Earth and Planetary Science Letters 296 (2010) 311–318
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Earth and Planetary Science Letters
j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / e p s l
Elasto-plastic modeling of volcano ground deformation
Gilda Currenti, Alessandro Bonaccorso, Ciro Del Negro ⁎, Danila Scandura, Enzo Boschi
Istituto Nazionale di Geofisica e Vulcanologia, Sezione di Catania, Italy
a r t i c l e
i n f o
Article history:
Received 12 October 2009
Received in revised form 6 May 2010
Accepted 12 May 2010
Available online 16 june 2010
Editor: L. Stixrude
Keywords:
ground deformation
elasto-plastic rheology
finite element method
Etna volcano
a b s t r a c t
Elasto-plastic models for pressure sources in heterogeneous domain were constructed to describe, assess, and
interpret observed deformation in volcanic regions. We used the Finite Element Method (FEM) to simulate the
deformation in a 3D domain partitioned to account for the volcano topography and the heterogeneous material
properties distribution. Firstly, we evaluated the extent of a heated zone surrounding the magmatic source
calculating the temperature distribution by a thermo-mechanical numerical model. Secondly, we included
around the pressurized magma source an elasto-plastic zone, whose dimension is related to the temperature
distribution. This elasto-plastic model gave rise to deformation comparable with that obtained from elastic and
viscoelastic models, but requiring a geologically satisfactory pressure. We successfully applied the method to
review the ground deformation accompanying the 1993–1997 inflation period on Mt Etna. The model
considerably reduces the pressure of a magma chamber to a few tens of MPa to produce the observed surface
deformation. Results suggest that the approach presented here can lead to more accurate interpretations and
inferences in future modeling-based assessments of volcano deformation.
© 2010 Elsevier B.V. All rights reserved.
1. Introduction
When modeling deformation in volcanic regions, the assumption
of elastic rheology can be an over simplification. The elastic
approximation is generally appropriate for small deformations of
crustal materials with temperatures below the brittle–ductile transition, which is between 600 K and 1000 K, depending principally on
composition and strain rate. Although elastic behavior well describes
the upper 15 km of the Earth's crust (Ranalli, 1995; Turcotte and
Schubert, 2002), in active volcanic areas the variation in brittle–
ductile transition may be related to the perturbation in geothermal
gradient due to the presence of intrusive bodies or varying saturation
state of fluid-filled fractured rock matrix (Mandal et al., 2007).
Materials surrounding long-lived magmatic sources are heated
significantly above the brittle–ductile transition and rocks no longer
behave in a purely elastic manner, but permanently deform because of
the plastic deformation (Fung, 1965; Ranalli, 1995). Therefore, the
thermal state makes the elastic approximation inappropriate and can
greatly influence the surface deformation field. Although mechanical
deformation models based on the assumption of elastic rheology have
been successfully and widely applied to interpret geodetic data
acquired on several volcanoes (e.g. Walsh and Decker, 1971; Yang et
al., 1992; Okada and Yamamoto, 1991; Bonaccorso and Davis, 1999;
Currenti et al., 2008a), in many cases elastic models seem unable to
reproduce the observed deformation unless unrealistic overpressures
⁎ Corresponding author.
E-mail address: [email protected] (C. Del Negro).
0012-821X/$ – see front matter © 2010 Elsevier B.V. All rights reserved.
doi:10.1016/j.epsl.2010.05.013
or lower effective rigidity moduli are considered (Newman et al.,
2001, 2006; Bonaccorso et al., 2005; Trasatti et al., 2005).
Under the assumption of elastic rheology, the amplitude of the
deformation field for a spheroidal pressure source is linearly related to
the intensity factor V∙ΔP/μ (V = volume, ΔP = pressure change, μ = rigidity modulus). Hence, the effect of changes in pressure and in volume
of a spheroidal source on the ground deformation cannot be separated
from the estimate of rheology parameters. Because of the close link
between crustal rigidity, source pressure and deformation, a lack of
insight into the rheology contributes to increase the uncertainty on
source volumes and associated pressures (Newman et al., 2001;
Bonaccorso et al., 2005). Differences between the static elastic modulus
and the dynamic elastic modulus deduced from seismic velocities can
furthermore alter these estimates (Ciccotti and Mulargia, 2004; Cheng
and Johnston, 1981). Therefore, the correct estimate of the magmatic
source pressure from the ground deformation is still an open issue.
The inclusion of an anelastic rheology also affects the estimate of
pressure change. Particularly, a viscoelastic shell around the magmatic
source requires less overpressure than a purely elastic model to
produce comparable deformation (Dragoni and Magnanensi, 1989;
Newman et al., 2001; Newman et al., 2006; Del Negro et al., 2009). A
further reduction of the overpressure could be obtained by considering
an elasto-plastic rheology for the material surrounding the magmatic
source. This behavior is expected to enhance deformation with respect
to both the viscoelastic and the elastic rheologies, and hence to require
a geologically satisfactory pressure (Trasatti et al., 2003).
In this work, we evaluated the temperature dependence of the
ground deformation using a thermo-mechanical numerical model.
Finite Element Method (FEM) was used to compute the temperature
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G. Currenti et al. / Earth and Planetary Science Letters 296 (2010) 311–318
distribution in the medium generated from a volcano chamber hosting
the magma. We developed an elasto-plastic model in which the
magmatic source, embedded in an elastic heterogeneous 3D medium,
is surrounded by a region where high temperature induces plastic
behavior. This model was applied to re-analyze the ground deformation
accompanying the 1993–1997 inflation period on Mt Etna that preceded
the 2001 and 2002–03 eruptions.
2. Temperature-dependent elasto-plastic numerical model
2.2. Elasto-plastic rheology
When a rock is strained beyond an elastic limit, Hooke's law no
longer applies. The behavior of rocks beyond their elastic limit is rather
complicated. Generally, when the stress exceeds a critical value (the
yield strength), the material will undergo plastic deformation. During
an infinitesimal increment of stress, strain changes ɛ can be split up
into elastic ɛe and plastic ɛ p components:
e
p
ð3Þ
dε = dε + dε
It is reasonable to assume that rocks near the magma source are
considerably heated and weakened beyond the brittle-ductile transition temperature, where elasto-plastic rheology is more appropriate to
describe the mechanical behavior of the rocks. This behavior has also
been proven by several laboratory experiments, conducted on rocks
under high temperature and low pressure, which show a strong
decrease in static modulus with increasing temperature (e.g. Rocchi et
al., 2002; Tuffen et al., 2008). In order to investigate the effects of the
plastic rheology, we used FEM to construct a temperature-dependent
elasto-plastic model in the region around the magma chamber. We
used the values of the rheological parameters estimated by Rocchi et al.
(2004) and Balme et al. (2004) from experiments under simulated
conditions on actual rocks from Etna. Firstly, we developed a
temperature model to derive the rheology behavior of the medium
from the computed temperature distribution. Secondly, we constructed an elasto-plastic deformation model in which the magmatic
source is embedded in an elastic medium and surrounded by a
temperature-dependent region of elasto-plastic material. Thirdly, we
compared the elastic and anelastic strain responses of the magmatic
host rocks using a simple 3D axi-symmetric model.
The plastic strain increments are related to the yield function (or
yield surface) F that specifies the stress conditions required for plastic
flow and prescribes the relationships among stress components
during flow. The yield surface F depends on the state of stress and
strain and on the history of loading. Plastic strain can occur only if the
stresses satisfy the general yield criterion:
p
Fðσ; ε ; κÞ = 0
ð4Þ
where κ is a work hardening parameter defining the plastic
deformation history of the material. If yielding occurs, we need further
information concerning the increment or rate of deformation in order
to complete the description of the material behavior. Drucker (1951)
showed that the plastic strain increment vector must be normal to the
yield surface at a regular point. The normality principle leads to the
associated flow rule (i.e., the stress/strain relation):
p
dεij = G
∂F ∂F
dσ
∂σij ∂σkl kl
ð5Þ
2.1. Temperature distribution
where
To derive the temperature profile, we numerically solved the heat
conduction equation:
G = −
∇ ·ðk∇TÞ = −AðzÞ
Different work hardening functions can be assumed on the basis of
stress–strain relationships fitting to experimental measurements on
rocks. A few of the more common forms of hardening functions are
illustrated in Fig. 1. A perfectly plastic material neglects the effect of
work hardening and the stress increments lie on the yield surface.
Several yield criteria have been developed to model the mechanical
behavior of rocks undergoing plastic deformation. Mohr–Coulomb law
and its approximation in 3D case, given by Drucker–Prager failure
criterion, are often used for modeling plastic behavior (Cattin et al., 2005).
Davis et al. (1974) showed that a zone around a magma chamber that
exhibits Mohr–Coulomb failure causes the effective pressure source to
expand outwards including both the magmatic zone and the enveloped of
failed rock. The net result is that for a given surface deformation the
stresses are much lower than required by elastic models only. Chery et al.
(1991) investigated a model in which the upper crust exhibits pressuredependent plasticity versus the lower crust, which exhibits temperaturedependent viscoplasticity with a von Mises yield criterion. With increasing
temperature, ductility increases and failure behavior changes from brittle
ð1Þ
where T is the temperature field, k is thermal conductivity, z is the depth,
and A(z) = AS exp(−z/b) is the crustal volumetric heat production,
where As is the volumetric rate of heat production at ground surface, and
b is a characteristic depth of the order of 10 ± 5 km. Since the
deformation timescales are much shorter than those over which the
magma chamber evolution takes place, the temperature distribution,
and hence the elasto-plastic volume, can be considered as steady. As
boundary condition at the ground surface, we assumed that the surface
is kept constant at atmospheric temperature, since the thermal
conductivity of the air is much smaller than that of the ground. We
assigned the geothermal temperature values at bottom and lateral
boundaries, because they are far enough to not be affected by the
magmatic source. A steady-state geothermal profile, which holds for the
upper lithosphere, was used (Ranalli, 1995; Turcotte and Schubert,
2002):
TðzÞ = Ts +
q z
m
+
k
2
As b
k
!
−z = b
1−e
∂F
∂εpmn
1
+
∂F ∂κ
∂κ ∂εpmn
∂F
∂σmn
ð6Þ
ð2Þ
where Ts is the surface temperature, and qm is the heat flow coming from
the mantle. A continuous refilling of the magma chamber was simulated
by setting the temperature on the source wall. Physically, this boundary
condition is equivalent to stating that the magma wall acts as a heating
source (Dragoni et al., 1997; Civetta et al., 2004).
Fig. 1. Examples of common types of hardening functions in the elasto-plastic rheology:
A) perfect plastic solid, B) linear strain hardening solid, and C) power-law hardening solid.
G. Currenti et al. / Earth and Planetary Science Letters 296 (2010) 311–318
Fig. 2. Temperature field distribution assuming a source wall temperature of
Tw = 1200 K.
fracture to ductile flow. The high temperature around a heated magmatic
source may lead the rocks behavior toward ductility rather than
brittleness, and yielding occurs mainly by ductile flow. At this point,
there is no longer any pressure dependence of the strength and the
behavior is fully ductile (Scholz, 2002). Therefore, the Mohr–Coulomb
criterion may apply to the most of the edifice where brittle failure takes
place, but not to the hot rock surrounding a magma chamber, where von
Mises plasticity criterion can better describe the ductile behavior of the
medium. The von Mises criterion is usually adopted as a suitable criterion
for the ductile domain of the lithosphere, where high temperatures induce
ductile deformation and plasticity controls the rock behavior (Cattin et al.,
2005). This criterion assumes that isotropic deformation is always related
elastically to the mean pressure, while the deviatoric strain is elastically
related to deviatoric stress until yielding is not reached, then plastic strain
takes place at constant deviatoric stress. The yielding function F=I2′−κ 2
depends only on the second invariant I2′=1/2σ′ij σ′ij of the deviatoric stress
σ′ij =σij −1/3I1δij, while the first invariant of deviatoric stress I1′=σ′kk is
identically zero. For a work hardening material, κ will be allowed to
change with strain history. For an ideal plastic material obeying von Mises
condition, κ is a constant independent of strain history and it is related to
the yield stress σy of the material in pure shear. Therefore, the material
exhibits linear elasticity as long as I2′≤σy2/3, and plasticity occurs when the
second invariant reaches the yield strength σy2 (Fung, 1965; Ranalli, 1995).
313
outermost lateral boundaries and on the bottom are fixed to zero,
while the boundary at the ground surface is stress-free.
Macroscopic experiments, which investigate the mechanical behavior
in triaxial tests at high temperatures, show two principal mechanical
effects: (i) thermal softening and (ii) thermally enhanced ductility.
Therefore, we computed the deformation field for three different cases:
(i) elastic, (ii) temperature-dependent elastic and (iii) elasto-plastic
rheology. The first model is a three-layered elastic half-space with
Poisson ratio ν= 0.25. A very soft layer with a Young's modulus of 2 GPa
was assumed for the upper part extending from the ground surface to
2 km. A Young's modulus of 40 GPa was assumed for the second layer
from 2 km to 6 km, whereas a Young's modulus of 80 GPa was used for
the third layer from 6 km to 50 km. The second and third models were
constructed introducing in the layered half-space a shell surrounding the
magmatic source. The thickness of the shell is dependent on the
temperature state of the magmatic source. The temperature field
distribution (Fig. 2) was computed solving the thermal model with
values of the model parameters reported in Table 1. The temperature on
the magma chamber wall was set to Tw =1200 K. Laboratory experiments on basaltic rocks have shown that the elastic modulus decreases
steadily with temperature and ductility is expected above 900 K at a
differential stress of about 10–15 MPa (Rocchi et al., 2004; Dingwell,
1998). Etnean rocks remain fully brittle up to 900 K, and above this
temperature the elastic modulus decreases reaching 10% of the original
value (Rocchi et al., 2004). Therefore, in the second model the rigidity
modulus varies linearly with the temperature (18 MPa/K) besides 900 K,
and is kept at 10% of the initial values above 1100 K. In the third model an
elasto-plastic behavior was supposed in the shell surrounding the source,
while the remaining domain was set as elastic. For this third model,
starting from the temperature distribution, we modified the properties of
the medium through the constitutive equations, allowing the element of
the computational domain to behave elastically or plastically in function
of the temperature distribution. We associated different rheologies to the
medium: (i) elastic behavior where the temperature values are below
900 K, and (ii) elasto-plastic behavior above this threshold. To simulate
the ductile behavior of the hot rocks surrounding the magma chamber,
we implemented the yield stress/strain laws considering an ideal plastic
behavior obeying to von Mises criterion. The yield strength of
surrounding rocks was assumed as σy =15 MPa, while the elastic
parameters of the medium are those of the first model previously
described. An overpressure of 100 MPa was applied on the source wall.
Model results are shown in Fig. 3. The layered elastic model gives a
2.3. Comparison between elastic and anelastic strain in a 3D axi-symmetric
model
We developed a 3D axi-symmetric model using FEM in order to
assess the effect of rheology on the surface deformation field.
Computations were carried out by the commercial software COMSOL
Multiphysics, version 3.3 (Comsol, 2006). In such a case a simpler twodimensional domain can be considered by exploiting the symmetries.
The source has a spherical geometry with a radius of 0.5 km and is
centered at 4 km depth. The axi-symmetric model is composed of
∼ 200,000 triangular elements, covering a rectangular half-space that
extends 50 km horizontally from the source centre and 50 km below
the ground surface. For boundary conditions, the displacements on the
Table 1
Thermal model parameters.
Thermal parameters
Ts
qm
k
As
b
Surface temperature
Heat flow
Thermal conductivity
Volumetric rate of heat production
Length scale for crustal radioactive
decay
300 K
0.03 Wm− 2
4 Wm− 1 K− 1
2.47 × 10−6 Wm− 3
14.170 km
Fig. 3. Comparison of the deformation expected from a pressurizing (100 MPa)
chamber centered at a 4 km depth in a layered medium for different rheologies:
A) elastic, B) elastic with varying temperature, C) elasto-plastic.
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G. Currenti et al. / Earth and Planetary Science Letters 296 (2010) 311–318
Table 2
Pressure sources from geodetic data inversion at Mt Etna (after Bonforte et al., 2008). All models are based on elastic analytical solutions except Bonaccorso et al. (2005), who also
used a numerical solution.
Period
Source
Lat
([UTM km])
Long
([UTM km])
Depth
([km])
ΔP ⁎ V
([N ⁎ m])
Rigidity
([GPa])
Reference
Data
Sept 1993–Jul 1994
Jun 1993–Oct 1995
Jul 1996–Jul 1997
Sept 1993–Jul 1997
Sept 1993–Jul 1997
Sept 1994–Sept 1998
Sept 1993–Jul 2000
Jul 1994–Jul 2001
Mogi
Yang
Mogi
Mogi
Davis
Mogi
Mogi
Mogi
4179
4177.67
4180.9
4179.45
4177.96
4181
4181.37
4180.5
500
499.89
497.61
496.96
500.7
500
496.93
499
3.8
4.8
9.3
6.8
4
5
8.1
6.2
2.19 × 1017
2.74 × 1017
13.69 × 1017
17.24 × 1017
9.6 × 1017
2.32 × 1017
28.84 × 1017
33.4 × 1017
not reported
30
not reported
30
heterogeneous
10
30
not reported
Puglisi et al. (2001)
Lundgren et al. (2003)
Puglisi and Bonforte (2004)
Palano et al. (2007)
Bonaccorso et al. (2005)
Obrizzo et al. (2004)
Palano et al. (2007)
Houlie´ et al. (2006)
GPS
InSAR
GPS
GPS
EDM, GPS
levelling
GPS
GPS
vertical uplift above the source center of about 7.5 cm. When the elastic
modulus is decreased with temperature, an enhancement to 9.0 cm is
obtained. The elasto-plastic model considerably enhances the ground
uplift to a 14.5 cm, which is about twice than that obtained for the
heterogeneous elastic model. These results show that the presence of a
plastic region can greatly amplify the strain response.
deformation accumulated during the long-lasting inflation phase from
1993 to 1997 indicates a permanent deformation of the volcano edifice
(Bonforte et al., 2008) and also suggests investigating an elasto-plastic
rheology. Therefore, we re-analyzed the ground deformation accompanying the 1993–1997 inflation period on Mt Etna assuming a heated
pressurized magma chamber embedded in an elasto-plastic heterogeneous medium.
3. Etna application and results
3.1. Etna 1993–97 recharging phase
3.2. Model application
On Mt Etna, between the 1993 and 1997, geodetic data collected by
different monitoring networks (EDM, GPS, and leveling) identified an
inflationary phase. This was characterized by a uniform and continuous
expansion of the overall volcano edifice that was not accompanied by
eruptive activity (Bonaccorso et al., 2005). The beginning of the
inflationary phase was detected from the comparison of synthetic
aperture radar (SAR) satellite images taken from 1993 to 1995. The
inversion of interferograms yielded results that were interpreted as a
spheroidal magmatic source located at about 5 km b.s.l. (Lundgren et al.,
2003). Also leveling data supported the presence of a point-like
pressurized source beneath the summit craters at 4.5 km b.s.l. (Obrizzo
et al., 2004). A recharging phase was also highlighted by GPS data (Puglisi
et al., 2001; Puglisi and Bonforte, 2004). Most interpretations of the
deformation on Mt Etna from 1994 to 2004 (Table 2) were based on a
homogeneous elastic half-space model. Despite the different geodetic
data, the methodologies used and the results achieved, the values of
intensity factor V∙ΔP/μ are similar for most of the models (Table 2), but it
is not possible to determine them unambiguously because of the tradeoff between the volume and overpressure of the source. Assuming a
homogeneous half-space with an elastic shear modulus of 30 GPa,
Lundgren et al. (2003) from SAR images estimated a pressure of about
5 MPa for a spheroidal source with a volume of 68.69× 109 m3, which is
definitely too high as it has never been revealed by seismic investigations.
Bonaccorso et al. (2005) interpreted the 1993–1997 GPS and EDM data
using an analytical model of an ellipsoidal source with a volume of
3 ×109 m3, an overpressure of 20 MPa and a low homogeneous shear
modulus of 1 GPa. They also evaluated the effect of topography and
medium heterogeneity on the deformation field using a numerical
simulation employing an elastic model. Since the average values of the
shear modulus used in the simulation, which had been estimated from
seismic velocity measurements, were larger than those used in the
homogeneous analytical model, the observed displacements in the
heterogeneous models were reproduced by increasing the overpressure
up to about 300 MPa. The source inferred by the simpler analytical model
is equivalent to a lower pressurized magma chamber surrounded by a
lower rigidity, which can be considered as a sort of “effective” rigidity
requiring the need to use an anelastic rheology (Bonaccorso et al., 2005).
Recently, Del Negro et al. (2009) reviewed the 1993–1997 inflation phase
on Mt Etna using a 3D temperature-dependent viscoelastic numerical
model, which allows producing deformation comparable with those
obtained from elastic model with a lower pressure of about 200 MPa. The
We adopted the source geometry determined by Bonaccorso et al.
(2005), which is an ellipsoid located 4.2 km bsl beneath the central
craters (latitude 4177.9 UTM km and longitude 500.7 UTM km). The
ellipsoid has a semi-major axis of 1854 m and the other two semi-axes
of 725 m and 544 m, respectively, with an orientation angle of 124° and
a dip angle of 77°. Our model domain is a large volume extending
100 × 100 × 50 km in order to avoid artifacts in the numerical solution
because of the proximity of the boundary. The mesh of the ground
surface was generated using a digital elevation model of Mt Etna from
the 90 m Shuttle Radar Topography Mission (SRTM) data and
bathymetry model from GEBCO database (http://www.gebco.net/).
The computational domain was represented by 77,475 arbitrarily
distorted tetrahedral elements connected by 13,634 nodes. The mesh
resolution is about 100 m around the ellipsoidal source, about 300 m in
the area surrounding the volcano edifice, and decreases to 10 km in the
far field.
We performed eight models to investigate how the ground
deformation is affected by the rheology parameters (elastic moduli,
yield strength, and temperature threshold) and thermal state (source
wall temperature). The elastic moduli were fixed to those inferred from
seismic tomography (Currenti et al., 2007, 2008b), whereas different
yield strength values were used in accordance with laboratory
measurements on basalts from Etna (Rocchi et al., 2004; Balme et al.,
2004). A vertical geothermal gradient of 22 K/km was assumed for the
areas surrounding the volcano edifice in agreement with the temperature measurements carried out in deep boreholes (AGIP, 1977). The
source wall temperature Tw was chosen to range between 1100 and
1200 K (Corsaro and Pompilio, 2004). The effect of the transition
Table 3
Elasto-plastic model parameters. The misfit function is shown in Fig. 4.
Model
σy
([MPa])
Tw
([K])
Tt
([K])
χ2
ΔP
([MPa])
A
B
C
D
E
F
G
H
10
15
20
10
10
10
15
15
1100
1100
1100
1200
1200
1100
1200
1200
900
900
900
800
900
800
800
900
1.19 × 104
1.19 × 104
1.19 × 104
1.15 × 104
1.17 × 104
1.16 × 104
1.16 × 104
1.18 × 104
95
98
102
46
69
59
53
75
G. Currenti et al. / Earth and Planetary Science Letters 296 (2010) 311–318
Fig. 4. Chi-square values obtained varying the overpressure from 0 to 120 MPa for
different parameters of elasto-plastic models (see Table 3). The chi-square value for the
elastic model using a pressure of about 300 MPa (Bonaccorso et al., 2005) is also
reported (horizontal thick line).
temperature Tt, which controls if the behavior of the medium is elastic or
elasto-plastic, was investigated in a range from 800 to 900 K where
laboratory experiments showed ductile deformation failure (Rocchi et
al., 2004). The parameters used in all models are reported in Table 3. For
each model, we performed different simulations varying the magma
overpressure from 1 MPa to 120 MPa (with a step of 1 MPa) and
computed the chi-square value χ2 as:
2
χ =
∑ðUxobs −Uxcalc Þ + ∑ðUyobs −Uycalc Þ
σH2
+
∑ðUzobs −Uzcalc Þ
σV2
ð7Þ
where Uobs and Ucalc are the observed and computed displacements
respectively, and σ is the standard deviation of the measurements
(Bonaccorso et al., 2005). The standard deviation affecting measurements ranges between 2 and 3 mm in the horizontal components and
between 5 and 6 mm in the vertical component. The χ2 values are
shown in Fig. 4 for all the models and the pressure changes
Fig. 5. Vertical uplifts for the elasto-plastic models D (gray lines) and G (black lines) at
OBS (solid lines), TDF (dotted lines), ESLN (dashed lines) stations (see Fig. 7 for the
positions) for increasing source pressure.
315
corresponding to the minimum χ2 that best fits the geodetic data are
reported in Table 3. The pressure change estimates range between
46 MPa (model D) and 102 MPa (model C). In all the models, as the
pressure increases, the deformation starts to grow linearly as far as the
yield condition is not satisfied. When the elastic limit is reached, the
ground deformation increases more rapidly since plastic deformation
prevails. In the models with lower values of the yield strength (A, D, E, F),
the medium fails plastically at lower values of the pressure change and
the ground deformation is enhanced with respect to the other models
(Fig. 5). Therefore, the pressure estimates are strongly dependent on the
model parameters. The value of pressure, which gives the minimum χ2
value, is tuned by the yield strength. The higher the yield strength, the
higher the pressure required to obtain the same amount of deformation.
Besides the yield strength, another sensitive parameter is the transition
temperature. As the threshold temperature decreases, the volume
participating to the elasto-plastic flow increases and gives more
contribution to the plastic deformation. A variation of 100 K in the
temperature is sufficient to vary the estimated pressure from 46 MPa
(model D) to 69 MPa (model E) when the other parameters are kept
constant.
Further simulations were performed assuming a decrease in
rigidity modulus with increasing temperature for the models A, B,
and C. Following the results of laboratory tests on basalt rock samples
from Etna (Rocchi et al., 2004), we assumed that the elastic modulus
decreases steadily at 900 K reaching 10% of the original values at
1100 K. The comparison among the different models shows that the
dependence of the elastic modulus on temperature does not affect the
simulation results very much (Fig. 6). A slight difference is obtained at
lower overpressure when the deformation is almost elastic. When the
plastic deformation prevails on the elastic deformation, it becomes the
dominant process and the effect of variations in the elastic modulus
due to temperature is negligible.
The minimum χ2 values of the elasto-plastic models are comparable
with each other and less than the χ2 value of the elastic model (Fig. 4).
On the basis of the χ2 values, all the considered models provide similar
fits to the data with fairly similar pressure changes, which are well
below the value obtained using an elastic rheology. The minimum χ2 is
obtained by the model D (Fig. 4), which also requires the minimum
pressure change (46 MPa). The fit to the data is slightly improved from
1.4 × 104 for the elastic model to 1.1× 104 for the elasto-plastic model
(Fig. 7). It is worth noting that the elasto-plastic model uses a pressure
change of only 46 MPa, whereas the elastic model needs an unrealistic
Fig. 6. Chi-square values obtained varying the overpressure from 0 to 120 MPa for
different elasto-plastic models: (i) A, B, and C (continuous lines) simulations with
rigidity modulus independent on temperature, (ii) AEt, BEt, CEt (dotted lines)
simulations using decreasing rigidity modulus with increasing temperature.
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G. Currenti et al. / Earth and Planetary Science Letters 296 (2010) 311–318
Fig. 7. Comparison between GPS observed (black) and computed deformation during the 1993–1997 period. The numerical computations are performed assuming a heterogeneous
elastic medium (red, after Bonaccorso et al., 2005) and the elasto-plastic model D (blue). Horizontal displacements (top) and vertical displacements (bottom) are calculated at GPS
stations. Assuming the same volume and rigidity, the elastic model requires an overpressure of ≈ 300 MPa, whereas the elasto-plastic model reaches similar deformation with an
overpressure of ≈ 46 MPa.
pressure of about 300 MPa to get comparable deformation. The elastic
model with a 46 MPa of pressure change would have caused ground
uplifts of few centimeters, whereas the elasto-plastic model reaches
tens of centimeters (Fig. 8). Interpretation of long-term deformation is
complicated by the coarse spatial resolution of the available geodetic
data, and the contribution of more than one mechanism to the observed
ground displacements. A significant contribution to the horizontal
displacements on the easternmost stations (MIL and GIA), affected by
flank instability, could be given by the effect of the sliding of the volcano
eastern flank (Bonaccorso et al., 2006), which could also alter the
estimate of the source parameters.
It is worth noting that our numerical models disregard the stress
induced by the topographic load. In the near-surface region, topography
loading can make the stress state different from the lithostatic stress
state (zero deviatoric stress), usually assumed for half-space model. Due
to the topography, the medium is initially in a non-zero deviatoric stress
G. Currenti et al. / Earth and Planetary Science Letters 296 (2010) 311–318
317
have perturbed the geothermal gradient and, hence, the mechanical
behavior of the surrounding rocks. Recently, Bonforte et al. (2008)
showed that the deformation observed during the long-lasting
inflation phase was not recovered in the following eruptive period,
indicating a permanent deformation of the volcano. The deformation
seems to be mainly accumulated following a non-elastic behavior. All
these evidences point to the elasto-plastic rheology as the more
probable behavior to perform more realistic numerical simulations.
Results suggest that the integration of the material rheology variations
into volcano deformation modeling is a critical and necessary
advancement toward more reliable predictions for eruptive activity.
Acknowledgments
Fig. 8. Comparison between the vertical uplift along the AB profile (Fig. 7) for the elastic
(dashed line) and elasto-plastic model D (solid line) using a pressure change of 46 MPa.
state and the elastic limit could even be reached with a lower source
overpressure.
4. Discussion and conclusions
Geodetic observations are useful to discriminate between different
deformation models (elastic, viscoelastic, and elasto-plastic) and may
provide valuable constraints on the rheological and mechanical
parameters of the medium. The elastic models usually require either
overpressures of several hundred MPa or exceedingly high source
volumes to justify the observed ground deformation. High overpressure values are unrealistic as the induced stresses would be so high
that rocks would fracture because of the low tensile strength of
common solid rocks (Balme et al, 2004; Haimson and Rummel, 1982;
Schultz, 1995; Amadei and Stephansson, 1997).
We showed that the overpressure can be lowered to geologically
satisfying values if elasto-plastic behavior is taken into account. We
tested the elasto-plastic behavior for the recharging phase occurred
between 1993 and 1997 on Etna volcano. Despite the clear evidence of
the overall volcano edifice expansion identified at Mt Etna by different
geodetic data (InSAR, leveling, GPS and EDM), the estimate of source
parameters is still under debate. The high overpressure foreseen by the
elastic models is incompatible with the low levels of both volcanic
activity and seismicity occurring in the analyzed period (Patanè et al.,
2003). The low seismicity associated with the recharging phase in the
time interval 1993–1997 would support the hypothesis that the
volume surrounding the pressurizing source could mainly have failed
by ductility rather than brittleness. Although the summit craters were
continuously degassing between March 1993 and July 1995, no fresh
magma reached the surface (Allard et al., 2006). Afterwards, a series of
paroxysmal eruptions occurred at the summit craters (Bocca Nuova
and North-East craters), whereas the South-East crater became active
in the autumn of 1996 after 5 years of repose. This volcanic activity was
very modest and only in the middle of 1998 significantly resumed with
a series of lava fountaining from the summit craters (Allard et al.,
2006). A shallow broad region of low Qp hot fluids has been recognized
on the west of an high rigidity body beneath the South-East flank of
Etna volcano (Martinez-Arevalo et al., 2005; De Gori et al., 2005;
Patanè et al., 2006), which concurs with the location of the estimated
ellipsoidal source inferred by Bonaccorso et al. (2005) and used in the
elasto-plastic model in this work. During the last decades, this area has
been a preferential pathway of magma rising and a region of
intermediate magma storage (Bonforte et al., 2008), which could
This study was undertaken with financial support from the V3-LAVA
and V4-FLANK projects (DPC-INGV 2007–2009 contract). This work was
developed in the frame of the TecnoLab, the Laboratory for the
Technological Advance in Volcano Geophysics organized by INGV-CT
and DIEES-UNICT. We thank the Editor Lars P. Stixrude and the
anonymous referees who provided constructive comments for improving the manuscript.
References
AGIP, S.p.A., 1977. Temperature sotterranee. Grafiche Fili Brugora, Milano.
Allard, P., Behncke, B., D'Amico, S., Neri, M., Gambino, S., 2006. Mount Etna 1993–2005:
Anatomy of an evolving eruptive cycle. Earth Sci. Rev. 78, 85–114.Amadei, B.,
Stephansson, O., 1997. Rock Stress and its Measurement. Chapman and Hall, London.
Amadei, B., Stephansson, O., 1997. Rock stress and its measurements. Chapman & Hall,
London, p. 490.
Martinez-Arevalo, C., Patanè, D., Rietbrock, A., Ibanez, J.M., 2005. The intrusive process
leading to the Mt. Etna 2001 flank eruption: Constraints from 3-D attenuation
tomography. Geophys. Res. Lett. 32, L21309. doi:10.1029/2005GL023736.
Balme, M.R., Rocchi, V., Jones, C., Sammonds, P.R., Meredith, P.G., Boon, S., 2004. Fracture
toughness measurements on igneous rocks using a high-pressure, high-temperature
rock fracture mechanics cell. J. Volcanol. Geoth. Res. 132, 159–172.
Bonaccorso, A., Davis, P.M., 1999. Models of ground deformationfrom vertical volcanic
conduits with application to eruptions of Mount St. Helens and Mount Etna. Geophys.
Res. L ett. 104 (B5), 10531–10542.
Bonaccorso, A., Cianetti, S., Giunchi, C., Trasatti, E., Bonafede, M., Boschi, E., 2005. Analytical
and 3D numerical modeling of Mt. Etna (Italy) volcano inflation. Geophys. J. Int. 163,
852–862.
Bonaccorso, A., Bonforte, A., Guglielmino, F., Palano, M., Puglisi, G., 2006. Composite
ground deformation pattern forerunning the 20042005 Mount Etna eruption.
J. Geophys. Res. 111, B12207. doi:10.1029/2005JB004206.
Bonforte, A., Bonaccorso, A., Guglielmino, F., Palano, M., Puglisi, G., 2008. Feeding system
and magma storage beneath Mt Etna as revealed by recent inflation/deflation cycles.
J. Geophys. Res. 113, B05406. doi:10.1029/2007JB005334.
Cattin, R., Cecile Doubre, C., de Chabalier, J.B., King, G., Vigny, C., Avouac, J.P., Jean-Claude
Ruegg, J.C., 2005. Numerical modelling of quaternary deformation and post-rifting
displacement in the Asal–Ghoubbet rift (Djibouti, Africa). Earth Planet. Sci. Lett. 239,
352–367. doi:10.1016/j.epsl.2005.07.028.Chiarabba, C., Amato, A., Boschi, E., Barberi,
F., 2000. Recent seismicity and tomographic modeling of the Mount Etna plumbing
system. J. Geophys. Res. 105, 10923–10938.
Comsol Multiphysics 3.3, 2006. Comsol AB, Stockholm, Sweden.
Ciccotti, M., Mulargia, F., 2004. Differences between static and dynamic elastic moduli
of a typical seismogenic rock. Geophys. J. Int. 157, 474–477.
Cheng, C.H., Johnston, D.H., 1981. Dynamic and static moduli. Geophys. Res. Lett. 8 (1),
39–42.
Chery, J., Bonneville, A., Vilotte, J.P., Yuen, D., 1991. Numerical modelling of caldera,
dynamical behaviour. Geophys. J. Int. 105 (2), 365–379. doi:10.1016/j.
epsl.2005.07.028.
Civetta, L., D'Antonio, M., De Lorenzo, S., Di Renzo, V., Gasparini, P., 2004. Thermal and
geochemical constraints on the ‘deep’ magmatic structure of Mt. Vesuvius. J. Volcanol.
Geotherm. Res. 133, 1–12.
Corsaro, R.A., Pompilio, M., 2004. Magma dynamics in the shallow plumbing system of
Mt. Etna as recorded by compositional variations in volcanics of recent summit
activity (1995–1999). J. Volcanol. Geotherm. Res. 137, 55–71. doi:10.1016/j.
jvolgeores.2004.05.008.
Currenti, G., Del Negro, C., Ganci, G., 2007. Modeling of ground deformation and gravity
fields using finite element method: an application to Etna volcano. Geophys. J. Int..
doi:10.1111/j.1365-246X.2007.03380.x
Currenti, G., Del Negro, C., Ganci, G., Scandura, D., 2008a. 3D numerical deformation
model of the intrusive event forerunning the 2001 Etna eruption. Phys. Earth
Planet. Inter.. doi:10.1016/j.pepi.2008.05.004
Currenti, G., Del Negro, C., Ganci, G., Williams, C., 2008b. Static stress changes induced
by the magmatic intrusions during the 2002–2003 Etna eruption. J. Geophys. Res.
113, B10206. doi:10.1029/2007JB005301.
318
G. Currenti et al. / Earth and Planetary Science Letters 296 (2010) 311–318
Davis, P.M., Hastie, L.M., Stacey, F.D., 1974. Stresses within an active volcano with
particular reference to Kilauea. Tectonophysics 22, 355–362.
De Gori, P., Chiarabba, C., Patanè, D., 2005. Qp structure of Mt Etna: constraints for the
physics of the plumbing system. J. Geophys. Res. 110. doi:10.1029/2003JB002875.
Del Negro, C., Currenti, G., Scandura, D., 2009. Temperature-dependent viscoelastic
modeling of ground deformation: application to Etna volcano during the 1993–1997
inflation period. Phys. Earth Planet. In. 172, 299–309. doi:10.1016/j.pepi.2008.10.019.
Dingwell, D.B., 1998. The glass transition in hydrous granitic melts. Phys. Earth Planet.
Inter. 107, 1–8.
Dragoni, M., Harabaglia, P., Mongelli, F., 1997. Stress field at a transcurrent plate boundary
in the presence of frictional heat production at depth. Pure Appl. Geophys. 150,
181–201.
Dragoni, M., Magnanensi, C., 1989. Displacement and stress produced by a pressurized,
spherical magma chamber, surrounded by a viscoelastic shell. Phys. Earth Planet.
Inter. 56, 316–328.
Drucker, D.C., 1951. A more fundamental approach to plastic stress strain relations.
Proc. 1st. U.S. Natl. Congress Appl. Mech. 487–491.
Fung, Y.C., 1965. Foundations of solid Mechanics. Prentice-Hall, Englewood Cliffs.
Haimson, B.C., Rummel, F., 1982. Hydrofracturing stress measurements in the Iceland
research drill hole at Reydarfjördur Iceland. J. Geophys. Res. 87, 6631–6649.
Houlie´, N., Briole, P., Bonforte, A., Puglisi, G., 2006. Large scale ground deformation of Etna
observed by GPS between 1994 and 2001. Geophys. Res. Lett. 33, L02309 . doi:10.1029/
2005GL024414.
Lundgren, P., Berardino, P., Coltelli, M., Fornaro, G., Lanari, R., Puglisi, G., Sansosti, E.,
Tesauro, M., 2003. Coupled magma chamber inflation and sector collapse slip
observed with synthetic aperture radar interferometry on Mt. Etna volcano.
J. Geophys. Res. 108 (B5), 2247. doi:10.1029/2001JB000657.
Mandal, P., Chadha, R.K., Raju, I.P., Kumar, N., Satyamurty, C., Narsaiah, R., Maji, A., 2007.
Coulomb static stress variations in the Kachchh, Gujarat, India: Implications for the
occurrences of two recent earthquakes (Mw = 5.6) in the 2001 Bhuj earthquake
region. Geophys. J. Int. doi:10.1111/j.1365-246X.2006.03301.x.
Newman, A.V., Dixon, T., Ofoegbu, G., Dixon, J., 2001. Geodetic data constraints on recent
activity at Long Valley Caldera, California: evidence for viscoelastic rheology. J. Volcan.
Geoth. Res. 105, 183–206.
Newman, A.V., Dixon, T.H., Gourmelen, N., 2006. A four-dimensional viscoelastic
deformation model for Long Valley Caldera, California, between 1995 and 2000.
J. Volcanol. Geotherm. Res. 150, 244–269.
Obrizzo, F., Pingue, F., Troise, C., De Natale, G., 2004. Bayesian inversion of 1994–98 vertical
displacements at Mt Etna; evidence for magma intrusion. Geophys. J. Int. 157 (2),
935–946. doi:10.1111/j.1365- 246X.2004.02160.x.
Okada, Y., Yamamoto, Y., 1991. Dyke intrusion model for the 1989 seismovolcanic
activity of Off Ito, Central Japan. J. Geophys. Res. 96, 10361–10376.
Palano, M., Puglisi, G., Gresta, S., 2007. Ground deformation at Mt. Etna: a joint
interpretation of GPS and InSAR data from 1993 to 2000. Boll. Geofis. Teor. Appl. 48
(2), 81–98.
Patanè, D., De Gori, P., Chiarabba, C., Bonaccorso, A., 2003. Magma ascent and the
pressurization of Mount Etna's volcanic system. Science 299, 2061–2063.
Patanè, D., Barberi, G., Cocina, O., De Gori, P., Chiarabba, C., 2006. Time-resolved seismic
tomography detects magma intrusions at mount etna. Science 313, 821–823.
Puglisi, G., Bonforte, A., Maugeri, S.R., 2001. Ground deformation patterns on Mt. Etna,
between 1992 and 1994, inferred from GPS data. Bull. Volcanol. 62, 371–384.
Puglisi, G., Bonforte, A., 2004. Dynamics of Mount Etna volcano inferred from static and
kinematic GPS measurements. J. Geophys. Res. 109, B11404. doi:10.1029/
2003JB002878.
Ranalli, G., 1995. Rheology of the Earth. Chapman and Hall, London, p. 413.
Rocchi, V., Sammonds, P.R., Kilburn, C.R.J., 2004. Fracturing of Etnean and Vesuvian rocks at
high temperatures and low pressures. J. Volcanol. Geotherm. Res. 132, 137–157.
Rocchi, V., Sammonds, P.R., Kilburn, C.R.J., 2002. Flow and fracture maps for basaltic
rock deformation at high temperatures. J. Volcanol. Geotherm. Res. 120, 25–42.
Scholz, C.H., 2002. The mechanics of earthquakes and faulting. Cambridge University
Press, Cambridge, p. 471.
Schultz, R.A., 1995. Limits of strength and deformation properties of jointed basaltic
rock masses. Rock Mechanics and Rock Engineering 28, 1–15.
Trasatti, E., Giunchi, C., Bonafede, M., 2003. Effects of topography and rheological layering
on ground deformation in volcanic regions. J.Volcanol. Geotherm. Res. 122, 89–110.
Trasatti, E., Giunchi, C., Bonafede, M., 2005. Structural and rheological constraints on
source depth and overpressure estimates at the Campi Flegrei caldera, Italy. J. Volc.
Geoth. Res. 144, 105–118.
Tuffen, H., Smith, R., Sammonds, P.R., 2008. Evidence for seismogenic fracture of silicic
magma. Nature 453 (7194), 511–515. doi:10.1038/nature06989.
Turcotte, D.L., Schubert, G., 2002. Geodynamics — applications of continuum physics to
geological problems, 2nd edition. Wiley, New York. pp. 528.
Walsh, J.B., Decker, R.W., 1971. Surface deformation associated with volcanism. J. Geophys.
Res. 76, 3291–3302.
Yang, X., Davis, P.M., Delaney, P.T., Okamura, A.T., 1992. Geodetic analysis of dike intrusion
and motion of the magma reservoir beneath the summit of Kilauea volcano, Hawaii:
1970–1985. J. Geophys. Res. 97, 3305–3324.