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8/21/2019 Petroleum Geostatistic - Caers Slides
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Stochastic inverse modeling underrealistic prior model constraintswith multiple-point geostatistics
Jef Caers
Petroleum Engineering DepartmentStanford Center for Reservoir Forecasting
Stanford, California, US
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c!nowledgements
would li!e to ac!nowledge the contri#utions
f the SCRF team, in particular ndre Journeland all graduate students
who contri#uted to this presentation
8/21/2019 Petroleum Geostatistic - Caers Slides
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$uote
“%heor& should #e as simple as possi#le#ut not simpler as possi#le….”
Albert EINSTEIN
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'verview
• (ultiple-point geostatistics • Why do we need it ?• How does it work ?• How do we defne prior odels with it ?
• Data integration• Inte!r"tion o# $ltiple types%s&"les o# d"t"• Ipro'eent on tr"dition"l ("yesi"n
ethods
• Solving general inverse pro#lems• )sin! prior odels #ro p !eost"tisti&s• Appli&"tion to history "t&hin!
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Part "
*$ltiple+point ,eost"tisti&
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)imitations of traditionalgeostatistics
*ariograms E+
0.4
0.8
1.2
10 20 30 400
0.4
0.8
1.2
10 20 30 400
3
12
*ariograms S
-point correlation is not enough to characteri.e connectivit& prior geological interpretation is re/uired
and it is '% multi-0aussian
1 2 3
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Stochastic se/uential simulation
• -efne " $lti+'"ri"te ,"$ssi"n/ distrib$tiono'er the r"ndo #$n&tion 0u/
• -e&opose the distrib$tion "s #ollows
1 1 N N
1 1 2 2 1 3 3 2 1 N N N 1 1
Pr(Z( ) z , ,Z( ) z )
Pr(Z( ) z ).Pr(Z( ) z | z ).Pr(Z( ) z | z ,z ) Pr(Z( ) z | z , , z )−
≤ ≤ =
≤ ≤ ≤ ≤
u u
u u u u
1
1 1
1 1 N N
1 1 2 2 1 3 3 2 1 N N N 1 1
Pr(Z( ) z , ,Z( ) z | (n))
Pr(Z( ) z | (n)).Pr(Z( ) z | z , (n)).Pr(Z( ) z | z ,z , (n)) Pr(Z( ) z | z , , z , (n))−
≤ ≤ =
≤ ≤ ≤ ≤
u u
u u u u
1
1 1
2r in its &ondition"l #or
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Practice of se/uential simulation
A
(3
(4
(5
167(38(48(59
:A;1/ 6 N8σ/
8σ !i'en by kri!in!8 depend on"$to&orrel"tion '"rio!r"/ #$n&tion
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(ultiple-point 0eostatistics
Reservoir2 well data
multiple-pointdata event
: A ; 1 / 3
Se/uentialsimulation
A
1
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E4tended ormal E/uations
k
k
k
n
1
S K {s , k 1, , K}
1 S( ) sPr(A 1 | S( ) s , ) ?
0
1 S( ) s , 1, , nD A
0
k
Attrib$te t"kin! possible st"tesAt lo&"tion 8 $nknown bin"ry r"ndo '"ri"ble
i#A
i# not
i#
i# not
α α
α αα
α=
=
== = = ∀α
= α == =
∏
u
uu
u
1
1
%raditionan
k k
1
n
k k 1
Pr(A 1 | S( ) s , ) E[A ] (1 E[A ])
Pr(A 1 | S( ) s , ) E[A ] (1 E[A ]) (1 E[A A ])
α α α αα=
α α α α β α βα= β
= = ∀α = + λ −
= = ∀α = + λ − + µ −
∑
∑ ∑
u
u
l !riging5 point statistics
E4tensions to three point statistics
$
$α
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Single ormal E/uation
k k k
k k k
k k k k
k
Pr(A 1 |S( ) s , ) Pr(A 1 | D 1) E[A ] (1 E[D])
E[A D] E[A ]E[D].Vr[D] !"#[A ,D]
E[D](1 E[D])
E[A D] E[A ]E[D]Pr(A 1 | D 1) E[A ]
E[D](1 E[D])
Pr(A 1 | D 1)
s$&h th"t
α α= = ∀α = = = = + λ −
−λ = ⇒ λ =
−
−= = = +
−
= = =
u
"n case of 6n789-point statistics
k k k
k k
E[A D] E[A ]E[D]E[A ]
E[D]
E[A D] Pr[A 1, D 1]
E[D] Pr[D 1]
−+
= == =
=
1a&es Rule :
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%he training image module
%raining image module 2standardi.ed analog model/uantif&ing geo-patterns
P 6 A ; 1 9 2 8 < = 2 (ud
SES"( algorithmRecogni.ing P6;19 for all possi#le ,
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%he SES"( algorithm
2
14 11
5 7
3 1 2 5 3 0
5 3
1 1
1 0 0 0 0 0
0 0 0 0 0
1 1 1 1 1 1
1 1 1 1 1
3 2
2
%raining imageData template
6data search neigh#orhood9
Search tree
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Pro#a#ilities from a Search %ree
2
14 11
5 7
3 1 2 5 3 0
5 3
1 1
1 0 0 0 0 0
0 0 0 0 0
1 1 1 1 1 1
1 1 1 1 1
3 2
2
u
Search neigh#orhood
Search tree
%raining image
5
4
3
2
j=1
i = 1 2 3 4 5
u
u
u
u
u
uu u uu
u
u u u
)evel > 6no CD9
)evel 8 68 CD9
)evel = 6= CD9
.
.
.
.
.
.
1 2
3 4
1 1 1 1
1 1 1 1 1 1 1 1
2 2 2 2 2 2 22
uu 1 1
2 2 3 3
u
1
2 3
4
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E4ample
=>> sample data
Reali.ation
%rue image
%raining image
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+here do we get a ?D %" 3
East
N o r t h
0. 0 500. 0000. 0
500. 000
shale
sand
%raining image re/uires @stationarit&@ 'nl& patterns 2 @repeated multipoint statistics@ can #e rep
C"lid tr"inin! i"!e Not C"lid
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(odular training image
*od$l"r ?
D no $nits D rot"tion+in'"ri"nt D "nity+in'"ri"nt
Tr"inin! I"!e *odels !ener"ted with snesi$sin! the S(E tr"inin! i"!e
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Properties of training image
Re/uired
• St"tion"rityF p"tterns by defnition repe"t• Er!odi&ityF to reprod$&e lon! r"n!e #e"t$re 6G l"r!e i
• iited to + &"te!ories
ot re/uired
• )ni'"ri"te st"tisti&s need not be the s"e "s "&t$"l fe• No &onditionin! to ANJ d"t"• Anity%rot"tion need not be the s"e
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Part ""
*$ltiple+point ,eost"tisti&s"nd d"t" inte!r"tion
Si l i diA l
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Simple /uestion, diAcultpro#lemB
A geologist belie'es b"sed on !eolo!i&"l d"t" th"t there isKBL &h"n&e o# h"'in! " &h"nnel "t lo&"tion M
A geoph&sicist belie'es b"sed on !eophysi&"l d"t" th"tthere is L &h"n&e o# h"'in! " &h"nnel "t lo&"tion M
A petroleum engineer belie'es b"sed on en!ineerin! d"t"th"t there is KL &h"n&e o# h"'in! " &h"nnel "t lo&"tion M
+hat is the pro#a#ilit& of having a channel at 3
%he essential data integration pro#lemB
P6;19
P6;C9
P6;D9
P6;1,C,D93
C #i i f
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Com#ining sources ofinformation
P(A | $,!) (P(A), P(A | $), P(A | !)) (P(A), P($ | A),P(! | A))
,
P(A | $, !) [0,1]
P(A | ( )) P(A | ( )) 1
! A % P(A | !, ( )) P(A | ( ))
! A % P
-esir"ble properties o#
3.
4.
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Conditional independence
P($, ! | A) P($ | A) P(! | A))
P(A) P($|A) P(!|A)P(A | $,!)
P($,!)
P($,!)
P(A | $, !) [0,1]
P(A | ( )) P(A | ( )) 1
! A % P(A | !, ( )) P(A | ( ))
!
3. ?
4.
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Correcting conditionalindependence
P(A | $,!) P(A | $,!) 1
P(A) P( $ | A) P(! | A) P(A) P($ | A) P(! | A) P($, !)
P(A) P($|A) P(!|A)P(A | $, !)
P(A) P( $ | A) P(! | A) P(A) P( $ | A) P(! | A)
P(A | $) P(A | !)
P(A)
P(A | $
+ = ⇒
× × + × × =
× ×=
× × + × ××
=
Solution5 Standardi.e the e4pression
Closure
) P(A | !) P(A | $) P(A | !)
P(A) P(A)
× ×+
%his e4pression honors all conditions 8-
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Permanence of ratios h&pothesis
P(A | $) P(A | !)
P(A)P(A | $, !)
P(A | $) P(A | !) P(A | $) P(A | !)
P(A) P(A)
1 & ' P(A | $, !)
1 & ' '
1 P(A) 1 P(A | $) 1 P(A | !) 1 P(A | $, !) ' &P(A) P(A | $) P(A | !) P(A | $,!)
&
&"n be resh"ped "s #ollows
or
with
×
=× ×+
− −= = =
+ +
− − − −= = = =
− ' ' is "
−= permanence of ratios h&pothesis
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dvantages of using ratios
• No ter :(8
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Simple pro#lemB
2
21 P(A | $, !, D) 1 & '= =+ +:A;(/ 6 B.KB
:A;
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E4ample reservoir
:A;C/
0.0
1.000
Tr"inin! i"!e:A;(/ Sin!le re"li>"tion
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P6;C9, 2 single-point :
:A;C/ Pe"li>"tion When &obin! :A;(/ #ro!eolo!y "nd :A;
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Concept of ('DU)R trainingimage
*od$l"r ?
D St"tion"ry p"tterns D rot"tion+in'"ri"nt D "nity+in'"ri"nt D no $nits
*od$l"r Tr"inin! I"!e *odels !ener"ted with snesi$sin! the S(E tr"inin! i"!e
)ocal rotation angle from
8/21/2019 Petroleum Geostatistic - Caers Slides
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)ocal rotation angle fromseismic
:A;C/ o&"l "n!le
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Results
2 realizations ith an!le itho"t an!le
Constrain to local Ichannel
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Constrain to local Ichannelfeatures
#eat"re $a%
Realization P6;C9
H"rd d"t"
#ro seisi&
So#t d"t"
#ro seisi&
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Part """
In'erse odelin! with$ltiple+point !eost"tisti&s
pplication to histor& matching
Production data does not inform
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Production data does not informgeological heterogeneit&
a
a
a
a
a
a
a
a
a
a
a
a
a
a
0.0
0.2
0.4
0.6
0.8
1.0
& a t e r ' " t
0 200 400 600 800 1000
t i$e( da)s
a a a a a a a a a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
aa
a a a
a a a
a a a
a a a a a
a a a a a a a a a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
aa
a
aa
aa
aa
a a a a
a a a a a a
a
a
*ni t ial +er$e a,
B
3
W " t e r & $ t
3BBB Tie d"ys/
East
N o r t h
0.0 50.00.0
50.0
0.0
200.0
400.0
600.0
800.0
1000.0
East
N o r t h
0.0 50.00.0
50.0
0.0
200.0
400.0
600.0
800.0
1000.0
East
N o r t h
0.0 50.00.0
50.0
0.0
200.0
400.0
600.0
800.0
1000.0
East
N o r t h
0.0 50.00.0
50.0
0.0
200.0
400.0
600.0
800.0
1000.0
A :etrole$ En!ineer,eolo!ist 3
,eolo!ist 4-is"!reein! ,eolo!ist?
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pproach
*ethodolo!y
-efne " non+st"tion"ry *"rko' &h"in th"t o'es" re"li>"tion to "t&h d"t"8 two properties
• At e"&h pert$rb"tion we "int"in !eolo!i&"lre"lis $se ter P6;19
•
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(ethodolog&5 two facies
1
0
1 *+ +*s s "-rs
( )0 *+ +*s s "-rs
=
u (&, /, z)=u
- 6 set o# histori& prod$&tion d"t" press$res8 Qows
Some notation5
Initi"l !$ess re"li>"tionF(")* ( ) ∀u u( )* ( ) ∀u ulPe"li>"tion "t iter"tion( )l
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DeKne a (ar!ov chain
( )
( 1)
* ( )
?
* ( )+
∀
⇓
∀
u u
u u
l
l
( 1) ( )
( 1) ( )
( 1) ( )
( 1) ( )
Pr{ ( ) 1 | D, * ( ) 0} ?
Pr{ ( ) 0 | D, * ( ) 1} ?
Pr{ ( ) 1 | D, * ( ) 1} ?
Pr{ ( ) 0 | D, ( ) 0} ?
+
+
+
+
= = =
= = =
= = =
= = =
u u
u u
u u
u u
l l
l l
l l
l l
DeKne a transition matri45
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%ransition matri4
( 1) ( )
( 1) ( )
( 1) ( )
( 1) ( )
D D
D
D
D
Pr{ ( ) 1 | D, * ( ) 0}
Pr{ ( ) 0 | D, * ( ) 1}
Pr{ ( ) 1 |
r Pr{( ) 1} r [0,1]
r (1 Pr{( ) 1})
1 r (1 Pr{(D, * ( ) 1}
Pr{ ( )
) 1})
1 r Pr{(0 | D,* ( ) 0 ) 1}}
+
+
+
+
= =
= =
= =
= =
= = ∈
= − =
= − − =
= − =
D
u u u
u
u
u
r i
u u
u
s a constant ove
u
u u
l l
l l
l l
l l
r the reservoir and depends on production D
4 R 4 tr"nsition "triR des&ribes the prob"bility o#&h"n!in! #"&ies "t lo&"tion u "nd we defne it "s#ollows
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Parameter rD
( 1)
( 1) ( )
D D
( 1) ( )
D D
A { ( ) 1}
Pr{ ( ) 1 | D,* ( ) 0} r Pr{( ) 1} r [0,1]
Pr{ ( ) 1 | D,* ( ) 1} 1 r Pr{( ) 0} r [0,1]
+
+
+
= =
= = = = ∈
⇔
= = = − = ∈
⇔
(l)
D
(l)
D
P(A | D) = r P(A) if i (u) =
P(A | D) = 1 ! r (1 ! P(A)) if i
u
u u u
u
(
u
u ) = 1
u
l
l l
l l
⇔ (l)D DP(A | D) = (1 ! r ) " i (u) # r " P(A)
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Determine rD
Use P6;D9 as a pro#a#ilit& model in multiple-point geostatistics
⇒
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rD determines a Ipertur#ation
initial $odel
East
N o r t h
0.0 50.0000.0
50.000
facies 0
facies 1
rD / 0.05
East
N o r t h
0.0 50.0000.0
50.000
rD / 0.1
East
N o r t h
0.0 50.0000.0
50.000
facies 0
facies 1
rD / 0.2
East
N o r t h
0.0 50.0000.0
50.000
rD / 0.5
East
N o r t h
0.0 50.0000.0
50.000
facies 0
facies 1
rD / 1
East
N o r t h
0.0 50.0000.0
50.000
trainin! i$a!e
East
N
o r t h
0.0 150.000
0.0
150.000
r-6B.B3Soe initi"l odel
r-
6B.3 r-
6B.4
r-6B. r-63ind r- th"t "t&hes best
the prod$&tion d"t"ind r- th"t "t&hes best
the prod$&tion d"t"6 one+diension"loptii>"tion
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•
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E4amples
Reference $odel
East
N o r t h
0.0 50.0000.0
50.000
facies 0
facies 1
trainin! i$a!e
East
N o r t h
0.0 150.0000.0
150.000
,ener"te 3B reser'oir odelth"t 3. Honor the two h"rd d"t" 4. Honor #r"&tion"l Qow 5. H"'e !eolo!i&"l &ontin$it
siil"r "s TII
:
0 4
0.45
reference$atch
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Single model*nitial $odel
East
N
o r t h
0.0 50.0000.0
50.000
facies 0
facies 1
*teration 1( rD/0.21
East
N
o r t h
0.0 50.0000.0
50.000
facies 0
facies 1
*teration 3( rD/0.52
East
N o r t h
0.0 50.0000.0
50.000
facies 0
facies 1
*teration 5( rD/0.50
East
N o r t h
0.0 50.0000.0
50.000
facies 0
facies 1
*teration 7( rD/0.31
East
N o r t h
0.0 50.0000.0
50.000
facies 0
facies 1
*teration 9( r D/0.24
East
N o r t h
0.0 50.0000.0
50.000
facies 0
facies 1
0 5 10 15 20 25 30 35 400
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
ti$este%s da)s
f
$atchinit
0 1 2 3 4 5 6 7 8 90
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
*terations
3 , 4 e c t i 5 e f " n c t i o n
1 2 3 4 5 6 7 8 90.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
*terations
5 a l " e o f r D
rD values,
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rD values,
single 8D optimi.ation
0 0.5 10.005
0.01
0.015
0.02
0.025
0.03
*teration1
0 0.5 10.008
0.01
0.012
0.014
0.016
0.018
0.02
0.022
0.024
*teration2
0 0.5 10.006
0.008
0.01
0.012
0.014
0.016
0.018
0.02
0.022
3 , 4 e c t i 5 e f " n c t i o n
*teration3
0 0.5 10.008
0.01
0.012
0.014
0.016
0.018
0.02
*teration4
0 0.5 14
5
6
7
8
9
10 10
- 3 *teration5
0 0.5 10.01
0.012
0.014
0.016
0.018
0.02
0.022
0.024
0.026
*teration6
0 0.5 11
1.5
2
2.5
3
3.5
4
4.5
5 10
- 3
t- al"e
*teration7
0 0.5 14
5
6
7
8
9
10
11
12 10
- 3
t- al"e
*teration8
0 0.5 10
0.002
0.004
0.006
0.008
0.01
0.012
t- al"e
*teration9
r- '"l$e
2 b T e & t i ' e # $ n & t i o n
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DiLerent geolog&
0 5 10 15 20 25 30 35 400
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
ti$este%s da)s
f
reference$atchinit
. ..
. ..
. ..
.
*teration 7
East
N o r t h
0.0 50.0000.0
50.000
facies 0
facies 1
Reference $odel
East
N o r t h
0.0 50.0000.0
0.000
facies 0
facies 1
0 5 10 15 20 25 30 35 400
0.05
0.1
0.15
0.2
0.25
0.3
0.35
ti$este%s da)s
f
reference$atchinit
Reference $odel
East
N o r t h
0.0 50.0000.0
.000
.
.
.
.
.
.
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*teration 3
East
N o r t h
0.0 50.0000.0
50.000
.
.
.
.
.
.
.
.
.
.
.
*nitial $odel100.000
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(ore wells
0 10 20 30 400
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
ti$este%s da)s
f
&ell 1
reference$atchinit
0 10 20 30 400
0.05
0.1
0.15
0.2
0.25
0.3
0.35
ti$este%s da)s
f
&ell 2
reference$atchinit
0 10 20 30 400
0.1
0.2
0.3
0.4
0.5
ti$este%s da)s
f
&ell 3
reference$atchinit
0 10 20 30 400
0.1
0.2
0.3
0.4
0.5
0.6
0.7
ti$este%s da)s
f
&ell 4
reference$atchinit
Reference $odel
East
N o r t h
0.0 100.0000.0
100.000
*teration 8
East
N o r t h
0.0 100.0000.0
100.000
East
N o r t h
0.0 100.0000.0
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Mierarchical matching
D irst &hoose fRed pere"bility per #"&ies8pert$rb #"&ies odel
D Then8 #or " fRed #"&iespert$rb the pere"bility within #"&ies$sin! tr"dition"l ethods8 ss&8 !r"d$"l de#or"
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E4ample
Reference $odel
East
N o r t h
0.0 50.0000.0
50.000
facies 0
facies 1
I
:
*nitial !"ess *teration 1
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*nitial !"ess
East
N o r t h
0.0 50.0000.0
50.000
/50$D
/500$D
*teration 1
East
N o r t h
0.0 50.0000.0
50.000
/50$D
/500$D
*teration 2
East
N o r t h
0.0 50.0000.0
50.000
/12$D
/729$D
*teration 3
East
N o r t h
0.0 50.0000.0
50.000
/12$D
/729$D
*teration 4
East
N o r t h
0.0 50.0000.0
50.000
/11.5$D
/694$D
Results
1low 6 B1hi!h 6 BB
1low 6 B1hi!h 6 BB
1low 6 341hi!h 6 4O
1low 6 341hi!h 6 4O
1low 6 331hi!h 6 UO
Reference $odel
East
N o r t h
0.0 50.0000.0
50.000
1low 6 3B1hi!h 6 B
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Results
*nitial "ess( :1/500( :0/50
East
N o r t h
0.0 50.0000.0
50.000
;fter i ter 1( :1/721( :0/12
East
N o r t h
0.0 50.0000.0
50.000
;ft er iter 4( : 1/694( :0/11.5
East
N o r t h
0.0 50.000
0.0
50.000
10 20 30 400
0.1
0.2
0.3
0.4
f
reference$atchinit
10 20 30 400
0.1
0.2
0.3
0.4reference$atchinit
10 20 30 400
0.1
0.2
0.3
0.4
ti$este%s da)s
f
reference$atchinit
10 20 30 400
0.1
0.2
0.3
0.4
ti$este%s da)s
f
reference$atchinit
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(ore realistic
East
N o r t h
0.0 50.0000.0
50.000
Pe#eren&e
East
N o r t h
0.0 50.0000.0
50.000
Initi"l odel
East
N o r t h
0.0 50.0000.0
50.000
0.0
100.000
200.000
300.000
400.000
"t&hed odel
East
N o r t h
0.0 50.0000.0
50.000
East
N o r t h
0.0 50.000
0.0
50.000
N o r t h
0.0 50.0000.0
000
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Conclusions
+hat can multiple-point statistics provide
• "r!e QeRibility o# prior odels8 no need #or "th.de#.
• A #"st8 rob$st s"plin! o# the prior
• A ore re"listi& d"t" inte!r"tion "ppro"&h th"n
tr"dition"l ("yesi"n ethods
• A !eneri& in'erse sol$tion ethod th"t honors priorin#or"tion
(ore on conditional
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(ore on conditionalindependence
$ $ ! !
$ !
$ $ ! !
$ $
P ($|A)?
$ + (A) ! + (A)
,P($ ' | A,! ) P( ' + (A) | + (A) , A)
P( ' + (A) | A), ',
P($|A)
siil"rly
Recommended