Generating multiple history matched models in metric space · History matching and uncertainty yWe know how to generate a history matched model yConstrain to multiple wells production

Generating multiple history matchedmodels in metric space

Generating multiple history matchedmodels in metric space

Jef Caers and Kwangwon ParkStanford University, USA

History matching and uncertainty

We know how to generate a history matched modelConstrain to multiple wells productionConstrain to geological information (MPS)Constrain to other data such as 3D/4D seismic

We do not know how to generate multiple history matched models

≠ just generating more history matched models

||

( | ) ( )( | )

( )

f ff

f= DM M

MDD

d m mm d

d

What do we do today?

Optimization (gradient, GDM)Start with an initial Earth modelUpdate this initial Earth model into a new modelUntil “criteria of matching” are met

Filtering (EnKf, genetic algorithms)Start with a set of modelsUpdate that initial set into a new setUntil “criteria of matching” are met

butartificial reduction of uncertainty

Sampling via McMC

Again, start with an initial Earth model mPropose a perturbation m* must be a sample of fM(m)

Accept that perturbation with probability α

|

|

( | *)α min 1 ,

( | )

f

f

⎧ ⎫⎪ ⎪= ⎨ ⎬⎪ ⎪⎩ ⎭

DM

DM

d m

d m

Internal consistent with Bayes’ rulebut too slow (1000s of flow simulations)

Proposal:Sampling based on an ensemble of models

Borrow the best of both worldsPerform sampling (not optimization)Work with ensemblesUpdate ONE model based on AN ENSEMBLE

Start with an ensemble of models

Sample an improved Earth model based on that ensemble

Repeat this “sampling” to get many Earth models

Let’s start with an example

Geology (training image)Geology (training image)

Sinuous channelsNE50 direction


Structure and well log dataStructure and well log data

310 ft x 310 ft rectangular reservoir

One injector and one producerSand facies at both wells



Production dataProduction data

Watercut history at the producer over 3 years


Generating an initial set of prior modelsGeology (training image)Geology (training image) Structure and well log dataStructure and well log data







Multiple-point geostatistical method:

SNESIMGenerate 100 prior models

Multiple-point geostatistical method:

SNESIMGenerate 100 prior models

Honoring prior informationHonoring prior information

Construct a metric space by defining a distance

Difference between responses of any two models

Difference between responses of any two models

Define a distanceDefine a distance

Responses from forward simulationsResponses from forward simulations

Construct a metric spaceConstruct a metric space

x1 x2 x3 …

x1 0 d(x1,x2) d(x1,x3) …

x2

d(x2,x1) 0 d(x2,x3) …

… … … … …

From the metric spaceFrom the metric space

Projection of metric space in 2D spaceProjection of metric space in 2D space

Projection of metric space by MDS

x1 x2 x3 …

x1

0 d(x1,x2) d(x1,x3) …

x2

d(x2,x1) 0 d(x2,x3) …

… … … … …

Mapping the “true Earth” in metric space

production data is the response of “true Earth”production data is the response of “true Earth”

Distance between the “true Earth” and any modelDistance between the “true Earth” and any model

Construct a new metric space including the “true Earth”Construct a new metric space including the “true Earth”

Parameterization

Any inverse modeling requires a parameterizationTo lower dimensionalityTo emphasize important and sensitive parameters

Traditional parameterization is that of a single Earth modelAllows for geologically consistency (PPM, GDM)

Here: parameterization is deduced from the ensembleAllows geological consistency (PPM, GDM)Allows internal consistency with Bayes’ rule

Parameterization in metric space

How to create a single new model from the ensemble ?

?

How?

Forward kernel transformation: First make the MDS plot “easier”

Model expansion: expansion entails creating a single model from an ensemble, consistent with the prior of that ensemble

Backward transformation: what is the Earth model corresponding to the model expansion?

Introducing a kernel

From metric space to kernel spaceFrom metric space to kernel space

Metric spaceMetric space Kernel spaceKernel space

Xm φ

Model expansion

[ ]1 2 L

1/2

Gaussian realizations , , ,

Euclidean distance matrix A

Dot‐product

KL‐expansion

T

T T

new

X

B XX V V

V

=⇓

⇓= = Λ⇓

= Λ

x x x

x y

K

Example“new Gaussian vectors”

KLexpansion

“Gaussian vectors”

1/2

KL‐expansion new V = Λx yDot‐product

T TB XX V V= = Λ

Model expansion in kernel space

Metric space Kernel space

Xm φ

Model expansion in kernel space

[ ]1 2 L

1/2

Gaussian realizations , , ,

Euclidean distance matrix A

Kernel matrix

KL‐expansion

T

T T

new

K V V

V

φ φ φ

Φ

Φ Φ Φ

Φ Φ

Φ =⇓

⇓=ΦΦ = Λ⇓Φ = Λ y

K

From model expansion to modelthe pre‐image problem

A sample y provides a model expansion in kernel space

The (short) vector y is the parameterization of any new Earth model based on the ensemble of models

Sampling of a y entails sampling from the prior

Pre‐image problem: how to go from y to an actual Earth model m (see next presentation)

Pre‐image problem

Metric space Kernel space

Associated with y

?

How ? See later presentation !

Back to the actual inverse problem: Introducing the post image problem

the “true Earth”the “true Earth”

1 2 3 4( , , , ,..., , )new newLf=x x x x x x y

Sampling problem

Find by sampling all possible y such that the model expansions generated with that xnew

maps at the location of the production data


We have established a link between a parameterization y and a new model xnew

Sampling problem

Find by sampling all possible y such that the model expansions generated with that xnew

maps at the location of the production data


Suitable sampling methodsgradual deformationmetropolis sampling

Sampling problem

Metric spaceMetric space Kernel spaceKernel space

y1,y2, y3, y4, …

Summary of the methodology

Generate a prior ensembleRun flow simulationCreate distances, metric and feature spaceConstruct KL‐expansionSolve pre‐image problem

Back to the example:Multiple posterior models

4 of 30 posterior models obtained by solving post-image and pre-image problems (197 forward simulations)4 of 30 posterior models obtained by solving post-image and pre-image problems (197 forward simulations)

4 of 30 posterior models obtained by rejection sampling (9,563 forward simulations)4 of 30 posterior models obtained by rejection sampling (9,563 forward simulations)

Posterior models matching data

Post-image and pre-image problems(197 forward simulations)

Post-image and pre-image problems(197 forward simulations)

Rejection sampling (9,563 forward simulations)Rejection sampling (9,563 forward simulations)

Spatial uncertainty in facies distribution

Post-image and pre-image problems (197 forward simulations)Post-image and pre-image problems (197 forward simulations)


PosteriorPosteriorPriorPrior

mean variance

mean variance

mean variance

mean variance

A new scenario: uncertainty modeling right after water breakthrough

Geology (training image)

Geology (training image)

Structure and well log data

Structure and well log data

Sinuous channels (NE50)Sinuous channels (NE50) 310 ft x 310 ft rectangular Sand facies at both wells

310 ft x 310 ft rectangular Sand facies at both wells

Nonlinear time-dependent dataNonlinear time-dependent data



Uncertainty is substantial

Prior is not likely to match the data(efficiency issue in the pre-image problem!)

Uncertainty is substantial

Prior is not likely to match the data(efficiency issue in the pre-image problem!)

More practical situation requiring uncertainty modeling

More practical situation requiring uncertainty modeling

Multiple posterior models

4 of 15 posterior models obtained by solving post-image and pre-image problems (238 forward simulations)4 of 15 posterior models obtained by solving post-image and pre-image problems (238 forward simulations)

4 of 15 posterior models obtained by rejection sampling (12,424 forward simulations)4 of 15 posterior models obtained by rejection sampling (12,424 forward simulations)

Spatial uncertainty in facies distribution

Post-image and pre-image problems (238 forward simulations)Post-image and pre-image problems (238 forward simulations)


PosteriorPosteriorPriorPrior

mean variance

mean variance

mean variance

mean variance

Conclusions

There is currently no practical approach for creating multiple history matches within a consistent framework

Approach borrows best of two worldsEnsemble method to span the prior space of uncertaintySampling methods for creating realistic uncertainty

Proposed approach has the potential tobe easily integrated in softwarebe efficient (10s‐100s of flow simulations)

Documents

Generating multiple history matched models in metric space · History matching and uncertainty yWe know how to generate a history matched model yConstrain to multiple wells production