Amplitude
Variation with
Offset
presented by
Roxy Frary
Theory
Just some background
…ok a lot of background
Snell’s Law
sin(Q1 ) sin(Q2 ) sin(F1 ) sin(F 2 )
=
=
=
VP1
VP 2
VS1
VS 2
Reflection
Coefficients
r2VP 2 - r1VP1
RC =
r2VP 2 + r1VP1
I P 2 - I P1
=
I P 2 + I P1
DI P
=
2I PA
Zoeppritz
Equations
(Aki & Richards, 1980)
R(q ) = f (r1, r2,VP1,VP2,VS1,VS2,q )
(Much-needed)
Simplifications
Aki & Richards, 1980
1
1
DVP
2 2 æ Dr ö
2 2 DVS
RPP (Qa ) » (1- 4 p VSa ) ç ÷ +
- 4 p VSa
2
2
VSa
è ra ø 2 cos (Q a ) VPa
attempt to separate the
density dependence, Pwave, and S-wave
…still complicated…
2
æ sinQ1 ö
VP 2 cosQ1 - VP1 cosQ2
2
2
RPP (Q1 ) »
+ 2ç
(V
V
S1
S2 )
÷
VP2 cosQ1 + VP1 cosQ2
è VP1 ø
(Much-needed)
Simplifications
Hilterman, 1983
Separates into
“acoustic/fluid” and “shear”
terms – by assuming
constant density
…still complicated…
é
Ds ù 2
1 DVP
2
2
RPP (q1 ) » R0 + ê A0 R0 +
sin
q
+
(tan
q
sin
q1 )
1
1
2ú
(1- s a ) û
2 VP
ë
(Much-needed)
1- 2s
Simplifications
A0 = B0 - 2(1+ B0 )
DVP
B0 =
DVP
VPa
1- s
VPa
+ Dr
Shuey, 1985
ra
Normal incidence reflection coefficient
Intermediate angles
Approaching the critical angle
Each term describes a
different angular range of
the offset curve
RPP (q1 ) » A + Bsin2 q1 + Csin 2 q1 tan2 q1
• A (or R0) is the normal
incidence, or “zerooffset” stack
Weighted
Stacking
(Geostack)
Smith and Gidlow, 1987
reducing the prestack
• B is the AVO “slope” or
“gradient”
information to AVO attribute
traces
compute local incident
angle at each time, then do
• 3rd term is the “faroffset” stack
a regression analysis
RPP (q1 ) » A + Bsin2 q1 + Csin 2 q1 tan2 q1
é
Ds ù 2
1 DVP
RPP (q1 ) » R0 + ê A0 R0 +
sin
q
+
(tan 2 q1 - sin 2 q1 )
1
2ú
(1- s a ) û
2 VP
ë
RPP (q1 ) » R0 cos q1 + 2.25Ds sin q1
2
2
• near-offset stack
images the P-wave
impedance contrasts
• far-offset stack images
Poisson’s ratio contrasts
The “Most
Simple”
Simplification
Hilterman, 1989
At small angles, R0
dominates
Δσ dominates at larger
angles
Poisson’s Ratio
Koefoed, 1955
( )
( )
VP
s=
2
VS
é VP
2ê V
S
ë
-2
2
ù
-1ú
û
Incidence Angle
Koefoed, 1955
Shuey, 1985
é
Ds ù 2
1 DVP
RPP (q1 ) » R0 + ê A0 R0 +
sin
q
+
(tan 2 q1 - sin 2 q1 )
1
2ú
(1- s a ) û
2 VP
ë
VP Contrast
Koefoed, 1955
Shuey, 1985
é
Ds ù 2
1 DVP
RPP (q1 ) » R0 + ê A0 R0 +
sin
q
+
(tan 2 q1 - sin 2 q1 )
1
2ú
(1- s a ) û
2 VP
ë
é
Ds ù 2
1 DVP
RPP (q1 ) » R0 + ê A0 R0 +
sin
q
+
(tan 2 q1 - sin 2 q1 )
1
2ú
(1- s a ) û
2 VP
ë
Rule #1
Theoretical Conclusions
from
Koefoed, 1955
modified by
Shuey, 1985
An increase (decrease) of
Poisson’s ratio for the
underlying medium
produces an increase
(decrease) in the reflection
coefficient at larger angles
of incidence
é
Ds ù 2
1 DVP
RPP (q1 ) » R0 + ê A0 R0 +
sin
q
+
(tan 2 q1 - sin 2 q1 )
1
2ú
(1- s a ) û
2 VP
ë
Rule #2
Theoretical Conclusions
from
1- 2s
A0 = B0 - 2(1+ B0 )
1- s
Koefoed, 1955
modified by
Shuey, 1985
When Poisson’s ratio of the
media are equal, an
increase (decrease) of
Poisson’s ratio causes an
increase (decrease) in
reflection coefficient at
larger angles of incidence
é
Ds ù 2
1 DVP
RPP (q1 ) » R0 + ê A0 R0 +
sin
q
+
(tan 2 q1 - sin 2 q1 )
1
2ú
(1- s a ) û
2 VP
ë
Rule #3
Theoretical Conclusions
from
Koefoed, 1955
modified by
Shuey, 1985
Interchange of the media
affects the shape of the
curves only slightly – RPP
simply changes sign when
the elastic properties are
interchanged – except at
large angles
Industry Use:
Gas Sands
Since 1982
Gas Sands
• Ostrander, 1984
• Hypothetical gas
model
But how do we
see this in seismic
data?
Ostrander, 1984
Sacramento Valley
Sand reservoir at 1.75 s
Fault at SP 95
Reservoir limits SP 75-135
CDP Gathers
Ostrander, 1984
offset increases to the left
A & B show an increase in
amplitude with offset –
change in Poisson’s ratio –
gas-saturated sand
C shows a decrease in
amplitude with offset –
uniform Poisson’s ratio –
no gas sand
Another Example
Ostrander, 1984
Nevada
Amplitude anomaly at 1.6
s
Decrease in amplitude
with offset on gathers –
uniform Poisson’s ratio –
BASALT
But different Gas Sands
have different signatures
Rutherford & Williams, 1989
• Class 1: high impedance
o gradient is usually greatest
• Class 2: near-zero
impedance contrast
o seem to suddenly appear at
larger offsets, when amplitudes
rise above noise level
• Class 3: low impedance
o large reflectivities at all offsets
Class 1 Gas Sand
Example
Rutherford & Williams, 1989
Arkoma Basin
Pennsylvanian-aged
Hartshorn sand
“dim out”
polarity change at midoffset
Class 2 Gas Sand
Example
Rutherford & Williams, 1989
Gulf of Mexico
Brazos area
mid-Miocene
not a classic “gas sand”
anomaly – 2.1 s
Class 2 Gas Sand
Example (Cont’d)
Rutherford & Williams, 1989
AVO effects are
pronounced in mid- and faroffset synthetics
constant reflection angle
display confirms synthetic
data
Class 3 Gas Sand
Example
Rutherford & Williams, 1989
Gulf of Mexico
High Island area
Pliocene
most typical – large
reflectivity at all offsets
Class 4 Gas Sand
Castagna & Swan, 1997
Low impedance as well, but
reflectivity decreases with
offset
Industry Use:
Fluid Identification
Since 1997
• Substituting and
neglecting secondorder perturbations
yields
B = (1- 8g 2 )A - 4gDg
B = (1- 8g 2 )A
VS
g = V
P
Fluid Line
Foster & Keys, 1999
plotting in the slopeintercept domain
R(q ) » A + Bsin2 q +...
DVPa Dra
A=
+
2VPa 2 ra
DVPa
VSa2 é Dra DVSa ù
B=
-4 2 ê
+
ú
2VPa
VPa ë 2 ra VSa û
Fluid Line
(Cont’d)
Foster & Keys, 1999
R(q ) » A + Bsin2 q +...
B = (1- 8g 2 )A - 4gDg
•
Reflections from wet sands/shales fall on the Fluid
Line (little contrast in γ) – hydrocarbon-bearing
sands do not
•
Abrupt decrease (increase) in γ causes the
reflection to fall above (below) the Fluid Line – like
the tops and bases of sands
B = (1- 8g 2 )A
g = VS V
P
B = (1- 8g 2 )A - 4gDg
B = (1- 8g 2 )A
g = VS V
P
Fluid Line and
Gas Sands
Foster & Keys, 1999
•
Class 1: high-impedance – below
Fluid Line, to the right of the slope
axis
•
Class 2: negligible impedance
contrast – intersection with slope
axis
•
Class 3: low-impedance –
negative intercept and slope
•
Class 4: even lower impedance –
negative intercept, slope is zero
or positive
Fluid Line, Gas
Sands, and Rock
Properties
B = (1- 8g 2 )A - 4gDg
B = (1- 8g 2 )A
g = VS V
P
•
Foster & Keys, 1999
Start with top of Class 3 gas sand
at point 1
•
To get to point 2:
•
•
Alternatively, to get to point 3:
•
•
reduce porosity
Point 4:
•
•
increase porosity
replace gas with brine
To get to point 5:
•
reduce porosity of brine
Fluid Line, Gas
Sands, and Rock
Properties
(Cont’d)
Foster, Keys & Lane, 1999
•
Point 1: at normal incidence, the
reflection is negative, and
becomes more negative with
increasing offset
•
Point 2: reflection is more negative,
but less variation with offset than
Point 1
•
Point 3: small amplitude at normal
incidence, but will be more
negative with increasing offset
(more than 1 or 2)
•
Point 4: small positive amplitude at
normal incidence, and decreases
with offset
•
Point 5: large positive amplitude,
decreases with offset (more than 4)
B = (1- 8g 2 )A - 4gDg
B = (1- 8g 2 )A
g = VS V
P
Fluid Line, Gas
Sands, and Rock
Properties
(Cont’d)
Foster, Keys & Lane, 2010
•
Increasing the shale content
increases acoustic impedance
by reducing porosity (solid brown
line) – must also decrease γ
because pure shale lies on the
Fluid Line
•
Adding clay past the critical
concentration reduces acoustic
impedance (dashed brown line)
B = (1- 8g 2 )A - 4gDg
B = (1- 8g 2 )A
g = VS V
P
AVO for hydrocarbon
detection
Foster, Keys & Lane, 2010
Evaluation of
potential to
differentiate
hydrocarbons
from water
Well 1: central structure
Well 2: west structure
•
Step 1 – forward model
the expected AVO
response for brine- and
hydrocarbon-filled sands
from well log information
Well information
Well 1: a & b
Well 2: c & d
a & c indicate the expected
AVO for individual sand units
b & d are derived from
synthetic gathers modeled
from the well logs
•
We should expect a
reflection from the top of a
gas sand to peak at zero
offset and become larger
with increasing angle
•
Amplitudes should
decrease downdip from a
gas/water contact
Note change in amplitude convention
•
Class 3 at the top of the
reservoir section, Class 2
deeper as porosity
decreases
Seismic data
3D prestack time-migrated
gathers
Blue points are background
data, containing wet sands
and shales – used to define
the Fluid Line
Red points are the reservoir
– predominantly Class 3
sand
Applying AVO
scheme to stacked
seismic data
dark-green over light-green: top
and bottom of Class 3 sand
purple (Class 2) sands seen at
depth
gas/water contact (AVO anomaly)
terminates downdip
Check with
structure in map
view
anomaly extends to the
eastern structure as well
AVO for lithology
discrimination
Foster, Keys & Lane, 2010
Evaluation of
potential to
differentiate
reservoir sands
Back to basics: thicker sands
in a main channel feeding a
turbidite fan, porosity
decreases further from the
sediment source
Class 2 sands (b) have lower
porosity than Class 3 sands
(a)
AVO extraction
to map view
Well A found a commercial
reservoir
Well B found poor porosity
• Fluids cannot support
shear, so maximum
value of σ is 0.5
• Typical values:
o 0.05 for very hard rocks
o 0.45 for loose,
unconsolidated
sediments
o Close to 0.0 for gas sands
• At 0.33, S-wave velocity
is half P-wave velocity
• As gas saturation
increases, Poisson’s
ratio decreases
More on
Poisson’s Ratio
( )
( )
VP
s=
2
VS
é VP
2ê V
S
ë
-2
2
ù
-1ú
û
• The slope of the
“background trend”
depends only on the
background γ
B = (1- 8g 2 )A
More on A & B:
plotting in the
slope-intercept
domain
Castagna, Swan & Foster, 1998
R(q ) » A + Bsin2 q +...
DVPa Dra
A=
+
2VPa 2 ra
DVPa
VSa2 é Dra DVSa ù
B=
-4 2 ê
+
ú
2VPa
VPa ë 2 ra VSa û
A – normal incidence
B – AVO gradient/slope
B = (1- 8g 2 )A - 4gDg
B = (1- 8g 2 )A
g = VS V
P
• Shale/brine sand and
shale/gas sand
reflections
More on A & B:
plotting in the
slope-intercept
domain (Cont’d)
Castagna, Swan & Foster, 1998
R(q ) » A + Bsin2 q +...
DVPa Dra
A=
+
2VPa 2 ra
DVPa
VSa2 é Dra DVSa ù
B=
-4 2 ê
+
ú
2VPa
VPa ë 2 ra VSa û
A – normal incidence
B – AVO gradient/slope
B = (1- 8g 2 )A - 4gDg
B = (1- 8g 2 )A
g = VS V
P
• Shale/brine sand and
shale/gas sand
reflections – laboratory
measurements
More on A & B:
plotting in the
slope-intercept
domain (Cont’d)
Castagna, Swan & Foster, 1998
R(q ) » A + Bsin2 q +...
A – normal incidence
B – AVO gradient/slope
B = (1- 8g 2 )A - 4gDg
B = (1- 8g 2 )A
g = VS V
P
Porosity differences account for
variation
Background velocity is different for
each sand, so they don’t all plot
on same trend
• A & B become more
negative by adding
hydrocarbons (decreasing
Poisson’s ratio)
More on A & B:
plotting in the
slope-intercept
domain (Cont’d)
Castagna, Swan & Foster,
1998
A – normal incidence
B – AVO gradient/slope
B = (1- 8g 2 )A - 4gDg
B = (1- 8g 2 )A
g = VS V
P
Top of said layer plots
below background trend
Bottom of said layer plots
above the background
trend
Can’t classify
sands based on
properties of the
sand alone – the
advent of Class 4
Castagna, Swan & Foster, 1998
Overlying unit is shale
Class 3
Overlying unit is tight
(calcareous)
Class 4
Key difference: Vs contrast
Case History
Gulf of Mexico Bright Spot
Nsoga Mahob, Castagna & Young, 1999
Amplitude Anomaly
AVO Inversion –
Brine Model
max changes:
VP – 1000 ft/s
VS – 3000 ft/s
layer thickness – 100 ft
density – 1.0 g/cm3
used near-offset inverted P-wave velocity
curve
S-wave and Poisson’s ratio curves related
AVO Inversion –
Brine Model
not really all that close…
AVO Inversion –
Gas Model
max changes:
VP – 1000 ft/s
VS – 3000 ft/s
layer thickness – 100 ft
density – 1.0 g/cm3
used near-offset inverted P-wave velocity
curve
constant Poisson’s ratio of 0.1 in pay zone
AVO Inversion –
Gas Model
decently close!
gas model, with appropriate
mechanical properties,
converges to the real
seismic data
Some Issues
• Thin-bed tuning
o Can cause amplitude to
increase/decrease with offset
depending on time-thickness
and frequency
• Attenuation
o Signal/noise decrease with
offset
• NMO errors
o Conventional velocity analysis
is not “perfect” enough
o Ambiguity between stacking
velocity and reflectivity
o Can be corrected with full
waveform inversion
Key Takeaway Conclusions
• Important AVO simplification:
• The Rules:
é
Ds ù 2
1 DVP
2
2
RPP (q1 ) » R0 + ê A0 R0 +
sin
q
+
(tan
q
sin
q1 )
1
1
2ú
(1- s a ) û
2 VP
ë
o An increase (decrease) of Poisson’s ratio for the underlying medium produces an
increase (decrease) in the reflection coefficient at larger angles of incidence
o When Poisson’s ratio of the media are equal, an increase (decrease) of Poisson’s
ratio causes an increase (decrease) in reflection coefficient at larger angles of
incidence
o Interchange of the media affects the shape of the curves only slightly – RPP simply
changes sign when the elastic properties are interchanged – except at large
angles
• Gas Sand Classification:
o
o
o
o
Class 1 – high impedance contrast, high gradient, polarity change, low porosity
Class 2 – near-zero impedance contrast, seem to suddenly appear at larger offsets
Class 3 – low impedance contrast, high reflectivity at all offsets
Class 4 – low impedance contrast, reflectivity decreases with offset, high porosity
• Lithology and fluid identification
Key Takeaway Conclusions
References
•
•
•
•
•
•
•
•
•
•
•
•
•
•
Aki & Richards, 1980
Hilterman, 1983
Shuey, 1985
Smith and Gidlow, 1987
Hilterman, 1989
Koefoed, 1955
Ostrander, 1984
Rutherford & Williams, 1989
Castagna & Swan, 1997
Foster & Keys, 1999
Foster, Keys & Lane, 2010
Castagna, Swan & Foster, 1998
Nsoga Mahob, Castagna & Young, 1999
Fatti, Smith, Vail, Strauss & Levitt, 1994