1. Pablo Sanz 1, David Pollard 2 and Ronaldo Borja 1 FINITE ELEMENT MODELING OF FRACTURES EVOLUTION DURING FOLDING OF AN ASYMMETRIC ANTICLINE 1 Department of Civil and Environmental Engineering, Stanford University 2 Department of Geological and Environmental Sciences, Stanford University July 2007 – Stanford, CA

2. Contour map of Sundance Formation base by Forster et al. (1996) Sheep Mountain Anticline Contour map: elevation of Sundance formation 3D elevation around the nose Elevation of cross sections

3. Sheep Mountain Anticline: fracture data and interpretation from Bellahsen et al., 2006. Perpendicular to bedding Observed throughout the fold Present before Laramide compression No common deformation mode Reactivated set I fractures in the forelimb Set I

4. Modeling Folding and Fracturing Rock Folding Fracturing Modeling Inelastic deformation Deterioration of tangent stiffness Very large movements Rigid body translation and rotation material geometric material contact Frictional sliding (also along beds) Gap Contact search and contact constraint Type of nonlinearity Large deformation frictional contact model (bulk plasticity) Objective: to simulate the evolution of existing fractures during folding

5. Nonlinear Contact Mechanics Undeformed configuration (t 0 ) Stick (t 1 ) Slip (t 2 ) Slip+gap (t 3 ) stick (elastic) region slip function Methodology Implementation: penalty method Coulomb friction law: suitable for geomaterials Formulation REFERENCES: Laursen and Simo, International Journal for Numerical Methods in Engineering, 1993 Wriggers, Computational Contact Mechanics, 2002

6. Load cases (i) Gravity loads (ii) Folding+contraction 4 layers (3 rock layers + 1 for bottom BC) 6,874 nodes, 12,775 CST elements 31 fractures + 2 bedding surfaces + bottom BC Outer layer (200 m) Inner layer (100 m) with vertical fractures E i = 2 GPa, i = 0.25 E o = 0.2, 0.4, 1, 2, or 4 GPa, o = 0.25  = 26 kN/m 3 p v = 40 MN/m 2 (1.5 km of rocks) r = E i / E o = 0.5, 1, 2, 5 or 10  v = 500 m (asymmetric anticline)  H = 200 m, 300 m, or 400 m  =  H /  V = 0.4, 0.6, or 0.8 Geometry Properties Finite element mesh Folding and Fracturing Asymmetric anticline Evolution of existing fractures Frictionless interface for bottom b.c. 50 + 500 km L o = 6,000 m REFERENCE: Sanz, Pollard and Borja, paper in preparation

7. Folding and Fracturing: evolution of fold  =  H /  V = 300 m / 500 m = 0.6 r = E i / E o = 1  x = 100%  x = 75%  x = 50%  x = 25%  V = 500 m  H = 300 m

8. Displacement on bottom boundary condition: interface 1 Interface 1  x = 200 m   = 0.4  x = 400 m   = 0.8

9. Fractures evolution: backlimb  =  H /  V = 300 m / 500 m = 0.6 r = E i / E o = 1

10. Fractures evolution: hinge  =  H /  V = 300 m / 500 m = 0.6 r = E i / E o = 1

11. Fractures evolution: forelimb  =  H /  V = 300 m / 500 m = 0.6 r = E i / E o = 1

12. Fractures evolution: forelimb  =  H /  V = 300 m / 500 m = 0.6 r = E i / E o = 1 10 cm Reverse fault of a pre-folding bed-perpendicular fracture at SMA from Bellahsen et al. (2005)

13. No frictional beds vs. frictional beds  = 0.6, r = 2 Stick Frictional

14. Fracture evolution: traction vector [h = 30%] Left lateral (-) Right lateral (-) #1#10#20#30 fracture bottom h = 30% h

15. Fracture evolution: traction vector [h = 70%] Left lateral (-) Right lateral (-) #1#10#20#30 fracture top h = 70% h

16. Fracture evolution: traction vector [h = 30%] Left lateral (-) Right lateral (-) #1#10#20#30 fracture bottom h = 30% h

17. Fracture evolution: traction vector [h = 70%] Left lateral (-) Right lateral (-) fracture top h = 70% h #1#10#20#30

18. Fracture evolution: traction vector [h = 30%] Left lateral (-) Right lateral (-) #1#10#20#30 fracture bottom h = 30% h

19. Fracture evolution: traction vector [h = 70%] Left lateral (-) Right lateral (-) #1#10#20#30 fracture top h = 70% h

20. Forelimb Hinge Fractures Fracture reactivation as: joints (hinge) reverse faults (limbs)

21. Maximum slip along bedding surface (interface 2) Interface 2

22. Slip (+) Slip (-) Interface 2 Slip along bedding surface (interface 2)

23. Interface 2 Slip along bedding surface (interface 2 & 3) Interface 3 Slip (+) Slip (-)

24. Existing fractures can be reactivated as faults, joints, or shear/opening mode depending in the location of the fracture. Opening mode fractures  along the hinge Fractures in the limbs are predominantly reactivated as reverse faults Shearing of fractures is more important along the forelimb than in the backlimb. We studied the effect of: Material properties (stratigraphy) Parallel slip along bedding surfaces Overall contraction Folding and Fracturing: summary and conclusions

25. stick (elastic) region slip function Coulomb friction law: suitable for geomaterials It is appropriate to analyze a number of geologic and geotechnical phenomena It is very simple  easy to implement Does not capture: tensile failure, slip weakening, cohesive end zone Current and Future Work