Transcript Transition from Pervasive to Segregated Fluid Flow in Ductile Rocks

Transition from Pervasive to Segregated Fluid Flow in Ductile Rocks
James Connolly and Yuri Podladchikov, ETH Zurich
A transition between “Darcy” and Stokes regimes
•
• Geological scenario
Review of steady flow instabilities => porosity waves
• Analysis of conditions for disaggregation
lithosphere
Mid-Ocean Ridge
Lithosphere
Basalt sills
Partially (3 vol %) molten
asthenosphere
Massive Dunites
Replacive
Dunites
Basalt dikes
Replacive Dunites = reactive transport channeling instability?
Basalt dikes = self propagating cracks?
Basalt sills = segregation caused by magical permeability barriers?
Massive Dunites = remobilized replacive dunite?
1D Flow Instability, Small f (<<1-f) Formulation, Initial Conditions
t=0
8
f = f , disaggregation condition
d
f
6
4
2
-250
-200
-150
-100
-50
0
-100
-50
0
z
1
p
0.5
0
-0.5
-1
-250
-200
-150
z
1
p
0.5
0
-0.5
-1
1
1.5
2
2.5
f
3
1D Movie? (b1d)
3.5
4
4.5
5
t = 70
5
4
f
1D Final
3
2
1
-350
-300
-250
-200
-150
-100
-50
0
-150
-100
-50
0
z
1
p
0.5
0
-0.5
-1
-350
-300
-250
-200
z
1
p
0.5
0
-0.5
-1
1
1.5
2
2.5
3
3.5
4
f
• Solitary vs periodic solutions
• Solitary wave amplitude close to source amplitude
• Transient effects lead to mass loss
4.5
5
2D Instability
Birth of the Blob
Bad news for Blob fans:
•
Stringent nucleation conditions
•
Small amplification, low velocities
•
Dissipative transient effects
Is the blob model stupid?
A differential compaction model
Dike Movie? (z2d)
The details of dike-like waves
Comparison movie (y2d2)
Final comparison
• Dike-like waves nucleate from essentially nothing
• They suck melt out of the matrix
• They are bigger and faster
• Spacing dc, width dd
But are they solitary waves?
Velocity and Amplitude
Blob model
Dike model
5.2
40
amplitude
velocity
amplitude
velocity
5
35
4.8
30
4.6
25
4.4
20
4.2
15
4
10
3.8
5
3.6
3.4
0
0
5
10
15
20
time / t
25
30
35
0
0.5
1
1.5
2
time / t
2.5
3
3.5
1D Quasi-Stationary State
Horizontal Section
Vertical Section
35
30
Phase Portrait
35
Pressure,
Porosity
30
25
25
20
20
Pressure,
Porosity
6
4
f1
2
15
p
15
10
10
5
5
0
0
-5
-5
f1
0
-2
-4
-6
-10
4.5
5
x/d
5.5
-10
-60
-40
-20
0
0
y/d
• Essentially 1D lateral pressure profile
• Waves grow by sucking melt from the matrix
•The waves establish a new “background”” porosity
• Not a true stationary state
10
20
f
30
40
So dike-like waves are the ultimate in porosity-wave fashion:
They nucleate out of essentially nothing
They suck melt out of the matrix
They seem to grow inexorably toward disaggregation
But
Do they really grow inexorably, what about 1-f?
Can we predict the conditions (fluxes) for disaggregation?
Simple 1D analysis
Mathematical Formulation
• Incompressible viscous fluid and solid components
• Darcy’s law with k = f(f), Eirik’s talk
• Viscous bulk rheology with
  vs  -f

peq

f
s
1 - f 
q 1
 fd - f 
2 - 2q
f f
m -1
(geological formulations ala McKenzie have   vs  -
pe
)
s
• 1D stationary states traveling with phase velocity w
Balancing ball
Balancing Ball
v
h
- g
t
x
Porosity Wave
v p
x f
tz
x
v
t

v
g h
x
v x
0  vdv  g
E
h
dx
x
v2
 hg
2
p
 f 0 f , w 
z
f
p
 - f1 (f )
z
ws
p
w H
- s
f
p f
0  pdp  ws

g  w s
U
H
df
f
p2
 ws H
2
H(omega)
Phase diagram
Sensitivity to Constituitive Relationships