magma migration applied to oceanic ridges geophysical porous media workshop project josh taron -...

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


1. Plate Tectonics Intro (Angel)

2. Magma Migration (Danica)

3. Solitary Waves (Selene)

4. Modeling Results (Josh)

Outline


Earthquakes


Volcanoes


Plate Tectonics Boundaries


Types of Boundaries


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.

QuickTime™ and aSorenson Video 3 decompressorare needed to see this picture.


Core

Crust andLithosphere(~100km)

(~3000km)


So, what makes magma migration strange?

Localized Flow


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.


Implications?

US

GS

Na

tion

al G

eo

gra

ph

ic

BB

C N

ew

s


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


Governing Equations


Compressible Flow Equations (No Shear, No Melting)


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.


Solitary Waves


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.


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.


€

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


€

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



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



( )( ) 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:


Modeling Results


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


QuickTime™ and aCinepak decompressor

are needed to see this picture.

Convection/Conduction Transport

•Homogeneous porosity•No mechanical considerations


Velocity Field



Low PorosityRegion

Convection/Conduction Transport in Heterogeneous Media

•A bit more exciting•No mechanical considerations


•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…


What about mechanical coupling? Does it dramatically change the system?

1.The elastic scenario (near surface)

2.The viscous scenario (way down there)


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)


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)


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.






3D Solitary Waves

From Wiggings & Spiegelman, 1994, GRL


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


What would we like for that to look like?


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…


Applying the level set method from before…

• Adding reaction (melting) the result becomes




Concluding Remarks (in picture form)

Fluid only Fluid only

Fluid/Mechanical Fluid/Reactive (melt)










The End…Questions?

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