A Dynamical Model of Seismogenic Volcanic Extrusion, Mount St. Helens, 2004-2005

Richard IversonU.S. Geological SurveyCascades Volcano Observatory

S. Schilling photoFeb. 22, 2005

Fact 1: extrusion rate of solid dacite plug is nearly constant when measured over timescales ranging from a few minutes to a few months

Time (month/day/year)

10/1/04 12/1/04 2/1/05 4/1/05 6/1/05 8/1/05 10/1/05 12/1/05

Ex

tru

de

d v

olu

me

(m3 )

0

20x106

40x106

60x106

80x106

steady extrusion at 1.5 m

3 /s

Dec. 2004 - D

ec. 2005

Fact 1: extrusion rate of solid dacite plug is nearly constant when measured over timescales ranging from a few minutes to a few months

S. Schilling photo

Fact 2: striated fault gouge that coats thesurface of the newly extruded dacite plugexhibits rate-weakening frictional strength

Displacement (mm)0 2 4 6 8

Sh

ear

stre

ss (

kPa)

68

69

70

71

72

Ap

pro

xim

ate

fric

tio

n c

oef

fici

ent

0.450

0.455

0.460

0.465

0.470

0.475

faster slip (3 x 10-6

m/s)

slower slip (1.5 x 10-6

m/s)

constant normal stress = 159 kPa

Fact 2: striated fault gouge that coats thesurface of the newly extruded dacite plugexhibits rate-weakening frictional strength

Example of 24 hours of seismicity, Dec. 1, 2005Fact 3: repetitive “drumbeat” earthquakesoccurred almost periodically (T ~ 100 s), hadmagnitudes ≤ 2, hypocenters < 1 km directly beneath the new dome, and mostly“hybrid” waveforms with impulsive onsets.

Magma compressibility α1

Conduit compliance α2

Magma density ρ

Constants

Parameters that evolveas prescribed functionsof dependent variablesor time

Dependent variablesthat evolve with time

Rock density ρr

1-D“SPASM” model

0

1du g pA u F

dt m t

1 2

1/dp V Au RB Q

dt

1

1 2

dV A u RB Q Q B

dt

01 01 1 exp[ ( )]

r r

R p p

10sgn( ) 1 sinh

ref

uF u mg c

u

where

and

1-D conservation of mass and momentum leads to

01(1 ) 2

( )

2

2

Kgtu du ud Kt D dt V V Q RB /Adt

/[( ) / ]u u Q RB A 0t = t /t0/V V V

1 20 1 2 0 0 0

00 0

[ ( ) ] 1

2

/m + V t t dFt K D K

A m m du

where

Obtain equation for damped, forced oscillations of normalized extrusion velocity

Find exact solutions, steady or oscillatory, if V´ =1and D is constant, but behavior is unstable for D < 0

Predicted free oscillation period of u' (linear theory)

0 0 1 20 1 2

[ ( )]2 2 2 [( ) ]r con plug

m VT t H H

A

Results for ρr=2000 kg/m3

Hcon = 8 km

Variable damping D arises from use of nonlinear rate-weakening friction rule for sliding at plug margins:

Relative velocity, u / uref

0 20 40 60 80 100

Rel

ativ

e fr

ictio

n co

effic

ient

, /

0

0.80

0.85

0.90

0.95

1.00

c = - 0.005

c = - 0.025

c = - 0.02

c = - 0.015

c = - 0.01

c = - 0.03

for u/uref <1, approximates linear rate dependence

for u/uref >1, approximates logarithmic rate dependence

10sgn( ) 1 sinh

ref

uF u mg c

u

If κ = 0, B = Q, and t0 is constant, behavior of numerical solutions depends almost entirely on D evaluated at the equilibrium slip rate u = u0= Q/A:

0

1/ 22

0 0 0 00

0 0

1 11

2 2u u ref ref

t gt u udFD c

m du u u u

00

0

1

2

gtD c

u

which simplifies to

if u0/uref >> 1

Computedstart-upbehavior withT =10 s, D =−0.01 and initial conditionsu = Q/A, p = p0,V = V0

Phase-plane representation of start-up behavior withD = −0.01 and initial conditions u=Q/A, p = p0, V = V0

Time series and phase-plane representations of stick-slip limit cycles computed for T =10 s and various values of D, with initial conditions u = 0, p = p0, V = V0

With D = -2, work done against friction during a slip cycle is 2×108 J, similar to energy release in a M 2.3 earthquake

Details for D = −2

For fixed D, sensitivity of limit cycles to choice of u0/uref in the friction rule is slight, provided that u0/uref ≥ 1

Results for D = −2

For fixed D,sensitivity of limit cycles tochoices of c and λis nil. That is,static friction andrate weakeninghave counter-balancing effects on dynamics.

Results for D = −2

Commensurate with7 × 107 N force dropduring slip event

Effect of disequilibrium initial condition(0.005% initial excess magma pressure)

Conclusions1. Stick-slip oscillations are inevitable as a consequence of momentum

conservation, driving force supplied by compressible magma, restoring force supplied by gravity, and rate-weakening plug boundary friction.

2. Use of realistic (i.e. best-guess) parameter values produces stick-slip oscillations with roughly the correct period, amplitude, and force drop to

produce repetitive “drumbeat” earthquakes at MSH.

3. Fluctuations in magma pressure during stick-slip cycles are very small, a few kPa, implying that departures from equilibrium are very slight.

4. Long-term, oscillatory behavior of the system is remarkably stable unless magma influx or composition changes or friction evolves.

5. Initial conditions far from equilibrium probably didn’t exist at MSH. If they had, a large pulse of motion would have occurred initially, irrespective of the type of frictional resistance.