The Rock Physics of AVO - Stanford University · PDF fileStanford Rock Physics Laboratory -...

Stanford Rock Physics Laboratory - Gary Mavko

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The Rock Physics of AVO


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Water-saturated40 MPa

Water-saturated10-40 MPa

Gas andWater-saturated10-40 MPa

L8


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N.1

More than 400 sandstone data points, with porositiesranging over 4-39%, clay content 0-55%, effectivepressure 5-40 MPa - all water saturated.

When Vp is plotted vs. Vs, they follow a remarkablynarrow trend. Variations in porosity, clay, and pressuresimply move the points up and down the trend.


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N.2

Variations in porosity, pore pressure, and shalinessmove data along trends. Changing the pore fluidcauses the trend to change.


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Different shear-related attributes.N.2


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In an isotropic medium, a wave that is incident on aboundary will generally create two reflected waves (oneP and one S) and two transmitted waves. The total sheartraction acting on the boundary in medium 1 (due to thesummed effects of the incident an reflected waves) mustbe equal to the total shear traction acting on the boundary inmedium 2 (due to the summed effects of thetransmitted waves). Also the displacement of a point inmedium 1 at the boundary must be equal to the displace-ment of a point in medium 2 at the boundary.

VP1, VS1, ρ1

VP2, VS2, ρ2

θ1

φ1

θ2φ2

Reflected P-wave

Incident P-wave

Reflected S-wave

Transmitted P-wave

Transmitted S-wave

N.4


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By matching the traction and displacement boundaryconditions, Zoeppritz (1919) derived the equationsrelating the amplitudes of the P and S waves:

sin θ1( ) cos φ1( ) − sin θ2( ) cos φ2( )−cos θ1( ) sin φ1( ) −cos θ2( ) − sin φ2( )sin 2θ1( ) VP1

VS1cos 2φ1( ) ρ2VS 2

2VP1ρ1VS1

2VP 2sin 2θ2( ) −

ρ2VS 2VP1ρ1VS1

2cos 2φ2( )

−cos 2φ1( ) −VS1VP1sin 2φ1( ) −

ρ2VP2ρ1VP1

cos 2φ2( ) −ρ2VS 2ρ1VP1

cos 2φ2( )

RppRpsTppTps

=

− sin θ1( )−cos θ1( )sin 2θ1( )

−cos 2φ1( )


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AVO - Shuey's Approximation

P-wave reflectivity versus angle:

Intercept Gradient

R θ( ) = R0 + ER0 +∆ν

1− ν( )2

sin2θ +

12

∆VPVP

tan2θ − sin2θ[ ]

R0 ≈12

∆VPVP

+∆ρρ

E = F − 2 1 + F( )1− 2ν1−ν

F =∆VP /VP

∆VP /VP + ∆ρ / ρ

∆VP = VP 2 − VP1( )∆VS = VS 2 − VS1( )∆ρ = ρ2 − ρ1( )

VP = VP2 + VP1( ) / 2VS = VS2 + VS1( ) / 2ρ = ρ2 + ρ1( ) / 2


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P-wave reflectivity versus angle:

Intercept Gradient

R θ( ) = R0 +12

∆VPVP

− 2VS2

VP2∆ρρ

+ 2 ∆VSVS

sin2θ

+12

∆VPVP

tan2θ − sin2θ[ ]

R0 ≈12

∆VPVP

+∆ρρ

∆VP = VP 2 − VP1( )∆VS = VS 2 − VS1( )∆ρ = ρ2 − ρ1( )

VP = VP2 + VP1( ) / 2VS = VS2 + VS1( ) / 2ρ = ρ2 + ρ1( ) / 2

AVO - Aki-Richard's approximation:


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AVO Response

P-Velocity Poisson ratio AVO response contrast contrast

negative negative increase negative positive decrease positive negative decrease positive positive increase


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Vp-Vs Relations

There is a wide, and sometimes confusing, variety ofpublished Vp-Vs relations and Vs prediction techniques,which at first appear to be quite distinct. However, mostreduce to the same two simple steps:

1. Establish empirical relations among Vp, Vs, and porosityfor one reference pore fluid--most often water saturated ordry.2. Use Gassmann’s (1951) relations to map these empiricalrelations to other pore fluid states.Although some of the effective medium models predict both Pand S velocities assuming idealized pore geometries, the factremains that the most reliable and most often used Vp-Vsrelations are empirical fits to laboratory and/or log data. Themost useful role of theoretical methods is extending theseempirical relations to different pore fluids or measurementfrequencies. Hence, the two steps listed above.


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N.5

0

1

2

3

4

5

6

7

0 0.5 1 1.5 2 2.5 3 3.5 4

SandstonesWater Saturated

Vp (k

m/s

)

Vs (km/s)

mudrockV s = .8621Vp- 1 . 1 7 2 4

Castagna et al. (1993)V s = .8042Vp- . 8 5 5 9

Han (1986)V s = .7936Vp- . 7 8 6 8

(after Castagna et al., 1993)

0

1

2

3

4

5

6

7

0 0.5 1 1.5 2 2.5 3 3.5 4

ShalesWater Saturated

Vp (k

m/s

)

Vs (km/s)

mudrockV s = .8621 Vp- 1.1724

Castagna et al. (1993)V s = .8042Vp- . 8 5 5 9

Han (1986)V s = .7936Vp- . 7 8 6 8


N.5

MudrockVs = .86 Vp - 1.17

Han (1986)Vs = .79 Vp - 0.79

Castagna et al. (1993)Vs = .80 Vp - 0.86


Han (1986)Vs = .79 Vp - 0.79



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012345678

0 0.5 1 1.5 2 2.5 3 3.5 4

LimestonesWater Saturated

Vp (k

m/s

)

Vs (km/s)

V s = Vp/ 1 . 9Pickett (1963)

Castagna et al. (1993)V s = -.05508 VP2 + 1.0168 Vp - 1.0305

water (after Castagna et al.,1993)

012345678

0 0.5 1 1.5 2 2.5 3 3.5 4

DolomiteWater Saturated

Vp (k

m/s

)

Vs (km/s)

Castagna et al. (1993)V s = .5832Vp - . 0 7 7 7 6

Pickett (1963)V s = Vp/ 1 . 8


N.6

Pickett(1963)Vs = Vp / 1.9

Castagna et al. (1993)Vs = -.055 Vp2 + 1.02 Vp - 1.03

Pickett(1963)Vs = Vp / 1.8



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0

1

2

3

4

5

6

0 0.5 1 1.5 2 2.5 3 3.5 4

Shaly SandstonesWater Saturated

Vp-sat c>.25Vp-sat c<.25

Vp

(km

/s)

Vs (km/s)

clay > 25 % Vs= .8423Vp-1 .099

clay < 25 %Vs= .7535Vp- .6566

mudrock Vs= .8621Vp-1 .1724

(Data from Han, 1986)

0

1

2

3

4

5

6

0 0.5 1 1.5 2 2.5 3 3.5 4

Shaly SandstonesWater Saturated (hf)

Vp-sat Phi>.15Vp-sat Phi<.15

Vp (k

m/s

)

Vs (km/s)

porosity > 15 % Vs = .7563Vp-.6620

porosity < 15 %Vs = .8533Vp-1.1374

mudrock Vs = .8621Vp-1.1724

(Data from Han, 1986)

N.7


Clay < 25%Vs = .75 Vp - 0.66

Clay > 25%Vs = .84Vp-1.10

porosity > 15%Vs = .76Vp - 0.66

porosity < 15%Vs = .85 Vp - 1.14



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Dry Poisson’s Ratio Assumption

02468

10121416

0 5 10 15 20 25 30 35

clay < 10%clay > 10%

Vs2

dry

Vp2 dry

ν = 0.01ν = 0.1

ν = 0.2

ν = 0.3

ν = 0.4

Shaly Sandstones - Dry

The modified Voigt Average Predicts linear moduli-porosity relations.This is a convenient relation for use with the critical porosity model.

These are equivalent to the dry rock Vs/Vp relation and the dry rockPoisson’s ratio equal to their values for pure mineral.

The plot below illustrates the approximately constant dry rockPoisson’s ratio observed for a large set of ultrasonic sandstonevelocities (from Han, 1986) over a large rance of effective pressures (5< Peff < 40 MPa) and clay contents (0 < C < 55% by volume).

N.8

Kdry = K0 1−φφc

µdry = µ0 1−

φφc

VSVP

dry rock

≈VSVP

mineral

νdry rock ≈νmineral


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Krief’s Relation (1990)The model combines the same two elements:1. An empirical Vp-Vs-φ relation for water-saturatedrocks, which is approximately the same as the criticalporosity model.2. Gassmann’s relation to extend the empiricalrelation to other pore fluids.Dry rock Vp-Vs-φ relation:

where β is Biot’s coefficient. This is equivalent to:

where

β and Kf are two equivalent descriptions of the pore spacestiffness. Determining β vs. φ or Kφ vs φ determines therock bulk modulus Kdry vs φ.

Krief et al. (1990) used the data of Raymer et al. (1980) toempirically find a relation for β vs φ:

Kdry = Kmineral 1− β( )

1Kdry

=1K0

+φKφ

1Kφ

=1v p

dv pdσ

PP = constant

; β =dv pdV

PP = constant

=φKdry

Kφ

1− β( ) = 1− φ( )m φ( ) where m φ( ) = 3 / 1−φ( )


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Assuming dry rock Poisson’s ratio is equal to themineral Poisson’s ratio gives

N.9

Kdry = K0 1− φ( )m φ( )

µdry = µ 0 1−φ( )m φ( )


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Expressions for any other pore fluids are obtained fromGassmann’s equations. While these are nonlinear, theysuggest a simple approximation:

where VP-sat, VP0, and Vfl are the P-wave velocities of thesaturated rock, the mineral, and the pore fluid; and VS-satand VS0 are the S-wave velocities in the saturated rockand mineral. Rewriting slightly gives

where VR is the velocity of a suspension of minerals in afluid, given by the Reuss average at the critical porosity.This modified form of Krief’s expression is exactlyequivalent to the linear (modified Voigt) K vs φ and µ vs φrelations in the critical porosity model, with the fluid effectsgiven by Gassmann.

which is a straight line (in velocity-squared) connecting themineral point ( ) and the fluid point ( ). A moreaccurate (and nearly identical) model is to recognize thatvelocities tend toward those of a suspension at highporosity, rather than toward a fluid, which yields themodified form

VP02 , VS0

2 Vfl2 , 0

VP− sat2 − Vfl

2

VS −sat2 =

VP 02 − Vfl

2

VS02

VP− sat2 − VR

2

VS −sat2 =

VP 02 − VR

2

VS02

VP− sat2 = Vfl

2 + VS− sat2 VP 0

2 − Vfl2

VS 02


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0

5

10

15

20

25

30

35

0 5 10 15 20

Vp-Vs Relation in Dry andSaturated Rocks

V p2 (km

/s)2

Vs2 (km/s)2

saturated

dry

Sandstones mineral point

fluid point

N.10


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N.11

0

10

20

30

40

50

0 4 8 12 16

Vp-Vs Relation in Sandstoneand Dolomite

V p2 (km

/s)2

Vs2 (km/s)2

Sandstone

Dolomite

mineralpoints

fluid points


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VS = 1

2 XiΣi = 1

LaijVP

jΣj = 0

Ni

+ XiΣi = 1

LaijVP

jΣj = 0

Ni –1 –1

Greenberg and Castagna (1992) have given empiricalrelations for estimating Vs from Vp in multimineralic, brine-saturated rocks based on empirical, polynomial Vp-Vsrelations in pure monomineralic lithologies (Castagna etal., 1992). The shear wave velocity in brine-saturatedcomposite lithologies is approximated by a simple averageof the arithmetic and harmonic means of the constituentpure lithology shear velocities:

Castagna et al. (1992) gave representative polynomialregression coefficients for pure monomineralic lithologies:

Regression coefficients for pure lithologies with Vp and Vsin km/s:

Xi = 1Σi = 1

L

VS = ai2VP2 + ai1VP + ai0 (Castagna et al. 1992)

whereL number of monomineralic lithologic constituent

Xi volume fractions of lithological constituentsaij empirical regression coefficientsNi order of polynomial for constituent iVp, Vs P and S wave velocities (km/s) in composite brine-

saturated, multimineralic rock

Lithology a i2 a i1 a i0S andstone 0 0.80416 -0.85588Limestone -0.05508 1.01677 -1.03049Dolomite 0 0.58321 -0.07775Shale 0 0.76969 -0.86735


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1

2

3

4

5

6

7

0 1 2 3 4 5

SandstoneL imestoneDo l om i t eS h a l e

Vp (

km/s

)

Vs (km/s)

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 anyother 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, withGassmann’s relation, to the measured Vp and the unknown Vsfor 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 empiricalregression.3. Do fluid substitution using VPb and VSb in the Gassmannequation to get VSf.4. With the calculated VSf and the measured VPf, use theGassmann relation to get a new estimate of VPb. Check withprevious value of VPb for convergence. If convergencecriterion is met, stop; if not go back to step 2 and continue.

N.12


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Coefficients for the equation ρb = aVp2 + bVp + c

L ithology a b c Vp Range (Km/s)

Shale - .0261 .373 1.458 1.5-5.0Sandstone - .0115 .261 1.515 1.5-6.0Limestone - .0296 .461 0.963 3.5-6.4Dolomite - .0235 .390 1.242 4.5-7.1Anhydrite - .0203 .321 1.732 4.6-7.4

Coefficients for the equation ρb = dVpf

L ithology d f Vp Range(Km/s)

Shale 1.75 .265 1.5-5.0Sandstone 1.66 .261 1.5-6.0Limestone 1.50 .225 3.5-6.4Dolomite 1.74 .252 4.5-7.1Anhydrite 2.19 .160 4.6-7.4

Both forms of Gardner’s relations applied to log and lab shale data, aspresented by Castagna et al. (1993)

Polynomial and powerlaw forms of the Gardner et al. (1974) velocity-densityrelationships presented by Castagna et al. (1993). Units are km/s andg/cm3.

N.15


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Both forms of Gardner’s relations applied tolaboratory dolomite data.

N.16


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Both forms of Gardner’s relations applied to laboratory limestone data. Notethat the published powerlaw form does not fit as well as the polynomial. wealso show a powerlaw form fit to these data, which agrees very well with thepolynomial.

Both forms of Gardner’s relations applied to log and lab sandstone data, aspresented by Castagna et al. (1993).

N.17

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