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8/14/2019 Ensemble-Based History Matching for Channelized Petroleum Reservoirs (slides)
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Ensemble-Based History Matching
for Channelized Petroleum ReservoirsMatei ene
Delft Institute of Applied Mathematics
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Motivation
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Petroleum Reservoir
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Water flooding
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2D Channelized Reservoirs
Y-channel reservoir
Rock type Permeability Porosity
Channelsand
100 mD 20%
Backgroundshale 0.1 mD 5%
Sat after 1 year Sat after 5 years
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Reservoir State
Rock properties
Flow variables
Production data
0, 1, likelihood that grid cell is in a channel
1 , ,
1, ,
0,1-
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History Matchinga.k.a. Data Assimilation
are poorly known a priori Let
represent the prior information
More information becomes available during production
Task:incorporate this information to obtain s.t. Realizations also show channelized structure Observations are verified by the estimate Proper representation of uncertainty
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Ensemble Kalman FilterKalman, 1960; Evensen, 1994; Burgers 1998
Monte Carlo approximation of and , , Forecast model
(+) () Observations
Update
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Multi-point GeostatisticsStrebelle, 2002
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EnKF Workflow
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Adapting the Results for Simulation
, where : , 1= linear, but not convex, combination
0,1-, but may break outside of 0,1- We use to compute and Negative permeabilities? Porosities > 1?
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(1) Truncation
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(2) logittransform
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Parameterized EnKF
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Research topic 1
Propose a parameterization that preserves the structure ofchannelized reservoirs over sequential assimilation steps.
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Polynomial Feature Space (1)
Coordinates are all monomials of the state variables up to
degree
Mapping Example: for 2grid cells and degree 2
image vector
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Polynomial Feature Space (2)
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Dealing with Dimensionality
The dimension of : 1
= 109 for 45 45cells and 3 - Hilbert space (Mercers theorem) Inner products are easy to compute:
, , () Use Principal Component Analysis!
=
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Principal Component AnalysisHotelling, 1933
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KernelPCASchlkopf, 1998
Let , , be a set of reservoir realizations The PC are the eigenvectors of their covariance matrix:
PCA on(),,() Keep the most significant of the Compute projections on the PC
: , where (), for any state vector
Solely through the kernel function!
Kernel
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The Preimage Problem
We need ()
Impossible, in general
Sarma et al (2008) approximate using fixed-point iterations: () () = 1 , 2 , () ()
()
Local optima!
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Analytical Solution
Let ,
If invertible, then where 0 0 1 0 0- (,) ( , )=
We propose the similar kernel
, 1 1
= Less computational expense!
() =
invertible for odd
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Preimage Experiments (1)Grid: 45 X 45 Training set: 500 samples trunc
Analytical
Iterative
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Preimage Experiments (2)Grid: 45 X 45 Training set: 4000samples trunc
Analytical
Iterative
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Preimage Experiments (3)Grid: 100 X 100 Training set: 500 samples trunc
Analytical
Iterative
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Preimage Experiments (4)Grid: 45 X 45 Training set: 500 samples logit
Analytical
Iterative
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Preimage Experiments (5)Grid: 200 X 200 Training set: 500 samples logit
Analytical
Iterative
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KPCA-EnKF
1 3 5
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Ensemble Collapse!
1 3 5
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Subspace EnKFSarma and Chen, 2013
Partition the ensemble into groups
Define a different parameterization for each group
Assumption: the EnKF update is equivalent to the steepest
descent equation
where is the mean squared error
Then, by the chain rule,
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Subspace EnKF Workflow
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Results (1)
1 3 5
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Ensemble Mean
EnKF KPCA-EnS, d=3 5-Subspace EnKF
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Ensemble Variability
EnKF KPCA-EnS, d=3 5-Subspace EnKF
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Research topic 2
Study the effect of the number of subspaces when using theSubspace EnKF for history matching channelized reservoirs.
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Experiment Setup
Sources of information
Ensemble: 100members Training set: 51500samples
Split the 7500training samples evenly over *2,10,50subspaces
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Ensemble Variability
2 subspaces 10 subspaces 50 subspaces
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Research topic 3
Develop a strategy to form the subspaces which takes intoaccount the prior information about the reservoir.
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Training Set Clustering
Generally applicable to any type of reservoir
It can create specialized subspaces
We used a separate set of 1400samples to train a KPCAorder 3 parameterization,
Applied it to the training set, ()And performed K-means clustering on the , in order to
partition the 7500training samples for *2,10,50subspaces
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Ensemble Variability
2 subspaces 10 subspaces 50 subspaces
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Ensemble Means
2 subspaces 10 subspaces 50 subspaces
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Recommendations
The analytical solution is generally preferable over approximate
preimage schemes
Normalization + logittransform is generally preferable to
truncaton when updating bounded variables When using the Subspace EnKF, the number of subspaces needs
to be balanced with the training set size.
Training set clustering seems to increase posterior variability,
especially when a large number of subspaces is used. One assimilation method is not generally better than the others;
the results need to be discussed with an expert
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Future research
What is the effect of polynomial KPCA when used to updatecontinuous variables?
Can we extend the faciesvariables to cases with more than 2
types of rock? (see Sebacher et al, 2013)
What is the benefit when using polynomial chaos expansionstogether with KPCA? (see Ma and Zabaras, 2011).
Is the Kalman update equivalent with the steepest descent
equation? (see Sarma and Chen, 2013).
Is it possible to adapt higher degree KPCA to the SubspaceEnKF framework? (see Sarma and Chen, 2013).
How do the presented assimilation methods scale to realistic
3D cases?
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Keywords
Water flooding
Channelized reservoir
State vector, facies
History matching
Ensemble Kalman Filter
Multi-point geostatistics
Adaptation methods
Parameterization
Feature space
Polynomial KPCA
Preimage problem
Ensemble collapse
Subspace EnKF
Training set clustering
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Cheat Slides
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Rock Properties
Porosity (%)
Permeability (mD)
flow effort
pore connectivity
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Computational Expense (1)
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Computational Expense (2)
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Normalization
1
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transform
: 0,1
+
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Y-channel Setup
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Ribbon Setup
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Exp2 Ensemble Means
2 subspaces 10 subspaces 50 subspaces
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Exp3 Production (1)
Prior 2 subspaces
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Exp3 Production (2)
10 subspaces 50 subspaces
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Exp3 50 subspaces members