JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 98, NO. B12, PAGES 21,809-21,821, DECEMBER 10, 1993 Statistical Physics Model for the Spatiotemporal Evolutionof Faults PATIENCE A. COWIE, 1CHRISTIAN VANNESTE, AND DIDIER SORNETIE Laboratoirede Physique de la Matibre Condensde, Universitd de Nice, Nice, France A statistical physics modelis used to simulate antiplane shear deformation and rupture of a tectonic plate with heterogeneous materialproperties. Ruptureoccurs when the chosen statevariable reaches a threshold value. After rupture, broken elementsare instantaneously healed and retain the original material properties. We document the spatiotemporal evolution of the rupture pattern in response to a constant velocity boundary condition. A fundamentalfeature of this model is that ruptures become strongly correlated in space and time, leading to the development of complex fractalstructures. Thesestructures, or "faults," are simplydefined by the loci wheredeformation accumulates. Repeated rupture of a fault occurs in events ("earthquakes") which themselves exhibit both spatial and temporal clustering. Furthermore, we observethat a fault may be active for long periods of time until the locus of activity spontaneously switches to a different fault. The formation of the faults andthe temporal variation of rupture activityis due to a complex interplay betweenthe random small-scale structure, long-rangeelastic interactions, and the threshold natureof rupture physics. The characteristics of this scalar model suggest that spontaneous self- organization of active tectonics doesnot result solely from the tensoffal natureof crustal deformation; that is, kinematic compatibility is not required for complex fault pattern formation. Furthermore, the localizationof the deformation is a dynamical effect ratherthan a consequence of preexisting structure or preferential weakening of faults compared to the surrounding medium. We present an analysis of scaling relationships exhibited by the fault pattern and the earthquakes in this model. INTRODUCTION A variety of "block-slider"models have been presented in the literatureas simulations for earthquake ruptures on a single fault surface[e.g., Burridge and Knopoff, 1967; Carlson and Langer, 1989; Bak and Tang, 1989; Nakanishi, 1990, 1991]. In these models, elastically coupled blocks resting on a frictional surface are driven, via a set of springs, by the movement of an adjacent rigid plate. Models of this type reach a self-organized critical state in which the cascade effect of nearest-neighbor interactions generates a power law distribution of earthquake magnitudes similar to the Gutenberg-Richter law. It has been argued theoretically that crustaldeformation is in fact self-organized on long (geologic) timescales as well as on seismic timescales as a consequence of the buildup of long-range elastic correlations [Sornette and Sornette, 1989; D. Sornette et al., 1990, Sornette and Virieux, 1992]. In this paper we develop this hypothesis further through the presentation of a simulation of crustal-scale deformation of a tectonic plate as opposed to a single fault. Specifically, we have brought together short and long timescale deformation using a statisticalrupture model which includes elastic deformation, a rupture criterion, and postrupture healing. The deformation is driven by a constant velocity which is applied along the plate boundary at a distance from the locusof the rupture activity. It is this feature of the model which allows us to investigate the effect of long- range elastic interactions. Two important features of the rupture behavior are immediately apparent. First, we find that the rupture pattern becomeslocalized and forms well-defined structures, which we call "faults," even though the initial structure is uniformly random. And second, the faults 1Now at Department of Geology and Geophysics, Edinburgh University, Edinburgh, Scotland. Copyright 1993 by the American Geophysical Union. Paper number 93JB02223. 0148- 0227/93/93 JB-02223 $05.00 accumulate displacement as a consequence of rupture events which themselvesexhibit spatial and temporal clustering on the faults andarethus analogous to earthquakes. In this model we have made no provisionfor effects directly associated with seismic rupture, that is, rate dependent friction, inertia, or radiation damping. Thus we treat the rupture dynamics as being instantaneous. However, we observe rupturephenomena at a range of different timescales. We relatd' the shortest timescale of this model to that of a single earthquake (simultaneousrupture of lattice elements). The intermediate timescale represents variations in seismic moment releaserate on a fault. The third and largesttimescale concerns the switchingof the locus of activity betweenfaults. The idea that the crust is in a critical state [Sornette and Sornette, 1989; Bak and Tang, 1989; D. Sornette et al., 1990] implies that in many locations the crust is at or close to rupture; that is, the stress level remains on average closeto the finite strength of the crust. In this state, deformation mechanisms such as stress corrosion and time-dependent damage can become important. Such mechanisms have been suggestedas explanations for foreshocks, aftershocks, and compoundearthquakes, all of which are instances of rupture threshold phenomena [e.g., Das and Scholz,1981' Yamashita and Knopoff, 1987; Main, 1991' Main and Meredith, 1991' Sornette et al., 1992]. In general, crustal deformation probably involves a rupture mechanism that depends on both the stress level and the accumulation of damage. Sornette et al. [1992] developed a model similar in some respects to that presented here, with rupture controlled by damage accumulation D, which has a power law dependence on applied stress,that is, dD/dt o• cr m. These workers apply a constant stress boundary condition to the model, and there is no postrupture healing. They observed irreversible development of rupture with a power law increasein event rate with time similar to Omori's law, which they compared to a foreshock sequence prior to a greatearthquake. The present model differs in several respects from the model of Sornette et al. [1992]' first, a ruptured element is healed 21,809