### 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:

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

(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.

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.