02. BasicConcepts

8/7/2019 02. BasicConcepts

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Stanford Rock Physics Laboratory - Tapan Muker

14

Basic Geophysical Concepts




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whereρ density

K bulk modulus = 1/compressibilityµ shear modulusλ Lamé's coefficientE Young's modulusν Poisson's ratioM P-wave modulus = K + (4/3) µ

P wave velocity

S wave velocity

E wave velocity

In terms of Poisson's ratio we can also write:

Body wave velocities have form: velocity= modulusdensity

Moduli from velocities:

µ = ρ V S

2K = ρ V

P

2 − 43

⎛ ⎝

⎞ ⎠V S

2⎛ ⎝ ⎜

⎞ ⎠⎟

E = ρ V E

2 M = ρ V

P

2

V P=

K + (4 /3)µ

ρ =

λ + 2µ

ρ

V S=

µ

ρ

V E =

E

ρ

V P

2

V S

2 = 2 1 −v

( )(1 − 2v )

V E

2

V P

2 = 1+v

( )(1−2v)

(1− v)v =

V P

2

−2V S

2

2(V P

2 −V S

2)

=V E

2

−2V S

2

2V S

2

Relating various velocities:

V E

2

V S

2 =

3V

P

2

V S

2− 4

V P

2

V S2

−1

V P

2

V S

2=

4 −V E

2

V S

2

3 −V E

2

V S2




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We usually quantify Rock Physics relations interms of moduli and velocities , but in the field

we might look for travel time or Reflectivity

ρ 1V 1

ρ 2V 2

The reflection coefficient of a normally-incident P-wave on a boundary is given by:

where ρV is the acoustic impedance. Therefore,anything that causes a large contrast in impedance

can cause a large reflection. Candidates include:

•Changes in lithology

•Changes in porosity

•Changes in saturation

•Diagenesis

R =ρ

2V

2− ρ 1V 1 ρ

2V

2+ ρ

1V

1




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AVOAmplitude Variation with Offset

VP1, VS1, ρ1

VP2, VS2, ρ2

θ1

φ1

θ2φ2

ReflectedP-wave

IncidentP-wave

ReflectedS-wave

Transmitted

P-wave

TransmittedS-wave

N.4

C D

Deepwater Oil Sand

Recorded CMP Gather Synthetic

In an isotropic medium, a wave that is incident on aboundary will generally create two reflected waves (oneP and one S) and two transmitted waves. The total sheartraction acting on the boundary in medium 1 (due to thesummed effects of the incident an reflected waves) must

be equal to the total shear traction acting on the boundary inmedium 2 (due to the summed effects of thetransmitted waves). Also the displacement of a point inmedium 1 at the boundary must be equal to the displace-ment of a point in medium 2 at the boundary.




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AVO - Aki-Richards approximation:

P-wave reflectivity versus incident angle:

Intercept Gradient

R0 ≈1

2

∆V P

V P

+∆ ρ ρ

⎛

⎝ ⎜ ⎞

⎠⎟

R(θ ) ≈ R0+

1

2

∆V P

V P

− 2V S

2

V P

2

∆ ρ ρ

+ 2∆V

S

V S

⎛

⎝ ⎜

⎞

⎠⎟

⎡

⎣⎢⎢

⎤

⎦⎥⎥sin2θ

+1

2

∆V P

V P

tan2θ − sin

2θ [ ]

In principle, AVO gives us information aboutVp, Vs, and density. These are critical foroptimal Rock Physics interpretation. We’llsee later the unique role of P- and S-wave

information for separating lithology,pressure, and saturation.




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Seismic Amplitudes

Many factors influence seismic amplitude:

• Source coupling

• Source radiation pattern

• Receiver response, coupling, and pattern

• Scattering and Intrinsic Attenuation

• Sperical divergence

• Focusing• Anisotropy

• Statics, moveout, migration, decon, DMO

• Angle of Incidence

…

• Reflection coefficient

Source Rcvr




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Intervals or Interfaces?Crossplots or Wiggles?

Rock physics analysis is usually applied to intervals, where

we can find fairly universal relations of acoustic properties tofluids, lithology, porosity, rock texture, etc.

Interval Vp vs. Vs

In contrast, seismic wiggles depend on interval boundaries

and contrasts. This introduces countless variations in

geometry, wavelet, etc.

Interval Vp vs. Phi

A

B




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Convolutional Model

Impedancevs. depth

Reflectivity

ConvolveWith

wavelet

Normal IncidenceSeismic

Normal incidence reflection seismograms can beapproximated with the convolutional model. Reflectivitysequence is approximately the derivative of theimpedance:

Seismic trace is “smoothed” with the wavelet:

R(t ) ≈ 12d

dt ln ρ V ( )

S(t ) ≈ w(t )∗ R(t )

Rock propertiesin each small

layer

Derivatives of

layerproperties

Smoothed image

of derivative ofimpedance

Be careful of US vs. European polarity conventions!




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Inversion

Two quantitative strategies to link intervalrock properties with seismic:

•Forward modeling•Inversion

•We have had great success in applyingrock physics to interval properties.

•For the most part, applying RP directly tothe seismic wiggles, requires a modeling

or inversion step.

We often choose a model-based study,calibrated to logs (when possible) to

•Diagnose formation properties•Explore situations not seen in the wells•Quantify signatures and sensitivities




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Problem of ResolutionLog-scale rock physics may be different

than seismic scale




Seismic properties (velocity, impedance,Poisson Ratio, etc)

… depend on pore pressure and stress

Units of Stress:

1 bar = 106 dyne/cm2 = 14.50 psi

10 bar = 1 MPa = 106 N/m2

1 Pa = 1 N/m2 = 1.45 10-4 psi = 10-5 bar

1000 kPa = 10 bar = 1 MPa

Stress always has units of force/area

Mudweight to Pressure Gradient

1 psi/ft = 144 lb/ft3

= 19.24 lb/gal

= 22.5 kPa/m

1 lb/gal = 0.052 psi/ft

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