Earth and Planetary Science Letters 434 (2016) 64–74
Contents lists available at ScienceDirect
Earth and Planetary Science Letters
www.elsevier.com/locate/epsl
On the mechanisms governing dike arrest: Insight from the 2000
Miyakejima dike injection
F. Maccaferri a,∗ , E. Rivalta a , L. Passarelli a , Y. Aoki b
a
b
GeoForschungsZentrum (GFZ), Potsdam, Germany
Earthquake Research Institute (ERI), University of Tokyo, Japan
a r t i c l e
i n f o
Article history:
Received 31 July 2015
Received in revised form 11 November 2015
Accepted 17 November 2015
Available online 2 December 2015
Editor: T.A. Mather
Keywords:
lateral dike propagation
dike arrest
dike–fault interaction
triggered seismicity
volcano deformation
a b s t r a c t
Magma stored beneath volcanoes is sometimes transported out of the magma chambers by means of
laterally propagating dikes, which can lead to fissure eruptions if they intersect the Earth’s surface. The
driving force for lateral dike propagation can be a lateral tectonic stress gradient, the stress gradient due
to the topographic loads, the overpressure of the magma chamber, or a combination of those forces.
The 2000 dike intrusion at Miyakejima volcano, Izu arc, Japan, propagated laterally for about 30 km
and stopped in correspondence of a strike-slip system, sub-perpendicular to the dike plane. Then the
dike continued to inflate, without further propagation. Abundant seismicity was produced, including five
M > 6 earthquakes, one of which occurred on the pre-existing fault system close to the tip of the dike,
at approximately the time of arrest. It has been proposed that the main cause for the dike arrest was the
fault-induced stress.
Here we use a boundary element numerical approach to study the interplay between a propagating
dike and a pre-stressed strike-slip fault and check the relative role played by dike–fault interaction and
topographic loading in arresting the Miyakejima dike. We calibrate the model parameters according to
previous estimates of dike opening and fault displacement based on crustal deformation observations. By
computing the energy released during the propagation, our model indicates whether the dike will stop
at a given location. We find that the stress gradient induced by the topography is needed for an opening
distribution along the dike consistent with the observed seismicity, but it cannot explain its arrest at
the prescribed location. On the other hand, the interaction of dike with the fault explains the arrest
but not the opening distribution. The joint effect of the topographic load and the stress interaction with
strike-slip fault is consistent with the observations, provided the pre-existing fault system is pre-loaded
with a significant stress, released gradually during the dike–fault interplay.
Our results reveal how the mechanical interaction between dikes and faults may affect the propagation
of magmatic intrusions in general. This has implications for our understanding of the geometrical
arrangement of rift segments and transform faults in Mid Ocean Ridges, and for the interplay between
dikes and dike-induced graben systems.
© 2015 Elsevier B.V. All rights reserved.
1. Introduction
A fundamental question regarding magmatic dike propagation
is how dikes are arrested. Several mechanisms have been proposed
to explain why intrusions stop: magma freezing due to the large
aspect ratio of dikes (Delaney and Pollard, 1981; Lister and Dellar,
1996), choked magma supply from the reservoir due to co-diking
pressure drop (Buck et al., 2006; Rivalta et al., 2015) or similarly magma volume loss in the dike tail (Taisne and Tait, 2009;
*
Corresponding author.
E-mail address: [email protected] (F. Maccaferri).
http://dx.doi.org/10.1016/j.epsl.2015.11.024
0012-821X/© 2015 Elsevier B.V. All rights reserved.
Rivalta et al., 2015), dikes reaching a level of neutral buoyancy
and therefore loosing their driving force (Taisne et al., 2011;
Maccaferri et al., 2011), stress heterogeneities exerting compression around the propagating tip, for example due to topographic
loads (Watanabe et al., 2002; Buck et al., 2006; Maccaferri et al.,
2011), structural discontinuities such as layering (Gudmundsson,
2002; Maccaferri et al., 2010; Geshi et al., 2012), or co-diking slip
on pre-existing fractures or faults (Rubin et al., 1998).
Probably in most cases dikes become arrested due to a combination of multiple factors, and while the mechanics of some
individual “arresting” factors for dikes has been investigated, their
interplay is poorly understood. A prominent example is the 2000
intrusion at Miyakejima. The dike, originated from a magma reser-
F. Maccaferri et al. / Earth and Planetary Science Letters 434 (2016) 64–74
voir below Miyake volcano, propagated laterally for tens of km before becoming arrested. The dike arrest occurred on 1 July, temporally coincident with a M l = 6.5 earthquake (local magnitude, from
the seismicity catalog of the Japan Meteorological Agency, JMA)
that struck close to the dike’s tip, and spatially coincident with
an inversion of the topography of the ocean floor (downhill propagation during the first four days and uphill during the last few
hours). After the arrest, the dike never resumed propagation but
continued to inflate, showing that propagation was not halted by
a lack of supply or by freezing. The dike volume increased, accompanied by massive induced seismicity that released a total seismic
moment of 2 × 1019 J roughly equivalent to a single event of magnitude 6.8. Two more M l > 6 earthquakes occurred in the area of
the 1 July event, one on 7 July and the other on 18 August, and
creep on the same fault has also been suggested (Hughes, 2010;
Nishimura et al., 2001). Despite the effort dedicated to understand
the Miyakejima intrusion, the questions of what arrested the dike
and what blocked further propagation remain unanswered. Continuous yielding on pre-existing faults (Hughes, 2010) and increasing
topographic load are the most plausible hypotheses for this case,
but what was the relative role played by each?
Efficient interaction of dikes with nearly-perpendicular active
faults has been inferred in other volcanic environments. Examples
include Mt. Etna volcano, in Italy, where the strike-slip Pernicana
fault has been shown to interact with dikes propagating along the
North–East rift (Ruch et al., 2013) and the transform Tjörnes Fracture Zone, specifically the 1976 Kópasker earthquake on the Grimsey Oblique Rift in the North Volcanic Zone of Iceland (Passarelli
et al., 2013).
The generation of seismicity and faulting by dike intrusions has
been investigated through crustal deformation (Aoki et al., 1999;
Wright et al., 2006), seismic (Rubin et al., 1998), theoretical (Rubin
and Gillard, 1998), and statistical studies (Traversa et al., 2010;
Passarelli et al., 2015), or combining observations and physical
models (Segall et al., 2013; Passarelli et al., 2015). These studies
clarified many aspects concerning the dynamics of faulting due to
dike-induced stresses but did not consider the reaction of the dike
to the faulting, as faulting was assumed as a passive process, not
a source of stress itself. However, seismicity and faulting release
elastic energy and are therefore expected to influence the dynamics of the dike, as confirmed by observations and models (Rubin
et al., 1998; Rubin and Gillard, 1998; Rivalta and Dahm, 2004;
Rivalta et al., 2015). Klein et al. (1987) observed that areas on the
East Rift Zone of Kilauea with persistent seismicity seem to act as
barriers for dike propagation. Rivalta and Dahm (2004) developed
a numerical model of a dike interacting with faults and fractures
with application to the 2000 Miyakejima intrusion. They found
that a dike will react to faulting by showing a shear component of
slip on the opening dike plane, and changing the propensity of the
dike propagation. Le Corvec et al. (2013) showed by means of laboratory experiments of air-filled crack propagating in pre-fractured
gelatin that both the direction and the velocity of propagating
dikes are affected by the presence of pre-existing faults.
Topographic loads have also been linked to changes in the velocity or propagation direction of dikes. Fiske and Jackson (1972)
recognized that gravitational stresses are a key factor in controlling
the direction of propagating dikes. They observed that lateral dikes
on the Hawaiian chain are distributed radially or cluster in welldefined rifts. Fialko and Rubin (1999) developed numerical and experimental models of dike propagation focusing on the effect of a
slope on the dynamics of the dikes, recognizing the control played
by the competition of gravitational stresses and dike driving pressure. Dahm (2000), Maccaferri et al. (2011, 2014), Sigmundsson et
al. (2014), Roman and Jaupart (2014), Corbi et al. (submitted for
publication) modeled the trajectories of dikes taking into account
65
the contribution of gravitational stresses caused by surface loads
or unloading due to surface mass redistribution.
Here we use a numerical model to study the interplay between
a propagating dike and a pre-existing, tectonically pre-loaded fault
system that the dike is approaching during propagation. We calibrate the model on the 2000 intrusion at Miyakejima based on
results by Hughes (2010) who constrained dike and fault parameters by combining seismicity and GPS observations. We calculate
the expected shape and reach of the dike and the slip on the fault
by considering the stress induced by: a) bathymetry, b) slip on a
pre-loaded fault, c) a combination of a) and b).
2. Tectonic setting and the 2000 dike intrusion at Miyakejima
Miyakejima volcano lies in the middle of the Izu-Bonin volcanic arc associated with subduction of the Pacific plate beneath
the Philippine Sea plate (Fig. 1a). The Philippine Sea plate collides
towards the Honshu mainland to the north and subducts from
Suruga and Sagami troughs, resulting in northwest–southeast compression and northeast–southwest extension (Ukawa, 1991).
Miyakejima is a ten-thousand year old stratovolcano with an
approximately circular shape of 8 km in diameter at sea level and
a basal diameter of about 25 km, the height of the volcanic edifice is 800 m a.s.l. and 1200 m above the sea floor. During the
last 500 years, basaltic dikes have fed 14 fissure eruptions on the
volcano flanks. The last four eruptions showed an approximately
regular recurrence interval of 20 years (i.e. 1940, 1962, 1983 and
2000) (Saito et al., 2005).
The 2000 dike intrusion started on 26 June 2000 at 18:30
Japan Standard Time (JST) with thousands of low magnitude earthquakes beneath the west coast of Miyakejima. During the first
18 hours the seismicity was confined below the volcanic edifice at about 3–5 km depth. The nucleation and initial propagation of the dike was rather complex, involving the bending of the dike from EW to SE–NW (Ueda et al., 2005). After
12 hours from the onset, the propagation regularized, with seismicity migrating at a nearly constant velocity of 0.05 m s−1 towards the NW in the direction of Kozushima (Ito and Yoshioka,
2002; Ozawa et al., 2004) (Fig. 1b). On 1 July the dike became
suddenly arrested, temporally coincident with a M l = 6.5 earthquake (16:01 JST, day 4.86 in Fig. 1b), the migrating seismic
cloud had by then reached Kozushima. The displacements measured by continuous GPS on Kozushima and Niijima were less
than 5 cm, suggesting an initially narrow dike (Ozawa et al., 2004;
Hughes, 2010).
During the two following months the crust continued deforming while the seismicity cloud remained stable. This indicates that
the dike was stalled, but continued to inflate (Ito and Yoshioka,
2002; Ozawa et al., 2004; Yamaoka et al., 2005). The majority of
the hypocenters remained confined on a subvertical plane extending from 10 km NW of Miyakejima to SE of Kozushima (Ozawa
et al., 2004; Hughes, 2010). The seismicity further branched off
the central seismic cloud, showing sustained activity north of
Kozushima and south of Miyakejima. In this phase four M l > 6
earthquakes occurred. Modeling of the deformation on Miyakejima
recorded by tiltmeters and GPS suggests the deflation of a 4–6 km
deep magma chamber (Ueda et al., 2005).
A caldera progressively formed at the summit of Miyakejima,
eventually reaching 1.6 km in diameter and a depth of 450 m
(Fujita et al., 2002). Erupted products were basaltic and amounted
to only 0.02 km3 (Saito et al., 2005), a volume significantly smaller
than the caldera itself. This indicates that most of the magma extracted from the magma reservoir beneath Miyakejima had fed the
crustal intrusion.
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F. Maccaferri et al. / Earth and Planetary Science Letters 434 (2016) 64–74
Fig. 1. a) Map of the Izu archipelago and Peninsula, topographic data from ETOPO1 (Amante and Eakins, 2009) are color coded. The black solid line represents the surface
projection of the daily centroid of the seismicity induced by the dike intrusion during the first five days of dike propagation phase. The arrows indicate the direction of
motion of the Philippine Sea Plate (PSP) with respect to the Eurasia plate (EUP), PP stands for Pacific Plate. In the inset map, the red box highlights the studied region. The
plates’ relative velocity is from Argus et al. (2010). b) Time vs. distance plot for the seismicity accompanying the 2000 Miyakejima dike intrusion. The distance is projected
along the N310E direction (parallel to the dike), time is in days from 26 June 2000 18:30 JST. The color code and the size of the circles refer to the local magnitude of
the earthquakes according to the Japan Meteorological Agency’s catalog. The solid line is the best fitting regression line calculated using the data from 12 hour from the
intrusion start. Dashed lines are ±5 km from the best fitting line indicating the fracturing processes induced by the propagating dike. The slope b (∼0.05 m s−1 ) of the
regression is the propagation velocity of the seismicity front and is approximately constant in time during the interval 0.5–5 days from intrusion onset. (For interpretation of
the references to color in this figure legend, the reader is referred to the web version of this article.)
3. Results from previous inversions of GPS displacements and
seismicity
The offshore nature of the dike intrusion and the consequent
non-optimal distribution of GPS data results in a poorly constrained geometry of the dike. All published studies agree on the
fact that a simple deflating magma reservoir with a propagating
dike cannot explain the observed cumulative deformation, confirming the hypothesis of a strong interaction of the dike with
pre-existing tectonic structures. Several models with different degrees of complexity have been proposed. The models assume at
least one dike intrusion and one deflating source beneath Miyakejima. Many incorporate also seismic and aseismic slips on faults
(Nishimura et al., 2001; Ito and Yoshioka, 2002; Ozawa et al., 2004;
Yamaoka et al., 2005).
The intrusion has been modeled by means of one or two tensile
dislocations inverting for the cumulative of the deformation signal
(Nishimura et al., 2001; Yamaoka et al., 2005) or performing time-
F. Maccaferri et al. / Earth and Planetary Science Letters 434 (2016) 64–74
67
Table 1
Rock and magma parameters used in the model.
Rock fracture toughness
(K c )
Shear modulus
(μ)
Poisson’s ratio
(ν )
Rock density
(ρ )
Magma density
( ρm )
Magma bulk modulus
(B)
[100–300] MPa m1/2
30 GPa
0.25
2900 kg m−3
2900 kg m−3
10 GPa
dependent inversion (Ito and Yoshioka, 2002; Toda et al., 2002;
Ozawa et al., 2004). The lateral dimensions of the dislocation
vary from 10 km to 20 km with downdip lengths of 1 km up
to 15 km and resulting dike widths ranging from 2 m to 28 m
(Nishimura et al., 2001; Ito and Yoshioka, 2002; Toda et al., 2002;
Ozawa et al., 2004; Yamaoka et al., 2005). The discrepancy between
the estimated dike volume (∼1 km3 ) and volume change at the
magma reservoir at Miyakejima (∼0.1 to 0.2 km3 ), has been explained by adding one additional magma chamber at Kozushima or
at halfway between Kozushima and Miyakejima (Nishimura et al.,
2001; Ito and Yoshioka, 2002; Murase et al., 2006; Yamaoka et al.,
2005). However, there is no indication in the co-diking deformation time series of a sudden brake of slope that could be related to
a new magma source contributing to the inflation (Hughes, 2010).
The discrepancy could be also explained by compressibility effects
(Johnson et al., 2000; Rivalta et al., 2015). In the following, we
will consider the reservoir below Miyakejima as the only magma
source.
Creeping faults have been introduced in several studies
(Nishimura et al., 2001; Ozawa et al., 2004; Yamaoka et al., 2005;
Murase et al., 2006). Nishimura et al. (2001) introduced a strikeslip fault at the northwestern tip of the second dike model and
inferred the fault to creep by about 10 m, equivalent to a M w 6.6
earthquake. A creeping source with strike-slip mechanism placed
in between Kozushima and Niijima by Yamaoka et al. (2005) gave
a higher seismic moment released corresponding to a M l 7.3 event.
Both models do not explain the far-field displacement observed in
at GPS sites in the Tokai area (Murase et al., 2006).
Hughes (2010) studied the dynamics of dike arrest and interaction with the SW–NE transform tectonic structures located between and around Kozushima and Niijima. For the inversion she
used daily displacement vectors at 61 GPS stations and considered a Mogi source underneath Miyakejima, a dike and a strike-slip
fault. She provided inversion results both for the full deformation
event and for the dike propagation phase, terminated with the 1
July earthquake and the dike arrest.
For the full deformation event, she developed four models ranging from highly to minimally constrained, testing different geometries based on the hypocenters of relocated seismicity and on geological and physical constraints. The geometry providing the best
fit prescribes the strike-slip fault (5.7 × 7.4 km at depth of 8 km)
to be sub-perpendicular to the dike, with the western tip of the
fault coinciding with the northern tip of the dike, 30 km long, extending for 11 km from a depth of 13 km, with a final opening
of 3.4 m.
For the propagation phase, Hughes (2010) assumed the same
fault geometry of her preferred model for the full event and tested
two models for the dike: 1) the dike height and depth are fixed to
11 km and 13 km, respectively, this resulted in a dike opening and
length of 0.18 m and 30 km, respectively; 2) all dike parameters
were kept free, this resulted in a dike 30 km long, 1 km high and
reaching a depth of 1.5 km, the total opening was 0.82 m.
Hughes (2010) also postulated that from deformation data at
the closer-to-the-intrusion GPS sites no creeping was detectable in
the fault zone between Kozushima and Niijima in the first week.
Conversely, only an aseismic deformation source could explain the
displacement field detected since 1 July at the GPS station of
Shinikejima.
4. Numerical model
Based on the observations and the available studies as highlighted above, we include in our model the following three elements:
1) The dike interacted with the pre-existing strike-slip fault,
triggering an earthquake and responding to it. We model the fault
and the dike as fully coupled structures. We use a boundary element model in plane strain approximation based on the displacement discontinuity method (Crouch et al., 1983). This type
of model is particularly suitable for treating interacting systems, as
it is straightforward to release the shear stress caused by the dike
on the fault plane and the stresses caused by the fault on the dike
plane. We define the fault as a pre-stressed, initially locked shear
crack on which the entire stress available is released when unlocked, and the dike as a fluid-filled crack with both opening and
shearing component of the slip
2) The magma influx rate from the magma reservoir supported
the propagation and the following inflation phase of the intrusion. In our model we prescribe a magma influx such that the dike
pressure remains high enough during the propagation in order to
fracture the rock ahead, according to an energetic criterion (see
below).
3) The stress gradient due to topographic loads affected the
magma pressure profile within the dike. A lateral stress gradient
linked to a declining topography profile above a dike will favor lateral propagation (Fialko and Rubin, 1999; Pinel and Jaupart, 2004;
Buck et al., 2006; Passarelli et al., 2013; Sigmundsson et al., 2014;
Roman and Jaupart, 2014). When during propagation the topographic slope changes sign, the increasingly compressive lateral
stress gradient induces dike deceleration and possibly arrest. The
lateral lithostatic pressure gradient induced by the topography can
be estimated as:
d P lit
ds
=ρ·g·
dh
ds
(1)
where s is the coordinate along the dike path, h is the topography
height above sea level, g is the acceleration due to gravity and ρ
is the rock density for h > 0 and the density difference between
rock and water when h < 0 (off-shore).
In summary, in our model the magma influx represents the
main engine supporting the dike propagation, the fault induced
stresses counter-acts it and the horizontal lithostatic stress gradient supports propagation while the dike is traveling downslope
and opposes propagation when the topographic slope at the tip
becomes inverted. We test three scenarios: i) the dike is only affected by the topography inversion, ii) the dike interacts with the
fault slip, with no topographic gradient, iii) the dike interacts both
with the topography and the fault.
We set the rock and magma parameters, fix the geometry and
calibrate the dynamic parameters (fracture energy, magma influx
rate, fault pre-stress) so that on 1 July our model outputs the maximum dike opening and the fault slip obtained by Hughes (2010)
(Table 1, Table 2 and Fig. 2). The model then produces the distributed dike opening and fault slip and the position of the dike
tip as a function of the dike volume change. We verify whether in
those conditions the dike will propagate and reach the dike opening and fault slip obtained by Hughes (2010) for 30 September, and
if an additional stressing rate on the fault is needed. Therefore the
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F. Maccaferri et al. / Earth and Planetary Science Letters 434 (2016) 64–74
Table 2
Preferred model by Hughes, 2010 used here as model parameters. The dike opening at the end of the propagation phase was used here to constrain the rock fracture
toughness.
End of propagation phase (1 July)
Entire deformation event (30 September)
Dike length
Dike opening
Fault length
Fault slip
30 km
30 km
0.82 m
3.4 m
5.7 km
5.7 km
0.39 m
4.4 m
Fig. 2. a) Map of the approximate trajectory of the dike (dashed red line) chosen for the numerical model. A is the location of the summit of Miyakejima, B is the knee point
in the simplified trajectory, C is where the dike comes close to the south-western tip of the strike-slip fault. The epicenters of the M w > 3 seismicity (moment magnitude
calculated using standard scaling relationship from the seismic moment given by the National Institute for Earth Science and Disaster Prevention, NIED. M l given by JMA
catalog and M w calculated from NIED differ of about 0.3 magnitude points) relative to the propagation phase (from 26 June to 2 July) are plotted as white circles. The
assumed geometry of the strike-slip fault is indicated as a black solid line, with arrows indicating left-lateral slip. The focal mechanism of the M l = 6.5 (M w = 6.1 using the
NIED catalog) event on 1 July as inverted by NIED is plotted close to location C. Inset: the red square indicates the focus area. b) Same as a) but including the seismicity from
2 July to 30 September and the focal mechanisms of the events with M w > 6.1. c) Topography profile along the dike trajectory. The profile has been obtained by averaging
the topographic data from ETOPO1 (Amante and Eakins, 2009), on a circle of 1 km radius along the dike trajectory (Fig. 1a). (For interpretation of the references to color in
this figure legend, the reader is referred to the web version of this article.)
simulation will be terminated only if the dike opening reaches the
assumed final value (3.4 m, Table 2) or if the dike tip propagates
more than 15 km further away from the expected arrest location
(point C and s = 0 in Fig. 2).
The fault length is 5.7 km, striking perpendicular to the dike
(Fig. 2), the dike length is not fixed and will result from the model.
We find that the observed fault slip values (Table 2) cannot be
reproduced only by means of interaction with the dike opening
(Table 2). This means that the fault must have been tectonically
pre-loaded and the earthquakes on the fault triggered (as opposed
to fully induced). Therefore, for the first slip event (0.39 m) and
for the following slip events (until reaching 4.4 m) we prescribe
on the fault a first tectonic stress release (τ ), and a further daily
shear stress release (τ ), that reflects the effect of the tectonic
stress released during the following 60 days by the triggered seismicity and fault creep. The values of τ and τ depend on the
scenario considered. Such release of pre-loaded shear stress can be
seen as the cumulative effect of the strain released by breaking
additional asperities on the fault zone, which in the model is simplified as a single fault. During the process, the left lateral motion
on the strike-slip fault induces extension perpendicular to the dike
plane along the segment BC, promoting opening there and compression further north–east (Fig. 2), that will inhibit further dike
propagation.
F. Maccaferri et al. / Earth and Planetary Science Letters 434 (2016) 64–74
Differently to our previous works (Maccaferri et al., 2010, 2011
and 2014), here we adopt the observed dike trajectory, without
investigating the reasons for the change in the direction of the
dike in the early stage of the dike propagation phase. We simplify
the trajectory of the dike to two segments striking respectively
∼N90◦ E and ∼N130◦ E, and concentrate on the dike arrest. Therefore the model starts when the dike tip has already traveled 20 km
on the first segment (AB in Fig. 2) and is 20 km distant from
the strike-slip fault. The dike and the fault are discretized in fracture elements of 0.1 km long, all interacting with each other. We
propagate the dike by adding an element at the tip. Dike propagation is allowed if the elastic energy release (E) due to the virtual
elongation overcomes the fracture energy threshold of the rock
(E c ) (Griffith, 1921). Provided the region of inelastic deformation
around the dike tip is small (Rubin, 1995), E is linked to the stress
intensity factor (K ) of the propagating dike and so are the fracture
energy threshold E c is linked to the rock fracture toughness (K c ):
K ( c) =
2 μ · E ( c)
1−ν
(2)
where μ is the rigidity of the rock and ν is the Poisson’s ratio
(Jaeger et al., 2009, eq. (10.116)). Therefore, our method is equivalent to requiring dike propagation for K K c .
The use of a fracturing criterion for dike propagation has been
an object of debate. The values for K c measured in the laboratory
for tensile fractures are small (K c ∼ 1–4 MPa m1/2 , corresponding
to about E c ∼ 10−4 MJ m−2 ) so that the elastic energy spent by
fracturing the brittle rock at the tip would be negligible in comparison to the energy dissipated in the viscous flow within the dike.
However, laboratory and field measurements on opening fractures
from cm to km scale indicate that the effective value of elastic energy dissipated in the elongation of opening fractures grows with
the fracture’s scale, so that the fracture energy for a km-scaled dike
is effectively 2–3 orders of magnitude larger than laboratory values for rock (see reviews by Rubin, 1995, pp. 322–323, and Rivalta
et al., 2015, sec. 4.4.2), mainly due to the massive amount of elastic energy released collaterally by faulting and fracturing. Here we
use E c = 0.2–0.7 MJ m−2 , in line with this reasoning.
To simulate the feeding of magma, we incrementally grow the
magma volume in the dike. This will increase E and K . If E > E c is
satisfied, the propagation of the next dike element will be allowed.
If not, we iterate the incremental volume increase. With this procedure, we do not gather any information on the time needed for
these steps, as everything is related to the path variable s, and we
obtain the volume of the intrusion as a function of the position
of the propagating tip. From a known (or assumed) magma influx
rate we can retrieve the position of the dike tip as a function of
time. Alternatively, we can constrain the velocity of the dike tip by
means of the induced seismicity and retrieve a rate for the magma
influx. Note that the pressure increase in the arrested dike will
possibly be accompanied by a vertical extension of the dike, and
this needs in general to be taken into account when estimating
the 3D volume influx from the areal volume we obtain from the
model.
We neglect the role played by the viscous flow of magma
within the dike. While this process certainly plays a large role during the nucleation and early propagation phase, we assume that
during the late propagation phase and after the dike arrest the
elastic energy released by faulting (as testified by M l > 6 earthquakes occurring) will dominate over the energy released by the
viscous flow within the dike, especially since the dike is large and
thick. We also neglect any additional stress gradient of tectonic
origin, or related to the local thermal or rheological structure of
the crust. Also, we only model the horizontal cross-section of the
dike and implicitly neglect the presence of the free surface (or the
ocean floor surface, free of shear stresses) or of non-plane-strain
69
deviations of the stress field induced by the fault or by the dike.
We postpone a 3D simulation to future studies.
We constrained the maximum of our distributed dike opening
equal to the uniform opening of a single dislocation element used
by Hughes (2010). Alternatively the average opening or the total
areal volume could have been constrained using Hughes (2010).
Constraining the total areal volume differs from the other two because of the different effective length of the dike in the three
scenarios. Given the 2D nature of our model, none of these alternatives would be fully mechanically consistent with the 3D inversion
by Hughes (2010), as the dike height changes over time. In spite
of this limitation, we expect that our specific choice may influence
some estimated parameter values (such as the rock strength and/or
the rigidity of the crust, which are in any case poorly constrained)
but not the most important outcome of our results, that is the
relative role of topographic/bathymetric loads and fault-induced
stresses on the dike arrest. The main limitation of our model is
however that it is 2D, while at least some 3D effects were certainly occurring especially during the inflation phase. While the
specific numerical values of the model parameters are certainly affected by the 2D nature of our model, we are confident that our
approach on the dike arrest is solid. In particular, a vertical extension of the dike during the inflation phase would contribute to
lower the overpressure and affect the lateral propagation similarly
for all scenarios, without altering the relative contribution of topography and fault to the dike arrest.
5. Results
5.1. CASE 1: Topography only
In the case where only topographic/bathymetric loads (Fig. 3a)
influence the dynamics of the dike, a fracture energy threshold of
0.2 MJ m−2 (equivalent to K c ∼ 125 MPa m1/2 ) is needed to reach a
maximum opening of 0.82 m with the dike tip at s = 0. During the
propagation phase, the magma influx sustained the propagation
with E E c , as prescribed (solid red curve, Fig. 3b). The initial
shape is that of a tear-drop (curve a1 in Fig. 3c). This is due to the
pressure gradient acting on the dike from the downhill topographic
slope. Additionally, the effect of the magma viscous flow, neglected
here, would have added a thin magma-filled channel between the
magma reservoir and the dike (Rubin, 1995). The dike advances
without changing its shape significantly (Fig. 3c, curves a2 and a3)
and as soon as the topography starts going upslope, the dike elongates both ahead and behind (Fig. 3c, curve a4). The areal volume, A, within the dike redistributes more symmetrically (Fig. 3c,
curve b), and the shape of the tips becomes increasingly blunt. The
dike reaches the opening u = 0.82 m (see Table 2) when the tip
is at s = −1.3 km with a dike length of 22.0 km (Fig. 3c, curve a5).
Its areal volume is ∼20% of the final areal volume (Fig. 3b, black
dashed curve). The dike continues propagating beyond Kozushima
(s > 0) and never reaches the final observed opening u = 3.4 m
(curve c1–c2–c3 in Fig. 3c). Therefore, for the set of parameters
used, our model shows that the topography alone cannot arrest
the dike. Note that with a higher value for E c the tip may become
arrested: for instance, for K c = 280 MPa m1/2 (E c = 1.0 MJ m−2 ),
the dike would reach its final opening of 3.4 m, but this would
happen when the dike tip is already at s = 8.5 km. This occurs
because the propagation with a higher K needs a higher overpressure within the dike and consequently a larger opening. Moreover,
in that case the dike opening would reach 0.82 m much earlier
than s(a5) (Fig. 3c), and the dike would be much shorter. Therefore we conclude that a model accounting only for the topographic
stress gradient cannot fit both the observed opening at the end of
the propagating and the inflation phase. In particular, if we cali-
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F. Maccaferri et al. / Earth and Planetary Science Letters 434 (2016) 64–74
Fig. 3. The result of the simulation considering the bathymetry only. Panel a: topography profile above the dike trace. The coordinate s = 0 correspond to point C
in Fig. 2. Panel b: energy release (red) and dike cross sectional area, A, normalized
with respect to the final dike cross section, A f , (black dashed curve) as a function of
the dike tip coordinate s. The black dashed curve represent the amount of magma
needed for the dike to reach a certain distance from C. The dike tip in this case
does not stop in proximity of s = 0 but propagates further (s > 0) and overcomes
the topographic high at the side of Kozushima. Panel c: dike opening distribution
at different stage of propagation. Curves a1 to a5 are relative to the propagation
phase, b is the opening distribution when the dike tip reach s = 0, and curves c1
to c3 represent the inflation phase. The model fails in reproducing the dike tip arrest. (For interpretation of the references to color in this figure legend, the reader is
referred to the web version of this article.)
brate the model to fit the propagation phase, we cannot explain
the dike arrest.
5.2. CASE 2: Dike–fault interaction only
Here we account for the interaction of the dike with the fault
but neglect the effect of the topographic stress gradient. During
Fig. 4. Results from the simulation considering dike–fault interaction only. Panels a,
b and c: Dike tip shape and fault-induced stress perpendicular to the dike plane
(compressive blue, tensile red), before and after the first earthquake on the fault
and right after the dike arrest, respectively. Panel d: energy release (red curve) and
dike cross sectional area, A, normalized with respect to the final dike cross section,
A f , (black dashed curve) as function of the dike tip coordinate, s. The black dashed
curve represent the amount of magma needed for the dike to reach a certain distance. (For interpretation of the references to color in this figure legend, the reader
is referred to the web version of this article.)
the propagation phase, differently from the case with topography
only, the magma influx needed to have E E c (Fig. 4d, red solid
curve) turns out to be approximately constant, with the magma
input into the dike increasing linearly with the propagated distance
(Fig. 4d, black dashed curve for s < 0). The value of E c needed to
obtain u = 0.82 in s = 0, is E c = 0.7 MJ m−2 (equivalent to K c
∼235 MPa m1/2 ).
The dike assumes and maintains an approximately elliptical
shape, elongating during the entire propagation phase (curves a1
to a5, Fig. 5a). This results from the absence of any pressure gradient on the dike plane. When the dike tip reaches close to the tip
of the strike slip fault (s = 0 in Fig. 4 and 5a, curve a5), we unlock
the fault. The stress drop required to have s = 0.39 m (Table 2
and Fig. 5b, curve b) is τ = 1.36 MPa.
The fault slip induces a change in the dike opening (Fig. 4a and
b and Fig. 5a), thereby suddenly increasing K (peak in the energy
Fig. 5. Distribution of the dike opening and fault slip in the course of a simulation considering the dike–fault interaction only. Panel a: Curves a1 to a5 relate to the
propagation phase, b is the opening distribution when the dike tip get first trapped by the fault stress in s = −0.2 km, and curves c1 to c3 represent the inflation phase.
Panel b: Fault slip at different stage of dike propagation, curve labels in panels a and b refer to the same model steps.
F. Maccaferri et al. / Earth and Planetary Science Letters 434 (2016) 64–74
71
assuming a constant dike vertical extension of 5 km). We find that
the interaction of the dike with the fault is very effective in arresting the dike. Under the above conditions, the dike tip propagates
beyond the first arrest location by only 2 elements, for a total of
200 m. At the end of the inflation phase the dike tip is located
500 m beyond the tip of the strike-slip fault, bending toward the
direction parallel to the fault plane (Fig. 4).
The competition between τ and φ controls whether the dike
tip becomes trapped at the expected location. We find that by
increasing φ by about 18% the dike tip escapes the compressive
stress induced by the fault slip and propagates further before u
reaches 3.4 m. Similarly, if we decrease τ by about 18%, we
obtain that the compressive stress exerted by the fault is not sufficient to trap the dike tip.
5.3. CASE 3: Topography and dike–fault interaction
Fig. 6. Results of the numerical calculation considering bathymetry and dike–fault
interaction. Panels a, b and c: dike tip shape and fault-induced stress normal to
the dike plane (compressive blue, tensile red), before (c) and after (b) the 1 July
earthquake, and at the dike arrest (c). Panel d: same as Fig. 4d. (For interpretation
of the references to color in this figure legend, the reader is referred to the web
version of this article.)
release, Fig. 4d). The stress induced by the fault slip, which is compressive on the s < 0 half-space, inhibits any further elongation
of the dike: the energy release drops rapidly under the fracture
threshold after the propagation of 3 fracture elements, for a total
of 300 m (Fig. 4c).
Next (dike inflation phase), we determine the constant-rate
shear stress release on the fault, τ , and the magma influx rate,
φ so that the final slip on the fault and the final dike opening (Table 2) are reached simultaneously (Fig. 5, curves c4). Here
the information about time is obtained by considering that the
inflation phase had a duration of about 60 days, corresponding
to 120 model steps. We obtain τ = 0.33 MPa day−1 and φ =
9.2 × 10−4 km2 day−1 (corresponding to φ = 4.6 × 10−3 km3 day−1
In a model accounting for the effect of both the topographic
load and the interaction with the strike-slip fault, we obtain E c =
0.2 MJ m−2 (K c ∼ 125 MPa m1/2 ) as the optimal value to match
the opening and slip constraints as in Table 2 (Fig. 6). We obtain
u = 0.82 m for s = −1.3 km (Fig. 7b, curve a5). Similarly to case
1 (topography only) the dike shape during propagation is affected
by a horizontal gradient, resulting in an approximately teardrop
shape (Fig. 7b, curves a1 to a4).
At the end of the propagation phase we unlock the fault and
start the dike–fault interaction. We require the sudden release of
a pre-loaded stress τ = 2.0 MPa. In this way the fault, with the
contribution of the shear stresses induced by the dike, will reach
the expected slip of 0.39 m (Fig. 7c, curve b). The “response” stress
change induced by the fault on the dike tip suddenly increases K
(see the peak in E, red curve, Fig. 6d) mainly due to the extension induced on the dike plane (Fig. 6b). The dike tip propagates
and stops again at s = 0.2 km (curve b in Fig. 7b). From now on
the inflation phase starts, the maximum opening is u = 0.78 m,
slightly less than 0.82 m, due to the last elongation caused by the
interaction with the fault, and the dike length is 23.2 km (curve b
in Fig. 7b and Fig. 6c).
As in case 2, here we calculate what constant-rate shear
stress release, τ , and magma influx, φ , are needed in order
Fig. 7. Distribution of the dike opening and fault slip for a simulation considering the bathymetry and dike–fault interaction. Panel a: topography profile above the dike
segments ABC (Fig. 2). Panel b, dike shape at different stages of propagation. Curves a1 to a5 are relative to the propagation phase, b is the opening distribution when the
dike tip gets first trapped by the fault stress in s = 0.2, and curves c1 to c3 represent the inflation phase. Panel c: Fault slip at different stage of dike propagation.
72
F. Maccaferri et al. / Earth and Planetary Science Letters 434 (2016) 64–74
Table 3
Summary of the dike and fault parameters resulting from our model.
Case 1
Case 2
Case 3
*
Dike
arrest
Final dike areal
volume*
Final
dike
length
Average
final dike
over-pressure
Magma influx
(φ )
Rock fracture
though
(K c )
Tectonic stress
released by
1st earthq
(τ )
Constant-rate
tectonic
stress release
(τ )
Total
tectonic
stress
released
NO
YES
YES
6.3 × 10−2 km2
7.5 × 10−2 km2
7.0 × 10−2 km2
>41 km
1.6 MPa
4.9 MPa
5.1 MPa
8.5 × 10−4 km2 day−1
9.2 × 10−4 km2 day−1
9.2 × 10−4 km2 day−1
125 MPa m1/2
235 MPa m1/2
125 MPa m1/2
x
1.36 MPa
2.0 MPa
x
0.33 MPa day−1
0.32 MPa day−1
x
21.2 MPa
21.2 MPa
30.5 km
26.7 km
To be multiplied by the dike vertical extension, which is not provided by our 2D model.
to match the observations on dike opening and fault slip. We
obtain φ = 9.2 × 10−4 km2 day−1 (corresponding to φ = 4.6 ×
10−3 km3 day−1 assuming a dike vertical extension of 5 km) and
τ = 0.32 MPa day−1 . During the inflation phase the dike tip
propagates by 4 elements, 400 m, bending toward the fault plane.
Also in this case the dike becomes trapped after the fault slips,
but again this requires a pre-loaded stress and additional stressing
beyond what is provided by the dike. Note that the model calibration resulted in a lower energy threshold than in case 2.
5.4. Summary of model results
With a mechanical model of dike propagation, we demonstrated
that:
Case 1 – Interaction with topography:
• The stress induced on the dike plane by an uphill topographic
slope has the effect of inhibiting the lateral propagation of intrusions and promote its continuous inflation. However, in the
case of the Miyakejima dike this contribution was not sufficient to explain the abrupt and effective stop in the lateral
growth of the dike.
Case 2 – Interaction with the strike slip fault:
• Interaction with a pre-existing fault perpendicular to the dike
on its path was on the contrary sufficient to arrest and hold
the dike, provided a sufficient amount of elastic strain was
stored in the fault (the cumulative tectonic shear stress released in about 60 days of triggered seismicity ≈20 MPa,
Table 3). The compression on the dike tip caused by the fault
slip trapped the dike.
• Continued slip or creep on the fault allowed the dike to reach
an opening consistent with that inverted from GPS data without the dike resuming propagation.
• If we neglect the influence of the topographic loads, the intrusion assumes an elongated shape and a significant opening along the whole ABC segment (Fig. 5, panel a). This is in
contrast with the observed deformation, showing a closure of
the dike tail, and seismicity, showing a concentrated cloud of
earthquakes propagating away from Miyakejima (Passarelli et
al., 2015).
of 0.39 and 4.4 m at the end of the propagation and inflation phases, respectively. A significant fault pre-stress and an
additional “stress drop” rate are necessary to obtain dike arrest and keep the dike arrested for the set of assumptions and
parameters we used. It is reasonable to assume that the fault
was pre-loaded when the dike hit it. An additional progressive
release of stress may be the manifestation of the cumulative
effect of the entire fault system and the various asperities
behind the massive triggered seismicity observed there. Alternatively, this might result from the fact that we constrained
the fault to a fixed length.
The condition for dike propagation we used (E > E c ) implies
that an important parameter for the arrest of the dike is the rock
fracture energy, E c , or fracture toughness, K c . We constrained Ec
by requiring that the observed dike opening at the end of the
propagation phase (0.84 m) was reached with the dike tip close
to the location of the fault. In cases 1 and 3 we found that
K c = 125 MPa m1/2 reproduces well the dike propagation phase.
K c < 125 MPa m1/2 would have led to a smaller opening of the
dike, and K c > 125 MPa m1/2 to a situation where the opening of
0.84 m is reached earlier on the segment BC. Once we fixed K c by
calibrating the propagation phase, there is no guarantee, that the
same value will be consistent with the arrest and inflation phase.
For case 2 we need to set K c to a larger value (235 MPa m1/2 ). Both
values are in the range of rock fracture toughness values expected
in nature for large dikes (Delaney and Pollard, 1981; Olson, 2003;
Rivalta et al., 2015, par. 4.4.2). Here we do not explore the possibility that K c might scale with the dike size. This may be a restrictive
hypothesis, given that massive seismicity was observed during the
inflation phase. However, given the epistemic uncertainties associated to realistic values for K c , we postpone a test to a future
study.
We also performed some tests (not reported here) with different orientations of the fault with respect to the dike, and obtained
that a fault perpendicular to the dike is most effective in converting the dike-induced stresses into slip and returning a compressive
stress to the dike’s tip. For angles in the range 90◦ to 135◦ we find
the mechanism is still effective. For lower angles (<90◦ ) the fault
becomes at least partially clamped by the compressive stresses induced at the sides of the dike, and therefore the interaction does
not promote additional opening in the dike.
6. Discussion and conclusions
Case 3 – Interaction with topography and the strike slip fault:
• Combining the effect of the stress gradient due to the topographic load, and the interaction with a pre-stressed fault
(case 3), results in the best fit with the observations. The
dike tip is trapped in correspondence of point C and the final dike opening is mostly concentrated on the segment BC
(Fig. 2), in agreement with the locations of the induced seismicity (Fig. 2b).
• Interplay of the dike with a non pre-loaded fault would have
not been sufficient to produce the required fault slip values
Our results reveal the importance of dike–fault coupling in
stopping a laterally propagating dike. The need of a pre-stress and
possibly an additional stressing rate on the fault showed that the
Miyakejima dike could not get arrested by interacting with a preexisting but stress-free fault around its tips, or by fully inducing
fracturing of the host rocks. However, in different conditions, and
for a different balance of the driving forces, induced fracturing
could play an important role in arresting an intrusion.
Faults sub-perpendicular to dikes or to rifts are very frequent
around the world, as seen on the bathymetry around Mid Ocean
F. Maccaferri et al. / Earth and Planetary Science Letters 434 (2016) 64–74
Ridges (MOR) and rift systems on volcanoes, such as Etna, where
the North rift system is sub-perpendicular to the Pernicana fault
(Ruch et al., 2013). Why these dike–fault systems are organized in
such a geometry is still poorly understood. Oldenburg and Brune
(1975) proposed that the perpendicularity of MOR and ridge transform faults results from a resistance to sliding on the transforms
lower than the tensile and shear strength of the plate. Gerya
(2010) showed that transform faults are developing as a result of
the dynamical instability of constructive plate boundaries. If rheologically strong transform faults were simply induced by the rifting
dikes, their orientation would be 130◦ to 150◦ , as predicted by the
maximum induced Coulomb stress change (Hill, 1977). Our models suggest that a perpendicular arrangement is the most efficient
configuration to confine the lateral propagation of dikes within a
rift segment bounded by weak transform faults. This would imply that such configuration is the most stable possible, offering a
mechanical process consistent with previous models and with the
recent recognition of the active role played by diking during rifting
(see below).
A ubiquitous case of dike–fault interaction occurs between
dikes and graben-like normal faults. In that case our 2D model applies to a cross section normal to the dike. Our results suggest that
the interaction with a normal fault, especially if the fault bends
towards horizontal at its deep root, may be effective in arresting
ascending dikes or preventing laterally propagating dikes to intersect the surface and erupt as fissures. Observations from the 2005
Manda-Hararo Dubbahu rifting event (Ethiopia) suggest that eruptions along rifts may be localized where the induced graben faults
are narrower (see Fig. 3b in Grandin et al., 2009). During its ascent, the 2009 intrusion at Harrat Lunayyir (Saudi Arabia) induced
the formation of a wedge-shaped graben on the surface. The depth
of the dike, estimated from inversion of crustal deformation data
(Pallister et al., 2010) varied along the dike length and correlated
with the graben width, showing a strong interaction between dike
and shallow faulting.
A correlation between topographic inhomogeneities and dike
velocity was observed for the 2014 dike propagation at Bardarbunga (Sigmundsson et al., 2014). The dike became slower or
temporarily arrested corresponding to a local increase of the lithostatic pressure (Sigmundsson et al., 2014, supplementary material).
Abundant magma was still flowing into the dike, and this helped
the dike overcoming the topographic steps, as long as they had
a short spatial wavelength. Once the dike reached the flank of
Askja volcano, it became ultimately arrested. The temporary slowing down of the Bardarbunga dike was also associated to slight
changes in the dike orientation (explained by Sigmundsson et al.,
2014, as due to a maximization of the total energy release, as previously suggested by Dahm, 2000 and Maccaferri et al., 2010, 2011)
and possibly to small offsets in the dike trace. These offsets are
currently not fully understood. Approaches similar to our numerical scheme may offer insight into large and small scale interactions
with topography and faults.
The modeling approach presented here may be used to further improve hazard maps in volcanic areas by providing a forecast on the timing of dike arrest and on the location of eruptive
vents.
Acknowledgements
We thank Tim Wright, an anonymous reviewer and the Editor
Tamsin Mather for their useful and constructive comments. This
work was funded by European Union through the ERC StG CCMPPOMPEI, grant agreement No. 240583, and the Supersite MED-SUV
project, grant agreement No. 308665.
73
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