Quasi-Dynamic 3D Modeling of the Generation and Afterslip of a Tohoku-oki Earthquake Considering Thermal Pressurization and Frictional Properties of the Shallow Plate Boundary

Abstract

The generation of the 2011 Tohoku-oki earthquake has been modeled by many authors by considering a dynamic weakening mechanism such as thermal pressurization (TP). Because the effects of TP on afterslip have not been investigated, this study develops a 3D quasi-dynamic model of the earthquake cycle to investigate afterslip of the Tohoku-oki earthquake, considering TP and the geometry of the plate boundary. We employ several velocity-weakening (VW) patches for M_w 7 class events, and two large shallow VW patches. The frictional properties are set as velocity-strengthening (VS) outside the VW patches. The results show that, during megathrust earthquakes, fast slip propagates to the surrounding VS regions near the VW patches owing to weakening by TP. Following M_w 9 events, large afterslips occur in regions below the northern shallow rupture area in the off-Fukushima region close to the Japan Trench, which is consistent with observations. In the VS region near the VW patches, during the early afterslip period, frictional behavior exhibits less VS with increasing slip velocity due to pore pressure reduction. We also consider the frictional properties of the shallow plate boundary fault off Tohoku, which exhibits a transition from VW to VS from low to high slip velocities. The results show the occurrence of slow slip events (SSEs) at intervals of a few decades at the shallow plate boundary. During megathrust events, the VW property at low slip velocity promotes slip along the shallow SSE region more than the case with VS property throughout the entire velocity range.

Appendix

We perform a quasi-dynamic analysis using the following equation (e.g. Rice 1993):

$$\tau_{i} = \sum\limits_{{i_{\text{s}} }} {k_{{i - i_{\text{s}} }} \left( {v_{\text{pl}} t - u_{{i_{\text{s}} }} } \right)} - \frac{G}{2\beta }\frac{{{\text{d}}u_{i} }}{{{\text{d}}t}},$$

(8)

where $k_{{i - i_{\text{s}} }}$ is the stiffness which is the stress at the center of gravity of triangular element $i$ caused by uniform slip over triangular element $i_{\text{s}}$, $G$ is the rigidity and $\beta$ is the shear wave velocity. To calculate the stiffness for the triangular elements, we use the program developed by Stuart et al. (1997). In the quasi-dynamic modeling, we add seismic radiation damping to the second term introduced by Rice (1993) to approximate the effects of inertia during earthquakes. The centers of gravity of the triangular elements are taken as the nodes (i.e., co-location points).

From Eqs. (1) and (8), we obtain the following equation:

$$\mu \left( {v,\varTheta } \right)\frac{{\sigma_{{{\text{n}},i}}^{\text{eff}} }}{G} = \sum {k^{\prime}_{{i - i_{\text{s}} }} } \left( {v_{\text{pl}} t - u_{{i_{\text{s}} }} } \right) - \frac{1}{2\beta }\frac{{{\text{d}}u_{i} }}{{{\text{d}}t}},$$

(9)

where $k^{\prime}_{{i - i_{\text{s}} }} = k_{{i - i_{\text{s}} }} /G$. From this equation, it is clear that we get the same solution for the same $\sigma_{{{\text{n}},i}}^{\text{eff}} /G$. In this calculation, $G$ is assumed to be 30 GPa. However, we get the same result when $G$ = 40 GPa and $\sigma_{\text{n}}^{\text{eff}}$ increases by a factor of 40/30. In this case,$\tau$, $T$ and $P$ also increase by a factor of $40/30$.

We solve Eqs. (1)–(3) and (8) using a fifth-order Runge–Kutta method with adaptive step-size control (Press et al. 1992), to obtain $v^{n + 1}$ and $\tau^{n + 1}$ at time step $n + 1$ using $P^{n}$ and $T^{n}$ at step $n$. We calculate the average slip rate $\underline{\nu }^{n + 1} = (v^{n + 1} + v^{n} )/2$ and shear stress $\underline{\tau }^{n + 1} = (\tau^{n + 1} + \tau^{n} )/2$ between steps $n$ and $n + 1$. Using the values of $\underline{\nu }^{n + 1}$ and $\underline{\tau }^{n + 1}$ to calculate the heat source ($\omega$), we obtain $P^{n + 1}$ and $T^{n + 1}$ by solving Eqs. (4) and (5) using a spectral method (Noda and Lapusta 2010). We then update $P^{n + 1}$ in Eq. (1). A few iterations are performed with a fixed step size to obtain the converged solutions.