A Finite Difference Method for Irregular Geometries: Application to Dynamic Rupture on Rough Faults (poster with former undergraduate SCEC intern)
David Belanger (Harvard) and Eric M. Dunham (Harvard)
We have developed a finite difference method to solve dynamic rupture problems in irregular geometries. Our objective is to connect properties of high frequency radiation produced during slip on rough faults to statistical measures of fault roughness. To handle irregular geometries, we transform the governing equations from a non-Cartesian coordinate system that conforms to the irregular boundaries of the physical domain to a Cartesian coordinate system in a rectangular computational domain, and solve the equations in the computational domain. To accurately capture the high frequency wavefield, we use a method that produces far smaller numerical oscillations than those plaguing conventional finite difference/element methods. The governing equations (momentum conservation and Hooke's law) are written as a system of first-order equations for velocity and stress, which are defined at a common set of grid points and time steps (i.e., there is no staggering in space or time). Time stepping is done using an explicit third-order Runge-Kutta method. The equations are hyperbolic and the fields can be decomposed into a set of waves (with associated wave speeds) propagating in each coordinate direction. Spatial derivatives are computed with fifth-order WENO (weighted essentially non-oscillatory) finite differences in the upwind direction associated with each wave [Jiang and Shu, J. Comp. Phys., 126(1), 202-228, 1996]. Rather than using data from a single stencil (i.e., set of grid points) to calculate the derivative, a weighted combination of data from several candidate stencils is used. The weights are assigned based on solution smoothness within each stencil, and stencils in which the solution exhibits excessive variations are given minimal weight. Consequently, numerical oscillations are suppressed, even in the vicinity of the rupture front and at wavefronts. Boundary conditions are implemented by again appealing to the hyperbolic nature of the governing equations. At each point on a boundary (or fault), the solution is decomposed into a set of waves propagating into and out of the boundary. The amplitudes of incoming waves are preserved, while those of outgoing waves are modified to satisfy the boundary conditions. On the fault, this amounts to solving the friction law together with an equation expressing shear stress as the sum of a load, the radiation damping response, and the stress change carried by the incoming waves.
Earthquake Ruptures with Thermal Weakening and the Operation of Faults at Low Overall Stress Levels (poster)
Eric M. Dunham (Harvard), Hiroyuki Noda (Caltech), and James R. Rice (Harvard)
We have conducted rupture propagation simulations incorporating flash heating of microscopic asperity contacts and thermal pressurization of pore fluid [Noda, Dunham, and Rice, in preparation, 2007-08]. These are arguably the primary weakening mechanisms at coseismic slip rates, at least prior to large slip accumulation. Ruptures on strongly rate-weakening faults take the form of slip pulses or cracks, depending on the background stress level. Self-sustaining slip pulses exist only within a narrow range of stresses; below this range, artificially nucleated ruptures arrest, and above this range, ruptures are crack-like. Certain features of our simulations lend support to the idea that faults operate at the minimum critical level required for propagation, such that natural earthquakes take the form of slip pulses. Using flash heating parameters measured in recent laboratory experiments, the critical range occurs when the ratio of shear to effective normal stress on the fault is 0.2-0.3 (a range that is only mildly influenced by the choice of thermal pressurization parameters, at least within a reasonable range of uncertainty around laboratory-measured values). This level is consistent with the low stress inferred to be acting on the San Andreas fault (SAF); a ratio of shear to effective normal stress of 0.24 was measured at 2.1 km depth in the SAFOD pilot hole [Hickman and Zoback, 2004], adding further support to other measurements indicating that the maximum horizontal compressive stress is nearly perpendicular to the SAF. While the overall background stress level is quite small, stresses concentrated at the rupture front are consistent with typical static (and low velocity) friction coefficients of 0.6-0.9; this stress concentration is required to initiate slip. Growing slip pulses have stress drops close to 3 MPa and feature slip increasing with propagation distance at a rate of about 0.14 m/km. These values are consistent with seismic inferences of stress drop and field constraints on slip-length scaling. On the other hand, cracks have stress drops of over 20 MPa, and slip at the hypocenter increases with propagation distance at a rate of about 1 m/km.
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Last updated: September 16, 2008