Magma Migration
Applied to Oceanic Ridges
Geophysical Porous Media Workshop Project
Josh Taron - Penn State
Danica Dralus - UW-Madison
Crust and
Lithosphere
(~100km)
Selene Solorza- UABC - Mexico
Jola Lewandowska - UJF France
Angel Acosta-Colon - Purdue
2 M.I.A.s
Core
(~3000km)
Advisors: Scott King & Marc Spiegelman
Magma Migration Applied to Oceanic Ridges
Outline
1. Plate Tectonics Intro (Angel)
2. Magma Migration (Danica)
3. Solitary Waves (Selene)
4. Modeling Results (Josh)
Magma Migration Applied to Oceanic Ridges
Earthquakes
Magma Migration Applied to Oceanic Ridges
Volcanoes
Magma Migration Applied to Oceanic Ridges
Plate Tectonics Boundaries
Magma Migration Applied to Oceanic Ridges
Types of Boundaries
Magma Migration Applied to Oceanic Ridges
Plate Tectonics Boundaries
•Earth is divided into dynamics rigid
plates.
•The plates are continuously created
and “recycled”.
•Magma migration affects the plates
evolution.
QuickTime™ and a
Sorenson Video 3 decompressor
are needed to see this picture.
•In ocean ridges, the magma will
control the geochemical evolution of
the planet and fundamentals of the
plate tectonics dynamics.
Magma Migration Applied to Oceanic Ridges
Crust and
Lithosphere
(~100km)
Core
(~3000km)
Magma Migration Applied to Oceanic Ridges
So, what makes magma migration strange?
Localized Flow
Magma Migration Applied to Oceanic Ridges
Supporting Evidence (an example)
•MORBs are typically undersaturated
in OPX.
•OPX is plentiful in the mantle and
dissolves quickly in undersaturated
mantle melts.
•Observations suggest MORBs travel
through at least the top 30 km of
oceanic crust without equilibrating
with residual mantle peridotite.
•MORBs are also not in equilibrium
with other trace elements.
Magma Migration Applied to Oceanic Ridges
USGS
National Geographic
BBC News
Implications?
Magma Migration Applied to Oceanic Ridges
What do we need for a working theory?
• At least 2 phases (melt and solid)
• Allow mass-transfer between phases (melting/reaction/crystallization)
• System must be permeable at some scale
• System must be deformable (consistency with mantle convection)
• Chemical Transport in open systems
Magma Migration Applied to Oceanic Ridges
Governing Equations
Magma Migration Applied to Oceanic Ridges
Compressible Flow Equations (No Shear, No Melting)
Magma Migration Applied to Oceanic Ridges
Dimensionless Compressible Flow Equations
That is, porosity only changes by dilation/compaction. The
compaction rate is controlled by the divergence of the melt flux
and the viscous resistance of the matrix to volume changes.
Magma Migration Applied to Oceanic Ridges
Solitary Waves
Magma Migration Applied to Oceanic Ridges
History
•On August 1834 the Scottish engineer John Scott Russell (1808-1882)
made a remarkable scientific discovery: The solitary wave.
•Russell observed a solitary wave in the Union Canal, then he reproduced
the phenomenon in a wave tank, and named it the “Wave of Translation.
Magma Migration Applied to Oceanic Ridges
History
Drazin and Johnson (1989) describe solitary wave as solutions of
nonlinear Ordinary Differential Equations which:
1.
2.
3.
Represent waves of permanent form;
Are localized, so that they decay or approach a constant at
infinity;
Can interact with other solitary waves, but they emerge from the
collision unchanged apart from a phase shift.
Magma Migration Applied to Oceanic Ridges
1-D Magmatic Solitary Wave
t  C
(1)
-  n Cx   C   n 
x
x
(2)
where  is porosity and C is the compaction rate.
Then substituting eq. (1) into (2), we have


t   n x -  n xt x  0

Magma Migration Applied to Oceanic Ridges
(3)
1-D Magmatic Solitary Wave
Assuming a solution of the form
 x,t   f x  ct   f  
(4)
Where  is the distance coordinate in a frame moving at

constant speed c.
By the chain rule
df
 t  c
 cf '
d
(5)
Magma Migration Applied to Oceanic Ridges
1-D Magmatic Solitary Wave
From eq. (1) and (5), the compaction rate satisfies
C  cf '
Thus, eqs. (1) and (2) are transformed into the non-linear
ODE
cf 'f 'cf f ''' 0
n
n
Magma Migration Applied to Oceanic Ridges
1-D Magmatic Solitary Wave
For n=3, using the second order Runge-Kutta numerical
method to solve the 1-D magmatic solitary wave eq. (4)
for periodic boundary conditions and initial conditions:
 0,40   3
 0,90   2
Animation of the collision of the solitary wave (From Spiegelman)
Magma Migration Applied to Oceanic Ridges
Modeling Results
Magma Migration Applied to Oceanic Ridges
Fluid-Mechanical Coupling
How do behaviors vary?
•The simplest case:
–Convection/Conduction transport – No
mechanical considerations (uncoupled)
•Coupled examples:
–Elastic systems:
–Viscous systems:
The Mendel-Cryer effect
The solitary wave
Magma Migration Applied to Oceanic Ridges
Convection/Conduction Transport
QuickTime™ and a
Cinepak decompressor
are needed to see this picture.
•Homogeneous porosity
•No mechanical considerations
Magma Migration Applied to Oceanic Ridges
Convection/Conduction Transport in
Heterogeneous Media
Low Porosity
Region
Velocity Field
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Cinepak decompressor
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•A bit more exciting
•No mechanical considerations
Magma Migration Applied to Oceanic Ridges
A bit about the method so far…
•Darcy flow with Convection/Conduction to
track magma location
•Level Set
Media
Magma
0 if   1
H    
1 if   1
P  x, y, t   P1   P2  P1   H   
Smoothing Function
Coupling:
Convection Velocity = Darcy Velocity
Why COMSOL? Starting from scratch…time constraints
Magma Migration Applied to Oceanic Ridges
What about mechanical coupling? Does it
dramatically change the system?
1.The elastic scenario (near surface)
2.The viscous scenario (way down there)
Magma Migration Applied to Oceanic Ridges
Elastic Systems: The Mendel-Cryer Effect
• Described by Biot Theory (Linear Poroelasticity)
• Verified in laboratory and at field scale
• Is well defined (unlike for a viscous medium) and pressure effects of a
similar response will alter behavior of fluid transport (coupled system)
Images from Abousleiman et al., (1996). Mandel’s Problem Revisited. Géotechnique, 46(2): 187-195.
Mandel, J. (1953). Consolidation des sols (étude mathématique). Géotechnique, 3: 287-299.
Skempton, A.W. (1954). The pore pressure coefficients A and B. Géotechnique, 4: 143-147.
Magma Migration Applied to Oceanic Ridges
And the viscous scenario…
• Recall the derivation for coupled flow and
deformation in a viscous porous medium

Neglects melting (reaction)
C
t
   nC  C     n k
• No need for level-set
• What are the mechanical effects?
– Remember the solitary wave
Magma Migration Applied to Oceanic Ridges
Fluid-Mechanical in a Viscous
Medium: Solitary Wave
QuickTime™ and a
Cinepak decompressor
are needed to see this picture.
QuickTime™ and a
Cinepak decompressor
are needed to see this picture.
The mathematics are well posed. Does this actually occur??
In the second video, the matrix is allotted a downward velocity.
Watch for the phase shift.
Magma Migration Applied to Oceanic Ridges
3D Solitary Waves
From Wiggings & Spiegelman, 1994, GRL
Magma Migration Applied to Oceanic Ridges
What would we like to do?
• Couple the reaction equation (mass transfer)…

  DaA ceqf  c f
Da(R)
A
cfeq-cf

= Damkohler Number (relation of reaction speed to velocity of flow)
= Area of Dissolving phase (matrix) available to reaction
= Distance of reacting solubility (i.e. melting solid fraction in molten flow)
from equilibrium
…to the fluid-mechanical viscous medium derivation
• System mimics the “salt on beads” interaction
Magma Migration Applied to Oceanic Ridges
What would we like for that to look
like?
Magma Migration Applied to Oceanic Ridges
What does it look like?
• What do we need to make it work?
1. Time 2. Bigger computer 3. Sanity 4. Siesta 5. Beer
• The backup plan…
Magma Migration Applied to Oceanic Ridges
Applying the level set method
from before…
• Adding reaction
(melting) the
result becomes
QuickTime™ and a
Cinepak decompressor
are needed to see this picture.
Magma Migration Applied to Oceanic Ridges
Concluding Remarks (in picture form)
Fluid only
Fluid only
QuickTime™ and a
Cinepak decompressor
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QuickTime™ and a
Cinepak decompressor
are needed to see this picture.
QuickTime™ and a
Cinepak decompressor
are needed to see this picture.
QuickTime™ and a
Cinepak decompressor
are needed to see this picture.
Fluid/Mechanical
Magma Migration Applied to Oceanic Ridges
Fluid/Reactive (melt)
The End…
Questions?
Magma Migration Applied to Oceanic Ridges