Nearly Magnitude-Invariant Stress Drops in Simulated
Crack-Like Earthquake Sequences on Rate-and-State
Faults with Thermal Pressurization of Pore Fluids
Stephen M. Perry
1
, Valère Lambert
1
, and Nadia Lapusta
1,2
1
Seismological Laboratory, California Institute of Technology, Pasadena, CA, USA,
2
Mechanical and Civil Engineering,
California Institute of Technology, Pasadena, CA, USA
Abstract
Stress drops, inferred to be magnitude-invariant, are a key characteristic used to describe
natural earthquakes. Theoretical studies and laboratory experiments indicate that enhanced dynamic
weakening, such as thermal pressurization of pore fluids, may be present on natural faults. At first glance,
magnitude invariance of stress drops and enhanced dynamic weakening seem incompatible since larger
events may experience greater weakening and should thus have lower final stresses and higher stress
drops. We hypothesize that enhanced dynamic weakening can be reconciled with magnitude-invariant
stress drops due to larger events having lower average prestress when compared to smaller events. We
conduct numerical simulations of long-term earthquake sequences in fault models with rate-and-state
friction and thermal pressurization, and in the parameter regime that results mostly in crack-like ruptures,
we find that such models can explain both the observationally inferred stress drop invariance and
increasing breakdown energy with event magnitude. Smaller events indeed have larger average initial
stresses than medium-sized events, and we find nearly constant stress drops for events spanning up to two
orders of magnitude in average slip, comparable to approximately six orders of magnitude in seismic
moment. Segment-spanning events have more complex behavior, which depends on the properties of the
arresting velocity-strengthening region at the edges of the faults.
1. Introduction
Stress drops and breakdown energy are important descriptors of natural earthquakes. Stress drops charac-
terize the average change in stress state from before to after the dynamic event (Kanamori & Anderson,
1975; Knopoff, 1958; Kostrov, 1974). The stress drop distribution varies along the fault and can be averaged
in several different ways in order to produce a single, representative value for an event (Section 3). There
is a fair amount of scatter in the inferred average values of stress drops of natural earthquakes, from about
0.1 MPa up to values around 100 MPa (Baltay et al., 2011; Kanamori & Brodsky, 2004). However, the inferred
values of stress drop are magnitude-invariant; most events have stress drops that fall between 1 MPa and
10 MPa, and this trend has been observed for events ranging nine orders of magnitude in seismic moment
(Abercrombie & Rice, 2005; Allmann & Shearer, 2009; Cocco et al., 2016; Ide & Beroza, 2001). The generality
of the inferred magnitude invariance of stress drops is still a topic of ongoing research, with some observa-
tions indicating that some individual earthquake sequences may exhibit mildly increasing trends in stress
drop with increasing moment (e.g., Cocco et al., 2016; Viesca & Garagash, 2015). The interpretation and
reliability of the stress drops estimates have been actively studied recently, with indications that the current
standard methods of estimating stress drops can introduce some significant discrepancies between the actual
and inferred stress drops (e.g., Kaneko & Shearer, 2014, 2015; Lin & Lapusta, 2018; McGuire & Kaneko, 2018;
Noda et al., 2013). However, there are no indications at present that the overall nearly magnitude-invariant
trend should be questioned.
Breakdown energy, a quantity analogous to fracture energy from singular and cohesive zone models of frac-
ture mechanics, is meant to capture the energy consumed near the rupture tip that controls the dynamics of
the rupture front (Cocco et al., 2004; Palmer et al., 1973; Rice, 1980). Breakdown energy is a part of the over-
all energy budget of a seismic event, with the total strain energy released (
Δ
W
) typically divided into the
breakdown energy
G
, radiated energy
E
R
, and other dissipation
E
D
(Kanamori & Rivera, 2013). It is a more
straightforward concept for shear stress versus slip behavior that follows slip weakening during dynamic
RESEARCH ARTICLE
10.1029/2019JB018597
Key Points:
• Thermal pressurization (TP) results
in continuous weakening and
increasing breakdown energy with
slip
• TP allows larger ruptures to
propagate into lower prestress
regions, leading to
magnitude-invariant stress drops
• Stress drops for large events are
impacted by the properties of the
arresting velocity-strengthening
regions into which they propagate
Correspondence to:
V. Lambert,
vlambert@caltech.edu
Citation:
Perry, S. M., Lambert, V., &
Lapusta, N. (2020). Nearly
magnitude-invariant stress drops
in simulated crack-like earthquake
sequences on rate-and-state faults
with thermal pressurization of pore
fluids.
Journal of Geophysical
Research: Solid Earth
,
125
,
e2019JB018597. https://doi.org/10.
1029/2019JB018597
Received 3 SEP 2019
Accepted 28 FEB 2020
Accepted article online 4 MAR 2020
©2020. The Authors.
This is an open access article under the
terms of the Creative Commons
Attribution License, which permits
use, distribution and reproduction in
any medium, provided the original
work is properly cited.
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Journal of Geophysical Research: Solid Earth
10.1029/2019JB018597
Figure 1.
Stress drop and breakdown energy implications for linear slip-weakening and rate-and-state friction with
additional dynamic weakening. (a) In linear slip-weakening laws, smaller and larger events weaken to the same
dynamic levels of shear resistance over the same slip. This leads to the same breakdown energies (dotted regions) and
similar stress drop (marked with stars). (b) If smaller and larger events both nucleate at the same levels of prestress,
and larger events weaken more than small events, one expects both larger breakdown energies and larger stress drops
for larger events. (c) However, if dynamic weakening allows larger events to propagate into areas of lower stress, then
the average prestress of these events may be lower than for smaller events. In this case, breakdown energies still
increase with event size, but stress drops may be magnitude-invariant.
rupture (Kanamori & Heaton, 2013; Kanamori & Brodsky, 2004; Rice, 2000). It is calculated by taking the
area underneath the stress-slip curve for a single event from initiation to the lowest dynamic level of stress
and then subtracting off the frictional energy dissipation (Figure 1 and Section 3). Breakdown energy is
inferred to increase with the event size in natural earthquakes (Abercrombie & Rice, 2005; Rice, 2006; Viesca
& Garagash, 2015).
It is clear that during dynamic rupture, the fault shear resistance overall decreases, resulting in a stress drop.
The exact nature of this evolution is currently an active area of research. Slip-weakening models, where
the shear stress decrease depends on the slip accumulated during the event, are commonly used (Ida, 1972;
Palmer et al., 1973). Linear slip weakening (LSW) is a simplified model where the shear resistance decreases
linearly with slip until it reaches a constant dynamic level (Section 4.1).
Significant insights into the physics of shear resistance during earthquakes have been obtained from the
laboratory, showing much richer behavior. At slip rates between
10
−
9
and
10
−
3
m/s, laboratory findings are
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10.1029/2019JB018597
well described by the so-called rate-and-state friction laws (Dieterich, 2015, and references therein). Stud-
ies using rate-and-state models have successfully reproduced a number of earthquake source observations,
including the decay of aftershocks (Dieterich, 1994), sequences of earthquakes on an actual fault segment
(Barbot et al., 2012), and repeating earthquakes (Chen & Lapusta, 2009).
At the same time, many experiments and theoretical studies have shown that enhanced dynamic weakening
can be a dominating effect during earthquakes (Di Toro et al., 2011). This type of weakening can be caused
by several different mechanisms, many of them due to shear heating. Thermal pressurization may be caused
by the shear heating of pore fluids during slip (Andrews, 2002; Rice, 2006; Sibson, 1973); if the pore fluid
is heated quickly enough and not allowed to diffuse away, it pressurizes and relieves normal stress on the
fault. Flash heating is another shear-heating effect of rapid weakening due to microcontacts between the
two sides of the fault melting at small scales and rapidly decreasing the effective friction coefficient (Goldsby
& Tullis, 2011; Passelégue et al., 2014; Rice, 1999). Other weakening mechanisms can act in the shear zone,
including the thermal decomposition of rocks (Han et al., 2007; Sulem & Famin, 2009), macroscopic melt-
ing (Goldsby & Tullis, 2002; Di Toro et al., 2004, 2011), elastohydrolubrication (Brodsky & Kanamori, 2001),
and silica gel formation (Brodsky & Kanamori, 2001; Di Toro et al., 2004; Goldsby & Tullis, 2002). Consid-
erations of heat production during dynamic rupture are a substantial constraint for potential fault models
as field studies rarely suggest the presence of melt and show no correlation between faulting and heat flow
signatures (Lachenbruch & Sass, 1980; Sibson, 1975).
Several numerical studies used these enhanced dynamic weakening effects to explain some observations for
natural earthquakes. Thermal pressurization of pore fluids can explain the inferred increase in breakdown
energy with the increasing event size (Rice, 2006; Viesca & Garagash, 2015); this has been shown using
simplified theoretical arguments. Models with dynamic weakening have been successful in producing fault
operation at low overall prestress and low heat production (Noda et al., 2009; Rice, 2006) as supported by
several observations (Brune et al., 1969; Hickman & Zoback, 2004; Williams et al., 2004; Zoback et al., 1987).
However, it is not clear whether enhanced dynamic weakening is consistent with magnitude-invariant stress
drops. In the following intuitive scenario, they are not. Let us assume that smaller and larger events nucle-
ate at nearly the same level of average prestress. The smaller event has less slip and thus weakens a smaller
amount. This results in a smaller breakdown energy (the dotted region) and a higher final stress. The larger
event weakens more and has a larger breakdown energy and lower final stress. In this scenario, larger events
would have systematically larger stress drops and larger breakdown energy (Figure 1b). However, this intu-
itive scenario may be incorrect, due to the following hypothesis which is illustrated and supported by the
simulations in this work. Both smaller and larger events would
nucleate
at locations with relatively high
prestress, matching the quasi-static frictional strength. But we must consider the average initial stress of all
points involved in the rupture, not just those involv
ed in nucleation. Larger events would have larger slips
and hence dynamically weaken more and may be able to propagate over areas of much less favorable (lower)
prestress conditions. This means that the initial stress averaged over the entire rupture area may be lower
for larger events than that for smaller events. Overall,
larger events would dynamically weaken more and
potentially arrest at a lower average final stress, but they would also have occurred with lower average ini-
tial stress. Thus, the average stress drop can be similar for smaller and larger events (Figure 1c). However,
the observed increase of the breakdown energy with event size is still preserved.
Here, we use fully dynamic simulations of earthquake sequences on rate-and-state faults to investigate this
hypothesis and study if enhanced dynamic weakening can indeed be compatible with magnitude-invariant
stress drops while also maintaining increasing breakdown energy with increasing event size. Different
dynamic weakening mechanisms produce different weakening behaviors, but here we focus on thermal
pressurization as a representative dynamic weakening mechanism that can lead to continuous fault weaken-
ing with earthquake-source slip. We consider the simplest scenario that allows us to explore this hypothesis,
that of a seismogenic fault segment with uniform properties of quasi-static fault strength. For heterogeneous
faults, the argument should still hold, since larger ruptures with larger slip and hence more pronounced
weakening should be able to propagate over larger areas of locally unfavorable prestress, as compared to
smaller ruptures, potentially still resulting in nearly magnitude-invariant stress drops, but with some scatter
due to heterogeneity. Such scenarios will be investigated in future work.
We indeed find that the hypothesis of lower average initial stress before larger events holds for a wide range of
events in our simulations that arrest within the seismogenic region, resulting in nearly magnitude-invariant
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