Petroleum Geostatistic - Caers Slides

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


2/54

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


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$uote

“%heor& should #e as simple as possi#le#ut not simpler as possi#le….”

Albert EINSTEIN


4/54

'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!


5/54

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


7/54

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


8/54

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


9/54

(ultiple-point 0eostatistics

Reservoir2 well data

multiple-pointdata event

: A ; 1 / 3

Se/uentialsimulation

A

1


10/54

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


11/54

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 ,


13/54

%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


15/54

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


17/54

(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


18/54

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


20/54

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


21/54

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.


22/54

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.


23/54

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-


24/54

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


26/54

Simple pro#lemB

2

21 P(A | $, !, D) 1 & '= =+ +:A;(/ 6 B.KB

:A;


27/54

E4ample reservoir

:A;C/

0.0

1.000

Tr"inin! i"!e:A;(/ Sin!le re"li>"tion


28/54

P6;C9, 2 single-point :

:A;C/ Pe"li>"tion When &obin! :A;(/ #ro!eolo!y "nd :A;


29/54

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


30/54

)ocal rotation angle fromseismic

:A;C/ o&"l "n!le


31/54

Results

2 realizations ith an!le itho"t an!le

Constrain to local Ichannel


32/54

Constrain to local Ichannelfeatures

#eat"re $a%

Realization P6;C9

H"rd d"t"

#ro seisi&

So#t d"t"

#ro seisi&


33/54

Part """

In'erse odelin! with$ltiple+point !eost"tisti&s

pplication to histor& matching

Production data does not inform


34/54

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?


35/54

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

•


36/54

(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


37/54

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


38/54

%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


39/54

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)


40/54

Determine rD

Use P6;D9 as a pro#a#ilit& model in multiple-point geostatistics

⇒


41/54

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


42/54

•


43/54

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


44/54

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


East

N o r t h

0.0 50.0000.0

50.000

facies 0

facies 1


East

N o r t h

0.0 50.0000.0

50.000

facies 0

facies 1


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,


45/54

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


46/54

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

.

.

.

.

.

.

.

.

.

.

.

. ..

.

.

.

.

.

.

.

.

.

.

.

.

. ..

.

.

.

.

.

.

.

.

.

.

.

.

. ..

.

.

.

.

.

.

.

.

.

.

.

.

*teration 3

East

N o r t h

0.0 50.0000.0

50.000

.

.

.

.

.

.

.

.

.

.

.

*nitial $odel100.000


47/54

(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


48/54

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"


49/54

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


50/54

*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


51/54

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


52/54

(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


53/54

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

Petroleum Geostatistic - Caers Slides

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