Implementation of quasistatic viscoelasticity and poroelasticity using memory variables into dynamic earthquake sequence simulation with SBIEM
Hiroyuki Noda, Kyoto University, Disaster Prevention Research Institute

In a dynamic earthquake sequence simulation with SBIEM, storage of Fourier-transformed slip rate history and computation of dynamic stress transfer are limiting factors, which determine the maximum affordable problem size for restricted numerical resources. Calculation of static traction change on the fault does not cost significantly for an elastic medium because it depends only on the current slip distribution and thus temporal convolution is not needed. In inelastic medium, however, the static stress depends not only on the current slip distribution, but also on its history. Calculation of static stress by conducting temporal convolution would require additional storage of slip rate history, and thus affect the ability of numerical resolution for a given numerical resources. In this talk, a way of going around the temporal convolution is introduced by raising a couple of examples, Maxwell viscoelasticity (Miyake and Noda, 2019) and poroelasticity (revision submitted). Key tricks are definition of auxiliary memory variables and reformulation of the integral expressions of Fourier-transformed traction change to ODEs of the memory variables. In the present method, we have only to integrate the ODEs each timestep, similarly to integration of state variables in a rate- and state-dependent friction law.

Fault-size dependent fracture energy, seismogenesis, and cascading rupture on multi-scale fault networks
Dmitry Garagash, Dalhousie University

Fracture energy fundamentally affects all aspects of earthquake rupture, including fault seismogenesis. Seismological inferences of fracture energy [1-3] are seen to increase with both slip and the size of fault source. To explain these observations, refs [3-5] invoke co-seismic shear heating as the mechanism leading to continuing fault co-seismic weakening with slip on all scales. In this case, the observed correlation of fracture energy G with source/fault size R can be simply a consequence of larger faults been capable of hosting larger slip, and can otherwise be independent of individual fault properties (such as the thickness of the fault’s principal slip zone known to depend on fault maturity, total accrued fault slip, and fault size). While theoretically feasible at larger slip permissible on larger faults, such proposed fault-invariance of the fracture energy is questionable at smaller slip and/or smaller faults and fractures. It is the latter, i.e. the fracture energy at smaller values of slip, that governs dynamic rupture nucleation and is relevant to understanding seismogenesis.

We, therefore, propose that the fracture energy can be decomposed:
G = Gc (R) + ΔG(δ), where  Gc (R) is the minimum, ‘small slip’ fracture energy that is a fault property linked to the fault size R or total fault accrued slip, while ΔG(δ) is the ‘large-slip’ and possibly fault-invariant part of the fracture energy that increases continuously with co-seismic slip. The two can be linked to distinct fault weakening mechanisms activated at different levels of co-seismic slip [3].

We revisit the compilation [3] of seismologically-inferred partial fracture energy G' vs. source size R varying from ~10 meters to ~10 km. We convert G’ estimates which assume zero dynamic stress over/undershoot to the full fracture energy estimates by incorporating the effect of co-seismic restrengthening evaluated on the basis of a circular crack-like rupture driven by flash-heated, rate-state friction. While doing so, we recover the emergent linear scaling of the minimum fracture energy  Gc with fault size. 

We apply this scaling to model mechanically viable composite earthquake ruptures occurring as a cascade over networks of faults/fractures of very different sizes [6]. Depending on the prestress orientation, we recover two distinct modes of a composite earthquake rupture behavior. The first is the ‘classical’ rupture on a well-oriented main fault activating poorly-oriented fractures in the damage zone (off-fault slip). The second mode is the earthquake rupture cascading on well-oriented multi-scale fractures in the damage zone without main fault activation. The latter has very distinct ‘signature’ characteristics potentially observable by seismological methods, including the misalignment of the focal mechanism with the main fault trace, naturally limited apparent rupture (cascading) speed, and complex moment-rate release in the intermediate-to-high frequency content.

References

[1] R. E. Abercrombie and J. R. Rice. Can observations of earthquake scaling constrain slip weakening? Geophys. J. Int., 162:406–424, 2005.
[2] P. M. Mai, P. Somerville, A. Pitarka, L. Dalguer, S. Song, G. Beroza, H. Miyake, and K. Irikura. On scaling of fracture energy and stress drop in dynamic rupture models: Consequences for near-source ground-motions. Geophysical Monograph-American Geophysical Union, 170:283, 2006.
[3] R. C. Viesca and D. I. Garagash. Ubiquitous weakening of faults due to thermal pressurization. Nature Geoscience, 8:875–879, 2015.
[4] J. R. Rice. Heating and weakening of faults during earthquake slip. J. Geophys. Res., 111:B05311, 2006.
[5] N. Brantut and Robert C Viesca. The fracture energy of ruptures driven by flash heating. Geophysical Research Letters, 44(13):6718–6725, 2017. 
[6] K. Palgunadi, A.-A. Gabriel, D. Garagash and P. M. Mai, Cascading Earthquake Rupture in A Multi-Scale Fracture Network (S22A-04), AGU Fall Meeting 2021.

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