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Multi-fidelity meta-models
for reservoir engineering
Arthur Thenon
IFP Energies nouvelles (IFPEN)
Véronique Gervais (IFPEN)
Mickaële Le Ravalec (IFPEN)
March 23, 2016 – MASCOT-NUM
Introduction Reservoir engineering context
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Reservoir characterization
Core sample data
Seismic data
Geological data
Reservoir model
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Log data
2
Production data
A. Thenon - Multi-fidelity meta-models for reservoir engineering - MASCOT-NUM - March 23, 2016 - Toulouse
Introduction Reservoir engineering context
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Goals of reservoir engineering
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Assessment of field production potential
Management of the field development
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Tools
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3D numerical models
Flow simulator
Reservoir
model
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Flow simulation
Simulated
production
data
Main issues
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Size of reservoir models
Building representative models
 time consuming flow simulations
 lots of uncertainties
A. Thenon - Multi-fidelity meta-models for reservoir engineering - MASCOT-NUM - March 23, 2016 - Toulouse
Introduction Reservoir engineering context
To handle / reduce uncertainties
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Identify the most influent uncertain parameters
Probabilistic forecasting
Incorporate all available data (dynamic data)
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History-matching process
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Flow simulation
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Reservoir
model
Simulated
production
data
Measured
production
data
Objective function
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A. Thenon - Multi-fidelity meta-models for reservoir engineering - MASCOT-NUM - March 23, 2016 - Toulouse
Introduction Reservoir engineering context

To handle / reduce uncertainties



Identify the most influent uncertain parameters
Probabilistic forecasting
Incorporate all available data (dynamic data)


Dealing with uncertainties and optimizing field production call for a huge
number of flow simulations
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5
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History-matching process
Prohibitive computation time
Solutions

Using reservoir models with coarser grids
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Approximate responses
Using meta-models (polynomial, kriging, etc…)
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Good context to use
multi-fidelity meta-models
Still requires a lot of simulations to be predictive
A. Thenon - Multi-fidelity meta-models for reservoir engineering - MASCOT-NUM - March 23, 2016 - Toulouse
Introduction Multi-fidelity meta-models
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Principle: use responses from different levels of fidelity to build a metamodel at the finest level
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We consider multi-fidelity co-kriging meta-models as introduced by
[Kennedy and O’Hagan, 2000] : 𝒁𝒕 𝒙 = 𝝆𝒕−𝟏 𝒁𝒕−𝟏 𝒙 + 𝜹𝒕 𝒙
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6
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Save computation time to get a predictive meta-model
𝒁𝒕 a Gaussian random process modeling the response at level t
𝜹𝒕 a Gaussian random process independent of 𝒁𝒕−𝟏 , … , 𝒁𝟏
𝝆𝒕−𝟏 a scale factor
R package MuFiCokriging [Le Gratiet, 2012]
A. Thenon - Multi-fidelity meta-models for reservoir engineering - MASCOT-NUM - March 23, 2016 - Toulouse
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Outline
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Introduction
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Numerical experiment
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Vectorial output modeling
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Sequential design strategy
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Conclusion and future work
A. Thenon - Multi-fidelity meta-models for reservoir engineering - MASCOT-NUM - March 23, 2016 - Toulouse
Outline
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Introduction
Numerical experiment
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PUNQ test case description
Numerical experiment description
Results
Discussion
Vectorial output modeling
Sequential design strategy
Conclusion and future work
A. Thenon - Multi-fidelity meta-models for reservoir engineering - MASCOT-NUM - March 23, 2016 - Toulouse
Numerical experiment – PUNQ description
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The case is inspired from PUNQ S3 [Floris et al., 2001]
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6 production wells (PRO-1, 4, 5, 11, 12 and 15)
Produced by depletion
Two reservoir models with different grid resolutions
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Coarse model: 19*28*5 grid blocks (average simulation time: 10s)
Fine model: 57*84*5 grid blocks (average simulation time: 180s)
0.30
PUNQ S3 top and well
positions
0.25
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0.20
0.15
0.10
Porosity in layer 3 for the
coarse and fine PUNQ models
0.05
0
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A. Thenon - Multi-fidelity meta-models for reservoir engineering - MASCOT-NUM - March 23, 2016 - Toulouse
Numerical experiment - Description
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7 uncertain parameters
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3 outputs per well
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Bottom hole pressure (BHP)
Gas/oil ratio (GOR)
Water cut (WCUT)
All outputs are used to compute the objective function (OF).
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3 critical saturations, 3 permeability multipliers and 1 aquifer permeability
𝐎𝐅 𝒙 =
𝒚 𝒙 −𝒚𝑟𝑒𝑓
𝝈
𝑟𝑒𝑓
2
𝒚 the outputs, 𝒚
the outputs for the reference simulation (standing for
production data) and 𝝈 the weights
Multi-fidelity meta-models (and kriging) are built to approximate the OF
from various size nested LHS (and LHS).
A. Thenon - Multi-fidelity meta-models for reservoir engineering - MASCOT-NUM - March 23, 2016 - Toulouse
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Numerical experiment - Results
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A. Thenon - Multi-fidelity meta-models for reservoir engineering - MASCOT-NUM - March 23, 2016 - Toulouse
© 2015 - IFP Energies nouvelles
Numerical experiment - Results
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The Q2 coefficient is computed for an
independent test sample (LHS of 200 points).
𝐧𝐟 = 𝐍𝐟 = number of points on the fine level
A. Thenon - Multi-fidelity meta-models for reservoir engineering - MASCOT-NUM - March 23, 2016 - Toulouse
Numerical experiment - Discussion
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Multi-fidelity meta-models show poor performance
when approximating the OF.

Similar behaviors do not imply a good correlation
between the OF contributions.
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Poor correlation between OF values for fine and coarse
levels
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A. Thenon - Multi-fidelity meta-models for reservoir engineering - MASCOT-NUM - March 23, 2016 - Toulouse
Outline
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Introduction
Numerical experiment
Vectorial output modeling
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Sequential design strategy
Conclusion and future work
© 2015 - IFP Energies nouvelles
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Methods
Results
Discussion
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A. Thenon - Multi-fidelity meta-models for reservoir engineering - MASCOT-NUM - March 23, 2016 - Toulouse
Vectorial output modeling - Methods
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Existing method from [Douarche et al., 2014] [Marrel et al., 2015]
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Apply a Proper Orthogonal Decomposition on a set of responses
Compute meta-models to approximate the coefficients of the reduced basis (scores)
Design of
experiment
(LHS)
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Set of
responses
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Get the scores
Build kriging
models
In multi-fidelity, we propose to compute coarse level scores by projecting
the coarse level responses on the fine level reduced basis.
Set of fine
level
responses
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Apply POD
Apply POD
Get the fine
level scores
Build
co-kriging
models
Design of
experiment
(nested LHS)
Set of coarse
level
responses
Projection
on the fine
reduced basis
Get the coarse
level scores
A. Thenon - Multi-fidelity meta-models for reservoir engineering - MASCOT-NUM - March 23, 2016 - Toulouse
Vectorial output modeling - Results
Using this approach we approximate the OF by modeling each output
involved in its definition (OF through output modeling).
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A. Thenon - Multi-fidelity meta-models for reservoir engineering - MASCOT-NUM - March 23, 2016 - Toulouse
Vectorial output modeling - Discussion
Example: GOR at well 1
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Example: BHP at well 4
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A. Thenon - Multi-fidelity meta-models for reservoir engineering - MASCOT-NUM - March 23, 2016 - Toulouse
Vectorial output modeling - Discussion
Example: BHP at well 4
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A. Thenon - Multi-fidelity meta-models for reservoir engineering - MASCOT-NUM - March 23, 2016 - Toulouse
Outline
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


Introduction
Numerical experiment
Vectorial output modeling
Sequential design strategy
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

Conclusion and future work
© 2015 - IFP Energies nouvelles
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Introduction
Algorithms
Results
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A. Thenon - Multi-fidelity meta-models for reservoir engineering - MASCOT-NUM - March 23, 2016 - Toulouse
Sequential design strategy - Introduction
How to choose the point location and the level of fidelity to be evaluated
to build better multi-fidelity meta-models of the objective function?
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A. Thenon - Multi-fidelity meta-models for reservoir engineering - MASCOT-NUM - March 23, 2016 - Toulouse
Sequential design strategy - Introduction
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Algorithm proposed by [Le Gratiet, 2015]
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A straightforward extension to our approach is difficult
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Meta-models are at the score levels (≠ one meta-model of the OF)
Computationally too expensive (numerous meta-models are involved)
We proposed a sequential algorithm strategy well-suited for our context
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Point location determined by co-kriging variance adjusted by the cross-validation errors (CVE)
Level of fidelity chose by comparison between the variance part from each level and with
respect of the time ratio
We can make the following hypotheses:
- only two levels of fidelity
- time ratio between the fine and coarse level simulations
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The coarse level of fidelity solely is evaluated until well-known
Co-kriging variance and CVE at the OF level can be computed from the co-kriging variance
and cross-validation predictions of the different scores
Point location determined using CVE and co-kriging variance
A. Thenon - Multi-fidelity meta-models for reservoir engineering - MASCOT-NUM - March 23, 2016 - Toulouse
Sequential design strategy - Algorithms

In simple fidelity context
Initial design of
experiment
(small size LHS)
Flow simulation
Meta-model of the OF
(output modeling)
Stopping
criterion
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Stopping
criterion
21
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Sufficient predictivity
or end of time budget
Compute the CVE on
the OF
Sampling of points in the
Voronoï cell with the
highest CVE
Selection of the point
maximizing the OF
variance
A. Thenon - Multi-fidelity meta-models for reservoir engineering - MASCOT-NUM - March 23, 2016 - Toulouse
Final meta-model of the
OF
Sequential design strategy - Algorithms

In multi-fidelity context
Initial design of
experiment
(small size nested LHS)
Coarse level
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Fine level
Stopping
criterion
Stopping
criterion
23
Flow simulation
Final meta-model of the
OF
Meta-model of the OF
(output modeling)
Stopping
criterion
Stopping
criterion
Compute the CVE on
the OF
Compute the CVE on
the OF
Sampling of points in the
Voronoï cell with the
highest CVE
Sampling of points in the
Voronoï cell with the
highest CVE
Selection of the point
maximizing the OF
variance
Selection of the point
maximizing the OF
variance
Sufficient predictivity
Sufficient predictivity
or end of time budget
A. Thenon - Multi-fidelity meta-models for reservoir engineering - MASCOT-NUM - March 23, 2016 - Toulouse
Sequential design strategy - Results
Both levels
evaluations
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Coarse level
evaluations
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A. Thenon - Multi-fidelity meta-models for reservoir engineering - MASCOT-NUM - March 23, 2016 - Toulouse
Sequential design strategy - Discussion
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The sequential strategy proposed can still be improved.
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The stopping criterion at the coarse level is arbitrary (𝑸𝟐𝒄𝒗 > 0.95).
Defining a criterion to choose the level to be evaluated (based on crossvalidation errors)
Limit of the PUNQ test case
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Small size LHS or nested LHS are sufficient to reach Q2 > 0.9.
Defining a more complex case
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A. Thenon - Multi-fidelity meta-models for reservoir engineering - MASCOT-NUM - March 23, 2016 - Toulouse
© 2015 - IFP Energies nouvelles
Outline
26

Introduction

Numerical experiment

Vectorial output modeling

Sequential design strategy

Conclusion and future work
A. Thenon - Multi-fidelity meta-models for reservoir engineering - MASCOT-NUM - March 23, 2016 - Toulouse
Conclusion and future work
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Conclusion
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© 2015 - IFP Energies nouvelles
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The use of multi-fidelity co-kriging meta-models in reservoir
engineering can save computation time, especially if the time
budget is very limited.
Suitable methodologies must be deployed.
Efficiency depends on the correlation between the fidelity
levels and the time ratio.
Future work
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
Application to a more complex test case (Brugge Field)
Improvement of the proposed sequential strategy
Sequential strategies for optimization
A. Thenon - Multi-fidelity meta-models for reservoir engineering - MASCOT-NUM - March 23, 2016 - Toulouse
References
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Kennedy, M., and O’Hagan, A. (2000). Predicting the output from a complex computer code
when fast approximations are available. Biometrika 87, 1–13.
Le Gratiet, L. (2012). MuFiKriging: Multi-fidelity co-kriging models. R package version 1.0.
Floris, F.J.T., Bush, M.D., Cuypers, M., Roggero, F., and Syversveen, A.-R. (2001). Methods
for quantifying the uncertainty of production forecasts: a comparative study. Petroleum
Geoscience 7, S87–S96.
Douarche, F., Da Veiga, S., Feraille, M., Enchéry, G., Touzani, S., and Barsalou, R. (2014).
Sensitivity Analysis and Optimization of Surfactant-Polymer Flooding under Uncertainties. Oil &
Gas Science and Technology – Revue d’IFP Energies Nouvelles, 69(4), 603-617.
Marrel, A., Perot, N., Mottet, C. (2015). Development of a surrogate model and sensitivity
analysis for spatio-temporal numerical simulators. Stoch Environ Res Risk Asses, 29, 959-974.
Le Gratiet, L., & Cannamela, C. (2015). Cokriging-Based Sequential Design Strategies Using
Fast Cross-Validation Techniques for Multi-Fidelity Computer Codes. Technometrics, 57(3),
418–427.
A. Thenon - Multi-fidelity meta-models for reservoir engineering - MASCOT-NUM - March 23, 2016 - Toulouse
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www.ifpenergiesnouvelles.com
A. Thenon - Multi-fidelity meta-models for reservoir engineering - MASCOT-NUM - March 23, 2016 - Toulouse