Transcript Dynamics of Volcanic Eruptions

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
†
Conduit flow during explosive
eruption
Schematic view
of the system
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.
t
x
Kamchatka steady
Barmin, 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.
Mass conservation equations (bubbly zone)
1      1   1  c     Vm  Qm
0
0
 gVg   m 1   1    cVm  Qg
0
m
0
c
 - volume concentration of gas (1-) - of condensed phase
 - volume concentration of crystals in condensed phase=const
 - 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
Momentum and bubble growth
m  c,  Vm
dps
  g 
2
dx
D
k   d pg
Vg  Vm  
m g dx
dR
R
Vm

pg  pm 

dx 4m m  c 
ps  1    pm   pg
k    k0
3.5
 - mixture density
 - resistance coefficient
(32 - pipe, 12 -dyke)
k() - permeability
mg- gas viscosity
p- pressure “s”- mixture,
“m”- condensed phase, “g”gas
Equations in gas-particle dispersion
 g0Vg   g0 bVm  Qg  gas
m 1    bVm  Qb  big particles
s,b- volume
fractions of particles
 - volume fraction
of gas in big
particles
m sVg  Qs  small particles
dVm
 m b 1   Vm
   m b 1    g  Fgb  Fsb
dx
dVg dp
0
0





V




 g m s  g dx dx  g  m s  g  Fgb  Fsb
 m   m0 1     c0  ;    s   b  1; p   g0 RT
F - interaction forces: ”sb” - between small and big particles
“gb” - between gas and big particles
Fragmentation wave
Conservation laws
 g0 Vg   g0 Vg   g0 b Vm  gas phase




1


V


V


1


V



m sg b
m  condensed phase
pm 1      pg    m 1    Vm2   g0 Vg2  mixture momentum
p   b   m 1      g0 Vm2    g0    m s Vg2
Vm  Vm  big particles momentum
Additional relations
pg  pm  p* ;     ;
1  m  s  m 1    b
Geometric parameters
L
D
d
 m0
 c0

c0
n0
m
T0
m
k
P *
kp
P n
conduit length
3000-15000 m
conduit diameter
20-50 m
diameter of big particles
0.1-10 mm
Densities and concentrations
magma melt density
2300 kg/m3
crystals density
2700 kg/m3
volume concentration of crystals
0-0.3
initial gas concentration in the melt
0.02-0.07
initial number density of bubbles
1010-1014 m-3
mass concentration of small particles
0-1
magma temperature
850-1000 0C
Other parameters and functions
magma viscosity
Hess & Dingwell
bubbly liquid permeability
k0= 0-10-11 m2
critical overpressure for fragmentation 1-10 MPa
solubility coefficient
4.11 10-6 Pa s
nucleation overpressure
0.5-2 MPa
Ascent velocity vs.
chamber pressure
0 gD
 c0 
V0 
; p0   
32m
k
2
2
Model of
vulcanian
explosion
generated by lava
dome collapse
(Kamchatka
transient)
Assumptions
• 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
Mechanical model
Conservation of mass and number density of bubbles:
 g  gV

 0,
t
x
l lV

 0,
t x
 n  nV

 0,
t x
No mass transfer: l  (1   ) 1     m   c  ,
 g   g0
Equilibrium mt: p  p    , c  k p
Momentum:
 V

t
  V 2  pm 
x
32mV

 g 
D2


 , pm  1    pl   pg , m  m  c 

Rayleigh equation
R
t
V
R R

pg  pl  ,

 x 4m m
pg   g0 RgT ,
Fragmentation condition: pg  pl  p*

4 3
Rn
3
Results of calculation (eq. case)
Discharge rate and fragmentation depth
equilibrium mt.
no mass transfer
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
Transient Problem
__ magma
discharge
__ water influx
__ fragmentation
front
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.