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

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

3 GROW Griffith Crack  Make the ellipse into a crack  Find the maximum tangential stress  Find the stress concentration    

4 GROW 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) Dip=60° Initiation angle Wing crack

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

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

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

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

9 GROW Model validation For closed flaws dipping 60°, wing crack initiation angles measured clockwise from tip: 110° Experimental & Numerical (Nemat-Nasser & Horii, 1982; Horii & Nemat- Nasser, 1985) 126° Experimental (Bobet & Einstein, 1998) 117° Numerical (Bobet, 2000) 110° Numerical (DeBremaecker & Ferris, 2004) Initiation angle Wing crack

10 GROW 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). Initiation angle Wing crack

11 GROW 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. Initiation angle Wing crack

12 GROW  Observation is ~126˚ for a closed flaw Work minimization is at an angle of 120°

13 GROW Uniaxial wing crack experiments uniaxial Min Work Max Work  Under biaxial stress opening- mode wing cracks grow from the fault tip

14 GROW Animation of crack growth under biaxial loading Bobet & Einstein (1998)

15 GROW Biaxial experiments Biaxial Min Work Max Work  Under biaxial stress, shearing cracks prefer to grow in-line.

16 GROW Animation of crack growth under biaxial loading De Braemecker and Ferris, 2004

17 GROW  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 > (W prop +W seis ) then continue propagation along the most efficient path  Stops propagation when fault intersects boundaries or another crack

18 GROW  Specify growth from an existing fault tip or specify a point and explore initiation of a fault

19 GROW Adding elements At a point within the body  Considers angles 0˚ to 180˚ At tip of a fault  Considers angles 45˚ to 135˚  Cannot intersect other elements

20 GROW Effect on External Work  Hypothesis: Fault growth increases the efficiency of the system Critical  W ext for fault initiation is smaller than the overall work effect of a complete fault surface. The change in work is non- linear with fault growth.  W ext =  S+ W seis + W fric

21 GROW Input File

22 GROW Bookkeeping parameters  Tolerance controls the frictional convergence

23 GROW Observation and Boundary Lines Dip=60° 160 x 160 mm 25 mm flaw 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.

24 GROW Specify the Fault  Sandbox version has dynamic weakening  Will incorporate this into Fric2d

25 GROW Can have fractures as well  Fractures are mode I Joints fluid-filled crack

26 GROW 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 inputFile angleIncrement criticalWork (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.

27 GROW 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

28 GROW Output file

29 GROW

30 GROW Other scenarios  Fault approaching a bedding plane  Growth of two faults

31 GROW Fault Approaching a bedding plane

32 GROW Interaction of two faults

33 GROW Download  ftp://eclogite.geo.umass.edu/pub/cooke/grow_p ack.zip ftp://eclogite.geo.umass.edu/pub/cooke/grow_p ack.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 Click link for software downloads

34 GROW 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

35 GROW 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