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A numerical program for steady flow through volcanic conduits Larry G. Mastin U.S. Geological...
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Transcript of A numerical program for steady flow through volcanic conduits Larry G. Mastin U.S. Geological...
a numerical program for steady flow through volcanic conduits
Larry G. Mastin
U.S. Geological Survey
Assumptions and simplifications• Steady-state• Vertical conduit• 1-dimensional• homogeneous flow• Equilibrium degassing (distributed
version)• No heat or mass flow through
conduit walls
Main features
• User-friendly, visual interface• Publicly-documented, open-source code
• See http://vulcan.wr.usgs.gov/Projects/Mastin• Calculates solubility & thermodynamics using
• Full MELTS equations (Ghiorso & Sack, 1995) for melts
• Full equation of state for gas (H2O)• Uses viscosity of bubbly melts formulation
based on Capillary number.• Fragmentation criterion based either on
volume fraction gas or strain rate.
Weaknesses of this model (and others)
• Poorly characterized rheology• Kinetics of degassing, crystallization• Magma-host rock interaction
• Conduit geometry
• Difficulty in comparing results with observations
ODE for 1-D flow
• Option 1: specify conduit geometry, solve for pressure
2
22
1 Mdz
dA
A
u
r
fug
dz
dp
Boundaryconditions
m a g m ab o d y
pressure
dia
met
er s
pe
cifi
ed
c alcula ted
p M o r
p
A=x-sectional area r=radiusf=friction factor =densityM=Mach #u=velocity
gravityfriction geometry
Mach #
ODE for 1D flow
• Option 2: Specify pressure gradient and calculate conduit geometry
r
ufgM
dz
dp
u
A
dz
dA 22
2)1(
p
p
Boundaryconditions
m a g m ab od y
pressuresp
ecif ied
dia
met
er c
alc
ula
ted
Pressure gradient
gravity friction
Friction Factor
Log Re2 7
f
.001
1
004.16
uD
f
16/Re
laminar turbulent
viscosity
Viscosity• Melt viscosity
• SiO2>70 wt%: Hess & Dingwell (1996)
• SiO2<70 wt%: Shaw (1972)
• Effect of Crystals• <40%: Roscoe-Einstein (Marsh calibration)
5.2
6.01
vm
Volume fraction crystals
Melt viscosity
Viscosity (cont’d)
• Effect of bubbles ngm v 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80
0.5
1
1.5
2
2.5
3
3.5
4
vg
/ m
n=-1
-0.5
0
0.5 1
-1<n<1
Depending on Capillary number
Relationship between n and Ca
))log(5(tan2 1 Can
-2 -1 0 1 2-1
0
1
Log10(Ca)
n
Calculating Capillary number
=Surface tension=0.34 N/m
r=avg. bubble radius = (3vg/(4Nvm))1/3 , where
Vg, vm are volume fractions gas and melt, respectively
N=nuclei per unit volume; log(N) =~0.2*wt%SiO2
=avg. shear-strain rate=~4u/3D under laminar flow
r
Ca m
m ea ndu/d r
Conduit flow profile
Demo of the program
Ongoing & future work
• Kinetics of magma degassing (with M. Mangan)
• Calculating bubble-size distributions• Improved fragmentation criterion• Improved rheology of bubbly magmas• Elastic coupling of fluid & host rock• 2-D, transient flow (with R. Denlinger)• Etc. etc.
Kinetics of magma degassing
• In collaboration with Margaret Mangan, Tom Sisson
0 100 2000
2
4
6w
t% H
O 2
e q uilib rium
no n-e q uilib rium ( = 8.5 M Pa /s)d p /d t
p re ssure , M Pa
Data from Mangan & Sisson (2001)
SampleResults
• Depend significantly on fragmentation criterion (vg=.75 or strain-rate)
0 100 200-8
-7
-6
-5
-4
-3
-2
-1
0
pressure, MPa
de
pth
, k
m
0 0.5 1-8
-7
-6
-5
-4
-3
-2
-1
0
vfgas-5 0 5
-8
-7
-6
-5
-4
-3
-2
-1
0
log velocity, m/s-10 0 10
-8
-7
-6
-5
-4
-3
-2
-1
0
log viscosity, Pa s
equilibriumdelayed