ensemble-based history matching for channelized petroleum reservoirs (slides)

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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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 Means



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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



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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

ensemble-based history matching for channelized petroleum reservoirs (slides)

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