Dr. Brittany A. Erickson has been granted the NSF Earth Sciences postdoctoral fellowship to carry out a research and education plan at Stanford University. Interdisciplinary research between the fields of geophysics and mathematics will be conducted in order to understand unknown phenomena emerging from earthquake simulations with certain physical features that have not been previously studied. This work will study dynamic models of fault systems in order to explore how small-scale nonplanar features influence single event rupture behavior, as well as long term interactions in fault networks. Single-event and long-term modeling of fault networks has never been studied with such an intricately detailed description of fault topography that will be considered. It has been suggested that faults with complex geometries, including bends and fault branching, affect such rupture properties as nucleation, propagation and arrest. Understanding how and why ruptures break multiple fault segments, or jump to nearby fault lines plays an important role in seismic hazard assessment. The goals of this project are: (1) to understand how the incorporation of such physical features as small-scale non-planar features and off-fault plasticity affect rupture dynamics, (2) to explore the path through which ruptures propagate in a geometrically complicated fault network, (3) to further develop computer codes to be able to solve the quasi-static problem and simulate long term dynamics, (4) to compute statistics associated with jumping and branching of faults with small scale roughness. High performance numerical coding of single event and long term earthquake dynamics on nonplanar faults will be developed and the code will be simulated on clusters of computers.
The results from this work will greatly affect seismic hazard estimates of earthquake nucleation, rupture behavior, jumping and branching in real fault networks. It will determine the regions more susceptible to greater earthquake damage. Furthermore, the possibility for an earthquake rupture to jump from one fault to another implies that the duration of the rupture process can be longer than expected if only one fault is involved and consequently increase the amount of damaging ground motion to structures and buildings. This project will properly quantify the statistics involved with earthquake rupture and allow for a proper assessment of the risk and damages associated with earthquake activities. The education plan includes sharing of mathematical knowledge in courses and workshops for the geophysicists involved in order to educate researchers in computational science.
Earthquake modeling presents great computational challenges because earthquakes are characterized by processes that vary over a wide range of scales in both space and time. Understanding the physics that govern earthquake nucleation, propagation, and arrest requires the study of these processes at all scales. Faults like the San Andreas in southern California creep at depth at a constant rate of about 35 mm/year with a recurrence interval for large earthquakes of about 150 years. Meters of slip accumulate during these large dynamic events, and slip velocities increase to values on the order of meters per second. Furthermore, faults can be hundreds of kilometers long, with frictional properties on the order of millimeters or smaller. Modeling the earthquake cycle with full spatial resolution over multiple time scales is thus very difficult even with modern computing capabilities. Most earthquake models are designed to capture only the earthquake - or ``dynamic rupture" itself - and they don't include the interseismic period between earthquakes. This is due to the fact that modeling the behavior of a fault over different time scales is very difficult computationally. But a dynamic rupture model needs to be given an initial condition that defines the stress state just prior to the earthquake. Because the stress in the earth remains largely unknown, earthquakes are usually initiated under the (unlikely) assumption of a spatially uniform background stress. Because of the highly heterogeneous residual stresses left by previous ruptures, it is unlikely that this present choice of initial conditions is realistic. For these reasons, we have developed a model capable of handling multiple times scales. The method captures the interseismic period in which initial conditions are generated from the effects of tectonic loading and cause earthquakes to occur spontaneously. Current earthquake cycle models make many simplifying assumptions: the first is that the material surrounding the fault is homogeneous; secondly, that the material behaves as an elastic solid, which means it cannot undergo permanent deformation. However, field observations of faults reveal that not only is the material highly heterogeneous, but there exist damage zones of highly fractured rock on the order of hundreds of meters surrounding the fault. Allowing a model to capture this ``inelastic", or ``plastic", response means that the material surrounding the fault can undergo permanent, irrecoverable deformation. Including off-fault plasticity, however, is challenging computationally, which is why it's rarely done. We have developed a numerical method capable of simulating cycles of multiple earthquakes that allows for both heterogeneous materials and off-fault plasticity. The efficiency of the method is due to an assumption we make regarding inertia effects being negligible during the interseismic period. This allows the method to integrate the mathematical equations quickly during the periods in which the system is effectively static. Once an earthquake begins and the fault begins to slide, the method takes much smaller numerical time steps in order to fully resolve the rupture. While the efficiency of methods in the past rely on simplifying assumptions made about the surrounding medium, our method considers the off-fault material in such a way as to allow for more complexity. The method allows for spatial complexity in the sense of variable material properties, but it also allows the system to respond inelastically. We included inelastic response by incorporating into the numerical integration scheme a method for allowing the system to experience ``yielding", in the sense that if it is stressed high enough it accrues damage. We tested the accuracy of our methods by ``manufacturing" an exact, known solution to the nonlinear equations. Our method generates a numerical approximation to this exact solution and we show that it converges to the correct solution if we consider a small enough spatial grid. For our geophysical applications simulations we vary internal friction parameters and show that our model can generate more realistic initial conditions prior to multiple earthquakes. For some parameters we consider, we see smaller events preceding the larger events. With off-fault plasticity, our preliminary results show that as the earthquake propagates, it can cause the system to accrue damage in the material surrounding the fault. This methodology will be used to study many important questions related to understanding how damage left by past earthquakes (in the form of plastic strain) affects subsequent earthquakes. The results from this work will greatly a?ect seismic hazard estimates of earthquake nucleation and behavior. Computational methods for cycle simulations have been developed. Interdisciplinary collaborations between mathematics and geophysics have prospered.
