A new paradigm in modeling pulselike ruptures: The pulse energy equation

Earthquake ruptures are complex processes both spatially and temporally. Seismic source inversions and numerical simulations show that ruptures can occur in one of two primary modes; the slip pulse mode and the expanding crack mode. In order to understand the long time behavior of fault systems sustaining repeated earthquake ruptures we have to run models through hundreds or thousands of events so that the system under consideration evolves into a statistically stable stressed state that is consistent with the assumed friction law. For systems that fail primarily in pulse-like ruptures, the ruptures and the prestress can become very complex at many length scales. With the current computational methods and resources, we are unable to deduce which prestresses are compatible with an assumed friction law. However, the spatial compactness of slip pulses may provide another alternative approach to simulating many complex events. Here we construct a nonlinear ordinary differential equation that relates the final slip in a pulselike event to the prestress that exists before that event. The differential equation is based on an energy conservation principle for the slip pulse. We illustrate our methodology within the framework of 1D spring block slider system with strong velocity weakening friction. In this system, as the slip pulse propagates, it changes the potential energy of the springs, it loses energy through frictional dissipation, and it possesses a nonzero kinetic energy. The energy conservation dictates that the difference between the change in potential energy and the frictional work dissipated has to be equal to the kinetic energy of the pulse. By expressing the change in the potential energy of the system exactly in terms of slip and prestress, and through approximating the frictional dissipation and pulse kinetic energy as power law functions in the pulse slip (an approximation that is motivated by the approximate self-similarity of the propagating pulse) we arrive at the nonlinear first order differential equation that relates the final slip to the prestress distribution. Our methodology reduces the complexity of the problem (at least in the framework of the spring block slider model) from solving a number of coupled differential equations equal to the number of blocks to solving a single nonlinear differential equation. This equation predicts the final slip distribution with a considerable saving in the computational time. The direct knowledge of the final slip facilitates modeling repeated ruptures in a much shorter time. We report on our attempts to produce long time statistics from our equation consistent with the long time statistics of the spring block system. Finally, we think that a similar approach for the more complicated case of ruptures propagating in the continuum may be possible. It is certainly more challenging due to the need of taking into consideration the effect of radiated energy through the transmitted wave field and also the nonlocalization of the potential energy density. However, it may be possible , since similar principles hold in both of the continuum and discrete systems, e.g. energy conservation and spatial localization of slip pulses suggesting that local interaction are of prime importance relative to longer range interaction.