Summary
We continued the development of a quantitative framework for studying the coupled evolution of earthquakes and faults in a rheologically-layered half space. Our model accounts for the creation of new fracture surfaces with arbitrary evolving geometry as dictated by current and past strain, healing of inactive faults, and associated seismicity patterns. The model is based on a thermodynamically-rooted version of damage rheology developed originally by Lyakhovsy and co-workers [e.g., Lyakhovsy et al., 1993]. Our last year studies provided additional developments and refinements to the damage rheology model which, among other things, cast the theoretical results in terms of parameters that can be measured directly in the lab [Lyakhovsy, Ben-Zion and Agnon, '97a; Agnon, Lyakhovsy, and Ben-Zion, '97]. We also improved the computational framework for simulating deformation in a 3D structure consisting of an upper seismogenic layer over a Maxwell viscoelastic substrate [Lyakhovsy, Ben-Zion and Agnon, '97b]. The seismogenic layer is treated in our model as an elastic medium where distributed damage, modifying the elastic stiffness, evolves as a function of the deformation. The formulation accounts in a self-consistent manner for evolving 3D deformation fields, surface topography, fault structures, aseismic energy release, and spatio-temporal seismicity patterns. In a related but different study, we conducted analytical and numerical investigations of the statistical mechanics of disordered fault systems in an elastic half space [Fisher, Dahmen, Ramanathan, and Ben-Zion, '97; Dahmen, Ertas, and Ben-Zion, '97] to obtain a more complete understanding of such systems. The statistical mechanics work leads to a two parameter phase diagram summarizing different types of earthquake statistics. Essential aspects of the works are discussed briefly below. We also completed other works related to SCEC projects of previous years (see publication list).
Coupled Evolution of Earthquakes and Faults
The starting framework for the large scale structure for our work on this subject is the generalized Elsasser model [Rice, '80; Lehner et al., '81], consisting of an elastic upper crust layer over a Maxwell viscoelastic lower crust. The deformation in the upper crust is approximated to the plane stress case, and the model employs variables which give vertically-averaged fields in the elastic upper layer. The employed plane stress formulation is appropriate for deformations with dominant wavelength comparable to or larger than the thickness of the elastic upper crust. Thus the model can be applied to simulate crust-transcending faults, and earthquakes in the magnitude range 6 < M < 8.5. A problem with the generalized Elsasser model is that displacement in that formulation decreases exponentially with normal distance from the fault x, rather than having the algebraic 1/x2 dependency characterizing true 3D deformation. This may not be a serious shortcoming for studies concerned with long term processes (which are governed mainly by the viscous component of deformation). However, for studies such as ours concerned with self-organization of earthquakes and faults, the correct 3D behavior of fields during rapid brittle processes should be incorporated in the solution. To overcome this problem we write the displacement as sum of elastic and viscous components. The viscous component satisfies the same relations as before; however, in contrast to Rice ['80] and Lehner et al. ['81], the elastic component is calculated using 3D Green's function. The resulting formulation provides an efficient framework for 3D calculations of stress and displacement fields in a vertically layered elastic/viscoelastic half space.
Our damage rheology model accounts for elastic deformation, aseismic creep, brittle failure, viscous relaxation, and processes of strength degradation and healing. Lyakhovsy et al. ['97a] give a detailed discussion of the physics and mathematical derivation leading to an equation for damage evolution in terms of measurable rock deformation parameters. The final equation for the damage evolution has two functional coefficients: (1) a "generalized internal friction" separating states associated with material degradation and healing, and (2) damage rate parameters, one for positive changes (degradation) and two for negative evolution (healing). The parameters are constrained by acoustic emission, fracture strength, rate- and state-dependent friction, and other measurements of rock-mechanics experiments.
The above developments allow us to simulate the coupled evolution of earthquakes and faults in an internally consistent framework. The initial simulations show extremely interesting phenomena. The results indicate that low generalized internal friction and fast healing rate, describing a relatively weak upper crust with relatively short memory, lead to the development of highly localized, geometrically regular, fault systems. The associated seismicity patterns are compatible with the characteristic earthquake distribution and quasi-periodic temporal occurrence of large events. Conversely, high generalized internal friction and slow healing rate, describing a more brittle upper crust with longer system memory, tend to lead to the development of a network of disordered fault systems (Figure 1). In such cases the corresponding frequency-size statistics of earthquakes are more power law like, and the temporal distributions of large events are random or clustered. The simulated seismicity patterns are non stationary in time and space, and the statistics depend on the sizes of the employed temporal and spatial domains. Model simulations with rheological parameters constrained by lab data exhibit (Figure 2) alternating overall switching of response, from periods of intense seismic activity to periods during which the deformation occurs aseismically. The amount of time spent in each mode is on the order of 1000 yr.
In another study, based on a mean field approximation to the model of Ben-Zion and Rice ['93] for heterogeneous fault system in elastic half-space, we attempted to clarify the interplay between the roles of disorder and dynamical effects on the seismic response of the system. Fisher et al. ['97] found a rigorous class of models which operate naturally at a critical point whose properties yield power law scaling of earthquake statistics. However, the analysis indicates that various dynamical effects expected to exist in nature can change the behavior to a distribution of small events combined with characteristic system size events. Additional studies in this topic [Dahmen et al., '97a,b] show that the different types of earthquake statistics can be characterized by a two parameter phase diagram. One of the parameters, D, is the ratio of "dynamic" stress drop during repeating failures at a given point in an event to the "static" stress drop that characterizes the first failure at the point. The other parameter, R, is the normal distance from the fault of the driving forces. In general, large D and small R tend to produce Gutenberg-Richter type statistics, while small D and large R lead to a "characteristic earthquake" distribution.
Remarkably, we find in this study too that under certain conditions the response switches spontaneously between characteristic earthquake behavior with quasi periodic fault-spanning ruptures, to a cascade of many small events following the Gutenberg-Richter statistics. The amount of time spent in each "mode" depends on the fault size and R. The size of the largest event in the Gutenberg-Richter distribution depends on the same quantities. For the characteristic earthquake distribution, the magnitude separation between the event terminating the power-law statistics and the characteristic event size depends also on D and on the amount of heterogeneities in the fault system.
The results discussed above may have profound implications for estimates of seismic hazard potential in different space-time domains and, more generally, the understanding of the seismic response of the crust to tectonic loading. Our proposed continuing studies will focus on clarifying the range of parameters leading to an overall mode switching of response, and other properties of regional and local spatio-temporal patterns of earthquakes and faults.
References
Agnon, A., Lyakhovsky, V. and Y. Ben-Zion, Localization of distributed damage and strain to stick-slip faults, Extended abstract in Symposium on Localization Phenomena and Dynamics of Brittle and Granular Systems, August, 1997.
Ben-Zion, Y. and J. R. Rice, Earthquake failure sequences along a cellular fault zone in a three-dimensional elastic solid containing asperity and nonasperity regions, J. Geophys. Res., 98, 14109-14131, 1993.
Dahmen, K., D. Ertas and Y. Ben-Zion, Phase Diagram of Earthquake Statistics in Simple Models of Heterogeneous Fault Systems, EOS Trans. Amer. Geophys. Union, 78, F464, 1997a.
Dahmen, K., D. Ertas and Y. Ben-Zion, Transitions between Gutenberg-Richter to characteristic earthquake behavior in simple mean-field models of heterogeneous faults, in preparation, 1997b.
Fisher, D. S., K. Dahmen, S. Ramanathan and Y. Ben-Zion, Statistics of Earthquakes in Simple Models of Heterogeneous Faults, Phys. Rev. Lett., 78, 4885-4888, 1997.
Lehner, F. K., V. C. Li, and J. R. Rice, Stress diffusion along rupturing plate boundaries, J. Geophys. Res., 86, 6155-6169, 1981.
Lyakhovsky, V., Y. Podladchikov, and A. Poliakov, Rheological model of a fractured solid. Tectonophysics, 226, 187-198, 1993.
Lyakhovsky, V., Y. Ben-Zion and A. Agnon, Distributed Damage, Faulting, and Friction, J. Geophys. Res., 102, 27635-27649, 1997a.
Lyakhovsky, V., Y. Ben-Zion and A. Agnon, Simultaneous evolution of earthquakes and faults in a rheologically layered half-space, in preparation, 1997b.
Rice, J. R., The mechanics of earthquake rupture, in Physics of the Earth's Interior, ed. A. M. Dziewonski and E. Boschi, pp. 555-649, Italian Physical Society / North Holland, Amsterdam, 1980.
Last Year Publications Supported by SCEC Studies
Papers:
Ben-Zion, Y. and J. R. Rice, Dynamic simulations of slip on a smooth fault in an elastic solid, J. Geophys. Res., 102, 17771-17784, 1997.
Dahmen, K., D. Ertas and Y. Ben-Zion, Transitions between Gutenberg-Richter to characteristic earthquake behavior in simple mean-field models of heterogeneous faults, in preparation, 1997.
Eneva, M. and Y. Ben-Zion, Techniques and parameters to analyze seismicity patterns associated with large earthquakes, J. Geophys. Res., 102, 17785-17795, 1997.
Eneva, M. and Y. Ben-Zion, Application of pattern recognition techniques to earthquake catalogs generated by models of segmented fault systems in three-dimensional elastic solids, J. Geophys. Res., 102, 24513-24528, 1997.
Fisher, D. S., K. Dahmen, S. Ramanathan and Y. Ben-Zion, Statistics of Earthquakes in Simple Models of Heterogeneous Faults, Phys. Rev. Lett., 78, 4885-4888, 1997.
Lyakhovsky, V., Y. Ben-Zion and A. Agnon, Distributed Damage, Faulting, and Friction, J. Geophys. Res., 102, 27635-27649, 1997.
Lyakhovsky, V., Y. Ben-Zion and A. Agnon, Simultaneous evolution of earthquakes and faults in a rheologically layered half-space, in preparation, 1997.
Abstracts:
Agnon, A., Lyakhovsky, V. and Y. Ben-Zion, Localization of distributed damage and strain to stick-slip faults, Extended abstract in Symposium on Localization Phenomena and Dynamics of Brittle and Granular Systems, August, 1997.
Dahmen, K., D. Ertas and Y. Ben-Zion, Phase Diagram of Earthquake Statistics in Simple Models of Heterogeneous Fault Systems, EOS Trans. Amer. Geophys. Union, 78, F464, 1997.
Lyakhovsky, V., Y. Ben-Zion and A. Agnon, Earthquakes and faults self-organization in a rheologically layered half-space, IASPEI meeting, 1997.
