# Rock Physics: Basic Concepts

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Stanford Rock Physics Laboratory - Gary Mavko

14

Basic Geophysical Concepts

Stanford Rock Physics Laboratory - Gary Mavko

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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 = VP

2 43

VS

2

E = VE2M = VP

2

VP2

VS2 =

2 1v( )(12v)

VE2

VP2 =

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

v = VP2 2VS2

2(VP2 VS

2 )=VE2 2VS2

2VS2

VP2

VS2 =

4 VE2

VS2

3 VE2

VS2

VE2

VS2 =

3VP2

VS2 4

VP2

VS2 1

VP =K + (4 / 3)

=

+ 2

VS =

VE =E

Stanford Rock Physics Laboratory - Gary Mavko

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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 impedancecan cause a large reflection. Candidates include:Changes in lithologyChanges in porosityChanges in saturationDiagenesis

We usually quantify Rock Physics relations interms of moduli and velocities, but in the fieldwe might look for travel time or Reflectivity

R =2V21V12V2+1V1

1V12V2

Stanford Rock Physics Laboratory - Gary Mavko

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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 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.

VP1, VS1, 1

VP2, VS2, 2

1

1

22

Reflected P-wave

Incident P-wave

Reflected S-wave

Transmitted P-wave

Transmitted S-wave N.4

AVOAmplitude Variation with Offset

Recorded CMP Gather Synthetic

Deepwater Oil Sand

Stanford Rock Physics Laboratory - Gary Mavko

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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 foroptimal Rock Physics interpretation. Wellsee later the unique role of P- and S-waveinformation for separating lithology,pressure, and saturation.

Intercept Gradient

R0 12

VPVP

+

R() R0 +12

VPVP

2VS2

VP2

+ 2 VSVS

sin2

+12

VPVP

tan2 sin2[ ]

Stanford Rock Physics Laboratory - Gary Mavko

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Seismic AmplitudesMany 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

Stanford Rock Physics Laboratory - Gary Mavko

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Intervals or Interfaces? Crossplots or Wiggles?

Interval Vp vs. Vs

A

B

Rock physics analysis is usually applied to intervals, wherewe can find fairly universal relations of acoustic properties tofluids, lithology, porosity, rock texture, etc.

In contrast, seismic wiggles depend on interval boundariesand contrasts. This introduces countless variations ingeometry, wavelet, etc.

Interval Vp vs. Phi

Stanford Rock Physics Laboratory - Gary Mavko

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Convolutional ModelImpedancevs. 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) 12ddtln V( )

S(t) w(t)R(t)Be careful of US vs. European polarity conventions!

Rock propertiesin each smalllayer

Derivatives oflayerproperties

Smoothed imageof derivative ofimpedance

Stanford Rock Physics Laboratory - Gary Mavko

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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 modelingor inversion step.

We often choose a model-based study,calibrated to logs (when possible) toDiagnose formation propertiesExplore situations not seen in the wellsQuantify signatures and sensitivities

Stanford Rock Physics Laboratory - Gary Mavko

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The Rock Physics Bottleneck

Seismic Attributes

TraveltimeVnmoVp/VsIp,IsRo, GAI, EIQanisotropyetc

Acoustic Properties

VpVsDensityQ

ReservoirProperties

PorositySaturationPressureLithologyPressureStressTemp.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 canonly give us three acoustic attributesOthers yield spatial or geometric information.

Stanford Rock Physics Laboratory - Gary Mavko

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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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