Mass-conservative numerical scheme of bubble growth in incompressible viscous magmas
paper-number : V11D-2318
Louis FORESTIER
1
COSTE
, Alain
2
BURGISSER ,
Francois
1
JAMES
abstract-number : 949668
1
MANCINI
et Simona
email: louis.forestier-coste@math.cnrs.fr
1
Laboratoire MAPMO - UMR 6628 - Université d’Orléans, UFR Sciences, Route de chartres, B.P. 6759 - 45067 Orléans cedex 2 - France
2
ISTO, CNRS - Université d’Orléans, 1A rue de la Férollerie - 45071 Orléans cedex 2 - France
Introduction
We are interested in modeling the growth of water vapor bubble in a constantly decompressed viscous magma because it partly influences on the style of volcanic eruption, whether explosive or effusive. We study a simplified
system of monodisperse bubble growth in a viscous incompressible magma by considering only the pressure of gas in the bubble, the bubble radius and weight fraction of volatiles (water) in the melt [Lensky et al. 2004].
Bubbles are assumed spherical. We propose a new numerical resolution of this classical model.
R
α S
Momentum conservation
Mass
conservation
Diffusion
S
√
3ρm
Σ
1
R
3
2
2
2
2
D
dt(P R ) = ΘD r D∂r C r=R
r ∂tC + R dt(R)∂r C = ΘD ∂r (r D∂r C) ; CR = KH P , ∂r C|r=S = 0
dimensionless eq.
dtR = ηef f ΘV P − Pa − R
R
3
ρm
ηeff(t)
D : Diffusivity
Di
normalized by
C : water concentration
ηef f : effective viscosity
ηi
σ : surface tension
ρi :
ρm : melt density
ρi
M
Pa(0) GT
R : bubble radius
Ri
M : water molar mass
Numerical scheme mass-conservative
P : bubble pressure
Pa(0)
G : gas constant
Pa : ambient pressure
Pa(0)
T : temperature
ΘV =
4ηi ∆P
(Pa (0))2
:
t : time
Pa (0)
∆P
viscous time scale
decompression time scale
KH : Henry’s constant
√1
S : influence radius
Ri
Pa (0)
ΘD =
Ri2 ∆P
Di Pa (0)
:
diffusion time scale
decompression time scale
Convergence to simplified cases
Explicitly conserves the total water mass, gaseous
and dissolved by
•a moving mesh depending on the radius ,
•resolution of the diffusion equation but the flux
at bubble boundary is set by the boundary condition.
• Simplified cases : ΘD and/or ΘV = 0 or ∞
Σ=
∆P : decompression rate
Pa(0)
2σ
Ri Pa (0)
r
P
C(t,r)
Pa(t)
: dimensionless surface tension
α : porosity
Comparison with data from laboratory decompression experiments
10
limit
0,1
case
0,001
ΘV 1E-05
equilibrium
1E-07
case
1E-09
1E-11
0,0000001
general
case
limit
case
0,00001
0,001
0,1
ΘD
10
Gardner, Hilton, Carroll (jul 1999)
Mangan, Sisson (2000)
Lyakhovsky, Hurwitz, Navon (1996) + Hurwitz, Navon (1994)
Mongrain, Larsen, King (2008)
Larsen, Gardner (2004)
Gardner (2007)
Gardner, Hilton, Carroll (feb 1999)
Iacono Marziano, Schmidt, Dolfi (2007)
Takeuchi, Tomiya, Shinohara (2009)
Takeuchi, Nakashima, Tomiya, Shinohara (2005)
Martel, Schmidt (2002)
Shea, Gurioli, Larsen, Houghton (2010)
Burgisser, Gardner (2005)
Distribution of laboratory experiments in terms of ΘV and ΘD . Data from 13 studies.
Mesh
coalescence cases
Larsen, Gardner (2004)
• few points required for the discretization of the
diffusion equation thanks to a moving 1D mesh
n+1 3
(ri )
=
n 3
(ri )
+ (R
n+1 3
n 3
1
9
Bubble radius evolution and convergence with respect to
ambient pressure.
2,00
3,00
4,00
5,00
6,01
7,01
7
calculated
0,6
α
calculated
6
R 5
5,00
5,09
5,33
bubble distance
5,73
6,29
6,98
2
Dynamics and energetics of bubble growth in magmas : Analytical formulation and
numerical modeling, Journal of Geophysical Research, 1998
bubble distance
0,4
3
Comparison with Proussevitch model
7,76
0,2
1
0
When the bubble radius increase, the mesh is refined at
the bubble boundary
Proussevitch model
current model
Constraint on the time step by physical consideration
• positivity of the radius R
• positivity of the water concentration C
• positivity of the pressure P
8
6
R
4
2
0
0
0,2
0,4
0,6
Pa
0,8
1
1,2
20
30
40
0
50
0,2
0,4
0,6
measured α
uncertain initial conditions
matching case
Lyakhovsky, Hurwitz, Navon (1996) + Hurwitz, Navon (1994)
Mangan, Sisson (2000)
Shea, Gurioli, Larsen, Houghton (2010)
Gardner, Hilton, Carroll (feb 1999)
Gardner, Hilton, Carroll (jul 1999)
Gardner (2007)
Burgisser, Gardner (2005)
0,8
1
1
20
Bubble radius R
with respect to
ambient pressure
P a.
• Difference is due to model assumption : constant
D and different viscosity.
• Proussevitch model is faster
• high temporal resolution possible with our model
10
measured R
12
Scheme stability
0
0
• comparison between model outputs from our
scheme and that of an existing scheme
10
When initial conditions are unknown, we
set the radius to 1µm and the porosity
to 0.1%. Radius rescaled with the BND
whenever possible.
0,8
8
4
1,00
Takeuchi, Nakashima, Tomiya, Shinohara (2005)
10
• convergence towards these simplified cases.
) − (R )
Takeuchi, Tomiya, Shinohara (2009)
18
0,8
16
14
calculated
12
0,6
α
10
8
0,4
calculated
6
R
4
0,2
2
0
0
0
10
20
30
measured R
40
50
0
0,5
measured α
1
• Model output reproduce observations to a satisfactory degree for
rhyolites.
• Neglecting the effect of coalescence
clearly limits model applicability.
This is most visible in less viscous
melts such as phonolites.
• Uncertainly in initial conditions
can affect both porosity and bubble
radius.
• See poster V11D-2316 for a bubble
growth model including coalescence.