Dike emplacement and flank instability at Mount Etna: Constraints from a poro-elastic-model of flank collapse

Journal of Volcanology and Geothermal Research 199 (2011) 153–164

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Dike emplacement and flank instability at Mount Etna: Constraints from aporo-elastic-model of flank collapse

M. Battaglia a,⁎, M. Di Bari b, V. Acocella c, M. Neri d

a Department of Earth Sciences, Sapienza, University of Rome, Italyb Department of Geology and Geophysics, University of Bari, Italyc Dipartimento di Scienze Geologiche, Università Roma Tre, Roma, Italyd Istituto Nazionale di Geofisica e Vulcanologia, Catania, Italy

⁎ Corresponding author.E-mail address: [email protected] (M. Battagl

0377-0273/$ – see front matter © 2010 Elsevier B.V. Aldoi:10.1016/j.jvolgeores.2010.11.005

a b s t r a c t

Article history:Received 12 April 2010Accepted 3 November 2010Available online 12 November 2010

Keywords:Etnadike intrusionflank instabilityporo-elasticityanalytical modelling

Many volcanic edifices are subject to flank failure, usually produced by a combination of events, rather thanany single process. From a dynamic point of view, the cause of collapse can be divided into factors thatcontribute to an increase in shear stress, and factors that contribute to the reduction in the friction coefficientμ of a potential basal failure plane. We study the potential for flank failure at Mount Etna considering aschematic section of the eastern flank, approximated by a wedge-like block. For such geometry, we perform a(steady state) limit equilibrium analysis: the resolution of the forces parallel to the possible basal failure planeallows us to determine the total force acting on the potentially unstable wedge. An estimate of the relativestrength of these forces suggests that, in first approximation, the stability is controlled primarily by thebalance between block weight, lithostatic load and magmatic forces. Any other force (sea load, hydrostaticuplift, and the uplift due to mechanical and thermal pore-fluid pressure) may be considered of second order.To study the model sensitivity, we let the inferred slope α of the basal surface failure vary between −10° and10°, and consider three possible scenarios: no magma loading, magmastatic load, and magmastatic load withmagma overpressure. We use error propagation to include in our analysis the uncertainties in the estimates ofthe mechanics and geometrical parameters controlling the block equilibrium. When there is no magmaloading, the ratio between destabilizing and stabilizing forces is usually smaller than the coefficient of frictionof the basal failure plane. In the absence of an initiating mechanism, and with the nominal values of thecoefficient of friction μ=0.7±0.1 proposed, the representative wedge will remain stable or continue to moveat constant speed. In presence of magmastatic forces, the influence of the lateral restraint decreases. If weconsider the magmastatic load only, the block will remain stable (or continue to move at constant speed),unless the transient mechanical and thermal pressurization significantly decrease the friction coefficient,increasing the instability of the flank wedge for αN5° (seaward dipping decollement). When the magmaoverpressure contribution is included in the equilibrium analysis, the ratio between destabilizing andstabilizing forces is of the same order or larger than the coefficient of friction of the basal failure plane, and theblock will become unstable (or accelerate), especially in the case of the reduction in friction coefficient.Finally, our work suggests that themajor challenge in studying flank instability atMount Etna is not the lack ofan appropriate physical model, but the limited knowledge of the mechanical and geometrical parametersdescribing the block equilibrium.

ia).

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© 2010 Elsevier B.V. All rights reserved.

1. Introduction

Massive, destructive edifice collapses have dramatically sculptedhundreds of stratovolcanoes, including Mount St. Helens and Augus-tine (United States), la Soufrière volcano (Guadalupe), Merapi(Indonesia), Mount Bandai (Japan) and Mount Etna (Italy) (Acocella,2005, and references therein). Although destabilizing shallow

intrusion of magma into the edifice accompanies most collapses(e.g., Mount St. Helens), others have occurred without eruption ofjuvenile magmatic materials (e.g., Bandai volcano; Reid, 2004).

Flank instability seems to be independent of volcanoes composi-tion, shape and geodynamic setting. The instabilities are characterizedby different velocities of the mobilized mass, from creep-like move-ments (velocity of 10−9–10−10 m/s; Froger et al., 2001), to cata-strophic fast-moving landslides (velocity ~102 m/s; Voight et al.,1981). Instabilities may occur suddenly (Cervelli et al., 2002) orconsist of accelerated movements within prolonged periods ofcreeping of the volcano flank (Neri et al., 2004). Mobilized volumes

154 M. Battaglia et al. / Journal of Volcanology and Geothermal Research 199 (2011) 153–164

vary enormously, even at the same volcano: in general, significantcollapses commonly mobilize volumes of 108–1011 m3 (Carracedoet al., 1999; Day et al., 1999; Hall et al., 1999; Masson et al., 2002). Thelarger the volume of the mobilized mass, the lower is the frequency ofcollapse (McGuire, 1996).

Several factors have been proposed to trigger the instability ofvolcanoes. These may act independently, or, more commonly,simultaneously. Magma emplacement is possibly the most commontriggering factor, mainly in the form of dike emplacement (Delaney etal., 1998; Elsworth and Day, 1999; Tibaldi, 2001; Acocella et al., 2003;Walter et al., 2005; Acocella and Neri, 2009). Dike emplacement mayinduce static and dynamic effects on the stability of the flank of avolcano. From a dynamic point of view, the emplacement of the dikedisplaces the host rock away and, if buttressing is limited, thedisplacement effect may propagate to a significant part of the volcano(Delaney and Delinger, 1999; Cervelli et al., 2002; Acocella et al.,2003). From a static point of view, the effect of a dike is to increase themagmastatic force of the magmatic column, represented by the dikeitself, and to increase the pore pressures in the groundwater withinthe volcano, whichmay induce failure (Elsworth and Voight, 1992). Inparticular, consideration of field and seismic evidence, together withsimple calculations, indicate that reductions in the strength, or in theeffective friction coefficient, of volcanic materials are mainly due tothe effect of high pore-fluid pressure relative to confining pressure,produced by a variety of mechanisms (Day, 1996): heating of confinedpore water by intrusions (e.g., Elsworth and Voight, 1995); degassingof intrusions; discharges of highly pressurized fluids from depththrough clastic dykes; and by deformation associated with dikeintrusion (e.g., Elsworth and Voight, 1992).

Sector collapses characterize the evolution of Mt. Etna, as testifiedby the Valle del Bove scar and by the occurrence of local Pleistoceneand Holocene debris-avalanche and debris-flow deposits exposed onthe eastern flanks of the volcano (Borgia et al., 1992; Rust and Neri,1996; Groppelli and Tibaldi, 1999; Neri et al., 2005, 2009). Despitethe large amount of studies and an acceptable definition of thesurface deformation induced by flank instability, a quantitativeanalysis of the static effects of dike injection on flank instability, inparticular on the role played by pore-fluid pressurization, has beenlacking.

Here, we study how the emplacement of dikes in one of the MountEtna rift zones can promote flank failure. We consider the efficacy ofshallow edifice intrusion, and mechanical and thermal pore-fluidpressurization, to destabilize large regions of the volcano (e.g.,Elsworth and Day, 1999). To determine the factors controlling flankfailure at Mount Etna, we have performed a limit equilibrium analysis(Elsworth and Voight, 1995). Pore pressures developed bymechanicaland thermal effects are readily evaluated using simple, but rigorous,models. Mechanically induced pore-fluid pressures are evaluatedthrough an analogy with a moving line dislocation within a saturatedporous-elastic medium (Elsworth and Voight, 1992). Thermallyinduced pore-fluid pressures are evaluated from a one-dimensionaladvective–diffusive solution for low Peclet transport, representingbehaviour around a plane feeder dyke of infinite extent (Delaney,1982; Elsworth and Voight, 1995). A key part of our analysis is tounderstand how the uncertainty in our knowledge of the geometricaland physical parameters of Mount Etna may influence the evaluationof the potential for flank failure.

2. Mount Etna

Mount Etna (Fig. 1) is a 3329 m high and ~40×60 km widevolcano, formed in the last 200 Ka (Romano, 1982; Corsaro et al.,2002; Branca et al., 2008, and references therein; Neri et al., 2008).Three prominent rift zones, characterized by eruptive fissures andparasitic cones radiating from the volcano summit (Acocella and Neri,2003), are currently active on Etna: the Northeast, South and West

Rifts. Much of the current volcanism here is characterized by thelateral movement ofmagma radiating from the central conduit (Allardet al., 2006; Acocella and Neri, 2009).

Several studies have highlighted the extensive sliding of theeastern and southern sectors of the volcano towards E and S,respectively (Borgia et al. 1992, 2000; Lo Giudice and Rasà 1992;Rust and Neri 1996; Froger et al. 2001; Neri et al., 2004, 2005, 2009).Flank instability is partly triggered by volcanic activity, as for exampleobserved during the dike-fed 2001 and 2002–2003 eruptions, when asignificant part of the E flank slipped eastward, locally reachingdisplacement rates of m/month (Behncke and Neri, 2003; Neri et al.,2004, 2009; Walter et al., 2005; Allard et al., 2006; Carbone et al.,2009).

The mobile eastern portion is bordered, to the north, by the E–Wtrending transtensive Pernicana fault system (PFS), with a left lateral–normal motion (Acocella and Neri, 2005 and references therein). Thesliding southern sector is confined, to the west, by the N–S trendingRagalna fault system (RFS), with a predominant dextral–normalmotion (Neri et al., 2007 and references therein). The spreading frontis in part characterized by compression, forming an anticline, whichinvolves the sub-volcanic sediments at the southern base of thevolcano (Borgia et al. 1992; 2000; Froger et al., 2001; Neri et al., 2009;Solaro et al., 2010).

Little is known on the continuation of the unstable flank at depth,in particular on the existence, location and geometry of anydecollement surface confining the sliding mass. The existence of thedecollement has been largely postulated from indirect constraints(stratigraphic, structural, seismic or modelling constraints; Borgiaet al., 1992; Tibaldi and Groppelli, 2002; Neri et al., 2004; Bonforte andPuglisi, 2006; Bonforte et al., 2008, 2009; Puglisi et al., 2008; Ruchet al., 2010), so that no direct evidence has been provided so far. Itslocation has been inferred to lie somewhere between 1–2 km a.s.l. and6 km b.s.l. (Kieffer, 1985; Borgia et al., 1992; Lo Giudice and Rasà,1992; Bousquet and Lanzafame; 2001; Neri et al., 2004; Ruch et al.,2010), with the possible occurrence also of multiple discontinuities(Tibaldi and Groppelli 2002; Bonforte and Puglisi, 2006; Bonforteet al., 2008; Puglisi et al., 2008; Bonforte et al., 2009). Finally, thegeometry of the decollement(s) is also subject of debate, as differentstudies postulate the presence of a W-dipping (Borgia et al., 1992;Neri et al., 2004) or of a E-dipping surface (Tibaldi and Groppelli 2002;Rust et al., 2005; Bonforte and Puglisi, 2006; Bonforte et al., 2008,2009; Puglisi et al., 2008; Ruch et al., 2010).

The role of the pore pressures on flank instability has not beenconsidered at Etna so far. Available constraints suggest the presence ofan open aquifer limited to the volcanic pile (which reaches amaximum depth of 1 km a.s.l. under the volcano summit; Rust andNeri, 1996), whose top is characterized by an inclination of ~3° (C.Federico, personal communication). No information is available onthe presence and extent of other aquifers within the sedimentarybasement, deeper than 1 km a.s.l.

3. Forces involved in flank failure

Flank failure is usually produced by a combination of events, ratherthan any single process. From a mechanical point of view, the cause ofcollapse can be divided into factors that contribute to an increase inshear stress, and factors that contribute to the reduction in the frictioncoefficient μ, or shear strength, of a potential basal failure plane(Voight and Elsworth, 1997).

To determine the factors controlling flank failure at Mt. Etna, wehave performed a limit equilibrium analysis (Elsworth and Voight,1995). We first determine the forces involved in the volcano flankfailure, then we evaluate the force budget. Many of the mathematicdetails of the steady state model presented below are discussed indepth by Elsworth and Voight (1992, 1995 and 1996) and in the

Fig. 1. Structural sketch map of Mt. Etna. Shaded area is the sector of the volcano affected by flank instability; black arrows indicate the directions of movement of the slide blockswithin that sector. The blocks showed in Figs. 2 and 3 are constructed along the blue line.

155M. Battaglia et al. / Journal of Volcanology and Geothermal Research 199 (2011) 153–164

Appendix. Unless explicitly stated, all our variables and parametersare dimensionless (see Tables 1 and 2 for notation).

3.1. Applied forces

To determine the forces involved, we consider a schematic sectionof the volcano flank (see Fig. 1), approximated by a wedge-like block(Fig. 2). For such geometry, the resolution of the force parallel to thebasal failure plane (the shear surface in Fig. 2) allows us to determinethe total force driving the motion of the potentially unstable wedge(Fig. 3). This force is countered by the friction along the basal failureplane and the force along the sides of the wedge. All the forces arenormalized with respect to the quantity 1/γwhs

3, where γw is the unitweight of water in MPa/m, and hs is the elevation above sea level in mof the block crest (Table 1).

3.1.1. The block weightThe weight force Mp acting on the block is

Mp≈12

1−nð ÞγrD hfD + hmD� �

lD + n lθD + lbD� �

bDh i

dD ð1Þ

with lDθ ≈ lD

b −aD tan β. Eq. (1) takes into account the porosity n of therock and the presence of water. For simplicity, we do not differentiatebetween the unit weight of the volcanics γD

r and the substrate, and we

consider negligible any difference in unit weight between fresh andsea water.

3.1.2. Dike emplacementThemagmatic forces Fm due to the emplacement of a ~N–S trending

dike strikingperpendicularly to thedirectionof the ~eastward slip of theblock (as commonly observed at Mt. Etna; Bonaccorso et al., 2002; Neriet al., 2005; Aloisi et al., 2006; Neri and Acocella, 2006; Currenti et al.,2008, 2009; Palano et al., 2008) are made up by two contributions: themagmastatic force Fms and the overpressure Fmo. The first results fromthe lateral pressure of the magmatic column, the second from themagmaoverpressure driving themagma conduitflow. Themagmastaticforce is function of the height of the dike hD

m and the unit weight ofmagma γD

m=γm/γw

Fms =12

hmD� �2γm

D dD ð2Þ

The overpressure force is equal to

Fmo = ε hmD� �2γm

D dD ð3Þ

where the product εhDm is the equivalent height of overpressure, i.e.the equivalent excess height of the magma column corresponding tothe overpressure magnitude; δ is a constant of proportionality used toexpress the equivalent height as a fraction of hDm.

Table 1Notation and parameters value (see Fig. 2).

Symbol Description Units Value Note

a Sea depth at the failing block toe m (1±1)·103

A Thermal strain 1 [1]b Height of piezometric surface m (3±3)·103

B Skempton's coefficient 0.7±0.1c Hydraulic diffusivity a m2/s 10−103

d Width of failing block m (1.0±0.5)·104

hf+hm Height of block rear scarp m 3350hm Height of magma columns at

dike contactm 0bhmb3350

hs Height of slope crest above sea level m 3350k0 Coefficient of earth pressure at rest 0.5±0.3 [1]Kb Undrained bulk skeleton modulus Pa 109 [1]

kμ Ratio of permeability and waterviscosity

m2

Pad s10−10–10−8 [2]

l Horizontal projection of thelength along the basal failure plane

m (1.0±0.5)·104

lb Length along the basalfailure plane

m (1.0±0.5)·104 lb≈ l

lθ Length along the piezometric surface m (0.95±0.5)·104

n Porosity 0.13±0.08 [1]U Dike advance rate m/s 0.30±0.20w Dike width m 3±1α Inclination of basal failure surface −10°bαb10°β Inclination of failing block 9°ε Relative height of magma column 0.75γm Unit weight of magma Pa/m (27±1)·103

γr Unit weight of volcanic rock Pa/m (26±3)·103

γω Unit weight of water Pa/m 10·103

κ Thermal diffusivity of saturated rock m2/s 10−6 [1]μ Coefficient of attrition 0.7±0.1μ′ Apparent coefficient of friction 0.49±0.14

[1] After Ellesworth and Voight (1995).[2] After Reid (2004).

a The hydraulic diffusivity is estimated from c=2(k/μ)G (1−v) /(1−2v) with shearmodulus G=3 GPa and Poisson’s ratio v=0.25 (Ellsworth and Voight, 1995).

156 M. Battaglia et al. / Journal of Volcanology and Geothermal Research 199 (2011) 153–164

3.1.3. Uplift force due to mechanical and thermal pore-fluid pressureThe total uplift force Fp due to the mechanical and thermal pore-

fluid pressures triggered by the intrusion of a planar dyke of infiniteextent is

Fp = Fpm + Fpt =wDUD

cosα∫

12dD

−12dD

∫lD cosα

0

PmD dxDdzD +

dDAD

cosαffiffiffiffiffitD

p ð4Þ

where Fpm is the mechanical contribution and Fpt the thermalcontribution. The mechanical pressure PD

m is controlled by the in-

Table 2Dimensionless parameters used in the evaluation of flank stability.

Symbol Definition Eq. Range

aD a/hs (6) 0.3±0.3AD AKb/(γwhs) (4) 30bD b/hs (7) 0.9±0.9dD d/hs (1) 3.0±1.5hDf +hD

m (hf+hm)/hs (1) 1hDm hm/hs (2) 0.75±0.25

lD l/hs (7) 3.0±1.5lDθ lθ/hs (1) 2.8±1.5tD κt/(πhs2) (4) 2.7·10−6

(3 years)wD

a (ws)/(πγwh2skμ) (4) 0.75±0.25

UD (Uhs)/(2c) (4) 0.51–51γDm γm/γw (2) 2.7±0.1

γDr γr/γw (8) 2.6±0.3

a Note that wD is independent of the permeability kμ (see Table 1).

trusion propagation velocity UD and the geometric variables RD

and xD

PmD = K0 UDRDð ÞeUDxD ð5Þ

where K0 is the modified Bessel function of the second kind of order

zero and RD =ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2D + z2D

q. The pressure PDm is defined as the difference

between the total fluid pressure following the intrusion and theinitial fluid pressure. The magnitude of the mechanical uplift forceFpm is modulated by the dike widthwD and the intrusion propagationvelocity UD; the cos α term accounts for the dip of the basal failureplane (Fig. 3). Because of the exponential term, Eq. (5) may becomenumerically unstable when the intrusion propagation velocity UD islarge, or when the hydraulic diffusivity c (or the ratio betweenpermeability and water viscosity kμ) is very small (see Table 1). Inour case, numerical instability starts when kμ≤10− 11 m2/Pa ⋅ s. Thissolution is steady state and PD

m is capped by the lithostatic pressure(see Fig. 4).

The simplified expression for the thermal uplift force Fpt is areasonable approximation for a porous medium with moderatepressure gradients and tDNdD/UD. According to Delaney (1982), thissimplified analysis may lead to an under-prediction of pressurechanges smaller than 20%. The magnitude of the thermal uplift forceFpt is modulated by the thermal forcing AD (see Table 2).

Rock thermal and hydraulic properties influence themagnitude andduration of fluid pressurization. Permeability and thus diffusivity canvary over many orders of magnitude in volcanic rocks, making it one ofthe most uncertain parameters controlling far-field pore-fluid pressur-ization. Intermediate hydraulic diffusivities allow fluid pressurization topropagate upward into the edifice without hydro-fracturing. If weconsider a permeability/viscosity range between 10−10bkμb10−8 m2/Pa⋅s, this range spans the transition from hydrothermal systemsdominated by conduction to those controlled by advection of heat(Reid, 2004). For flow simulations with this intermediate range ofhydraulic properties, the time lag between magma intrusion at depthand elevated pressures reachingmidway into the edifice varies between2 weeks for higher-diffusivity rocks and 3.3 yr for lower-diffusivityrocks. Maximum elevated pressures high within the edifice may occurtens to hundreds of years after intrusion (Reid, 2004).

3.1.4. The seawater loadThe force Fs, applied by the seawater load on the submerged part

of the block, is

Fs =12

cot βa2DdD ð6Þ

where aD is the depth of the sea at the block toe.

3.1.5. Uplift force due to static groundwater pressureThe uplift force Fps due to static groundwater pressure is

Fps≈12n bD + aDð ÞlDdD ð7Þ

3.1.6. Lithostatic forceFl is the lateral force due to the effective lithostatic pressure

Fl≈212k0γ

rD hfD + hmD� �2

lD ð8Þ

The lateral force resulting from lithostatic pressure is defined interms of k0, denoted as the coefficient of earth pressure at rest andgiven by the ratio of vertical effective stress to horizontal effectivestress.

dDα

Dike

wD

UD

hfD

zD

xD

yD

lD

hsD

aD

βl bD

hmD

θ

piezometric surface

sea surface

shear surface

bDl θ

D

Fig. 2. Geometry of the unstable flank block. The dikewidth iswD and itmoveswith velocityUD. The shear is applied along a basal failure plane inclined atα degrees to the horizontal axis.

157M. Battaglia et al. / Journal of Volcanology and Geothermal Research 199 (2011) 153–164

3.2. Block geometry and aquifers

3.2.1. Block geometryThe block is described by the width dD; the underlying basal failure

surface lDb with dip α; the slope β of the block (for simplicity, we

assume that this is equal to the slope of the volcano flank); and thetotal height of the rear scarp hD

f +hDm (Fig. 2).

Unfortunately, many of the parameters (see Table 1) necessary tostudy the slope instability are not well constrained. The height abovesea level of the volcano is approximately 3350 m, while the on-shore

α

zD

xD

yD

Mp

Fl

Fms

Fmo

Fps

Fpt

Fpm

Fig. 3. Forces taken into account in the equilibrium analysis. All forces are defined as positivforce Mp , the magmastatic force Fms and the overpressure Fmo , the uplift force due to mechadue to static groundwater pressure Fps , and the lithostatic force Fl . The total force driving

portion of the volcanic edifice has a radius l of about 20 km. This givesa β≈9∘ for the subaerial portion of the slope. The slope α of the basalshear surface is highly debated (see above). To study the modelsensitivity to α, we consider a range of values going from−10° (deepseated configuration) to 10° (shallow flank failure).

3.2.2. Volcanic aquiferThe presence of an aquifer has three different effects in our model: a

fluid in the porous rocks affects the calculation of theweight force actingon the block; it can generate an uplift force due to the hydrostatic pore-

dD

βFs

Fl

e. The angle α is positive for shallow failure. Forces acting on the system are the weightnical and thermal pore-fluid pressure Fpm and Fpt , The seawater load Fs , the uplift forcethe wedge motion is F∥, while the resisting force is F⊥.

Fl

Fs

Fps

−10 −5 0 5 10

α (deg)

For

ce/M

p

1

0.1

0.01

(A)

Fms

−10 −5 0 5 10

α (deg)

For

ce/M

p

1

0.1

0.01

Fl

Fs

Fps

Fpm high kμ

Fpm low kμ

Fpt

(B)

−10 −5 0 5 10

α (deg)

For

ce/M

p

1

0.1

0.01

Fl

Fs

Fps

Fpm high kμ

Fpm low kμ

Fpt

Fm

(C)

2.5(D)

FptFpm high kμ

Fpm low kμ

FmFlFpsFsMp

0

0.5

1

1.5

2

Fig. 4. Relative strength of forces acting on the system: (A) no intrusion (hDm=0); (B) dike intrusion (hDm=0.75±0.25 and ε=0); (C) dike intrusion with overpressure (hDm=0.75±0.25 and ε=0.75). (D) Uncertainty (hDm=0.75±0.25, ε=0.75). Forces acting on the system are the weight forceMp , the magmastatic force Fms and the overpressure Fmo , the upliftforce due to mechanical and thermal pore-fluid pressure Fpm and Fpt , The seawater load Fs , the uplift force due to static groundwater pressure Fps , and the lithostatic force Fl . See alsoFig. 3.

158 M. Battaglia et al. / Journal of Volcanology and Geothermal Research 199 (2011) 153–164

fluid pressure; finally, the intrusion of a dike in wet volcanic rocks canincrease the pore pressure. The aquifer hydrostatic uplift is controlled bythe porosity n and the height bD. In our study area (e.g., Brusca et al.,2001), the aquifer is hosted by the Etnean volcanics (fractured basalticrocks), lying above a less permeable basement (sandstone and claysmaking the flysch deposits). The presence of a deeper confined aquiferwithin the basement deposits is possible, although there is no clearevidence (Bellomo; INGV Palermo, www.pa.ingv.it/sorveglianza/etna/falde.html). There is a very limited knowledge of the hydraulic pa-rameters of theaquifers. FollowingReid (2004),weassume intermediatevalues for the permeability with 10−14bkμb10−11 m2/Pa⋅s.

3.3. Relative strength

In Fig. 4, we compare the relative strength of the different forcesacting on the block. For the standard parameters proposed in Tables 1and 2, the largest forces (in addition to the block weight Mp) are themagmatic, Fm, and lithostatic, Fl, forces. The sea load, Fs, and uplift dueto the static groundwater pressure, Fps, are one order of magnitudesmaller (Fig. 4). The uplift force due to the increase of the mechanicalpore-fluid pressure triggered by the dike intrusion, Fpm, depends fromthe hydraulic parameters of the volcanic edifice through the rate ofintrusion UD (see Table 2). Permeability and thus diffusivity can varyover many orders of magnitude in volcanic rocks, making it one of themost uncertain parameters controlling pore-fluid pressurization. If weconsider a permeability/viscosity range between 10−10bkμb10−8 m2/Pa⋅s (this is an intermediate range of hydraulic properties for geothermalreservoirs and hydrothermal aquifers; Reid, 2004), the mechanical

pressurization force Fpm can vary between the value of the hydrostaticuplift Fps (lower permeability range) and approximately one-half of thelithostatic load Fl for higher permeability (Fig. 4B and C). The proposedpermeability range spans the transition from hydrothermal systemsdominated by conduction to those controlled by advection of heat. Forsimulations with this intermediate range of hydraulic properties and atime lag of approximately 3 years, the uplift force due to thermalpore pressure Fpt is two orders of magnitude smaller than the blockweightMp, or one order of magnitude smaller than the mechanical forceFpm (Fig. 4B and C).

These results suggest that, in first approximation, the stability iscontrolledprimarily by thebalancebetweenblockweightMp, lithostaticload Fl and magmatic forces Fm. The mechanical force Fpm may play aminor role when the permeability/viscosity is large (in our example,kμ∼10−8 m2/Pa⋅s). The remaining forces (sea load Fs, hydrostatic upliftFps, and the uplift due to thermal pore-fluid pressure Fpt) shouldprobably be considered negligible.

4. Limit equilibrium analysis

We investigate flank failure due to an increase in shear stress, or tothe reduction of the friction coefficient μ of a potential basal failureplane. A summary of the geometrical and physical parameters thatmight be representative of the mechanical and thermal properties ofEtna's flank are presented in Tables 1 and 2. A key part of our analysis isto understand how the uncertainty in our knowledge of the geometricaland physical parameters of Mount Etna may influence the evaluation ofthe potential for flank failure. This can be done by propagating the

159M. Battaglia et al. / Journal of Volcanology and Geothermal Research 199 (2011) 153–164

parameters uncertainties in the equations for the forces and the limitequilibrium analysis (see Appendix A for mathematical details).

4.1. Increase in shear stress

For the geometry illustrated in Figs. 2 and 3, the force componentsparallel to the shear surface, F||, promote motion (destabilizing forces).This force is countered by the friction force, μF⊥, proportional to forcesnormal to the possible shear surface and to the sides of the wedge(stabilizing forces). The friction coefficient μ is assumed to be a constantmaterial property. Resolving the force components parallel and normalto the faults, we obtain the forces driving and resisting failure

F jj = Mp sinα + Fm cosα + Fs sin β−αð Þ ð9Þ

μF⊥ = μ Mp cosα−Fm sinα−Fps−Fpm−Fpt + Fl + Fs cos β−αð Þh i

ð10Þ

The overbar denotes the non-dimensional magnitude of the forces.The block becomes unstable when the shear force F|| overcomes thefrictional force μF⊥

F∥F⊥

N μ ð11Þ

In our calculation, we assume μ=0.7±0.1 (Table 1). Experimentson basaltic rocks show a value of the coefficient of friction μbetween0.6 and 0.8 (Ramana and Gotge, 1989). Shear tests and normal faultinganalogue experiments on various dry granular materials suggest avalue of μ≈0.65 for natural rocks (Schellart, 2000). According toByerlee (1978), at normal stresses up to 0.2 GPa the shear required tocause sliding is given approximately by the equation

F∥ = 0:85F⊥ ð12Þ

These equations are valid for initially finely ground surfaces,initially totally interlocked surfaces or on irregular faults produced ininitially intact rocks. Rock types have little or no effect on friction. Ifhowever, the sliding surfaces are separated by large thicknesses ofgouge composed of minerals such as montmorillonite or vermiculite,the friction can be significantly lower. Since natural faults oftencontain gouge composed of alteration minerals, the friction of naturalfaults may be strongly dependent on the composition of the gouge.

4.2. Reduction in the friction factor

The tendency of many fault gouge minerals to take on adsorbed orinterlayer water may strongly influence their frictional strength andexplain the anomalously low strength of certain natural fault gouges(Morrow et al., 2000).We can qualitatively describe the effect of fluidson the friction coefficient by defining an apparent friction factor as(Beeler et al., 2000)

μ′ = μ 1−Bð Þ; ð13Þ

reducing the failure criterion (Eq. 11) to

F∥F⊥

N μ′ = μ 1−Bð Þ ð14Þ

with

μ′F⊥ = μ 1−Bð Þ Mp cosα−Fm sinα + Fl + Fs cos β−αð Þh i

: ð15Þ

Poromechanical and thermal forces (Fpm, Fpt) are not incorporatedin Eq. (15), to avoid superimposing them again onto the solution. The

parameter B is the Skempton's coefficient, where 0bBb1. Experi-mental determinations of B for volcanic rocks indicate a range from0.7 to 0.8 for tuffs (Fredrich et al., 1995) and a value B=0.12 for solid(no fractures) Handford basalt (Wang, 2000, Table C.1). Calculationsof poro-elastic parameters for fractured, water saturated basalticrocks show 0.17bBb0.53 (Zencher et al., 2006). Here, we assume a(fractured, water saturated) basalt lithology for the aquifer andB=0.3±0.2.

4.3. The potential for failure

The stability of the system, defined by Eqs. (11) and (14), iscontrolled by several parameters. Fourteen dimensionless coefficientsdescribe the stability in the case of shear stress increase

F∥F⊥

= f AD; dD;D; hfD + hmD ;h

mD ; k0; lD;wD;UD;α;β; ε;γ

mD ;γ

rD

� �ð16Þ

Ten dimensionless coefficients describe the stability when there is areduction in the friction coefficient

F∥F⊥

= f dD;hfD + hmD ;h

mD ; k0; lD;α;β; ε;γm

D ;γrD

� �ð17Þ

Poromechanical and thermal forces are not included in the forcebudget (17).

Three parameters (dD, hDf +hDm and lD,) represent the geometry of

the wedge and define the width, depth and length of the failure planerespectively. The mechanical influence of the dyke is representedthrough the dike height hD

m, the dike width wD and the dykepropagation rate UD. When thermal effects are included, the systemis additionally controlled by the magnitude of the thermal forcing AD,and the ratio of thermal and hydraulic diffusivities D. Finally, if theinfluence of the magma overpressure is taken into account, theequivalent height of overpressure εhDm must be accommodated. Theparameters aD and bD describe the sea load and the hydrostatic upliftforces. Since these two forces are several times smaller than otherforces involved (see Fig. 4), they are not expected to play a major rolein initiating failure.

4.4. Sensitivity evaluation

Using the standard values proposed in Tables 1 and 2, three casesof rear scarp loading are used for sensitivity evaluation (Fig. 5)

(1) Zero magma loading (hDm=0)(2) Dike intrusion (hDm=0.75±0.25 and ε=0)(3) Dike intrusion with overpressure (hDm=0.75±0.25 and

ε=0.75)

The assumption above implies that the dike height hDm ranges

between hDs /2 and hD

s . We also let the slope of the basal surface failure(an unknown parameter) vary between −10∘ and 10∘. For simplicity,we assume hD

f +hDm=1.

For flank failure in the absence of dike intrusion, the stability isconditioned by the ratio between the lateral restraint Fl and the blockweight Mp and the slope α of the basal failure plane

F∥F⊥

≈ sinαcosα + Fl =Mp

N μ ð18Þ

All the other parameters are constrained by the assumed geometryand the narrow range of the density parameters (see Table 2). Usingthe proposed ratio for Fl =Mp (Fig. 4A), stability is evaluated fordifferent values of the slope α (Fig. 5A). In volcanoes, a coefficient ofearth pressure at rest k0=0.5 is considered reasonable (Voight andElsworth, 1997). In the absence of an initiating mechanism, and with

−10 −5 0 5 100

0.5

1

1.5

2

α (deg)

−10 −5 0 5 10

α (deg)

−10 −5 0 5 10

α (deg)

−10 −5 0 5 10

α (deg)

−10 −5 0 5 10

α (deg)

−10 −5 0 5 10

α (deg)

Fric

tion

coef

ficie

nt

0

0.5

1

1.5

2

Fric

tion

coef

ficie

nt

0

0.5

1

1.5

2

Fric

tion

coef

ficie

nt

0

0.5

1

1.5

2

Fric

tion

coef

ficie

nt

0

0.5

1

1.5

2

Fric

tion

coef

ficie

nt

0

0.5

1

1.5

2

Fric

tion

coef

ficie

nt

µhigh kμlow kμ

(A) shear increase

(B) shear increase

(C) shear increase

µ'F||/F⊥

(A) friction reduction

(B) friction reduction

(C) friction reduction

Fig. 5. Limit equilibrium analysis. (A) no intrusion (hDm=0); (B) dike intrusion (hDm=0.75±0.25 and ε=0); (C) dike intrusion with overpressure (hDm=0.75±0.25 and ε=0.75).The gray rectangles indicate the range of the friction coefficient μ and the apparent friction coefficient μ′. Markers with error bars indicate the value of F∥/F⊥ (see Appendix A fordetails on error propagation). The block is unstable when F∥/F⊥Nμ or F∥/F⊥Nμ′.

160 M. Battaglia et al. / Journal of Volcanology and Geothermal Research 199 (2011) 153–164

the nominal values of the coefficient of friction μ proposed, therepresentativewedge characterized by Figs. 2 and 3 remains stable (orwill continue to move at a nearly constant speed), because of theinfluence of the stabilizing force due to the lateral restraint. When theeffects of lateral restraint Fl are discounted (the coefficient of earthpressure at rest k0→0) then the stability condition reduces to

F∥F⊥

≈ tanα N μ ð19Þ

and the flank failure will be controlled by the slope α only.For flank failure in the absence of magma overpressure but in

presence of magmastatic forces Fms stability is controlled by the heightof the magma column hD

m as well (Fig. 4B). When the mechanicallyinduced pore-fluid pressures are added to the static behaviour, controlshould include two additional parameters: the dimensionless intrusionrate UD and the dimensionless dike width wD, see Eq. (4). For theproposed permeability range and parameters (Tables 1 and 2),dimensionless intrusion rates UD are in the order of 10−1−10 and

the dimensionless dike width wD∼10−1. The stability condition (11)reduces to

F∥F⊥

≈sinα + cosα Fms =Mp

cosα + Fl =Mp− sinα Fms

=Mp−Fpm =Mp

N μ ð20Þ

Although the influence of the lateral restraint Fl decreases, nomajor decrease in stability occurs with the increase in the destabiliz-ing forces (Fig. 5B, left plot). The situation changes when we considerthe possibility of a 30% decrease in the friction coefficient from 0.7±0.1 to 0.49±0.14 (Table 1) due to either poro-elastic effects(Section 4.2) or other feasible mechanism like weathering of clayminerals (Fig. 5B, right plot). In this case, the flankwedgemay becomeunstable (or accelerate), when αN5∘ (shallow flank failure).

In addition to the hydrostatic pressure from the magma column,magma overpressure may drive conduit flow. Such overpressuregenerates an additional lateral force Fmo. The larger resulting force Fmis the sum of the two components Fms and Fmo. The distribution of

161M. Battaglia et al. / Journal of Volcanology and Geothermal Research 199 (2011) 153–164

stabilizing and destabilizing forces is similar to that introduced inEq. (20)with themagmastatic force Fms replaced by the largermagmaticforce Fm. Figs. 4C and 5C represent behaviour consistent with theintrusion of an overpressured dike. It is worth noting that when theforce frommagmaoverpressure is significant (e.g., we assume ε=0.75),the wedge becomes unstable, both for the simulation considering anincrease in the shear stress (Fig. 5C, left plot) and that taking intoaccount a decrease in the friction coefficient (Fig. 5C, right plot).

5. Discussion

We have performed a limit equilibrium analysis to study the factorscontrolling flank instability at Mount Etna. In our model, the unstableflank is approximated by a wedge-like geometry (Fig. 2). Our analysisevaluates the influence of several forces (Fig. 3): block weight, lateralconstraint from the lithostatic load, push on the rear scarp of the wedgebecause of the dike intrusion, sea load, and uplift due poro-elasticeffects. Our results show that uplift forces due to poro-elastic effects aremost probably secondorder. Therefore, upward pressures related to anypressurization of the aquifer during dike emplacement are not expectedto increase the instability of the slope of the volcano. Block weight,magmatic forces, and lateral (lithostatic) constraint remain the mainfactors controlling flank instability (Fig. 4).

When there is no dike intrusion (Figs. 4A and 5A), the block does notvary its equilibrium condition (i.e., the block will remain stable orcontinue to move at a nearly constant speed). This latter case may berepresentative of the recent activity of the eastern flank ofMt. Etna: this,in absence of any dike intrusion, has been showing in the last decades anoverall continuous eastward movement, in the order of 1–2 cm/yr(Froger et al., 2001; Palano et al., 2008; Neri et al., 2009). This steady, yetmoderate motion may be representative of a longer-term behaviour ofthe flank of the volcano, without any variation of the equilibriumconditions and in absence of dike intrusion (Section 4.4).

Dike emplacement is here studied considering equilibrium (static)conditions, neglecting any dynamic component, related to the lateralpush induced by the emplacement of the dike. With these assumptions,both the lateral pressure of the magmatic column (Fig. 5B) and magmaoverpressure driving magma conduit flow (Fig. 5C) increase thepossibility to vary the equilibrium conditions, leading to acceleratedflank instability. Such apossibility is further enhancedby thepresence ofa shallow decollement (eastward dipping; with αN5∘) (Fig. 5B).However, our analysis shows that the most effective process possiblyaffecting flank instability during dike emplacement in static conditionsmay be related to the decrease in the coefficient of friction along thepotential decollement. Although thermal and mechanical poro-elasticforces due to dike intrusion are second order (Fig. 4B and C), they mayhave an important long-term effect: hydrothermal alteration andweakening of the basal failure surface. This effect is ideally representedin our model through the Skempton's coefficient B and the apparentcoefficient of friction μ′of the basal failure surface; see Eq. (13). In ourmodel, we have assumed a sandstone lithology for the aquifer (B=0.3±0.2). When we consider the possibility of a reduction in the coefficient offriction, dike intrusion may increase the pore pressures, increasinghydrothermal alteration, decreasing the coefficient of friction, in turnvarying the equilibrium conditions and enhancing the instability of theflank of the volcano. (Fig. 5B and C). This process appears mostly feasibleon the mid-to long term, and may explain the continuous but moderateslip of the eastern flank of the volcano (Froger et al., 2001; Allard et al.,2006) along hydrothermally weakened decollement surfaces.

On the shorter term, during episodes of dike emplacement alongthe S and NE Rifts, as in 2001 and 2002, the movement of the flanksignificantly accelerated, locally reaching several tens of cm/yr (Neriet al., 2004, 2005, 2009; Puglisi et al., 2008; Solaro et al., 2010). Theobserved acceleration occurred contemporaneously to dike emplace-ment, suggesting that the lateral push induced by the dike increasedthe instability of the flank of the volcano. Therefore, it appears that,

during major episodes of dike emplacement, dynamic processes areresponsible for flank instability. Any effect, indirectly or directlyresulting from the considered static conditions (including anydecrease of the coefficient of friction along a decollement), appearsof minor importance.

Our study also shows that the challenge in any analysis of flankstability atMount Etna is not the sophistication of themodel(s) applied,but the limited knowledge of the model(s) physical parameters. Wehave little or no idea ofwhat are the bounds on the coefficient of friction,slope of the basal failure plane, coefficient of earth pressure at rest,Skempton's parameter, permeability and extension of the aquifers.Notwithstanding these uncertainties, we can use our theoretical model(together with error propagation) to gain a general assessment of thestability of the easternflank ofMount Etnawith some acceptable degreeof confidence (i.e., error bars in Fig. 5).

6. Conclusions

Our results show that when there is no dike intrusion (Fig. 6A), theeastern flank of Mt. Etna does not vary its equilibrium conditions, thatis, the block will remain stable or, if alreadymoving, it will continue tomove at a similar rate. These conditions have been usually met at Mt.Etna in the last decades.

The possibility of block instability increases with dike intrusion(Fig. 6B). While block weight, magmatic forces, and lateral (litho-static) constraint play a major role on the stability of the E flank, upliftforces due to poro-elastic effects are most probably second order.Although poro-elastic forces are second order, they may have animportant longer-term effect, consisting of the hydrothermal alter-ation and weakening of the basal failure surface. This effect is ideallyrepresented in our model through the Skempton's coefficient and theapparent coefficient of friction.

However, during the recent episodes of dike intrusion of 2001and 2002, the observed increased instability of the flank probablyresulted from the dynamic effect (i.e. lateral push) due to the em-placement of the dikes. Any hydrothermal weakening of a decolle-ment plane, as acting on a longer-term, must have been of minorimportance.

Finally, our ability to apply this (or any other model) to preciselypinpoint in time and space the potential for flank instability is limitednot by the mathematics but by the large uncertainties in the estimateof the geomechanical parameters (e.g., Table 1).

Acknowledgements

The authors warmly thank Cinzia Federico for providing informa-tion on the aquifer of Mt. Etna, T. Apuani, C. Newhall and G. Puglisi foruseful discussions. Comments by D. Elsworth greatly helped to im-prove the manuscript.This work was funded by Istituto Nazionale diGeofisica e Vulcanologia (INGV) and the Italian Dipartimento per laProtezione Civile (DPC) (DPC-INGV project V4 “Flank”).

Appendix A. Error propagation

The symbol δx is the uncertainty or error in a measurement of x.

The block weight

ðδMpMp Þ2 =

δnn

� �2

+δγr

D hfD + hmD� �

lDh i2

+ γrDδ hfD + hmD� �

lDh i2

+ γrD hfD + hmD� �

δlDh i2

γrD hfD + hmD� �

lD + lθD + lbD� �

bDh i2 +

++ δ lθD + lbD

� �bD

h i2+ lθD + lbD

� �δbD

h i2γrD hfD + hmD� �

lD + lθD + lbD� �

bDh i2 +

δdDdD

� �2

Fig. 6. Three-dimensional scheme of Mount Etna (view from southeast) showing a schematic section of the unstable sector analyzed through limit equilibrium analysis. Stabilityconditions with: (A) no dike intrusion; (B) dike emplacement. In the first case, the unstable block does notmove or continues tomove (white arrow)with similar rates. In the secondcase, the unstable block accelerates its movement (yellow arrow), mainly as a result of the hydrothermal alteration and weakening of the basal failure surface induced by the dikeintrusion.

162 M. Battaglia et al. / Journal of Volcanology and Geothermal Research 199 (2011) 153–164

Dike emplacement

δFmsF ms

� �2

= 4δhmDhmD

� �2

+δγm

D

γmD

� �2

+δdDdD

� �2

δFmoFmo

� �2

= 4ε2δhmDhmD

� �2

+δγm

D

γmD

� �2

+δdDdD

� �2

Uplift force due to mechanical and thermal pore-fluid pressure

δFpmFpm

!2

=δwD

wD

� �2+

δUD

UD

� �2+

sinαd δαcosα

� �2

δFptFpt

!2

=δdDdD

� �2+

δAD

AD

� �2+

sinαd δαcosα

� �2

The seawater load

δFsFs

!2

= 4δaDaD

� �2+

δdDdD

� �2

Uplift force due to static groundwater pressure

δFpsFps

!2

≈ δnn

� �2+

δb2D + δa2DbD + aDð Þ2 +

δlDlD

� �2+

δdDdD

� �2

Lithostatic force

δFlFl

!2

≈ δk0k0

� �2+

δγrD

γrD

� �2

+ 4δ hfD + hmD� �hfD + hmD

24

352

+δlDlD

� �2

Increase in shear stress

δF jj� �2

= sinαd δMp

� �2+ Mpd cosα⋅δa� �2

+ cosα⋅δFm� �2

+ Fmd sinα⋅δa� �2 + + δFsd sin β−αð Þ� �2

+ Fs⋅ cos β−αð Þ⋅δa� �2

163M. Battaglia et al. / Journal of Volcanology and Geothermal Research 199 (2011) 153–164

δ F⊥ð Þ½ �2 = cosα⋅δMp

� �2+ Mpd sinαd δa� �2

+ sinαd δFm� �2

+ Fmd cosαd δa� �2 + + δFpm

� �2+ δFps� �2

+ δFpt� �2

+ δFl� �2 + δFsd cos β−αð Þ� �2 + Fsd sin β−αð Þd δa� �2

Reduction in the effective friction factor

δ F⊥ð Þ½ �2 = cosαd δMp

� �2+ Mpd sinαd δa� �2

+ sinαd δFm� �2

+ Fmd cosαd δa� �2 + + δFl

� �2 + δFsd cos β−αð Þ� �2+ Fsd sin β−αð Þ⋅ δa� �2

Limit equilibrium condition

δF∥ =F⊥F∥ =F⊥

� �2=

δF∥F∥

� �2+

δF⊥F⊥

� �2

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