Source: https://pangea.stanford.edu/courses/gp262/Notes/2.Basic%20Concepts.pdf

1. Stanford Rock Physics Laboratory - Gary Mavko
14
Basic Geophysical Concepts

2. Stanford Rock Physics Laboratory - Gary Mavko
15
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= modulus
density
Moduli from velocities:
µ = ρVS
2
K = ρ VP
2
−
4
3



VS
2



E = ρVE
2
M = ρVP
2
VP
2
VS
2 =
2 1−v( )
(1−2v)
VE
2
VP
2 =
1+ v( )(1−2v)
(1− v)
v =
VP
2
−2VS
2
2(VP
2
−VS
2
)
=
VE
2
−2VS
2
2VS
2
VP
2
VS
2 =
4 −
VE
2
VS
2
3 −
VE
2
VS
2
VE
2
VS
2 =
3
VP
2
VS
2
− 4
VP
2
VS
2
−1
VP =
K + (4 /3)µ
ρ
=
λ + 2µ
ρ
VS =
µ
ρ
VE =
E
ρ

3. Stanford Rock Physics Laboratory - Gary Mavko
16
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
We usually quantify Rock Physics relations in
terms of moduli and velocities, but in the field
we might look for travel time or Reflectivity
R =
ρ2
V2
−ρ1
V1
ρ2
V2
+ρ1
V1
ρ1
V1
ρ2
V2

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In an isotropic medium, a wave that is incident on a
boundary will generally create two reflected waves (one
P and one S) and two transmitted waves. The total shear
traction acting on the boundary in medium 1 (due to the
summed effects of the incident an reflected waves) must
be equal to the total shear traction acting on the boundary in
medium 2 (due to the summed effects of the
transmitted waves). Also the displacement of a point in
medium 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
θ2
φ2
Reflected
P-wave
Incident
P-wave
Reflected
S-wave
Transmitted
P-wave
Transmitted
S-wave
N.4
AVO
Amplitude 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 about
Vp, Vs, and density. These are critical for
optimal Rock Physics interpretation. We’ll
see later the unique role of P- and S-wave
information for separating lithology,
pressure, and saturation.
Intercept Gradient
R0 ≈
1
2
∆VP
VP
+
∆ρ
ρ


 


R(θ) ≈ R0 +
1
2
∆VP
VP
− 2
VS
2
VP
2
∆ρ
ρ
+ 2
∆VS
VS














sin2
θ
+
1
2
∆VP
VP
tan
2
θ − sin
2
θ[ ]

6. Stanford Rock Physics Laboratory - Gary Mavko
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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

7. 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, 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

8. Stanford Rock Physics Laboratory - Gary Mavko
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Convolutional Model
Impedance
vs. depth
Reflectivity
Convolve
With
wavelet
Normal Incidence
Seismic
Normal incidence reflection seismograms can be
approximated with the convolutional model. Reflectivity
sequence is approximately the derivative of the
impedance:
Seismic trace is “smoothed” with the wavelet:
R(t) ≈
1
2
d
dt
ln ρV( )
S(t) ≈ w(t)∗ R(t)
Be careful of US vs. European polarity conventions!
Rock properties
in each small
layer
Derivatives of
layer
properties
Smoothed image
of derivative of
impedance

9. Stanford Rock Physics Laboratory - Gary Mavko
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Inversion
Two quantitative strategies to link interval
rock properties with seismic:
•Forward modeling
•Inversion
•We have had great success in applying
rock physics to interval properties.
•For the most part, applying RP directly to
the 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

10. Stanford Rock Physics Laboratory - Gary Mavko
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The Rock Physics Bottleneck
Seismic
Attributes
Traveltime
Vnmo
Vp/Vs
Ip,Is
Ro, G
AI, EI
Q
anisotropy
etc
Acoustic
Properties
Vp
Vs
Density
Q
Reservoir
Properties
Porosity
Saturation
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 seismic
attributes we observe, inversions can
only give us three acoustic attributes
Others yield spatial or geometric information.

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Problem of Resolution
Log-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