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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