Lecture 3Lecture 3

• Governing equations for multiphase flows. Continuum hypothesis.

• Fragmentation mechanisms.• Models of conduit flows during

explosive eruptions and results.• Volcanic plume dynamics in the

atmosphere.

Dynamics of dispersed systemsDynamics of dispersed systems

Mixture properties:

0

mass of componentBulk density =

volume of mixture

mass of componentPhase density =

volyme of component

volume of componentVolume fraction =

volume of mixture

mass frac

ii

mixture

ii

i

ii

mixture

m

m

mass of component

tion = mass of mixture

;

ii

mixture

mixture i mixture i

mX

m

m m

Bubbles

Particles

Mixture properties (continue)Mixture properties (continue)

component velocity =

mixture velocity =

ij iji

i

i i

mixture

mixture i

m VV

m

V

Continuity equations Mass fluxes

Momentum equations Momentum exchange

Energy equations Heat fluxes

Conduit flow during explosive eruptionConduit flow during explosive eruption

Schematic view of the system

xt

Flow regimes and boundaries. Homogeneous from magma chamber until

pressure > saturation pressure. Constant density, viscosity and velocity, laminar.

Vesiculated magma from homogeneous till magma fragmentation. Bubbles grow due to exsolution of the gas and

decompression. Velocity and viscosity increases. Flow is laminar with sharp gradients before

fragmentation due to viscous friction. Fragmentation zone or surface (?).

Fragmentation criteria. Gas-particle dispersion from fragmentation

till the vent. Turbulent, high, nonequilibrium velocities. subsonic in steady case, supersonic in transient.

Modelling strategyModelling strategy

Equations

• Mass conservation for liquid and gas phases– intensity of mass transfer, bubble nucleation and diffusive growth

• Momentum equations– gravity forces, conduit resistance, inertia

– momentum transfer between phases

• Energy equations– energy transfer between phases

– dissipation of energy by viscous forces

• Bubble growth equation - nonequilibrium pressure distribution

• Physical properties of magma - density, gas solubility, viscosity

• Fragmentation mechanism

• Boundary conditions - chamber, atmosphere, between flow zones

Models of fragmentationModels of fragmentation FP - fragmentation at fixed porosity.

SR - critical elongation strain-rate

OP- critical overpressure in a growing bubble p

gpm

pp mg

RR

4

3

RRRR

22

2

small

Hydrostatic vs. Lithostatic Hydrostatic vs. Lithostatic pressure gradientpressure gradient

Chocked flowsChocked flows

FlowHigh pressure

Low pressure

Q Chocked

low highp p

Boundary conditionsBoundary conditions

Magma chamber:

pressure, temperature

initial concentration of dissolved gas - calculate volume fraction of bubbles

Atmosphere:

Pressure is equal to atmospheric if flow is subsonic

Chocked flow conditions - velocity equal to velocity of sound

Need to calculate discharge rate

Slezin (1982,1983,1992)Slezin (1982,1983,1992)Main assumptions:Conduit has constant cross-section areaMagma - Newtonian viscous liquid, =constBubbles do not rise in magmaWhen = 0.7 - fragmentation, porous foamAfter fragmentation = 0.7, all extra gas goes to

interconnected voids.When concentration of gas in voids = 0.4 -

transition to gas particle dispersion.Particles are suspended (drag force=weight)

Slezin (results)Slezin (results)

WoodsWoods,, Koyaguchi Koyaguchi (1994) (1994)• Gas escape from ascending magma through the

conduit walls.

• Fragmentation criteria = *.

Magma ascends slowly - looses its gas - no fragmentation - lava dome extrusion.

Magma ascends rapidly - no gas loss - fragmentation - explosive eruption.

• Contra arguments: Magma permeability should be > rock permeability. Vertical pressure gradient to gas escape through the magma.

Barmin, Melnik (2002)Barmin, Melnik (2002)

• Magma - 3-phase system - melt, crystals and gas.

• Viscous liquid (concentrations of dissolved gas and crystals).

• Account for pressure disequilibria between melt and bubbles.

• Permeable flow through the magma.

• Fragmentation in “fragmentation wave.”

• 2 particle sizes - “small” and “big.”

0 0

0 0

00

1 1 1

1 1

m c m m

g g m m g

m m

c V Q

V cV Q

nV n V

Mass conservation equations (bubbly zone)Mass conservation equations (bubbly zone)

- volume concentration of gas (1-) - of condensed phase

- volume concentration of crystals in condensed phase

- densities, “m”- melt, “c”- crystals, “g” - gas

c - mass fraction of dissolved gas = k pg1/2

V - velocities, Q - discharge rates for “m”- magma, “g” - gas

n - number density of bubbles

Momentum equations in bubbly zoneMomentum equations in bubbly zone

2

3.50

,

1

ms

gg m

g

s m g

c Vdpg

dx Dd pk

V Vdx

p p p

k k

- mixture density

- resistance coefficient

(32 - pipe, 12 -dyke)

k() - permeability

g- gas viscosity

p- pressure “s”- mixture, “m”- condensed phase, “g”-gas

Rayleigh equation for bubble growthRayleigh equation for bubble growth

4m g m

m

dR RV p p

dx c

Additional relationships:

3 04, R

3 g gR n p T

0 0

1

g g g b m g

m b m b

m s g s

gas

big particles

V V Q

V Q

V Q small particles

Equations in gas-particle dispersionEquations in gas-particle dispersion

0 0

0 0 0

1 1

1 ; 1;

mm b m m b gb sb

gg m s g g m s gb sb

m m c s b g

dVV g F F

dxdV dp

V g F Fdx dx

p RT

F - interaction forces:”sb” - between small and big particles

“gb” - between gas and big particles

Fragmentation waveFragmentation wave

0 0 0

2 0 2

0 2 0 2

Conservation la

1 1

1 1

ws

1

g g g g g b m

m s g b m

m g m m g g

b m g m g m s g

m m

V V V

V V V

p p

gas phase

condensed phase

mixturemomentum

big particle

V V

p V V

s mV V mo

*

Additional relat

; ;

1 1

ions

g m

s b

entu

p p p

m m

m

Steady discharge vs. chamber pressureSteady discharge vs. chamber pressure

Pressure profiles in the conduitPressure profiles in the conduit

Model ofModel of vulcanian vulcanian

explosion generated explosion generated by lava dome by lava dome

collapse collapse

AssumptionsAssumptions

• Flow is 1D, transient

• Velocity of gas and condensed phase are equal

• Initial condition - V = 0, pressure at the top of the conduit > patm, drops down to patm at t =0

• Two cases of mass transfer: equilibrium (fast diffusion), no mass transfer (slow diffusion)

• Pressure disequilibria between bubbles and magma

• No bubble additional nucleation

Mechanical modelMechanical model

0

2

Conservation of mass and number density of bubbles:

No mass transfer:

Equilibrium mt:

Moment

0, 0, 0,

(

um

1 ) 1 ,

,

32

:

g g l l

l m c g g

m

V V n nV

t x t x t x

p p c k p

V pV Vg

t x D

2

0 3

*

Rayleigh equation

Fragmentation condition

, 1 ,

4, ,

4 3

:

m l g

g l g gm

g l

p p p c

a a aV p p p RT a n

t x

p p p

Results of calculation (eq case)Results of calculation (eq case)

Discharge rate and fragmentation depthDischarge rate and fragmentation depth

(eq case)(eq case)

Pulsing fragmentationPulsing fragmentation

Seismic record of eruptionSeismic record of eruption

Results of simulations (no mt case)Results of simulations (no mt case)Discharge rate and fragmentation depth

Parameter Calculated Observed*

Duration 100 –600 s 60-300 s Max velocity 118-142 m/s 120-130 m/s Fragmentation depth 200-1400 m 200-1000 m Volume of material 105- 106 m3 2 105- 106 m3

*Druitt et. al. (2001)

Volcanic plumesVolcanic plumesPlinian Collapsing

High - comes to stratosphere

Ash fallout, climate change

Acid rains, aviation hazards

Pyroclastic flow generation

Unsolved problemsUnsolved problems

• Physical properties of magma– Magma rheology for high strain-rates and high bubble and

crystal content • Bubbly flow regime

– Incorporation of bubble growth model into the conduit model

– Understanding bubble interaction for high bubble concentrations

– Understanding of bubble coalescence dynamics, permeability development

– Thermal effects during magma ascent - viscous dissipation, gas exsolution

Unsolved problems (cont)Unsolved problems (cont)

• Fragmentation– Fragmentation in the system of partly interconnected bubbles

– Partial fragmentation, structure of fragmentation zone, particle size distribution

• Gas-particle dispersion – Momentum and thermal interaction in highly concentrated gas-

particle dispersions

Unsolved problems (cont.)!Unsolved problems (cont.)!• General

– Coupling of conduit flow model with a model of magma chamber and atmospheric dispersal model

– Deformation of the conduit walls during explosive eruption

• Visco-elastic deformation

• Erosion– Interaction of magma conduit flow with permeable water saturated layers -

phreato-magmatic eruptions