8.velocitydispersion

8/14/2019 8.VelocityDispersion

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Stanford Rock Physics Laboratory - Gary Mavko

Velocity Dispersion and Q

189

Velocity Dispersion and

Wave Attenuation (Q)


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190

(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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191

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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192

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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193

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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194

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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195

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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196

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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197

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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198

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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199

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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200

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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201

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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202

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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203

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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204

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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205

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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206

1/Q

M

log( )

Constant Q Model

Attenuation is constant for all frequencies, andvelocity always increases with frequency.

I.7


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207

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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208

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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209

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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211

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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212

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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213

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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214

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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215

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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216

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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217

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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218

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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219

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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220

Constructing the Unrelaxed Moduli

I.10


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221

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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222

I.12


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223

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).

8.velocitydispersion

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