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Dynamic earthquake sequence simulation with an SBIEM accounting for interseismic poroelastic rebound

Abstract

Afterslip inside a coseismic slip patch is rarely observed, though some previous studies suggest that it is driven by poroelastic rebound (PER). These studies assume constant frictional strength, whereas time-dependent strengthening (healing) of a fault is expected from laboratory experiments, which provide a basis for a rate- and state-dependent friction law (RSF). In this study, quasistatic poroelasticity (PE) was implemented in a dynamic earthquake sequence simulation using a spectral boundary integral equation method, and the effect of PER on the behavior of a fault governed by RSF was examined. Spatio-temporal convolution for PE would significantly affect the resolution of the numerical simulation affordable. This problem has been resolved by numerical approximation of the time dependency of Green’s function of PE in the wavenumber domain, definition of memory variables, and reformulation of the temporal convolution into ordinary differential equations of them. In the novel method, the additional numerical costs due to PE are negligible. A planar fault with a rate-weakening patch embedded in the rate-strengthening region was simulated. Because it is the healing of the fault that competes against PER, both the aging law and slip law were examined, which have different characteristics in the evolution of the fault strength. The simulation results indicate that PER causes postseismic loading to the patch, but the healing efficiently suppresses afterslip not only for the aging law, but also for the slip law. When cases with different friction laws are compared, the healing is more significant for the aging law, which has log-\(t\) strengthening at a limit of \(V\to 0\). However, the effect of PER on the slip rate is minor for the slip law. The slip law yields additional healing if the fault is accelerated by loading owing to PER. The simulation results are consistent with the absence of afterslip within the coseismic slip patches in the observations.

Graphical Abstract

Introduction

Afterslip, which follows a dynamic earthquake rupture and can cause aftershocks (Hsu et al. 2006), takes place in the vicinity of a coseismic slip patch (e.g., Miyazaki et al. 2004). The afterslip inside the patch has rarely been confirmed. Geodetic inversion sometimes yields an afterslip distribution that apparently overlaps with a coseismic slip distribution, but the resolution check (e.g., the checkerboard test) indicates that it may be an artifact (e.g., Meneses-Gutierrez et al. 2019). The distribution of small repeating earthquakes (e.g., Kato and Igarashi 2012) more clearly shows that regions of afterslip and coseismic slip are separated. Earthquake sequence simulation for a fault governed by the rate- and state-dependent friction law (RSF) (Dieterich 1979; Ruina 1983) in an elastic medium typically yields the invasion of afterslip into the patch, but the core of the patch remains locked until the next earthquake (e.g., Tse and Rice 1986; Scholz 1998; Lapusta et al. 2000).

On the other hand, some previous theoretical and numerical studies have suggested the possibility of afterslip even inside the coseismic slip patch driven by the poroelastic effect. Rapid, undrained deformation due to earthquake rupture causes heterogeneity in the volumetric strain and pore pressure \(p\) in the surrounding poroelastic medium. This causes fluid flow, diffusion of \(p\), and associated poroelastic deformation, referred to as poroelastic rebound (PER) (e.g., Peltzer et al. 1996). For an isotropic medium, the undrained and drained responses have the same shear modulus \(\mu\) and different Poisson’s ratios \({\nu }_{\text{u}}>{\nu }_{\text{d}}\), where the subscripts u and d indicate the undrained and drained conditions, respectively.

Detournay and Cheng (1991) demonstrated that instantaneous opening of a drained crack of a fixed length (“mode 1” in their paper) is followed by additional time-dependent opening due to PER. The similarity between an in-plane shear crack and an open crack in the static limit suggests an afterslip due to the PER (Fig. 1). Coseismic, rapid stress drop \(\Delta \sigma\) causes slip distribution scaled by the inverse of the stiffness in the undrained limit \({\mu }_{\text{u}}^{^{\prime}}\). For example, a two-dimensional (2-D) linear shear crack of length \(2R\) in an in-plane problem hosts a maximum slip of \(2R\Delta \sigma /{\mu }_{\text{u}}^{^{\prime}}=2\left(1-{\nu }_{\text{u}}\right)R\Delta \sigma /\mu\). In the limit of a sufficiently large time, the stiffness in the drained limit \({\mu }_{\text{d}}^{^{\prime}}\) is important. In the example of the 2-D crack (Knopoff 1958), the maximum slip is given by \(2R\Delta \sigma /{\mu }_{\text{d}}^{^{\prime}}=2\left(1-{\nu }_{\text{d}}\right)R\Delta \sigma /\mu\). The difference between them 2 \(\left({\nu }_{\text{u}}-{\nu }_{\text{d}}\right)R\Delta \sigma /\mu\) is generated as an afterslip. Yamashita (2007) and Yamashita and Suzuki (2009) studied the case with differences in poroelastic properties across the fault, and suggested afterslip even within the coseismic slip patch and postseismic crack extension due to PER (Table 1).

Fig. 1
figure 1

Difference in the undrained and the drained responses for uniform stress drop \(\Delta \sigma\). Afterslip due to PE expected for an inplane shear crack with constant frictional strength is indicated in grey

Table 1 Physical properties used in the simulations

These models assume a constant interface strength, whereas laboratory friction experiments show an increase in the fault strength (healing) with the time of stationary contact after slip ceases (Dokos 1946; Dieterich 1972), as is realized by the RSF (e.g., Marone 1998). In this study, quasistatic poroelasticity (PE) was implemented in a dynamic earthquake sequence simulation (e.g., Lapusta et al. 2000) accounting for elastic wave propagation using a spectral boundary integral equation method (SBIEM), and its consequences on the fault behavior were investigated. Note that there are a number of recent studies on poroelastic effect on a fault governed by the RSF. Jha and Juanes (2014) developed a quasistatic model for RSF faults embedded in poroelastic medium with multiphase flow. Torberntsson et al. (2018) carried out quasidynamic earthquake sequence simulations for an RSF fault in poroelastic medium with a permeable shear zone, and demonstrated an earthquake rupture induced by fluid injection. A similar finite difference approach was taken by Heimisson et al. (2019) to study aseismic transients in a shape of slow slip pulses under the quasidynamic assumption. A linearized stability analysis of a RSF fault in a poroelastic medium allowing dilatancy and compaction in the fault zone was presented in Heimisson et al. (2021). Although PER should operate actively in just after earthquakes, postseismic behavior has not been reported in detail.

Linear PE is assumed here for simplicity. Linearity allows us to calculate the quasistatic deformation due to PE by spatio-temporal convolution of the slip history and Green’s function. On the other hand, in the dynamic earthquake sequence simulation developed by Rice and Ben-Zion (1996) for an elastic medium, dynamic, wave-mediated stress transfer is calculated by spatio-temporal convolution of the slip history and Green’s function of elastodynamics. The computational costs and storage of history for this calculation are limiting factors for the problem size given the restricted computational resources. The additional spatiotemporal convolution for PE significantly affects the resolution of the fault affordable. Miyake and Noda (2019) addressed this problem for a Maxwell-viscoelastic medium by defining a memory variable and reformulating the temporal convolution into an ordinary differential equation (ODE) of the memory variable. Note that Ruina (1983) took a similar approach in formulating the rate- and state-dependent friction law. This method enabled the implementation of quasistatic viscoelastic relaxation in a dynamic earthquake sequence simulation without significant additional cost. In this study, PE was implemented using a similar method by introducing 18 memory variables for each discrete wavenumber without conducting a temporal convolution.

A simple fault model with one rate-weakening patch embedded in the rate-strengthening region was simulated. The postseismic strengthening of the patch is a key process that competes against PER. Therefore, both the aging law and slip law were tested, which have different characteristics in the state evolution (Marone 1998). A fault within a uniform poroelastic medium was considered in this paper. The short-time response within a coseismic time scale was assumed to be given by the undrained response. Such assumption cannot be realistic for a fault between dissimilar materials (e.g., Rudnicki and Rice 2006; Dunham and Rice 2008). Note that attenuation and dispersion of elastic waves due to wave-induced fluid flow are neglected. Undrained cases were first simulated, and then simulations with PE with different characteristic diffusion times were conducted. Based on the simulation results, the possibility of enhancement of afterslip due to PER is discussed.

Methodology

Poroelastic Green’s function in the SBIEM

A planar fault embedded in a linearly poroelastic infinite medium is considered. To concentrate our focus on the effect of PER, dilatancy and compaction due to deformation of fault gouge (e.g.,, Heimisson et al. 2021) are neglected. Because it is the volumetric strain \({\epsilon }_{kk}\) that causes the pore pressure change \(p\) and the subsequent fluid flow, PE is not important in anti-plane problems. Therefore, the focus of this study was on in-plane problems. The new method proposed here can be readily applied to the 3-D problem of a planar fault, because it the convolution kernel for such a problem can be constructed by the combination of those for 2-D in-plane and anti-plane problems (Geubelle and Rice 1995; Lapusta and Liu 2009).

In an elastodynamic problem with a fault, shear traction on a fault \(\tau\) at location \(x\) and time \(t\) can be expressed as follows (Cochard and Madariaga 1994; Geubelle and Rice 1995):

$$\tau \left(x,t\right)={\tau }_{0}\left(x,t\right)+\phi -\frac{\mu }{2{c}_{\text{s}}}V\left(x,t\right) ,$$
(1)

where \({\tau }_{0}\) is the traction realized at a reference state without any slip on the fault, \(\phi\) is the wave-mediated stress transfer due to previous fault motion, \(V\) is the slip rate, \(\mu\) is the shear modulus, and \({c}_{\text{s}}\) is the shear wave speed. The final term in Eq. (1) represents the impedance effect, which is sometimes referred to as the radiation damping effect (Rice 1993). Rice and Ben-Zion (1996) split \(\phi\) into the following two terms:

$$\phi ={\phi }_{\text{st}}+{\phi }_{\text{dy}} ,$$
(2)

where \({\phi }_{\text{st}}\) is the static traction change, which would be achieved if the fault were instantaneously welded up and elastic waves radiated out of the system. \({\phi }_{\text{dy}}\) is simply defined as \({\phi }_{\text{dy}}=\phi -{\phi }_{\text{st}}\). In the elastic case, \({\phi }_{\text{st}}\) depends only on the current slip distribution, and \({\phi }_{\text{dy}}\) is calculated by the spatiotemporal convolution of \(V\) and Green’s function with truncation in terms of the delay time. The time window for the convolution \({t}_{\text{w}}\) is taken as the time for which the shear wave travels several times the system size (coseismic timescale). In an in-plane problem, the Fourier transform of \({\phi }_{\text{st}}\) is expressed as follows:

$${\Phi }_{\text{st}}\left(k,t\right)=\int\limits_{-\infty }^{\infty }{\phi }_{\text{st}}\left(x,t\right){\text{e}}^{-\text{i}kx}dx=-\frac{\left|k\right|{\mu }^{^{\prime}}}{2}D\left(k,t\right) ,$$
(3)

where \(\mu^{\prime}\) is expressed by \(\mu\) and the Poisson ratio \(\nu\) as

$${\mu }^{^{\prime}}=\frac{\mu }{1-\nu },$$
(4)

and \(D\) is the Fourier transform of slip the \(\delta\)

$$D\left(k,t\right)= \int \limits_{-\infty }^{\infty }\delta \left(x,t\right){\text{e}}^{-\text{i}kx}dx .$$
(5)

In this study, it was assumed that the fluid flow was negligible in the coseismic timescale; \({\phi }_{\text{dy}}\) was calculated in the same manner as in previous studies for an elastic medium (e.g., Lapusta et al. 2000) with the undrained properties. \({\phi }_{\text{st}}\) is modified to \({\phi }_{\text{st}}^{\text{PE}}\) to account for the PE. The constitutive law of linear poroelasticity is as follows:

$${\sigma }_{ij}={C}_{ijkl}{\epsilon }_{ij}-{\alpha }_{ij}p ,$$
(6)
$$p=M\left(\zeta -{\alpha }_{ij}{\epsilon }_{ij}\right) ,$$
(7)
$${q}_{i}=-{\kappa }_{ij}{p}_{,j} ,$$
(8)
$$\frac{\partial \zeta }{\partial t}+{q}_{i,i}=0 .$$
(9)

Equations (6) and (7) are the stress–strain relations accounting for the changes in \(p\) and in the fluid content \(\zeta\), where \({\sigma }_{ij}\) is the stress, \({\epsilon }_{ij}\) is the strain, \({C}_{ijkl}\) is the drained elastic modulus, \({\alpha }_{ij}\) is the Biot stress coefficient, and \(M\) is the Biot modulus. Equation (8) is Darcy’s law, where \({q}_{i}\) is volumetric flux of the fluid, \({\kappa }_{ij}\) is the mobility of the fluid given by the permeability of the medium divided by the viscosity of the fluid \({k}_{ij}/\eta\). Equation (9) is the mass conservation law for the pore fluid. For simplicity, the medium is assumed to be isotropic. In this case, the degrees of freedom of the tensorial rock properties are significantly reduced, such that \({C}_{ijkl}\) is expressed in terms of the shear modulus \(\mu\) and the drained Poisson ratio \({\nu }_{\text{d}}\), \({\alpha }_{ij}=\alpha\), and \({\kappa }_{ij}=\kappa =k/\eta\).

Cheng and Detournay (1998) presented a spatiotemporal stress change due to a sudden, spatially concentrated slip \(\delta \left(x,t\right)={\delta }_{\rm D}\left(x\right)H\left(t\right)\) (Appendix D.11 in their paper). \({\delta }_{\rm D}\) is the Dirac’s delta function, and \(H\) is the Heaviside function. Evaluating the corresponding shear traction on the fault, we obtain Green’s traction function for a concentrated slip \({\tau }^{\text{G}}\) as follows:

$$\begin{array}{l}{\tau }^{\text{G}}\left(x,t\right)=\frac{H\left(t\right)}{2\uppi {x}^{2}}\left\{{\mu }_{\text{u}}^{^{\prime}}+\Delta {\mu }^{^{\prime}}\left[3\frac{1-{e}^{-{x}^{2}/4ct}}{{x}^{2}/4ct}-2{e}^{-{x}^{2}/4ct}\right]\right\} .\end{array}$$
(10)

where \({\mu }_{\text{u}}^{\prime}\) and \(\Delta {\mu }^{\prime}\) are the effective on-fault rigidity under undrained conditions and the difference between the drained and undrained conditions, respectively, and are given by

$${\mu }_{\text{d}}^{^{\prime}}=\frac{\mu }{1-{\nu }_{\text{d}}}, {\mu }_{\text{u}}^{^{\prime}}=\frac{\mu }{1-{\nu }_{\text{u}}}, \Delta {\mu }^{^{\prime}} ={\mu }_{\text{d}}^{^{\prime}}-{\mu }_{\text{s}}^{^{\prime}} ,$$
(11)

where \({\nu }_{\text{u}}\) is the undrained Poisson ratio

$${\nu }_{\text{u}}=\frac{1}{2}\left(1-\frac{\mu \left(1-2{\nu }_{\text{d}}\right)}{M{\alpha }^{2}\left(1-2{\nu }_{\text{d}}\right)+\mu }\right) ,$$
(12)

where \(c\) is the diffusion coefficient of \(p\) given by

$$c=\frac{2\kappa \mu \left({\nu }_{\text{u}}-{\nu }_{\text{d}}\right)\left(1-{\nu }_{\text{d}}\right)}{{\alpha }^{2}\left(1-{\nu }_{\text{u}}\right){\left(1-2{\nu }_{\text{d}}\right)}^{2}} .$$
(13)

Using Green’s function \({\tau }^{\text{G}}\), the static part of the traction change accounting for PE \({\phi }_{\text{st}}^{\text{PE}}\) can be expressed as follows:

$${\phi }_{\text{st}}^{\text{PE}}\left(x,t\right)=\left[\int \limits_{-\infty }^{\infty }\text{d}{x}^{^{\prime}}\int \limits_{-\infty }^{t}\text{d}{t}^{^{\prime}}\right]V\left({x}^{^{\prime}},{t}^{^{\prime}}\right){\tau }^{\text{G}}\left(x-{x}^{^{\prime}},t-{t}^{^{\prime}}\right) .$$
(14)

Fourier transformation leads to

$${\Phi }_{\text{st}}^{\text{PE}}\left(k,t\right)=\int \limits_{-\infty }^{\infty }{\phi }_{\text{st}}^{\text{PE}}\left(x,t\right){\text{e}}^{-\text{i}kx}\text{d}x=\left[\int\limits_{-\infty }^{t}\text{d}{t}^{^{\prime}}\right]\dot{D}\left(k,{t}^{^{\prime}}\right){T}^{\text{G}}\left(k,t-{t}^{^{\prime}}\right) ,$$
(15)

where a dot represents time derivative, and \({T}^{\text{G}}\) is the Fourier transform of \({\tau }^{\text{G}}\)

$$\begin{array}{l}{T}^{\text{G}}\left(k,t\right)=\int \limits_{-\infty }^{\infty }{\tau }^{\text{G}}\left(x,t\right){\text{e}}^{-\text{i}kx}\text{d}x=-\frac{\left|k\right|{\mu }_{\text{u}}^{^{\prime}}}{2}\left[1+\frac{\Delta {\mu }^{^{\prime}}}{{\mu }_{u}^{^{\prime}}}\left(1-F\left(ct{k}^{2}\right)\right)\right] .\end{array}$$
(16)

Note that \(H\left(t\right)\) was eliminated, because \({T}^{G}\left(k,t\right)\) only in \(t>0\) is important. \(F\) is a universal function that is independent of the material properties:

$$F\left(s\right)= \left(1+2s\right)\text{erfc}\left(\sqrt{s}\right)-\frac{2}{\sqrt{\uppi }}\sqrt{s}{\text{e}}^{-s}.$$
(17)

Song and Rudnicki (2017) derived a Green’s function for a dislocation in a poroelastic medium with “leaky” fault plane, which has a resistance (fluid flux per pore pressure gap across the fault plane). It is a generalization of the solution by Cheng and Detournay (1998), and may substitute the Green’s function used here to study the effect of impermeable, caly-rich fault.

Implementation using memory variables

A direct evaluation of Eq. (15) requires storage of the slip rate history in the past, which requires significant additional computational resources. Miyake and Noda (2019) resolved this problem by reformulating the temporal convolution into an ODE of a memory variable, the Fourier-transformed effective slip \({D}_{\text{eff}}\). In this study, the effective slip was defined based on the undrained mechanical response as follows:

$$\begin{array}{l}{\Phi }_{\text{st}}^{\text{PE}}\left(k,t\right)=-\frac{\left|k\right|{\mu }_{\text{u}}^{^{\prime}}}{2}{D}_{\text{eff}}\left(k,t\right) .\end{array}$$
(18)

From Eqs. (15), (16), and (18), \({D}_{\text{eff}}\) is expressed as follows:

$${D}_{\text{eff}}=\int\limits_{-\infty }^{t}\dot{D}\left(k,{t}^{^{\prime}}\right)\left[1+\frac{\Delta {\mu }^{^{\prime}}}{{\mu }_{\text{u}}^{^{\prime}}}\left(1-F\left(c\left(t-{t}^{^{\prime}}\right){k}^{2}\right)\right)\right]d{t}^{^{\prime}} .$$
(19)

For Maxwell viscoelasticity in antiplane problems, Miyake and Noda (2019) found a simple ODE of \({D}_{\text{eff}}\). A similar formulation is possible for Maxwell viscoelasticity in in-plane problems (under preparation) and most likely for many other linear rheologies. For poroelasticity, an ODE equivalent to Eq. (19) could not be found so that a numerical approximation was developed.

Figure 2a, b shows \(F\left(s\right)\) in linear and logarithmic vertical scales, respectively. Note that \(F\left(s\right)\) represents the transition between the undrained (\(F\left(0\right)=1\)) and drained (\(F\left(\infty \right)=0\)) responses. \(F\left(s\right)\) is a monotonically decaying function and becomes smaller than \({10}^{-6}\) by the non-dimensional time \(s=10\). If it can be approximated as follows:

Fig. 2
figure 2

Plots of a \(F\left(s\right)\) and b \({\text{log}}_{10}\left(F\left(s\right)\right)\), which represent transition between undrained to drained responses

$$F\left(s\right)\approx \sum \limits_{i=1}^{n}{a}_{i}\text{exp}\left(-\frac{s}{{s}_{i}}\right) ,$$
(20)

Equation (19) then becomes

$$\begin{array}{l}{D}_{\text{eff}}\approx \int \limits_{-\infty }^{t}\dot{D}\left(k,{t}^{^{\prime}}\right)\left[1+\frac{\Delta {\mu }^{^{\prime}}}{{\mu }_{\text{u}}^{^{\prime}}}\left(1-{\sum }_{i=1}^{n}{a}_{i}\text{exp}\left(-\frac{c\left(t-{t}^{^{\prime}}\right){k}^{2}}{{s}_{i}}\right)\right)\right]d{t}^{^{\prime}}\\ =\left(1+\frac{\Delta {\mu }^{^{\prime}}}{{\mu }_{\text{u}}^{^{\prime}}}\right)D\left(k,{t}^{^{\prime}}\right)-\frac{\Delta {\mu }^{^{\prime}}}{{\mu }_{\text{u}}^{^{\prime}}}{\sum }_{i=1}^{n}{a}_{i}{D}_{i}\left(k,t\right) ,\end{array}$$
(21)

where \({D}_{i}\) represents additional memory variables defined as follows:

$${D}_{i}\left(k,t\right)=\int\limits_{-\infty }^{t}\dot{D}\left(k,{t}^{^{\prime}}\right)\text{exp}\left(-\frac{c\left(t-{t}^{^{\prime}}\right){k}^{2}}{{s}_{i}}\right)d{t}^{^{\prime}} .$$
(22)

Equation (22) can be reformulated into an ODE for each \(i\) as follows:

$${\dot{D}}_{i}\left(k,t\right)={\dot{D}}\left(k,t\right)-\frac{c{k}^{2}}{{s}_{i}}{D}_{i}\left(k,t\right) .$$
(23)

This can be easily integrated every time step.

Each memory variable \({D}_{i}\) has its weight of contribution \(-{a}_{i}\Delta {\mu }^{^{\prime}}/{\mu }_{\text{u}}^{^{\prime}}\) to the Fourier-transformed effective slip \({D}_{\text{eff}}\) and its non-dimensional characteristic decay time \({s}_{i}\). \({a}_{i}\) and \({s}_{i}\) for \(i=1, 2,\dots ,n\) are determined by least-squares fitting of Eq. (20) to discretely sampled data \(\left\{F\left(0.01\right), F\left(0.02\right), \dots ,F\left(10\right)\right\}\) under the constraint \({\sum }_{i=1}^{n}{a}_{i}=1\). The residual decreases as \(n\) increases (Fig. 3), and the absolute approximation error becomes less than \({10}^{-6}\) with \(n=18\), which is adopted in the applications described later in this paper.

Fig. 3
figure 3

\(F\left(s\right)\) and the residual of the numerical approximation of \(F\left(s\right)\) (Eq. (20)). The absolute error was less than \({10}^{-6}\) with \(n=18\)

There were 18 memory variables for each wavenumber, and the number of collocation points in the wavenumber domain was approximately the same as the number of spatial grid points. The uniform mode does not contribute to the shear stress and need not be considered. Noda and Lapusta (2010) used approximately 120 memory or state variables for each point on the fault to express the frictional heating and thermal pressurization of the pore fluid. The time integration of these variables does not result in a significant additional computational cost. Indeed, in the examples shown later, the computational time for a sequence of 20 earthquakes may be shorter in cases with PE relative to the undrained case. This is because PER decreases the recurrence interval. The proposed method was found to be quite efficient.

The time integrations of slip \(\delta\), its Fourier transform \(D\) and a state variable in a rate- and state-dependent friction law in the dynamic earthquake sequence simulation by Noda and Lapusta (2010) are based on the assumption of piecewise constant \(V\) over time. Here, the time integration of memory variables \({D}_{i}\) is performed exactly in the same manner, with an unconditionally stable second-order accurate predictor–corrector method based on an exponential time differencing method. Note that the characteristic decay time \({s}_{i}/{k}^{2}c\) is so small for high-wavenumber components that explicit time integration based on a constant time derivative is unrealistic in the simulation of interseismic periods. For details, please refer to Noda and Lapusta (2010), in which a similar technique was adopted to simulate the diffusion of temperature and pore pressure normal to the fault.

Problem setting of example problems

System geometry

Let us select length, speed, and stress as the three independent dimensions and consider appropriate scales relevant to natural earthquakes. Earthquakes occur in a vastly wide range of length scales, as recognized by the Gutenberg–Richter law. In addition, the permeability \(\kappa\) has the dimension of squared length and is known to vary for orders of magnitude depending on the rock type. Therefore, the length scale of the problem shall not be specified at this point. The variation in the elastic property of rock is minor compared to them. Therefore, the scales of speed and stress are specified in the example problems described below by setting \({c}_{\text{s}}=3 \,\text{km}/\text{s}\) and \(\mu =30\,\text{ GPa}\). Biot’s stress coefficient \(\alpha\), undrained and drained Poisson’s ratios \({\nu }_{\text{u}}\) and \({\nu }_{\text{d}}\) were set \(\alpha =0.47\), \({\nu }_{\text{u}}=0.34\), \({\nu }_{\text{d}}=0.25\), respectively, based on the values for granite in Wang (2001). The parameters used in the example simulations are listed in Table 1.

The effect of PER was investigated for a simple problem setting: a planar fault governed by RSF (Dieterich 1979; Ruina 1983) with a rate-weakening patch. The frictional resistance at steady state is as follows:

$$\begin{array}{l}{\tau }_{\text{ss}}={\tau }_{*}+\left(A-B\right)\text{ln}\left(\frac{V}{{V}_{*}}\right) ,\end{array}$$
(24)

where \({\tau }_{*}\) is the steady-state frictional resistance at the reference \(V={V}_{*}\), and \(A-B\) represents the steady-state rate dependency. Note that the value of \({\tau }_{*}\) is insignificant, as a linear poroelastic material is considered; spatial variation in \({\tau }_{*}\) is absorbed by definition of the initial or reference state from which the slip \(\delta\) is measured. \(A\) and \(B\) are the amounts of direct and evolution effects, respectively. The rate-weakening patch is expressed using a smoothed boxcar function (The SCEC/USGS Spontaneous Rupture Code Verification Project, TPV105; Noda and Lapusta 2010):

$$\begin{array}{l}\text{sbox}\left(x;W,w\right)=\left\{\begin{array}{ll}1& \left(\left|x\right|\le W\right)\\ \frac{1}{2}\left[1+\text{tanh}\left(\frac{w}{\left|x\right|-W-w}+\frac{w}{\left|x\right|-W}\right)\right]& \left(W\le \left|x\right|\le W+w\right)\\ 0& \left(\left|x\right|\ge W+w\right)\end{array}\right. ,\end{array}$$
(25)

as (Fig. 4)

Fig. 4
figure 4

Distributions of frictional parameters \(A\) and \(B\)

$$A=1 \text{MPa}, B=1.5 \text{sbox}\left(x;R,0.1R\right) \text{MPa} .$$
(26)

These values are consistent with effective normal stress about 100 MPa.

The fault is driven by prescribing a long-term constant slip rate \({V}_{\text{pl}}={10}^{-9} \text{m}/\text{s}\) in \(\left|x\right|>2R\). \({V}_{\text{pl}}\) constrains the long-term slip rate of the fault and was adopted as the reference, \({V}_{*}={V}_{\text{pl}}\). Noda (2021) developed a dynamic earthquake sequence simulation using an SBIEM without a periodic boundary condition based on the method of Cochard and Rice (1997). This method is only applicable if the source of the traction change is supported within a certain region. Here, the effective elastostatic source is the inverse Fourier transform of \({D}_{\text{eff}}\), which may not be constrained within \(\left|x\right|<2R\). Therefore, removal of the periodic boundary condition may be difficult. In the example problems, periodic boundaries are assumed every \(16R\), four times the length of the segment governed by the RSF law. This is sufficiently long such that the effect of the periodic boundary is modest (Lapusta et al. 2000; Noda 2021). The time window \({t}_{\text{w}}\) for the calculation \({\phi }_{\text{dy}}\) was set to \(12R/{c}_{\text{s}}\).

Friction law

Both the aging law and the slip law are examined as follows:

$$\begin{array}{l}\begin{array}{l}\tau ={\tau }_{*}+A\text{ln}\left(\frac{V}{{V}_{*}}\right)+B\text{ln}\left(\theta \right)\\ \dot{\theta }=\frac{V}{L}\left(\frac{{V}_{*}}{V}-\theta \right)\end{array} \left(\text{aging law}\right),\end{array}$$
(27)

and

$$\begin{array}{l}\begin{array}{l}\tau ={\tau }_{*}+A\text{ln}\left(\frac{V}{{V}_{*}}\right)+\Theta \\ \dot{\Theta }=\frac{V}{L}\left({\tau }_{\text{ss}}-\tau \right)\end{array} \left(\text{slip law}\right).\end{array}$$
(28)

where \(L\) is the characteristic slip of the state evolution, \(\theta\) is the state variable in the aging law representing recent nondimensional slowness, and \(\Theta\) is the state variable in the slip law representing the frictional strength relative to the reference (\(\tau -{\tau }_{*}\) at \(V={V}_{*}\)). The aging law provides more efficient healing of the fault than the slip law as follows (Marone 1998):

$$\underset{V\to 0}{\text{lim}}\dot{\theta }=\frac{{V}_{*}}{L} , \underset{V\to 0}{\text{lim}}\dot{\Theta }=0 .$$
(29)

At the rupture front, the aging law results in an almost linear slip weakening of the slope \(-\text{d}\tau /\text{d}\delta \approx B/L\), while the slip law causes exponential slip weakening towards the residual frictional strength (Ampuero and Rubin 2008), which causes a much larger \(-\text{d}\tau /\text{d}\delta\). In preliminary simulations with constant \(L\) (not shown in this paper), the dynamic earthquake sequence simulation with the slip law requires large computational resources and produces a series of supershear ruptures, which may not be realistic. To produce comparable earthquake sequences of sub-Rayleigh ruptures between the aging law and slip law, and to enable parameter study in terms of the fluid mobility \(\kappa\) for affordable computational costs, \(L\) is increased at the coseismic slip rate for the slip law, while it is kept constant for the aging law as follows:

$$L=\left\{\begin{array}{ll}{L}^{\text{ag}}& \left(\text{Aging law}\right)\\ {L}^{\text{sl}}+V{t}_{\rm c}& \left(\text{Slip law}\right)\end{array}\right. .$$
(30)

This modification leads to a time-weakening model at high enough slip rate \(V\gg {L}^{\text{sl}}/{t}_{\rm c}\) for the slip law with the characteristic time \({t}_{\rm c}\).The fracture energy increases as the peak slip rate increases, and the length of the process zone is kept finite even after a rupture grows extensively. Andrews (2004) used a time-weakening friction in simulation of dynamic rupture to account for increasing fracture energy with rupture size. Note that this modification is not meant to make the simulation “realistic” regarding complex physical and chemical effects activated at coseismic slip rates (e.g., Di Toro et al. 2011). The healing of the fault after an earthquake rupture may rule out afterslip driven by the PER. Thus, a comparison of the effect of PER is important between a fault governed by the aging law and that governed by the slip law at a low quasistatic slip rate.

\({L}^{\text{ag}}\) is selected such that the nucleation radius \({R}_{\text{c}}\) (Rubin and Ampuero 2005) at the undrained condition becomes as follows:

$${R}_{\text{c}}=\frac{1}{\uppi }\frac{B}{{\left(B-A\right)}^{2}}\frac{\mu }{1-{\nu }_{\text{u}}}{L}^{\text{ag}}=0.15R .$$
(31)

This yields \({L}^{\text{ag}}=1.73\times {10}^{-6}R\). This is small enough for the rate-weakening patch to accommodate nucleation, so that a compact region of acceleration appears prior to an earthquake rupture (e.g., Chen and Lapusta 2009; Noda and Hori 2014). The length scale of the process zone (Lapusta and Liu 2009) is as follows:

$${\Lambda }_{0}^{\text{ag}}=\frac{9\pi }{32}\frac{\mu }{1-{\nu }_{\text{u}}}\frac{{L}^{\text{ag}}}{B}=4.63\times {10}^{-2}R .$$
(32)

To resolve the process zone of the rupture front, the interval of the grid points \(\Delta {x}^{\text{ag}}\) is selected as \(\Delta {x}^{\text{ag}}=4R/512=7.81\times {10}^{-3}R\) such that \({\Lambda }_{0}^{\text{ag}}/\Delta {x}^{\text{ag}}=5.93\). The interval between the periodic boundaries, \(16R\) is discretized by \(\text{2,048}\) grid points.

In example problems with the slip law, \({L}^{sl}\) is selected as \({L}^{sl}=2\times {10}^{-6}R\). The characteristic time \({t}_{\text{c}}\) is selected as follows:

$${t}_{\text{c}}=\frac{{L}_{\text{sl}}}{1\, \text{m}/\text{s}} .$$
(33)

The Rayleigh wave speed for the undrained Poisson’s ratio \({\nu }_{\text{u}}=0.34\) is approximately as follows:

$${c}_{\text{R}}=9.34\times {10}^{-1}{c}_{\text{s}} .$$
(34)

For a rupture of this terminal speed, the corresponding length scale of the process zone is as follows:

$${\Lambda }_{\text{R}}={c}_{\text{R}}{t}_{\text{c}}=5.60\times {10}^{-3}R .$$
(35)

Ampuero and Rubin (2008) showed that a length scale inversely proportional to \(B\) plays an important role in nucleation with the slip law. With the present parameter selection, it becomes as follows:

$${L}_{b}=\frac{\mu }{1-{\nu }_{\text{u}}}\frac{{L}^{\text{sl}}}{B}=6.06\times {10}^{-2}R .$$
(36)

If a point on the fault is in a steady state at \({V}_{\text{pl}}\) and accelerates owing to stress concentration up to \({V}_{\text{dy}}\), then Eqs. (28) and (30) yield the maximum slope of the slip weakening as follows:

$$\begin{array}{l}-\frac{\text{d}\tau }{\text{d}\delta }=\text{ln}\left(\frac{{V}_{\text{dy}}}{{V}_{\text{pl}}}\right)\frac{B}{{L}^{\text{sl}}+{V}_{\text{dy}}{t}_{\text{c}}}=\text{ln}\left(\frac{{V}_{\text{dy}}}{{V}_{\text{pl}}}\right){\left(1+\frac{{V}_{\text{dy}}}{1\, \text{m}/\text{s}}\right)}^{-1}\frac{B}{{L}^{\text{sl}}} . \end{array}$$
(37)

The corresponding on-fault length scale of the process zone is as follows:

$${\Lambda }_{0}^{\text{sl}}=\frac{9\pi }{32}{L}_{b}{\left(\text{ln}\left(\frac{{V}_{\text{dy}}}{{V}_{\text{pl}}}\right)\right)}^{-1}\left(1+\frac{{V}_{\text{dy}}}{1 \,\text{m}/\text{s}}\right) .$$
(38)

This has the minimum value achieved at \({V}_{\text{dy}}=5.92\times {10}^{-2} \,\text{m}/\text{s}\), generating:

$${\Lambda }_{0}^{\text{sl}}\ge 5.23\times {10}^{-2}{L}_{b}=3.17\times {10}^{-3}R .$$
(39)

Compared to Eq. (35), modification of \(L\) (Eq. (30)), and the selection of the present frictional parameters makes the resolution of the propagating rupture front easier than the nucleation of the earthquake ruptures. The interval of the grid points \(\Delta {x}^{\text{sl}}\) is selected as \(\Delta {x}^{\text{sl}}=4R/\text{4,096}=9.77\times {10}^{-4}R\) so that \({\Lambda }_{0}^{\text{sl}}/\Delta x=3.25\). The interval between the periodic boundaries, \(16R\) is discretized by \(\text{16,384}\) grid points.

Fluid transport property

Among the physical parameters, the permeability of the medium \(k\) is the most poorly constrained, thus the mobility of the fluid \(\kappa\) or the diffusion coefficient \(c\) are difficult to constrain. \(k\) varies by several orders of magnitude depending on many factors, such as rock type, conditions of consolidation, and degree of fracturing or damage. It is most likely heterogeneous and is known to show scale dependency (Schulze-Makuch et al. 1999). Accounting for these complexities is beyond the scope of this study. Here, \(\kappa\) was assumed to be uniform in the medium, and a parameter study was conducted.

First, the undrained limit (\(\kappa =0\)) was simulated for reference purposes. A sequence of 20 earthquakes was calculated, and the intervals of the 19th and 20th earthquakes were selected as the characteristic timescales of the seismic cycle \({T}_{\text{u}}\). Simulations with different \(\kappa\) values were conducted to see the effects of PER. The timescale of PER, \({T}_{\text{PE}}\) is given by an appropriate length scale to the second power divided by the diffusion coefficient \(c\). Adopting \(R\) as the length scale as follows:

$${T}_{\text{PE}}=\frac{{R}^{2}}{c}=\frac{1}{2}\frac{{R}^{2}}{\kappa \mu }\frac{{\alpha }^{2}\left(1-{\nu }_{\text{u}}\right){\left(1-2{\nu }_{\text{d}}\right)}^{2}}{\left({\nu }_{\text{u}}-{\nu }_{\text{d}}\right)\left(1-{\nu }_{\text{d}}\right)} .$$
(40)

Simulations with \({T}_{\text{PE}}/{T}_{\text{u}}\) from \({10}^{-7}\) to \({10}^{1}\) were conducted in the parameter study.

Initial conditions

The distribution of the initial loading on fault \({\tau }_{0}\) was selected consistently with a steady-state solution at \(V={V}_{*}={V}_{\text{pl}}\) and \(\tau ={\tau }_{*}\) as follows:

$${\tau }_{*}={\tau }_{0}-\frac{\mu }{2{c}_{\text{s}}}{V}_{\text{pl}} .$$
(41)

Initial perturbations in the state variables are given as follows:

$$\theta \left(x,0\right)=10\left(1+0.1\frac{x}{R}\right) \left(\text{aging law}\right) ,$$
(42)
$$\Theta \left(x,0\right)=A\left(1+0.1\frac{x}{R}\right) \left(\text{slip law}\right) .$$
(43)

The fault was initially stronger than in the steady-state solution to generate the first events shortly after the start of the simulation. Mirror symmetry at \(x=0\) was broken in the initial state to avoid perfectly symmetric solutions that are unstable. If numerical error including inevitable round-off error is controlled perfectly symmetric, then a numerical solution for a symmetric problem is symmetric. However, if numerical error is not symmetric, then the anti-symmetric component may grow and a numerical solution deviates from the symmetric solution unless it is stable. Because we are interested in a stable solution (attractor), there is no point simulating the growth of the numerical error from a very small amplitude.

Results

For presentation purposes, the length scale was set as \(R=1\,\text{ km}\) in visualizing the results, which yielded \({L}^{\text{ag}}=1.73\, \text{mm}\) and \({L}^{\text{sl}}=2.00\, \text{mm}\). If a reader is interested in a longer/shorter fault by a factor, the conversion is easily achieved by multiplying by the same factor the values having dimension of length (e.g., fault length and slip) and time, that is, the length times the speed (e.g., event intervals).

Aging law, undrained case

Figure 5 shows the simulation results for the undrained case. Seismic events are defined by a threshold of \(0.1 \,\text{m}/\text{s}\) in the spatially maximum slip rate. Coseismic slip distributions are indicated by red lines in Fig. 5a, and there was almost no gap between earthquakes in the central part of the rate-weakening patch. In the core of the seismogenic patch, the seismic coupling was almost one, and significant afterslip did not occur. Although the location of earthquake nucleation alternated between the positive and negative \(x\) sides of the patch, the recurrence interval decays to a stable value of \({T}_{\text{u}}=5.13\times {10}^{8} \,\text{s}\). The elastodynamic time window \({t}_{w}=4\, \text{s}\) corresponds to \(7.80\times {10}^{-9}\,{T}_{\text{u}}\).

Fig. 5
figure 5

Simulation result in the undrained case with the aging law. For presentational purposes, the length scale is selected as \(R=1 \,\text{km}\). a Cumulative slip distribution, blue: every \({10}^{8} \,\text{s}\), red: every \(0.02 \, \text{s}\) during earthquakes. Integers indicate ordinal numbers of earthquakes in the simulated sequence. b Shear stress (red) and strength (bule) of the fault relative to the reference at \(x=0.5\Delta {x}^{\text{ag}}=R/256\) as a function of time after the 19th earthquake. c Slip rate near the center, \(x=0.5\Delta {x}^{\text{ag}}\) as a function of time after the 19th earthquake

Figure 5b shows the interseismic shear stress and fault strength relative to the reference \({\tau }_{\text{u}}-{\tau }_{*}\) and \(B\text{ln}\left({\theta }_{\text{u}}\right)\) at the center (\(x=0.5\Delta x\)) as a function of time after the 19th earthquake \({t}_{\text{pos}}\). The subscript \({}_{\text{u}}\) indicates the solution in the undrained case, and \({t}_{\text{pos}}=0\) is defined by a threshold at the spatially maximum slip rate of \(0.1 \, \text{m}/\text{s}\). Equation (27) indicates that \(\text{ln}\left(V\right)\) is proportional to the stress minus strength, \(\left({\tau }_{\text{u}}-{\tau }_{*}\right)-B\text{ln}\left({\theta }_{\text{u}}\right)\) (Fig. 5c). Thus, the comparison of \({\tau }_{\text{u}}-{\tau }_{*}\) and \(B\text{ln}\left({\theta }_{\text{u}}\right)\) helps us to understand the mechanism of acceleration and deceleration of the fault (Fig. 5c). After the 19th earthquake, \({V}_{\text{u}}\) was still sufficiently large, even below the threshold, to decrease \({\tau }_{\text{u}}\) in the patch until approximately \({t}_{\text{pos}}={10}^{-9}{T}_{\text{u}}\). Then, \({\tau }_{\text{u}}\) at the center increased due of aseismic slip outside the locked patch and penetration of creep fronts into it, significantly after about \({t}_{\text{pos}}={10}^{-1}{T}_{\text{u}}\). \(B\text{ln}\left({\theta }_{\text{u}}\right)\) shows a log-\(t\) increase with the slope of \(B\text{ln}\left(10\right)\), which is characteristic of the aging law in the limit of zero slip rate after the cut-off time due to initially non-zero \({\theta }_{\text{u}}\). During the interseismic period, \({V}_{\text{u}}\) varied by more than \(10\) orders of magnitude. The change in \(\text{ln}\left({V}_{\text{u}}\right)\) was first dominated by the change in strength before \({t}_{\text{pos}}/{T}_{\text{u}}\) becomes a good fraction of \(1\), and then by a change in stress. The increase in \(B\text{ln}\left({\theta }_{\text{u}}\right)\) leads to the deceleration of the fault, and \({V}_{\text{u}}\) becomes smaller than the long-term slip rate \({V}_{\text{pl}}={10}^{-9} \,\text{m}/\text{s}\) at \({t}_{\text{pos}}=2.86\times {10}^{-6}{T}_{\text{u}}\).

Aging law, effect of PER

The recurrence interval of the undrained case \({T}_{\text{u}}=5.13\times {10}^{8}\, \text{s}\) and Eq. (40) yielded \(\kappa =k/\eta =1.75\times {10}^{-14}{\left({T}_{\text{PE}}/{T}_{\text{u}}\right)}^{-1} {\text{m}}^{2}/\text{Pas}\). For a typical value of water viscosity \(\eta ={10}^{-4} \,\text{Pas}\), \(k=1.75\times {10}^{-18}{\left({T}_{\text{PE}}/{T}_{\text{u}}\right)}^{-1} {\text{m}}^{2}\). The investigated range of \({T}_{\text{PE}}/{T}_{\text{u}}\) from \({10}^{-7}\) to \({10}^{1}\) corresponds to the range of \(k\) from \(1.75\times {10}^{-11} {\text{m}}^{2}\) to \(1.75\times {10}^{-19} {\text{m}}^{-2}\).

An example of the simulation result with PER (\({T}_{\text{PE}}={10}^{-6}{T}_{\text{u}}\)) is shown in Fig. 6. It is apparent from the plot of cumulative slip (Fig. 6a) that there was no significant afterslip in the middle of the rate-weakening patch. This was the instance for all simulations conducted.

Fig. 6
figure 6

Example simulation with PE for a fault governed by the aging law. a Cumulative slip distribution, blue: every \({10}^{8} \,\text{s}\), red: every \(0.02 \,\text{s}\) during earthquakes. Integers indicate ordinal numbers of earthquakes in the simulated sequence. b Shear stress (red) and strength (bule) of the fault relative to the reference at \(x=0.5\Delta {x}^{\text{ag}}=R/256\) as a function of time after the 19th earthquake. c Slip rate near the center, \(x=0.5\Delta {x}^{\text{ag}}=R/256\) as a function of time after the 19th earthquake

The evolution of the fault strength \(B\text{ln}\left(\theta \right)\) almost traces that in the undrained case, while \(\tau -{\tau }_{*}\) increased at approximately \({t}_{\text{pos}}={10}^{-7}{T}_{\text{u}}=0.1 {T}_{\text{PE}}\) by approximately \(1 \text{MPa}\). The PER loaded the fault in the post-seismic period, but the amplitude was not sufficient to enhance the afterslip significantly. Because \(A=1 \text{MPa}\), the additional loading due to PER increased \(V\) by a factor of \(3\) (Fig. 6c), and this effect persisted until the next earthquake. Despite an increase in \(V\), the time-dependent strengthening without slip increment (the first term in Eq. (27)) dominated, such that PER did not affect \(\theta\).

Cases with different \({T}_{\text{PE}}\) are compared in Fig. 7. The stress buildup due to PER occurred at approximately \({t}_{\text{pos}}=0.1 {T}_{\text{PE}}\), and its amplitude was smaller for a larger \({T}_{\text{PE}}\) (Fig. 7a). With \({T}_{\text{PE}}={10}^{1}{T}_{\text{u}}\), PER required a longer time than the recurrence interval of the earthquake, and its effect was hardly recognized. The effect of PER on the fault strength \(B\text{ln}\left(\theta \right)\) is not clear (Fig. 7b), and thus the effect on the slip rate (Fig. 7c) is explained by that in \(\tau\).

Fig. 7
figure 7

Effect of PE at \(x=0.5\Delta {x}^{\text{ag}}\) with the aging law and various fluid mobility as a function of the time after the 19th earthquake. Poroelastic diffusion time \({T}_{\text{u}}={R}^{2}/c\) is indicated by color as shown in a. a Shear stress, b shear strength, and c Slip rate relative to the undrained case

Slip law, undrained case

The undrained case with the slip law is shown in Fig. 7. Although \(L\) at a low slip rate was longer than in the aging law, the nucleation size was much smaller (Ampuero and Rubin 2008). Similar to the case with the aging law, the afterslip was difficult to recognize in Fig. 8a. The recurrence interval was \({T}_{\text{u}}=4.71\times {10}^{8} \,\text{s}\). The elastodynamic time window \({t}_{w}=4\, \text{s}\) corresponds to \(8.49\times {10}^{-9}\,{T}_{\text{u}}\).

Fig. 8
figure 8

Simulation result in the undrained case with the slip law. For presentational purposes, the length scale is selected as \(R=1 \, \text{km}\). a Cumulative slip distribution, blue: every \({10}^{8}\, \text{s}\), red: every \(0.02 \, \text{s}\) during earthquakes. Integers indicate ordinal numbers of earthquakes in the simulated sequence. b Shear stress (red) and strength (bule) of the fault relative to the reference at \(x=0.5\Delta {x}^{\text{sl}}=R/2048\) as a function of time after the 19th earthquake. c Slip rate near the center, \(x=0.5\Delta {x}^{\text{sl}}\) as a function of time after the 19th earthquake

The evolution of the stress relative to the reference \({\tau }_{\text{u}}-{\tau }_{*}\) (Fig. 8b) is similar to that in the case with the aging law (Fig. 5b): \({\tau }_{\text{u}}-{\tau }_{*}\) decreased until approximately \({t}_{\text{pos}}={10}^{-9}{T}_{\text{u}}\), and then increased significantly after approximately \({t}_{\text{pos}}={10}^{-1}{T}_{\text{u}}\). The fault strength \({\Theta }_{\text{u}}\) showed a log-\(t\) increase, even though it was not predicted in the limit of \(V\to 0\) (Eq. (28)). The evolution of \({\Theta }_{\text{u}}\) requires a slip increment for the slip law. The strengthening rate was smaller than \(B\text{ln}\left(10\right)\), as mentioned by Marone (1998) in the context of a slide-hold-slide friction experiment. \({V}_{\text{u}}\) decreased with time more slowly than in the case of the aging law and became smaller than \({V}_{\text{pl}}\) at \({t}_{\text{pos}}=4.81\times {10}^{-4}{T}_{\text{u}}\). It continued to decrease until \({t}_{\text{pos}}\) became a good fraction of \({T}_{\text{u}}\), and increased thereafter (Fig. 8c). \({\Theta }_{\text{u}}\) also increased sharply during this acceleration period.

Slip law, effect of PER

The recurrence interval \({T}_{\text{u}}=4.71\times {10}^{8}\, \text{s}\), Eq. (40), and \(\eta ={10}^{-4} \,\text{Pas}\) yield \(k=1.91\times {10}^{-18}{\left({T}_{\text{PE}}/{T}_{\text{u}}\right)}^{-1} {\text{m}}^{2}\). The investigated range of \({T}_{\text{PE}}/{T}_{\text{u}}\) from \({10}^{-7}\) to \({10}^{1}\) corresponds to the range of \(k\) from \(1.91\times {10}^{-11} {\text{m}}^{2}\) to \(1.91\times {10}^{-19} {\text{m}}^{-2}\).

An example case with the slip law and PER (\({T}_{\text{PE}}={10}^{-6}{T}_{\text{u}}\)) is shown in Fig. 9, on the same vertical scale as that with the aging law (Fig. 6). Again, afterslip was hardly visible in the plot of the cumulative slip (Fig. 9a). This was the instance for all simulations conducted. Therefore, the present simulations do not support significant afterslip in the core of the seismogenic patch due to PER for the slip law or for the aging law.

Fig. 9
figure 9

Example simulation with PE for a fault governed by the aging law. a Cumulative slip distribution, blue: every \({10}^{8} \, \text{s}\), red: every \(0.02 \, \text{s}\) during earthquakes. Integers indicate ordinal numbers of earthquakes in the simulated sequence. b Shear stress (red) and strength (bule) of the fault relative to the reference at \(x=0.5\Delta {x}^{\text{sl}}=R/2048\) as a function of time after the 19th earthquake. c Slip rate near the center, \(x=0.5\Delta {x}^{\text{sl}}\) as a function of time after the 19th earthquake

\(\tau -{\tau }_{*}\) increased due to PER by approximately \(1\text{ MPa}\) at approximately \({t}_{\text{pos}}=0.1 {T}_{\text{PE}}\), and \(\Theta\) also increased relative to the undrained case almost at the same time (Fig. 9b). PER increased \(V\) (Fig. 9c), but this effect was minor compared to the case with the aging law (Fig. 6c) owing to the increase in \(\Theta\).

Cases with different \({T}_{\text{PE}}\) are compared in Fig. 10. Loading due to PER increased \(\tau\) at approximately \({t}_{\text{pos}}=0.1 {T}_{\text{PE}}\) (Fig. 10a). When \({T}_{\text{PE}}={10}^{1} {T}_{\text{u}}\), PER did not affect \(\tau\) significantly. \(\Theta\) also increased at approximately \({t}_{\text{pos}}=0.1 {T}_{\text{PE}}\) by a smaller but comparable amount (Fig. 10b). Consequently, the increase in \(V\) relative to the undrained case was only modest (Fig. 10c).

Fig. 10
figure 10

Effect of PE at \(x=0.5\Delta {x}^{\text{sl}}\) with the slip law and various fluid mobility as a function of the time after the 19th earthquake. Poroelastic diffusion time \({T}_{\text{u}}={R}^{2}/c\) is indicated by color as shown in a. a Shear stress, b shear strength, and c Slip rate relative to the undrained case

Discussion

The series of simulations conducted in this study indicate that PER causes postseismic loading to the patch of coseismic slip, but the enhancement of afterslip due to PER is negligible. In the idealized case of uniform stress drop and constant frictional strength (Fig. 1), the afterslip normalized by the coseismic slip is \({\mu }_{\text{u}}^{^{\prime}}/{\mu }_{\text{d}}^{^{\prime}}-1=\left(1-{\nu }_{\text{d}}\right)/\left(1-{\nu }_{\text{u}}\right)-1=1.36\times {10}^{-1}\). Compared to this, Figs. 6a and 8a clearly show that the postseismic healing of a fault efficiently suppresses the afterslip at the center of the rate-weakening patch. This is the case not only in cases with the aging law but also with the slip law. The simulation results were consistent with the absence of afterslip inside the coseismic slip patches in observation.

Marone et al. (1995) inferred a very long cut-off healing time of approximately \(100 \text{days}\approx {10}^{7}\, \text{s}\) based on the observation of small repeating earthquakes. The long cut-off time delays healing and may produce afterslip due to PER. However, by what means to realize such a long cut-off time in RSF is an open question. In the aging law, the cut-off time is given by \(L\) divided by the coseismic slip rate well behind the rupture front. Stress drop \(\Delta \sigma\) of \(3 \text{MPa}\) and the impedance of \(\mu /{c}_{\text{s}}\) of \(10 \, \text{MPa}{\left(\text{m}/\text{s}\right)}^{-1}\) yields coseismic slip rate of the order of \(0.1 \, \text{m}/\text{s}\) in self-similar dynamic rupture (Broberg 1978). For a cut-off time of \({10}^{7} \, \text{s}\), \(L\) must be on the order of \({10}^{6}\text{ m}\). Evidently, the nucleation size (Eq. (31)) is too large to model a realistic earthquake.

The intrinsic cut-off time of the state evolution \({t}_{\text{cx}}\) (Nakatani and Scholz 2006) may also produce a long cut-off time. If it is implemented in the aging law, the friction law becomes as follows:

$$\begin{array}{l}\begin{array}{l}\tau ={\tau }_{*}+A\text{ln}\left(\frac{V}{{V}_{*}}\right)+B\text{ln}\left(\theta +1\right)\\ \dot{\theta }=\frac{V}{L}\left(\frac{{V}_{*}}{V}-\theta \right)\\ {V}_{*}=L/{t}_{\text{cx}}\end{array} .\end{array}$$
(44)

The steady-state shear strength is given as follows:

$${\tau }_{\text{ss}}={\tau }_{*}+A\text{ln}\left(\frac{V{t}_{\text{cx}}}{L}\right)+B\text{ln}\left(\frac{L}{V{t}_{\text{cx}}}+1\right) ,$$
(45)

and the rate dependency, as follows:

$$\begin{array}{l}\frac{\text{d}{\tau }_{\text{ss}}}{\text{dln}\left(V\right)}=A-\frac{B}{1+V{t}_{\text{cx}}/L} .\end{array}$$
(46)

Therefore, the friction law becomes rate strengthening at \(V>\left(B/A+1\right)L/{t}_{\text{cx}}\equiv {V}_{\text{c}}\). Laboratory-measured values of \(L<1\, \text{mm}\) (e.g., Blanpied et al. 1998) and the large cutoff time \({t}_{\text{cx}}\approx {10}^{8} \, \text{s}\) yield \({V}_{\text{c}}<{10}^{-11} \, \text{m}/\text{s}\), which is too small for frictional instability to occur at typical long-term slip rate. The reconciliation of the long cut-off time and seismogenic fault behavior deserves future study.

Conclusions

Quasistatic poroelasticity (PE) was successfully implemented in a dynamic earthquake sequence simulation for a fault governed by a rate- and state-dependent friction law (RSF) using a spectral boundary integral equation method. In this method, the numerical approximation of the time dependency of Green’s function in the wavenumber domain and the definition of memory variables are the keys to avoid temporal convolution for PE. The additional computational cost relative to the elastic simulation was negligible. A simple model of a planar fault with a rate-weakening patch embedded in a rate-strengthening region was simulated to investigate the effects of poroelastic rebound (PER). As for the RSF, not only the aging law, but also the slip law is investigated with modifications to increase the characteristic state-evolution slip distance at coseismic slip rates for tractability of the numerical simulation. The simulation results indicate that PER causes postseismic loading to the region of stress drop of approximately \(0.1{R}^{2}/c\) after an earthquake, where \(R\) is the half length of the coseismic slip patch and \(c\) is the diffusivity of the pore fluid pressure. For both the aging law and slip law, however, the healing of a fault efficiently suppresses afterslip inside a coseismic slip patch, while it was suggested by previous theoretical studies for a crack of constant strength. The simulation results are consistent with the absence of afterslip within the coseismic slip patches in the observations. When the cases with the two different friction laws are compared, healing is more significant in the cases with the aging law, but the effect of PER on the slip rate \(V\) is minor in the cases with the slip law. In the interseismic period, the log-\(t\) healing term is dominant for the aging law. Therefore, the change in \(V\) hardly affects the evolution of the fault strength. On the other hand, for the slip law, healing is driven by the slip increment, and thus the increase in \(V\) results in additional strengthening, which works as a negative feedback.

Availability of data and materials

No observational data were used for this study. The simulation code used in this study is available upon reasonable request to the author.

Abbreviations

FFT:

Fast Fourier transformation

ODE:

Ordinary differential equation

PE:

Poroelasticity

PER:

Poroelastic rebound

RSF:

Rate- and state-dependent friction law

SBIEM:

Spectral boundary integral equation method

References

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Acknowledgements

The author would like to thank Y. Iio for a discussion useful in motivating the present study and T. Nishimura for a discussion on the interpretation of geodetic inversion results. Two reviewers (P. Romanet and E. R. Heimisson) gave very useful comments in improving the manuscript. In the review process, Heimisson informed the author of his manuscript and an online preprint on a similar problem submitted to JGR on the same day with the present paper.

Funding

The present study was supported by the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan under its Second Earthquake and Volcano Hazards Observation and Research Program (Earthquake and Volcano Hazard Reduction Research) and by JSPS KAKENHI Grant Numbers 21H05201, 19K04038.

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HN developed the present method, coded the simulation program, analyzed the results, and composed the original draft. The author read and approved the final manuscript.

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Correspondence to Hiroyuki Noda.

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Noda, H. Dynamic earthquake sequence simulation with an SBIEM accounting for interseismic poroelastic rebound. Earth Planets Space 74, 89 (2022). https://doi.org/10.1186/s40623-022-01649-8

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Keywords

  • Poroelasticity
  • Numerical simulation
  • Earthquake sequence
  • SBIEM
  • Fault healing
  • Afterslip