Dual chamber-conduit models of non-linear dynamics behaviour at Soufrière Hills Volcano, Montserrat

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1 Dual chamber-conduit models of non-linear dynamics behaviour at Soufrière Hills Volcano, Montserrat Oleg Melnik 1, Antonio Costa 2 1 Institute of Mechanics, Moscow State University, Moscow, Russia, and Earth Science Department, University of Bristol, Bristol, UK. 2 Environmental Systems Science Centre, University of Reading, Reading, UK, and Istituto Nazionale di Geofisica e Vulcanologia, Naples, Italy. To be submitted to the special volume The Eruption of Soufriere Hills Volcano, Montserrat from 2000 to 2010, Editors: G. Wadge, R. Robertson, B. Voight, Memoir of the Geological Society of London. 1. Introduction Lava dome eruptions are typically characterized by large fluctuations in discharge rate showing cyclic patterns of seismicity, ground deformation and volcanic activity. Time-scales associated with these fluctuations range from hours-days (short-term cycles) to years-decades (long-term cycles) with intermediate of 5 to 7 weeks as, for example, observed on the Soufrière Hills Volcano (SHV), Montserrat. Concerning short-term cycles, cyclic patterns of seismicity, ground deformation and volcanic activity have been documented at Mount Pinatubo, Philippines, in 1991 and Soufrière Hills Volcano, Montserrat, British West Indies, in At SHV, periodicity in seismicity and tilt ranged from ~3 to 30 hours (Voight et al., 1999; Odbert and Wadge, 2010). Several models have been proposed to explain the observed cyclicity in the last decade. In both Denlinger and Hoblitt (1999) and Wylie et al. (1999) models the lower part of the conduit acts like a capacitor that allows magma to be stored temporally in order to release it during the intense phase of the eruption. Costa et al.

2 (2012) generalized Denlinger and Hoblitt (1999) model to a more realistic system made of a compressible magma flowing through an elastic dyke evolving to a shallower cylindrical conduit. Lensky et al. (2008) explained short-term cyclic behaviour at the SHV as a result of gas diffusion into growing bubbles and filtration through the bubble network in stagnated magma column before the critical overpressure is reached followed by magma motion, depressurisation and stagnation of the plug at the top of the conduit. Regarding long-term cycles Costa et al. (2007a;b) considered an elastic dyke evolving to a cylinder toward the surface coupled with a shallow (~5 km) magma chamber. They showed that degassing-induced crystallization within the conduit coupled with the wallrock elasticity play a major control on process for the long-term cyclicity during lava dome building eruptions. In particular, they showed that there is a regime where the period of pulsations is controlled by the elasticity of the dyke (~weeks to months) and a regime where the period is controlled by the volume of the magma chamber (~years). Intermediate regimes are possible. The geometry proposed by Costa et al. (2007a;b) is also consistent with inversion of strain data for the March 2004 SHV explosion that highlight the presence of a spherical pressure source and a shallow dyke (Linde et al., 2010, Hautmann et al., 2009). Most of the existing conduit flow models for lava dome building eruptions assume a system consisting of a magma chamber (a large volume of magma located in elastic wallrocks) connected to the surface through a conduit. Magma chamber is assumed to be fed from below with a constant rate with a fresh magma. In few models (Dirksen et al, 2006; Maeda, 2000) the influx rate into the magma chamber is specified as a function of time. Modern geophysical data recorded during lava dome building eruptions indicate the presence of interconnected multiple magma storage regions. Concerning the SHV system different geometries have been proposed on the basis of both mineralogical and geodetic observations. Elsworth et al. (2008) proposed a model with two stacked magma reservoirs connected by vertical conduits (the shallow chamber was centered at 6 km and the deep at 12 km below sea level). Surface efflux and GPS station velocities were inverted to calculate crustal magma transfer. Results from the inversion also indicated that the eruptive episodes deplete the lower reservoir only, and not the upper reservoir.

3 A vertically elongated chamber idealized as a prolate ellipsoid was proposed by Voight et al. (2010). This chamber was assumed stratified with upper parts compressible due to exsolved gas phases (magma sponge), and fed at the base by deep influx estimated of the order of 2 m 3 /s. The top of the ellipsoid was assumed at 5 km below sea level and its centroid near 10 km. Whether SHV system is better represented by a single prolate chamber or a two chamber model is still not resolved. However Foroozan et al. (2010) showed that the deformations of GPS stations located near or far from the volcano are better explained by a two chamber model. In a more recent paper Foroozan et al. (2011) simultaneously inverted geodetic data and measured magma efflux rates with the additional constraint of constant basal influx. In contrast to Elsworth et al. (2008) they used data from 6 to 13 stations (versus 4 previously) and a compressible magma phase (versus incompressible previously). Their results suggest a basal supply of 1.2 m 3 /s for the activity from Centroids of the chambers were estimated to depths of about 5 and 19 km. In contrast to Elsworth et al. (2008), Foroozan et al. (2011) found that the data are characterized by synchronous deflation then inflation of both shallow and deep chambers. In order to account for the large-strain effects and modulus softening that are unrepresented in the extrapolated 1-D seismic velocity data, Foroozan et al. (2011) decreased the shear modulus of the wallrock G by a factor of 10, estimated on the basis of 1-D seismic velocity profile (Shalev et al., 2010). Rather than this arbitrary approach, we will use, following Costa et al. (2009; 2011), an empirical relationship between static and dynamic values of the Young modulus E of rocks (Wang, 2000): E static (GPa) = E dynamic (GPa) On the other hand the analysis of data for the period from the borehole strainmeter network at SHV gives good constraints on where the pressure changes occur. These observations are also consistent with a system made of a deep reservoir coupled with a shallow chamber connected to the surface through a shallow dyke (Sacks et al., 2011). Most numerical models of these eruptions assume a single magma chamber fed from below with a constant or prescribed time dependent influx rate and connected to the Earth surface through a conduit. Here we present a development of the model of

4 extrusive eruption by Costa et al. (2007a,b) considering a system made of two magma chambers located in elastic rocks and connected by a dyke between each other. The deep chamber is fed from below with a constant influx rate. For sake of comparison with previous studies (e.g., Hautmann et al., 2009; 2010) we use locations and volumes of the magma chambers inferred for SHV, Montserrat, from ground deformation studies by Hautmann et al. (2010) and also the more recent estimations obtained integrating seismic tomography with numerical models of the shallow magma chamber by Paulatto et al. (2012). The estimations of both chamber locations and volumes contain large uncertainties that need to be systematically investigated but here we will focus on exploring the control of two magma chambers on the dynamics of lava dome extrusion only. In order to study the role of a dual magma chamber system, however, we explore a large range of chamber volumes and two different chamber shapes (spheroidal and prolate) for both deep and shallow magma chamber. The model shows cyclic behaviour with a period that depends on the intensity of the influx rate and the chamber connectivity (described as the horizontal extent of the dyke connecting the two chambers). For a weak connectivity the overpressure in the lower chamber stays nearly constant during the cycle and the influx of fresh magma into the shallow chamber is also nearly constant. For a strong connectivity between the chambers their overpressures increases or decreases during the cycle in a synchronous way. Influx into the shallow chamber stays close to the extrusion rate at the surface. 2. A model for magma transfer between multiple reservoirs and dykes We consider a system like that proposed by Hautmann et al. (2010) represented in Fig. 1. The system is composed by a deep magma chamber fed from below and connected to a shallow magma chamber through a dyke. The shallow magma chamber is connected to the surface via a volcanic conduit made of dyke that evolves to a cylinder approaching to the surface as described Costa et al. (2007a,b). The system of equations for the description of magma transport from the deep to the shallow chamber is:

5 p z = ρg 4μ a2 + b 2 V a 2 b 2 S t + SV z = 0; S = πab a = a 0 + ΔP [ 2E (1 2ν)a 0 + 2(1 ν)b 0 ] ΔP b = b0 + [ 2(1 ν ) a0 (1 2ν ) b0 ]. 2E Here μ denotes magma viscosity, ΔP = p p e (z) is a magmatic overpressure, p is (1) 141 the pressure inside the dyke, p e (z) = ρ r gz σ t denotes the lithostatic pressure (g is acceleration due to gravity) minus σ t, i.e. tensile stress along the dyke due to the presence of magma chamber and an extensional/compressional far-field stress. S is the cross-section area of the dyke which is assumed to have an elliptical shape, a and b are the semi-axis of the ellipse. Wallrocks are assumed to be purely elastic with Young modulus E and Poisson ratio ν. The Muskhelishvili solution (Muskhelishvili, 1963) is used for calculation of the relation between magmatic overpressure and the deformation of the dyke. Densities of magma ρ and wallrocks ρ r, in the deeper part of the magmatic system, are assumed to be constant, z is a vertical coordinate increasing upwards, V is the ascent velocity. In order to simplify the equation system, since typically a 0 >> b 0 relative changes of a are negligible, thus, we can assume a constant and the initial value of b equal to zero, i.e. b 0 = 0. Moreover, at typical depths of several kilometers, we can reasonably neglect the presence of the free gas phase and assume incompressible magma of constant viscosity. For CO 2 rich magma this assumption will be violated due to deep exsolution of dissolved volatiles, but for the case of SHV, even assuming the upper bound estimation of CO 2 content in the magma (600ppm wt%, Edmonds et al., 2012) is exsolved, magma density variation is negligible, i.e. from 2520 to 2600 kg/m 3 when pressure changes from 240 to 300 MPa (moreover for such CO 2 content at 300 MPa magma bulk

6 modulus is ~3GPa, only few times smaller than rock rigidity at that depth). Hence we have: z (p + ρgz) = 4μ V b 2 b t + bv z = 0 (2) b = ΔP E (1 ν)a 0 = αδp where α = (1 ν)a 0 parabolic equation: ΔP t E. The above system can be rewritten as a non-linear α 2 ΔP 4 α 2 (ΩΔP 3 ) = 0 (3) 16μ z 2 4μ z where Ω= [ z p e (z) + ρgz]. In order to solve eq. (3) we impose the following boundary conditions: d dt ΔP = Q Q ch, d ( in, d ) P V ch, d ch, d 0 with Q = π a bv = [ ΔP + pe( z) + ρgz] 1 πa b 0 being the volumetric outflux from the 4μ z deep magma chamber and Q in,d being the influx into deep magma chamber from a deeper magmatic source. Q in,d will be considered as a given constant parameter in this study, although it can be influenced by pressure variations in the magmatic source, local tectonic stress and thermal development of the feeding system.

7 Eq. (3) is solved together with the system of equations that describes the flow from the shallow magma chamber through a dyke shaped conduit that evolves to a cylinder in the vicinity of the surface. These equations are described in detail in Costa et al. (2007a,b) and are summarized in the Appendix. FIG. 1 HERE Fig. 1 Sketch of the investigated system. Modified after Hautmann et al. (2010). 3. Parametric study Here we present results of two separate studies. First, we analyze the effects of a farfield stress and a pressurized deep magma chamber on the steady state solution of the flow equation in the upper conduit. Second, assuming a neutral far-field stress we present results on the role of the connectivity between deep and shallow magma chambers during an extrusive eruption on the transient behaviour of the system. 3.1 Effect of far-field stress The presence of a deep reservoir affects the dynamics and the cyclicity of the shallow part of the system, but it also has an important effect in perturbing the stress field around it. A pressurized magma chamber under the effect of a far-field stress can modify the tensile stress along the axis of the conduit (see Costa et al., 2011). This stress is calculated using the general analytical solution by Gao (1996) obtained in the limit of a plane 2D geometry (for the limitations of this assumption see Costa et al., 2011). We explore the effect of far field stresses, typical of arc volcanism environment, from compressional (-10 MPa) to extensional environment (10 MPa) (Hautmann et al., 2010; Roman et al., 2006) on the steady state solution of the flow equation in the upper conduit. Fig. 2 reports a set of steady-state solutions for fixed deep reservoir overpressure of 10 MPa and shows the control of the far-field stress field on the system sketched in Fig. 1. A compressional far-field stress tends to move the steady state solution toward larger pressures, whereas an extensional far-field stress moves the solution to lower

8 pressures allowing larger flow rates at low pressures. The conduit cross-section area is larger in the case of extensional environment, thus the transition to the upper branch moves to higher values of discharge rate because ascent velocity for large is smaller for the same discharge rate and crystals grow more efficiently (see Fig. 12b in Melnik and Sparks, 2005 for more detailed explanations). FIG. 2 HERE Fig. 2. A set of steady state solutions: dependence of discharge rate on chamber pressure for three different far-field stresses: compressional (-10 MPa; dashed-dotted line), neutral (full line), and extensional (10 MPa; dotted line). The remaining parameters are set as in Costa et al. (2007b). 3.2 Effect of coupling the two reservoirs In this section we explore the role of the connectivity between deep and shallow magma chamber during an extrusive eruption. We consider only the effect due to the two pressurized magma chambers and will set 0. As initial condition we assume that both magma chambers have zero overpressure with respect to the lithostatic pressure, the flow rate is zero everywhere and the dyke between the two chambers has a negligible thickness. At time t = 0 influx of magma into the deep magma chamber starts with a constant intensity Q in,d. Pressure in the deep chamber starts to grow leading to magma rising and opening of the pre-existing dyke through the rocks between two chambers. When the dyke reaches the shallow chamber, influx of the fresh magma leads to pressure growth and initiation process of the dome extrusion. Figs. 3a,b show the evolution of the chamber overpressures normalized to their maximum values (upper plots) and the influx into the shallow magma chamber Q in,s and discharge rate at the surface Q out (lower plots), for two cases: i) weak connectivity between the chambers (when the value of a 0 in Eq. (2) is small, i.e. 100 m) and ii) strong connectivity (when the value of a 0 in Eq. (2) is small, i.e. a 0 = 600 m).

9 In the case of weak connectivity the overpressure in the deep chamber increases to a high value (~70 MPa) and stays nearly constant while the overpressure in the shallow chamber shows a pronounced cyclic activity. In this case influx into the shallow chamber is almost constant although its variation is shifted with respect Q out. In the case of strong connectivity both chambers inflate and deflate simultaneously. Influx rate into the shallow chamber increases when the pressure in the shallow magma chamber decreases. Q out follows changes in Q in,s. For the intermediate values of a 0 the system stays in between these two end member cases. The case of strong connectivity is consistent with reconstructions of Foroozan et al. (2011). Behaviour for the weak connectivity is not supported by the recent analysis and interpretation of geodetic observations (Foroozan et al., 2011). In this case high chamber overpressure suggests a horizontal extension of the dyke and a consequent improvement of connectivity between two magma chambers. FIG. 3 HERE Fig 3. Time evolution of the chamber overpressures normalized to their maximum values in the upper plots, and of influx into the shallow magma chamber Q in,s (thicker lines) and discharge rate at the surface Q out (thinner lines) in the lower plots, for two cases: (a) weak connectivity between the chambers (a 0 = 100 m) and (b) strong connectivity (a 0 = 600 m). Π denotes the normalized to its maximum overpressure value for the deep magma chamber (thicker lines) and Π denotes the normalized overpressure for the shallow magma chamber (thinner lines). In order to quantify the inflation/deflation phase shift between two chambers we introduce a variable characterizing the value of offset as:,,, where T is the period of discharge rate variation. In the case when the two chambers work in phase the value of the offset is equal to 0 while an offset of 1 corresponds to completely opposite directions of chambers evolution. Fig. 4 shows the dependence of the offset on the intensity of deep magma chamber influx Q in,d. Curves are shown for the values of a 0 = 100 (short dashed line), 200, 300, 400, 500 and 600 m (solid line).

10 The offset progressively decreases with an increase of the horizontal dyke extent a 0 and increases with Q in,d because for larger influx intensity the pressure in the deep chamber reaches higher values (see Fig. 5) and the dyke cross-section area increases. This improves the connectivity between the chambers. FIG. 4 HERE Fig. 4. Dependence of the offset on the intensity of deep magma chamber influx Q in,d for the magma chamber volumes suggested by Hautmann et al. (2010), i.e. V ch,s = 4 km 3, V ch,d = 32 km 3, and different a 0. Fig. 5 shows the maximum overpressure P d that is reached in the deep chamber (a) and the amplitude of overpressure variation during the cycle ΔP d (b) as a function of Q in,d for different a 0. Small values of a 0 result in unrealistically high overpressures and small amplitude of pressure variations. However in real situations as the overpressure increases above a rock strength it is expected that a fracture will propagate at the tips of the dyke increasing the dyke dimension and moving the system to lower more realistic pressures. Increase in Q in,d leads to improvement of the connectivity between the chambers and thus amplitude of pressure variation. FIG. 5 HERE Fig. 5. Maximum overpressure P d that is reached in the deep chamber (a) and the amplitude of overpressure variation during the cycle ΔP d (b) as a function of Q in,d for V ch,s = 4 km 3, V ch,d = 32 km 3, as in Hautmann et al. (2010), and different a 0. The time lag between the beginning of the influx into the deep chamber and the start of the eruption is shown in Fig. 6 as a function of Q in,d for different a 0. It varies from ~3 months to more than a year. Larger influx intensities and better chamber connectivity lead in shortening of this interval. Mattioli et al. (2010) show that ground deformation and extrusion of the magma at the surface are closely correlated in time. This supports a hypothesis of good chamber connectivity for the SHV. FIG. 6 HERE

11 Fig. 6. Time lag between the beginning of the influx into the deep chamber and the start of the eruption for V ch,s = 4 km 3, V ch,d = 32 km 3, as in Hautmann et al. (2010), and different a Period of discharge rate variations In the case of weak connectivity between the chambers (Fig. 7) for fixed values of the shallow and deep magma chamber volumes (e.g., V ch,s = 4 km 3 and V ch,d = 32 km 3 ) the period of pulsations reaches a minimum value for some intermediate value of Q in,d corresponding to an unstable branch of the steady-state solution similar to the case of a single magma chamber (see Fig. 7 in Melnik and Sparks, 2005). When the connectivity is strong the period monotonically increases with increase in the influx rate into the deep chamber. In order to evaluate the role of magma chamber volumes on the period of discharge rate variation we performed a series of runs varying aspect ratios and volumes of both shallow (Fig. 8) and deep magma chambers (Fig. 9). FIG. 7 HERE Fig. 7. Period of pulsations as function of the influx into the deep chamber for V ch,s = 4 km 3, V ch,d = 32 km 3, as in Hautmann et al. (2010), and different a 0. Informed by the volume estimates of Paulatto et al. (2012), we varied the volume of the shallow magma chamber between 4 and 30 km 3 for different values of the deep magma chamber (from 30 to 60 km 3 ) fixing Q in,d = 2 m 3 /s, a value that represents a realistic estimation for the deep magma influx (Voight et al., 2010; Odbert and Wadge, 2010). Keeping fixed the volume of the deep chamber to V ch,d = 32 km 3, for V ch,s 20 km 3 the period monotonically increase from 125 to 150 days, but above 20 km 3 there is a bifurcation and the solution becomes multi-periodic. The case of V ch,s = 30 km 3 results in a double period behaviour with periods for small amplitude oscillations of order of 150 days whereas the large peak of increase in discharge rate occurs every 400 days (Fig. 8). Similar double periodicity behaviour was discovered first in Costa et al. (2007). FIG. 8 HERE

12 Fig 8. Change in the period of discharge rate variation with respect to the volume of the shallow magma chamber. In this case the volume of the deep magma chamber is fixed at 32 km 3 as estimated by Hautmann et al. (2010). The variation of discharge rate within one period of pulsation is shown in insets. Fig. 9 shows the dependence of the period and the shape of the pulsations as a function of the deep magma chamber volume. The volume of the shallow magma chamber is fixed at 15 km 3 in accord to Paulatto et al., (2012). Two shapes of the shallow magma chamber (spherical and prolate) are studied comparing results for aspect ratio of 1 and 1.5 respectively. For small volumes of the deep magma chamber (V ch,d < 37 km 3 ) the period does not strongly depend on the shallow chamber geometry and pulsations have a unimodal shape pattern. For larger magma chambers the pattern during the cycle consists of peaks in discharge rate separated by intervals of periodic variation in discharge rate with much lower amplitude that characterize major part of the cycle. FIG. 9 HERE Fig 9. Change in the period of discharge rate variation with respect to the volume of the deep magma chamber. The volume of the shallow magma chamber is fixed at 15 km 3. The patterns of variation of discharge rate within one period of pulsation is shown in the insets. We have discovered that a prolate shape of the shallow magma chamber leads to much longer periods of pulsations due to stronger influence of the chamber pressure on the opening of the dyke that connects it to the surface. The opposite effect was observed for the aspect ratio of deep magma chamber. A more spherical deep magma chamber tends to produce fluctuations with longer periodicity. Implications to SHV Observations at SHV suggest long-term cyclic behaviour with a period of 2 to 3 years. This period, for example, can be obtained for a prolate shallow magma chamber with a volume of ~15 km 3 and a deep magma chamber of ~50 km 3. Fig. 10 represents the variation in extrusion rate with time for these parameters.

13 Although with these magma chamber values we can obtain reasonable periodicity the simulated patterns of the flow rate variations, alternations between phases of high and low discharge rate, are not in agreement with observations for the SHV. Typical behaviour of the volcano involves repetitive periods of extrusion separated with repose periods with nearly no activity. Moreover, during the first three cycles the periods of extrusion were longer (~years) with respect the fourth and fifth cycle that show extrusion activity having a duration of months similar to the simulations showed in Fig. 10. The behaviour of the first three cycles is better reproduced assuming a narrow shallow dyke, d s (a 0 ~150m), as shown in Fig. 11. It is worth noting that this cyclic behaviour is closer to observations, and a systematic parametric investigation involving also the shallow part of the system is an object of ongoing research. It is also likely that magma rheology can play a key role on the behaviour of the extrusion pattern. For example, we would expect that for a Bingham rheology discharge rate between the two major pulses is zero until a critical chamber overpressure is reached (Melnik and Sparks, 2005). Moreover a more realistic description of the effective viscosity of magma during lava dome eruptions should account for the coupling with energy loss, viscous dissipation, two-dimensional and stick-slip effects (e.g., Costa and Macedonio, 2005; Costa et al. 2007c; 2012). However, due to the limitations of the current transient numerical code, we cannot investigate effect of complex rheologies. FIG. 10 HERE Fig. 10. Variation of discharge rate with time (V ch,s = 15 km 3, V ch,d = 50 km 3 ) for a shallow dyke width as in Costa et al. (2007a). FIG. 11 HERE Fig. 11. Variation of discharge rate with time (V ch,s = 15 km 3, V ch,d = 50 km 3 ) for a shallow dyke of 300m width. For comparison variation of discharge rate with time for a single chamber geometry and a shallow dyke of 300m (V ch,s = 30 km 3 ) is also shown.

14 Discussion and conclusion The models show that presence of two magma chambers connected by an elastic dyke significantly influences dynamics of magma extrusion in comparison with a case of a single magma chamber fed with a constant influx rate. The dynamics is strongly influenced by the amount of connectivity between the chambers measured as a length of large semi-axis of the dyke a 0. When a 0 is small (weak connectivity) the pressure in the deep chamber reaches high values and the influx into the shallow chamber is close to a constant value. This case, in terms of surface extrusion, is similar to the case of a single magma chamber but geodetic signal in this case will be different. At the initial stage of the eruption deep source will inflate producing wide spread deformation pattern. Later the pressure in the deep reservoir will come to a high constant value and ground deformation will be controlled by changes in pressure in the shallow chamber. In the case of strong connectivity (large a 0 ) initial ground deformation pattern will be controlled by an inflation of the deep magma chamber. Later in the eruptions both pressure sources work in phase. For the intermediate a 0 there is a transition between two end-members. It is clear that assumption of constant value of a 0 greatly oversimplifies the physics. In case of large overpressures, stress at dyke tips will exceed fracture toughness of the rocks and the dyke will expand horizontally. The maximum dyke extent will be controlled by horizontal chamber dimensions (Costa et al., 2011). When deep magma chamber deflates overpressure in the dyke will decrease and, as flow rate decreases, magma at dyke tips can solidify leading to a decrease in a 0 (Kavanagh and Sparks, 2011). Another strong simplification of the model comes from assumed Newtonian rheology of the magma. Although the system produces periodic behaviour the shape of discharge rate evolution during the cycle is different from observed at the SHV. Measurements of the dome volume suggest relatively slow initiation of the eruption with progressively increasing or nearly constant extrusion rate at later stages of the cycle followed by a rapid drop in discharge rate leading to stagnation of the eruption. Cycles produced by our model show rapid increase in discharge rate with more gradual reduction in eruption intensity. In the simulations with a shallow dyke of about 500m, as in Costa et al. (2007a), discharge rate remains always larger than zero

15 leading to continue extrusion of magma, whereas simulations using a narrower shallow dyke (~300m) show a better agreement. Finally, in order to understand if simulations can give insights about the issue single versus dual magma chamber system, we carried out also some simulations assuming only the presence of the shallow chamber. Results show that for a shallow dyke of ~300m we can have 2-3 periodicity assuming a chamber volume of about 30 km 3, implying that model results alone do not allow us to prefer a dual rather than a single chamber geometry. Predictions of the model must be verified against measurements of ground deformation, dome volume, gas flux and other geophysical data that provides constraints on the possible scenarios of the eruption. Only a robust approach that is based on an integrated system of geophysical observations will be able to give strong constraints to the model and give some insight into the geometry and dynamics of the system Acknowledgments. O.M. acknowledges the support from the ERC grant no (VOLDIES) and the Russian Foundation for Basic Research ( ). A.C. was supported by NERC research grant reference NE/H019928/1. We thank R.S.J. Sparks and G. Wadge for very useful comments on an earlier version of the manuscript

16 Appendix Governing equations in the shallow conduit (elliptical dyke to cylinder). The ascent of magma is simulated by 1-D transport equations. We assume that the conduit has an elliptical cross-section with area S = ab, with a and b major and minor semi-axes, respectively. Their vertical variation occurs at length-scales that are much larger than the dyke width. The set of cross-section averaged equations is presented below S t (Sρ ) + 1 m S x (Sρ V ) = G G m mc ph 1 S t (Sρ ) + 1 mc S x (Sρ V ) = G mc mc 1 S t (Sρ ) + 1 ph S x (Sρ V ) = G ph ph 1 S t (Sρ ) + 1 d S x (Sρ V ) = J d 1 S t (Sρ ) + 1 g S x (Sρ V ) = J g g (A1) (A2) (A3) (A4) (A5) Here t denotes time, x the vertical coordinate, ρ m, ρ ph, ρ mc, ρ d and ρ g are the densities of melt, phenocrysts, microlites, dissolved gas and exsolved gas respectively, and V and V g are the velocities of magma and gas, respectively. G ph, G mc represent the mass transfer rate due to crystallization of phenocrysts and microlites, respectively, and J

17 the mass transfer rate due to gas exsolution. Eq. (A1) represents the mass conservation for the melt phase, eqs. (A2) and (A3) are the conservation equations for microlites and phenocrysts respectively, eqs. (A4) and (A5) represent the conservation of the dissolved gas and of the exsolved gas respectively. Two momentum equations for the mixture as a whole (A6) and for the free gas phase (A7) describe the magma motion: p x = ρg 4μ a2 + b 2 a 2 b 2 V (A6) 501 V g V = k μ g p x (A7) Here p is the pressure, ρ the bulk density of magma, g the gravity acceleration, μ is the magma viscosity, k is the magma permeability and μ g is the gas viscosity. The main source of the magma temperature change is the release of the latent heat of crystallization (A8) S t (SρC mt ) + 1 S x (SρC mvt ) = L * (G mc + G ph ) (A8) Here C m is the bulk specific heat of magma, T is the bulk flow-averaged temperature, L * is latent heat of crystallization (viscous heating and heat flux to the host rocks are neglected). For details of equation of state and mass fluxes related to gas exsolution and crystallization see Costa et al. (2007a,b) For parameterizations of magma permeability k and magma viscosity μ we use:

18 k = k( α)= k 0 α j (A9) μ = μ m ( c,t)θ( β)η α,ca ( ) (A10) where k depends on bubble volume fraction of bubbles α, μ depends on water content c, temperature T, crystal content β, bubble fraction and capillary number Ca (Costa et al., 2007a,b). Variations of dyke semi-axis, a and b, with pressure are characterized by eq. (1). In order to get a smooth transition from the dyke at depth to a cylindrical conduit the value of a 0 is parameterized as: 524 x L a ( x) = A arctan + A T wt (A11) Here L T and w T are the height and the vertical extent of the cylinder to ellipse transition zone and constants A 1 and A 2 are calculated to satisfy conditions a 0 (L) = R and a 0 (0) = a 0, where R is the radius of the cylindrical part of the conduit and a 0 is the length of major dyke semi-axis at the inlet of the dyke. The value of b 0 is calculated in order to conserve the cross-section area of the unpressurized dyke, although it can also be specified independently References Costa, A., Macedonio G. (2005), Viscous heating in fluids with temperaturedependent viscosity: triggering of secondary flows, J. Fluid Mech., 540, Costa A., Melnik O., Sparks R.S.J. (2007a), Controls of conduit geometry and wallrock elasticity on lava dome eruptions. Earth Planet. Sci. Lett. 260: doi: /j.epsl

19 Costa A., Melnik O., Sparks R.S.J., Voight B. (2007b), The control of magma flow in dykes on cyclic lava dome extrusion. Geophys. Res. Lett. 34: L doi:1029/2006gl Costa, A., Melnik, O., Vedeneeva, E., (2007c), Thermal effects during magma ascent in conduits. J. Geophys. Res., 112, doi: /2007jb Costa, A., Sparks, R.S.J. Macedonio, G., Melnik, O. (2009), Effects of wall-rock elasticity on magma flow in dykes during explosive eruptions. Earth Planet. Sci. Lett. 288, , doi: /j.epsl Costa A., Gottsmann J., Melnik O., Sparks R.S.J. (2011), A stress-controlled mechanism for the intensity of very large magnitude explosive eruptions, Earth Planet. Sci. Lett., doi: /j.epsl Costa A., Wadge G., Melnik O. (2012), Cyclic extrusion of a lava dome based on a stick-slip mechanism. Earth Planet. Sci. Lett., in press Denlinger R.P., Hoblitt R.P. (1999), Cyclic eruptive behaviour of silicic volcanoes. Geology 27 (5): Dirksen O., Humphreys M.C.S., Pletchov P., Melnik O., Demyanchuk Y., Sparks R.S.J., Mahony S. (2006), The dome-forming eruption of Shiveluch Volcano, Kamchatka: Observation, petrological investigation and numerical modelling. J. Volcanol. Geotherm. Res. 155: doi: /j. jvolgeores Edmonds, M., M. Humphreys, E. Hauri, R.A. Herd, G. Wadge, H. Rawson, R. Ledden, M. Plail, J. Barclay, A. Aiuppa, T. Christopher, G. Giudice, R. Guida (2012), Pre-eruptive vapour and its role in controlling eruption style and longevity at Soufrière Hills Volcano, submitted to this issue. Elsworth, D., G. Mattioli, J. Taron, B. Voight, and R. Herd (2008), Implications of magma transfer between multiple reservoirs on eruption cycling, Science, 322(5899), , doi: /science

20 Foroozan, R., D. Elsworth, B. Voight, and G. Mattioli (2010), Dual reservoir structure at Soufriere Hills Volcano inferred from continuous GPS observations and heterogeneous elastic modeling, Geophys. Res. Lett., 37, L00E12, doi: /2010gl Foroozan, R., D. Elsworth, B. Voight, and G. S. Mattioli (2011), Magmatic metering controls the stopping and restarting of eruptions, Geophys. Res. Lett., 38, L05306, doi: /2010gl Hautmann S., Gottsmann J., Sparks R.S.J., Costa A., Melnik O., Voight B. (2009), Modelling ground deformation response to oscillating overpressure in a dyke conduit at Soufriere Hills Volcano, Montserrat, Tectonophysics, Vol. 471: 87-95, doi: /j.tecto Hautmann, S., Gottsmann J., Sparks R.S.J., Mattioli G.S., Sacks I.S., Strutt M.H. (2010), Effect of mechanical heterogeneity in arc crust on volcano deformation with application to Soufrière Hills Volcano, Montserrat, West Indies, J. Geophys. Res., 115, B09203, doi: /2009jb Kavanagh, J.L., R.S.J. Sparks (2011), Insights of dyke emplacement mechanics from detailed 3D dyke thickness datasets J. Geol. Soc., 168, , doi: / Lensky, N.G., Sparks. R.S.J., Navon. O. and Lyakhovsky, V. (2008), Cyclic activity at Soufrière Hills volcano, Montserrat. In Lane, S.J. & Gilbert, J.S., ((eds) Fluid Motions in Volcanic Conduits: A Source of Seismic and Acoustic Signals. Geological Society, London, Special Publications, 307, Linde, A. T., et al. (2010), Vulcanian explosion at Soufrière Hills Volcano, Montserrat on March 2004 as revealed by strain data, Geophys. Res. Lett., 37, L00E07, doi: / 2009GL Maeda I. (2000), Nonlinear visco-elastic volcanic model and its application to the recent eruption of Mt. Unzen. J. Volcanol. Geotherm. Res. 95: 35 47

21 Mattioli, G. S., R. A. Herd, M. H. Strutt, G. Ryan, C. Widiwijayanti, and B. Voight (2010), Long term surface deformation of Soufrière Hills Volcano, Montserrat from GPS geodesy: Inferences from simple elastic inverse models, Geophys. Res. Lett., 37, L00E13, doi: /2009gl Melnik, O., and R.S.J. Sparks, (2005), Controls on conduit magma flow dynamics during lava dome building eruptions, J. Geophys. Res., 110, B02209, doi: /2004jb Muskhelishvili, N. (1963), Some Basic Problems in the Mathematical Theory of Elasticity, Noordhof, Leiden, The Netherlands. Odbert, H.M., G. Wadge, (2009), Time series analysis of lava flux, J. Volcanol. Geotherm. Res., 188 (4), Paulatto, M., C. Annen, T.J. Henstock, E. Kiddle, T.A. Minshull, R.S.J. Sparks, B. Voight (2012). Magma chamber properties from integrated seismic tomography and thermal modelling at Montserrat, submitted to Geochem. Geophys. Geosyst. Roman, D. C., J. Neuberg, and R. R. Luckett (2006), Assessing the likelihood of volcanic eruption through analysis of volcanotectonic earthquake fault-plane solutions, Earth Planet. Sci. Lett., 248, , doi: /j.epsl Sacks, S., A. Linde, D. Hidayat (2011), Change in the Soufriere Hills Volcano magma system, Scientific Conference Soufriere Hills Volcano 15 years on, Montserrat, West Indies, 4-8 April Shalev, E., et al. (2010), Three dimensional seismic velocity tomography of Montserrat from the SEA CALIPSO offshore/onshore experiment, Geophys. Res. Lett., 37, L00E17, doi: /2010gl Voight, B., et al. (1999), Magma flow instability and cyclic activity at Soufriere Hills Volcano, Montserrat, Science, 283, Voight, B., C. Widiwijayanti, G. Mattioli, D. Elsworth, D. Hidayat, and M. Strutt (2010), Magma-sponge hypothesis and stratovolcanoes: Case for a compressible

22 reservoir and quasi-steady deep influx at Soufriere Hills Volcano, Montserrat, Geophys. Res. Lett., 37, L00E05, doi: /2009gl Wang, Z., (2000), Dynamic versus static elastic properties of reservoir rocks, SEG Books, Vol. 19, Pags Wylie J.J., Voight B., Whitehead J.A. (1999) Instability of magma flow from volatiledependent viscosity. Science 285:

23 FIGURE CAPTIONS Fig. 1 Sketch of the investigated system. Modified after Hautmann et al. (2010). Fig. 2. A set of steady state solutions: dependence of discharge rate on chamber pressure for three different far-field stresses: compressional (-10 MPa; dasheddotted line), neutral (full line), and extensional (10 MPa; dotted line). The remaining parameters are set as in Costa et al. (2007b). Fig 3. Time evolution of the chamber overpressures normalized to their maximum values in the upper plots, and of influx into the shallow magma chamber Q in,s (thicker lines) and discharge rate at the surface Q out (thinner lines) in the lower plots, for two cases: (a) weak connectivity between the chambers (a 0 = 100 m) and (b) strong connectivity (a 0 = 600 m). Π denotes the normalized to its maximum value overpressure for the deep magma chamber (thicker lines) and Π denotes the normalized overpressure for the shallow magma chamber (thinner lines). Fig. 4. Dependence of the offset on the intensity of deep magma chamber influx Q in,d for the magma chamber volumes suggested by Hautmann et al. (2010), i.e. V ch,s = 4 km 3, V ch,d = 32 km 3, and different a 0. Fig. 5. Maximum overpressure P d that is reached in the deep chamber (a) and the amplitude of overpressure variation during the cycle ΔP d (b) as a function of Q in,d for V ch,s = 4 km 3, V ch,d = 32 km 3, as in Hautmann et al. (2010), and different a 0. Fig. 6. Time lag between the beginning of the influx into the deep chamber and the start of the eruption for V ch,s = 4 km 3, V ch,d = 32 km 3, as in Hautmann et al. (2010), and different a 0.

24 Fig. 7. Period of pulsations as function of the influx into the deep chamber for V ch,s = 4 km 3, V ch,d = 32 km 3, as in Hautmann et al. (2010), and different a 0. Fig 8. Change in the period of discharge rate variation with respect to the volume of the shallow magma chamber. In this case the volume of the deep magma chamber is fixed at 32 km 3 as estimated by Hautmann et al. (2010). The variation of discharge rate within one period of pulsation is shown in insets. Fig 9. Change in the period of discharge rate variation with respect to the volume of the deep magma chamber. The volume of the shallow magma chamber is fixed at 15 km 3. The patterns of variation of discharge rate within one period of pulsation is shown in the insets. Fig. 10. Variation of discharge rate with time (V ch,s = 15 km 3, V ch,d = 50 km 3 ). Fig. 11. Variation of discharge rate with time (V ch,s = 15 km 3, V ch,d = 50 km 3 ) for a shallow dyke of 300m width. For comparison variation of discharge rate with time for a single chamber geometry and a shallow dyke of 300m (V ch,s = 30 km 3 ) is also shown.

25 Q out cylindrical part Dyke ds (connecting shallow magma chamber with the shallow cylindrical conduit) Dyke d d (connecting deep with shallow reservoir) P Q in,s Chamber Cs (shallow magma chamber centered at ~6 km depth, spherical geometry, volume V ch,s ) P Chamber C d (deep magma reservoir, centered at ~13 km depth, prolate geometry, volume V ch,d ) Q in,d (deep supply)

26 Discharge Rate (m 3 /s) Pressure, P (MPa) s σ =-10MPa ff σ = 0 MPa ff σ =10 MPa ff ΔP Deep Chamber = 10 MPa Extensional Compressional

27 Π Π 6 6 Q ins, Q out (m 3 /s) 4 Π Π Q ins, Q out (m 3 /s) time (days) time (days) (a) (b)

28 offset Q in,d (m 3 /s )

29 Δ (a) (b)

30 t beg (days) Q in,d (m 3 /s )

31 period (days) Q in,d (m 3 /s)

32 period (days) V ch,s (km 3 )

33 Aspect ratio period (days) V ch,d (km 3 )

34 8 6 Q out (m 3 /s) time (days)

35 12 1 chamber 2 chambers 8 Q out (m 3 /s) time (days)