2. basic concepts - stanford earth concepts.pdf · stanford rock physics laboratory - gary mavko 15...
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Stanford Rock Physics Laboratory - Gary Mavko
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Basic Geophysical Concepts
Stanford Rock Physics Laboratory - Gary Mavko
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where ρ density K bulk modulus = 1/compressibility µ shear modulus λ Lamé's coefficient E Young's modulus ν Poisson's ratio M 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:
Relating various velocities:
Body wave velocities have form: velocity= modulusdensity
Moduli from velocities:
µ = ρVS2 K = ρ VP
2 −43
VS
2
E = ρVE2M = ρVP
2
VP2
VS2 =
2 1−v( )(1−2v)
VE2
VP2 =
1+ v( )(1−2v)(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)µ
ρ=
λ + 2µρ
VS =µρ
VE =Eρ
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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 impedancecan cause a large reflection. Candidates include:•Changes in lithology•Changes in porosity•Changes in saturation•Diagenesis
We usually quantify Rock Physics relations interms of moduli and velocities, but in the fieldwe might look for travel time or Reflectivity
R =ρ2V2−ρ1V1ρ2V2+ρ1V1
ρ1V1ρ2V2
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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, ρ1
VP2, VS2, ρ2
θ1
φ1
θ2φ2
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
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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. We’llsee later the unique role of P- and S-waveinformation for separating lithology,pressure, and saturation.
Intercept Gradient
R0 ≈12
∆VPVP
+∆ρρ
R(θ) ≈ R0 +12
∆VPVP
− 2VS2
VP2∆ρρ
+ 2 ∆VSVS
sin2θ
+12
∆VPVP
tan2θ − sin2θ[ ]
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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
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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
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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 ρV( )
S(t) ≈ w(t)∗R(t)Be careful of US vs. European polarity conventions!
Rock propertiesin each smalllayer
Derivatives oflayerproperties
Smoothed imageof derivative ofimpedance
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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 modelingor 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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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.
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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