Elastic scattering by planar fractures

$Page 1: Elastic scattering by planar fractures$
Intro Theory Experiments Direct excitation

Elastic scattering by planar fractures

Thomas E. Blum1, Roel Snieder2, Kasper van Wijk1, and MarkE. Willis3

1Physical Acoustics LabBoise State University

Boise, ID

2Center for Wave PhenomenaColorado School of Mines

Golden, CO

3Formerly at ConocoPhillips, Houston, TX,now at Halliburton, Houston, TX

September 21, 2011

T. E. Blum, R. Snieder, K. van Wijk, M. E. Willis Elastic scattering by planar fractures


Outline

1 Introduction

2 Scattering by a plane crack

3 Laboratory experiments

4 Direct excitation



Faults and fractures

Controls fluid flow: hydrocarbons, water, magma. . .

Characterization of fracture properties with elastic waves

Active or passive monitoring

1 mm

http://makel.org/fractures/ http://phobos.ramapo.edu/ http://http://geotripper.blogspot.com/



Theoretical expression and laboratory modeling

Single fracture

Linear-slip model: [ui ] = ηirTr , with [u] displacementdiscontinuity, η compliance (1/stiffness) and T traction

Born approximation: scattered field small compared toincident field

Frequency domain: f (t) =∫F (ω)e−iωtdω

Previous work: small fractures ⇒ effective medium (Crampin,1981; Hudson, 1981),or large fractures ⇒ reflection coefficients (Pyrak-Nolte et al.,1990; 1992)




Single fracture








Single fracture








Single fracture








Single fracture








Fractured plastic samples

Ultrasonic frequencies (100 kHz - 10 MHz) ⇒ λ ∼ 10−4 -10−2 m

Laser generation and detection of body waves

Units are cm



Ultrasonic laser receiver

Wide bandwidth

Absolute displacement

Non-contact and small footprint compared to the wavelength

No moving parts

Scanning system

Laser source

receiver

Laser



General expression

Decomposition of the compliance η:ηij = ηN fi fj + ηT (δij − fi fj)

Displacement as a function of the scattering amplitude:

u(P)n (x) = fPP(η) e

ikαR

R mn

y

z

ˆ f

ˆ n

ˆ m

x

σ stressω angular frequencyα P-wave velocityρ density of the materialkα wavenumberR distance to the fracture



General expression

Decomposition of the compliance η:ηij = ηN fi fj + ηT (δij − fi fj)

Displacement as a function of the scattering amplitude:

u(P)n (x) = fPP(η) e

ikαR

R mn

y

z

ˆ f

ˆ n

ˆ m

x

σ stressω angular frequencyα P-wave velocityρ density of the materialkα wavenumberR distance to the fracture



Scattering amplitude of a circular plane crack

For the experimental geometry:

fP,P(ψ, θ) =ωa

2ρα3(sinψ − sin θ)J1(ωaα

(sinψ − sin θ))

×[ηN{

(λ+ µ)2 + (cos 2ψ + cos 2θ)(λ+ µ)µ

+µ2(cos 2ψ cos 2θ)}

+ηTµ2 (sin 2ψ sin 2θ)

].

⇒ term in ηN , and term in ηT non-zero for ψ 6= 0

y

z

ˆ f

ˆ n

ˆ m

x

Source

Receiver

ω angular frequencyα P-wave velocityρ density of the materialλ, µ Lame parametersa fracture radius



Experimental setup

Sample: PMMA cylinder (transparent plastic material), 150mm high x 50.8 mm diameter

Piezoelectric transducer source, 5 MHz, 400 V pulse

Laser ultrasonic receiver: wide bandwidth (20 kHz – 20 MHz),absolute vertical displacement, small footprint, sensitivity in A

Fixed source-fracture angle ψ and moving receiver (θ changes)

f^

ψsourcePZT

θ

z

x

ring

of re

ceiv

ers

δ

fracture

PZT source

Top view

x

z

y



Experimental setup: geometry

f^

ψsourcePZT

θ

z

x

ring

of re

ceiv

ers

δ

fracture

PZT source

Top view

x

z

y



Non-fractured sample

velocities α = 2600 m/s and β = 1400 m/s

ρ = 1190 kg/m3 ⇒ Lame parameters λ = 3.4 GPa andµ = 2.3 GPa

f-k filter to remove surface waves

δ (°)

Tim

e (

µs)

90 180 270 360

0

5

10

15

20

Dis

pla

cem

en

t (n

m)

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

Rayleigh

Direct P

δ (°)

Tim

e (

µs

)

90 180 270 360

0

5

10

15

20

Dis

pla

ce

me

nt

(nm

)

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4



Non-fractured sample

velocities α = 2600 m/s and β = 1400 m/s

ρ = 1190 kg/m3 ⇒ Lame parameters λ = 3.4 GPa andµ = 2.3 GPa

f-k filter to remove surface waves

δ (°)

Tim

e (

µs)

90 180 270 360

0

5

10

15

20

Dis

pla

cem

en

t (n

m)

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

Rayleigh

Direct P

δ (°)

Tim

e (

µs

)

90 180 270 360

0

5

10

15

20

Dis

pla

ce

me

nt

(nm

)

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4



Fractured sample: data

δ (°)

Tim

e (

µs

)

90 180 270 360

0

5

10

15

20

Dis

pla

cem

en

t (n

m)

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

Scattered P

δ (°)

Tim

e (

µs

)

90 180 270 360

0

5

10

15

20

Dis

pla

ce

me

nt

(nm

)

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

Scattered P

f^

θ

z

x

δ

Top view

PZT source

ψ = 0

f^

ψ

θ

z

x

δ

PZT source

Top view

= 50ψ o



Fractured sample: data

δ (°)

Tim

e (

µs

)

90 180 270 360

0

5

10

15

20

Dis

pla

cem

en

t (n

m)

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

Scattered P

δ (°)

Tim

e (

µs

)

90 180 270 360

0

5

10

15

20

Dis

pla

cem

en

t (n

m)

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

Scattered P

f^

θ

z

x

δ

Top view

PZT source

ψ = 0

f^

ψ

θ

z

x

δ

PZT source

Top view

= 50ψ o



Fractured sample: scattering amplitudes, ψ = 0◦

90 180 270 3600

0.1

0.2

0.3

0.4

0.5

θ (°)

Sc

att

ere

d a

mp

litu

de

(n

m)

Influence of ηN :experimental amplitude




90 180 270 3600

0.1

0.2

0.3

0.4

0.5

θ (°)

Sc

att

ere

d a

mp

litu

de

(n

m)

Influence of ηN :experimental amplitudeηN = 10−11 m/Pa




90 180 270 3600

0.1

0.2

0.3

0.4

0.5

θ (°)

Sc

att

ere

d a

mp

litu

de

(n

m)

Influence of ηN :experimental amplitudeηN = 10−11 m/PaηN = 0.5 10−11 m/Pa




90 180 270 3600

0.1

0.2

0.3

0.4

0.5

θ (°)

Sc

att

ere

d a

mp

litu

de

(n

m)

Influence of ηN :experimental amplitudeηN = 10−11 m/PaηN = 0.5 10−11 m/PaηN = 2 10−11 m/Pa




90 180 270 3600

0.1

0.2

0.3

0.4

0.5

θ (°)

Sc

att

ere

d a

mp

litu

de

(n

m)

Influence of ηT :experimental amplitude




90 180 270 3600

0.1

0.2

0.3

0.4

0.5

θ (°)

Sc

att

ere

d a

mp

litu

de

(n

m)

Influence of ηT :experimental amplitudeηT = 10−12 m/Pa




90 180 270 3600

0.1

0.2

0.3

0.4

0.5

θ (°)

Sc

att

ere

d a

mp

litu

de

(n

m)

Influence of ηT :experimental amplitudeηT = 10−12 m/PaηT = 10−13 m/Pa




90 180 270 3600

0.1

0.2

0.3

0.4

0.5

θ (°)

Sc

att

ere

d a

mp

litu

de

(n

m)

Influence of ηT :experimental amplitudeηT = 10−12 m/PaηT = 10−13 m/PaηT = 10−11 m/Pa



Fracture scattering: summary

Analytic expression for the scattering amplitude

Good agreement between theory and laboratory data

Estimation of the compliance ηN ≈ 10−11 m/Pa

Same range of compliance as found the literature(Pyrak-Nolte et al., 1990, Worthington, 2007)

Low sensitivity to ηT



Direct excitation of a fracture



Experimental setup

Same fractured sample

Direct excitation by laser-induced thermal expansion

Pulsed infrared laser source

sourceLaser

y

x

Receiver

Fracture

Sample rotation (

θ )



Results

sourceLaser

y

x

Receiver

Fracture

Sample rotation (

θ )

θ (°)

Tim

e (

µs

)

Surface excitation

−90 0 90 180 270

0

10

20

30

40



Results

sourceLaser

y

x

Receiver

Fracture

Sample rotation (

θ )

θ (°)

Direct excitation

−90 0 90 180 270

0

10

20

30

40



Results

θ (°)

Tim

e (

µs

)

Surface excitation

−90 0 90 180 270

0

10

20

30

40

Laser noise

θ (°)

Direct excitation

−90 0 90 180 270

0

10

20

30

40



Tip diffractions ⇒ radius estimation a = 3.5 mm

θ (°)

Tim

e (

µs

)

Surface excitation

−90 0 90 180 270

16

18

20

22

24

sourceLaser

y

x

Receiver



Tip diffractions ⇒ radius estimation a = 3.5 mm

θ (°)

Direct excitation

Tim

e (

µs

)

−90 0 90 180 270

6

8

10

12

14

sourceLaser

y

x

Receiver



Aknowledgements

We thank ConocoPhillips, especially Phil Anno, and Samik Sil forsupporting this research. We also thank John Scales and FilippoBroggini from the Colorado School of Mines and fellow membersof the Physical Acoustics Laboratory at Boise State University fortheir constructive ideas and comments.

www.earth.boisestate.edu/pal


www.earth.boisestate.edu/pal

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Elastic scattering by planar fractures