1. Work Budget 2. Boundary Element Method 3. GROW

1. 1. Work Budget2. Boundary Element Method3. GROW Michele Cooke Department of Geosciences

2. Work min = limit analysis ? • Civil structures • Attention to the most efficient mode of failure • Efficient = least load at failure = min max load • Geologic structures • Is the Earth lazy? • Most efficient fault grows… or doesn’t Photo by Mike Gross

4. Fault Evolution: San Gorgonio Knot Up to ~500 ky Mission Creek Strand 500 ky -> ~120 ky Mill Creek Strand Reactivate San Gorgonio 120 ky -> Present Day San Bernardino Strand Garnet Hill Fault Reactivate Banning Modified from Matti et al, 1992

5. Work min = limit analysis ? • Civil structures • Attention to the most efficient mode of failure • Efficient = least load at failure = min max load • Geologic structures • Is the Earth lazy? • Most efficient fault grows… or doesn’t Photo by Mike Gross

6. Ways to understand fault growth • Field Evidence: • Secondary fractures reveal fault history • Empirical Criterion: • Laboratory tests on intact rock • Theory: • Linear Elastic Fracture Mechanics Corona fault, San Francisco

7. Ways to understand fault growth • Field Evidence: • Secondary fractures reveal fault history • Empirical Criterion: • Laboratory tests on intact rock • Theory: • Linear Elastic Fracture Mechanics Normal faults in Moab, UT Valley of Fire, NV Myers and Aydin, 2004, JSG

8. Ways to understand fault growth • Field Evidence: • Secondary fractures reveal fault history • Empirical Criterion: • Laboratory tests on intact rock • Theory: • Linear Elastic Fracture Mechanics Image from EP solutions • Measure strength at different confining pressures -> Mohr-Coulomb Criterion • |t| = c + ms

9. Ways to understand fault growth • Field Evidence: • Secondary fractures reveal fault history • Empirical Criterion: • Laboratory tests on intact rock • Theory: • Linear Elastic Fracture Mechanics Failure whenG >= Gc • Faults grow by coalescence of cracks • For faults Gc not well-constrained • Micromechanics • Seismologic

10. How do faults grow and evolve? • Minimization of work considers the behavior of the entire fault system whatever Active faults of southern California (from Southern California Earthquake Center) Is the Earth Lazy?

11. How does the Earth know that it is lazy? • A ball rolling downhill doesn’t know that it is lazy but still follows the path of least resistance.

12. Rymer, 2000 Evidence of Work Minimization Geometry of spreading centers [Sleep, 1979] and mudcracks reflects work minimization accommodate shrinkage with minimum new fracture surface • Faults become more smooth with greater slip • Strike-slip traces [e.g. Wesnousky, 1988], extensional fault traces [Gupta et al., 1998], and lab [Scholz, 1990].

13. Applications of Work Minimization: Normal fault arrays Antithetic faults are favored over synthetic faults [Melosh & Williams, 1989] Synthetic Antithetic Photo by Marli Miller

14. Applications of Minimum Work: fabric evolution • Code Elle uses minimization of average local work rate to simulate the evolution of microstructures during deformation and metamorphism [ e.g. Lebensohn et al., 2008, Griera et al, 2011] Griera et al., 2011

15. Applications of Minimum Work: fold and thrust belts • Growth of critical tapered wedges [e.g. Masek and Duncan, 1998], duplexes [Mitra and Boyer, 1986] and folds [Ismat, 2009] • Burbidge and Braun [2002]: use work analysis to explain the accretion-underthrust cycle • Work minimization to predict fault evolution [Maillot & Leroy, 2003; Souloumiac et al., 2008; Cubas et al, 2008] from Dahlen, et al., 1984 From Cubas et al., 2008

16. Mechanical work: Force * Distance • Deformation – stored work ½ stress * strain • Potential Energy weight * distance • Frictional Heat Shear stress * slip • Acoustic/Seismic Energy Shear stress drop * slip • Fracture energy Gibb’s free energy * surface area reversible irreversible Cooke & Murphy, 2004

17. Work Budget: Wint + Wgrav + Wfric + Wseis + Wprop = Wext tectonic Cooke & Murphy, 2004

18. Work Budget: Wint + Wgrav + Wfric + Wseis + Wprop = Wext tectonic deformation Cooke & Murphy, 2004

19. Work Budget: Wint + Wgrav + Wfric + Wseis + Wprop = Wext uplift against gravity tectonic deformation Cooke & Murphy, 2004

20. Work Budget: Wint + Wgrav + Wfric + Wseis + Wprop = Wext uplift against gravity tectonic heat deformation Cooke & Murphy, 2004

21. Work Budget: Wint + Wgrav + Wfric + Wseis + Wprop = Wext uplift against gravity tectonic heat ground shaking deformation Cooke & Murphy, 2004

22. Work terms associated with weakening • Seismologists divide as EF, G and ER Savage & Cooke, 2010 Cooke & Murphy, 2004

23. Work Budget: Wint + Wgrav + Wfric + Wseis + Wprop = Wext Lab:10-104 J/m2 (Wong, 1982, 1986; Cox & Scholz, 1988; Lockner et al., 1992). Field: 105-106 J/m2 (Wilson et al 2005; Pittarello et al, 2008). uplift against gravity new fault surfaces tectonic heat ground shaking deformation Cooke & Murphy, 2004

24. Fric2D • Two-dimensional Boundary Element Method code • Continuum mechanics • Discretize boundaries and faults into linear dislocation elements • Crack/fault propagation via addition of elements • Static friction along faults • Non-linear behavior requires iterative convergence • Other features not presented here • Growth of fault damage (e.g. Savage & Cooke, 2010)

25. Analog models provide direct observation of fault growth from Ask & Morgan, 2010 from Adam et al., 2005 from Cubas et al., 2010

26. New faults grow during accretion • Accretion: new forethrust • Underthrusting Wedge thickening • Accretion: new forethrust

27. Sandbox experiments • Particle Image Velocimetry (PIV) records the development of accreting forethrust with 2.2 cm of contraction Adam et al. 2005 Henry Cadell ~1880

28. Forethrust Model Set-Up • Boundary Element Method (Fric2d) • Simulate %0.5 cm of contraction • Frictional slip along faults • Medium sand • E = 10 MPa;  = 1732 kg/m3

29. Thrust Sheet Growth • Total work increases during underthrusting • With addition of the forethrust, work decreases • Increased Wint is offset by decreased Wfric Del Castello and Cooke, 2007

30. Energy of Fault Growth Wint + Wfric + Wgrav Wprop + Wseis Del Castello and Cooke, 2007

31. Location and vergence of most efficient thrust Test a suite of locations and vergence 30˚ dipping forethrusts ahead of the wedge are more efficient than 40˚ dipping backthrusts The preferred location and dip match the sandbox Del Castello and Cooke, 2007

32. Force drop with fault growth observed in sandbox From Cubas et al., 2008 Nieuwland et al, 2001

33. Evolution of force during accretion sandbox experiment at Stanford (Cruz et al, 2010) sandbox experiment from Université de Cergy-Pontoise

34. MeasuringWprop+Wseis ½ ΔF Δd = ΔW ΔW = γΔS + Wseis+ Wfric Cost of fault growth We can use the observed change in work per unit fault area to predict fault growth 80 mJ/m2

35. Calibration Stiff model approximates first 4 cm Soft model matches past 6 cm Basal friction 0.5 static 0.35 dynamic within range of Souloumiac et al. ( 2012, EGU and JSG)

36. Timing of fault growth • Hypothesis: The development of faults is more productive at peak loading than prior to peak The addition of a fault to the stiffer sand produces greater change in work than the softer sand. Early compaction of the sand facilitates the development of faults. Work Minimization Analog Experiments Numerical Simulations Conclusions

37. What does this mean for fault growth? • Can we use the energy of fault growth to predict timing of fault development in the sandbox? • How much energy does it take to grow a fault in the crust? • Lab:10-104 J/m2 (Wong, 1982, 1986; Cox & Scholz, 1988; Lockner et al., 1992). • Field: 105-106 J/m2 (Wilson et al 2005; Pittarello et al, 2008). • Need more constraints • If Wprop were negligible then faults would not be long-lived. Lazy?