Chapter 16
Transient Models of Conduit
Flows during Volcanic
Eruptions
Oleg Melnik, Stephen Sparks
Magma flow in volcanic conduits involves complicated physico-chemical transformations during ascent. It includes gas exsolution, bubble nucleation and
growth, gas escape from the magma, and magma fragmentation (in the case
of explosive eruptions). These changes are accompanied by large changes in
the rheological properties of magma. The structure of the flow can change
from homogeneous liquid flow at depth to gas-particle dispersion flow in the
upper part of a conduit. There are two distinct zones of the flow: the zone
where the liquid is a continuous phase and flow is mainly controlled by viscous resistance, and the zone with continuous gas phase where the flow is
dominated by inertia. These zones are separated by a fragmentation front,
whose position must be determined during the solution of the flow dynamics.
This makes modelling of conduit flows a difficult problem and requires strict
constraints on the accuracy and stability of the numerical method.
The bibliography related to the modelling of conduit flow processes during explosive eruptions contains many tens of papers. Good overviews can
be found in Mader (this volume), Melnik (2000), Papale (1998), Sahagian
(2004), Slezin (2003), Sparks et al. (1997) and Woods (1995). Most of the
models presented in the literature describe conduit flow during volcanic eruptions as a 1-dimensional steady-state process based on the assumption that
the length of the conduit is much larger than its radii and timescales of parameter variations is much longer than the resident time of an individual
1
parcel of magma inside the conduit. These models are described by systems
of ordinary differential equations (ODE) which relate observable variables
(such as discharge rate, exit pressure and velocity) to the processes occurring
inside the conduit. In recent years these models have been further developed
to allow more than one dimension (transient 1D and 2D models). These
models are described by systems of partial differential equation (PDE) and
allow a consideration of the evolution of the eruption with time. The goal
of this paper is to discuss approaches to modelling of the transient flows in
volcanic conduits and show the advantages of using PDE systems in studies of conduit flow dynamics. Only a few models that consider transient
evolution of the eruption are based on the solution of PDE systems. The
first model was presented by Ramos (1995). He considered a homogeneous
(one-velocity) flow in the conduit after disruption of a lava plug at its top.
Several numerical methods were suggested and compared with the stabilized
solution from Dobran (1992). In Barmin and Melnik (1996) a model for nonhomogeneous transient flow was considered. The evolution of the explosive
eruption intensity due to changes in magma chamber pressure was explored.
Several transient models have been published in a special volume of Journal
of Volcanology and Geothermal Research as a result of the Volcanic Eruption Mechanism Modelling Workshop (November 14-16, 2002 - University of
Hew Hampshire, Durham, New Hampshire, USA). Proussevitch and Sahagian (2004) consider simultaneous solution of the bubble growth equations,
diffusion equation for volatiles around growing bubbles, and transient conduit
flow equations that describe variations in pressure-temperature conditions.
In Dufek and Bergantz (2004) the influence of closure models for the granular
stress in a gas-particle dispersion on the evolution of eruption is considered
based on a 2D transient code. In Mason et al. (2004) the role of the intensity
of diffusion on transient eruption dynamics is considered, based on a quasistatic solution of the diffusion equation. Barmin et al. (2003) and Starostin
et al. (2004) account for the magma interaction with water from the aquifer
surrounding the conduit during phreatomagmatic eruptions.
Modelling studies of conduit flows during lava dome building eruptions
have been presented in Barmin et al. (2002), Denlinger and Hoblitt (1999),
Melnik and Sparks (1999, 2002), and Wylie et al. (1999). In Barmin et
al. (2002) and Melnik and Sparks (1999, 2002) the decompression induced
crystallization during magma ascent is investigated.
In this paper, approaches to modelling of transient conduit flow will
be considered for both explosive (based on Mason et al. 2004) and effusive
eruption (based on Melnik and Sparks 2004) styles of activity. Time series
of data that are obtained from numerical calculations can be analysed statistically similar to natural datasets. Comparison of statistically meaningful
parameters obtained from modelling and observations will allow verification
2
of model outputs and the use of models as forecasting tools.
16.1
Transient Model of Conduit Flow
during Explosive Eruptions
Active silicic volcanoes are commonly in a state of slow magma ascent which
feeds either lava domes or subsurface intrusions (cryptodomes, see section
1.3 in Mader, this volume). The conduit is filled with magma which ascends slowly and decompresses, resulting in nucleation and growth of bubbles
driven by diffusion of the gas dissolved in the melt. The extruding dome or
cryptodome maintains a pressure difference between the magma ascending
in the conduit and atmosphere, which can be related to variations in dome
height and degassing processes that cause rheological stiffening (Melnik and
Sparks 1999; Stasiuk et al. 1993). Lava dome extrusion or cryptodome
intrusion can turn to explosive eruption by a sudden decompression. A common cause of a sudden decompression is dome collapse, as at the Soufrire
Hills volcano, Montserrat (Robertson et al. 1998), and edifice collapse, as at
Mount St. Helens in 1980 (Voight et al. 1983). After collapse the pressure
at the top of the conduit decreases rapidly. A rarefaction wave propagates
down the conduit reducing the pressure. Bubbles in the magma respond to
the pressure change by expanding, but viscous resistance results in an excess
pressure. If this overpressure exceeds a critical threshold for fragmentation,
then explosive disruption can be initiated. Fragmented bubbly liquid forms a
gas-particle dispersion. The fragmentation front propagates down following
the rarefaction wave and the gas-particle dispersion zone expands. In the gas
particle dispersion zone the fragmented material accelerates easily because
the mixture viscosity is negligible. We here develop a transient model to describe the dynamics of an explosive eruption, which is triggered by a sudden
decompression.
16.1.1
Governing Equations
The present model develops the transient model of Melnik and Sparks (2002)
which describes a flow generated by a rapid decompression at the top of
the conduit and considers two end-member cases for the intensity of mass
transfer between melt and growing bubbles: one in which diffusion into the
growing bubbles is fast enough to maintain the system close to equilibrium,
and the other in which diffusion is so slow that bubbles, that existed prior
to the onset of explosive activity, expand without further significant mass
transfer. Dimensional analysis provides criteria for these cases based on the
Peclet number, P e , which is the ratio of the characteristic time of diffusion
3
of dissolved gas and the characteristic time of decompression (Navon and
Lyakhovski 1998).
If P e ≫ 1 the mass transfer between the melt and bubbles is negligible. This situation might be appropriate to the initial stages of an explosive
eruption when the fragmentation front propagates with a velocity typically
in the range of tens to over one hundred m/s (Spieler et al. 2004). If P e ≪ 1
diffusion maintains the system close to equilibrium. This case describes a
situation when the velocity of the fragmentation wave is sufficiently slow
so that magma beneath the fragmentation front can grow bubbles close to
equilibrium.
Our model assumes a single nucleation event, and neglects interaction
between bubbles and variations of the mixture temperature and crystal content. We assume that the relative velocities between bubbles and liquid and
between particles and gas are small in comparison with the mixture velocity.
We also neglect changes in crystal content due to microlite crystallization as
this process is very slow and is not expected to occur on the time scale of an
explosive eruption. With these assumptions the system of flow equations for
both the bubbly liquid and gas-particle dispersion zones can be written:
∂
∂
ρm +
ρm v
∂t
∂x
∂
∂
ρg +
ρg v
∂t
∂x
∂
∂
ρv 2 + P
ρv +
∂t
∂x
ρ
= −λJ
(16.1)
= λJ
(16.2)
32µav v
d2
ρg + ρd + ρm + ρc ;
P
α
RT
(1 − α)(1 − β)cav ρ0m
(1 − α)(1 − β)(1 − cav )ρ0m
(1 − α)βρ0c
= −ρg − λ
(16.3)
=
(16.4)
ρg =
ρd =
ρm =
ρc =
Here ρ are densities (subscript: g− exsolved gas; d− dissolved gas; m− melt;
c− crystal; no subscript- mixture; superscript: 0− phase; no superscriptbulk), d is the conduit diameter, v is the mixture velocity, P is the mixture
pressure, α is the gas volume fraction, β is the crystal content, T is the temperature, R is the gas constant, t is the time and x is the vertical coordinate
with x = 0 corresponding to the top of the magma chamber and positive in
the upward direction. The coefficient λ indicates the flow regime: λ = 1 for
bubbly liquid, and λ = 0 for gas-particle dispersion. The subscript ”av” relates to the dissolved water concentration, c, and viscosity, µ, averaged over
the melt shells near the bubbles. The system consists of continuity equations
for the melt phase (16.1) and exsolved gas component (16.2), the momentum
4
equation for the mixture as a whole (16.3), and equations of state (16.4). The
continuity equations for exsolved gas and melt accounts for a mass flux, J,
due to diffusion. The momentum equation takes into account gravity forces
and conduit resistance. Because the flow is laminar (typical Reynolds numbers are less than 1000) the conduit resistance is proportional to the mixture
viscosity µav and the ascent velocity which is known as a Poiseuille solution
(Poiseuille, 1848).
16.1.2
Diffusion of Volatiles into Bubbles
In order to describe diffusion we use the quasi-steady bubble growth model
(Lensky et al. 2004; Navon and Lyakhovski 1998) which gives an exact distribution of concentration around a solitary bubble. For multiple bubbles we
assume that each bubble is surrounded by a spherical shell of melt, which
provides the bubble with water and expands according to mass conservation
(see Lensky et al. 2004, fig. 1, for details). Close to the bubble surface the
water concentration is at equilibrium. A diffusion equation with corresponding boundary conditions can be written as:
∂c
1 ∂ ∂c
= D 2 r2
∂t
r ∂r ∂r
(16.5)
√
c(a) = Cf P
c0 ρ0m
=
ρ0g
+
4πρ0m
Z
S
r 2 c(r)dr
a
4
πna3 = α
3
4
πnS 3 = 1
3
(16.6)
Here c is the mass fraction of dissolved gas (water), c0 is the initial concentration, D is the diffusion coefficient, Cf is the water solubility coefficient in a
silicate melt. Equations (16.6) specify boundary conditions for the diffusion
equation (16.5) at the bubble radius a (equilibrium concentration of water)
and mass conservation of water in the whole shell with radius S . The mass
flux, J, into a bubble is related to the concentration gradient on its border:
2
0 ∂c J = 4πa nDρm
(16.7)
∂r r=a
Here n is a number density of bubbles per unit volume of the mixture. When
the Peclet number is small, the partial time derivative in the right hand
side of (16.5) can be omitted, and an approximate analytical solution for the
5
concentration distribution around the bubble can be written down (Navon
and Lyakhovski 1998):
c(r, t) = C1 (t) + C2 (t)
1
r
(16.8)
Here functions C1 (t) and C2 (t) have to be determined from the boundary
conditions (16.6). This solution allows calculating the concentration gradient
and, therefore, the mass flux into the bubble.
16.1.3
Fragmentation Criteria
Following Barmin and Melnik (1993), Melnik (2000) and Melnik and Sparks
(2002) we assume that magma fragmentation occurs when the overpressure
in growing bubbles, due to the viscous resistance to the growth, reaches a
critical value or fragmentation threshold (Spieler et al. 2004). The evolution
of gas overpressure is controlled by the Raleigh-Lamb equation, which in the
case when inertia terms are negligibly small, and has the form (Navon and
Lyakhovski 1998):
4µef f ∂a
∂a
Pg − Pm = ∆P =
+v
a
∂t
∂x
Z S
µ(c(r))
dr
µef f = 3a3 ς
r4
a
−2.5
β
ς =
1−
β∗
(16.9)
A coefficient ς accounts for the influence of crystals on the effective viscosity µef f , β∗ is the critical concentration of crystals at which they become
interconnected. The value of β∗ depends on the crystal shapes and size distribution. The model with two pressures in the mixture (Melnik 2000) shows
that ∆P ≪ ∆Pcr for most of the flow and becomes comparable with ∆Pcr
only prior to the fragmentation. This relationship allows us to consider conduit flow with a one-pressure model.
16.1.4
Initial and boundary and conditions
We solve the transient problem in a conduit of length L + hd , where hd is the
height of the dome prior to the collapse. We use the steady-state solution
for an extrusive flow for x ∈ [0, L] with P (L) = Pexit > Patm as an initial
condition. The region x ∈ [L, L + hd ] is treated as a gas-particle dispersion
zone with atmospheric conditions fixed at P (L + hd ) = Patm .
6
As the eruption develops the magma chamber feeds magma into the
conduit. However, for most explosive eruptions the erupted volume in the
initial transient stages is much smaller than the chamber volume; hence we
assume the chamber pressure is fixed. The pressure at x = L + hd is fixed to
atmospheric if exit conditions are subsonic, otherwise no boundary conditions
are necessary.
16.1.5
Numerical Method
The Lax-Friedrichs conservative method (Jiang and Tadmor 1998) is applied
for unsteady flow in the conduit. The system of equations (16.1 - 16.3) can
be represented in the conservative form (no independent variables appear
outside the derivatives). This form guarantees automatic satisfaction of conservation laws in numerical approximation.
∂u ∂f (u)
+
= ψ(u)
∂t
∂x
T
u = (ρ, ρe , ρv)T ; f (u) = ρv, ρe v, P + ρv 2
T
32µav
(16.10)
ψ(u) =
0, J, −ρg − λ 2 v
d
Here the superscript T means transposition. The method works on the
equally-spaced x-grid with step ∆x . We choose a timestep ∆tn to satisfy stability conditions (Courant and Friedrichs 1976). The x-grid has to
= xni + 0.5∆x. At
be staggered on 0.5∆x on the odd time step ∆tn : xn+1
i
any time step the vector function u(x) is fitted by a piecewise-linear approximation. In order to account for discontinuities we use the minmod function
′
MM, which chooses the most appropriate slope ui (xni , tn ) from numerical
derivatives for the approximation:

∃k, l : dk dl ≤ 0
 0
mini (di ) ∀k
dk < 0
MM (d1 , d2 , d3 ) =
(16.11)

maxi (di ) ∀k
dk > 0
u (xi , tn ) − u (xi−1 , tn )
d1 =
∆x
u (xi+1 , tn ) − u (xi , tn )
d2 =
2∆x
u (xi+1 , tn ) − u (xi , tn )
d3 =
∆x
Here, instead of parameters di , left, right, and central derivatives are substituted. The principal method formulas follow from the approximation of the
equation (16.10) in the integral form on the rectangle
n
(xi , tn ) , (xni + ∆x, tn ) , xn+1
+ 0.5∆x, tn+1 .
i
7
The method has two semi-steps, a predictor that approximates the values,
u (xni + 0.5∆tn ), and a corrector that calculates the values on the staggered
grid, u (xni + 0.5∆x, tn+1 ).
u (xi , tn + .5∆tn ) = u (xi , tn ) +
∆tn ′
f
∆x
(u (xi , tn )) + u (xi + .5∆x, tn+1 )
= 21 (u (xi , tn ) + u (xi+1 , tn )) +
1
8
′
′
u (xi , tn ) − u (xi+1 , tn ) +
∆tk
∆x
(f (u (xi+1 , tn + .5∆tn )) − f (u (xi , tn + .5∆tn )))
(16.12)
Because the method uses a staggered grid, special treatment of the boundary
conditions is required. If the scheme has executed an odd time step, two
vectors have to be calculated additionally for the next step:
u (0, tn ) , u (L + hd , tn )
after an even time step four vectors have to be calculated:
u (−0.5∆x, tn ) , u (0.5∆x, tn )
u (L + hd − 0.5∆x, tn ) , u (L + hd + 0.5∆x, tn )
To obtain these values we make the linear projections of pressure, gas volume
and discharge rate and restore vectors, u, on the boundaries using appropriate
boundary conditions. To estimate the accuracy of this method the transient
problem was solved until stabilization and stabilized profiles were compared
with profiles from the steady-state solution obtained from integration of the
ODE system. For 500 cells in the x−grid, the steady and stabilized pressure
profiles coincided to within ±0.1%.
16.1.6
Results of Numerical Simulations
The model calculations (Figure 16.1) produce pulse-like eruptions for the
set of parameters listed in Table 16.1. The fragmentation front descends
in a series of steps (Figure 16.1 (b)). Initially it follows the rarefaction
wave generated after dome collapse. The fragmentation process stops when
the overpressure in the bubbles becomes less than the critical value ∆Pcr .
With no fragmentation the interface between bubbly magma and gas particle
dispersion ascends with the velocity of the flow. Fragmentation starts again
when the critical overpressure is reached and the interface moves downwards.
Each fragmentation event induces a pulse in discharge rate (Figure 16.1 (a)).
8
We compare the end member cases, equilibrium and no mass transfer
(Melnik and Sparks 2002), with the results for finite values of the diffusion
coefficient. We consider variants of the most and the least intensive mass
transfer that might be relevant to natural systems with a diffusion coefficient
of D = 10−11 m2 s−1 , and a bubble number density per unit volume of the
mixture n = 1016 m−3 corresponding to fast and D = 10−14 m2 s−1 , n =
1012 m−3 slow diffusion.
There is little difference in eruptive behaviour over the first 8 seconds
(Figure 16.1). The first peak in the discharge rate develops in the first several seconds with results being similar for all cases. After about 8 seconds
the fragmentation front evolves differently. For the no-mass-transfer case
the fragmentation front stops at 1.3 km depth and the level of unfragmented
magma then tends to return to the initial position. For the equilibrium-masstransfer case fragmentation propagates further down to 2 km and then after
a pause descends step by step until reaching the magma chamber. Over a
30-minute period the no-mass-transfer case has only one discharge pulse, the
equilibrium-mass-transfer case has several pulses. The new results, which
take account of mass transfer processes, show intermediate behaviour. For
the fast diffusion case the fragmentation front stops at 2 km, then the bubbly
liquid level rises with the flow over several minutes till the second pulse of
fragmentation starts at 33 minutes. For the slow diffusion case the fragmentation front first descends to 1.5 km depth and then ascends gradually.
Rapid changes in eruption intensity can be observed on seismic records
of some eruptions. Figure 16.2 shows seismic energy release during the subPlinian eruption on the Soufrière Hills Volcano, Montserrat. Pronounced
peaks of seismic activity can be related to multiple fragmentation events
that lead to significant changes in pressure distribution inside the conduit.
16.2
Transient Model of Conduit Flow
during Extrusive Eruptions
Lava dome eruptions commonly display fairly regular alternations between
periods of high and low or no activity with time scales typically of weeks
to years and sudden transitions between effusive and explosive activity (Figure 16.3). In this case magma fragmentation does not occur and the conduit
is filled by slowly-ascending bubbly magma. Because the timescale of magma
ascent is much longer than for the case of an explosive eruption (days instead
of minutes) different physical processes become important. Due to pressure
decrease and gas exsolution the liquidus temperature of the magma increases
significantly leading to crystal nucleation and growth. Increase in crystal
content results in rheological stiffening of magma (Mader, this volume) and
9
to the release of latent heat that increases the temperature. Exsolved gas
can escape from the ascending magma either in a vertical direction through
magma column or laterally to the wallrocks (Melnik and Sparks 1999, 2002,
2004).
16.2.1
Governing Equations
We have modelled the ascent of magma along the conduit from the chamber
with the following 1D equations:
∂
∂
∂
∂
ρm +
ρm V = −G;
ρc +
ρc V = G
∂t
∂x
∂t
∂x
∂
∂
∂
∂
ρd +
ρd V = −J;
ρg +
ρg Vg = J
∂t
∂x
∂t
∂x
32µ (c, β) V
∂
P = −ρg −
∂x
d2
k (α) ∂
P
Vg − V = −
µg ∂x
∂
∂
ρCm
T +V
T
= L∗ G
∂t
∂x
(16.13)
(16.14)
(16.15)
(16.16)
(16.17)
We follow notations from the previous section. Additionally, Vg is the velocity of the exsolved gas phase, G is a mass flux related to crystallization,
k(α) is the permeability coefficient, µg is the viscosity of the gas phase, L∗ is
the latent heat of crystallization, Cm is the heat capacity of magma. Equations (16.13) represent conservation of mass for melt and crystals, (16.14) for
dissolved and exsolved gas, respectively. Equation (16.15) is a momentum
equation for the mixture as a whole, in which the inertial term is negligibly small and pressure drop occurs due to gravity and viscous resistance.
Equation (16.16) is Darcy’s law for the gas flow through the system of interconnected bubbles. Equation (16.17) is the energy equation accounting for
release of latent heat of crystallization.
Following Hort (1998) crystal growth and nucleation rates were introduced as functions of undercooling, where effective liquidus temperature depends both on concentration of dissolved gas and amount of crystallized
material (Cashman and Blundy 2000).
G=
3σρ0c
(1 − β) (1 − α) U(t)
Z
t
I(ω)
0
Z
t
U(η)dη
ω
2
dω
(16.18)
Here I is the nucleation rate (m−3 s−1 ), which defines the number of newly
nucleated crystals per cubic metre per second, U is the linear crystal growth
rate (m s−1 ), ω and η are integration variables. The outer integral in equation
10
(16.18) determines the amount of crystals that appear in the time interval
[0, t]. The internal integral shows the change of surface area for the same
interval of time. By multiplying the integrals by 3σU(t), where σ is a crystal
shape factor ( σ = 1 for a spherical crystal), we obtain the increase in volume. Both U(t) and I(t) are functions of magma undercooling (which is the
difference between the magma temperature and the melting temperature).
The mass flux due to gas exsolution is described by equation (16.7). Parameterization of crystal growth and nucleation rates was done by comparison
of calculated results with experimental data (Couch et al. 2003).
16.2.2
Boundary Conditions
Equations (16.13)-(16.17) are solved numerically between the top of the
magma chamber and the top of the lava dome. Flow in the dome is represented by a continuation of the conduit, with the same diameter for the
active zone of flow within the dome and extrusion of new lava at the summit, consistent with observations (Young et al. 1998) for the Soufrière Hills
Volcano. As the extrusion rate is subsonic, we assume that the pressure on
the top of the dome is equal to atmospheric. We assume that the magma
chamber is located in elastic rocks and is fed from below with new magma.
The relation between pressure at the top of the magma chamber Pch and
intensity of influx Qin and outflux Qout of magma from the chamber is given
by (Barmin et al. 2002; Melnik 2000; Woods and Koyaguchi 1994):
dpch
4EhKi
=
(Qin − Qout ) ;
dt
hρiVch (3hKi + 4E)
hKi = hρi
∂p
∂ρ
(16.19)
Here Vch is the volume of the magma chamber, hρi and hKi are the average
density and bulk modulus of the magma, respectively, and E is the elastic
modulus of the surrounding rocks. The average bulk modulus of magma is
controlled by the presence of bubbles and pressure distribution inside the
chamber (Woods and Huppert 2003). We assume that the volume concentration of crystals and mass transfer between the melt and bubbles in the
magma chamber are in equilibrium:
βch = βeq (pch , Tch ) , αch = αeq (pch , Tch )
Magma and gas velocities at the conduit entrance are equal because the
magma is impermeable at the low volume fraction of bubbles.
16.2.3
Numerical Method
The integration interval x ∈ [0, L] is divided in a non-uniform mesh containing n points with the step of the mesh decreasing towards the top of
11
the conduit where the gradients of variables reach their maximum. System
(16.13)-(16.17) can be represented as F (Uj−1 , Uj ) = 0, where Uj is a vector
function that contains values of dependent variables on the j-th interval of
the mesh. By taking the first member of the Taylor series, we obtain:
0 = F (Uj−1 , Uj )
∂F ∗
∗
= F Uj−1 , Uj +
∂U U∗
j−1
∂F ∆Uj−1 +
∆Uj
∂U U∗
(16.20)
j
Here the symbol * represents the value of the function from the previous
iteration, and ∆U is the increment. Solution of (16.20) for ∆Uj leads to the
following matrix equation.
∆Uj − Pj ∆Uj−1 = Qj
!−1
∂F •
P =
∂U U∗
j−1
j
!−1
∂F
Q = F U∗j−1 , U∗j •
∂U U∗
∂F ∂U U∗
(16.21)
j
System (16.21) is solved by means of the two-point chaser method as shown
below:
∆Un =
=
=
=
Pn ∆Un−1 + Qn
Pn (Pn−1 ∆Un−2 + Qn−1 ) + Qn
...
Ψ2 ∆U1 + Θ2
(16.22)
Equation (16.22) relates the value of increments of dependent variables in
the first and the last mesh points. First we calculate matrixes Ψ and Θ,
and, with the help of boundary conditions
ϕ∆Un + θ = 0,
find the values of ∆U1 . Later, using equation (16.21), the calculation of all
flow parameters is performed in a new iteration. Solution in the new time
step is considered to converge when the sum of ∆U2j is smaller than a given
value.
16.2.4
Results of Numerical Simulations
The set of parameters used in the calculations listed in Table 16.1 is based on
studies of the Soufrière Hills Volcano, Montserrat (Melnik and Sparks 1999,
12
2002). The plots of discharge rate as a function of chamber pressure and time
are shown on Figure 16.4. As shown previously (Barmin et al. 2002; Melnik
and Sparks 1999, 2002) the steady-state solution of a boundary value problem
may not be unique; for certain fixed parameters in the magma chamber, there
can be up to three steady-state regimes of extrusion. The uppermost regime
(”U”) is characterized by high discharge rate and either no or low rates
of crystallization, so that conduit resistance is controlled by the relatively
low magma viscosity, allowing high ascent velocity. In the lowermost regime
(”L”), crystallization during magma ascent is significant, and conduit friction
is high due to the relatively high viscosity of magma in comparison with the
magma that failed to crystallize during ascent. In the intermediate regime
(”I”), conduit resistance decreases with increase in discharge rate due to
the decrease in crystal content and, therefore, the relative viscosity of the
magma.
The asymptotic behaviour for t → ∞ of the transient solution depends
on the value of Qin . If Qin corresponds to the upper or the lower regime, the
discharge rate will stabilize with time with
Q = Qin , dpch /dt = 0.
However, if Qin corresponds to the intermediate regime, periodic variation
in the discharge rate can occur. Here we compare magma chamber volumes of 1 km3 and 10 km3 , with the influx corresponding to the intermediate
regime. For Vch = 10 km3 the system behaves quasi-statically: the eruption
starts within the regime with low discharge rate, closely following it up to
the transition point. According to the boundary condition (Equation 16.19)
the chamber pressure has to increase further (Qin > Qout ) but there is no
steady-state solution with low discharge rate for higher chamber pressures,
and therefore the system jumps to the uppermost regime. In this regime,
Qin < Qout , and the chamber pressure decreases. The eruption follows the
uppermost regime until the second transition point and jumps down to the
lowermost regime. The transition between the regimes occurs with nearly
constant chamber pressure. This behaviour leads to periodic variations in
discharge rate with time (Figure 16.4 (b)).
For V ch = 1 km3 the system starts in the lower regime and progressively
departs from the steady-state solution, reaching a slightly higher chamber
pressure in the first cycle, moves to high discharge rates and lower chamber
pressure, and then the discharge rate decreases significantly. The second and
subsequent cycles are nearly identical and the amplitudes of pressure and
discharge rate variations are larger in comparison with the case for Vch =
10 km3 (Figure 16.2 (a)).
13
Influence of total concentration of dissolved gas on ascent dynamics
Typical melt water contents c0 are taken to be 4 - 7 wt% for lava dome eruptions. Total gas content and chamber pressure control the volume fraction of
bubbles inside the magma chamber and, therefore, magma compressibility.
For pressures less than the saturation pressure, according to the solubility
law, there is a free gas phase in bubbles. A set of steady-state solutions corresponding to different initial water contents is shown in Figure 16.5 (a). With
increase in c0 the average density of magma inside the conduit decreases and,
therefore a lower chamber pressure is required for the same discharge rate.
The discharge rates at the transition points depend only slightly on the value
of c0 if the magma is supersaturated in the magma chamber. For low values
of c0 (< 4.5 wt%) the magma in the conduit is initially undersaturated and
remains as a homogeneous liquid for a large vertical part of the conduit. This
means that decompression-induced crystallization occurs only at the upper
part of the conduit and variations in crystal content with discharge rate are
small. Therefore, the interval of discharge rates, where oscillatory behaviour
is possible, shrinks as c0 decreases. Figure 16.5 (b) shows the period of oscillations as a function of chamber volume and water content. For c0 = 4.5
and 5 wt% the volume fraction of bubbles in the chamber is very small so
that mixture compressibility is low and is determined by the compressibility
of pure magma. Therefore, the period of pulsation is small and depends linearly on the chamber size. For c0 = 7 wt% the volume fraction of bubbles is
significant for all magma chamber volumes leading to high magma compressibility and long periods of oscillation. In the case c0 = 6 and 6.5 wt% for a
small magma chamber the compressibility is high and the period is longer
than for c0 = 5 wt%. With increase in the chamber size the proportion of the
chamber occupied by bubbly magma and average compressibility decreases
and the period of pulsation is much smaller than for c0 = 7 wt%.
Influence of non-Newtonian properties on eruption behaviour
High crystal content magmas deviate from classical Newtonian rheological
laws showing more complicated behavior (presence of yield strength and shear
rate dependent viscosity, see Mader, this volume for details). We compare
the dynamics of magma extrusion in the cases of Newtonian and Bingham
rheology. We will assume that yield strength appears when the concentration
of crystals reaches a critical value:
τb f or β > βcr
τ=
(16.23)
0 f or β ≤ βcr
Figure 16.6 (a) shows a set of steady-state solutions for different values of
τb . Values of τb and βcr depend on crystal shape, crystal size distribu14
tion, magma temperature and other properties. To illustrate the influence
of Bingham rheology the value of βcr = 0.65 was chosen so that for the uppermost regime, the magma has Newtonian rheology. The higher the value
of τb the larger the chamber pressure that is necessary to start the eruption.
Figure 16.6 (b) shows the influence of these two rheological models on the dynamics of magma extrusion. In the case of Bingham rheology, the discharge
rate between the two pulses is zero until the critical chamber overpressure
is reached. Then the discharge rate increases rapidly with the decrease in
crystal content, leading to a significant reduction of both magma viscosity
and shear force associated with yield strength. The system transits to the
uppermost flow regime and the pressure then decreases quickly. Because the
pressure at the onset of the pulse was significantly larger than in the case
of a Newtonian liquid, the resulting discharge rate in the case of Bingham
rheology is also significantly higher.
16.3
Concluding Remarks
We have presented here two cases for application of principles of fluid dynamics and numerical methods to transient flows in volcanic conduits. Calculations allow us to investigate the principal controls on the dynamics of
eruptions and give estimates of the values of governing parameters, such as
magma chamber volume or conduit diameter, by allowing us to compare
calculated results with observations.
Our models indicate that magmatic systems can be very sensitive to
small changes in parameters. This sensitivity is most marked when the
system is close to the cusps of the steady-state solutions. We have illustrated sensitivity of results by varying only one parameter at a time on plots
of chamber pressure and discharge rate. However, magmatic systems have
many controlling parameters that may vary simultaneously. Furthermore,
some controlling parameters are likely to be interdependent (such as temperature, volatile content and phenocryst content, for example) and others
may be independent (such as magma temperature and conduit dimensions).
An eruption can be expected to move through n-parameter space, making
simulation and its parameter depiction difficult. Our results are simplified,
so system sensitivity and behaviour in the real world is likely to be yet more
complex. A volcanic system may be quite predictable when it is within a
stable regime, but may become inherently unpredictable when variations in
the parameters move the system towards transition points and flow regime
boundaries.
The full simulation of any particular volcanic eruption in such a nonlinear and sensitive system may appear a hopeless task. However, some
reduction in uncertainties will help to make the models more realistic. Ad15
vances in understanding the controls on magma input into an open-system
chamber would be beneficial, since the delicate balance between input and
output is a prime control on periodic behaviour. Further model development
includes 2D effects, elastic deformation effects in dyke-fed domes and coupling
magma chamber and conduit flow dynamics. Even with such improvements,
large parameter uncertainties and modelling difficulties will remain. In such
circumstances the logical approach will be to start quantifying the uncertainties and sampling from them to produce probabilistic outputs based on
ensemble models where numerical models of the kind discussed here can be
run many times. A future challenge for numerical models will also be to
produce simulated outputs which compare, in detail, with observations, for
particular time series of discharge rates.
Further Reading
For general reviews of eruption processes and models see the volumes edited by
Gilbert and Sparks (1998) and Freundt and Rosi (1998). Detailed descriptions of
lava dome building eruptions can be found in Fink (1990) for Mount St Helens and
in Druitt and B.P. Kokelaar (2002) for the Soufrire Hills Volcano (Montserrat). An
excellent general text covering many aspects of volcanology, including monitoring
and hazard assessment but excluding statistics, is the Encyclopedia of Volcanology
edited by Sigurdsson et al. (2000).
Acknowledgements
This work was supported by following grants: NERC (GR3/13020), EC INTAS
(01-0106), EC MULTIMO and by the Russian Foundation for Basic Research
(RFBR 05-01-00228). RSJS acknowledges a Royal Society-Wolfson Award.
16
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20
Parameter
Conduit length
Pressure at magma chamber
Pressure at base of the dome
Melt water content
Conduit diameter
Magma temperature
Magma crystal content
Gas constant for H2 0
Density of melt
Density of crystals
Solubility coefficient
Diffusion coefficient
Number density of bubbles
Specific heat
Latent heat of crystallization
Gas viscosity
Symbol
L
Pch
Pend
c0
d
Tch
R
ρ0m
ρ0x
Cf
D
n∗
Cm
L∗
µg
Unit
m
MPa
MPa
wt%
m
K
β
J K−1
kg m−3
kg m−3
1
Pa− 2
m2 s−1
m−3
J kg−1 K−1
J kg−1
Pa(s)
Explosive
5000
130
10
5
30
1123
0.6
461
2300
2700
4.1 × 10−6
−14
10
- 10−11
10
10 - 1014
N/A
N/A
N/A
Table 16.1: Parameters for numerical simulations
21
Extrusive
5000
from BC
0.1
3-7
30
1123
equilibr.
461
2300
2700
4.1 × 10−6
10−12
1012
1.2 × 103
3.5 × 106
1.5 × 10−5
fragmentation depth (km)
discharge rate x 106 (kgs-1)
(b)
(a)
chamber volume (km3)
chamber pressure (MPa)
Figure 16.1: Dependence of discharge rate (a) and fragmentation depth (b)
on time for different intensities of mass transfer (mt). Curves correspond to
different rates of mass transfer of gas dissolved in the melt to the bubbles.
2
3
Figure 16.2: One-minute RSAM seismic record over an 8 hour period for 17
September 1996 subPlinian explosive activity. During the subPlinian phase
three peaks of seismic activity are marked.
22
(a)
(b)
1981 1982 1983 1984 1985 1986
1915
1935
20
40
1955
1975
1995
16
II
III
2
discharge rate (m3/s)
discharge rate (m3/s)
I
12
8
4
0
1.5
1
0.5
0
0
500
1000
1500
2000
60
80
100
t (years)
time (days)
Figure 16.3: Variation of discharge rate with time for Mount St Helens (a)
and Santiaguito (bf b) volcanoes. In the eruption of Mount St Helens (USA)
in 1980-1986 more than 20 short episodes of dacite dome growth, lasting 2-7
days, alternated with longer periods of no growth. There were two sequences
of periodic dome growth with an intervening episode of near continuous dome
growth lasting 368 days (Swanson and Holcomb 1990). The Santiaguito
dacite dome (Guatemala) has been continuously active since 1922. Dome
growth is characterised by periods of 3-5 years of relatively high discharge
rate alternating with periods of 3-11 years of low discharge rate (Harris et
al. 2003).
23
(a)
(b)
Figure 16.4: Dependence of discharge rate on chamber pressure (a) and time
(b). Numbers represent different chamber volumes, arrows - directions of
system evolution with time. The steady state solution (solid line in (a) is
sigmoidal.
(a)
(b)
Figure 16.5: Dependence of discharge rate on chamber pressure (a) and
period of pulsations on chamber volume (b). Values of initial water content
(in wt%) are labeled on the plot.
24
(a)
(b)
Figure 16.6: Dependence of discharge rate on chamber pressure (a) and time
(b). The values of τb (in MPa) are shown in the legend.
25