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