defining a random function (rf) from a given set of...

Céline Scheidt, Kwangwon Park and Jef Caers

Stanford University

Defining a Random Function (RF) From a Given Set of Model Realizations

Introduction

Uncertainty in reservoir modeling requires the construction of ensembles of models

Many applications are “goal”-oriented and thus require the construction of ensembles having particular features

E.g. data conditioning, model updating, optimization, etc.

Ineffective: realizations are generated without the response function in mind

2 SCRF Affiliate Meeting - April 30, 2009

Motivation Example

Set of Initial Realizations

One realization matching production data

0 500 1000 15000

2

4

6

8x 10

4

Time

Pro

duc

tion

Da

ta

Data Acquisition

How to generate efficiently a new set of realizations that match data ?


Goals

To generate new realizations without re-running the algorithm/sampler that generated them (Model expansion)

To identify a small set of variables which describe the uncertainty (Parameterization)(useful for inverse modeling, response

analysis, optimization, etc…)


General Approach

Use of dissimilarity distances and MDSIntegrate information specific to the applicationOptimization or uncertainty quantification problems more efficient

Use of Karhunen-Loeve expansion (KL-expansion) in Feature space

Preserve the spatial variability and the hard data conditioning


Use of Application-Specific Distances

SCRF Affiliate Meeting - April 30, 20096

In the motivation example previouslyDistance represents dissimilarity of desired properties of the model (here production data)Wish to construct new realizations with small distances to the “reference”

xClose to reference

Far to reference

K-L Expansion for Gaussian Realizations

2/1 newnew V yx Λ=N(0,1)~newy

[ ]L21 ,,, xxx K=X

New realization:

From Jef Caers


[ ]

V

VVXXB

X

new

TT

yx

xxx

2/1

L21

expansion -KL

product -Dot

,,, nsrealizatioGaussian

Λ=⇓

Λ==⇓

= K

MDSUsing K

featurespace

Model expansion

ector)Gaussian v standard a is (L

1 with )(

:expansionLoeve-Karhunen

y

ybbx KVΦ ==ϕ

ΦX)( ii aa ⇒xφx

Φ

21/KKVΦ Λ=

L)(L Matrix Gramor Kernel

)()(

),(

×⇒

⎟⎟⎠

⎞⎜⎜⎝

⎛ −−−==

K

expkK jiT

j,dijiij σ

xxxxxx

TKKK VVK Λ=

Model Expansion in Feature SpaceFrom Jef Caers

8SCRF Affiliate Meeting - April 30, 2009

Feature Space

F

)](,),([ ,1, Ldd xx ϕϕ K=Φ

Metric Space

M

Definition of a RF for Non-Euclidean Distances and non-Gaussian Realizations

ϕ

MDS ],,[ ,1, Lddd xxX K=

newnew Φbx =)( ϕ

ϕ-1

],,[ 1 LxxX K=

CNℜ∈ix

Distance Matrix δ:

Non-EuclideanApplication

tailoredPre-image

Reconstruction of

realizations


The Pre-image ProblemHow to map back realizations from feature space F to metric space M ?

Feature Space

F

Metric Space

M

ϕ-1

newΦb

The Pre-image Problem

newK

newnewnewd

newd V-Φ

newd

ybbxxx L

1 with )ˆ(minargˆ2

2== ϕ

Model expansion is defined in high-dimensional space F

Mapping function ϕ is unknown, non-linearand non-uniqueOnly approximate solutions are possible

Optimization problem

newΦb

Feature Space

F

Metric Space

Mϕ ϕ-1

newΦb


ˆnewdx

The Pre-image ProblemPre-image problem can be solved using an iterative procedure, called fixed point algorithm (Schoelkopf and Smola, 2002)

: non-linear combination of the realizations in Metric space

∑∑

′

′= =+

),ˆ(),ˆ(

ˆ,

,1 ,,

,1,

idnnew

di

L

i ididnnew

dinnewd kb

kbxxxxx

x

1 with ˆL

1,

L

1

== ∑∑== i

optiid

i

opti

newd ββ xx


1,ˆ +nnewdx

? ϕ

Fixed point algorithm:Initial guess is defined as linear combination of ’s, with random coefficientsDefine new point using equation from previous slideIterate until convergence


0,ˆ newdx id ,x

2,ˆ newdx nnew

d,x̂

1,ˆ newdx

newΦb0,ˆ newdx new

dx̂

Metric Space Feature Space

FM


idi

opti

newd β ,

L

1

ˆ xx ∑=

=


1 with ˆL

1,

L

1== ∑∑

== i

optiid

i

opti

newd ββ xx

Pre-image solution:

newdx̂

Summary

Metric Space

M


Creating a new modelHow to construct the corresponding new realization in physical space?

newdx̂

Metric Space

M

Creation of new models in physical space

Unconstrained optimizationApply the same weights as in the previous equation ( )

Ensures hard data conditioning:

Feature constrained optimizationConstrain explicitly the solutions of the inverse problem

Geological constrained optimizationUse of probability mapsUse of Gradual Deformation Method (Hu, 2001) or Probability Perturbation Method (Caers, 2002)

3 different options explored:

1L

1=∑

=i

optiβ

optiβ


Examples

Present 3 examples of different types of realizations

Continuous, Boolean and Multi-Point

RF generated as follow:Initial ensemble contains 300 realizations

Definition of a connectivity distance between realizations (Park and Caers, 2008)

Gaussian rbf Kernel for the definition of the feature space F

Pre-imageUnconstrained, feature constrained and geologically constrained


Unconstrained Pre-Image - ExampleDirect sequential simulation (dssim)

Uniform conditional distribution, exp. variogram

Initial Real.:

Connectivity distance

4

3

2

0

2

3

MDS

Kernel techniques

K-L expansion

Pre-image:

Physical realizations

2D projection of modelsFrom metric space

2D projection of modelsFrom Feature space


)( newnewd Φbx =ϕ


Initial Real.New Real.

∑=

=L

idi

opti

newd β

1,xx

∑=

=L

ii

opti

new β1

xx

ix

Unconstrained Pre-Image - Example

Analysis of the new generated realizations

Same variability between realizations19 SCRF Affiliate Meeting - April 30, 2009

300 Boolean realizations

Feature Constrained Optimization

3 Initial real.Connectivity

distance

MDS

Model expansion

Pre-image

Solve pre-image problem

optiβ

0

5

0

5

5

2

5




Can we apply weights directly to binary images?

Reconstruction of realizations from the pre-image weights


Difficult to apply non-linear weights on binary images

optiβ


How can we preserve the shape of the objects?

Boolean realizations


Channel objects are clearly identifiable

3 New real.

Weights obtained

previously (Unconstrained

pre-image)

InverseProximity Transform

Solution: Use a proximity distance

Proximity Transform

optiβ


5

0

5

5

2

5



Geologically Constrained OptimizationExample 1

optiβ

Unconstrained pre-image

Weights :3 Initial real.

Multiple-Point realizations (snesim)

0

0

0

0

0

0



Probability maps -> input for snesim as soft data

Geologically Constrained Opt.

Smoothing algorithm

Generation of real. using snesim

Geologically Constrained Optimization

Multiple-Point realizations (snesim)Reconstruction of 300 new realizations

3 Initial real. 3 New real.Metric space based on connectivity distance


Example 2

Optimization of the pre-image problem using

PPM

Geologically Constrained Optimization

Multiple-Point realizationsReconstruction of 1 realization close to one having desired properties

x

oooo

oxoo

Metric space


Conclusions

We presented a method to generate new realizations by solving the pre-image problem using an ensemble of existing realizations

By construction, new realizations share the same conditioning data and spatial continuityproperties as the existing ones.

Method applies to all types of input geostatistical model (Gaussian-type, Boolean and multi-point techniques).


Application of this technique is shown in other presentations (Scheidt a

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