[PPT] - Using Bayesian Model Probability with ensemble methods to quantify PowerPoint Presentation

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Using Bayesian Model Probability with ensemble methods to quantify uncertainty in reservoir modelling with multiple prior scenarios

Sigurd Ivar Aanonsen, NORCE Energy Svenn Tveit, NORCE Energy Mathias Alerini, Equinor ASA EnKF Workshop, Voss, June 3-5, 2019

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Outline

◮ Motivation ◮ Introduction to Bayesian model averaging and selection

◮ Bayesian Model Average (BMA) ◮ Bayesian Model Probability (BMP) ◮ Model Likelihood/Model Evidence (BME) ◮ Bayes Factor (BF)

◮ Calculating BME/BMP

◮ Challenges

◮ Examples ◮ Summary and conclusions

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Motivation

◮ Uncertainty quantification in history matching is typically based on a

single prior-model scenario. However, often, several alternative models or scenarios are viable a priori.

◮ Geological scenarios, flow scenarios, alternative seismic

interpretations, etc.

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Motivation

◮ Uncertainty quantification in history matching is typically based on a

single prior-model scenario. However, often, several alternative models or scenarios are viable a priori.

◮ Geological scenarios, flow scenarios, alternative seismic

interpretations, etc.

◮ Ensemble-based data assimilation methods, like EnKF or ES, does

not handle alternative prior models or scenarios. Handling complex model uncertanty is challenging.

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Motivation

◮ Uncertainty quantification in history matching is typically based on a

single prior-model scenario. However, often, several alternative models or scenarios are viable a priori.

◮ Geological scenarios, flow scenarios, alternative seismic

interpretations, etc.

◮ Ensemble-based data assimilation methods, like EnKF or ES, does

not handle alternative prior models or scenarios. Handling complex model uncertanty is challenging.

◮ Bayesian theory for models provides a framework for this:

◮ “Total” uncertainty through Bayesian Model Averaging (BMA). ◮ Selecting models or scenarios based on comparing Bayesian Model

Probabilities (BMP’s), i.e., probability for a given model or scenario to be correct given the data.

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Motivation, cont’d

◮ Bayesian model averaging and model probability rely on the calculation of

Bayesian Model Evidence (BME) or Bayes Factors (BF), which have a long history within a number of fields for model comparison and model selection.

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Motivation, cont’d

◮ Bayesian model averaging and model probability rely on the calculation of

Bayesian Model Evidence (BME) or Bayes Factors (BF), which have a long history within a number of fields for model comparison and model selection.

◮ Few applications to petroleum industry/reservoir modelling (Park et al.,

2013, Elsheikh et al., 2014, Hong et al, 2018).

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Motivation, cont’d

◮ Bayesian model averaging and model probability rely on the calculation of

Bayesian Model Evidence (BME) or Bayes Factors (BF), which have a long history within a number of fields for model comparison and model selection.

◮ Few applications to petroleum industry/reservoir modelling (Park et al.,

2013, Elsheikh et al., 2014, Hong et al, 2018).

◮ Recently, it has been shown that, for weakly nonlinear models, Bayesian

model selection can be efficiently coupled with ensemble-based data assimilation methods (Carrassi et al., 2017).

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Motivation, cont’d

◮ Bayesian model averaging and model probability rely on the calculation of

Bayesian Model Evidence (BME) or Bayes Factors (BF), which have a long history within a number of fields for model comparison and model selection.

◮ Few applications to petroleum industry/reservoir modelling (Park et al.,

2013, Elsheikh et al., 2014, Hong et al, 2018).

◮ Recently, it has been shown that, for weakly nonlinear models, Bayesian

model selection can be efficiently coupled with ensemble-based data assimilation methods (Carrassi et al., 2017).

◮ The methodology can be applied as a “simple” post-processing after having

applied standard ensemble-based data assimilation methods to the various scenarios.

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Motivation, cont’d

◮ Bayesian model averaging and model probability rely on the calculation of

Bayesian Model Evidence (BME) or Bayes Factors (BF), which have a long history within a number of fields for model comparison and model selection.

◮ Few applications to petroleum industry/reservoir modelling (Park et al.,

2013, Elsheikh et al., 2014, Hong et al, 2018).

◮ Recently, it has been shown that, for weakly nonlinear models, Bayesian

model selection can be efficiently coupled with ensemble-based data assimilation methods (Carrassi et al., 2017).

◮ The methodology can be applied as a “simple” post-processing after having

applied standard ensemble-based data assimilation methods to the various scenarios.

◮ However, the use of these methods are disputed. The calculations may be

very challenging with respect to e.g. stability, and further investigations of the applicability to reservoir modeling and updating are necessary.

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BMA/BMP/BME/BF

◮ Bayesian Model Average (BMA):

P (∆|D) =

k

P(∆|D, Mk)P(Mk|D)

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BMA/BMP/BME/BF

◮ Bayesian Model Average (BMA):

P (∆|D) =

k

P(∆|D, Mk)P(Mk|D)

◮ Bayesian (posterior) Model Probability (BMP):

P (Mk|D) = P(D|Mk)Ppri(Mk)

j P(D|Mj)Ppri(Mj) =

1

j

P(D|Mj) P(D|Mk) Ppri(Mj) Ppri(Mk)

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BMA/BMP/BME/BF

◮ Bayesian Model Average (BMA):

P (∆|D) =

k

P(∆|D, Mk)P(Mk|D)

◮ Bayesian (posterior) Model Probability (BMP):

P (Mk|D) = P(D|Mk)Ppri(Mk)

j P(D|Mj)Ppri(Mj) =

1

j

P(D|Mj) P(D|Mk) Ppri(Mj) Ppri(Mk) ◮ Model Likelihood/Model Evidence (BME):

P (D|Mk) =

P(D|θ, Mk)P(θ|Mk) dθ

Denominator in the “normal” Bayes formula for model k.

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BMA/BMP/BME/BF

◮ Bayesian Model Average (BMA):

P (∆|D) =

k

P(∆|D, Mk)P(Mk|D)

◮ Bayesian (posterior) Model Probability (BMP):

P (Mk|D) = P(D|Mk)Ppri(Mk)

j P(D|Mj)Ppri(Mj) =

1

j

P(D|Mj) P(D|Mk) Ppri(Mj) Ppri(Mk) ◮ Model Likelihood/Model Evidence (BME):

P (D|Mk) =

P(D|θ, Mk)P(θ|Mk) dθ

Denominator in the “normal” Bayes formula for model k.

◮ Bayes factor (BF):

BFj-k = P(D|Mj) P(D|Mk)

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Some alternatives for calculating BME

1. Gauss-linear approximation

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Some alternatives for calculating BME

1. Gauss-linear approximation

P (D|Mk) = N(Gkθk, Ck) = ((2π)nd det Ck)−1/2 exp

−1

2(D − Gkθk)TC −1

k

(D − Gkθk)

Ck

= CD + GkCθkG T

k

P (D|Mj) P (D|Mk) = det Ck det Cj 1/2 exp

− 1

2

(D − Gjθj)TC −1

j

(D − Gjθj) − ( D − Gkθk)TC −1

k

(D − Gkθk)

where θk is prior mean and Ck is prior covariance matrix.

Utilizing the ensemble representation of the pdf’s, the calculations may be performed in a space of dimension equal to the ensemble size.

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Some alternatives for calculating BME

1. Gauss-linear approximation
2. “Inverted Bayes”, i.e.,

P (D|Mk) = P(D|θ, Mk)P(θ|Mk) P(θ|D, Mk) using e.g. posterior mean or MAP estimate for θ.

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Some alternatives for calculating BME

1. Gauss-linear approximation
2. “Inverted Bayes”, i.e.,

P (D|Mk) = P(D|θ, Mk)P(θ|Mk) P(θ|D, Mk) using e.g. posterior mean or MAP estimate for θ.

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Alternatives for calculating BME

1. Gauss-linear approximation
2. “Inverted Bayes”, i.e.,

P (D|Mk) = P(D|θ, Mk)P(θ|Mk) P(θ|Mk, D) , (1)

using e.g. posterior mean or MAP estimate for θ.

3. Importance sampling with posterior ensemble as importance sampler,

i.e., averaging Eq. (1) over posterior ensemble.

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Alternatives for calculating BME

1. Gauss-linear approximation
2. “Inverted Bayes”, i.e.,

P (D|Mk) = P(D|θ, Mk)P(θ|Mk) P(θ|Mk, D) , (1)

using e.g. posterior mean or MAP estimate for θ.

3. Importance sampling with posterior ensemble as importance sampler,

i.e., averaging Eq. (1) over posterior ensemble.

4. Harmonic average of likelihoods over posterior ensemble

(approximation to the importance sampling above): P (D|Mk) ≈

1

Ne

i=1

1 P(D|θi, Mk) −1 , (2)

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Alternatives for calculating BME

1. Gauss-linear approximation
2. “Inverted Bayes”, i.e.,

P (D|Mk) = P(D|θ, Mk)P(θ|Mk) P(θ|Mk, D) , (1)

using e.g. posterior mean or MAP estimate for θ.

3. Importance sampling with posterior ensemble as importance sampler,

i.e., averaging Eq. (1) over posterior ensemble.

4. Harmonic average of likelihoods over posterior ensemble

(approximation to the importance sampling above): P (D|Mk) ≈

1

Ne

i=1

1 P(D|θi, Mk) −1 , (2)

5. Eq. 2 with only one realization, i.e., simply the likelihood function

(with posterior mean, e.g.).

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Challenges

◮ Stability issues: BMP very sensitive to “everything” (amount of data,

data mismatch, prior and data covariance matrices, quality of posterior, degree of nonlinearity, ...).

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Challenges

◮ Stability issues: BMP very sensitive to “everything” (amount of data,

data mismatch, prior and data covariance matrices, quality of posterior, degree of nonlinearity, ...).

◮ Gauss-linear: Nonlinear forward model (BMP mainly depends on

prior properties).

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Challenges

◮ Stability issues: BMP very sensitive to “everything” (amount of data,

data mismatch, prior and data covariance matrices, quality of posterior, degree of nonlinearity, ...).

◮ Gauss-linear: Nonlinear forward model (BMP mainly depends on

prior properties).

◮ Inverted bayes (single-value or average): Requires calculating 3

probabilities, which may be small away from the mean.

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Challenges

◮ Stability issues: BMP very sensitive to “everything” (amount of data,

data mismatch, prior and data covariance matrices, quality of posterior, degree of nonlinearity, ...).

◮ Gauss-linear: Nonlinear forward model (BMP mainly depends on

prior properties).

◮ Inverted bayes (single-value or average): Requires calculating 3

probabilities, which may be small away from the mean.

◮ Harmonic average: Requires large ensemble size; very sensitive to

individual ensemble members with small likelihood values (unstable).

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Challenges

◮ Stability issues: BMP very sensitive to “everything” (amount of data,

data mismatch, prior and data covariance matrices, quality of posterior, degree of nonlinearity, ...).

◮ Gauss-linear: Nonlinear forward model (BMP mainly depends on

prior properties).

◮ Inverted bayes (single-value or average): Requires calculating 3

probabilities, which may be small away from the mean.

◮ Harmonic average: Requires large ensemble size; very sensitive to

individual ensemble members with small likelihood values (unstable).

◮ Likelihood function: Does not take into account prior uncertainty.

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Some approaches to handle these challenges

◮ Multidimensional scaling and kernel density estimation of likelihood

(Park et al., 2013).

◮ Localization (Metref et al., 2018). ◮ Multilevel methods (Hoel et al., 2016, Fossum et al., 2019)

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Examples

From S.I. Aanonsen, S. Tveit and M. Alerini: Using Bayesian Model Probability for Ranking Different Prior Scenarios in Reservoir History Matching SPEJ, 2019

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Example 1

Measurements of one, single parameter Bimodal prior

6 4 2 2 4 6 x 0.0 0.2 0.4 0.6 0.8 1.0 Probability density Prior Prior-1 Prior-2 6 4 2 2 4 6 x 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Probability density Data

Prior: Mean = -3.0 / 2.0 Data: Mean = 0.0, Std = 2.0

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Example 1

Bayesian average vs “Total EnKF”

6 4 2 2 4 6 x 0.0 0.2 0.4 0.6 0.8 1.0 Probability density Prior Prior-1 Prior-2 6 4 2 2 4 6 x 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Probability density Data 6 4 2 2 4 6 x 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Probability density Bayesian average Analytical EnKF_Post-1 EnKF_Post-2 6 4 2 2 4 6 x 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Probability density Total EnKF

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Example 1

Bayesian average with repeated measurements

6 4 2 2 4 6 x 0.0 0.2 0.4 0.6 0.8 1.0 Probability Density

Update 1 P2 = 64%

Analytical EnKF_Post-1 EnKF_Post-2 6 4 2 2 4 6 x 0.0 0.2 0.4 0.6 0.8 1.0 Probability Density

Update 2 P2 = 75%

Analytical EnKF_Post-1 EnKF_Post-2 6 4 2 2 4 6 x 0.0 0.2 0.4 0.6 0.8 1.0 Probability Density

Update 3 P2 = 82%

Analytical EnKF_Post-1 EnKF_Post-2 6 4 2 2 4 6 x 0.0 0.2 0.4 0.6 0.8 1.0 Probability Density

Update 4 P2 = 88%

Analytical EnKF_Post-1 EnKF_Post-2 6 4 2 2 4 6 x 0.0 0.2 0.4 0.6 0.8 1.0 Probability Density

Update 5 P2 = 91%

Analytical EnKF_Post-1 EnKF_Post-2 6 4 2 2 4 6 x 0.0 0.2 0.4 0.6 0.8 1.0 Probability Density

Update 6 P2 = 93%

Analytical EnKF_Post-1 EnKF_Post-2

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Example 2

Estimating top reservoir surface from 4D seismic data Synthetic model inspired by a real North Sea oil field

Seismic interpretations

500 1000 1500 2000 x 990 1000 1010 1020 1030 1040 1050 1060 1070 1080 Depth

GI-0 GI-1 OP-0 OP-1 OP-2

Top interpretation 1 Top interpretation 2 Top interpretation 3 True top and base

Prior realizations

5 10 15 20 25 30 35 40 Grid cell 1000 1020 1040 1060 1080 Depth model 1 model 2 model 3

GI-0 GI-1 OP-0 OP-1 OP-2

Gas injection into undersaturated oil reservoir. 2D cross section Data: Gas-cap thickness No of parameters: 40 No of ensemble members: 100

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