Eric M. Dunham

Earthquake ruptures with thermal weakening and the operation of major faults at low overall stress levels

We model ruptures on faults that weaken in response to flash heating of microscopic asperity contacts (within a rate-and-state framework) and thermal pressurization of pore fluid. These are arguably the primary weakening mechanisms on mature faults 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. Self-sustaining slip pulses exist within a narrow range of stresses: below this range, artificially nucleated ruptures arrest; above this range, ruptures are crack-like. Natural earthquakes will occur as slip pulses if faults operate at the minimum stress required for propagation. Using laboratory-based flash heating parameters, propagation is permitted when the ratio of shear to effective normal stress on the fault is 0.2–0.3; this is mildly influenced by reasonable choices of hydrothermal properties. The San Andreas and other major faults are thought to operate at such stress levels. While the overall stress level is quite small, the peak stress at the rupture front is consistent with static friction coefficients of 0.6–0.9. Growing slip pulses have stress drops of ∼3 MPa; slip and the length of the slip pulse increase linearly with propagation distance at ∼0.14 and ∼30 m/km, respectively. These values are consistent with seismic and geologic observations. In contrast, cracks on faults of the same rheology have stress drops exceeding 20 MPa, and slip at the hypocenter increases with distance at ∼1 m/km.

Evidence for a Supershear Transient during the 2002 Denali Fault Earthquake

Elastodynamic considerations suggest that the acceleration of ruptures to supershear velocities is accompanied by the release of Rayleigh waves along the fault from the stress breakdown zone. These waves generate a secondary slip pulse trailing the rupture front, but manifest almost entirely in ground motion perpendicular to the fault in the near-source region. We construct a spontaneously propagating rupture model exhibiting these features and use it to explain ground motions recorded during the 2002 Denali fault earthquake at pump station 10, located 3 km from the fault. We show that the initial pulses on both the fault normal and fault parallel components are due to the supershear stress release on the fault, whereas the later- arriving fault normal pulses result from the trailing subshear slip pulse on the fault. Online material: MPEG movies of rupture history and ground motion.

Earthquake Ruptures with Strongly Rate-Weakening Friction and Off-Fault Plasticity, Part 1: Planar Faults

Abstract Observations demonstrate that faults are fractal surfaces with deviations from planarity at all scales. We study dynamic rupture propagation on self-similar faults having root mean square (rms) height fluctuations of order 10 -3 to 10 -2 times the profile length. Our 2D plane strain models feature strongly rate-weakening fault friction and off-fault Drucker–Prager viscoplasticity. The latter bounds otherwise unreasonably large stress concentrations in the vicinity of bends. Our choice of a cohesionless yield function prevents tensile stress states and thus fault opening. A consequence of strongly rate-weakening friction is the existence of a critical background stress level above which self-sustaining rupture propagation, in the form of self-healing slip pulses, first becomes possible. Around this level, at which natural faults are expected to operate, ruptures become extremely sensitive to fault roughness and exhibit substantial fluctuations in rupture velocity. Except for shallow inclinations of the maximum compressive stress to the fault (less than about 20°), the fluctuations are anticorrelated with the local fault slope. These accelerations and decelerations of the rupture, together with naturally emerging slip heterogeneity, excite waves of all wavelengths and result in ground acceleration spectra that are flat at high frequency, consistent with observed strong motion records.

Near-source ground motion from steady state dynamic rupture pulses

Consider two identical isotropic linear elastic half-spaces, having a shear modulus μ and Pand S-wave speeds cp and cs, joined along the interface y = 0. Along this fault a steady state two-dimensional rupture pulse propagates in the positive x direction at a constant velocity V . The fields are a function of x− V t and y only, and for convenience we shall take t = 0 such that x−V t→ x in the following expressions, equivalent to transforming into a frame of reference moving with the rupture. We shall take the origin of the coordinate system to coincide with the rupture front. On the fault, we specify the length of the slip zone L and the shear traction within the slip zone τ(x). The mixed boundary value problem is defined formally as

Simulation of Dynamic Earthquake Ruptures in Complex Geometries Using High-Order Finite Difference Methods

We develop a stable and high-order accurate finite difference method for problems in earthquake rupture dynamics in complex geometries with multiple faults. The bulk material is an isotropic elastic solid cut by pre-existing fault interfaces that accommodate relative motion of the material on the two sides. The fields across the interfaces are related through friction laws which depend on the sliding velocity, tractions acting on the interface, and state variables which evolve according to ordinary differential equations involving local fields.The method is based on summation-by-parts finite difference operators with irregular geometries handled through coordinate transforms and multi-block meshes. Boundary conditions as well as block interface conditions (whether frictional or otherwise) are enforced weakly through the simultaneous approximation term method, resulting in a provably stable discretization.The theoretical accuracy and stability results are confirmed with the method of manufactured solutions. The practical benefits of the new methodology are illustrated in a simulation of a subduction zone megathrust earthquake, a challenging application problem involving complex free-surface topography, nonplanar faults, and varying material properties.

Strong Ground Motion Prediction Using Virtual Earthquakes

Sedimentary basins increase the damaging effects of earthquakes by trapping and amplifying seismic waves. Simulations of seismic wave propagation in sedimentary basins capture this effect; however, there exists no method to validate these results for earthquakes that have not yet occurred. We present a new approach for ground motion prediction that uses the ambient seismic field. We apply our method to a suite of magnitude 7 scenario earthquakes on the southern San Andreas fault and compare our ground motion predictions with simulations. Both methods find strong amplification and coupling of source and structure effects, but they predict substantially different shaking patterns across the Los Angeles Basin. The virtual earthquake approach provides a new approach for predicting long-period strong ground motion. Ambient seismic noise helps predict the ground motion associated with future large earthquakes. Noise in Motion A large earthquake along the southern San Andreas Fault has the potential to cause serious damage to the city of Los Angeles, USA. Earthquake simulations in this region, which lies in a sedimentary basin capable of amplifying shaking, predict strong ground motion but they lack validation with observational data. Denolle et al. (p. 399) developed an independent method to predict ground motion using virtual earthquakes and information gleaned from background seismic noise. This ambient seismic field—generated by sources such as the oceans and atmosphere—produces differences in ground motion in the Los Angeles Basin compared to simulations, but suggests that locally shaking could on average be 3 times larger than the surrounding areas.

Predicting fault damage zones by modeling dynamic rupture propagation and comparison with field observations

We use a two-dimensional plane strain dynamic rupture model with strongly rate-weakening friction and off-fault Drucker-Prager plasticity to model damage zones associated with buried second-order thrust faults observed in the SSC reservoir. The modeling of ruptures propagating as self-sustaining pulses is performed in the framework of continuum plasticity where the plasticity formulation includes both deviatoric and volumetric plastic strains. The material deforming inelastically due to stress perturbations generated by the propagating rupture is assumed to be the damage zone associated with the fault. Dilatant plastic strains are converted into a fracture population by assuming that the dilatant plastic strain is manifested in the form of fractures. The cumulative effect of multiple slip events is considered by superposition of the plastic strain field obtained from individual slip events. The relative number of various magnitude slip events is chosen so as to honor the Gutenberg-Richter law. Results show that the decay of fracture density (F) with distance (r) from the fault can be described by a power law F = F0r− n. The fault constant F0 represents the fracture density at unit distance from the fault. The decay rate (n) in fracture density is approximately 0.85 close to the fault and increases to ~1.4 at larger distances (>10 m). Modeled damage zones are approximately 60–100 m wide. These attributes are similar to those observed in the SSC reservoir using wellbore image logs and those reported in outcrop studies. Considering fault roughness affects local damage zone characteristics, these characteristics are similar to those modeled around planar faults at a scale (~10 m) that affects bulk fluid-flow properties.

Interaction of Waves with Frictional Interfaces Using Summation-by-Parts Difference Operators: Weak Enforcement of Nonlinear Boundary Conditions

We present a high-order difference method for problems in elastodynamics involving the interaction of waves with highly nonlinear frictional interfaces. We restrict our attention to two-dimensional antiplane problems involving deformation in only one direction. Jump conditions that relate tractions on the interface, or fault, to the relative sliding velocity across it are of a form closely related to those used in earthquake rupture models and other frictional sliding problems. By using summation-by-parts (SBP) finite difference operators and weak enforcement of boundary and interface conditions, a strictly stable method is developed. Furthermore, it is shown that unless the nonlinear interface conditions are formulated in terms of characteristic variables, as opposed to the physical variables in terms of which they are more naturally stated, the semi-discretized system of equations can become extremely stiff, preventing efficient solution using explicit time integrators.The use of SBP operators also provides a rigorously defined energy balance for the discretized problem that, as the mesh is refined, approaches the exact energy balance in the continuous problem. This enables one to investigate earthquake energetics, for example the efficiency with which elastic strain energy released during rupture is converted to radiated energy carried by seismic waves, rather than dissipated by frictional sliding of the fault. These theoretical results are confirmed by several numerical tests in both one and two dimensions demonstrating the computational efficiency, the high-order convergence rate of the method, the benefits of using strictly stable numerical methods for long time integration, and the accuracy of the energy balance.

Distinguishing barriers and asperities in near-source ground motion

We investigate the ground motion produced by rupture propagation through circular barriers and asperities in an otherwise homogeneous earthquake rupture. Using a three-dimensional finite difference method, we analyze the effect of asperity radius, strength, and depth in a dynamic model with fixed rupture velocity. We gradually add complexity to the model, eventually approaching the behavior of a spontaneous dynamic rupture, to determine the origin of each feature in the ground motion. A barrier initially resists rupture, which induces rupture front curvature. These effects focus energy on and off the fault, leading to a concentrated pulse from the barrier region and higher velocities at the surface. Finally, we investigate the scaling laws in a spontaneous dynamic model. We find that dynamic stress drop determines fault-parallel static offset, while the time it takes the barrier to break is a measure of fracture energy. Thus, given sufficiently strong heterogeneity, the prestress and yield stress (relative to sliding friction) of the barrier can both be determined from ground motion measurements. In addition, we find that models with constraints on rupture velocity have less ground motion than constraint-free spontaneous dynamic models with equivalent stress drops. This suggests that kinematic models with such constraints overestimate the actual stress heterogeneity of earthquakes.

Vibrational modes of hydraulic fractures: Inference of fracture geometry from resonant frequencies and attenuation

Oscillatory seismic signals arising from resonant vibrations of hydraulic fractures are observed in many geologic systems, including volcanoes, glaciers and ice sheets, and hydrocarbon and geothermal reservoirs. To better quantify the physical dimensions of fluid-filled cracks and properties of the fluids within them, we study wave motion along a thin hydraulic fracture waveguide. We present a linearized analysis, valid at wavelengths greater than the fracture aperture, that accounts for quasi-static elastic deformation of the fracture walls, as well as fluid viscosity, inertia, and compressibility. In the long-wavelength limit, anomalously dispersed guided waves known as crack or Krauklis waves propagate with restoring force from fracture wall elasticity. At shorter wavelengths, the waves become sound waves within the fluid channel. Wave attenuation in our model is due to fluid viscosity, rather than seismic radiation from crack tips or fracture wall roughness. We characterize viscous damping at both low frequencies, where the flow is always fully developed, and at high frequencies, where the flow has a nearly constant velocity profile away from viscous boundary layers near the fracture walls. Most observable seismic signals from resonating fractures likely arise in the boundary layer crack wave limit, where fluid-solid coupling is pronounced and attenuation is minimal. We present a method to estimate the aperture and length of a resonating hydraulic fracture using both the seismically observed quality factor and characteristic frequency. Finally, we develop scaling relations between seismic moment and characteristic frequency that might be useful when interpreting the statistics of hydraulic fracture events.