Pablo Sanz1, David Pollard2 and Ronaldo Borja1

FINITE ELEMENT MODELING OF FRACTURES EVOLUTION DURING FOLDING

OF AN ASYMMETRIC ANTICLINE

1Department of Civil and Environmental Engineering, Stanford University2Department of Geological and Environmental Sciences, Stanford University

July 2007 – Stanford, CA

Contour map of Sundance Formation base by Forster et al. (1996)

Sheep Mountain AnticlineContour map: elevation of Sundance formation

3D elevation around the nose Elevation of cross sections

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

Modeling Folding and Fracturing

Rock

Folding

Fracturing

Modeling

Inelastic deformation

Deterioration of tangent stiffness

Very large movements

Rigid body translation and rotation

material

geometric

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

Nonlinear Contact Mechanics

Undeformed configuration (t0)

Stick (t1)

Slip (t2)

Slip+gap (t3)

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

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)

Outer layer (200 m)Inner layer (100 m) with vertical fractures

Ei = 2 GPa, i = 0.25

Eo = 0.2, 0.4, 1, 2, or 4 GPa, o = 0.25

= 26 kN/m3

pv = 40 MN/m2 (1.5 km of rocks)

r = Ei / Eo = 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 0.8Geometry

PropertiesFinite element mesh

Folding and FracturingAsymmetric anticline

Evolution of existing fractures

Frictionless interface for bottom b.c.

50 +

500

km

Lo = 6,000 m

REFERENCE: Sanz, Pollard and Borja, paper in preparation

Folding and Fracturing: evolution of fold=H / V = 300 m / 500 m =

0.6r = Ei / Eo = 1

x = 100%

x = 75%

x = 50%

x = 25%

V = 500 m

H = 300 m

Displacement on bottom boundary condition: interface 1

Interface 1

Interface 1

x = 200 m = 0.4

x = 400 m = 0.8

Fractures evolution: backlimb

=H / V = 300 m / 500 m = 0.6

r = Ei / Eo = 1

Fractures evolution: hinge

=H / V = 300 m / 500 m = 0.6

r = Ei / Eo = 1

Fractures evolution: forelimb

=H / V = 300 m / 500 m = 0.6

r = Ei / Eo = 1

Fractures evolution: forelimb

=H / V = 300 m / 500 m = 0.6

r = Ei / Eo = 1

10 cm

Reverse fault of a pre-folding bed-perpendicular fracture at SMA

from Bellahsen et al. (2005)

No frictional beds vs. frictional beds

= 0.6, r = 2Stick

Frictional

Fracture evolution: traction vector [h = 30%]

Left lateral (-)

Right lateral (-)

#1 #10 #20 #30

fracture bottom

h = 30%h

Fracture evolution: traction vector [h = 70%]

Left lateral (-)

Right lateral (-)

#1 #10 #20 #30

fracture top

h = 70%h

Fracture evolution: traction vector [h = 30%]

Left lateral (-)

Right lateral (-)

#1 #10 #20 #30

fracture bottom

h = 30%h

Fracture evolution: traction vector [h = 70%]

Left lateral (-)

Right lateral (-)

fracture top

h = 70%h

#1 #10 #20 #30

Fracture evolution: traction vector [h = 30%]

Left lateral (-)

Right lateral (-)

#1 #10 #20 #30

fracture bottom

h = 30%h

Fracture evolution: traction vector [h = 70%]

Left lateral (-)

Right lateral (-)

#1 #10 #20 #30

fracture top

h = 70%h

0.00

0.02

0.04

0.06

0.08

0.10

0 1 2 3 4 5 6 7 8 9 10

r = Ei/Eo

Max

imu

m f

ract

ure

sli

p (

m/m

)

Frictional bedsFrictional bedsFrictional bedsNo frictional bedsNo frictional bedsNo frictional beds

=0.4=0.6

=0.8

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0 1 2 3 4 5 6 7 8 9 10

r = Ei/Eo

Max

imu

m g

ap (

m/m

)

Frictional bedsFrictional bedsFrictional bedsNo frictional bedsNo frictional bedsNo frictional beds

=0.4

=0.6

=0.8

Forelimb

Hinge

Fractures

Fracture reactivation as:

• joints (hinge)

• reverse faults (limbs)

-90

-70

-50

-30

-10

10

30

0 1 2 3 4 5 6 7 8 9 10

r = Ei/Eo

Max

imu

m s

lip

(m

)

Backlimb

Backlimb

Backlimb

Forelimb

Forelimb

Forelimb

=0.6

=0.4

Forelimb =0.8

=0.6

=0.4

=0.8

Backlimb

Maximum slip along bedding surface (interface 2)

Interface 2

-80

-60

-40

-20

0

20

0 1000 2000 3000 4000 5000 6000

Bed

din

g s

urf

ace

slip

(m

)

x = 100%

Interface 2

X (m)

x = 50%

x = 25%

Slip (+)

Slip (-)

Interface 2

Slip along bedding surface (interface 2)

Interface 2

Slip along bedding surface (interface 2 & 3)

Interface 3

-80

-60

-40

-20

0

20

0 1000 2000 3000 4000 5000 6000

Bed

din

g s

urf

ace

slip

(m

)

Interface 2

Interface 3

X (m)

Slip (+)

Slip (-)

• 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

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