8.VelocityDispersion
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Stanford Rock Physics Laboratory - Gary Mavko
Velocity Dispersion and Q
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Velocity Dispersion and
Wave Attenuation (Q)
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(1) Seismic velocities almost always increase withfrequency, and
(2) Seismic waves are always attenuated as they
travel through rocks.
These two observations are usually intimately related.Both usually increase from dry to fluid saturatedconditions, and both usually decrease with increasingeffective pressure.
These effects complicate the comparison of laboratoryand field data, but they also reveal details about thepore space and the pore fluids it contains.
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Kramers-Kronig Relations
Q1() =
||
MR()
MR() MR(0)
d
MR () MR(0)=
Q1()MR ()
||
d
Causality leads to a very specific relationrequired between Q and modulus dispersion
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In most rocks and sediments, the velocity tends toincrease with frequency. This is accompanied byattenuation. Attenuation tends to be highest infrequency range where velocity is increasing most
rapidly.
I.1
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In real materials, we expect that multiple mechanismsof attenuation are present, each having its owncharacteristic frequency and magnitude.
In fact, we might expect a fairly constant level ofattenuation over wide frequency bands.
Thermoelastic
Fluid squirtBiot
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The difference between dry and saturated velocities and the disagreementwith the low frequency Gassmann theory often increases with fluid
viscosity. Again the differences are greatest at low pressures.Data from Winkler (1985).
I.3
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frequency
dry rock
3-D Seismic ultrasoniclab
Ve
locity
saturateddispersion
The most common recipe for applying ultrasonic core
data to field conditions is to use velocities measured
on (nearly) dry cores and then use Gassmann to add
the fluids. The basic assumption is that velocitydispersion is smaller for dry or nearly dry rocks, so
that the ultrasonic dry velocities are good estimates
of the low frequency dry velocities.
Gassmann
Measuredultrasonic
Measuredultrasonic
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Failure of Gassmann's theory to predict saturated ultrasonic velocitiesrelative to dry velocities. Navajo sandstone data from Coyner (1984).
I.4
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E1
E2
ViscoelasticityWe have been talking about elastic materials where
stress is proportional to strain.
volumetric
shear
general
Viscoelastic materials also depend on rate or history.
Maxwell model
Voigt model
Standard linear
solid
11 + 22 + 33
3= K(11 + 22 + 33)
ij = 2ij
ij = ij + 2ij
ij=
ij
2+
ij
2
ij = 2ij + 2ij
ij + E1 + E2( )ij = E2 ij + E1ij( )
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Wave Propagating in a ViscoelasticSolid
At any point, the stress and strain are out of phase:
The ratio of stress to strain is the complex modulus.
u(x,t) = u0exp ( )x exp i t kx( )
=
0exp i t
kx( )[ ]= 0exp i t kx ( )[ ]
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low Q: large dissipation
high Q: small dissipation
Different views of Q:energy dissipated per wave
cycle
peak strain energy of the wave
velocity
frequency
phase delay
amplitude loss per cycle
1
Q=
W
2W
1
Q=
Vf
1
Q
1
lnu(t)
u(t+)
1
Q= tan()
Quality Factor Q
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Spectral Ratio MethodWe can think of Q-1 as the fractional loss perwavelength or per oscillation. Therefore over a fixed
distance there is a tendency for shorter wavelengths
to attenuate more:
or
If we propagate the wave
Then we can compare the amplitudes at two different
distances:
1
QV
f
fVQ
u= u0exp x[ ]
lnux2
ux1
= x2 x1( )
lnux2ux1
=
f
QVx2 x
1( )
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Standard Linear Solid
If we assume sinusoidal motion
Then we can write:
with the complex, frequency-dependent modulus
In the limits of low frequency and high frequency
= 0e
it
= 0e
it
0 = M()0
M() =E
2E
1 +i( )
E1 + E2 + i
=
M M0 + i
r
M0M
M + i
r
M0M
M0=
E2E
1
E1+ E
2
, 0
M= E
2
,
Re M ( ){ } =M
0M 1+
r
2
M +
r
2
M0
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Standard Linear SolidSimilarly, we can write Q as a function of frequency:
where
The maximum attenuation
occurs at
1
Q=
MI ( )
MR ( )=
r
M0M M M0( )
M0M 1+
r
2
1
Q=
E2
E1E1+ E
2( )
r
1+
r
2
r=
E1E1+ E
2( )
1
Q
max
=
1
2
E2
E1(E1 + E2)
1
Q
max=
1
2
M M0M0M
1
2
M
M
=r
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Standard Linear Solid Model
Attenuation and velocity dispersion tend to be mostlocalized in frequency. Attenuation is largest wherevelocity is changing most rapidly with frequency.
Peak attenuation and modulus dispersion are relatedby:
I.5
1
Q
max
=
1
2
M
M
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Nearly Constant Q Model
Attenuation is nearly constant over a finite range offrequencies. It is sometimes interpreted as a super-position of individual (Standard Linear Solid)attenuation peaks. The broadening of the attenuation
peak is accompanied by a broadening of the range offrequency where velocity increases.
I.6
1
Q
1
2
M
M
ln 2/
1( )
2
1
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Finally
Nearly Constant Q ModelLiu, et al. (1976) considered a model in which simple
attenuation mechanisms are combined such that the
attenuation is nearly a constant over a finite range of
frequencies.
We can then write
which relates the velocity dispersion within the band
of constant Q, to the value of Q and the frequency.
We can express as:
Expanding for small and substituting in:M /M
1
Q
1
2
M
M1
ln 2/
1( )
1
2
M
M1
1
Q ln 2 /1( )
V(2)
V(1) 1+
M
M1
V(2) /V(1)
V(2)
V(1)=1+
1
Qln 2 /1( )
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1/Q
M
log( )
Constant Q Model
Attenuation is constant for all frequencies, andvelocity always increases with frequency.
I.7
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As with the nearly constant Q model, we can simplifythis for large Q, giving:
Constant Q Model
Kjartansson (1979) considered a model in which Q is
strictly constant. In this case the complex modulus
and Q are related by:
where
=
1
arctan
1
Q
1
Q
12
M
M1
ln 2/
1( )
M() = M1i
1
2
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Fontainebleau Sandstone
Just as velocity increases with effective pressure, so doesQ. The strong pressure dependence is a clue that cracksare important for the physical mechanism of attenuation.From Nathalie Lucet, 1989, Ph.D. dissertation, Univ. ofParis/IFP.
1000
1500
2000
2500
3000
3500
4000
4500
5000
0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 0
Velocity
BR-EBR-SU S - PU S - S
Velocity(m/s)
Pressure (MPa)
0
5 0
1 0 0
1 5 0
2 0 0
0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 0
Attenuation
BR -EBR -SU S - PU S - S
1000/Q
Pressure (MPa)
0 . 5
0 . 6
0 . 7
0 . 8
0 . 9
1
1 . 1
1 . 2
0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0
Velocity Dispersion
ExtensionTorsionCQ ModelNCQ Model
Vbr/Vus
1000/Qbr I.8
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Some values of Q in rocks and sediments, summarizedby Bourbi, Coussy, and Zinszner, 1987, Acoustics of
Porous Media, Gulf Publishing Co.
Location Type of rock Depth (m) Measurement
frequency (Hz)
Limon (Colorado) Pierre shale 0-225 50-450 32
Gulf Coast (30 km
south of Houston)
Loam/sand/clay
Sands and shales
Sandy clay
Clay/sand
0-3
3-30
30-150
150-300
50-400
50-400
50-400
50-400
2
181
75
136Offshore-
Lousiana
(Pleistocene)
Southeast Texas
Southeast Texas
Southeast Texas
Clay/sand
Sands and shales
Same but more sandy
Sandbanks, silty shale
Mostly shale
Sand (23%) and clay
Sand (20%) and clay
Limestone and chalk
Sand (45%) and clay
Sand (24%) and clay
1170-1770
1770-2070
2070-2850
900-1560
1560-1800
1800-2100
600-1560
1590-1755
660-1320
>1020
125
125
125
80
80
80
80
80
15-40
40-70
67
>273
28
52
>273
30
41
>273
28
55
Beaufort Sea
(Canada)
549-1193
945-1311
125
425
Offshore
Baltimore
Siliceous chalk
Siliceous chalk with
porcellanite joints
278-442
442-582
5000-15000
5000-15000
68 on
ave.
287
on
ave.an d b ank s,silty shale
McDonald et al.
(1958)
Tullos and Reid
(1969)
Hauge (1981)
Ganley, Kansewich(1980)
Golberg (1958)
from Carmichael (1984) and Goldberg (1985)
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In fully-saturated rock, squirt flow between
the stiff and soft parts of the pore space
In partially saturated rock, gas gives the
viscous liquids more mobility
SQUIRTFLOW
Wave-induced fluid motion in the rock
appears to be the dominant source of
attenuation and dispersion:
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Q and Gas SaturationLab data and field observations indicate
that Q may be used to detect gas-
saturated reservoirs
Murphy's (1982) experiments show that(a) attenuation in gas saturated rocks is larger
than in dry rocks,
(b) attenuation peaks at low gas saturation.
2 0
3 0
4 0
5 0
6 0
7 0
0 0.5 1
1000
/Qe
MASSILON SANDSTONE
EXTENSIONAL LOSS
811 - 846 Hz
571 - 647 Hz
S w
1000/Qe
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Klimentos (1995) reports, based on well log data,
that P-wave attenuation in gas-saturated sandstone
is larger than in liquid-saturated sandstone.
5
10
15
20
1.5 1.6 1.7 1.8
P-WaveAttenuation(dB/m)
Vp/Vs
Gas +Condensate
Oil +WaterKlimentos '95
Well LogsMediium Porosity SS
~ 2500 m5
10
15
20
1.5 1.6 1.7 1.8
S-WaveAttenuation(dB/m)
Vp/Vs
Gas +
Condensate
Oil +
Water
Klimentos '95
Well LogsMediium Porosity SS
~ 2500 m
Q and Gas SaturationLab data and field observations suggest
that Q might help to detect gas
P-Attenuation(dB/cm
)
S-Attenuation(dB/cm
)
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Biot TheoryBiot developed a macroscopic theory to attempt to model
the behavior of fluid-saturated poroelastic systems.
His generalized form of Hookes law:
where and are the dry rock moduli, and the fluid
pressure P is linearly related to the normal stresses (andnot the shears) by a new constant . Similarly, the
increment of fluid content in an elementary cell of solid is
linearly related to the pore pressure and the solid
volumetric strain . These describe essentially the samemechanical problem as the Gassmann theory for coupling
the fluid and solid.
The equations of motion are:
where uw describes an inertial coupling between the solid
and the fluid, and is a dissipation term.
ij = ij + 2ij Pij
= 1M
P +
ij
x j
=
2
ui
t
2+ uw
2
wi
t
2
P
x i= uw
2ui
t2+ w
2wi
t2+
1
wi
t
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Biot TheoryThe high frequency limiting velocities and , aregiven by [in Geertsma and Smits (1961) notation]:
VSVP
where
Kfr, fr bulk and shear moduli of dry rock frame
K0 bulk modulus of mineralKfl effective bulk modulus of pore fluid
porosity0 mineral densityfl fluid density
low frequency density of saturated composite:
a-1 tortuosity
The low frequency limiting velocities are the same aspredicted by Gassmanns relations.
= 1 0 + fl
VP =1
01 ( ) +fl 1 a
1( )Kfr +
4
3fr
+
fla1 + 1
Kfr
K0
1
Kfr
K0
2a1
1Kfr
K0
1
K0+
Kfl
1
2
VS =fr
01 ( )+ fl 1 a
1( )
1
2
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Biot Theory
Biot and later Stoll (1977) considered adding frameattenuation on top of fluid effects. This plot by Stollshows attenuation vs. frequency for two extremecases and for a typical sand showing how framelosses and fluid losses combine to control the overallresponse.
I.9
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Squirt Flow
When a rock is compressed by the stress of a passing
wave, increments of pore pressure are induced in the pore
fluid. At very low frequencies there is time for the pore
pressure to equilibrate throughout the pore space, and the
fluid effect is described by the Gassmann theory.
However, at high frequencies we expect that unequal pore
pressures are induced on the microscale of individual
pores--larger increments in the soft, crack-like porosity and
smaller increments in the stiffer, equi-dimensional pores. If
these do not equilibrate, the rock will be stiffer, and the
velocities will be faster, than at low frequencies when they
do equilibrate.
This frequency-dependent distribution of pore pressure
leads to velocity dispersion, and the tendency for the fluid
to flow and adjust leads to attenuation.
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Estimating the High FrequencySquirt Flow Modulus
We compute the high frequency bulk modulus in 2 steps:
1. the unrelaxed bulk modulus of the wetted frame where
liquid is trapped in the thinnest cracks and the remaining
space is dry, is given by
pore pressure in the ith thin crack:
Combining gives:
P4
P1
P2
P3
So trapping water in the thinnest cracks is approximately the
same as closing the cracks under high pressure.
2. Finally the remaining pore space is saturated using
Gassmann with Khigh fused as the dry rock modulus.
1
Khigh f
1
Kmineral
=
1
Kfluid
1
Kmineral
iPi
low P
high P
Pi
1
1+1
Kfluid
1
Kmineral
i / i /( )dry
1
Khigh f
1
Kdry
highP
+
1
Kfluid
1
Kmineral
soft ( ) +K
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Squirt Flow
The shear modulus:
Comparing with the bulk modulus:
1
high f
1
dry
4
15
i /( )dry
1+1
Kfluid
1
Kmineral
i / i /( )dry
lowP
highP
1
high f
1
dry
4
15
1
Khigh f
1
Kdry
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Constructing the Unrelaxed Moduli
I.10
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I.11
In these plots, the dry data are taken as inputs. The ultrasonicwater-saturated data are compared with predictions by Gassmann,the high frequency Biot limit, and the high frequency squirt limit.
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I.12
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Most physical mechanisms of dispersion and attenuationhave a characteristic frequency where attenuation islargest and velocity is changing most rapidly withfrequency. It also separates the low frequency relaxedbehavior from the high frequency unrelaxed behavior.
It is very difficult to predict the characteristic frequency
very accurately, because it depends on idealized modelassumptions, and details of the rock microstructure thatare not well known.
Nevertheless, here are some rough estimates:Biot:
patchy saturation:
viscous shear in crack:
squirt:
fBiot=
2f
fsquirt =K
0
3
fpatchy =Kf
L2
fvisc.crack =
2
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Biot Theory
Compiled in Bourbi, Coussy, and Zinszner,1987, Acoustics of Porous Media, Gulf Publishing Co.
Parameter Porosity
(%)
Permeability
(mD)
Characteristic frequency
Sample Water
(h = 1cP)
(4)
Normal oil
(h = 10-50
cP) (4)
Heavy oil
(h = 100-
500 cP) (4)
Fontainebleausandstone (1)
5 0.1 80 MHz 800-4000MHz
8-40 GHa
Fontainebleau
sandstone (1)
20 1000 30 kHz 300-1500
kHz
3-15 MHz
Tight sand (2) 8 0.02 1 GHz 10-50 GHz 100-500
GHz
Cordova
Cream
limestone (2)
24.5 9 4.5 MHz 45-230
MHz
450-300
MHz
Sintered glass 28.3 1000 42 kHz 420-2100
kHz
4.2-21 MHz
(1) Bourbi and Zinszner (1985)
(2) Carmichael (1982)
(3) Plona and Johnson (1980)
(4) Viscosity is expressed in centipoises (1 cP = 1 mPa. s).