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

Partia lly (3 vol % ) m oltenasthenosphere

Basalt d ikes

Basalt s ills

M assive D unites

R eplacive D unites

R eplacive D unites = reactive transport channeling instability?

Basalt d ikes = se lf propagating cracks?

Basalt s ills = segregation caused by m agica l perm eability barriers?

M assive D unites = rem obilized replacive dunite?

M id-O cean R idge

lithosphere

1D Flow Instability, Small (<<1) Formulation, Initial Conditions

-250 -200 -150 -100 -50 0

2

4

6

8t = 0

z

-250 -200 -150 -100 -50 0-1

-0.5

0

0.5

1

z

p

1 1.5 2 2.5 3 3.5 4 4.5 5-1

-0.5

0

0.5

1

p =

d, disaggregation condition

1D Movie? (b1d)

1D Final

-350 -300 -250 -200 -150 -100 -50 0

1

2

3

4

5t = 70

z

-350 -300 -250 -200 -150 -100 -50 0-1

-0.5

0

0.5

1

z

p

1 1.5 2 2.5 3 3.5 4 4.5 5-1

-0.5

0

0.5

1

p

• Solitary vs periodic solutions

• Solitary wave amplitude close to source amplitude

• Transient effects lead to mass loss

2D Instability

Birth of the Blob

• Stringent nucleation conditions

• Small amplification, low velocities

• Dissipative transient effects

Bad news for Blob fans:

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 c, width d

But are they solitary waves?

Velocity and Amplitude

0 5 10 15 20 25 30 353.4

3.6

3.8

4

4.2

4.4

4.6

4.8

5

5.2

time /

Blob model

amplitudevelocity

0 0.5 1 1.5 2 2.5 3 3.50

5

10

15

20

25

30

35

40

time /

Dike model

amplitudevelocity

1D Quasi-Stationary State

4.5 5 5.5-10

-5

0

5

10

15

20

25

30

35

x/

Horizontal Section

-60 -40 -20 0-10

-5

0

5

10

15

20

25

30

35

y/

Vertical Section

0 10 20 30 40

-6

-4

-2

0

2

4

6

p

Phase Portrait

Pressure,Porosity

Pressure,Porosity

• 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

1

1

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?

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(), Eirik’s talk

• Viscous bulk rheology with

• 1D stationary states traveling with phase velocity

es

s

2 2q

11

d

1

q

mq

pv

f

f

es

s

pv

(geological formulations ala McKenzie have )

Balancing ball

gv h

t x

v p

x

t z

0 ,p

fz

xv

t

1( )s

pf

z

v g h

x v x

sp H

p

0h

vdv g dxx

0 s

Hpdp d

2

2

vE hg

2

2 sp

U H sg

Porosity WaveBalancing Ball

H(omega)

Phase diagram

Sensitivity to Constituitive Relationships