of 28
1.
Introduction
Recently, the role of fluids in faults has received great interest for two main reasons: first, by the discov
-
ery of a strong causal link between fluid injection and induced seismicity (e.g., Ellsworth,
2013
); second,
by the mounting evidence that slow slip and tremor are generated at high ambient fluid pressures (e.g.,
Bürgmann,
2018
). A topic of notable recent interest in studies of induced seismicity is the role of poroe
-
lasticity. The slow slip and tremor literature has been significantly influenced by the idea of dilatancy and
how dilatancy can stabilize fault slip and generate slow slip events. Recently, it has become clear that the
topics of slow slip and aseismic transients in nature and human-induced seismicity are closely linked. For
example, Bhattacharya and Viesca (
2019
) and Viesca and Dublanchet (
2019
) have shown how spontaneous
aseismic and slow slip transients arise on faults subject to pore-pressure changes. Torberntsson et al. (
2018
)
investigated slow and fast slip in response to fluid injection near a fault in a poroelastic solid. Further, dila
-
tancy as a stabilizing mechanism for faults subjected to fluid injection has been studied recently (Ciardo &
Lecampion,
2019
). This study combines both poroelasticity and dilatancy to understand frictional sliding in
a fully coupled sense, where pore pressure changes of the shear zone influence the bulk and vice versa. In
this introduction, we start by discussing poroelasticity, then we review the concept of dilatancy, and finally
we provide an overview of the paper.
Biot's theory of poroelasticity has gained much interest in the study of induced seismicity (Segall & Lu,
2015
)
because fluid injection does not only change pore pressure, but also induces long-ranging stress interactions
through the coupling of fluid pressure and straining of the porous rock. It is well established that the crust
behaves as a poroelastic solid (Jónsson et al.,
2003
) and thus Biot's theory of poroelasticity offers a more
realistic way to model the earth's crust than simple elasticity.
The role of poroelasticity in the propagation of shear cracks and frictional sliding has been a subject of in
-
terest for decades (Dunham & Rice,
2008
; Heimisson et al.,
2019
; Rice & Cleary,
1976
; Rice & Simons,
1976
;
Abstract
Faults in the crust at seismogenic depths are embedded in a fluid-saturated, elastic, porous
material. Slip on such faults may induce transient pore pressure changes through dilatancy or compaction
of the gouge or host rock. However, the poroelastic nature of the crust and the full coupling of inelastic
gouge processes and the host rock have been largely neglected in previous analyses. Here, we present a
linearized stability analysis of a rate-and-state fault at steady-state sliding in a fully-coupled poroelastic
solid under in-plane and anti-plane sliding. We further account for dilatancy of the shear zone and
the associated pore pressure changes in an averaged sense. We derive the continuum equivalent of the
analysis by Segall and Rice (1995,
https://doi.org/10.1029/95jb02403
), and highlight a new parameter
regime where dilatancy stabilization can act in a highly diffusive solid. Such stabilization is permitted
since the time scale of flux through the shear zone and diffusion into the bulk can be very different.
A novel aspect of this study involves analyzing the mechanical expansion of the shear layer causing
fault-normal displacements, which we describe by a mass balance of the solid constituent of the gouge.
This effect gives rise to a universal stabilization mechanism in both drained and undrained limits. The
importance of the mechanism scales with shear-zone thickness and it is significant for wider shear zones
exceeding approximately 1 cm. We hypothesize that this stabilization mechanism may alter and delay an
ongoing shear localization process.
HEIMISSON ET AL.
© 2021. The Authors.
This is an open access article under
the terms of the
Creative Commons
Attribution-NonCommercial
License,
which permits use, distribution and
reproduction in any medium, provided
the original work is properly cited and
is not used for commercial purposes.
Dilatancy and Compaction of a Rate-and-State Fault in a
Poroelastic Medium: Linearized Stability Analysis
Elías Rafn Heimisson
1,2
, John Rudnicki
3
, and Nadia Lapusta
1,4
1
Seismological Laboratory, California Institute of Technology, Pasadena, CA, USA,
2
Swiss Seismological Service, ETH
Zurich, Zurich, Switzerland,
3
Department of Civil and Environmental Engineering and Department of Mechanical
Engineering, Northwestern University, Evanston, IL, USA,
4
Department of Mechanical and Civil Engineering,
California Institute of Technology, Pasadena, CA, USA
Key Points:
We analyze stability of a rate-and-
state fault in a poroelastic solid with
fully coupled dilatancy
We show that dilatancy stabilization
can also occur in a highly diffusive
bulk if shear zone permeability is
low
We identify a new stabilizing
mechanism associated with the
mechanical expansion of the shear
zone
Correspondence to:
E. R. Heimisson,
eheimiss@caltech.edu
;
elias.heimisson@sed.ethz.ch
Citation:
Heimisson, E. R., Rudnicki, J., &
Lapusta, N. (2021). Dilatancy and
compaction of a rate-and-state
fault in a poroelastic medium:
Linearized stability analysis.
Journal
of Geophysical Research: Solid Earth
,
126
, e2021JB022071.
https://doi.
org/10.1029/2021JB022071
Received 17 MAR 2021
Accepted 24 JUN 2021
10.1029/2021JB022071
RESEARCH ARTICLE
1 of 28
Journal of Geophysical Research: Solid Earth
Rudnicki & Koutsibelas,
1991
; Rudnicki & Rice,
2006
). Perhaps the most intriguing aspect of this problem
is the role of the pore pressure changes during in-plane, or mode II, sliding. Such sliding induces volumet
-
ric stress change on both sides of the fault plane, whereas anti-plane or mode III sliding does not induce
volumetric stress. During in-plane sliding, the volumetric stress change is compressive on one side and ex
-
pansive on the other, with a discontinuity across the plane. This raises an important question of which pore
pressure should be used to compute the effective normal stress at the frictional interface. Field observations
of faults suggest that the principal slip zone often lies at the boundary of the damage zone and the fault core
(F. M. Chester et al.,
1993
,
2004
; Dor et al.,
2006
). The fault core generally has a much lower permeability
than the damage zone (Wibberley & Shimamoto,
2003
). In models that idealize the fault core as an imper
-
meable surface, the relevant pore pressure is often taken to be the value at an infinitesimal distance from
the shear zone (presented as
p
+
or
p
in Figure
1a
but no core depicted) (Dunham & Rice,
2008
; Heimisson
et al.,
2019
; Rudnicki & Koutsibelas,
1991
; Rudnicki & Rice,
2006
). Another view was presented by Jha and
HEIMISSON ET AL.
10.1029/2021JB022071
2 of 28
Figure 1.
We explore various ways in which the deformation of a leaky and pressurized thin shearing layer of thickness 2
ε
and mobility
κ
c
, which we consider
to be the shear zone, may couple to the surrounding poroelastic medium of mobility
κ
. (a) In-plane shear across the thin layer (indicated by horizontal arrows)
compresses the bulk material on one side of the shear crack tip and dilates the material on the other. Due to poroelastic coupling, this increases pore pressure
on the compressive side of the layer and decreases the pore pressure on the dilation side. This case, in which changes from pore pressure arise only from
slip
δ
x
(
x
,
t
), was explored by Heimisson et al. (
2019
). (b) Processes in the thin layer, such as injection or inelastic dilation/compaction, may cause the layer to
contract or expand (as indicated by vertical arrows), which would cause pore pressure changes in the surrounding medium. For example, expansion of the
layer (outward facing arrows) would compress the bulk (as indicated by the word “compression”) and raise pore pressure in the bulk. (c) Internal pore pressure
decrease can occur in the layer,
p
c
(
x
,
t
) < 0, perhaps due to inelastic dilation. The flow of pore fluids into the layer from the surrounding medium would cause
compression adjacent to the layer in the bulk. (d) An example of a situation that combines changes in the pore pressure
p
c
(
x
,
t
) in the layer (an increase in this
case, e.g., due to fluid injection) and bulk effects of shear across the layer. The bulk material adjacent to the fault may undergo both compression and dilation
(due to slip) and dilation due to pore pressure flow from the fault to the bulk if pressure
p
c
(
x
,
t
) exceeds the slip induced pressure changes at the boundary, as
shown.
y
y=0
compression/dilation
dilation
d
Pore pressure change
0
y
y=0
Leaky layer
Distance from layer cente
r
compression
dilation
Distance along layer
x
0
y
y=0
Distance from layer cente
r
compression
dilation
Distance along layer
x
Pore pressure
at y fixed
change
y
y=0
compression
compression
Distance along layer
x
Pore pressure change for x fixed
0
Pore pressure change for x fixed
0
Pore pressure change for x fixed
Distance along layer
x
c
a
b
Journal of Geophysical Research: Solid Earth
Juanes (
2014
), in which shear localization occurs preferentially in the fault core where effective normal
stress is low and thus the relevant pore pressure is where it is highest on either side of the fault core. How
-
ever, such a model requires the shear localization zone to be able to change sides dynamically in the core
depending on how the normal stress evolves. We conclude that significant uncertainty remains regarding
how slip-induced pore pressure changes interact with the shear zone and/or fault core and dynamically
change the effective normal stress.
Here, we introduce a somewhat conservative and simplified view and select the average pore pressure
through the shear zone as the relevant pore pressure for computing the effective normal stress. We allow
the shear zone to have a different permeability than the host rock. This choice of the relevant pore pressure
implies that the shear-zone width is initially at steady state and not localizing or delocalizing at any relevant
frictional or diffusional time scale. As we explain in more detail in the next section, the problem of selecting
the appropriate pore pressure for shear of a finite-width fault zone in a poroelastic medium remains largely
unsolved and likely needs explicit modeling.
When sheared and perturbed, for example, due to changes in slip speed, the fault gouge can dilate or com
-
pact. The process changes the void volume fraction of the gouge, which is also approximately the porosity
of the gouge. If the volume change occurs faster than the fluid pressure can equilibrate, then the changes
in the void volume fraction can dramatically alter the pore pressure. Much like other processes of frictional
interfaces, the influence of these volume changes on frictional strength has not been derived from first
principles. The related models and theory have been largely derived and developed based on empirical
observations (e.g., Lockner & Byerlee,
1994
; Marone et al.,
1990
; Proctor et al.,
2020
). Nevertheless, the
process can be understood as the result of continuous rearranging and deformation of grains in the gouge
to accommodate sliding. Based on experimental results (Marone et al.,
1990
), Segall and Rice (
1995
) postu-
lated, following the critical state concept in soil mechanics, the existence of a steady-state void volume (or
porosity) which establishes itself eventually for sliding at steady state with a given constant slip velocity. If
the slip speed increases or decreases, the granular structure dilates or compacts, respectively. Dilatancy and
compaction are well established from laboratory frictional experiments spanning three decades (e.g., Lock
-
ner & Byerlee,
1994
; Marone et al.,
1990
; Proctor et al.,
2020
) and have been attributed to strain-rate harden
-
ing of visco-plastic asperity contacts in simulations of rough interfaces (Hulikal et al.,
2018
) as well as the
dynamics of grains in simulations of granular media without viscoplasticity (Ferdowsi & Rubin,
2020
). In
addition to being induced by shear of granular materials, dilatancy is well known to accompany inelastic
deformation of brittle rocks (Brace et al.,
1966
) and can be induced by earthquake nucleation and rupture
(Lyakhovsky & Ben-Zion,
2020
; Templeton & Rice,
2008
). Dilatancy accompanying earthquake nucleation
and rupture will likely happen at a different scale than the granular dilatancy and further increase the com
-
plexity of pore-pressure changes in the vicinity of the shear-zone (Viesca et al.,
2008
). In this paper we will
not further consider this type of dilatancy.
Segall and Rice (
1995
) used the laboratory observations of Marone et al. (
1990
), which documented porosity
changes in a velocity-stepping experiment under drained conditions, to propose a model for the observed
porosity changes. They postulated the existence of a steady-state porosity which depends on the slip velocity
and to which the porosity evolves with slip:
 


 






0
0
log,
VV
LV
(1)
where
φ
0
is the steady-state porosity at the reference slip speed
V
0
,
L
is the characteristic state evolution
distance, and
γ
is an empirical dilatancy coefficient. Segall and Rice (
1995
) also proposed a related dilatancy
model in which the porosity depends on the frictional state variable that reflects the evolution of the sliding
surface(here Equation
33
). Near steady-state sliding, the two models behave the same, but some differences
occur away from the steady state. Recent experiments have suggested that a state-variable formulation may
be more appropriate (Proctor et al.,
2020
). We emphasize that even though
γ
is referred to as a dilatancy
coefficient, the formulations by Segall and Rice (
1995
) describe both dilatancy and compaction, or alterna
-
tively void volume changes.
HEIMISSON ET AL.
10.1029/2021JB022071
3 of 28
Journal of Geophysical Research: Solid Earth
Segall and Rice (
1995
) then coupled the dilatancy model with a simple single-degree-of-freedom spring slid
-
er system and a membrane diffusion model (Rudnicki & Chen,
1988
) and carried out a linearized stability
analysis and numerical simulations. This work was revisited by Segall et al. (
2010
) who expanded previous
work on the spring-slider stability analysis and explored a more elaborate homogenous diffusion model.
However, the main goal of Segall et al. (
2010
) was to explore dilatancy as a mechanism that can quench
earthquake instability and generate slow slip. Models using dilatancy for stabilization have found agree
-
ment with observed behavior of subduction zone slow-slip events (e.g., Dal Zilio et al.,
2020
; Liu,
2013
;
Segall et al.,
2010
). These models go beyond the spring-slider analysis and explore a rate-and-state fault with
dilatancy coupled to an elastic continuum. However, to date, dilatancy coupled to a poroelastic bulk, as we
do here, has not been explored.
In this study, we formulate a closed system of equations and carry out a linearized stability analysis of a
rate-and-state fault with dilatancy coupled to a poroelastic bulk. Further, we allow the shear zone to have
different diffusivity from the bulk.
The paper starts by discussing the governing equations, boundary conditions, and various effects that may
arise from frictional sliding and dilatancy or compaction in a poroelastic solid. That section concludes by
presenting solutions for stresses and pore pressures at the fault in a joint Fourier-Laplace transform domain.
The following section derives various constitutive relationship for the shear layers and presents the rate-
and-state friction law. The section concludes with the mathematical formulation of the linearized stability
analysis. Finally, we present the results and derive several simple approximations that characterize stability
in certain limiting cases. The section concludes by comparing these approximations to the full solutions to
the characteristic equation obtained through a standard root-finding algorithm.
2.
Problem Statement and Boundary Conditions
We consider two poroelastic half spaces with interface at
y
= 0 that are uniformly sliding past each other
with slip rate
V
0
across the interface which is spatially and temporally uniform.
V
0
is small enough such
that inertial effects and wave-mediated stress transfer can be ignored. The interface is at a uniform shear
stress
τ
0
and effective normal stress
σ
0
and thus friction coefficient
f
0
=
τ
0
/
σ
0
. The pore pressure
p
is also at
equilibrium and spatially uniform. At time
t
= 0, this steady-state configuration is perturbed by introducing
a Fourier mode slip perturbation
δ
x
=
e
st+
kx
, with the total slip for
t
> 0 being
V
0
t
+
δ
x
. This non-uniform (or
heterogeneous) slip excites spatial variation in slip speed, shear stress, pore-pressure, and normal stress.
The displacements
u
i
and pressure changes
p
relative to an equilibrium pressure state are governed four
coupled partial differential equations. These are (e.g., Detournay & Cheng,
1995
)

,
,,
12
i kk
k ki
i
G
Gu
u
p
(2)
and


,,
,
1
,
t
kk
k kt
pp
u
M
(3)
where
u
i
are displacements and we have assumed that body forces are negligible. The equations are present
-
ed in the index notation. Subscript “,
t
” indicates the partial derivative with respect to time, subscript “,
i
indicates the partial derivative with respect to the spatial coordinate
i
. Index
i
= 1 refers to the
x
axis, which
lies in the fault plane. Index
i
= 2 refers to the
y
axis that is, perpendicular to the fault plane. Finally, index
i
= 3 corresponds to the
z
axis, but all fields will be assumed invariant in that direction since we will con
-
duct a plane-strain analysis. Repeated indices such as “
kk
” represent sum over all spatial indices. Finally,
the material parameters are denoted as follows,
G
: shear modulus,
ν
: drained Poisson's ratio,
α
: Biot-Willis
parameter,
M
: Biot modulus. Finally,
κ
is the mobility, which is defined as the ratio between the permeabil
-
ity and fluid viscosity. Later we shall replace some of these parameters with other poroelastic parameters
for more compact and intuitive expressions. In Appendix
A
, we provide expressions for converting between
poroelastic parameters and Table
A1
with important fixed parameters.
HEIMISSON ET AL.
10.1029/2021JB022071
4 of 28