Michele CookeDepartment of Geosciences

1. Work Budget2. Boundary Element

Method3. GROW

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

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Fault Evolution: San Gorgonio Knot

Modified from Matti et al, 1992

Up to ~500 kyMission Creek Strand

500 ky -> ~120 kyMill Creek StrandReactivate San Gorgonio

120 ky -> Present DaySan Bernardino Strand Garnet Hill FaultReactivate Banning

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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

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Ways to understand fault growth

Field Evidence:• Secondary fractures

reveal fault history Empirical Criterion:

• Laboratory tests on intact rock

Theory:• Linear Elastic Fracture

MechanicsValley of Fire, NV

Myers and Aydin, 2004, JSG

Normal faults in Moab, UT

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Ways to understand fault growth

Field Evidence:• Secondary fractures

reveal fault history Empirical Criterion:

• Laboratory tests on intact rock

Theory:• Linear Elastic Fracture

Mechanics• Measure strength at different confining

pressures -> Mohr-Coulomb Criterion

= c +

Image from EP solutions

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Ways to understand fault growth

Field Evidence:• Secondary fractures

reveal fault history Empirical Criterion:

• Laboratory tests on intact rock

Theory:• Linear Elastic Fracture

Mechanics

• Faults grow by coalescence of cracks• For faults Gc not well-constrained

• Micromechanics• Seismologic

€

G =(1−ν 2)

EK I

2 +(1−ν 2)

EK II

2 +(1+ ν )

EK III

2

Failure when G >= Gc

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How do faults grow and evolve?

Is the Earth Lazy?

whatever

Active faults of southern California (from Southern California Earthquake Center)

Minimization of work considers the behavior of the entire fault system

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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.

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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].

Rymer, 2000

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Applications of Work Minimization: Normal fault arrays

Antithetic faults are favored over synthetic faults [Melosh & Williams, 1989]

Photo by Marli Miller

Antithetic

Synthetic

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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

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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

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Mechanical work: Force * Distance

Deformation – stored work ½ stress * strain

Potential Energyweight * distance

Frictional HeatShear stress * slip

Acoustic/Seismic EnergyShear stress drop * slip

Fracture energyGibb’s free energy * surface area

reversible

irreversible

Cooke & Murphy, 2004

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Work Budget: Wint + Wgrav + Wfric + Wseis + Wprop = Wext

Cooke & Murphy, 2004

tectonic

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Work Budget: Wint + Wgrav + Wfric + Wseis + Wprop = Wext

deformation

€

W int =1

2σ ijε ijdV∫∫∫

Cooke & Murphy, 2004

tectonic

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Work Budget: Wint + Wgrav + Wfric + Wseis + Wprop = Wext

uplift againstgravity

deformation

€

Wgrav = ρgdz (z)dV∫∫∫

Cooke & Murphy, 2004

tectonic

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Work Budget: Wint + Wgrav + Wfric + Wseis + Wprop = Wext

uplift againstgravity

deformation

heat

€

W fric = ∫ τ ε a( )slip ε a( )dε adS∫∫

Cooke & Murphy, 2004

tectonic

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Work Budget: Wint + Wgrav + Wfric + Wseis + Wprop = Wext

uplift againstgravity

ground shakingdeformation

heat

€

Wseis = ∫ Δτ (ε a )slip(ε a )dε adS∫∫

Cooke & Murphy, 2004

tectonic

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Work terms associated with weakening

Seismologists divide as EF, G and ER

Cooke & Murphy, 2004Savage & Cooke, 2010

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Work Budget: Wint + Wgrav + Wfric + Wseis + Wprop = Wext

uplift againstgravity

ground shakingdeformation

new faultsurfaces

heattectonic

W prop (1 2)

EK Ic

2 S

W prop S

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).

Cooke & Murphy, 2004

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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)

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Analog models provide direct observation of fault growth

from Ask & Morgan, 2010

from Adam et al., 2005

from Cubas et al., 2010

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New faults grow during accretion

a) Accretion: new forethrust

b) UnderthrustingWedge thickening

c) Accretion: new forethrust

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Sandbox experiments

Particle Image Velocimetry (PIV) records the development of accreting forethrust with 2.2 cm of contraction

Adam et al. 2005Henry Cadell ~1880

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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

Forethrust

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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

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Energy of Fault Growth

Wint

+ Wfric

+ Wgrav

Wprop

+ Wseis

Del Castello and Cooke, 2007

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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

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Force drop with fault growth observed in sandbox

From Cubas et al., 2008

Nieuwland et al, 2001

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Evolution of force during accretion

sandbox experiment from Université de Cergy-

Pontoise

sandbox experiment at Stanford (Cruz et al, 2010)

3434

½ ΔF Δd = ΔW ΔW = γΔS + Wseis + Wfric

Cost of fault growth

80 mJ/m2 We can use the observed

change in work per unit fault area to predict fault growth

Measuring Wprop+Wseis

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Calibration

Stiff model approximates first 4 cm

Soft model matches past 6 cm

Basal friction0.5 static0.35 dynamic

within range of Souloumiac et al. ( 2012, EGU and JSG)

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Timing of fault growth

Work Minimization Analog Experiments Numerical Simulations Conclusions

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.

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What does this mean for fault growth?

Lazy?

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.