Mathematical models of conduit flows during explosive eruptions

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• Mathematical models of conduit flows during
  explosive eruptions
• (Kamchatka steady, transient, phreatomagmatic)
• Oleg Melnik, Alexander Starostin,
• Alexey Barmin, Stephen Sparks, Rob Mason
• Institute of Mechanics, Moscow State University
• Earth Science Department, University of Bristol

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Conduit flow during explosive eruption

• Schematic view of the system

• Flow regimes and boundaries.
• Homogeneous from magma chamber until pressure gt
  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.

x
t
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Kamchatka steadyBarmin, Melnik (2002)

• Magma - 3-phase system - melt, crystals and gas.
• Viscous liquid m (concentrations of dissolved gas
  and crystals).
• Permeable flow through the magma.
• Account for pressure disequilibria between melt
  and bubbles.
• Fragmentation due to critical overpressure.
• 2 particle sizes after fragmentation.

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Mass conservation equations (bubbly zone)
a - volume concentration of gas (1-a) - of
condensed phase b - volume concentration of
crystals in condensed phaseconst r - 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
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Momentum and bubble growth
r - mixture density l - resistance coefficient
(32 - pipe, 12 -dyke) k(a) - permeability mg-
gas viscosity p- pressure s- mixture, m-
condensed phase, g-gas
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Equations in gas-particle dispersion
as,ab- volume fractions of particles q - volume
fraction of gas in big particles
F - interaction forces sb - between small and
big particles
gb - between gas and big particles
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Fragmentation wave
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(No Transcript)
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Ascent velocity vs. chamber pressure
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Model of vulcanian explosion generated by lava
dome collapse(Kamchatka transient)
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Assumptions

• Flow is 1D, transient
• Velocity of gas and condensed phase are equal
• Initial condition - V 0, pressure at the top of
  the conduit gt 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

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Mechanical model
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Results of calculation (eq. case)
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Discharge rate and fragmentation depth
equilibrium mt. no mass transfer
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Model of phreatomagmatic eruption
Model of the magma flow in the conduit with
influx from the porous layer
Model of magma flow in the conduit
Model of water flow in the porous layer
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Transient Problem
__ magma discharge __ water influx __
fragmentation front
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Conclusions

• Set of models for steady-state and transient
  conduit flows.
• Realistic physical properties of magma.
• New fragmentation criteria.
• Explanation of transition between explosive and
  extrusive eruptions, intensity of volcanic
  blasts, cyclic variations of discharge rate
  during phreatomagmatic eruptions.