Magma Migration Applied to Oceanic Ridges
Geophysical Porous Media Workshop Project
Josh Taron - Penn State
Danica Dralus - UW-Madison
Selene Solorza- UABC - Mexico
Jola Lewandowska - UJF France
Angel Acosta-Colon - Purdue
2 M.I.A.s
Advisors: Scott King & Marc Spiegelman
Core
Crust andLithosphere(~100km)
(~3000km)
Magma MigrationApplied to Oceanic Ridges
Magma Migration Applied to Oceanic Ridges
1. Plate Tectonics Intro (Angel)
2. Magma Migration (Danica)
3. Solitary Waves (Selene)
4. Modeling Results (Josh)
Outline
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.
•In ocean ridges, the magma will control the geochemical evolution of the planet and fundamentals of the plate tectonics dynamics.
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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
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
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
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
HistoryDrazin and Johnson (1989) describe solitary wave as solutions of nonlinear Ordinary Differential Equations which:
1. Represent waves of permanent form;2. Are localized, so that they decay or approach a constant
at infinity;3. Can interact with other solitary waves, but they emerge
from the collision unchanged apart from a phase shift.
Magma Migration Applied to Oceanic Ridges
€
φt = C (1)
- φnCx( ) x
+ C = φn( )
x(2)
Then substituting eq. (1) into (2), we have
€
φt + φn( )
x- φnφxt( )
x= 0 (3)
ς
1-D Magmatic Solitary Wave
where is porosity and C is the compaction rate.φ
Magma Migration Applied to Oceanic Ridges
€
φ x, t( ) = f x − ct( ) = f ς( ) (4)
By the chain rule
€
φt = −cdf
dς≡ −cf ' (5)
Where is the distance coordinate in a frame moving at constant speed c.
Assuming a solution of the form
1-D Magmatic Solitary Wave
Magma Migration Applied to Oceanic Ridges
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 n( )'+c fn f ' '( )'= 0
1-D Magmatic Solitary Wave
Magma Migration Applied to Oceanic Ridges
( )( ) 290,0
340,0
=
=
φ
φ
Animation of the collision of the solitary wave (From Spiegelman)
1-D Magmatic Solitary WaveFor 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:
Magma Migration Applied to Oceanic Ridges
How do behaviors vary?•The simplest case:
–Convection/Conduction transport – No mechanical considerations (uncoupled)
•Coupled examples:–Elastic systems: The Mendel-Cryer effect–Viscous systems: The solitary wave
Fluid-Mechanical Coupling
Magma Migration Applied to Oceanic Ridges
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Convection/Conduction Transport
•Homogeneous porosity•No mechanical considerations
Magma Migration Applied to Oceanic Ridges
Velocity Field
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Low PorosityRegion
Convection/Conduction Transport in Heterogeneous Media
•A bit more exciting•No mechanical considerations
Magma Migration Applied to Oceanic Ridges
•Darcy flow with Convection/Conduction to track magma location•Level Set
( ) 0 if 1
1 if 1H
λλ
λ
<⎧= ⎨
>⎩
( ) ( ) ( )1 2 1, ,P x y t P P P H λ= + − ×
Media Magma
Smoothing FunctionCoupling:
Convection Velocity = Darcy Velocity
Why COMSOL? Starting from scratch…time constraints
A bit about the method so far…
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
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.
• 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)
Magma Migration Applied to Oceanic Ridges
And the viscous scenario…
• Recall the derivation for coupled flow and deformation in a viscous porous medium
• No need for level-set• What are the mechanical effects?
– Remember the solitary wave
kn n
Ct
C C
φ
φ φ
∂=
∂−∇⋅ ∇ + =∇⋅
Neglects melting (reaction)
Magma Migration Applied to Oceanic Ridges
Fluid-Mechanical in a Viscous Medium: Solitary Wave
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.
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Magma Migration Applied to Oceanic Ridges
What would we like to do?
• Couple the reaction equation (mass transfer)…
…to the fluid-mechanical viscous medium derivation
• System mimics the “salt on beads” interaction
( )f feqDaA c cΓ = −
Da(R) = Damkohler Number (relation of reaction speed to velocity of flow)A = Area of Dissolving phase (matrix) available to reactioncf
eq-cf = Distance of reacting solubility (i.e. melting solid fraction in molten flow) from equilibrium
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
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Magma Migration Applied to Oceanic Ridges
Concluding Remarks (in picture form)
Fluid only Fluid only
Fluid/Mechanical Fluid/Reactive (melt)
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