SEAS models capture the entire earthquake cycle, i.e. tectonic loading, nucleation, rupture, and afterslip, within one physical model (Erickson et al., 2020). A fault is idealized as an infinitesimally thin fault surface embedded in linear elastic media, and on-fault behaviour is described via laboratory-derived rate and state friction laws. Friction couples the slip $S$ , normal stress $\sigma_n$ , and shear traction $\tau$ on the fault surface $\Gamma_F$ , cf. the following conceptual illustration of a normal fault.

normal_fault

The rate and state friction relations are given by

\begin{aligned} -\tau_i &= \sigma_nf(|V|,\psi) V_i / |V| \\ \frac{dS_i}{dt} &= V_i\\ \frac{d\psi}{dt} &= g(|V|,\psi) \end{aligned}

The first equation states that shear traction is proportional to normal stress times a coefficient of friction $f$ , where $f$ depends on the slip-rate and a single state variable. Moreover, slip-rate $V$ and shear stress $\tau$ are anti-parallel. The evolution of state is controlled by $g$ .

The friction equations are coupled through the equations of linear elasticity in the domain $\Omega$ , i.e.

-\frac{\partial\sigma_{ij}(\bm{u})}{\partial x_j} = 0.

(Sums over indices appearing twice are implied.) A slip boundary is imposed in the linear elasticity problem, that is,

\llbracket u_i\rrbracket = T_{ij}S_j \text{ on } \Gamma_F,

where $T_{ij}(\bm{n})$ is a $D \times (D-1)$ matrix, $D$ being the space dimension, which contains a tangential basis of a fault segment with normal n.

The linear elasticity problem omits modelling of seismic waves, which are relevant during an earthquake but can be neglected otherwise. In order to get a stable formulation, the outflow of energy due to seismic waves is approximated with the damping term $\eta V_i$ in the frictional relation. (Rice, 1993)

-\tau_i = \sigma_nf(|V|,\psi) V_i / |V| \color{red} + \eta V_i

Adding the constitutive relation, Dirichlet and Neumann boundary conditions, and the damping term we get the following system of equations:

Linear elasticity with slip BC

\begin{aligned} -\frac{\partial\sigma_{ij}(\bm{u})}{\partial x_j} &= F_i & \text{ in } & \Omega\\ \sigma_{ij}(\bm{u}) &= c_{ijkl}\epsilon_{kl}(\bm{u}) & \text{ in } & \Omega\\ u_i &= g_i& \text{ on } & \Gamma_D \\ \sigma_{ij}(\bm{u})n_j &= 0 & \text{ on } & \Gamma_N \\ \llbracket u_i\rrbracket &= T_{ij}S_j & \text{ on } & \Gamma_F \end{aligned}

( $F$ : body force, $c$ : stiffness tensor, $n$ : unit normal)

Rate and state friction on $\Gamma_F$

\begin{aligned} -\tau_i &= \sigma_nf(|V|,\psi) V_i / |V| + \eta V_i \\ \frac{dS_i}{dt} &= V_i\\ \frac{d\psi}{dt} &= g(|V|,\psi)\\ \tau_i &= T_{ji}\sigma_{jk}(\bm{u})n_k \\ \sigma_n &= n_i\sigma_{ij}(\bm{u})n_j \end{aligned}

( $\delta$ : Kronecker symbol, $n$ : unit normal)

Although seismic waves are neglected, tectonic loading, nucleation, rupture, and afterslip can be observed in a SEAS model:

seas

The above shows a 2D simulation of a normal fault (vertical axis) over 1500 years. Slip profiles are plotted along the horizontal axis, and displaced by time in the in-screen direction. An earthquake (in red) occurs about every hundred years.

Time-steps vary strongly: In the interseismic phase time-steps of days to months are possible, whereas in the coseismic phase time-steps in the order of milliseconds are required.

References

Erickson, B. A., Jiang, J., Barall, M., Lapusta, N., Dunham, E. M., Harris, R., Abrahams, L. S., Allison, K. L., Ampuero, J. P., Barbot, S., Cattania, C., Elbanna, A., Fialko, Y., Idini, B., Kozdon, J. E., Lambert, V., Liu, Y., Luo, Y., Ma, X., … Wei, M. (2020). The Community Code Verification Exercise for Simulating Sequences of Earthquakes and Aseismic Slip (SEAS). Seismological Research Letters, 91(2A), 874–890. https://doi.org/10.1785/0220190248
Rice, J. R. (1993). Spatio-temporal complexity of slip on a fault. Journal of Geophysical Research: Solid Earth, 98(B6), 9885–9907. https://doi.org/10.1029/93JB00191