By combining a 3-D boundary element model, frictional slip theory, and fast computation method, we propose a new tool to improve fault slip analysis that allows the user to analyze a very large number of scenarios of stress and fault mechanical property variations through space and time. Using both synthetic and real fault system geometries, we analyze a very large number of numerical simulations (125 000) using a fast iterative method to define for the first time macroscopic rupture envelopes for fault systems, referred to as “fault slip envelopes”. Fault slip envelopes are defined using variable friction, cohesion, and stress state, and their shape is directly related to the fault system 3-D geometry and the friction coefficient on fault surfaces. The obtained fault slip envelopes show that very complex fault geometry implies low and isotropic strength of the fault system compared to geometry having limited fault orientations relative to the remote stresses, providing strong strength anisotropy. This technique is applied to the realistic geological conditions of the Olkiluoto high-level nuclear waste repository (Finland). The model results suggest that the Olkiluoto fault system has a better ability to slip under the present-day Andersonian thrust stress regime than for the strike-slip and normal stress regimes expected in the future due to the probable presence of an ice sheet. This new tool allows the user to quantify the anisotropy of strength of 3-D real fault networks as a function of a wide range of possible geological conditions and mechanical properties. This can be useful to define the most conservative fault slip hazard case or to account for potential uncertainties in the input data for slip. This technique therefore applies to earthquake hazard studies, geological storage, geothermal resources along faults, and fault leaks or seals in geological reservoirs.
Better understanding of the mechanical interplay between fault slip, 3-D
fault geometry, stresses, and rock mechanical properties (e.g. Byerlee,
1978; Morris et al., 1996; Lisle and Srivastava, 2004; Moeck et al.,
2009) is an actual and future scientific challenge in geosciences because
(1) conventional failure or plasticity laws derived from rock testing does
not apply to large rock volumes at geological conditions and timescale
(e.g. Brantut et al., 2013) and (2) because fault slip has increasing
societal applications (e.g. slip hazard; seismicity; hydraulic fracturing;
fault mechanical seal; rock stability; unconventional resources; and storage
of hydrocarbon gases,
Although general knowledge on the geometry, constitution, and
behavior of fault zones is improving (e.g. Holdsworth, 2004; Faulkner et
al., 2006; Wibberley et al., 2008), it is clear that the large-scale
strength of a faulted rock volume is poorly known (e.g. Colettini et al.,
2009; McLaskey et al., 2012). Laboratory tests on sampled rocks or fault
rocks partly resolve this problem in giving a range of mechanical properties
and friction laws. The strength of rocks has been classified under several
types of behavior defined by rupture or plasticity envelopes with respect to
rock type (Mohr–Coulomb, Byerlee, Griffith, Cam Clay types), which describe
typically the elastic domains of small, intact or precut rock samples
(Byerlee, 1978; Rutter and Glover, 2012). For pre-existing fault surfaces,
fault stability is generally described by the Mohr–Coulomb theory, in which
the shear strength (
Slip tendency analysis is a well-known method providing tools considering the ratio of resolved shear to normal stresses to model the likelihood for slip on pre-existing surfaces of all possible orientations relative to a regional stress field (e.g. Arthaud, 1969; Morris et al., 1996; Lisle and Srivastava, 2004; Collettini and Trippetta, 2007; Lejri et al., 2017). Beyond its successful application to many cases of fault slip hazard or induced seismicity (e.g. Moeck et al., 2009; Yukutake et al., 2015), this method does not provide the possibility to analyze together large numbers of geological conditions such as variations in stress state, orientation, friction, cohesion, or fluid pressure. Fault slip hazard has generally to be analyzed thoroughly with respect to potential variation through space and time of such important parameters, which actually requires full and time-consuming parametric modeling, and therefore fast calculation techniques. Such a development is, however, critically needed in the new age of data science and numerical geology featured by an increasing availability of 3-D numerical fault system data, in situ rock properties, stress measurements, and high-speed computers. It is also a way to account for potential uncertainties in the input data and to define the most conservative fault slip hazard case.
An improvement of the slip tendency analysis tool, or other equivalent
numerical method (Neves et al., 2009; Alvarez del Castillo et al., 2017),
would be to incorporate a 3-D mechanical model, allowing the user to analyze the DFN (discrete
fault or fracture network) subjected to multiple cases of stress states, and
in which fault strength is resolved using a complete static frictional
behavior (including cohesion). Although well accepted, Mohr–Coulomb theory
has been recently regarded more critically to explain fault initiation under
polyaxial loading or in situations where
In this paper, we use a 3-D boundary element numerical model (iBem3D; e.g. Maerten et al., 2014) in which a Coulomb frictional law is resolved on DFN surfaces to quantify fault system static strength as a function of variable mechanical parameters in a range consistent with geological conditions, and to assess zones having potential for fault slip. Using a fast-calculation iterative method allowing the user to analyze a very large number of numerical simulations (125 000), we define macroscopic fault slip envelopes of rock volumes containing faults as a function of variable stress orientation, 3-D fault geometry and frictional properties. This technique, applied to the case study of the Olkiluoto fault system, allows for analyzing fault-slip hazard for multiple geological scenarios, including variable triaxial stress profiles through space and time and fault mechanical properties in the range of potential uncertainties derived from mechanical tests.
Fault slip envelope of a simple-planar
elliptical fault of 60
We propose to calculate fault slip envelopes for both synthetic and real
fault system geometry using the 3-D numerical model iBem3D, a quasi-static
iterative boundary element model (Maerten et al., 2014). In iBem3D, faults
are discretized using triangulated surfaces of frictional behavior (Eq. 1)
in a heterogeneous, isotropic elastic whole- or half-space also allowing
mechanical interaction between each triangular element when the fault
surfaces slip (Maerten et al., 2002). For the first part of this study
aiming to define fault slip envelopes, the effective stress state is
simplified as a horizontal simple-uniaxial stress (
Fault slip can occur in places where the Coulomb criterion is reached on
fault surfaces. In other words, slip occurs along preferred orientations of
fault surfaces with respect to the amount of friction, cohesion, and resolved
shear and normal stresses on fault planes computed following the Cauchy
equations (e.g. Pollard and Fletcher, 2005; Jaeger and Cook, 1979).
Quasi-static fault displacement (net slip) can be computed on fault planes
using linear elasticity (see Thomas, 1993; Maerten et al., 2010, 2014, 2018;
for full explanation), taking into account static friction and cohesion,
mechanical interaction due to stress perturbation between faults (e.g. Crider and Pollard, 1998; Kattenhorn et al., 2000; Soliva et al., 2008;
Maerten et al., 2014), and using Young's modulus (
We studied first a synthetic fault geometry characterized by a 60
Synthetic 3-D fault geometries and their fault slip envelopes.
Examples of 3-D fault system geometry from a simple to a very
complex case, and related fault slip envelopes.
We apply this technique to study the potential of fault reactivation in the
fault system of Olkiluoto Island (Finland), where a deep geological
high-level nuclear waste repository is being built and which also is a site
for two operational nuclear power plants. The site is located in
Paleoproterozoic amphibolite-facies metasedimentary rocks and
tonalitic-granodioritic-granitic gneisses, cut by a complex 3-D geometry of
brittle faults, spanning in age from 1.7 to 1.0
The present-day stress state (Fig. 6a, 0
Based on present-day stresses (
We calculated the triaxial stress profiles due to an increase in vertical
load and its subsequent confining pressure such as
Stress permutations are expected due to strong variations in the stress
distribution at depth (Fig. 3a). A hybrid thrust-fault and strike-slip
regime is measured in the actual conditions with no ice sheet, with a
prominent proportion of thrust-fault regime above 1
The frictional behavior of the fault zones is also a major process to
consider as a variable. Measurements of friction and cohesion values have
been done on the Olkiluoto-fault core rocks (Hudson et al., 2008;
Mönkkönen et al., 2013). These estimations give effective
macroscopic values of static friction in the range of
In the same way as previously shown, 50 values of each variable, i.e. ice
thickness, friction, and cohesion, have been chosen and the fault slip
envelope obtained for the Olkiluoto Island fault system separates the
parametric domain (here
The fault slip envelope for a simple-elliptical fault (Fig. 1) appears
mainly sinuous in shape in the direction of
For more complex fault geometries, the fault slip envelopes are sinuous in
the direction of
Similar results can be found for the real fault systems (Fig. 3, see
references in Sect. 2.1 for the source data). In simple fault geometry
such as the Landers or Chimney Rock examples (Fig. 3a and b), the common
segmentation or conjugate geometries frequently found in fault systems
provide quite constant fault orientation through space. This therefore
results in significant spatial anisotropy of strength, expressed by a local
inflection of the fault slip envelope as a function of
In both synthetic and real fault system geometries (Figs. 1, 2, and 3), the
degree of irregularity of the fault slip envelope appears to be inversely
correlated with the degree of orientation anisotropy of the 3-D fault system.
The fault slip envelopes appear mainly sinuous in shape in the direction of
Examples of 3-D quasi-static fault displacement distributions on
the Landers model for different mechanical conditions and uniaxial stress
orientation.
The plot of computed displacement distribution along fault is a way to
analyze in which place the fault is prone to slip with respect to different
parametric conditions. Some examples of computed displacement occurring on
preferentially oriented parts of the fault surfaces are shown in Fig. 4
for the Landers 3-D fault geometry. Computed quasi-static fault displacement
distribution is shown (blue stars) for end-member conditions of friction,
cohesion, and stress orientation with respect to the position of the fault
slip envelope (note that the color bar scale of displacement is
logarithmic). These plots show how large values of friction coefficient
allow faults to slip, revealing a very different amount and distribution of
displacement as a function of
Examples of quasi-static stress distribution of
The 3-D shape of the fault slip envelope obtained using stress variations as proposed in Sect. 2.2 is quite simple in its geometry with a curvature in the upper corner where cohesion and ice thickness are low and friction is relatively high (Fig. 6c). This reveals that fault slip is promoted for small ice thicknesses (or vertical load) in thrust-fault regime, especially for low cohesion, allowing slip even for relatively high friction values. Other envelopes (pink) are shown in the slip domain of the diagram. These surfaces are not fault slip envelopes but envelopes of values of equal maximum quasi-static displacement computed along faults, calculated in each model. These envelopes, which depict two specific values of computed maximum displacement for different mechanical and loading conditions, mimic the shape of the fault slip envelope in the slip domain. They confirm the shape of the slip envelope and reveal that the largest ability of fault to accumulate displacement is expected for models with unrealistic conditions of no fault friction, no cohesion, and no ice sheet. Since quasi-static displacement takes into account fault interaction through the stress field, and that fault slip envelope does not, the similar shape of these three envelopes reveals the lesser influence of fault mechanical interaction compared to the effect of varying friction, cohesion, and stresses in the ranges considered in this study.
Case study of the Olkiluoto fault system.
Examples of 3-D quasi-static fault displacement distribution on the Olkiluoto model for different loading and fault property conditions indicated on the fault slip diagram by blue stars. Streamlines on fault surfaces are slickenlines. The color bar scale for displacement is logarithmic.
Quasi-static displacement distributions along fault slipping patches are
shown in Fig. 7 in colored areas (the color bar scale is logarithmic)
containing streamlines representing the orientation of fault slip, referred
to as “slickenlines”. Displacements are computed for individual models of
parametric conditions shown on the envelope with blue stars. We chose to
show end-member cases, i.e. far and progressively closer to the main fault
slip envelope, with roughly different parametric conditions. Displacement
distribution varies from one model to another and is heterogeneous within a
same model as a function of fault plane orientation, friction, and applied
stress state with changing ice thickness. Consistent with the shape of the
envelopes, the ability of slip to initiate and accumulate is enhanced for
conditions of low friction, low cohesion, and no ice sheet cover for which most
of the faults are slipping (maximum displacement lower than 0.7
We defined fault slip envelope and therefore quantified the strength of
large rock volumes containing faults as a function of friction, cohesion, and
remote stresses. In the first order, our parametric study of simple to
complex fault systems reveals that their strength can be assessed as a
function of their degree of geometric complexity. The more complex is the
geometry, the simpler the fault slip envelopes. Complex fault systems
always have optimally oriented fault surfaces that can slip with respect to
the boundary stress conditions applied. In contrast, fault systems having
limited fault orientations relative to the principal remote stresses
provide strong strength anisotropy such as revealed by strong curvature of
the fault slip envelope in the direction of
The case study of the Olkiluoto nuclear waste repository site allowed us to
apply this technique on a fault system subjected to realistic stress loading
conditions. The resulting fault slip envelope for the Olkiluoto system shows
the importance of the 3-D geometry of the fault system, but also the critical
importance of the applied geological stresses. For the present-day context
of no ice sheet (interglacial period), this fault system is subjected to a
thrust-fault stress regime governed by the E–W push from the Mid-Atlantic
Ridge. This geological context seems to provide the best conditions for
fault slip because of the highest resolved shear stress on low dipping fault
surfaces. Such fault surfaces are optimally oriented to slip under a
thrust-fault stress regime with
Although the actual conditions provide the largest resolved shear stress on
fault plane, the main results obtained are in agreement with the actual
conditions observed at the Olkiluoto repository site, where no slip is
monitored or observed along faults under the actual conditions. Although
probably variable along fault surfaces, the fault rock mechanical properties
derived from mechanical tests suggest a static friction and cohesion larger
than the conditions computed to allow fault slip. The worst case scenario
would corresponds to the upper-right 3-D model result shown in Fig. 7 (
The increase in thickness of an ice sheet implies progressive stress
permutation to the strike-slip regime in the stress profiles (Fig. 6a).
The general low dip of the faults (non-optimal orientation) combined with a
low differential stress, inherent to this strike-slip regime, provide
conditions for low resolved shear stresses on faults, and therefore better
general strength of the fault system (also see Johnston, 1987). The planar
and vertical shape of the fault slip envelope in this lower part of the
diagram reveals the little dependence of fault strength on the vertical load
increase, here the value of
A potential limitation of the proposed technique relies on the uncertainty
and biases of the 3-D fault surface discretization. In the example of
Olkiluoto, uncertainties in fault surface geometry were estimated from a
significant amount of data available from bore hole, seismic profiles,
tunnel wall observations, and outcrop measurements (Mattila et al., 2008).
Truncation bias is here defined by not considering the faults smaller than
100
A new tool referred to as “fault slip envelope” is proposed in order to
provide complementary analysis to the conventional methods used for fault
slip or slip tendency:
Fault slip is calculated along simple or very complex fault geometry on DFN
using the resolved shear and normal stresses with respect to the Mohr–Coulomb frictional slip theory and quasi-static elastic behavior. This
method allows for considering friction and cohesion as potential variables
through space and time. Combining a 3-D boundary element model and fast computation method allows the user to
run thousands of forward simulations in a very short time, and therefore to
provide a full parametric study with a wide range of variable mechanical
conditions such as stress orientation and magnitude. This technique allows the user, for the first time, to propose “fault slip envelopes”
which quantify fault system strength magnitude and anisotropy as a function
of important parameters, which are either unknown and/or considered
variable through time and space. This can be useful to address uncertainties
in input data for hazard assessment. We also calculate fault displacement based on a quasi-static elastic solution
allowing mechanical interaction through the stress field in places where
the Coulomb's criterion is reached along each fault of a DFN.
The quantification of the strength of fault systems in 3-D underlines the
importance of accuracy in deterministic studies of geological structures and
stresses. Beyond its societal application to fault slip hazard, geological
storage, geothermal systems, and reservoir leaks, this technology also
provides new considerations and perspectives in the analysis of fault
systems and Earth's crust strength. Major earthquakes at plate boundaries
occur on relatively simple fault systems, such as large strike-slip faults
or subduction plate megathrusts (Berryman et al., 2012; Chester et al.,
2013; also see Fig. 3a), where the strength is definitely anisotropic and
thoroughly depends on stress orientation and fault zone properties (e.g. Fig. 3d). It is nevertheless also well known that large earthquakes can
occur on more complex fault geometries, as for example in the Kaikoura and
Darfield fault systems in New Zealand's South Island
(Beavan et al., 2012; Hamling et al.,
2017; Ulrich et al., 2019) or in the Sierra Madre–Cucamonga thrust fault
system in southern California (Anderson et al., 2003). In such a case, fault
system strength is probably more isotropic and fault slip depends more on
static friction along faults than on stress orientation (e.g. Fig. 3f).
This difference in domain of stability allows for quantifying the bulk strength
of the brittle crust, which is lower for complex fault geometry rather
than a simple one for equivalent frictional and remote stress conditions, as
recently observed in experimental modeling of a complex versus simple
subduction interface (Van Rijsingen et al., 2019). As much as frictional
properties or pore pressure, the degree of complexity of a fault system
constitutes the basic premise for easier crustal stress relaxation and
prevention of major slip events. Consequently, the precise definition and
quantification of the strength in the brittle crust relies on the precise
knowledge of 3-D fault geometry, constitution, and stresses at each study
site. Significant progress in this field poses a challenge for future
geosciences.
Data and scientific
reports from the Olkiluoto nuclear site are archived and available online at
the following URL:
RS conceptualized the fault slip envelopes, ran models, analysed the results, wrote the article, and did the revisions. FM wrote the numerical codes, ran models, and participated in the analysis of the results and article writing. LM ran some early models, provided 3-D fault system data, and participated in the analysis of the results. JM provided data from the Olkiluoto nuclear repository site and participated in the article writings.
A first version of this article was rejected from the journal
The authors wish to thank the associate editor Cristiano Colettini and the two anonymous reviewers for their constructive comments, which helped improve the quality of the article.
This work was initiated during a research program funded by POSIVA and IGEOSS companies, and pursued with the financial support of Geoscience Montpellier and Schlumberger.
This paper was edited by Cristiano Collettini and reviewed by two anonymous referees.