a two-phase model for magma flowing in a volcanic conduit · l > x two phase flows in...
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Céline Acary-Robert 1, Didier Bresch 1,
Alain Burgisser 2, Marielle Collombet2,
Gladys Narbona-Reina3
1 LAMA, Université Savoie Mont Blanc - CNRS, Chambéry, France2 ISTerre, Université Savoie Mont Blanc - CNRS, Chambéry, France
3 Dpto. Matemática Aplicada I, Universidad de Sevilla
A two-phase model for magma
flowing in a volcanic conduit
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ANAK KRAKATAU VOLCANO, INDONESIA
VULCANIAN EXPLOSIONS
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PRINCIPLE
Pre-explosive
conditions
Explosion
trajectory
Nature
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INSIDE A PIECE OF MAGMA
Quenched
piece of magma
bubble
melt
crystal
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INSIDE A PIECE OF MAGMA
Quenched
piece of magma
bubble
melt
crystal
Model with fluid dynamics and two phases
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LIFE CYCLE OF BUBBLES IN MAGMAS
Collapse
Coalescence
Growth
Nucleation
Life stage
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Mechanisms:
gas expansion + water diffusion
BUBBLE GROWTH
+ w
ate
r d
iffu
sio
n in
me
lt
-
- melt viscosity +
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LAGRANGIAN BUBBLE GROWTH
Forestier Coste et al. (2012)
Experiment
Model
Lots of Lagrangian models
Few Eulerian models
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Many bubbles connect with each other → gas velocity ≠ melt velocity
Connected volume 1 path connecting top & base
3D images of experimental magmas (1 mm across)
BUBBLE COALECENCE & COLLAPSE
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Permeable gas flow
Permeable gas flow
Permeable gas flow
EULERIAN BUBBLE GROWTH:
TRANSITION BETWEEN ERUPTIVE REGIMES
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Permeable gas flow
Permeable gas flow
Permeable gas flow
EULERIAN BUBBLE GROWTH:
TRANSITION BETWEEN ERUPTIVE REGIMES
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NAVIER – STOKES EQUATIONS
{
Incompressible
momentum conservation
mass conservation
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NAVIER – STOKES EQUATIONS
{
Incompressible
Compressible
{
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Two-phase compressible systemgas = phase +
liquid = phase -
volume fractions sum to one
TWO PHASE FLOWS
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Two-phase compressible systemgas = phase +
liquid = phase -
volume fractions sum to one
TWO PHASE FLOWS
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Two-phase compressible systemgas = phase +
liquid = phase -
volume fractions sum to one
7 equ.
8 unknowns
TWO PHASE FLOWS
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7 equ.
8 unknowns
1) Algebraic closure
M. Ishii (1975), D.A. Drew & S.L. Passman (1998),
H.B. Stewart & B. Wendroff, Two-phase flow: models and methods, J. Comput.Phys. 56 (1984).
Two-phase compressible system
TWO PHASE FLOWS
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Bresch, D, Hillairet, M. A compressible multifluid system with new
physical relaxation terms. 2016. <hal-01262617>
2) PDE closure
Well-posed in specific cases such as constant h
Two-phase compressible system
7 equ.
8 unknowns
TWO PHASE FLOWS
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Two-phase system in magmagas
liquid
volume fractions sum to one
TWO PHASE FLOWS IN VOLCANOLOGY
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2) PDE closure
Two-phase compressible/incompressible system5 equ.
6 unknowns
TWO PHASE FLOWS IN VOLCANOLOGY
gas/liquid drag
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2) PDE closure
Two-phase compressible/incompressible system5 equ.
6 unknowns
work done by the
interface pressure pi
TWO PHASE FLOWS IN VOLCANOLOGY
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2) PDE closure
Two-phase compressible/incompressible system5 equ.
6 unknowns
?
?
work done by the
interface pressure pi
TWO PHASE FLOWS IN VOLCANOLOGY
pi = pg ?
pi = pl ?
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2) PDE closure
Two-phase compressible/incompressible system5 equ.
6 unknowns
?
?
Guillemaud (2007),
Gallouët, Hérard & N. Seguin (2004) Numerical modelling of two-phase flows using the
two-fluid two-pressure approach. Mathematical Models and Methods in Applied Sciences, 14:663–700.
work done by the
interface pressure pi
TWO PHASE FLOWS IN VOLCANOLOGY
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2) PDE closure
Two-phase compressible/incompressible system5 equ.
6 unknowns
?
?
Guillemaud (2007),
Gallouët, Hérard & N. Seguin (2004) Numerical modelling of two-phase flows using the
two-fluid two-pressure approach. Mathematical Models and Methods in Applied Sciences, 14:663–700.
TWO PHASE FLOWS IN VOLCANOLOGY
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2) PDE closure
Two-phase compressible/incompressible system5 equ.
6 unknowns
TWO PHASE FLOWS IN VOLCANOLOGY
𝜕𝑡𝜑 + 𝛻 ∙ 𝜑𝑢𝑙 =𝑝𝑔 − 𝑝𝑙
𝜀Two bubble growth mechanisms:
1. by gas expansion Lensky et al. (2004) J. Volcanol. Geotherm. Res.
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Two-phase compressible/incompressible system6 equ.
Two bubble growth mechanisms:
1. by gas expansion
2. by mass addition
6 unknowns
Lensky et al. (2004) J. Volcanol. Geotherm. Res.
TWO PHASE FLOWS IN VOLCANOLOGY
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Two-phase compressible/incompressible system6 equ.
6 unknowns
Two bubble growth mechanisms:
1. by gas expansion
2. by mass addition
Mancini, Forestier-Coste, Burgisser, James, Castro (2016)
J. Volcanol. Geotherm. Res.
TWO PHASE FLOWS IN VOLCANOLOGY
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Calculation of the total energy of the system
Our system is well posed if it dissipates energy.
𝑢𝑔 ∙ momentum conservation of g
momentum conservation of l
mass conservation of g
mass conservation of l
𝑢𝑙 ∙add
combine to get:
𝜕𝑡 𝜑𝜌𝑔𝑢𝑔
2
2+ ⋯ + 𝛻 ∙ 𝜑𝜌𝑔𝑢𝑔
𝑢𝑔2
2+ ⋯ = − 1 − 𝜑 𝑝𝑔 − 𝑝𝑙
𝑅
𝜌𝑙−⋯
𝜕𝑡𝐸 + 𝛻𝐹 = 𝐺
E = total energy (kinetic, potential, intern)
F = energy flux
G = source term; G<0 ↔ dissipative
TWO PHASE FLOWS IN VOLCANOLOGY
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Calculation of the total energy of the system
𝑢𝑔 ∙ momentum conservation of g
momentum conservation of l
mass conservation of g
mass conservation of l
𝑢𝑙 ∙add
combine to get:
𝜕𝑡 𝜑𝜌𝑔𝑢𝑔
2
2+ ⋯ + 𝛻 ∙ 𝜑𝜌𝑔𝑢𝑔
𝑢𝑔2
2+ ⋯ = − 1 − 𝜑 𝑝𝑔 − 𝑝𝑙
𝑅
𝜌𝑙−⋯
𝜕𝑡𝐸 + 𝛻𝐹 = 𝐺
E = total energy (kinetic, potential, intern)
F = energy flux
G = source term; G<0 ↔ dissipative
𝑅 = 𝐴 1 − 𝜑 𝜌𝑙 𝐶𝑙 − 𝑘ℎ 𝑝𝑙 A, kh given
Cl new variable, the mass of which is conserved
TWO PHASE FLOWS IN VOLCANOLOGY
Our system is well posed if it dissipates energy.
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Calculation of the total energy of the system
𝑢𝑔 ∙ momentum conservation of g
momentum conservation of l
mass conservation of g
mass conservation of l
𝑢𝑙 ∙add
combine to get:
𝜕𝑡 𝜑𝜌𝑔𝑢𝑔
2
2+ ⋯ + 𝛻 ∙ 𝜑𝜌𝑔𝑢𝑔
𝑢𝑔2
2+ ⋯ = − 1 − 𝜑 𝑝𝑔 − 𝑝𝑙
𝑅
𝜌𝑙−⋯
𝜕𝑡𝐸 + 𝛻𝐹 = 𝐺
E = total energy (kinetic, potential, intern)
F = energy flux
G = source term; G<0 ↔ dissipative
TWO PHASE FLOWS IN VOLCANOLOGY
Our system is well posed if it dissipates energy.
Gives us dissipations conditions such as if R > 0, then pg – pl > 0
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Calculation of the total energy of the system
Our system is self-consistent if it dissipates energy.
𝑢𝑔 ∙ momentum conservation of g
momentum conservation of l
mass conservation of g
mass conservation of l
𝑢𝑙 ∙add
combine to get:
𝜕𝑡𝐸 + 𝛻𝐹 = 𝐺
E = total energy (kinetic, potential, intern)
F = energy flux
G = source term; G<0 ↔ dissipative
𝑅 = 𝐴 1 − 𝜑 𝜌𝑙 𝐶𝑙 − 𝑘ℎ 𝑝𝑙 A, kh given
Cl new variable, the mass of which is conserved
Gives us dissipations conditions such as if R > 0, then pl > x
TWO PHASE FLOWS IN VOLCANOLOGY
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Work in progress:
• Asymptotic cases thanks to a drift flux formulation
• 1D case
• Equilibrium states
• Numerical resolution
very few published 1D systems to study conduit flow seem to be dissipative
TWO PHASE FLOWS IN VOLCANOLOGY
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Work in progress:
• Asymptotic cases thanks to a drift flux formulation
• 1D case
• Equilibrium states
• Numerical resolution
Current conclusion:
very few published 1D systems to study conduit flow seem to be dissipative
TWO PHASE FLOWS IN VOLCANOLOGY
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END
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2) PDE closure
Two-phase compressible/incompressible system6 equ.
7 unknowns
TWO PHASE FLOWS IN VOLCANOLOGY
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TRANSITION BETWEEN ERUPTIVE REGIMES
{
{