Wang
(2005) found that the earthquakes occurred non-periodically.

In order to interpret earthquake recurrences, Shimazaki and Nakata (1980)
proposed three simple phenomenological models. Each model has a constantly
increasing tectonic stress that is controlled by a critical stress level,
*σ*_{c}, for failure and a base stress level, *σ*_{b}. The
three models are (1) the perfectly periodic model (with constant *σ*_{c},
*σ*_{b}, and Δ*σ*); (2) the time-predictable model
(with constant *σ*_{c}, variable *σ*_{b}, and variable
Δ*σ*); and (3) the slip-predictable model (with variable
*σ*_{c}, constant *σ*_{b}, and variable Δ*σ*).
For the first model, both *T*_{R} and *D* of next earthquake can be predicted from
the values of *T*_{R} or *D* of previous ones. For the second model, *T*_{R} of
next earthquake can be predicted from the values of *D* of previous ones. For
the third model, *D* of next earthquake can be predicted from the values of
*T*_{R} of previous ones. However, debates about the three models have been
made for a long time. Some examples are given below. Ando (1975) suggested
that the second model worked for post-1707 events but not for pre-1707 ones
in the Nankai Trough, Japan. Wang (2005) assumed that the second model could
describe the earthquakes occurred on the Chelungpu fault, Taiwan, in the past
1900 years. For the Parkfield earthquake sequence, Bakun and McEvilly (1984)
took different models, while Murray and Segall (2002) considered the failure
of the second model. From laboratory results, Rubinstein et al. (2012)
assumed the failure of the time- and slip-predictable models for earthquakes.

Some models, for instance the crack model and dynamical spring-slider model, have been developed for fault dynamics, even though the seismologists have not a comprehensive model. There are several factors in controlling fault dynamics and earthquake ruptures (see Bizzarri, 2009; Wang, 2017b). Among the factors, friction (Nur, 1978; Belardinelli and Belardinelli, 1996) and viscosity (Jeffreys, 1942; Spray, 1993; Wang, 2007) are two significant ones.

Modeling earthquake recurrence based on different models has been long made
and is reviewed by Bizzarri (2012a, b) and Franović et al. (2016). Among
the models, the spring-slider model has been used to study fault dynamics
and earthquake physics (see Wang, 2008). Burridge and Knopoff (1967) proposed
the one-dimensional *N*-body model (abbreviated as the 1-D BK model
henceforth). Wang (2000, 2012) extended the 1-D model to 2-D one. The one-,
two-, three-, and few-body models with various friction laws have also been
applied to approach fault dynamics (see Turcotte, 1992). The studies for
various friction laws based on spring-slider models are briefly described
below: (1) for rate- and state-dependent friction (e.g., Rice and Tse, 1986;
Ryabov and Ito, 2001; Erickson et al., 2008, 2011; He et al., 2003; Mitsui
and Hirahara, 2009; Bizzarri, 2012a; Abe and Kato, 2013; Kostić et al.,
2013a; Bizzarri and Crupi, 2014; Franović et al., 2016); (2) for
velocity-weakening friction (e.g., Carlson and Langer, 1989; Huang and
Turcotte, 1992; Brun and Gomez, 1994; Wang and Hwang, 2001; Wang, 2003;
Kostić et al., 2013b); (3) for simple static–dynamic friction (e.g.,
Abaimov et al., 2007; Hasumi, 2007).

Some results concerning earthquake recurrence are simply explained below.
Erickson et al. (2008) suggested that aperiodicity in earthquake dynamics is
due to either the nonlinear friction law (Huang and Turcotte, 1990) or the
heterogeneous stress distribution (Lapusta and Rice, 2003). Wang and Hwang (2001)
emphasized the importance of heterogeneous frictional strengths.
Mitsui and Hirahara (2009) pointed out the effect of thermal pressurization.
Dragoni and Piombo (2011) found that variable strain rate causes
aperiodicity of earthquakes. Bizzarri and Crupi (2014) found that *T*_{R} is
dependent on the loading rate, effective normal stress, and characteristic
distance of the rate- and state-dependent friction law.

As mentioned previously, numerous studies have been made for exploring the frictional effect on earthquake recurrence. However, studies concerning the viscous effect on earthquake recurrence are rare. In the following, we will investigate the effects of slip-weakening friction due to thermal-pressurization and viscosity on earthquake recurrence based on the one-body spring-slider model.

Figure 2 displays the one-body spring-slider model. In the model, *m*, *K*, *σ*_{n}, *F*,
*η*, *u*, *v* (= d*u*∕d*t*), *v*_{p}, and *u*_{o} = *v*_{p}*t* denote, respectively, the
mass of the slider, the stiffness (or spring constant) of the leaf spring,
the normal stress, the frictional force between the slider and the moving
plate, the damping coefficient (to represent viscosity as explained below),
the displacement of the slider, the velocity of the slider, the plate moving
speed, and the equilibrium location of the slider. The frictional force *F*
(with the static value of *F*_{o}) is usually a function of *u* or *v*. Viscosity
results in the viscous force, Φ, between the slider and the moving
plate, and Φ is in terms of *v*. A driving force, *K**v*_{p}*t*, caused by the
moving plate through the leaf spring pulls the slider to move. The equation
of motion of the model is

When *K**v*_{p}*t* ≥ *F*_{o}, *F* changes from static frictional force to dynamic
one and thus makes the slider move. Among the physical models to approach
earthquake faults, the single spring-slider model, which can represent a
single fault, is actually the simplest one. However, based on this simple
model in the presence of thermal-pressurized friction and viscosity we can
obtain good simulations of earthquake recurrences along a single fault.
Results can exhibit the frictional and viscous effects on earthquake recurrence.

The frictional force *F*(*u*, *v*) is controlled by several factors (see Wang, 2016, and
cited references therein). An effect combined from temperature and fluids in
a fault zone can result in thermal pressurization (abbreviated as TP hereafter)
which would yield a shear stress (resistance) on the fault plane (Sibson,
1973; Lachenbruch, 1980; Rice, 2006; Wang, 2009, 2011, 2016, 2017a–c;
Bizzarri, 2009). Rice (2006) proposed two end-member models of TP, i.e., the
adiabatic undrained deformation (AUD) model and slip-on-a-plane (SOP) model.
Since the characteristic distance of the SOP model cannot be associated with
the wear process, the SOP model is not used in this study. The AUD model is
related to a homogeneous simple strain *ε* at a constant normal
stress *σ*_{n} on a spatial scale of the sheared layer. Its shear
stress–slip function, *τ*(*u*), is *τ*(*u*) = *μ*_{f}(*σ*_{n} − ${p}_{\mathrm{o}}\left)\mathrm{exp}\right(-u/{u}_{\mathrm{c}}$)
(Rice, 2006), which decreases exponentially with
increasing *u*. The characteristic displacement is *u*_{c} = *ρ*_{f}*C*_{v}*h*∕*μ*_{f}Λ,
where *ρ*_{f}, *C*_{v}, *h*, *μ*_{f}, and Λ
are, respectively, the fluid density, heat capacity
(in J ^{∘}C^{−1} kg^{−1}), the thickness, frictional strength, and the undrained
pressurization factor of the fault zone. The parameter Λ is
(*λ*_{f} − *λ*_{n})∕(*β*_{f} + *β*_{n}), where
*β*_{f} is isothermal compressibility of the pore fluid,
*β*_{n} is isothermal compressibility of the pore space,
*λ*_{f} is isobaric volumetric thermal expansion coefficient for the pore
fluid, and *λ*_{n} is isobaric volumetric thermal expansion
coefficient for the pore space.

Based on the AUD model, Wang (2009, 2016, 2017a–c) took a simplified slip-weakening friction law (denoted by the TP law hereafter):

The value of *F*(*u*) at *u* = 0 is *F*_{o}, i.e., the static friction force. An example
of the plot of *F*(*u*) versus *u* for five values of *u*_{c}, i.e., 0.1, 0.3, 0.5, 0.7,
and 0.9 m when *F*_{o} = 1*N* m^{−2}, which are taken from Wang (2016), is
displayed in Fig. 3. *F*(*u*) decreases with increasing *u* and its decreasing rate,
*γ*, decreases with increasing *u*_{c}. The force drop is lower for
larger *u*_{c} than for smaller *u*_{c} for the same final slip. When
*u* ≪ *u*_{c}, $\mathrm{exp}(-u/{u}_{\mathrm{c}})$ ≈ 1 − *u*∕*u*_{c}, thus indicating that
${u}_{\mathrm{c}}^{-\mathrm{1}}$ is almost *γ* at small *u*. This TP law is used in this study.

A detailed description about viscosity and the viscous force Φ(*v*) can be
found in Wang (2016), and only a brief explanation is given below. Jeffreys (1942)
first and then numerous authors (Byerlee, 1968; Turcotte and
Schubert, 1982; Scholz, 1990; Rice et al., 2001; Wang, 2016) emphasized the
viscous effect on faulting due to frictional melts. The viscosity
coefficient, *υ*, of rocks is influenced by *T* (see Turcotte and
Schubert, 1982; Wang, 2011). Spray (2005, and cited references therein)
observed a decrease in *υ* with increasing *T*. He also stressed that
frictional melts with low *υ* could produce a large volume of
melting, thus reducing the effective normal stress. This behaves like fault
lubricants during seismic slip.

The physical models of viscosity can be found in several articles (e.g.,
Cohen, 1979; Hudson, 1980). The stress–strain relationship is
*σ* = *E**ε*, where *σ* and *E* are, respectively, the stress and
the elastic modulus for an elastic body; and *σ* = *υ*(d*ε*∕d*t*),
where *υ* is the viscosity coefficient for a
viscous body. Two simple models with a viscous damper and an elastic spring
are often used to describe the viscous materials. A viscous damper and an
elastic spring are connected in series, leading to the Maxwell model, and in
parallel, resulting in the Kelvin–Voigt model (or the Voigt model). According
to Hudson (1980), Wang (2016) proposed that the latter is more suitable than
the former for seismological problems and thus the Kelvin–Voigt model, whose
constitution law is *σ*(*t*) = *E**ε*(*t*) + *υ*d*ε*(*t*)∕d*t*,
is taken here and displayed in Fig. 2. The viscous stress is *υ**v*.

In order to investigate the viscous effect in a dynamical system, Wang (2016)
defined the damping coefficient, *η*, based on the phenomenon
that an oscillating body damps in viscous fluids. According to Stokes' law,
*η* = 6*π**R**υ* for a sphere of radius *R* in a viscous fluid of
*υ* (see Kittel et al., 1968). Hence, the viscous force in Eq. (1) is represented by Φ = *η**v*. Note that the unit of *η* is
N(m/s)^{−1}. Since *υ* decreases with increasing *T*, *η*
decreases with increasing *T*. Hence, *η* can vary with time during
faulting. This point has been studied by Wang (2017b) for the generation of
nuclear phase before an earthquake ruptures. In this study, constant *η*
is considered for each case.

Some authors (Knopoff et al., 1973; Cohen, 1979; Rice, 1993; Xu and Knopoff, 1994; Knopoff and Ni, 2001; Bizzarri, 2012a; Dragoni and Santini, 2015) considered that viscosity plays a role on causing seismic radiation to release strain energy during faulting.

Putting Eq. (2) and Φ = *η**v* into Eq. (1) leads to

Equation (3) is normalized for easy numerical computations based on the
normalization parameters, which is dimensionless:
*D*_{o} = *F*_{o}∕*K*, *ω*_{o} = $(K/m{)}^{\mathrm{1}/\mathrm{2}}$, *τ* = *ω*_{o}*t*, *U* = *u*∕*D*_{o},
and *U*_{c} = *u*_{c}∕*D*_{o}. The normalized velocity, acceleration, and
driving velocity are *V* = d*U*∕d*τ* = $[{F}_{\mathrm{o}}/(mK{)}^{\mathrm{1}/\mathrm{2}}{]}^{-\mathrm{1}}$d*u*∕d*t*,
*A* = d^{2}*U*∕d*τ*^{2} = $({F}_{\mathrm{o}}/m{)}^{-\mathrm{1}}$d^{2}*u*∕d*t*^{2}, and
*V*_{p} = *v*_{p}∕(*D*_{o}*ω*_{o}), respectively.
Ω = *ω*∕*ω*_{o} is defined as the dimensionless angular frequency, and
thus the phase *ω**t* becomes Ω*τ*. For the purpose of
simplification, $\mathit{\eta}/(mK{)}^{\mathrm{1}/\mathrm{2}}$ is denoted by *η* below. Substituting
all normalization parameters into Eq. (3) leads to

In order to numerically solve Eq. (4), we define two new parameters, i.e.,
*y*_{1} = *U* and *y*_{2} = d*U*∕d*τ*. Equation (4) can be rewritten
as two first-order differential equations:

We can numerically solve Eq. (5) by using the fourth-order Runge–Kutta method
(Press et al., 1986). In general, the values of *D*_{o} are several meters
and *ω*_{o} are in the range of 0.1 Hz to few Hz (see Wang, 2016).
This leads to that *D*_{o}*ω*_{o} has an order of magnitude of 1 m s^{−1}.
The value of *V*_{p} must be much smaller than 1 because of
*v*_{p} ≈ 10^{−10} m s^{−1}. Since the value of *V*_{p} mainly influences the recurrence
time, *T*_{R}, between two events and can only make a very small influence on
the pattern of time variations in velocities and displacements of events. In
order to study long-term earthquake recurrence, there must be numerous
modeled events with clear and visualized time functions of displacements and
velocities for an event in the computational time period. If
*V*_{p} = 10^{−10} is considered, *T*_{R} is very long and thus
*τ*_{D} is much shorter than *T*_{R}. This makes the time function of an
event displayed in the long-term temporal variation in slip looks like just
a step function for the displacements and an impulse for the velocities.
Hence, in order to get fine visualization a larger value of *V*_{p} is
necessary. The value of *V*_{p}*τ* is usually very small during an event
and cannot influence the rupture because of a very tiny value of *V*_{p}.
Numerical test shows that when *V*_{p} > 10^{−2}, the value of
*V*_{p}*τ* is not small during an event and can influence the rupture.
Hence, *V*_{p} = 10^{−2} is taken in this study. The backward slip is not
allowed in the simulations because of common behavior of forward earthquake ruptures.

A phase portrait, which is a plot of a physical quantity, *y*, versus another,
*x*, i.e., *y* = *f*(*x*), is commonly used to represent nonlinear behavior of a dynamical
system (Thompson and Stewart, 1986). The intersection point between *f*(*x*) and the
bisection line of *y* = *x* is defined as the fixed point, that is, *f*(*x*) = *x*. If
*f*(*x*) is continuously differentiable in an open domain near a fixed point
*x*_{f} and $\left|{f}^{\prime}\right({x}_{\mathrm{f}}\left)\right|$ < 1, attraction can appear at the fixed point.
Chaos can also be generated at some attractors. The details can be seen in
Thompson and Stewart (1986). In this study, the phase portrait is the plot
of *V*∕*V*_{max} versus *U*∕*U*_{max}.

Numerical simulations lead to the temporal variations in particle velocities
and displacements as displayed in Fig. 1. The values of *V*_{m} and *D*,
respectively, represent the peak value of velocity and final slip for each
event. Since four cases related to our values of a particular model
parameter, there are four values of *V*_{m} and *D* in a figure. In order to
plot the temporal variations in both normalized displacements and
velocities, the maximum values of *V*_{m} and *D* of the modeled events, i.e.,
*V*_{max} and *U*_{max}, respectively, are taken into account. The values of
*V*_{m} and *D* usually appear in panel (a) of a figure.
Simulation results are shown in Figs. 4–12. The temporal variations in
*V*∕*V*_{max} (displayed by thin solid lines) and cumulative slip
Σ*U*∕*U*_{max} (displayed by solid lines) are displayed in the left
panels. The normalization scales to plot the temporal variations in slip and
velocity are *V*_{max} for the velocities and the final value of
Σ*U*∕*U*_{max} for the displacements in the computational time. Hence, the upper-bound scale is “1” for the two temporal variations. Hence, only the
patterns of temporal variations of velocity and cumulative slip are
concerned in these figures.

Simulation results displayed in these figures show that the maximum values
of both *V* and *U* decrease from case (a) to (d) in each figure. Hence, the
maximum velocity and maximum displacement, which are denoted by *V*_{max} and
*U*_{max}, respectively, for case (a) can be taken as the scaled factor to
normalize the waveforms from case (a) to (d). This makes it easy to
compare the waveforms of the four cases in each figure.

The cases excluding the viscous effect, i.e., *η* = 0, are first
simulated and results are shown in Fig. 4 for four values of *U*_{c}: (a) for
*U*_{c} = 0.2; (b) for *U*_{c} = 0.4; (c) for *U*_{c} = 0.8; and (d) for
*U*_{c} = 1.0. The results of the cases including viscosity, i.e.,
*η* ≠ 0, are displayed in Figs. 5–7 for four values of *η*: (a) for
*η* = 0.20; (b) for *η* = 0.40; (c) for *η* = 0.6; and (d) for
*η* = 0.8. The values of *U*_{c} are 0.2 in Fig. 5, 0.5 in Fig. 6, and
0.8 in Fig. 7.

The left panels in Fig. 4 with *η* = 0 show that the peak
velocity of an event, *V*_{m}, and final slip, *D*, with the respective maximum
values in case (a) as mentioned above, for all simulated events decrease
with increasing *U*_{c}. From Fig. 3, the force drop, Δ*F*, decreases
with increasing *U*_{c} for a certain final slip, thus indicating that larger
Δ*F* yields higher *V*_{m} and larger *D*. This interprets the negative
dependence of *V*_{m} and *D* on *U*_{c}. When the viscous effect is absent,
i.e., *η* = 0, the value of *τ*_{D} increases with *U*_{c}, while
*T*_{R} decreases with increasing *U*_{c}. When *U*_{c} = 1, *V*_{m} and *D* are
both very small and the system behaves like creeping of a fault. In the
right panels, there are two fixed points for each case: one is
called the non-zero fixed point at larger *V* and larger *U* and the other the
zero fixed point at *V* = 0 and *U* = 0. The slope at a fixed point is defined to
be d$(V/{V}_{\mathrm{max}})/$d(*U*∕*U*_{max}) = (*U*_{max}∕*V*_{max})(d*V*∕d*U*). The
absolute values of slope at the two fixed points decrease with increasing
*U*_{c}, thus suggesting that the fixed point is not an attractor for small
*U*_{c} and could be an attractor for larger *U*_{c}. The phase portrait for
*U*_{c} = 1 is very tiny, because the final slip for *U*_{c} = 1.0 is much
smaller than those for *U*_{c} = 0.2, 0.4, and 0.8. Hence, *U*_{c} = 1 will
not be taken into account in the following simulations.

The left panels in Figs. 5–7 show that *V*_{m} and *D* decrease
when either *U*_{c} or *η* increases, while *τ*_{D} increases with
*η* and *U*_{c}. Meanwhile, *T*_{R} increases when either *η* increases or
*U*_{c} decreases. The right panel shows that the phase
portraits coincide for all simulated events for a certain *η*. The
absolute values of slope at the two fixed points decrease when either
*U*_{c} or *η* increases. This suggests that the fixed point is not an
attractor for small *U*_{c} and low *η*, and can be an attractor for
large *U*_{c} and high *η*. Like Fig. 4, the final slip decreases with
increasing *U*_{c}.

In Figs. 5–7 we can see that the temporal variation in cumulative slip
can be described by the perfectly periodic model as mentioned above. Hence,
when *U*_{c} and *η* do not change with time, the rate of cumulative slip
retains a constant in the computational time period. This is similar to the
simulation results with the periodical earthquake occurrences obtained by
some authors (e.g., Rice and Tse, 1986; Ryabov and Ito, 2001; Erickson et
al., 2008; Mitsui and Hirahara, 2009) based on the one-body model with rate-
and state-dependent friction or velocity-weakening friction. However, the
present result is inconsistent with the simulation results, from which
either the time-predictable model or the slip-predictable model cannot
interpret the temporal variation in cumulative slip, based on the same model
obtained by others (e.g., He et al., 2003; Bazzarri, 2012b; Bizzarri and
Crupi, 2014; Kostić et al., 2013a, b; Franović et al., 2016). The
differences between the two groups of researchers might be due to distinct
additional constrains in respective studies. Although the detailed
discussion of such differences is important and significant, it is out of
the scope of this study and ignored here.

The phase portraits in Figs. 5–7 exhibit two kinds of fixed points as
mentioned above. The absolute values of slope at the non-zero fixed point
are higher than 1 and decreases with increasing *η*. This means that
larger *η* is easier to generate an attractor than small *η*.
However, the reducing rate of absolute value of slope decreases with
increasing *U*_{c}. The absolute values of slope of the zero fixed point are
higher than 1 and decrease with increasing *η*. This suggests that the
zero fixed points can be an attractor. This behavior becomes weaker when
*U*_{c} increases.

Figures 4–7 show that when *U*_{c} and *η* are constants during the
computational time periods, the general patterns of temporal variations in
cumulated slip do not change. Some of the previous studies (e.g., Bizzarri,
2012a, b; Franović et al., 2016) suggest that the patterns of
temporal variations in cumulated slip can change with time. The changes of
*U*_{c} and *η* with time should play the main roles. From
*u*_{c} = *ρ*_{f}*C*_{V}*h*∕*μ*_{f}Λ of the TP model (see
Rice, 2006), the width of the slipping zone, *h*, where the maximum deformation is
concentrated (Bizzarri, 2009), is a significant parameter in this study.
From geological surveys, Rathbun and Marone (2010) observed that *h* is not
spatially uniform even within a single fault. Hull (1988) and Marrett and
Allmendinger (1990) found that the wear processes occurring during faulting
could widen *h*, and thus *h* could vary with time. According to the results
gained by several authors (e.g., Power et al., 1988; Robertson, 1983;
Bizzarri, 2010), Bizzarri (2012b) proposed a linear dependence of *h* on the
cumulated slip, *S*(*t*) = ∑*D*(*t*), i.e., *h*(*t*) = *C**S*(*t*), where *C* is a dimensionless increasing
rate of *h* with *S* and is considered to be a constant in each case. Based on
*h*(*t*) = *C**S*(*t*), the more mature the fault is, the thicker its slip zone is. Since
*u*_{c} is proportional to *h* and *U*_{c} = *u*_{c}∕*D*_{o}, *U*_{c} is related to
*C*. Here, we assume that *U*_{c} varies with cumulative slip in the following
way: *U*_{c} = *U*_{co} + *C*∑*U*(*t*), where *U*_{co} is the initial value of
*U*_{c}. Simulation results for four values of *C* are shown in Figs. 8–12:
(a) for *C* = 0.0001, (b) for *C* = 0.001, (c) for *C* = 0.01, and (d) for *C* = 0.05 when
*η* = 0 in Figs. 8–10; (a) for *C* = 0.0001, (b) for *C* = 0.001, (c) for
*C* = 0.01, and (d) for *C* = 0.038 when *η* = 1 in Fig. 11; and (a) for
*C* = 0.0001, (b) for *C* = 0.001, (c) for *C* = 0.01, and (d) for *C* = 0.0136 when
*η* = 1 in Fig. 12. The initial values of *U*_{c} are 0.1 for Fig. 8,
0.5 for Fig. 9, 0.9 for Fig. 10, 0.1 for Fig. 11, and 0.5 for Fig. 12. Note
that the value *U*_{c} varies with time due to time-varying *h*.

The left panels of Figs. 8–12 show that *V*_{m}, *D*,
*τ*_{D}, and *T*_{R} are all similar when
*C* ≤ 0.001. However, in general *V*_{m} and *D* decrease with increasing
*C*,
*T*_{R} slightly decreases with increasing *C*, and *τ*_{D} slightly
increases with *C*. A decrease in *D* is particularly remarkable when *C* ≥ 0.01.
When *h* is wider than a critical value with *C* = 0.05 for *η* = 0, normal
earthquakes cannot occur and only creeping may happen. The critical value of
*h* decreases when the viscous effect is present with *η* = 1 in this
study. This decrease is also influenced by *U*_{c}: *C* = 0.038 when the initial
value of *U*_{c} is 0.1 and *C* = 0.0136 when the initial value of *U*_{c}
is 0.5. Obviously, *T*_{R} decreases with increasing *C*, thus leading to a
decrease in *T*_{R} with increasing *h*. This is similar to the result obtained
by Bizzarri (2010, 2012b). However, the viscous effect was not included in his studies.

The right panels of Figs. 8–12 show that the phase portraits
for *C* = 0.001 are slightly different from those for *C* = 0.0001 even though the
patterns of their variations in *V* and *U* are similar, while the phase portraits
for *C* > 0.001 are different from those for *C* ≤ 0.001. An increase
in *h* due to an increase in *C* with cumulative slip enlarges *U*_{c}. This can be
explained from Fig. 3, which shows that larger *U*_{c} yields a lower
Δ*F* than smaller *U*_{c} for the same final slip. Hence, an increase in
*U*_{c} produces a decrease in Δ*F*, thus resulting in low *V*_{m} and
small *D*. In addition, an increase in *U*_{c} makes $\mathrm{exp}(-U/{U}_{\mathrm{c}})$ approach unity,
especially for small *U*, thus reducing the nonlinear effect caused by TP friction.

Unlike Figs. 4–7, the size of phase portraits in the right
panels of Figs. 8–12 decreases with increasing *C*. This reflects a decrease
in both *T*_{R} and *D* of events with increasing *C* as mentioned previously. The
absolute values of slope at the non-zero fixed point are higher than 1 and
only slightly decrease with time when *C* < 0.01, while the values
remarkably decrease with time when *C* ≥ 0.01. The absolute values of slope
at the zero fixed point are higher than 1 and only slightly decrease with
time when *C* < 0.01, while those decrease with time when *C* ≥ 0.01.
Results suggest that the non-zero fixed points for all cases in the study are
not an attractor, and those at the zero fixed points can evolve to an
attractor with time when *C* ≥ 0.01. The phenomenon is particularly
remarkable for *C* = 0.05 in Figs. 8–10, *C* = 0.0380 in Fig. 11, and
*C* = 0.0136 in Fig. 12, and the evolution is faster for large *U*_{c} than for
small *U*_{c}.

The simulation results as mentioned previously demonstrate that when
*U*_{c} and *η* are constants during the computational time periods, the
general patterns of temporal variations in cumulated slip cannot change. In
order to investigate the effect of time-varying *η* and *U*_{c} on the patterns
of temporal variations in cumulated slip, we must consider changes of
*U*_{c} and *η* with time. The viscosity coefficient can actually vary
immediately before and after the occurrence of an earthquake (see Spray,
1883, 2005; Wang, 2017b, c). However, a lack of long-term variation in *η*
does not allow us to explore its possible effect on the change of general
patterns of temporal variations in cumulated slip. Here, only the possible
effect due to time-varying *U*_{c}.

As mentioned above, the equality *U*_{c} = *u*_{c}∕*D*_{o} leads to
*U*_{c} = *ρ*_{f}*C*_{V}*h*∕*μ*_{f}*D*_{o}Λ. Obviously,
*U*_{c} is controlled by six factors, i.e., *ρ*_{f},
*C*_{V}, *h*, *μ*_{f}, *D*_{o}, and Λ. Since the
tectonics of a region is generally stable over a long time, the value of
*D*_{o} = *F*_{o}∕*K* could not change too much and thus would not influence
*U*_{c}. The Debye law (cf. Reif, 1965) gives *C*_{v} ∼ (*T* + 273.16)^{3},
where 273.16 is the value to convert temperature from
Celsius to kelvin, at low *T* and *C*_{v} ≈ constant at high *T*. The
threshold temperature, from which *C*_{v} begins to approach a constant, is
200–300 ^{∘}K. In this study, *C*_{v} is almost a constant because of
*T* > 250 ^{∘}C = 523.16 ^{∘}K, which is the average ambient
temperature of fault zone with depths ranging from 0 to 20 km. Hence,
*C*_{v} is almost constant during a long time and thus cannot influence *U*_{c}.

The frictional strength, *μ*_{f}, is influenced by several factors
including humidity, temperature, sliding velocity, strain rate, normal
stress, and thermally activated rheology (Marone, 1998; Rice, 2006), and
thus can change with time (Sibson, 1992; Rice, 2006). Hirose and Bystricky (2007)
observed that serpentine dehydration and subsequent fluid
pressurization due to coseismic frictional heating may reduce *μ*_{f}
and thus promote further weakening in a fault zone. The pore fluid pressure
exists in wet rocks, yet not in dry rocks. Clearly, the time variation in
*μ*_{f} can affects the earthquake recurrences. However, a lack of
long-term observations of *μ*_{f} during a seismic cycle makes the
study of its effect on earthquake recurrence be impossible.

The fluid density *ρ*_{f} and the porosity *n* depend on *T* and *p*. Although
there are numerous studies on such dependence (Lachenbruch, 1980; Bizzarri,
2012b; and cited references therein), observed data and theoretical analyses
about the values of *ρ*_{f} and *n* during a seismic cycle are rare.

The porosity is associated with the permeability, *κ*. Bizzarri (2012c)
pointed out that the time-varying permeability, *κ*(*t*), and
porosity of a fault zone (cf. Mitsui and Cocco, 2010; Bizzarri, 2012b) can
reduce *T*_{R}. One of the Kozeny–Carman relations (Costa, 2006, and
references cited therein) is *κ*(*t*) = *κ*_{C}*φ*^{2}(*t*)d^{3}(*t*)∕[1 − *φ*(*t*)]^{2},
where *κ*_{C} is a dimensionless
constant depending on the material in consideration; *φ* is
*V*_{voids}∕*V*_{tot}, where *V*_{voids} and *V*_{tot} are, respectively, the pore
volume and the total volume of the porous materials; and *d* is the (average)
diameter of the grains, ranging between 4 × 10^{−5} and
1 × 10^{−4} m (Niemeijer et al., 2010). Usually, *κ*,
*φ*, and d can vary in the fault zone (Segall and Rice, 1995). After faulting
*κ* and *φ* would change and d becomes smaller because of refining
of the grains. According to this relation, Bizzarri (2012b) found that
*κ*(*t*) could significantly reduce *T*_{R} in comparison with the base
model with constant *κ*. The reason is explained below. An increase in
permeability can result in an increase in pore pressure, *p*_{f}. This can
reduce the frictional resistance from *τ* = *μ*(*σ*_{n} − *p*_{f})
and thus could trigger earthquakes earlier. Hence, the
time-varying permeability can change *T*_{R}. Nevertheless, we cannot
investigate its influence on earthquake recurrence here because there is a
lack of a long-term observation of hydraulic parameters during a seismic cycle.

It is significant to explore the factors that can yield a non-perfectly
periodic seismic cycle. The width of the slipping zone, *h*, can be a candidate
as pointed out by some authors (e.g., Bizzarri, 2009; Rathbun and Marone,
2010). Since the displacement along a fault is controlled by the fault
rheology, *h* should depend on the rheology on the sliding interface. The wear
processes occurring during faulting could widen *h* (Hull, 1988; Marrett and
Allmendinger, 1990). According to the results gained by several authors
(e.g., Power et al., 1988; Robertson, 1983; Bizzarri, 2010), simulation
results for various values of *C* and the results are shown in Figs. 8–10 with
*η* = 0 and in Figs. 11 and 12 with *η* = 1. Results show that when
*C* > 0.0001, the wear process affects the recurrence of slip and the
effect increases with *C* and when *C* is larger than an upper-bound value,
larger-sized events cannot occur and the earthquake recurrence does not
exist. Both *T*_{R} and *D* decrease when the fault becomes more mature due to a
thicker slip zone. Meanwhile, the viscous effect can also play a secondary
role on the earthquake recurrence because it makes upper-bound value become
smaller. Although either the time- or slip-predictable model can describe
the temporal variations of cumulative slip of events occurring in the
earlier time period, they cannot interpret those of events in the later
parts. This might suggest that it is more difficult to produce large
earthquakes along a fault when it becomes more mature, especially for the
cases with viscosity. This implicates that seismic hazard is higher for a
young fault than a mature one. Hence, it is significant and important to
identify the width of slip zone of an earthquake fault for seismic hazard estimates.

To study the frictional and viscous effects on earthquake recurrence,
numerical simulations of the temporal variations in cumulative slip have
been conducted based on the normalized equation of a one-body model in the
presence of thermal-pressurized slip-weakening friction and viscosity. The
wear process, which is included in the friction law, is also taken into
account. The model parameters of friction and viscosity are represented,
respectively, by *U*_{c} and *η*, where *U*_{c} = *u*_{c}∕*D*_{o} is the
normalized characteristic distance and *η* is the normalized damping
coefficient. Numerical simulation of the time variations in *V*∕*V*_{max} and
cumulative slip Σ*U*∕*U*_{max} as well as the phase portrait of *V*∕*V*_{max} versus
*U*∕*U*_{max} are made for various values of *U*_{c} and *η*.

Results show that both friction and viscosity remarkably affect
earthquake recurrences. The recurrence time, *T*_{R}, increase when *η*
increases or *U*_{c} decreases. The final slip, *D*, and the duration
time of slip, *τ*_{D}, of an event slightly decrease when *η* or
*τ*_{D} increases and slightly increases with *U*_{c}. Considering the
effect due to wear process, the thickness of slip zone, *h*, that depends on the
cumulated slip, *S*(*t*) = ∑*D*(*t*), i.e., *h*(*t*) = *C**S*(*t*) (*C* is a dimensionless constant), is an
important factor in influencing earthquake recurrences. *U*_{c} increases
with ∑*U* with an increasing rate of *C*. When *C* > 0.0001, the wear
process influences the recurrence of slip and the effect increases with *C*.
When *C* is larger than an upper-bound value, larger-sized events cannot occur
and the earthquake recurrence does not exist. If the slip zone becomes
thicker, the fault is more mature. This makes *T*_{R} and *D* become shorter.
This might suggest that it is more difficult to produce large earthquakes
along a fault when it becomes more mature. This phenomenon becomes
remarkable when the viscous effect exists because the upper-bound value
becomes smaller. The temporal variation in slip cannot be interpreted by the
time-predictable or slip-predictable model when the fault is affected by
wear process with large *C*. In addition, the size of phase portrait of
*V*∕*V*_{max} versus *U*∕*U*_{max} decreases with increasing *C*. This again reflects
decreases in both *T*_{R} and *D* of events with increasing *C* as inferred from
the temporal variations in cumulative slip.

No data sets were used in this article.

The author declares that he has no conflict of interest.

The author would like to thank Filippos Vallianatos (Editor of NHESS) and two
anonymous reviewers for their valuable comments and suggestions to help me to
substantially improve this article. The study was financially supported by
Academia Sinica and the Ministry of Science and Technology (grant no. MOST-106-2116-M-001-005).

Edited by: Filippos Vallianatos

Reviewed by: two anonymous referees

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