Stanford Rock Physics Laboratory - Gary Mavko
Basic Geophysical Concepts
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Stanford Rock Physics Laboratory - Gary Mavko
Body wave velocities have form:
K + (4 / 3)µ
=
ρ
VP =
λ + 2µ
ρ
velocity= modulus
density
µ
VS =
ρ
E
VE =
ρ
where
ρ
K
µ
λ
E
ν
M
P wave velocity
S wave velocity
E wave velocity
density
bulk modulus = 1/compressibility
shear modulus
Lamé's coefficient
Young's modulus
Poisson's ratio
P-wave modulus = K + (4/3) µ
Moduli from velocities:
2
S
 2  4  2
K = ρ VP −   VS 


3
E = ρVE2
µ = ρV
M = ρVP2
In terms of Poisson's ratio we can also write:
VP2 2(1− v )
2 =
VS
(1− 2v)
VE2 (1+ v )(1− 2v)
VP2 − 2VS2 VE2 − 2VS2
v=
2 =
2
2 =
VP
(1− v)
2(VP − VS )
2VS2
Relating various velocities:
VP2
3 2 −4
VE2
VS
=
VP2
VS2
−1
VS2
VE2
4− 2
VP2
VS
=
VE2
VS2
3− 2
VS
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Stanford Rock Physics Laboratory - Gary Mavko
We usually quantify Rock Physics relations in
terms of moduli and velocities, but in the field
we might look for travel time or Reflectivity
ρ1V1
ρ2V2
The reflection coefficient of a normally-incident Pwave on a boundary is given by:
ρ2V2 −ρ1V1
R = ρ V +ρ V
2 2 1 1
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
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Stanford Rock Physics Laboratory - Gary Mavko
AVO
Amplitude Variation with Offset
φ1
V P1, VS1, ρ1
θ1
Deepwater Oil Sand
Reflected
S-wave
Reflected
P-wave
Incident
P-wave
Transmitted
P-wave
φ2
VP2, V S2, ρ2
θ2
Transmitted
S-wave
N.4
Recorded CMP Gather
Synthetic
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 displacement of a point in medium 2 at the boundary.
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Stanford Rock Physics Laboratory - Gary Mavko
AVO - Aki-Richards approximation:
P-wave reflectivity versus incident angle:
Intercept
Gradient
 1 ∆VP
VS2  ∆ρ
∆VS   2
R(θ ) ≈ R0 + 
−2 2
+2
  sin θ

2
V
V
V
ρ

P
P
S 

1 ∆VP
2
2
+
tan θ − sin θ
2 VP
[
]
1  ∆VP ∆ρ 
R0 ≈
+
2  VP
ρ
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.
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Stanford Rock Physics Laboratory - Gary Mavko
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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Stanford Rock Physics Laboratory - Gary Mavko
Intervals or Interfaces?
Crossplots or Wiggles?
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.
Interval Vp vs. Phi
Interval Vp vs. Vs
In contrast, seismic wiggles depend on interval boundaries
and contrasts. This introduces countless variations in
geometry, wavelet, etc.
A
B
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Stanford Rock Physics Laboratory - Gary Mavko
Convolutional Model
Impedance
Reflectivity
vs. depth
Rock properties
in each small
layer
Normal Incidence
Seismic
Convolve
With
wavelet
Derivatives of
layer
properties
Smoothed image
of derivative of
impedance
Normal incidence reflection seismograms can be
approximated with the convolutional model. Reflectivity
sequence is approximately the derivative of the
impedance:
1 d
R(t) ≈
ln( ρV )
2 dt
Seismic trace is “smoothed” with the wavelet:
S(t) ≈ w(t)∗ R(t)
Be careful of US vs. European polarity conventions!
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Stanford Rock Physics Laboratory - Gary Mavko
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
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Stanford Rock Physics Laboratory - Gary Mavko
The Rock Physics Bottleneck
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.
Seismic
Attributes
Traveltime
Vnmo
Vp/Vs
Ip,Is
Ro, G
AI, EI
Q
anisotropy
etc
Acoustic
Properties
Vp
Vs
Density
Q
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Reservoir
Properties
Porosity
Saturation
Pressure
Lithology
Pressure
Stress
Temp.
Etc.
Stanford Rock Physics Laboratory - Gary Mavko
Problem of Resolution
Log-scale rock physics may be different
than seismic scale
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Stanford Rock Physics Laboratory - Gary Mavko
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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