Modélisation analogique et numérique des processus plutoniques: comment ? Pourquoi?

1. Modélisation analogique et numérique des processus plutoniques: comment ? Pourquoi? Catherine Annen Thierry Menand

2. Modelling: why ? • What we try to do: • Identify physical processes • Identify conditions required for an observed process • What we do not do: • Reproduce nature complexity

3. Modelling: why ? « Better to be approximately correct than precisely wrong » Warren Buffet

4. Examples • What are the conditions for: • a thermal aureole to reach a certain size ? • the nucleation of dykes ? • intrusion of sill ? • convection ? • the formation of a magma chamber ?

5. Analogical modelling: why ? • Verify a model • Identify physics (force balance) associated with a mechanism Complements analytical or numerical modelling

6. Example • Conditions for turbulence: Reynolds number http://www.flometrics.com/products/fluids_lab/reynold1_transition2.jpg http://pen.physik.uni-kl.de/medien/MM_Videos/reynolds/figure2.jpg

7. Analogical modelling: how ? • Choice of materials and scaling in order to respect forces balance. • Using low-viscosity fluid to model a magma is correct if the flow regime is respected (for example laminar) • Non-dimensionalisation and results analysis

8. Analogical modelling: limits • Scaling • Material needs to be appropriate for the studied process and the size of experiments. Force balance same as for the natural process. • Choice is limited

9. Analogical modelling: limits (van Wyk de Vries and Merle, 1996) • Example • Granular material for faults (friction). • Limit : no cohesion (no tensile fault or dykes).

10. Analogical modelling: limits • Example • Solid gelatine to model elastic deformations and tensile faults or dykes. • Limit : high friction (no other types of faults). (Kavanagh et al., 2006)

11. Analogical modelling: limits • Border effects • Experiments have a finite size. Borders can affect the results. Natural processes are uncontained.

12. Analogical modelling: example • Elongated laccolith composed of numerous stacked sills. • Trachyte Mesa laccolith, Henry Mountains, Utah: (Horsman et al., 2006)

13. Analogical modelling: example What are the mechanisms controlling sill emplacement? • Hypothesis • Edifice loading:Can an edifice induce a stress regime favourable for the formation of sills? • Buoyancy:Can density contrasts control the emplacement of sills? • Lithology:Can lithological discontinuities and the rigidity contrasts between adjacent layers play a role in sill formation?

14. External reservoir Gelatine solid Analogical modelling: example • Density and rigidity of gelatine, • Density of fluid, • Layering of gelatine solid, • Pressure of injection, • Load can be applied on gelatine surface. Injection point

15. Test 1: Edifice loading • Edifice loading can lead to vertical arrest of dyke underneath the edifice. • Once vertical propagation ceases, dyke propagates horizontally until edifice loading becomes negligible, at which point vertical propagation resumes. sédifice (Kavanagh et al., 2006)

16. and Stress perturbation induced by edifice loading Sill formation requires sver = s3. Edifice loading alone leads to sver > shor.

17. (Lister & Kerr, 1991) Test 2: Density contrasts • Density contrasts alone cannot control sill emplacement. • Density contrasts may lead to vertical arrest of dykes but intrusions continue as horizontal dykes not as sills. (Kavanagh et al., 2006)

18. Test 3: Rigidity contrasts • Sill formed when feeder dyke intersected the interface between a more rigid (stronger) layer overlying a less rigid (weaker) layer. Sill intruded at the interface layer. • Sill lifted upper layer when its length was ~ 2-3 times larger than the thickness of upper layer, resembling a laccolith. (Kavanagh et al., 2006)

19. Numerical modelling: why? • Answer to a closed question (yes or no): • Can a giant sill of basalt heat up several cubic km of upper crust above 950°C ? • Answer to an open question (quantification): • What is the emplacement rate of a pluton that is compatible with large-scale convection ?

20. Numerical modelling: why? Note the distinction between: • Analytical modelling • Numerical simulation

21. Analytic solution Flux de magma dans la chambre: 4.5 x 10-4 km3/an Baehr and Stephan, 2006 Annen et al. 2008

22. Numerical simulation • Discretisation of space and time • Calculation at the nodes of a grid

23. Numerical simulation: flexibility Volume constant: Sphère Dyke Sill Corps qui croît: r: éruption/intrusion Accrétion de dykes Accrétion de sills Annen et al. 2008

24. Annen et al. 2008

25. Numerical simulation: flexibility • Volume Vcrit above 875°C (0.3 km3) • Injection geometry • Duration

26. Sills 4 km long Injection rate: 6 cm/yr; 7.5 x 10-4 km3/yr Incubation: 50 kyrs; Intruded volume : 48 km3; Eruptable volume: 10 km3 Injection rate: 4 cm/yr; 5 x 10-4 km3/yr Volume erupted: 34 eruptions of 0.1-0.5 km3 (Fichaut et al., 1989) =~3.4-17 km3 in 13,500 yrs = 2.5-13 x 10-4 km3/yr

27. Numerical techniques • Finite differences • Finite elements • Boundary elements • Latice-Boltzman

28. Variation de la temperature Chaleur latente de cristallisation/fusion H Flux de chaleur q Finite differences H c : chaleur spécifique, k: conductivité thermique, L: chaleur latente, T: temperature, t: temps, x: distance, X: degré de fusion, r: masse volumique

29. Finite differences 1-D H Stability criteria: Dt<Dx2/2k

30. Finite differences 2-D i-1,j i,j-1 i,j i,j+1 Stability criteria: Dt<Dx2/4k i+1,j

31. Finite differences 3-D Stability criteria: Dt<Dx2/6k

32. Pseudo-3-D Dt<Dr2/4k

33. Finite differences Forward: Dt<Dx2/2k Backward No stability criteria Series of equation to solve simultaneously

34. Finite elements • Partial differential equations • Used for deformation • Variable mesh http://upload.wikimedia.org/wikipedia/commons/8/80/Example_of_2D_mesh.png

35. Boundary elements • Discretisation and resolution at the boundary of the system • Reduction of dimension in comparison with finite elements

36. Latice-Boltzman • Used in fluid dynamics • Particles move and collide • High computing power (parallel computing)

37. Numerical modelling • Initial conditions : important only for systems that do not reach equilibrium • Boundary conditions: • Fixed: Dirichlet. Example: Fixed T • Fixed derivate: Neumann. Example: Fixed heat flux • Boundary conditions are not important if the size of the system under study is much smaller that the numerical domain.

38. Numerical simulation

39. Numerical simulation • Steady-state or transient • Determinist or stochastic • General or case study

40. Steady-state At equilibrium, time independant Example: What is the influence of a magma chamber and/or of a magma conduit on the surface heat flux ? Rinderknecht et al., 2007

41. Transient Time dependant Example: what is the emplacement rate for an igneous body to become a magma chamber ?

42. Proportion of eruptible magma 0.1 m/yr over 50,000 yrs Temperatures Melt fractions

43. Incubation time depends on sillaccretion rate • Min accretion rates: • for a mush: > 2 cm/yr • an eruptiblechamber: 3 cm/yr

44. Numerical simulation • Steady-state or transient • Determinist or stochastic • General or case study

45. Non stochastic • Fixed dimension and orientation of intrusions. • Choice of orientation based on observations • Parametric analysis of dimensions

46. Annen and Sparks, 2002 Dashed line: 10 and 50 m thick sills; plain line: 500 m thick sills. Curves of 10 and 50 m sills overlap and cannot be distinguished.

47. Stochastic Dufek & Bergantz, 2005 Dykes orientation and length, and time intervals are randomly chosen.

48. Numerical simulation • Steady-state or transient • Determinist or stochastic • General or case study

49. General:generation of silicic magmas • What are the crust and mantle respective contributions in generation of calc-alkaline (I-type) melts? Annen et al., 2006

50. General:generation of silicic magmas Over-accretion Under-accretion Distributed Annen et al., 2008