Michele Cooke Betsy Madden Jess McBeck

1. Growth by Optimization of Work A Boundary Element Method tool for exploring fault evolution using the principle of work minimization Michele Cooke Betsy Madden Jess McBeck

2. Project overview: I. Current work:Automated search for the most efficient growth direction of a single flaw. II. Future work:Study coalescence patterns of multiple flaws. Min Work Max Work

3. Griffith Crack s1 s2 • Make the ellipse into a crack • Find the maximum tangential stress • Find the stress concentration shh

4. First step: Model validation For closed flaws dipping 60° under uniaxial compression, wing crack growth has been observed at: (angles measured clockwise from tip) • 110°Experimental & Numerical (Nemat-Nasser & Horii, 1982; Horii & Nemat-Nasser, 1985) • 126°Experimental(Bobet & Einstein, 1998) • 117°Numerical (maximum tangential stress)(Bobet, 2000) • 110°Numerical (max opening)(DeBremaecker & Ferris, 2004) Wing crack Initiation angle Dip=60°

5. Wing cracks observed at outcrop scale Bear Creek, Sierra Nevada, California August, 2007 Pollard & Segall (1983), Figure 7a

6. Wing cracks observed at outcrop scale Bear Creek, Sierra Nevada, California Tension 1. Jointing 2. Shear Reactivation 3. Wing cracks! Pollard & Segall (1983), Figures 7a, 13

7. Wing cracks created at laboratory scale • Loaded inuniaxial compression • First: to 20 kN(8.8 MPa) • Then: in 5 kN steps (2.2MPa) at 0.002 mm/s = 12.7mm Bobet & Einstein (1998), Figures 4a, 6

8. Wing cracks created at laboratory scale Closed flaw Open flaw Bobet & Einstein (1998), Figures 11, 13

9. Validation against direction of maximum opening • DeBremaecker and Ferris (2004) use maximum opening e.g. That a crack will grow in the direction that maximizes opening (normal displacement discontinuity). Wing crack Initiation angle

10. How is our work different? • Different criteria for crack growth have been utilized in previous studies. e.g. That a crack will grow in the direction that maximizes: • Strain energy; • Shear slip; • Opening. • But we think that the Earth is lazy, e.g. That a crack will grow in the direction that minimizes work. Wing crack Initiation angle

11. Work minimization is at an angle of 120° • Observation is ~126˚ for a closed flaw

12. Uniaxial wing crack experiments uniaxial Min Work Max Work • Under biaxial stress opening-mode wing cracks grow from the fault tip

13. Animation of crack growth under biaxial loading Bobet & Einstein (1998)

14. Biaxial experiments Biaxial Min Work Max Work • Under biaxial stress, shearing cracks prefer to grow in-line.

15. Animation of crack growth under biaxial loading De Braemecker and Ferris, 2004

16. GROW is a perl wrapper that calls fric2d • Runs models with different orientations of potential propagation • User specifies angle increment.. Resolution • Work Criterion: change in work produced by growing crack > (Wprop +Wseis) then continue propagation along the most efficient path • Stops propagation when fault intersects boundaries or another crack

17. Specify growth from an existing fault tip or specify a point and explore initiation of a fault

18. Adding elements At a point within the body At tip of a fault Considers angles 45˚ to 135˚ Cannot intersect other elements • Considers angles 0˚ to 180˚

19. Effect on External Work • Hypothesis: Fault growth increases the efficiency of the system The change in work is non-linear with fault growth. DWext = gS+ Wseis + Wfric Critical DWext for fault initiation is smaller than the overall work effect of a complete fault surface.

20. Input File

21. Bookkeeping parameters • Tolerance controls the frictional convergence

22. Observation and Boundary Lines 25 mm flaw 160 x 160 mm Dip=60° • For work minimization need to apply displacement boundary conditions. • If tractions are applied then the solution follows the maximum work because the most efficient fault systems under a given stress will produce the greatest displacement.

23. Specify the Fault • Sandbox version has dynamic weakening • Will incorporate this into Fric2d

24. Can have fractures as well • Fractures are mode I • Joints • fluid-filled crack

25. Searching for fault that minimizes work • Step 1 • Option A: flag a fault to grow • Option B: Specify to search at a point • Call GROW from the command line • grow inputFileangleIncrementcriticalWork(topo file) • If you are providing topography then add the file name, In this case sandbox.c is run instead of fric2d.c • If no topo file then fric2d will assign y=0 as the Earth’s surface if the gravity flag is turned on.

26. Files that creates • *.raw • Lists the angles tested and the resultant work • Updated as the fault propagates • *.eff • Is the input file for the most efficient fault • *.work • More detailed report of work • only created after the model is done

27. Output file

29. Other scenarios • Fault approaching a bedding plane • Growth of two faults

30. Fault Approaching a bedding plane

31. Interaction of two faults

32. Download • ftp://eclogite.geo.umass.edu/pub/cooke/grow_pack.zip • Unzip the file • Compile the fric2d and sandbox codes • Move the executable files up one directory so that they are in the same folder as the input files • Fric2d documentation • www.geo.umass.edu/faculty/cooke • Click link for software downloads

33. What is in the bundle • Directories with executables • Fric2d_stuff • sandbox_stuff • *.in  input files for toy models • GROW_4_12.pl <- script to continue running fric2D or sandbox and fie the most efficient fault propagation path • external_Work_New.pl <- script to calculate work

34. Making movies in matlab • Two matlab scripts can make movies of the growth • GROW.m <- from one end • GROW_from2ends.m <-shows growth from two ends • STEPS • COPY the Fault lines from the *.eff file created by GROW into matlab • Run the scrip • The range of the plot may need to be shifted for your model results