Principles of ‘ balanced ’ cross sections

1. Principles of ‘balanced’ cross sections • The true section is retrodeformable. • Any retrodeformable section is a possible model. Not unique however in general. • The ‘space’ of retrodeformable section is large. We need assumptions to limit possibilities: • Conservation of mass is often assumed to convert to conservation of volume (not correct in case of compaction or metamorphism). • if plane strain is assumed or if the structure is cylindrical, conservation of volume converts to conservation of area NB: Balanced = retrodeformable

2. Fold classification (Axial plane dip angle , Fold Tightness of folding Fold style, Thickness of folded beds) Axial plane dip angle: Upright Overturned Recumbent Fold tightness: Gentle (>90°), Open (90°), Tight, (<90) Isoclinal (~0°) Parallel gentle Fold style: parallel (=concentric) Similar N: Fold style classification relies on the relative thickness of the layers in the hinge vs the limbs, or equivalently on the relative curvature of the of the upper and lower bounding surfaces. Similar

3. Parallel folding Parallel folds commonly form by a deformation mechanism called flexural slip, where folding is accommodated by motions on minor faults that occur along some mechanical layering -- usually bedding. Flexural-slip surfaces, which can be observed in core or outcrop, may vary in spacing from a few millimeters to several tens of meters in spacing. In that case of flexural slip folding there is conservation of bed length and of bed thickness

4. First define section and assemble the data. • Two popular methods can then be used to construct balanced cross sections • The Busk Method (Busk, 1929) • - The Kink Method

5. The Busk’ method… parallel and concentric folds

6. The Busk’ method…

7. The Kink method for a parallel fold… The kink method is based on the assumption of flexural slip folding in the limit where dip angles varie only across axial surfaces (it is equivalent to the Busk method with infinite curvature within dip domains and zero curvature within axial surface. γ γ • the axial surface bisects the angle between the fold limbs Axial angle :γ

8. Constructing a balanced cross-section from the kink method • Assemble data (surface and subsurface observations) • Define dip domains, positions and dip angles of axial surface. • Extrapolate at depth by trials and errors. (you will need an eraser, experience will help). • Test that the section is indeed retrodeformable.

9. The Kink method for a parallel fold… • where two axial surfaces intersect, a new axial surface is formed. Its dip angle bisects the dip angles of the adjacent dip domains

10. The Kink method for a parallel fold… • Dip angles are constant within dip domains separated by axial surfaces. • the axial surface bisects the angle between the fold limbs

11. The Kink method for a parallel fold… Holland, 1914

12. The kink method is more general than the Busk method (any curve can be divided in straight segments).

13. Decollement

14. Minimum Shortening: 17.5km This section is retrodeformable and is thus a plausible model (but not a unique solution). A balanced cross-section generally yields a lower bound on the amount of shortening (because of erosion)

15. Fault-bend folding (Rich, 1934)… Flat: Fault is parallel to bedding Ramp: Fault cuts trough bed

16. Fault-bend folding (Rich, 1934)… The non planar fault has a flat-ramp-flat geometry. Translation of the thrust sheet along that fault requires axial surfaces. Note the difference between the active and passive axial surfaces

17. Constructing a balanced cross-section from the kink method • Assemble data (surface and subsurface observations) • Define dip domains, positions and dip angles of axial surface. • Extrapolate at depth by trials and errors. (you will need an eraser, experience will help). • Test that the section is indeed retrodeformable.

18. Two possible interpretations of structural measurements at the surface and along an exploration well. Which one is most plausible?

19. Two possible interpretations of structural measurements at the surface and along an exploration well. ‘Flat’ in hanging wall does not match ‘Ramp’ in footwall ‘Flat’ in hanging wall matches ‘Flat’ in footwall

20. - A balanced cross section might be checked from comparing hangingwall and footwall cutoff angles. - A balanced cross section might be checked and eventually retrodeformed based on the principle of conservation of area and conservation of bed length

21. Curvimetric Shortening & planimetric shortening Curvimetric Shortening: Sc= Lc-l Planimetric shortening: Sa=Asr/h Structural relief: Asr Undeformed depth of decollement below bed: h Area of Shortening: A Conservation of area implies: A=Asr= Sc*h. NB: Both quantities refer to a particular bed.

22. Curvimetric Shortening & planimetric shortening Curvimetric Shortening: Sc= Lc-l Structural relief: Asr Depth of decollement below bed: h Area of Shortening: A Conservation of area implies: A=Asr= Sc*h Planimetric shortening: A/h=Asr/h Conservation of area and bed length implies : - curvimetric shortening = planimetric shortening - Asr should increase linearly with elevation above decollement.

23. Sc= 0,95km Asr/h=2.6km Sc= 0,95km Asr/h=0.95km If area and bed length is preserved during folding then b is the most plausible solution …or a is correct but some diapirism is involved

24. Equality of curvilinear and planimetric shortening can be used to either check the section or predict the depth to the decollement. Asr= Sc*h

25. Structural relief: Asr Equality of curvilinear and planimetric shortening can be used to either check the section or predict the depth to the decollement. NB: Be careful with the possibility of diapirism

26. Structural elevation in balanced cross sections:the problem of filling space

27. Fault-bend-folding +Imprication/Duplex…

28. Balanced cross-section across the Pine Mountain (Southern Appalachians) (Suppe, 1983)

29. Structural elevation in balanced cross sections:the problem of filling space

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31. Klippe (Robinson, Geosphere, 2008)

35. Move-3D