2.basicconcepts

8/14/2019 2.BasicConcepts

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Stanford Rock Physics Laboratory - Gary Mavko

14

Basic Geophysical Concepts


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where density K bulk modulus = 1/compressibility shear modulus Lam's coefficient E Young's modulus Poisson's ratio

M P-wave modulus = K + (4/3)

P wave velocity

S wave velocity

E wave velocity

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

Relating various velocities:

Body wave velocities have form: velocity=modulusdensity

Moduli from velocities:

=

VS2

K = VP2

4

3

VS

2

E= VE

2M = V

P

2

VP

2

VS

2=

2 1v( )(12v)

VE

2

VP

2 =

1+ v( )(12v)(1 v)

v =V

P

22V

S

2

2(VP

2V

S

2)=

VE

22V

S

2

2VS

2

VP

2

VS

2=

4VE

2

VS

2

3VE

2

VS

2

VE

2

VS

2=

3V

P

2

VS

2 4

VP

2

VS

21

VP

=

K+ (4 / 3)

=

+ 2

VS=

VE=

E


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The reflection coefficient of a normally-incident P-

wave on a boundary is given by:

where V is the acoustic impedance. Therefore,

anything that causes a large contrast in impedance

can cause a large reflection. Candidates include:

Changes in lithologyChanges in porosity

Changes in saturation

Diagenesis

We usually quantify Rock Physics relations in

terms of moduli and velocities, but in the fieldwe might look for travel time or Reflectivity

R = 2V21V1

2V2+

1V1

1V1

2V2


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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) mustbe equal to the total shear traction acting on the boundary inmedium 2 (due to the summed effects of the

transmitted 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.

VP1, VS1, 1

VP2, VS2, 2

1

1

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ReflectedP-wave

IncidentP-wave

ReflectedS-wave

Transmitted

P-wave

TransmittedS-wave

N.4

AVOAmplitude Variation with Offset

Recorded CMP Gather Synthetic

Deepwater Oil Sand


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AVO - Aki-Richards approximation:

P-wave reflectivity versus incident angle:

In principle, AVO gives us information aboutVp, Vs, and density. These are critical for

optimal Rock Physics interpretation. Well

see later the unique role of P- and S-wave

information for separating lithology,

pressure, and saturation.

Intercept Gradient

R01

2

VP

VP

+

R() R0 +1

2

VP

VP

2V

S

2

VP

2

+ 2

VS

VS

sin2

+

1

2

VP

VP

tan2 sin2[ ]


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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?

Interval Vp vs. Vs

A

B

Rock physics analysis is usually applied to intervals, where

we can find fairly universal relations of acoustic properties to

fluids, lithology, porosity, rock texture, etc.

In contrast, seismic wiggles depend on interval boundaries

and contrasts. This introduces countless variations in

geometry, wavelet, etc.

Interval Vp vs. Phi


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Convolutional Model

Impedance

vs. depthReflectivity

Convolve

With

wavelet

Normal Incidence

Seismic

Normal incidence reflection seismograms can be

approximatedwith the convolutional model. Reflectivity

sequence is approximately the derivative of the

impedance:

Seismic trace is smoothed with the wavelet:

R(t) 1

2

d

dtln V( )

S(t) w(t)R(t)

Be careful of US vs. European polarity conventions!

Rock properties

in each small

layer

Derivatives of

layerproperties

Smoothed image

of derivative ofimpedance


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Inversion

Two quantitative strategies to link intervalrock properties with seismic:

Forward modelingInversion

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 wellsQuantify signatures and sensitivities


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The Rock Physics Bottleneck

SeismicAttributes

TraveltimeVnmo

Vp/Vs

Ip,Is

Ro, G

AI, EIQ

anisotropy

etc

Acoustic

Properties

Vp

Vs

Density

Q

Reservoir

Properties

PorositySaturation

Pressure

Lithology

Pressure

Stress

Temp.

Etc.

At any point in the Earth, there are only 3(possibly 4) acoustic properties: Vp, Vs,

density, (and Q).

No matter how many seismicattributes we observe, inversions can

only give us three acoustic attributesOthers yield spatial or geometric information.


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Problem of ResolutionLog-scale rock physics may be different

than seismic scale


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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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