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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 pressure

simply 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 shear traction acting on the boundary in medium 1 (due to thesummed effects of the incident an reflected waves) must

be 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

, 1

VP2 , V S2 , 2

1

1

22

ReflectedP-wave

IncidentP-wave

ReflectedS-wave

TransmittedP-wave

TransmittedS-wave

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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 1( ) cos 1( ) sin 2( ) cos 2( )cos 1( ) sin 1( ) cos 2( ) sin 2( )sin 2 1( ) V P 1

V S 1cos 2 1( ) 2V S 2

2 V P 1 1V S 1

2V P 2sin 2 2( ) 2V S 2V P 1

1V S 12

cos 2 2( )

cos 2 1( ) V S 1V P 1

sin 2 1( ) 2V P 2 1V P 1

cos 2 2( ) 2V S 2 1V P 1

cos 2 2( )

Rpp

Rps

Tpp

Tps

=

sin 1( )cos 1( )sin 2 1( )

cos 2 1( )

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AVO - Shuey's Approximation

P-wave reflectivity versus angle:

Intercept Gradient

R ( ) = R 0 + ER 0 +

1 ( )2

sin 2 +

1

2 V

P

V P

tan 2 sin 2 [ ]

R0

1

2

V PV

P

+

E = F 2 1 + F ( )1 2

1

F = V

P/ V

P

V P

/ V P

+ /

V P

= V P 2

V P 1( )

V S = V S 2 V S 1( ) = 2 1( )

V P

= V P 2

+ V P 1) / 2

V S = V S 2 + V S 1( )/ 2 = 2 + 1( )/ 2

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P-wave reflectivity versus angle:

Intercept Gradient

R ( ) = R 0 +1

2

V PV

P

2V

S

2

V P

2

+ 2V S V

S

sin 2

+1

2

V PV

P

tan2 sin 2 [ ]

R0

12

V PV

P

+

V P

= V P 2

V P 1( )

V S = V S 2 V S 1( ) = 2 1( )

V P

= V P 2

+ V P 1) / 2

V S = V S 2 + V S 1( )/ 2 = 2 + 1( )/ 2

AVO - Aki-Richard's approximation:

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AVO Response

P-Velocity Poisson ratio AVO responsecontrast contrast

negative negative increasenegative positive decreasepositive negative decreasepositive 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 Gassmanns (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 measurement

frequencies. Hence, the two steps listed above.

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N.5

0

1

2

3

4

56

7

0 0.5 1 1.5 2 2.5 3 3.5 4

SandstonesWater Saturated

V p

( k m

/ s )

Vs (km/s)

mudrockV s = .8621V p - 1 . 1 7 2 4

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

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

(after Castagna et al., 1993)

0

1

2

3

4

56

7

0 0.5 1 1.5 2 2.5 3 3.5 4

ShalesWater Saturated

V p

( k m

/ s )

Vs (km/s)

mudrockV s = .8621 V p - 1.1724

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

Han (1986)V s = .7936V p - . 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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0

1

2

3

4

5

67

8

0 0.5 1 1.5 2 2.5 3 3.5 4

LimestonesWater Saturated

V p

( k m

/ s )

Vs (km/s)

Vs = Vp / 1 . 9Pickett (1963)

Castagna et al. (1993)V s = -.05508 V P 2

+ 1.0168 V p- 1.0305

water (after Castagna et al.,1993)

0

1

2

3

4

5

67

8

0 0.5 1 1.5 2 2.5 3 3.5 4

Dolomite

Water Saturated

V p

( k m

/ s )

Vs (km/s)

Castagna et al. (1993)V s = .5832V p - . 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 Vp 2

+ 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 Sandstones

Water Saturated

Vp-sat c>.25Vp-sat c 25 %V s = . 8 4 2 3 V p - 1 . 0 9 9

clay < 25 %V s = . 7 5 3 5 V p - . 6 5 6 6

mudrockV s = . 8 6 2 1 V p - 1 . 1 7 2 4

(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 Sandstones

Water Saturated (hf)

Vp-sat Phi>.15Vp-sat Phi 15 %Vs = .7563Vp-.6620

porosity < 15 %Vs = .8533Vp-1.1374

mudrockVs = .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 Poissons Ratio Assumption

0

2

46

8

10

12

14

16

0 5 10 15 20 25 30 35

clay < 10%clay > 10%

V s

2

d r y

Vp 2 dry

= 0.01

= 0.1

= 0.2

= 0.3

= 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 rockPoissons ratio equal to their values for pure mineral.

The plot below illustrates the approximately constant dry rockPoissons ratio observed for a large set of ultrasonic sandstonevelocities (from Han, 1986) over a large rance of effective pressures (5< P

eff < 40 MPa) and clay contents (0 < C < 55% by volume).

N.8

K dry = K 0 1

c

dry= 0 1

c

V S V P

dry rock

V S V P

mineral

dry rock

mineral

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Kriefs Relation (1990)The model combines the same two elements:

1. An empirical Vp-Vs- relation for water-saturatedrocks, which is approximately the same as the criticalporosity model.2. Gassmanns relation to extend the empiricalrelation to other pore fluids.

Dry rock Vp-Vs- relation:

where is Biots coefficient. This is equivalent to:

where

and K f are two equivalent descriptions of the pore spacestiffness. Determining vs. or K vs determines the

rock bulk modulus K dry vs .Krief et al. (1990) used the data of Raymer et al. (1980) toempirically find a relation for vs :

K dry = K mineral 1 ( )

1

K dry=

1

K 0+

K

1

K =

1

v p

dv pd

PP = constant

; =dv pdV

PP = constant

=

K dryK

1 ( ) = 1 ( )m ( )

where m ( ) = 3 / 1 ( )

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Assuming dry rock Poissons ratio is equal to themineral Poissons ratio gives

N.9

K dry = K 0 1 ( )m ( )

dry= 0 1 ( )m ( )

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

where V P-sat , VP0 , and V fl are the P-wave velocities of thesaturated rock, the mineral, and the pore fluid; and V S-satand V S0 are the S-wave velocities in the saturated rockand mineral. Rewriting slightly gives

where V R is the velocity of a suspension of minerals in a

fluid, given by the Reuss average at the critical porosity.This modified form of Kriefs expression is exactlyequivalent to the linear (modified Voigt) K vs and vs relations 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 more

accurate (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

V P sat 2 V fl

2

V S sat 2

=

V P 02 V fl

2

V S 02

V P sat

2 V

R

2

V S sat

2=

V P 0

2 V

R

2

V S 0

2

V P sat 2

= V fl 2

+ V S sat 2 V P 0

2 V fl 2

V S 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 p

2 ( k m

/ s ) 2

Vs

2 (km/s) 2

saturated

dry

Sandstones mineralpoint

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 p

2 (

k m / s ) 2

Vs2

(km/s)2

Sandstone

Dolomite

mineralpoints

fluid points

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VS = 12 Xii = 1

L a ijVPjj = 0

Ni+ X i

i = 1L a ijVPj

j = 0

N i 1 1

Greenberg and Castagna (1992) have given empirical

relations 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:

X i = 1

i = 1

L

VS = a i2VP2 + a i1VP + a i0 (Castagna et al. 1992)

whereL number of monomineralic lithologic constituent

Xi volume fractions of lithological constituents

a ij 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 .85588

Limestone -0 .05508 1 .01677 -1 .03049Dolomite 0 0 .58321 -0 .07775S ha le 0 0 .76969 -0 .86735

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1

2

3

4

5

6

7

0 1 2 3 4 5

S a n d s t o n eL i m e s t o n eD o l o m i t e

S h a l e

V p

( k m

/ 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,Gassmanns 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, withGassmanns 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 V Pb .2. Calculate V Sb corresponding to V Pb from the empiricalregression.3. Do fluid substitution using V

Pband V

Sbin the Gassmann

equation to get V Sf .4. With the calculated V Sf and the measured V Pf , use theGassmann relation to get a new estimate of V Pb . Check withprevious value of V Pb for convergence. If convergencecriterion is met, stop; if not go back to step 2 and continue.

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Coefficients for the equation b = aVp 2 + bVp + c

Lithology 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 4.5-7.1

Anhydrite -.0203 .321 1.732 4.6-7.4Coefficients for the equation b = dVp f

L it ho lo gy 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 4.5-7.1

Anhydrite 2.19 .160 4.6-7.4

Both forms of Gardners 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 and

g/cm 3.

N.15

1.359 .386

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Both forms of Gardners relations applied tolaboratory dolomite data.

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

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