Robert Shcherbakov

Statistical physics approach to understanding the multiscale dynamics of earthquake fault systems

[1] Earthquakes and the faults upon which they occur interact over a wide range of spatial and temporal scales. In addition, many aspects of regional seismicity appear to be stochastic both in space and time. However, within this complexity, there is considerable self-organization. We argue that the occurrence of earthquakes is a problem that can be attacked using the fundamentals of statistical physics. Concepts of statistical physics associated with phase changes and critical points have been successfully applied to a variety of cellular automata models. Examples include sandpile models, forest fire models, and, particularly, slider block models. These models exhibit avalanche behavior very similar to observed seismicity. A fundamental question is whether variations in seismicity can be used to successfully forecast the occurrence of earthquakes. Several attempts have been made to utilize precursory seismic activation and quiescence to make earthquake forecasts, some of which show promise.

Hydraulic Fracturing and Seismicity in the Western Canada Sedimentary Basin

The development of most unconventional oil and gas resources relies upon subsurface injection of very large volumes of fluids, which can induce earthquakes by activating slip on a nearby fault. During the last 5 years, accelerated oilfield fluid injection has led to a sharp increase in the rate of earthquakes in some parts of North America. In the central United States, most induced seismicity is linked to deep disposal of coproduced wastewater from oil and gas extraction. In contrast, in western Canada most recent cases of induced seismicity are highly correlated in time and space with hydraulic fracturing, during which fluids are injected under high pressure during well completion to induce localized fracturing of rock. Furthermore, it appears that the maximum-observed magnitude of events associated with hydraulic fracturing may exceed the predictions of an often-cited relationship between the volume of injected fluid and the maximum expected magnitude. These findings have far-reaching implications for assessment of inducedseismicity hazards.

Model for the Distribution of Aftershock Interoccurrence Times

In this work the distribution of interoccurrence times between earthquakes in aftershock sequences is analyzed and a model based on a nonhomogeneous Poisson (NHP) process is proposed to quantify the observed scaling. In this model the generalized Omoris law for the decay of aftershocks is used as a time-dependent rate in the NHP process. The analytically derived distribution of interoccurrence times is applied to several major aftershock sequences in California to confirm the validity of the proposed hypothesis.

A Modified Form of Båth's Law

Baths law states that the differences in magnitudes between mainshocks and their largest aftershocks are approximately constant, independent of the magnitudes of mainshocks. In our modified form of Baths law we introduce the notion of the inferred “largest” aftershock from an extrapolation of the Gutenberg-Richter frequency-magnitude statistics of the aftershock sequence of a given mainshock. To illustrate the application of this modified law we consider 10 large earthquakes that occurred in California between 1987 and 2003 with magnitudes equal to or greater than m ms ≥ 5.5. The mean difference in magnitudes between these mainshocks and their largest detected aftershocks is 1.16 with a standard deviation σ Δ m = 0.46 (Baths law). Our estimated mean difference in magnitudes between the mainshocks and the inferred “largest” aftershocks is 1.11 with σ Δ m * = 0.29. The scaling associated with the modified Baths law implies that the stress transfer responsible for the occurrence of aftershocks is a self-similar process. We also estimate the partitioning of energy during a mainshock-aftershock sequence and find that about 96% of the energy dissipated in a sequence is associated with the mainshock and the rest is due to aftershocks. We suggest that the observed partitioning of energy could play a crucial role in explaining the physical origin of Baths law. Manuscript received 6 August 2003.

Damage and self-similarity in fracture

Abstract Consider applications of damage mechanics to material failure. The damage variable introduced in damage mechanics quantifies the deviation of a brittle solid from linear elasticity. An analogy between the metastable behavior of a stressed brittle solid and the metastable behavior of a superheated liquid is established. The nucleation of microcracks is analogous to the nucleation of bubbles in the superheated liquid. In this paper we have applied damage mechanics to four problems. The first is the instantaneous application of a constant stress to a brittle solid. The results are verified by applying them to studies of the rupture of chipboard and fiberglass panels. We then obtain a solution for the evolution of damage after the instantaneous application of a constant strain. It is shown that the subsequent stress relaxation can reproduce the modified Omori’s law for the temporal decay of aftershocks following an earthquake. Obtained also are the solutions for application of constant rates of stress and strain. A fundamental question is the cause of the time delay associated with damage and microcracks. It is argued that the microcracks themselves cause random fluctuations similar to the thermal fluctuations associated with phase changes.

Scaling Properties of the Parkfield Aftershock Sequence

Aftershock sequences present a unique opportunity to study the physics of earthquakes. Important questions concern the fundamental origin of three widely applicable scaling laws: (1) Gutenberg–Richter frequency–magnitude scaling, (2) Omori’s law for aftershock decay rates, and (3) Bath’s law for the difference between the magnitude of the largest aftershock and a mainshock. The high-resolution Parkfield seismic network provided the opportunity for detailed studies of the aftershock sequence following the 28 September 2004, M 6.0 Parkfield earthquake. In this article it is shown that aftershocks satisfy the Gutenberg–Richter scaling relation only for relatively large times after the mainshock. There is a systematic time delay for the establishment of this scaling law. The temporal evolution of the rates of occurrence of aftershocks is quantified using the generalized Omori’s law. This scaling law contains two characteristic times c and τ . The analysis suggests that the parameter c plays the role of a characteristic time for the establishment of Gutenberg– Richter scaling. This time increases systematically with a decreasing lower magnitude cutoff. The systematic time delay is attributed to a cascade of energy from long wavelengths to short wavelengths. The parameter τ is a measure of the average time until the first aftershock occurs. We find that τ slightly varies with the lower magnitude cutoff of the sequence. We also note that the largest aftershock inferred from an extrapolation of Gutenberg–Richter scaling, M 5.0, is equal to the largest observed aftershock. This scaling associated with the universal applicability of Bath’s law is attributed to a constant partitioning of energy between a mainshock and its associated aftershock sequence. We also give in this article the distribution of interoccurrence times between successive aftershocks. We show that this distribution is well approximated by a nonhomogeneous Poissons process driven by the modified Omori’s law. The self-consistency between interoccurrence statistics and decay rates is taken as further evidence for the applicability of our studies.

A Damage Mechanics Model for Aftershocks

Abstract — All earthquakes are followed by an aftershock sequence. A universal feature of aftershock sequences is that they decay in time according to the modified Omori’s law, a power-law decay. In this paper we consider the applicability of damage mechanics to earthquake aftershocks. The damage variable introduced in damage mechanics quantifies the deviation of a brittle solid from linear elasticity. We draw an analogy between the metastable behavior of a stressed brittle solid and the metastable behavior of a superheated liquid. The nucleation of microcracks is analogous to the nucleation of bubbles in the superheated liquid. In this paper we obtain a solution for the evolution of damage after the instantaneous application of a constant strain to a rod. We show that the subsequent stress relaxation can reproduce the modified Omori’s law. It is argued that the aftershocks themselves cause random fluctuations similar to the thermal fluctuations associated with phase transitions.

A simulation-based approach to forecasting the next great San Francisco earthquake

In 1906 the great San Francisco earthquake and fire destroyed much of the city. As we approach the 100-year anniversary of that event, a critical concern is the hazard posed by another such earthquake. In this article, we examine the assumptions presently used to compute the probability of occurrence of these earthquakes. We also present the results of a numerical simulation of interacting faults on the San Andreas system. Called Virtual California, this simulation can be used to compute the times, locations, and magnitudes of simulated earthquakes on the San Andreas fault in the vicinity of San Francisco. Of particular importance are results for the statistical distribution of recurrence times between great earthquakes, results that are difficult or impossible to obtain from a purely field-based approach.

DYNAMICS OF EULERIAN WALKERS

We investigate the dynamics of Eulerian walkers as a model of self-organized criticality. The evolution of the system is subdivided into characteristic periods which can be seen as avalanches. The structure of avalanches is described and the critical exponent in the distribution of first avalanches

Complexity and Earthquakes

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