Dynamics of magma flow inside volcanic conduits with bubble overpressure buildup and gas loss through permeable magma

Journal of Volcanology and Geothermal Research 143 (2005) 53 – 68 www.elsevier.com/locate/jvolgeores

Dynamics of magma flow inside volcanic conduits with bubble overpressure buildup and gas loss through permeable magma O. Melnika,b,T,2, A.A. Barmina,1, R.S.J. Sparksb,2 b

a Institute of Mechanics, Moscow State University, 1-Michurinskii prosp., Moscow, 119192, Russia Centre for Environmental and Geophysical Flows, Department of Earth Sciences, University of Bristol, Wills Memorial Building, Queen’s Road, Bristol, BS8 1RJ, UK

Received 16 January 2004; accepted 1 September 2004

Abstract Many volcanic eruptions show transitions between extrusive and explosive behaviour. We develop a new generic model that considers concurrence between pressure buildup in the bubbles due to the viscous resistance to their growth and gas escape through the bubble network as they become interconnected. When the pressure difference between bubbles and magma reaches the strength of the material fragmentation occurs. The effect of grain size distribution on the flow in gasparticle dispersion is modelled by two populations of particles which strongly influence the velocity of sound in the mixture. Solutions to the steady-state boundary value problem show non-uniqueness. There are at least two regimes for the fixed parameters in the magma chamber. In the low discharge rate regime, fragmentation does not occur and magma rises with partial gas escape. This regime corresponds to extrusive activity. The upper regime corresponds to explosive activity. The simulations using the parameters defined at the workshop produced the following results for a rhyolitic magma composition: discharge rate 5.5
107 kg/s; fragmentation at depth of 2585 m with magma vesicularity of 0.74; exit gas velocity varies from 200 to 450 m/s depending on the mass fraction of small particles in the fragmented mixture; exit pressure is in the range 1.5 to 3 MPa. Variation of conduit diameter d in the range 40 to 70 m gives a mass flow rate Q which depends on the diameter as d 2.8, less strongly than for the case of viscous flow of Newtonian liquid in a cylindrical pipe where Q~d 4. With the increase in conduit diameter, fragmentation happens later in the flow and conduit resistance remains high. Changes in magma temperature from 700 to 950 8C lead to increase in discharge rate only by a factor of 4 whereas viscosity decreases by more then 8000 times. D 2005 Elsevier B.V. All rights reserved. Keywords: magma; explosive eruption; fragmentation front; bubble overpressure; gas permeability

T Corresponding author. Journal of Volcanology and Geothermal Research. Tel.: +7 95 939 5286; fax: +7 95 939 01 65. E-mail addresses: [email protected] (O. Melnik)8 [email protected] (A.A. Barmin)8 [email protected] (R.S.J. Sparks). 1 Tel.: +7 95 939 5286; fax: +7 95 939 01 65. 2 Tel.: +44 117 954 5419; fax: +44 117 925 3385. 0377-0273/$ - see front matter D 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.jvolgeores.2004.09.010

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1. Introduction A notable feature of many volcanic eruptions of high-viscosity magmas is the sudden transition between explosive and effusive eruptions. The transition can be in either direction and can be quite unexpected. For example, Lascar volcano, in Chile had an intense explosive eruption on 18 and 19 April 1993 with a generation of a 25-km-high column and a numerous fountain-fed pyroclastic flows (Matthews et al., 1997). The explosive activity took place after 9 years of dome extrusion and occasional short-lived Vulcanian explosions. Many long-lived dome extrusions are characterized by alternations between extrusive and explosive activity (e.g. Denlinger and Hoblitt, 1999; Nakada et al., 1999; Sparks and Young, 2002). These transitions are unrelated to the variation of magma composition or gas content; in several cases, magma composition, mineral composition and estimated volatile contents of the source magma do not vary between explosive and extrusive products. Such observations lead to the notion that ascending magmas become permeable due to vesiculation in the upper regions of conduits, allowing exsolving gases to escape (Taylor et al., 1983; Eichelberger et al., 1986). An essential feature of this concept is that the eruptive style is controlled by a competition between gas pressure increase, which leads to conditions for explosive fragmentation (Alidibirov and Dingwell, 1996), and permeability development and gas escape, which inhibits the generation of large overpressures and can allow magma densification and generation of degassed lavas. Several modelling studies have considered the problem of gas escape through permeable magma in volcanic conduits in the context of understanding the transition between extrusive and explosive eruption (Slezin, 1983, 1984, 2003; Barmin and Melnik, 1990; Jaupart and Alle`gre, 1991; Woods and Koyaguchi, 1994; Jaupart, 1998). In Slezin’s papers gas escapes vertically through a partly broken foam formed after fragmentation at fixed porosity (75% bubbles). All the gas above 75% porosity forms a free gas phase that can move through the system of particles. According to Slezin (1983, 2003) when the volumetric concentration of free gas becomes equal to 40%, the transition from a partly broken foam to a gas-particle dispersion occurs. To calculate the relative velocity of the gas, the balance between the drag force and the particle weight is used.

The relative velocity in this case remains finite even if the volume fraction of free gas tends to zero. In Barmin and Melnik (1990) the model of Slezin was extended so that the flow of the gas was treated using Darcy’s law with permeability depending on the volume fraction of free gas. Jaupart and Alle`gre (1991) and Woods and Koyaguchi (1994) assumed gas escape to the conduit wallrocks. In these models large magma column overpressures are required for the extrusive regime to cause a gas flux into permeable wallrocks. Also the permeability of the magma should be higher than the wallrock permeability. Otherwise only the layer of magma near the conduit walls will be strongly degassed. With a large vertical pressure gradient and high magma permeability the main gas flux can also be vertical either in magma column itself or in the permeable region around the conduit (Gonnermann and Manga, 2003). Barmin and Melnik (1993) and Melnik (2000) developed models for conduit flow and obtained multiple steady-state solutions, providing an explanation for the transition between explosive and extrusive regimes. These studies assumed that large overpressure develops in growing bubbles with respect to the surrounding liquid. When the overpressure reaches a critical value fragmentation occurs. Gas escape was not taken into account in these models. Consequently, in the regime with low a discharge rate fragmentation occurred near the conduit exit, but the velocity of the gas-particle dispersion after the fragmentation was too low to produce explosive activity. These models predicted the possibility of catastrophic eruption intensification with the decrease of chamber pressure during magma evacuation from the chamber. The contribution of this paper is to combine the processes of overpressure development in growing bubbles during magma ascent and vertical gas escape (with reduction of overpressure) through the magma. Thus the modelling study develops and builds on previous studies, but for the first time, integrates two key processes that may govern regime transitions between explosive and effusive eruptions. Additionally two particle sizes were considered in the zone of gasparticle dispersion to make simulations closer to natural conditions that are characterized by a wide range of particle sizes after magma fragmentation. The presence of fine particles strongly changes the velocity of sound in the mixture and, therefore, influences conditions at the conduit outlet.

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To illustrate the model and study the sensitivity of the results to the main parameters, a standard set of parameters for rhyolitic magmas was used. The standard set was chosen by participants at the Volcanic Eruption Mechanism Modeling Workshop (November 14–16, 2002—University of New Hampshire, Durham, New Hampshire 03824, USA). The detailed description of this parameter set is given in the introductory paper to this volume (Sahagian, 2005-this issue). Deviations from the standard set of parameters are stated in the text.

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volcanic cloud p = patm or Vg = Vs

2. Mathematical model of magma flow in volcanic conduit gas-particle

2.1. Flow regimes in volcanic conduit The model treats the flow in a volcanic conduit with a transition from homogeneous magma to a gasparticle dispersion as magma ascends through the conduit and pressure decreases. The magma chamber is located in the Earth’s crust and is connected by a cylindrical conduit to the surface (Fig. 1). The magma chamber contains melt, crystals with volume fraction b, and the dissolved gas (assumed to be water) with mass fraction c 0 at pressure p ch (description of all the notations that are used in this paper can be found in Table 1). Flow in the conduit can be divided into three zones. In the lowest zone, the pressure is higher than the nucleation pressure, p nuc. Here, for a given initial concentration of dissolved gas c 0 and solubility coefficient k c ( pNp nuc=c 02/k c2Dp nuc), the flow is homogeneous and the usual model of a viscous liquid can be applied. In the intermediate zone, where pbp nuc, flow of a bubbly liquid takes place. These two zones are divided by a nucleation region in which bubbles are formed with a number density n that is assumed to be constant thereafter. For most of the calculations we assume that nucleation occur heterogeneously and the supersaturation pressure Dp nuc required for the nucleation is of order 1–2 MPa (Hurwitz and Navon, 1994). The effect of homogeneous nucleation on eruption dynamics will be also considered. As magma rises, bubble growth occurs as a consequence of gas exsolution and decompression. Due to viscous resistance, the pressure in the growing bubble, p g, decreases more slowly than the pressure in

dispersion gas escape bubbly liquid homogeneous

fragmentation pg-pm = ∆p*

nucleation p = pnuc

Fig. 1. Schematic view of the flow in volcanic conduit. If pressure in the magma chamber is higher than the nucleation pressure homogeneous magma enters the conduit. After nucleation conditions are reached bubbles start to grow and partly coalesce due to the exsolution of volatiles and decompression. Gas escapes through the system of interconnected bubbles. After fragmentation bubbly liquid transforms into a gas-particle dispersion.

the surrounding melt; p m. This can result in a large overpressure in the growing bubble Dp=p gp m, providing the magma ascent rate and magma viscosity are suitably high. When Dp exceeds a critical value, fragmentation of bubbly media occurs (Barmin and Melnik, 1993, Alidibirov and Dingwell, 1996). A competing process is the coalescence of the bubbles with development of a permeable porous structure and outflow of gas from the magma through a system of interconnected pores. This process reduces gas pressure and can also lead to a collapse of the porosity to form dense magma.

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Table 1 List of notations Symbol

Unit

Description

a c d F gl, F fl

m – m N

g k(a) kc m mk n p Q

m/s2 m2 Pa1/2 – – m3 Pa kg/s

R T V

J kg1 K1 K m/s

Vs x a a f, a l Dp* Dp nuc b k m h q

m/s m – – Pa Pa – – Pa s – kg/m3

x(b)

–

Bubble radii Mass fraction of dissolved water, index b0Q initial value Conduit diameter Interaction forces between gas and large particles and fine and large particles, respectively Gravity acceleration Permeability coefficient Solubility coefficient Mass fraction of fine particles Power law exponent in permeability coefficient k(a) Number density of bubbles Pressure, indexes: bnucQ—nucleation, bchQ—chamber, bgQ—gas, bmQ—melt Discharge rate, indexes: bmQ—melt, bgQ—gas, blQ—large particles, bfQ—fine particles Gas constant Temperature Velocity, indexes: bmQ—melt or large particles, bgQ—gas, V m0 melt velocity without bubbles Speed of sound Vertical coordinate, x nuc—position of the nucleation level Gas volume fraction Volume fractions of fine and large particles Critical overpressure for fragmentation Critical oversaturation for nucleation Crystals volume fraction Friction coefficient Viscosity, indexes: bmQ—melt, bgQ—gas Porosity of large particles Density, indexes: mQ—melt, bgQ—gas, bcQ—crystals, no index—mixture, superscript b0Q—density of the pure phase Einstein correction to the viscosity due to crystals

Fragmentation processes are complicated and in many respects determine the character of an explosive eruption. A narrow region of fragmentation separates a zone of high-density, high-viscous magma from a zone of low-density gas-particle dispersion, the resistance of which is determined by turbulent viscosity of the gas phase and is negligibly small. Therefore, total resistance of the conduit and average weight of a mixture are determined by the position of the fragmentation region.

homogeneous zone x nuc, speed of magma ascent V m and pressure is given analytically:
 klm ðc0 ÞðbÞVm ð1aÞ pch  pnuc ¼ qm g þ xnuc d2 logðlm ðcÞÞ ¼  3:545 þ 0:833lnðcÞ þ

9601  2368lnðcÞ T  ð195:7 þ 32:25lnðcÞÞ

ð1bÞ

qm ¼ q0m ð1  bÞ þ q0c b; xðbÞ ¼ ð1  0:67bÞ2:5 ð1cÞ

2.2. The homogeneous zone If p chNp nuc, magma flows out of the chamber containing no bubbles with constant density and viscosity. Thus, the relation between the length of

Here p ch is the chamber pressure, q m, q m0, q c0 are the densities of magma, melt and crystals, respectively, b is the volume fraction of crystals, g is acceleration due to gravity, l m is the melt viscosity

O. Melnik et al. / Journal of Volcanology and Geothermal Research 143 (2005) 53–68

calculated according to Hess and Dingwell (1996), T is the temperature, x(b) is the Einstein correction coefficient representing the influence of crystals on viscosity and being valid for bV0.5 (Marsh, 1981), d is the conduit diameter or width of a crack, x is the vertical coordinate relative to the chamber where x=0, k is a friction coefficient equal to 32 for a conduit with circular cross-section and k=12 for a dyke. Here further crystallization of magma during ascent is neglected (b=const), which is a consequence of the high ascent velocity of magma during explosive eruptions. However, we note that calculations for the extrusive regime may not necessarily be correct at low flow rates, because crystallization can occur during ascent leading to large viscosity increases (Cashman, 1992; Melnik and Sparks, 1999). 2.3. Bubbly flow zone Writing the system of equations for the bubbly melt, we accept the following simplifying assumptions, in detail justified in Melnik (2000). Magma flow is assumed to be laminar (Re m =q m V m d/ l m~103–10), with the viscosity dependent on concentration of the dissolved gas (Hess and Dingwell, 1996; Eq. (1b)) and crystal content (Eq. (1c)). The flow is assumed to be isothermal, an approximation that has been justified in previous studies (e.g. Wilson et al., 1980; Melnik, 2000). The ascent speed of a rising bubble is negligibly small in comparison with magma ascent velocity. Conduit resistance will be taken in the form of the Poiseuille flow as for an incompressible liquid of constant viscosity. The mass transfer between magma and growing bubbles occurs and is equilibrium, which means that the diffusion delay of gas exsolution from the magma is neglected. This particular assumption may not always be valid. The inertia of the melt around a growing bubble is neglected in comparison with the viscous stress (Navon and Lyakhovsky, 1998). The permeability of the magma is determined by the volume fraction of bubbles a. The dependence of permeability coefficient k(a) on a is taken in a form obtained by processing the results of experiments with cold magma samples (e.g. Eichelberger et al., 1986; Klug and Cashman, 1996). The inertial terms in the momentum equations for the liquid and gas phases are neglected in comparison with forces of conduit resistance.

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We write the system of the equations for the bubbly melt flow taking into account the assumptions listed above:   ð1  aÞ q0m ð1  bÞð1  cÞ þ q0c b Vm ¼ Qm ð2aÞ q0g aVg þ q0m ð1  aÞð1  bÞcVm ¼ Qg ; nVm ¼ n0 Vm0 ð2bÞ  d  kl ðcÞxðbÞVm ð1  aÞpm þ apg ¼  qg  m dx d2   k ðaÞ d apg Vg  Vm ¼  lg dx Vm

 da a  ¼ pg  pm dx 4lm ðcÞ

ð2cÞ

ð2dÞ ð2eÞ

  q ¼ ð1  aÞ q0m ð1  bÞ þ q0c b þ aq0g ; pg ¼ q0g RT ; a ¼

4 3 pffiffiffiffiffi pa n; c ¼ kc pg ; k ðaÞ ¼ k0 amk 3 ð2f Þ

System (2) contains the equations of mass conservation for melt (Eq. (2a)), gas phases and number density of bubbles (Eq. (2b)), the equation of momentum for the mixture as a whole (Eq. (2c)) and Darcy’s law for the gas phase (Eq. (2d)), the RayleighLamb equation (Eq. (2e)) for bubble growth in an infinite volume of incompressible liquid (Nigmatulin, 1987); (Eq. (2f)) gives definitions of parameters included in the equations. In system (2) q 0m , q c0 and q g0 are the densities of pure melt, crystals and gas, respectively, V m and V g, Q m and Q g are velocities and discharge rates per unit area for magma and gas, V m0 is the velocity of magma without bubbles, a and n are volumetric and numerical concentration of bubbles, a is bubble radius, c is mass fraction of the dissolved gas, p g and p m are pressures in the bubbles and surrounding liquid, respectively. The viscosity of the gas phase, l g, is assumed to be constant. Permeability coefficient k 0 can vary from 0 to 1011 m2 and the power law exponent m k is from 2 to 4. 2.4. Gas-particle dispersion zone A prominent feature of gas-particle dispersions formed as a result of destruction of a bubbly magma

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is the wide range of particle sizes. The sizes of most particles vary from a few microns (very fine ash) up to several centimeters (lapilli). Larger particles (volcanic bombs) are present in small amounts and cannot be considered as part of a continuous phase. Presence of a spectrum of particle sizes strongly influences the dynamics of the mixture, in particular, the speed of disturbance (sound) propagation. A similar model which takes into account several particle sizes was also considered in Papale (2001). Let us consider that the gas-particle dispersion contains two populations of particles bfineQ, moving with the speed of gas, and blargeQ. The fine particles do not contain bubbles and their density is equal to the density of magma. The large particles have porosity h equal to the volume fraction of bubbles at fragmentation. The flow of the gas-particle dispersion is modeled by means of a two-speed continuum containing a pseudo-gas mixture, consisting of gas with fine particles, and monodisperse large particles. We take into account an interphase exchange of momentum in the form of the interaction force between gas and particles and interaction force between fine and large particles following Neri and Macedonio (1996). In the above assumptions the system of equations for the gas-particle dispersion can be written as: q0g aVg þ q0g al hVm ¼ Qg ; qm ð1  hÞal Vm ¼ Ql ; qm af Vg ¼ Qf ;

ð3aÞ

qm al ð1  hÞVm

dVm ¼  qm al ð1  hÞg þ Fgl þ Ffl dx ð3bÞ



 dV dp g ¼ þ q0g a þ qm af Vg dx dx    q0g a þ qm af g  Fgl  Ffl

ð3cÞ

qm ¼ q0m ð1  bÞ þ q0c b; a þ al þ af ¼ 1; p ¼ q0g RT

ð3dÞ

Here a, a f, a l are volume fractions of gas, fine and large particles, h is the porosity of large particles, F gl and F fl are the interaction forces between gas and large particles, and between fine and large particles, respectively, Q l and Q f are discharge rates of large and fine

particles. In this system (Eq. (3a)) are the equations of mass conservation for the gas, accounting for free gas and gas in large particles, mass of fine and large particles, (Eq. (3b) and (3c)) are the equations of momentum for large particles and a mixture of the gas and fine particles, Eq. (3d) are definitions of parameters included in the equations. Forces of interphase interaction are defined by formulas given in Neri and Macedonio (1996). 2.5. Fragmentation wave As the transition from a bubbly melt to gas particle dispersion occurs in a narrow region in comparison with the length of the conduit (Melnik, 2000) we consider that fragmentation region is a discontinuity, on which laws of conservation of mass for components (Eq. (4a) and (4b)) and momentum for the mixture as a whole (Eq. (4c)) are satisfied:   0þ þ þ 0þ þ q0 g a Vg ¼ qg a Vg þ qg ab hVm

ð4aÞ

ð1  a ÞVm ¼ as Vgþ þ ab ð1  hÞVmþ

ð4bÞ

 2 pm ð1  a Þ þ pg a þ qm ð1  a ÞVm2 þ q0 g a Vg   2þ ¼ p þ ab qm ð1  hÞ þ q0þ g h Vm   þ þ q0þ a þ q a Vg2þ s m ð4cÞ g

Here the index bQ corresponds to the values in the bubbly melt directly below the zone of fragmentation, and b+Q—to the gas-particle dispersion above. Instead of the law of momentum conservation for one of the components we assume a continuity of the velocity of large particles due to their large inertia V m =V m+. For the evolutionary behaviour of the fragmentation wave it is necessary to express three additional relationships as boundary conditions [see Jeffrey and Taniuti (1964) for the description of the theory]. We assume, as in Barmin and Melnik (1993), that below the discontinuity overpressure in a bubble is equal to the critical Dp* which can be a function of the volume fraction of bubbles in unfragmented magma. We assume porosity of large particles is equal to the volume fraction of bubbles before fragmentation (h=a  ). Some further expansion of the particle is, of course, possible but if

O. Melnik et al. / Journal of Volcanology and Geothermal Research 143 (2005) 53–68

bubbles are strongly connected the gas will prefer to escape from the particle. The mass fraction of fine particles, m, is assumed as a free parameter since no theory yet exists to predict its value. Values of m can be reconstructed by analysis of particle size distribution for particular eruptions. 2.6. Numerical method The pressure p ch and initial concentration of the dissolved gas c 0 are fixed in the magma chamber. If p ch is higher than the nucleation pressure p nuc then the length of the homogeneous zone is calculated using the equation set (1). The equations of bubbly melt (Eq. (2)) are solved below the fragmentation level or before pressure in the magma decreases below atmospheric. At the approach of fragmentation conditions, the system (4) is solved and the parameters of the gasparticle dispersion are calculated. Further we solve the system of Eq. (3) up to reaching a pressure equal to atmospheric at subsonic flow conditions, or the local velocity of sound (the choked flow condition). To calculate the velocity of sound we can rewrite equations in the gas-particle dispersion as follows: A

 T dU ¼ F; U ¼ as ; ab ; Vm ; Vg ; p dx

To solve these equations the condition det(A)p0 should be p satisfied, whichpffiffiffi leads to the equation: ffiffiffi   Vg  A  B Vg  A þ B p0 where coefficients A and B in general form are complicated functions of flow parameters. In the case of the absence of blargeQ particles these coefficients simplify significantly and the equation has a clear physical meaning:    Vg  Vs Vg þ Vs ¼ 0; q0g  Vs2 ¼ RT  a q0m ð1  aÞ þ q0g a When a=1, the speed of sound V s is equal to the speed of sound in a pure gas phase and remains much lower for ab1. Calculations were carried out until V g=0.99 V s because at the choked flow condition the derivative dV g/dx tends to infinity. The calculated value of discharge rate differs less then 1% from the case when calculations were interrupted at V g=0.999 V s.

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Calculations of discharge rate were carried out by the shooting method. With an initial guess of discharge rate the Cauchy problem was solved until the upper boundary conditions were satisfied. Then the calculated length of the conduit was compared with that given and a new guess of discharge rate was estimated. The systems of ordinary differential equations were solved by a standard solver for stiff ODE systems (Deuflhard, 1983). If pressure in the magma chamber varies slowly in comparison with magma ascent time, it is possible to investigate the evolution of eruption with time assuming steady conditions inside the conduit.

3. Results 3.1. Eruption dynamics for the standard set of parameters As a test case for the model we will use the standard set of parameters for the rhyolitic magma discussed in Sahagian (2005-this issue). Parameter values and range of their variations in parametric studies are listed in Table 2. Because the code uses a square root water solubility law (see 1f) we have determined the best fit for the solubility law (Zhang, 1999) with k c=3.98 106 Pa1/2. The corresponding initial concentration of dissolved water is 5.85 wt.% if saturation conditions are assumed inside the chamber located at a depth of 8 km with a pressure of 200 MPa. We fix the critical overpressure for fragmentation to be 3 MPa in most of the calculations or use the critical overpressure as a function of volume fraction of bubbles Dp*=1.3 MPa/a as obtained experimentally in Spieler et al. (2004). The influence of this parameter on eruption dynamics will be investigated later. Fig. 2 represents the relationship between discharge rate and chamber pressure for the standard parameter set and different magma permeability coefficients (k 0). Chamber pressure can decrease below the lithostatic pressure (200 MPa for the standard parameter set) as material being erupted and magma chamber being emptied. Prior to the eruption p ch might be higher than lithostatic by 10–30 MPa to provide an energy for initial rapture of the rocks and formation of a conduit. For a fixed chamber pressure there can be up to three stationary solutions, with the discharge rates differing

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Table 2 Parameters for the simulations Parameter Default value

Range in the paper

c0 d dp

0.0585 50 m 200 Am

0.01–0.08 40–70 m –

k0

1011 m2

kc L mk

3.98 106 Pa1/2 8 km 3.5

p ch Dp*

200 MPa 3 MPa

Dp nuc

2 MPa

T b q0c

850 8C 0 2700 kg m3

q0m

2200 kg m3

Description

Initial water content Conduit diameter Particle size for large particles 0–1010 m2 Permeability coefficient – Solubility coefficient – Conduit length – Power law exponent in permeability coefficient 10–230 MPa Chamber pressure 1–3 MPa Critical overpressure for fragmentation 2, 100 MPa Critical oversaturation for nucleation 700–950 8C Temperature 0–0.4 Crystal content – Density of pure crystal phase – Density of pure melt phase

by orders of magnitude. In the solutions with smaller discharge rates (bottom curves) fragmentation does not occur and bubbly magma with forward gas escape reaches the surface; this is the extrusive regime. For solutions with high discharge rate (top curves) the flow of the gas-particle dispersion has an exit velocity equal to the local speed of a sound as it exits from the conduit; this is the explosive regime. The transition from the bottom to the top branch cannot be made by a continuous change of parameters along the stationary solution. The right boundary of the extrusive regime (e.g. point A) is the point at which fragmentation conditions are met in the conduit; the left boundary of the explosive regime (e.g. point B) is a point at which fragmentation stops. The position of these points depends on the magma permeability. For higher magma permeability, the transition to explosive regime shifts to higher discharge rates due to more efficient gas escape from the ascending magma. Discharge rate in the explosive regime weakly depends on the magma permeability because gas escape is a slow process. The choice for the critical bubble overpressure is unimportant in the explosive regime because near the fragmentation level bubble overpressure grows rap-

idly. The choice of the particular value of critical overpressure changes the position of the fragmentation level and, therefore, discharge rate insignificantly (see Fig. 2, calculations with Dp* equal to 1 and 3 MPa). At low discharge rates in the extrusive regime discharge rate decreases with the increase in chamber pressure. At higher chamber pressures the density of the magma feeding into the conduit from the chamber increases and remains high due to efficient gas escape through the magma. Therefore, discharge rate must decrease to reduce the conduit resistance at increasing magma chamber pressure. For the non-permeable magma reduction in density with decrease in chamber pressure is the only effect so that discharge rate is a monotonic descending function of descending chamber pressure. The model may not be strictly applicable to the lower part of the extrusive regime because it neglects magma crystallization during ascent. This process can become significant at low ascent rates (Melnik and Sparks, 1999). Calculated discharge rate for 200 MPa chamber pressure (saturation pressure at 8 km depth) is 5.5
107 kg s1, fragmentation occurs at a depth of 2585 m with magma vesicularity after fragmentation equal to 74%. Exit gas velocities are calculated to range 200 to 450 m/s and exit pressures are from 1.5 to 3 MPa, depending on the mass fraction of fine particles in the fragmented mixture. As the chamber pressure decreases, fragmentation occurs deeper (Fig. 2) because, due to increasing magma viscosity, the bubble overpressure builds up quicker. The conduit at low chamber pressures is mostly filled by the gas-particle dispersion with low weight and resistance. Therefore, p ch can reach very small values and caldera collapse is plausible for the chosen set of the governing parameters. Fig. 3 shows variations in volume fraction of bubbles at fragmentation with chamber pressure. This parameter can be directly compared with observations if we assume that there is no further particle expansion after fragmentation. This assumption will be valid for high-viscosity magma with partly interconnected bubbles when gas can easily escape from the particles after fragmentation. Gas volume fraction after fragmentation increases with decrease in chamber pressure but varies in a relatively narrow range (from 70 to 90%) meanwhile the volume fraction of bubbles at the top of the chamber (dashed line) has much larger

O. Melnik et al. / Journal of Volcanology and Geothermal Research 143 (2005) 53–68

m3s-1 (DRE) 104

explosive regime 1 MPa

discharge rate (kg s-1)

B 3 MPa 6

10

n

3 MPa h pt de

3000

1 MPa

tio

500

4000

a nt

e

gm

fra

100

A

5000

5

10

extrusive regime

10 104

6000

1

fragmentation depth (m)

107

61

7000

103 0 50

-11

10 100

10-10 150

8000 200

chamber pressure (MPa) Fig. 2. Discharge rate (solid lines for explosive regime and short-dashed lines for extrusive) and fragmentation depth (long-dashed lines) versus chamber pressure for the standard set of parameters for rhyolitic magma composition. For the explosive regime two values critical overpressures (1 and 3 MPa) are presented for k 0=1011 m2. Different curves for the extrusive regime correspond to different values of magma permeability coefficients k 0, as labeled on the figure. For fixed chamber pressure up to three steady-state regimes are possible.

variations. Values corresponding to different critical bubble overpressures differ only by about 10% because bubble overpressure develops very rapidly and fragmentation occurs nearly at the same pressure in the conduit (see Fig. 2). As chamber pressure decreases discharge rate also decreases and the gas pressure follows the pressure in the melt phase more closely. Therefore, fragmentation occurs at lower pressures and higher viscosities leading to higher volume fractions of the bubbles. Decrease in temperature leads to higher magma viscosity and overpressure in growing bubbles increases more rapidly. This reduces volume fraction at fragmentation to ~60% for T=780 8C and to 65% for T=800 8C for the standard set of parameters. Volume fraction of bubbles increases up to 85% for T=950 8C. 3.2. Extrusive regime The extrusive regime for the standard parameter set can occur at chamber pressures much lower then

lithostatic and can be possible as a result of significant pressure reduction in the explosive regime when very low values of p ch are reached. Maximum discharge rate corresponding to the transition from extrusive to explosive regime is a strong function of magma permeability varying from 115 m3 s1 (DRE) for nonpermeable magma up to 212 m3 s1 for k 0=1011 m2. Unrealistically high values of volume fraction of the bubbles (up to 95%) are reached at the transition point for Dp*=3 MPa. These values are much higher than observed discharge rates for the lava dome building eruptions. For example, for Mount St Helens peak discharge for the dome growth was recorded at ~20 m3 s1 (Swanson and Holcomb, 1990) and for the Soufrie´re Hill Volcano, Montserrat, the maximum value was about 10 m3 s1 (Sparks et al., 1998). Fig. 4 shows the relationship between the discharge rate along the extrusive regime curves presented at Fig. 2 and the gas overpressure at the top of the conduit for different permeability coefficients k 0. As discharge rate increases the gas overpressure also increases. The critical overpressure of 3 MPa is used

O. Melnik et al. / Journal of Volcanology and Geothermal Research 143 (2005) 53–68

1

gas volume fraction

0.8

3 MPa

∆p*=1.3/α (MPa)

0.6

1 MPa

0.4

0.2

0

50

100

150

200

chamber pressure (MPa) Fig. 3. Variation of the volume fraction of gas at fragmentation with chamber pressure for 3 values of fragmentation thresholds (1 and 3 MPa and variable with the volume fraction of bubbles according to Spieler et al., 2004) and k 0=1011 m2. Values of volume fraction of gas at the bottom of the conduit are shown with long-dashed curve.

for this calculations as the end of extrusive regime. If a different value of the gas overpressure is chosen the transition point to the explosive regime will slide along the calculated curves. Critical gas overpressure taken in the form of Spieler et al. (2004) (shown by dashed line with crosses) leads to the decrease in discharge rate corresponding to a transition point at 7.6–19 m3 s1 for 0bk 0b1011 m2. At these discharge rates a series of Vulcanian explosions occurred at the Soufrie´re Hill Volcano, Montserrat (Druitt et al., 2002) and several explosive eruptions occurred on Mount St Helens (Swanson and Holcomb, 1990). 3.3. Parameters distribution along the conduit The pressure profile for the basic set of parameters and gas and melt pressures before the fragmentation (in inset) are shown on Fig. 5. As already shown in previous studies (Papale, 1999, 2001; Melnik, 2000) the pressure gradient increases strongly before fragmentation mainly due to the rapid increase in viscosity close to the fragmentation level and consequent increase in conduit resistance. Rapid pressure and velocity changes lead to development of high bubble overpressures less than 50 m below the fragmentation

level. At fragmentation, according to the chosen criterion, the overpressure reaches its critical value. Immediately above the fragmentation level, pressure in the gas-particle dispersion is closer to the gas pressure before the fragmentation because highpressure gas is released after the fragmentation of the bubbly melt (see the inset in Fig. 5). The gas velocity profile is shown on Fig. 6. Velocity of the condensed phase differs negligibly from the gas velocity, except in a narrow region (~100 m) prior to the fragmentation where, due to a high pressure gradient, the relative velocity becomes large (see Eq. (2d)). Also near the conduit exit gas velocity increases very rapidly due to reaching the choked flow conditions at the top of the conduit [dV g/dx~(V s V g)1]. Due to the inertia of large particles, particle velocity changes slowly and there is a significant velocity disequilibrium between gas and large particles at the conduit exit. The velocity of sound is a strong function of the mass fraction of fine particles m. In this model the velocity of sound (see the frame at Fig. 6) is equal to the velocity of sound of a pure gas [V s=(RT)1/2=720 m s1] in the case of no fine particles. For m=0.05 velocity of sound is signifi-

bubble overpressure (MPa)

62

2 0 10- 11 10-10

∆p*=1.3/α (MPa)

1

1

10

100

discharge rate (m3s-1) Fig. 4. Changes of bubble overpressure as a function of discharge rate for the extrusive regime with different values of magma permeability coefficients k 0 as shown on Fig. 2. Transition points to the explosive regime in the case of Dp*( a) (Spieler et al., 2004) are shown with dashed line and crosses. Discharge rate corresponding to the transition to explosive regime is a strong function of critical bubble overpressure Dp*.

O. Melnik et al. / Journal of Volcanology and Geothermal Research 143 (2005) 53–68

120

24

20

16

xf 2580

80

2560

2540

depth (m)

40

6000

4000

2000

0

depth (m) Fig. 5. Pressure profile inside the conduit for the basic set of parameters. Gas and melt (dashed) pressures before fragmentation are shown in the inset. Significant pressure disequilibria occur only about 20 m below the fragmentation level. Due to the high-pressure gas release after fragmentation mixture pressure is closer to the gas pressure than to the melt pressure prior to the fragmentation.

cantly smaller (460 m s1) and decreases to 163 m s1 for m=0.95. The exit velocity of the gas is equal to the velocity of sound and, therefore, decreases with the increase in m. Exit pressure increases with increase of m from 1.5 to 4 MPa. This leads to changes in gas volume fraction at the conduit exit from 91 to 81%. After the expansion of the jet inside the crater the resulting mixture velocity can become supersonic. 3.4. Homogeneous vs. heterogeneous nucleation Fig. 7 represents the influence of bubble nucleation dynamics on the discharge rate. Two cases were considered. For heterogeneous nucleation, bubbles start to grow when the pressure drops below the saturation pressure by the value of 2 MPa (Hurwitz and Navon, 1994). Homogeneous nucleation requires much larger supersaturation. For this calculation supersaturation pressure is taken to be 100 MPa (Mangan and Sisson, 2000). If the chamber pressure is less then nucleation pressure in both cases bubbles will nucleate inside the magma chamber and from the point of view of the conduit flow model there will be no differences in input conditions. Therefore,

500

200

160

120

4 400 3

300

200

2

exit pressure (MPa)

pressure (MPa)

160

for p chb115 MPa for the chosen set of parameters, the solutions do not depend on the nucleation mechanism. In the case of homogeneous nucleation a large homogeneous flow zone appears in the conduit at higher chamber pressures. The presence of a homogeneous zone increases the overall weight of the magma in the conduit, but at the same time decreases viscous friction because magma contains a large amount of dissolved gas for a much larger part of the conduit. When the critical supersaturation is reached the volume fraction of bubbles immediately increases to its value at the nucleation pressure. The value of a is very closed to the value for the heterogeneous nucleation condition (see the inset on Fig. 7). Later on the growth of bubbles occurs in a very similar conditions and, therefore, fragmentation occurs nearly at the same level. Resulting discharge rates are also very similar for the standard parameter set. The current model oversimplifies the nucleation kinetics and also assumes equilibrium mass transfer to growing bubbles. In natural situations delay in mass transfer is likely to lead to smaller values of a after homogeneous nucleation and, therefore, delayed

exit velocity (m s-1)

pressure (MPa)

28

gas velocity (m s-1)

200

63

100 0

80

0.2

0.4

0.6

0.8

1

mass fraction of fine particles

40

0 8000

6000

4000

2000

0

depth (m) Fig. 6. Profile of gas velocity inside the conduit for the basic set of parameters. Particle velocity differs significantly only at the conduit exit (see inset). In the inset: influence of mass fraction of fine particles on exit gas (solid) and particles (short-dashed line) velocities and exit pressure (dashed line). Presence of fine particles significantly reduces the velocity of sound in the mixture and, therefore, exit gas velocity for chocked flow conditions.

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1000

discharge rate

2000

40

3000

20

0 8000

107

6000

4000

depth (m)

f ra

e gm

nt

o at i

n

de

pt

h

4000

5000

depth (m)

discharge rate (kg s-1)

α (%)

60

6000

nucleation depth 106

7000

8000 50

100

150

200

chamber pressure (MPa) Fig. 7. Discharge rate, fragmentation and nucleation depths as a function of chamber pressure for heterogeneous (solid) and homogeneous nucleation of bubbles (dashed lines). In the frame profiles of volume fraction of bubbles for p ch=200 MPa are shown. Much shorter zone of bubbly flow corresponds to homogeneous nucleation case.

fragmentation and smaller discharge rate. If required supersaturation pressures are very high nucleation and magma fragmentation can happen in a narrow region.

Therefore, the pressure in the growing bubbles follows the pressure in the melt phase more closely and the overpressure required for fragmentation is achieved later.

3.5. Influence of conduit diameter on eruption dynamics

3.6. Influence of the initial magma water content

The most uncertain parameter in the basic set is the conduit diameter. Fig. 8a represents the relation between the conduit diameter, discharge rate and the fragmentation level position. As the conduit diameter increases discharge rate also increases. In the classical Poiseuille solution for the cylindrical pipe discharge rate is proportional to d 4. Here the best fit for the model calculation results shows that Q~d 2.8. The cause of the slower increase in discharge rate with respect to increase in conduit diameter is that fragmentation occurs later in the flow and overall conduit resistance remains high for large conduit diameters. For larger d conduit resistance is lower and pressure in the melt phase decreases slower.

Fig. 8b shows discharge rate and the fragmentation level position as a function of initial concentration of dissolved gas, c 0, a for a chamber pressure equal to 200 MPa. For low values of c 0 only the extrusive regime with low discharge rate is possible because the viscosity of magma is extremely high. At higher values of c 0 both explosive and extrusive regimes (short dashed line) exist. If c 0N3.5 wt.% than only explosive eruption can occur for this set of parameters. The fragmentation depth increases slightly as magma becomes more volatile rich because increase in discharge rate makes bubble growth more disequilibrium. Porosity at fragmentation level decreases from 77% to 69% with decrease in c 0 from 8 to 2.5 wt.%.

2400

0.8

2800 0.4

(a)

65

2000 explosive 2200

0.1

2400 0.01

2600 extrusive

(b)

fragmentation depth (m)

Q ~ d 2.8

discharge rate x 108 (kg s-1)

1.2

2000

fragmentation depth (m)

discharge rate x 108 (kg s-1)

O. Melnik et al. / Journal of Volcanology and Geothermal Research 143 (2005) 53–68

2800

0.001 50

60

70

0.02

2000 0.6 4000 0.4

(c)

6000

discharge rate x 108 (kg s-1)

0.8

0.2 700

0.04

0.06

0.08

water content

fragmentation depth (m)

discharge rate x 108 (kg s-1)

conduit diameter (m) 0.5

2600 0.4 3200 0.3 3600

0.2 0.1

4000

(d)

fragmentation depth (m)

40

0 750

800

850

900

950

temperature (°C)

0

0.1

0.2

0.3

0.4

crystal content

Fig. 8. (a) Influence of conduit diameter on discharge rate (solid) and fragmentation depth (dashed line). Dependence of discharge rate on conduit diameter is weaker then for the case of Newtonian incompressible liquid in the pipe. (b) Influence of initial magma water content on discharge rate (solid) and fragmentation depth (dashed line). For low water contents extrusive regime is possible (short dashed line). (c) Influence of magma temperature on discharge rate (solid) and fragmentation depth (dashed line). Viscosity variations due to the temperature changes are more than two orders of magnitude but increase in discharge rate is less than one order. Very shallow fragmentation occurs for high magma temperatures. Above 950 8C explosive regime is not possible. (d) Influence of magma crystal content on discharge rate (solid) and fragmentation depth (dashed line). For b=0.4 magma viscosity increases by more than 20 times but due to the deepening of the fragmentation level decrease in discharge rate is only by a factor of 5.

3.7. Influence of magma temperature and crystal content Fig. 8c shows the same parameters for different magmatic temperatures. Changes in temperature from 700 to 950 8C result in changes in viscosity by a factor of around 104 but consequent changes in discharge rate are only from 2
107 to 8
107 kg s1. For lower temperatures the fragmentation condition in ascending magma is reached much earlier due to the high viscosity of magma and, therefore, high viscous resistance to bubble growth. At temperature

of 950 8C fragmentation occurs less than 300 m from the top of the conduit. For higher temperatures fragmentation conditions are not satisfied inside the conduit. Decrease in the fragmentation level with increasing temperature leads to a higher average weight of the mixture and, therefore, the increase in discharge rate is not as might be expected as a consequence of lower viscosity. Similar results occur for the variation of initial crystal content in the magma (Fig. 8d). Increase in crystal content to 40% leads to increase in viscosity by a factor of 21 but decrease in discharge rate by a factor less then 7.

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4. Conclusions and discussion The model developed in this paper has two novel features: simultaneous gas filtration through the system of interconnected bubbles and overpressure build-up due to the viscous resistance to bubble growth. The model provides an explanation of the abrupt transition between extrusive and explosive eruption regimes based on bubble growth dynamics. Predicted parameters for both regimes are in the range of observed values. Wider parametrical studies are necessary to check the sensitivity of the results. Similar non-unique solutions were earlier recognised by Slezin (1983, 1984, 2003), Jaupart and Alle`gre (1991) and Woods and Koyaguchi (1994) with other assumptions on the fragmentation mechanism and gas outflow from the magma. Dependence of discharge rate on the chamber pressure essentially differs topologically from these earlier studies. The solution between explosive and extrusive regimes is absent, whereas in Slezin (1983, 1984, 2003) and Woods and Koyaguchi (1994) the dependence is Sshaped. Transition to a catastrophic explosive eruption from a moderate explosive regime found in Barmin and Melnik (1993) does not occur because gas escape through the magma leads to later fragmentation of the magma and fragmentation level descends slower with a decrease in chamber pressure. The model produces a dependence of discharge rate on conduit diameter, temperature, water and crystal contents that is much weaker than for a simple viscous conduit flow. This is due to feedbacks between vesiculation, viscosity and fragmentation criteria which tend to counteract one another as these parameters change. Of a particular interest for explosive eruptions is the result that, for rhyolite magmas, hot and dry magmas will have fragmentation at shallow levels whereas cold wet magmas will fragment much deeper. Porosity of the magma at fragmentation varies in a narrow range (65 to 80%). It only slightly depends on the particular choice of the critical overpressure because the overpressure grows very rapidly prior to the fragmentation and bubble expansion is limited due to large viscous resistance to the bubble growth. There are several limitations of the current model that should be overcame in future. First, the model assumes equilibrium mass transfer between the melt

and growing bubbles. For high ascent velocity this assumption is not valid (Mangan and Sisson, 2000). The model, therefore, overestimates the amount of gas that takes part in explosive eruptions. Second, at a current stage the model is isothermal. There are several processes that can contribute to the temperature variation in ascending magma including viscous dissipation of heat, latent heats of water exsolution and crystallization, gas expansion and heat loss to the surrounding wallrocks. Temperature variation will lead to changes in magma rheology and, therefore, to changes in pressure loss and fragmentation parameters. Due to radial temperature variation the Poiseuille formula for the friction force is also not strictly valid (Costa and Macedonio, 2003). Third, the model assumes a single nucleation event and cannot explain the range of bubble size distribution commonly observed in natural magma samples (Cashman and Mangan, 1994). Fourth, there is an important and not yet theoretically solved problem of interphase interaction in concentrated multiphase systems. Because the volume fraction of the gas phase changes from zero to nearly one there is a significant flow region where concentration of both phases are comparable. This problem includes issues on bubble interaction, coalescence and permeability development in ascending magma together with momentum and energy exchange in concentrated gas-particle dispersions. Some of the issues discussed above are already addressed in the models presented in this volume. Further development of the model together with a new knowledge of physical properties of magma and geometrical constraints of volcanic systems will allow to gain a better understanding of this complicated natural phenomena.

Acknowledgements This work was supported by grants by Russian Foundation for Basic Research (RFBR 02-01-00065), NERC grant GR3/13020 and EC INTAS (01-0106) and EC MULTIMO. RSJS acknowledges the Royal Society-Wolfson Award. We would like to thank the reviews: A. Folch, A. Prusevich, J. Blower, G. Wadge for interesting suggestions that allowed to improve the readability of the paper.

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