Adaptive Spatial Resampling as a McMC Method for Uncertainty Quantification

in Seismic Reservoir Modeling

Cheolkyun Jeong*, Tapan Mukerji, and Gregoire Mariethoz

Stanford Center for Reservoir Forecasting

How to quantify uncertainty of models?

Why quantify uncertainty?

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Key issues

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1. We make decisions under uncertainty2. Modeling subsurface reservoir is a uncertain process

1. In a Bayesian framework, sampling posterior distribution can quantify the uncertainty

2. Rejection sampler is a theoretically perfect method but inefficient

A critical issue is to sample posteriors efficiently: A Markov chain Monte Carlo method as an equivalent posterior sampler

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Sampling efficiencyKey issues

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Rejection Sampler Markov Chain MC Reference

dobs dpredict

Proposed model

Forward modeling

dpredict

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d

a. Generate and evaluate its likelihood L().

b. Select a subset points .c. Generate a proposal

model d. Evaluate L() and accept or

reject by acceptance criterion.

e. Iterate b. ~ d.

𝒎𝟏

a

𝒓𝟏

bc d 𝒎𝟐

d

e

𝒓𝟐

d *e

Creating a Markov chain: Iterative Spatial Resampling (ISR)Methodology

5

Creating a Markov chain: ISR

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Methodology

a. Generate and evaluate its likelihood L().

b. Select a subset .c. Generate a proposal

model d. Evaluate L() and accept or

reject by acceptance criterion.

If L() L() orL() L() with P(L()/ L())

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ASR algorithm in acoustic impedance

Randomly sampled subset points

Adaptively sampled subset points

Randomly sampled subset points

Adaptively sampled subset points

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(𝒅¿¿𝒐𝒃𝒔−𝒈 (𝒎¿¿𝒆𝒍𝒂𝒔))¿¿

Methodology – Adaptive Spatial Resampling

spatial error map

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ASR algorithm in seismic section

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Methodology – Adaptive Spatial Resampling

Seismogram: obtained data

Seismogram: predicted model

Cross correlation coefficient in each trace

time

time

corr

elat

ion

coef

ficie

nt

CDP

Higher correlationHigher chance

Lower correlationMore perturbation

subset

Reference: faciesReference

Iterative Spatial Resampling Adaptive Spatial Resampling

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ASR algorithm in acoustic impedance

Methodology – Adaptive Spatial Resampling

Log1

0 RM

SE

Log1

0 RM

SE

Iteration Iteration

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1. Fraction rate in ASR

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Methodology – Parameter sensitivity

Log1

0 RM

SE

Iterations

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2. Number of traces in seismic section

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Methodology – Parameter sensitivity

Log1

0 SSE

Iterations

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1. Acoustic impedance for lithofacies characterization

Reference: facies Well dataWells

Predicted seismic data

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Illustration

Seismic dataacoustic impedance

CDP 25 125

MRayls

MRayls

Vp

𝝆

Bivariate pdf Rockphysics

Reference: facies

Etype of priors

Etype of sampled posteriors (RS)

Variance of sampled posteriors (RS)

100,000 priors

125 posteriors

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1. Acoustic impedance: Rejection SamplerIllustration

Reference: facies

1. Acoustic impedance: Results

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Etyp

eVa

rianc

e

125 posteriors (100,000 eval.)

21 posteriors (500 eval.)

94 posteriors(500 eval.)

Rejection sampling Iterative Spatial Resampling Adaptive Spatial Resampling

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Illustration

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1. Acoustic impedance: ASRIllustration

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2. Seismograms for facies characterization

Reference: facies Well dataWells

Predicted seismic data

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Seismic dataseismograms

CDP 25 125

Vp

𝝆

Bivariate pdf Rockphysics

Illustration

Reference: facies

2. Seismogram: Results

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Etyp

eVa

rianc

e

140 posteriors (100,000 eval.)

29 posteriors (500 eval.)

51 posteriors(500 eval.)

Rejection sampling Iterative Spatial Resampling Adaptive Spatial Resampling

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2. Seismogram

Illustration

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2. Seismogram results using MDS projectionIllustration

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1st principal coordinate

2nd p

rinci

pal c

oord

inat

e

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3. Verification using MDS projectionIllustration

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3. Finding facies not seen in well data

Reference: facies Well dataWells

Predicted seismic data

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Seismic dataseismograms

CDP 25 125

Vp

𝝆

Bivariate pdf Rockphysics

Oilsand

Brinesand

Shale

*Not detected oilsand distribution is generated by Gassmann’s equation

Facies Actual Logs

Vp 𝝆 One model in priorsOne model in posteriors

Illustration

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24 posteriors (50,000 eval.) 43 posteriors (1000 eval.)

3. Finding facies not seen in well data

Probability of Oil Sand Probability of Oil Sand

CDP CDP

Probability

Oilsand

Brinesand

Shale

Rejection sampling Adaptive Spatial Resampling

Reference: facies

Illustration

4. ASR as an optimizer

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Log1

0 RM

SE

Iterations

Reference: facies

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Keep only better models in a Markov chain: L() L()

Illustration

4. ASR as an optimizer

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Illustration

2. The adaptive spatial resampling (ASR) is a good approximation of rejection sampler as a posterior sampler, and it’s more efficient.

1. Sampling posteriors in seismic inverse modeling can be a good uncertainty quantification tool for decision making.

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3. Depending on the acceptation/rejection criterion, it is possible to obtain a chain for sampling posterior or calibrating the most likely earth model.

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Summary

1. Application in actual dataset: West Africa dataset

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Ongoing and Future work

3 wells, Near and Far offset seismic data

<Courtesy to Hess>

Geological Observation

Rockphysics model (Dutta, 2009)Facies 1: Channel DepositionFacies 2: Near channel leveesFacies 3: Medial-distal levees

What we have

1. Application in actual dataset: West Africa dataset

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Ongoing and Future work

2D slice : Acoustic Impedance

Geological ObservationBuild Training images

3D study

What we need

2. The adaptive spatial resampling (ASR) is a good approximation of rejection sampler as a posterior sampler, and it’s more efficient.

1. Sampling posteriors in seismic inverse modeling can be a good uncertainty quantification tool for decision making.

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3. Depending on the acceptation/rejection criterion, it is possible to obtain a chain for sampling posterior or calibrating the most likely earth model.

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Summary

1. Application in actual dataset: West Africa dataset

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Ongoing and Future work

Multiple subsurface scenarios

1. P(Tis | Seismogram) using pattern validation

Geologist (1)

Geologist (2)

Geologist (3)

Pattern Validation for finding distances between seismogram images

Generate priors, m Forward model, g(m)

Multiple subsurface scenarios

2. P(RPs | Seismic data) using pattern validation

Rockphysics (1)

Pattern Validation for finding distances between seismogram

Forward model, g(m)

3. Multiple subsurface scenarios

Rockphysics (2)

Generate priors, m

2. The adaptive spatial resampling (ASR) is a good approximation of rejection sampler as a posterior sampler, and it’s more efficient.

1. Sampling posteriors in seismic inverse modeling can be a good uncertainty quantification tool for decision making.

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3. Multiple subsurface scenarios help to choose the most applicable setting for unknown reservoir modeling.

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Summary

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3. Multiple subsurface scenarios1. P(Ti | Seismic data) using pattern validation

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3. Multiple subsurface scenarios

[301x301]

1. P(Ti | Seismic data) using pattern validation

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3. Multiple subsurface scenarios

Ti1 = 0.0014 at the data location, Ti2 was 2.4498, and Ti3 was 5.6447

According to Bayesian theorem and Park(2011), P(Ti2|data) = 30% and P(Ti3|data) = 70%

Ti2 Ti3

1. P(Ti | Seismic data) using pattern validation

2. P(RPs | Seismic data) using pattern validation

Rockphysics (1)

Pattern Validation for finding distances between seismogram

Forward model, g(m)

3. Multiple subsurface scenarios

Rockphysics (2)

Generate priors, m

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3. Multiple subsurface scenarios2. P(RPs | Seismic data) using pattern validation

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24 posteriors (50,000 eval.) 43 posteriors (1000 eval.)

3. Finding facies not seen in well data

Probability of Shale Probability of Shale

Probability of Brine Sand Probability of Brine Sand

Probability of Oil Sand Probability of Oil Sand

CDP CDP

Probability

Oilsand

Brinesand

Shale

Rejection sampling Adaptive Spatial Resampling

Illustration

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Appendix III : Ti