Stanford Rock Physics Laboratory - Gary Mavko
AVO
The Rock Physics of AVO
302
Stanford Rock Physics Laboratory - Gary Mavko
AVO
Water-saturated
40 MPa
Water-saturated
10-40 MPa
Gas and
Water-saturated
10-40 MPa
L8
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Stanford Rock Physics Laboratory - Gary Mavko
AVO
N.1
More than 400 sandstone data points, with porosities
ranging over 4-39%, clay content 0-55%, effective
pressure 5-40 MPa - all water saturated.
When Vp is plotted vs. Vs, they follow a remarkably
narrow trend. Variations in porosity, clay, and pressure
simply move the points up and down the trend.
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Stanford Rock Physics Laboratory - Gary Mavko
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N.2
Variations in porosity, pore pressure, and shaliness
move data along trends. Changing the pore fluid
causes the trend to change.
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Stanford Rock Physics Laboratory - Gary Mavko
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Different shear-related attributes.
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N.2
Stanford Rock Physics Laboratory - Gary Mavko
AVO
φ1
V P1, VS1, ρ1
θ1
Reflected
S-wave
Reflected
P-wave
Incident
P-wave
Transmitted
P-wave
φ2
θ2
Transmitted
S-wave
VP2, V S2, ρ2
N.4
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
By matching the traction and displacement boundary
conditions, Zoeppritz (1919) derived the equations
relating the amplitudes of the P and S waves:
 sin(θ1 )
cos(φ1 )
 − cos(θ )
sin(φ1 )
1

VP1

cos(2φ1 )
 sin(2θ1 )
VS1


VS1
sin(2φ1 )
 − cos(2φ1 ) −
VP1


− sin(θ 2 )
cos(φ 2 )
  Rpp  − sin(θ1 ) 
− cos(θ 2 )
− sin(φ 2 )



ρ2VS22VP1
ρ2VS 2VP1
  Rps  − cos(θ1 ) 
=
sin(2θ 2 ) −
cos(2φ 2 ) 
ρ1VS12VP 2
ρ1VS12
Tpp

  sin(2θ1 ) 



  Tps   − cos(2φ1 )
ρ 2VP 2
ρ2VS 2
−
cos(2φ2 )
−
cos(2φ 2 ) 
ρ1VP1
ρ1VP1

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Stanford Rock Physics Laboratory - Gary Mavko
AVO
AVO - Shuey's Approximation
P-wave reflectivity versus angle:
Intercept
Gradient


∆
1 ∆VP
ν
2
2
2

sin
R(θ ) = R0 +  ER0 +
θ
+
tan
θ
−
sin
θ]
[
2
2
V

P
(1− ν ) 
1  ∆VP ∆ ρ 
R0 ≈ 
+
2  VP
ρ 
E = F − 2(1 + F )
1− 2ν
1− ν
∆VP / VP
F=
∆VP / VP + ∆ρ / ρ
∆VP = (VP 2 − VP1)
∆VS = (VS 2 − VS1 )
∆ρ = ( ρ2 − ρ1 )
VP = (VP 2 + VP1) / 2
VS = (VS2 + VS1 ) / 2
ρ = (ρ 2 + ρ1 ) / 2
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Stanford Rock Physics Laboratory - Gary Mavko
AVO
AVO - Aki-Richard's approximation:
P-wave reflectivity versus angle:
Intercept
Gradient
 1 ∆VP
VS2  ∆ρ
∆VS   2
R(θ ) = R0 + 
−2 2
+2
  sin θ
VP  ρ
VS  
 2 VP
1 ∆VP
+
tan2 θ − sin2 θ ]
[
2 VP
1  ∆VP ∆ ρ 
R0 ≈ 
+
2  VP
ρ 
∆VP = (VP 2 − VP1)
∆VS = (VS 2 − VS1 )
∆ρ = ( ρ2 − ρ1 )
VP = (VP 2 + VP1) / 2
VS = (VS2 + VS1 ) / 2
ρ = (ρ 2 + ρ1 ) / 2
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Stanford Rock Physics Laboratory - Gary Mavko
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AVO Response
P-Velocity
contrast
negative
negative
positive
positive
Poisson ratio
contrast
negative
positive
negative
positive
311
AVO response
increase
decrease
decrease
increase
Stanford Rock Physics Laboratory - Gary Mavko
AVO
Vp-Vs Relations
There is a wide, and sometimes confusing, variety of
published Vp-Vs relations and Vs prediction techniques,
which at first appear to be quite distinct. However, most
reduce to the same two simple steps:
1. Establish empirical relations among Vp, Vs, and porosity
for one reference pore fluid--most often water saturated or
dry.
2. Use Gassmann’s (1951) relations to map these empirical
relations to other pore fluid states.
Although some of the effective medium models predict both P
and S velocities assuming idealized pore geometries, the fact
remains that the most reliable and most often used Vp-Vs
relations are empirical fits to laboratory and/or log data. The
most useful role of theoretical methods is extending these
empirical relations to different pore fluids or measurement
frequencies. Hence, the two steps listed above.
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Stanford Rock Physics Laboratory - Gary Mavko
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7
Sandstones
Water Saturated
6
Vp (km/s)
5
Mudrock
mudrock
.8621V
.1724
VVs
= .86
Vpp --11.17
s =
4
3
Han(1986)
(1986)
Han
868
VVs
s == .7936V
p - .-70.79
.79 Vp
2
Castagna
Castagnaetetal.al.(1993)
(1993)
V s = .8042Vp - . 8 5 5 9
1
Vs = .80 Vp - 0.86
(after Castagna et al., 1993)
0
0
0.5
1
1.5
2
2.5
3
3.5
4
Vs (km/s)
7
Shales
Water Saturated
6
Vp (km/s)
5
Mudrock
mudrock
.86 Vp
Vp -- 1.17
1.1724
VVs
s == .8621
4
3
Han (1986)
(1986)
Han
=
.7936V
7868
V
Vss = .79
Vpp-- .0.79
2
Castagna etetal.
Castagna
al.(1993)
(1993)
V s = .8042Vp - . 8 5 5 9
Vs = .80 Vp - 0.86
1
(after Castagna et al., 1993)
0
0
0.5
1
1.5
2
2.5
3
3.5
N.5
Vs (km/s)
313
N.5
4
Stanford Rock Physics Laboratory - Gary Mavko
AVO
8
Limestones
Water Saturated
7
Vp (km/s)
6
Castagna
et al. (1993)
Castagna et al. (1993)
Vs
= -.055 Vp2
V s = -.05508 VP 2
+ 1.02
Vp
+ 1.0168 Vp
- 1.03
- 1.0305
5
4
3
2
VPickett(1963)
s = Vp / 1 . 9
Pickett
Vs = Vp(1963)
/ 1.9
1
water
(after Castagna et al.,1993)
0
0
0.5
1
1.5
2
2.5
3
3.5
4
Vs (km/s)
8
Dolomite
Water Saturated
7
Pickett (1963)
Pickett(1963)
= V/p /1.8
1.8
Vs V=s Vp
Vp (km/s)
6
5
4
Castagna et al. (1993)
Castagna
et al. (1993)
V s = .5832Vp - . 0 7 7 7 6
Vs = .58 Vp - 0.078
3
2
1
(after Castagna et al., 1993)
0
0
0.5
1
1.5
2
2.5
3
3.5
N.6
Vs (km/s)
314
4
Stanford Rock Physics Laboratory - Gary Mavko
AVO
6
Shaly Sandstones
Water Saturated
Vp (km/s)
5
mudrock
Mudrock
= . 8Vp
6 2 1-V1.17
p-1.1724
Vs =V s.86
4
clay >
> 25%
25 %
Clay
Vs=.8423Vp-1.099
3 Vs = .84Vp-1.10
clay < 25 %
Clay < 25%
Vs=.7535Vp-.6566
2
Vs = .75 Vp - 0.66
Vp-sat c>.25
Vp-sat c<.25
1
(Data from Han, 1986)
0
0
0.5
6
1.5
2
2.5
Vs (km/s)
3
3.5
4
Shaly Sandstones
Water Saturated (hf)
5
Vp (km/s)
1
mudrock
Mudrock
Vs ==.86
.8621Vp-1.1724
Vs
Vp - 1.17
4
porosity
%
porosity<<15
15%
Vs
=
.8533Vp-1.1374
Vs = .85 Vp - 1.14
3
porosity
%
porosity>>15
15%
Vs = .7563Vp-.6620
2
Vs = .76Vp - 0.66
1
Vp-sat Phi>.15
Vp-sat Phi<.15
(Data from Han, 1986)
0
0
0.5
1
1.5
2
2.5
Vs (km/s)
315
3
3.5
N.7
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Stanford Rock Physics Laboratory - Gary Mavko
AVO
Dry Poisson’s Ratio Assumption
The modified Voigt Average Predicts linear moduli-porosity relations.
This is a convenient relation for use with the critical porosity model.

φ
K dry = K0 1− 
 φc 

φ
µdry = µ0 1− 
 φc 
These are equivalent to the dry rock Vs/Vp relation and the dry rock
Poisson’s ratio equal to their values for pure mineral.
 VS 
 VS 
≈
 V 
 V 
P dry rock
P mineral
νdry rock ≈ ν mineral
The plot below illustrates the approximately constant dry rock
Poisson’s ratio observed for a large set of ultrasonic sandstone
velocities (from Han, 1986) over a large rance of effective pressures (5
< Peff < 40 MPa) and clay contents (0 < C < 55% by volume).
16
Shaly Sandstones - Dry
14
ν = 0.1
clay < 10%
clay > 10%
12
Vs2 dry
ν = 0.01
ν = 0.2
10
ν = 0.3
8
6
ν = 0.4
4
2
0
0
5
10
15
20
2
Vp dry
316
25
30
35
N.8
Stanford Rock Physics Laboratory - Gary Mavko
AVO
Krief’s Relation (1990)
The model combines the same two elements:
1. An empirical Vp-Vs-φ relation for water-saturated
rocks, which is approximately the same as the critical
porosity model.
2. Gassmann’s relation to extend the empirical
relation to other pore fluids.
Dry rock Vp-Vs-φ relation:
K dry = K mineral (1− β )
where β is Biot’s coefficient. This is equivalent to:
1
1
φ
=
+
K dry K 0 Kφ
where
dv p
1 dv p
1
=
; β=
Kφ v p dσ P = constant
dV
P
PP = constant
φKdry
=
Kφ
β and Kf are two equivalent descriptions of the pore space
stiffness. Determining β vs. φ or Kφ vs φ determines the
rock bulk modulus Kdry vs φ.
Krief et al. (1990) used the data of Raymer et al. (1980) to
empirically find a relation for β vs φ:
(1− β) = (1− φ )
m (φ )
where m(φ ) = 3 / (1− φ )
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Stanford Rock Physics Laboratory - Gary Mavko
AVO
Assuming dry rock Poisson’s ratio is equal to the
mineral Poisson’s ratio gives
K dry = K 0 (1− φ )
µdry = µ 0 (1− φ )
m(φ )
m(φ )
N.9
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Stanford Rock Physics Laboratory - Gary Mavko
AVO
Expressions for any other pore fluids are obtained from
Gassmann’s equations. While these are nonlinear, they
suggest a simple approximation:
VP2− sat − V fl2 VP20 − V fl2
=
2
VS −sat
VS02
where VP-sat, VP0, and Vfl are the P-wave velocities of the
saturated rock, the mineral, and the pore fluid; and VS-sat
and VS0 are the S-wave velocities in the saturated rock
and mineral. Rewriting slightly gives
VP2− sat
2
2

V
−
V
P0
fl 
2
2
= V fl + VS− sat 
 V 2 
S0
which is a straight line (in velocity-squared) connecting the
2
2 ) and the fluid point ( 2
mineral point ( VP0
, VS0
Vfl , 0). A more
accurate (and nearly identical) model is to recognize that
velocities tend toward those of a suspension at high
porosity, rather than toward a fluid, which yields the
modified form
2
2
2
2
VP − sat − VR VP 0 − VR
=
VS2−sat
VS02
where VR is the velocity of a suspension of minerals in a
fluid, given by the Reuss average at the critical porosity.
This modified form of Krief’s expression is exactly
equivalent to the linear (modified Voigt) K vs φ and µ vs φ
relations in the critical porosity model, with the fluid effects
given by Gassmann.
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Stanford Rock Physics Laboratory - Gary Mavko
AVO
Vp-Vs Relation in Dry and
Saturated Rocks
35
Sandstones
mineral
point
30
25
Vp2 (km/s)2
saturated
20
15
dry
10
5
fluid point
0
0
5
10
15
2
2
Vs (km/s)
320
20
N.10
Stanford Rock Physics Laboratory - Gary Mavko
AVO
Vp-Vs Relation in Sandstone
and Dolomite
50
mineral
points
Vp2 (km/s)2
40
Dolomite
30
20
Sandstone
10
fluid points
0
0
4
8
Vs2 (km/s)2
321
12
16
N.11
Stanford Rock Physics Laboratory - Gary Mavko
AVO
Greenberg and Castagna (1992) have given empirical
relations for estimating Vs from Vp in multimineralic, brinesaturated rocks based on empirical, polynomial Vp-Vs
relations in pure monomineralic lithologies (Castagna et
al., 1992). The shear wave velocity in brine-saturated
composite lithologies is approximated by a simple average
of the arithmetic and harmonic means of the constituent
pure lithology shear velocities:
VS = 1
2
L
Σ
i=1
where
L
Xi
aij
Ni
Vp, Vs
L
Ni
j
Xi Σ a ijVP
Σ
i=1
j=0
+
Ni
L
j
X i Σ a ijVP
Σ
i= 1
j= 0
–1 –1
Xi = 1
number of monomineralic lithologic constituent
volume fractions of lithological constituents
empirical regression coefficients
order of polynomial for constituent i
P and S wave velocities (km/s) in composite brinesaturated, multimineralic rock
Castagna et al. (1992) gave representative polynomial
regression coefficients for pure monomineralic lithologies:
Lithology
S a ndstone
Lime stone
D olomite
S ha le
a i2
a i1
a i0
0
-0 .0 5 5 0 8
0
0
0 .8 0 4 1 6
1 .0 1 6 7 7
0 .5 8 3 2 1
0 .7 6 9 6 9
-0 .8 5 5 8 8
-1 .0 3 0 4 9
-0 .0 7 7 7 5
-0 .8 6 7 3 5
Regression coefficients for pure lithologies with Vp and Vs
in km/s:
(Castagna et al. 1992)
VS = a i2VP2 + ai1VP + a i0
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Stanford Rock Physics Laboratory - Gary Mavko
AVO
7
Vp (km/s)
6
5
4
3
Sandstone
Limestone
Dolomite
Shale
2
1
0
1
2
3
Vs (km/s)
4
5
N.12
Note that the above relation is for 100% brine-saturated rocks.
To estimate Vs from measured Vp for other fluid saturations,
Gassmann’s equation has to be used in an iterative manner.
In the following, the subscript b denotes velocities at 100%
brine saturation and the subscript f denotes velocities at any
other fluid saturation (e.g. this could be oil or a mixture of oil,
brine, and gas). The method consists of iteratively finding a
(Vp, Vs) point on the brine relation that transforms, with
Gassmann’s relation, to the measured Vp and the unknown Vs
for the new fluid saturation. the steps are as follows:
1. Start with an initial guess for VPb.
2. Calculate VSb corresponding to VPb from the empirical
regression.
3. Do fluid substitution using VPb and VSb in the Gassmann
equation to get VSf.
4. With the calculated VSf and the measured VPf, use the
Gassmann relation to get a new estimate of VPb. Check with
previous value of VPb for convergence. If convergence
criterion is met, stop; if not go back to step 2 and continue.
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Stanford Rock Physics Laboratory - Gary Mavko
AVO
Polynomial and powerlaw forms of the Gardner et al. (1974) velocity-density
relationships presented by Castagna et al. (1993). Units are km/s and
g/cm3.
Coefficients for the equation ρ b = aVp2 + bVp + c
L ithology
a
Shale
- .0261
Sandstone - .0115
Limestone - .0296
Dolomite
- .0235
Anhydrite - .0203
Coefficients for
L ithology
d
Shale
Sandstone
Limestone
Dolomite
Anhydrite
1.75
1.66
1.50
1.74
2.19
b
c
V p Range
(Km/s)
.373
1.458
.261
1.515
.461
0.963
.390
1.242
.321
1.732
the equation ρ b = dVpf
f
.265
.261
.225
.252
.160
1.5- 5.0
1.5- 6.0
3.5- 6.4
4.5- 7.1
4.6- 7.4
V p Range
(Km/s)
1.5- 5.0
1.5- 6.0
3.5- 6.4
4.5- 7.1
4.6- 7.4
N.15
Both forms of Gardner’s relations applied to log and lab shale data, as
presented by Castagna et al. (1993)
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Stanford Rock Physics Laboratory - Gary Mavko
AVO
N.16
Both forms of Gardner’s relations applied to
laboratory dolomite data.
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Stanford Rock Physics Laboratory - Gary Mavko
AVO
Both forms of Gardner’s relations applied to log and lab sandstone data, as
presented by Castagna et al. (1993).
Both forms of Gardner’s relations applied to laboratory N.17
limestone data. Note
that the published powerlaw form does not fit as well as the polynomial. we
also show a powerlaw form fit to these data, which agrees very well with the
polynomial.
326