Geophysical Porous Media Workshop Project

1. Crust and Lithosphere (~100km) Core (~3000km) Magma MigrationApplied 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 Magma Migration Applied to Oceanic Ridges

2. Outline • Plate Tectonics Intro (Angel) • Magma Migration (Danica) • Solitary Waves (Selene) • Modeling Results (Josh) Magma Migration Applied to Oceanic Ridges

3. Earthquakes Magma Migration Applied to Oceanic Ridges

4. Volcanoes Magma Migration Applied to Oceanic Ridges

5. Plate Tectonics Boundaries Magma Migration Applied to Oceanic Ridges

6. Types of Boundaries Magma Migration Applied to Oceanic Ridges

7. 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. Magma Migration Applied to Oceanic Ridges

8. Crust and Lithosphere (~100km) Core (~3000km) Magma Migration Applied to Oceanic Ridges

9. So, what makes magma migration strange? Localized Flow Magma Migration Applied to Oceanic Ridges

10. 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

11. Implications? BBC News USGS National Geographic Magma Migration Applied to Oceanic Ridges

12. 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

13. Governing Equations Magma Migration Applied to Oceanic Ridges

14. Compressible Flow Equations (No Shear, No Melting) Magma Migration Applied to Oceanic Ridges

15. Dimensionless Compressible Flow Equations Solitary Waves 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

16. Solitary Waves Magma Migration Applied to Oceanic Ridges

17. 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

18. History • Drazin and Johnson (1989) describe solitary wave as solutions of nonlinear Ordinary Differential Equations which: • 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

19. where is porosity and C is the compaction rate. 1-D Magmatic Solitary Wave Then substituting eq. (1) into (2), we have Magma Migration Applied to Oceanic Ridges

20.  Where is the distance coordinate in a frame moving at constant speed c. 1-D Magmatic Solitary Wave Assuming a solution of the form By the chain rule Magma Migration Applied to Oceanic Ridges

21. 1-D Magmatic Solitary Wave From eq. (1) and (5), the compaction rate satisfies Thus, eqs. (1) and (2) are transformed into the non-linear ODE Magma Migration Applied to Oceanic Ridges

22. 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: Animation of the collision of the solitary wave (From Spiegelman) Magma Migration Applied to Oceanic Ridges

23. Modeling Results Magma Migration Applied to Oceanic Ridges

24. Fluid-Mechanical Coupling • 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 Magma Migration Applied to Oceanic Ridges

25. Convection/Conduction Transport • Homogeneous porosity • No mechanical considerations Magma Migration Applied to Oceanic Ridges

26. Convection/Conduction Transport in Heterogeneous Media Low Porosity Region Velocity Field • A bit more exciting • No mechanical considerations Magma Migration Applied to Oceanic Ridges

27. A bit about the method so far… • Darcy flow with Convection/Conduction to track magma location • Level Set Magma Media Smoothing Function Coupling: Convection Velocity = Darcy Velocity Why COMSOL? Starting from scratch…time constraints Magma Migration Applied to Oceanic Ridges

28. 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

29. 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) 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. Magma Migration Applied to Oceanic Ridges

30. 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 And the viscous scenario… Neglects melting (reaction) Magma Migration Applied to Oceanic Ridges

31. 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. Magma Migration Applied to Oceanic Ridges

32. 3D Solitary Waves From Wiggings & Spiegelman, 1994, GRL Magma Migration Applied to Oceanic Ridges

33. Couple the reaction equation (mass transfer)… …to the fluid-mechanical viscous medium derivation System mimics the “salt on beads” interaction What would we like to do? Da(R) = Damkohler Number (relation of reaction speed to velocity of flow) A = Area of Dissolving phase (matrix) available to reaction cfeq-cf = Distance of reacting solubility (i.e. melting solid fraction in molten flow) from equilibrium Magma Migration Applied to Oceanic Ridges

34. What would we like for that to look like? Magma Migration Applied to Oceanic Ridges

35. What do we need to make it work? Time 2. Bigger computer 3. Sanity 4. Siesta 5. Beer The backup plan… What does it look like? Magma Migration Applied to Oceanic Ridges

36. Applying the level set method from before… • Adding reaction (melting) the result becomes Magma Migration Applied to Oceanic Ridges

37. Concluding Remarks (in picture form) Fluid only Fluid only Fluid/Mechanical Fluid/Reactive (melt) Magma Migration Applied to Oceanic Ridges

38. The End…Questions? Magma Migration Applied to Oceanic Ridges