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 5.5107 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: melnik@imec.msu.ru (O. Melnik)8 barmin@imec.msu.ru (A.A. Barmin)8 Steve.Sparks@bristol.ac.uk (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
54
O. Melnik et al. / Journal of Volcanology and Geothermal Research 143 (2005) 53–68
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 Allègre, 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 Allègre (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.
O. Melnik et al. / Journal of Volcanology and Geothermal Research 143 (2005) 53–68
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.
55
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.
56
O. Melnik et al. / Journal of Volcanology and Geothermal Research 143 (2005) 53–68
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.
57
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
58
O. Melnik et al. / Journal of Volcanology and Geothermal Research 143 (2005) 53–68
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.
59
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
60
O. Melnik et al. / Journal of Volcanology and Geothermal Research 143 (2005) 53–68
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
5.5107 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
Soufriére 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
Soufriére 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.
64
O. Melnik et al. / Journal of Volcanology and Geothermal Research 143 (2005) 53–68
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 2107 to 8107 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.
66
O. Melnik et al. / Journal of Volcanology and Geothermal Research 143 (2005) 53–68
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
Allègre (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.
O. Melnik et al. / Journal of Volcanology and Geothermal Research 143 (2005) 53–68
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