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Theory 1-1

PRACTICAL AVOPRACTICAL AVO

Part 1 Rock Physics &Part 1 Rock Physics &

Fluid Replacement odelin!Fluid Replacement odelin!

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Theory 1-2

Introduction

AVOAVOstands "orAmplitude Variations with OffsetAmplitude Variations with Offset# orAmplitude VersusAmplitude VersusOffsetOffset$

TheAVOAVOtechni%ue uses the amplitude ariations o" prestack seismic

re"lections to predict reseroir "luid e""ects$

In this course# 'e 'ill look atAVOAVOmodelin!# reconnaissance and

inersion techni%ues$

(e"ore discussin!AVOAVO# 'e 'ill hae a look at the essentials o" rock

physics$

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Theory 1-3

(asic Rock Physics

TheAVOAVOresponse is dependent on the properties o" P)'ae elocity *VVPP+#,)'ae elocity *VV,,+# and density *+ in a porous reseroir rock$ As sho'n

-elo'# this inoles the matri. material# the porosity# and the "luids "illin!

the pores/

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Theory 1-4

0ensity

DensityDensitye""ects can -e modeled 'ith the "ollo'in! e%uation/

)S1(S)1( whcwwmsat ++=

.subscriptswater,nhydrocarbo

matrix,saturated,w,sat,m,hc

,saturationwaterwS

porosity,

density,where:

=

=

=

=

This is illustrated in the ne.t

!raph$

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Theory 1-5

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Theory 1-6

Velocity

nlike density# 'hich is simply mass diided -y unit olume# velocityvelocityinoles the de"ormation o" a rock as a "unction o" time$ Let us "irst

consider the 'ays in 'hich a s%uare o" rock can -e moed or de"ormed/

(a) Contraction (b) Lengthening

(c) Rotation (d) Translation

(e) Shear

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Theory 1-7

,tress and ,train

In the preious slide# cases *a+# *-+# and *e+ are called strainsstrains# since therock chan!es its si2e or shape# -ut *c+ and *d+ are simply displacements$

The "orces that create this chan!e are called stresses$ Let3s look at *a+

and *e+ in more detail/

For the compressivecompressivecase# takin!

the ratio o" the t'o s%uares leads

to a strain o" *

u.4

. 5

uy4

y+

For the shearshearcase# takin!

the ratio o" the t'o s%uares leads

to a strain o" *

u.4

y 5

uy4

.+

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Theory 1-8

6ooke3s La'

,mall stresses and strains *the linear case+ are related -y Hookes LawHookes Law/

cep=where: p= stress = force per unit area,

c = an elastic constant,

and: e = strain

For a pure compressie stress 7case *a+8# the elastic constant is called

the ulk modulusulk modulus# !!"

For a pure shear stress 7case *e+8# the elastic constant is called the shearshear

modulusmodulus# $

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Theory 1-9

The ,tress Tensor

There are 9 possi-le stressesstresseson a cu-e o" rock# -ut only : areindependent# since/ p.y; py.# p.2; p2.# and py2; p2y$ This is sho'n -elo'#

-oth mathematically and physically$

=

zzzyzx

yzyyyx

xzxyxx

ppp

ppp

ppp

p

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Theory 1-10

The ,train Tensor

As 'ith stress# there are 9 possi-le strainsstrains on a cu-e o" rock# -ut only :are independent# since/ e.y; ey.# e.2; e2.# and ey2; e2y$ This is sho'n

-elo' in mathematical "orm$

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Theory 1-11

The >enerali2ed ,tress),train

Relation

The !enerali2ed relationship -et'een stressstressand strainstrainin the "ull

anisotropic elastic case inoles =1 components in the elastic moduluselastic modulus

matri#matri## as sho'n -elo'$

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Theory 1-12

The Isotropic ,tress),train Relation

For the isotropicisotropiccase# the situation is much simpler# inolin! only t'o

uni%ue alues# 'hich are called the Lam$ parametersLam$ parameters and /

+++

=

xy

xz

yz

zz

yy

xx

xy

xz

yz

zz

yy

xx

ee

e

e

e

e

0000000000

00000

0002

0002

0002

pp

p

p

p

p

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Theory 1-13

0eriin! the Velocities

Trans"ormin! the static stress)strain relationship into the dynamic e""ectso" velocityvelocityinoles t'o steps/

) introducin! momentum ia %ewtons law%ewtons law/ & = ma

) introducin! density# since mass is the product o" density times

olume$

The deriation 'ill not -e done here# -ut the "inal "orm is the wavewave

e'uatione'uation/

2

2

22

2

2

2

2

2

tu

V1

zu

yu

xu =++

: , , ,where V velocity a function of and =

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Theory 1-14

Velocity ?%uations usin! and

6ere are the e%uations "or velocityvelocityderied in their most -asic "orm usin!the Lam$ coefficientsLam$ coefficients/

2VP +=

=sV

: , the Lame parameters

: density.

where

and

=

=

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Theory 1-15

Velocity ?%uations usin! @ and

Another common 'ay o" 'ritin! the velocityvelocitye%uations is 'ith ulkulkandshear modulusshear modulus/

3

!

VP

+

=

=sV

nd

: the b!" mod!s,

23

the shear mod!s

#2 Lame parameter

where K

and

=

= +

=

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Theory 1-16

Poisson3s Ratio

A common 'ay o" lookin! at the ratio o" V(to V) is to use (oissons ratio(oissons ratio#de"ined as/

22

2

=

2

S

P

V

V:where

=

The inerse to the a-oe "ormula# allo'in! you to derie V(or V)"rom# is !ien -y/

12

22

=

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Theory 1-17

There are seeral alues o" (oissons ratio(oissons ratioand V(*V)

ratio that should -e noted/

A plot o" (oissons ratio(oissons ratioersus velocity ratiovelocity ratiois sho'n on the ne.t

slide$

I" V(*V)= +# then =

I" V(*V)= -".# then = "-*/as 0ase/as 0ase+

I" V(*V)= +# then = -*1*2et 0ase2et 0ase+

I" V(*V

)= # then = ".*VV

,,= = +

Poisson3s Ratio

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Theory 1-18

Vp4Vs s Poissons Ratio

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0 1 2 3 4 5 6 ! " 10

Vp4Vs

PoissonAs

Ratio

>as Case Bet Case

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Theory 1-19

Velocity in Porous Rocks

VelocityVelocitye""ects can -e modeled -y the -ulk aera!e e%uation as seen-elo' and in the ne.t "i!ure/

)S1(tSt)1(tt whcwwma ++=

1$where : t V =

n"ortunately# the a-oe e%uation does not hold "or !as sands# and

this lead to the deelopment o" other e%uations$

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Theory 1-20

Velocity s ,' 'ith Volume A!$ ?%$

Por ; D# Voil ; 1EE m4s# V!as ; EE m4s

1000

1500

2000

2500

3000

3500

0 0.1 0.2 0.3 0.4 0.5 0.6 0. 0.! 0." 1

Bater ,aturation

Velocity*

m4sec+

#il $as

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Theory 1-21

Other empirical e%uations hae -een proposed/

"#m

2

P VV)1(V +=

6o'eer# the -est "it to o-seration has -een o-tained 'ith the 3iot43iot4

/assmann e'uations/assmann e'uations$

$1%.2&3.'&.)s*m(VP = $%&.1&1.2.3)s*m(VS =

+aymer et a#.

-an et a#, where: $ Vo#ume $#ay $ontent

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Theory 1-22

The (iot)>assmann ?%uations

Independently# /assmann/assmann*191+ and 3iot3iot*19:+# deeloped the theory o"'ae propa!ation in "luid saturated rocks# -y deriin! e.pressions "or the

saturated -ulk and shear modulii# and su-stitutin! into the re!ular

e%uations "or P) and ,)'ae elocity/

sat

satsat

P3

!

V

+= sat

satsV

=

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Theory 1-23

(iot)>assmann ) ,hear odulus

In the 3iot4/assmann3iot4/assmanne%uations# the shear modulusshear modulusdoes not chan!e "oraryin! saturation at constant porosity/

drysat =

shear mod!s o% satrated ro&"

shear mod!s o% dry ro&"

where :sat

dry

=

=

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Theory 1-24

(iot)>assmann ) ,aturated (ulk odulus

The 3iot4/assmann ulk modulus e'uation3iot4/assmann ulk modulus e'uationis as "ollo's/

2

m

dry

m"#

2

m

dry

drysat

!

!

!

1

!

)!

!1(

!!

+

+=

ako et al# in 5he 6ock (hysics Handook5he 6ock (hysics Handook, re)arran!ed the a-oe

e%uation to !ie a more intuitie "orm/

)!!(

!

!!

!

!!

!

"#m

"#

drym

dry

satm

sat

+

=

2here sat = saturated rock, dry = dry frame, m = rock matri#, fl = fluid,

and

= porosity"

(1)

(2)

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Theory 1-25

The )aturated 3ulk 7odulus)aturated 3ulk 7odulus*@sat+ is a""ected -y/

Rock "rame -ulk modulus *@dry+

Porosity

Fluid -ulk modulus *@"l+

) ,aturation

) Temperature

) Pore Pressure

?""ectie Pressure

) Oer-urden Pore pressure

ineral -ulk modulus

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Theory 1-26

(iot)>assmann ) ,hear (ulk odulus

& 0ensity)aturated )hear 7odulus)aturated )hear 7odulus*sat+

Is ?%ual to Rock "rame shear modulus *dry+

Porosity

?""ectie Pressure

Oer-urden Pore pressure

)aturated Density)aturated Density*sat+ depends on

Rock matri. density *m+

PorosityFluid density

) ,aturation

) Temperature

) Pore Pressure

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Theory 1-27

The Rock atri. (ulk odulus

The -ulk modulus o" the solid rock matri.# !mis usually taken "rom

pu-lished data that inoled measurements on drill core samples$

Typical alues are/

!sandstone / 0Pa,

!limestone '/ 0Pa.

Be 'ill no' look at ho' to !et estimates o" the arious -ulk modulus

terms in the 3iot4/assmann3iot4/assmann e%uations# startin! 'ith the -ulk modulus

o" the solid rock matri.$ Values 'ill -e !ien in 8i8a(ascals8i8a(ascals9/(a9/(a#

'hich are e%uialent to ----dynes*cmdynes*cm++$

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Theory 1-28

The Fluid (ulk odulus

The fluid ulk modulusfluid ulk moduluscan -e modeled usin! the "ollo'in! e%uation/

hc

w

w

w

fl K

1

K

K

1 +=

b!" mod!s o% 'ater

b!" mod!s o% hydro&arbon.

where : K !w

Khc

=

=

?%uations "or estimatin! the alues o" -rine# !as# and oil -ulk modulii

are !ien in (at2le and Ban!# 199=# )eismic (roperties of (ore &luids)eismic (roperties of (ore &luids#>eophysics# G# 19:)1HE$ Typical alues are/

!8as /./21 0Pa, !oil /.& 0Pa, !w 2.3% 0Pa

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Theory 1-29

?stimatin! @dry

For kno'n V(# -ut unkno'nV)# !drycan -e estimated *>re!ory# 19GG+ -y

assumin! the dry rock (oissons ratio(oissons ratiodry$ >re!ory sho's that e%uation

*1+ can -e re'ritten as/

For kno'n V) and V(, !drycan -e calculated -y "irst calculatin! !sat

and then usin! 7avkos e'uation7avkos e'uation$

2

m

dry

m"#

2

m

dry

drysat

!

!

!

1

!

)!

!1(

+

+=

)1(

)1(3S:and

,S!3!

,3!:where

dry

dry

drydrydry

satsat

+

=

=+=

+=

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Theory 1-30

A"ter a lot o" al!e-ra# the preious e%uation can -e 'ritten as the"ollo'in! %uadratic e%uation "or a term that inoles !dry$ ,olin! "or

!ies the solution$

0"a 2 =++

=

+

=

=

==

1K

K

K

#c

K#1

KK"

!1a

!K

K1tcoefficien%iotthe:where

fl

&

&

sat

&

sat

fl

&

&

dry

?stimatin! @dry

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Theory 1-31

Porosity Chan!e

Porosity a""ects the dry rock -ulk modulus# and this e""ect can -ecomputed -y usin! the "ollo'in! e%uation/

&dry' K

1

K

1

K=

where: !(= pore ulk modulus

I" 'e assume that the pore -ulk modulus stays constant "or a ran!e o"

porosities# -ut the dry rock -ulk modulus chan!es as a "unction o"

porosity# 'e can compute a ne' dry rock -ulk modulus "or a di""erent

porosity usin! the "ollo'in! re)arran!ed ersion o" the a-oe e%uation/

&'

new

new(dry K

1

KK

1+=

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Theory 1-32

0ata ?.amples

In the ne.t "e' slides# 'e 'ill look at the computed responses "or -oth a!as)saturated sand and an oil)saturated sand usin! the 3iot4/assmann3iot4/assmann

e'uatione'uation$

Be 'ill look at the e""ect o" saturation on -oth elocity *V(and V)+ and

(oissons 6atio(oissons 6atio$

@eep in mind that this model assumes that the !as is uni"ormly

distri-uted in the "luid$ Patchy saturation proides a di""erent "unction$

*,ee ako et al/ 5he 6ock (hysics Handook5he 6ock (hysics Handook$+

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Theory 1-33

Velocity s ,' ) >as Case# Por ; D

@s ; HE# @!as ; $E=1# @dry ; $=# u ; $ >Pa

1000

1200

1400

16001!00

2000

2200

2400

2600

0 0.1 0.2 0.3 0.4 0.5 0.6 0. 0.! 0." 1

,'

Velocit

y*m4s+

%& %s

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Theory 1-34

PoissonAs Ratio &s Bater ,aturation

>as Case

0

0.1

0.2

0.3

0.4

0.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0. 0.! 0." 1

,'

PoissonAsRatio

'oissons Ratio

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Theory 1-35

0 2 4

**CT #* +,TR S,TR,T#/

'-+,% %L#CT (sec)

'#SS#/SR,T#

$as Sand ( 'hi 33 )0.5

0.4

0.3

0.2

0.1

0

0505

"0

"4

"6

"!

""

100

Another 'ay o" displayin! the data is on a t'o parameter

plot$ 6ere# (oissons ratio(oissons ratiois plotted a!ainst P)'ae elocity$

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Theory 1-36

Velocity &s ,' ) Oil Case

Porosity ; D# @oil ; 1$E Pa

1000

1500

2000

2500

3000

0 0.1 0.2 0.3 0.4 0.5 0.6 0. 0.! 0." 1

,'

Velocity*m4s+

%s %&

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Theory 1-37

Poissons Ratio s Bater ,aturation

Oil Case

0

0.1

0.2

0.3

0.4

0.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0. 0.! 0." 1

,'

PoissonAsRatio

'oissons Ratio

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Theory 1-38

' VV

12

22

=

This 'ill -e illustrated in the ne.t "e'

slides$

The udrock Line

The mudrock linemudrock lineis a linear relationship -et'een V(and V)deried -y Casta!na et al *19+/

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Theory 1-39

A60Os ori8inal mudrock derivationA60Os ori8inal mudrock derivation

*Casta!na et al# >eophysics# 19+

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Theory 1-40

The udrock Line

0

2000

2000

4000

6000

1000 3000 40000

1000

3000

5000

VP *m4s+

V,*m4s+

udrock Line

>as ,and

Th d k Li

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Theory 1-41

The udrock Line

0

2000

2000

4000

6000

1000 3000 40000

1000

3000

5000

VP *m4s+

V,*m4s+

udrock Line

>as ,and

; 14

or

VP4V,; =

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Theory 1-42

The udrock Line

0

2000

2000

4000

6000

1000 3000 40000

1000

3000

5000

VP *m4s+

V,

*m4s+

udrock Line

>as ,and

; 14

or

VP4V,; =

; E$1 or

VP4V,; 1$

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Theory 1-43

Finally# here is a display o" the udrock line and the dry

rock line on a (oissons ratio versus (4wave velocity(oissons ratio versus (4wave velocity plot$

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Theory 1-44

Tips "or sin! o" >assmann3s

?%uation@m/ ineral Term

JTe.t -ookK alues hae -een measured on pure mineralsamples *crystals+$

ineral alues can -e aera!ed usin! Reuss aera!in! to

estimate @m"or rocks composed o" mi.ed litholo!ies$

@dry/ Rock Frame

Represents the incompressi-ility o" the rock "rame *includin!

cracks and pores+$O"ten pressure dependent due to cracks closin! 'ith increasede""ectie pressure$

0i""icult to o-tain accurate alues in many cases$

La-oratory measurements o" representatie core plu!s underreseroir pressure may -e the -est source o" data$

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Theory 1-45

CATIO

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Theory 1-46

Fluid Replacement odelin! *FR+

?stimates VP# V,and density chan!es that occur 'hen saturationchan!es$

FR re%uires/

Top and -ottom depth o" the reseroir

P 'ae elocity lo!

Porosity and4or density in"ormation

,hear 'ae elocity in"ormation *lo! or estimate+

,aturation in"ormation *consistent 'ith input 'ell lo!s+

Rock matri. in"ormation *"rom mineral ta-les+

Fluid properties *From ()B "luid calculator+

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Theory 1-47

FR operates on the lo! data on a sample -y sample -asis$

Areas 'ith lo' porosity# or hi!h shale content should -e e.cluded usin!

!amma ray# density or porosity cut)o""s

0ensity and porosity in"ormation are re%uired$ This in"ormation must -e

consistent$

FR can accept/

) 0ensity lo! 'ith saturation data# matri. and "luid densities *porosity

is calculated+

) Porosity lo! 'ith saturation data# matri. and "luid densities *density

lo! is calculated+) 0ensity and porosity lo!s 'ith saturation data and "luid densities

*matri. densities are calculated+

FR can -e sensitie to poor %uality or inconsistent lo! data$

Input P 'ae and 0ensity In"ormation/

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Theory 1-48

,hear 'ae in"ormation is re%uired to calculate @dry"rom the saturated P'ae lo! in"ormation$

,hear 'ae in"ormation can come "rom/

0ipole ,hear 'ae sonic lo!s

?stimated ,)'ae elocity lo!s usin! the ARCO mudrock line0ry rock Poisson3s ratio *try alues "rom $1= to $= "or sandstones+

The udrock line underestimates , 'ae elocities in unconsolidated#

hi!hly porous sands$ This may result in incorrect estimates o" the dry

rock Poisson3s ratio and @dry$

In that case# su!!est/ replace the estimated , 'ae elocities "or these

sands in a synthetic , 'ae lo! 'ith a VP4V,o" =$E$

,hear Bae In"ormation/

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Theory 1-49

Bater saturation "or the initial reseroir conditions may -e proided as aconstant alue or as a lo!$

,aturation in"ormation must a!ree 'ith the recorded sonic lo! and

density alues$

The sonic tool measures the "astest trael path "rom source to receier$In many cases# the sonic elocity represents the "lushed 'ell -ore

annulus rather than the hydrocar-on saturation "ormation$

Petrophysicists can proide 'ater saturation lo!s that represent the

conditions o" the inaded re!ion$

Flushed re!ions o"ten e.hi-it patchy saturation$

Bater ,aturation In"ormation/

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Theory 1-50

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Theory 1-51

=+ Calculate input P 'ae modulus/

4

3

& & K = +

( ) += 1&"rwet

0etailed ,teps Assumin! Casta!na3s ?%uation "or

Bet ,ands/

1+ Calculate density "or 1EED -rine saturation/

2

'# V = + Calculate matri. P 'ae modulus/

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Theory 1-52

:+ Calculate V,'et"rom VP'et

() * ( ) *

fl "r

& & fl & "r

# K K

d # # # K # K = +

(1

&wet

## d

d=

+

wet

wet'

wet

#V

=

( wet

wet V V

=

H+ AdMust P 'ae modulus to 1EED 'ater/

+ Calculate VP'et

G+ Calculate V,input"rom V,'et

'wet c wet cV ) V %= +

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Theory 1-53

1E+ Calculate @sat'ith ne' "luid/

2

( +V =

2 4

( (3'

K V =

( ) *

fl

& & fl

K Ka

K K K K=

(1

dry &a

K Ka

=+

() *

out out dry fl

out out out & dry & fl

K Ka

K K K K= +

(

1

out&

aK K

a

=

+

+ Calculate @ and m "rom input data/

9+ O-tain @dry/

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Theory 1-54

11+ >et ne' density/

( ( )1 *out out out out fl & = +

4

3 +

out out

out'

out

K

V

+

=

outout

out

V

=

1=+ Finally the ne' elocitiesN

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Theory 1-55

uality Control o" the FR Result

Check dry rock Poisson3s ratio o" "irst sample on last FR panel$

se C plot option on FR "inal panel to produce displays$

0isplay error lo! to check "or reported pro-lems$

0ry rock Poisson3s ratio should -e/

Ran!e ean

Clastics E$E to E$=

Limestones E$= to E$ E$1

0olomites E$1: to E$= E$=G

0ry rock -ulk modulus should -e/

Ran!e ean

Clastics = to =E

Limestones = to :E

0olomites = to :E H

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Theory 1-56

Bhen pro-lems occur# check "or the "ollo'in!/

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Theory 1-57

Bhen multiple pore "luids are present# @"lis usually calculated-y a Reuss aera!in! techni%ue/

Kfl vs Sw and Sg

E

E$

1

1$=

=$

E E$= E$ E$G 1

Water saturation (fraction)

Bulkmodulus(Gpa)

5his avera8in8

techni'ue assumes

uniform fluiddistriution;

4/as and li'uid must

e evenly distriuted

in every pore"

1 *w o

fl w o *

K K K K= + +

This method heaily -iases compressi-ility o" the com-ined

"luid to the most compressi-le phase$

The JFi22 BaterK Issue

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Theory 1-58

Bhen "luids are not uni"ormly mi.ed# e""ectie modulus alues cannot-e estimated "rom Reuss aera!in!$

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Theory 1-59

Bhen patch si2es are lar!e# 'ith respect to the seismic 'aelen!th#Voi!t aera!in! !ies the -est estimate o" @"l*0omenico# 19G:+$

fl w w o o * *K K K K= + +Bhen patch si2es are o" intermediate si2e# >assmann su-stitution

should -e per"ormed "or each patch area and a olume aera!e

should -e made *0orkin et al# 1999+$

This can -e appro.imated -y usin! a po'er)la' aera!in! techni%ue

*(rie et al# 199+/

Patchy ,aturation/

( ) *

e

w*wfl KKKK +=

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Theory 1-60

Patchy ,aturation/

>assmann predicted elocities

nconsolidated sand matri.

Porosity ; ED

1EED >as to 1EED (rine saturation

1.5

1.7

1.

!.1

!."

!.5

# #.!5 #.5 #.75 1

Water Saturation (fraction)

$p(km%s)

&atc'

$oigt

euss

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Theory 1-61

Accordin! to a paper -y 6an and (at2le# The Leadin!

?d!e# April# =EE=/

the JFi22 BaterK e""ect is !reatly dependent on the pressure o" the

"ormation$

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Theory 1-62

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Theory 1-63

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Conclusions

An understandin! o" rock physics is crucial "or the interpretation o"AVOAVOanomalies$

The olume aera!e e%uation can -e used to model density in a 'ater

sand# -ut this e%uation does not match o-serations "or elocities in

a !as sand$

The3iot4/assmann3iot4/assmanne%uations match o-serations 'ell "or

unconsolidated !as sands$

Bhen dealin! 'ith more comple. porous media 'ith patchysaturation# or "racture type porosity *e$!$ car-onates+# the 3iot43iot4

/assmann/assmanne%uations do not hold$

TheA60O mudrock lineA60O mudrock lineis a !ood empirical tool "or the 'et sands

and shales