model-based control and optimization in reservoir engineering

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Delft Center for Systems and Control

Model-based control and optimization in reservoir engineering

Paul M.J. Van den Hof

Delft Center for Systems and ControlDelft University of Technology

SPE Workshop Closed-loop Reservoir Management, Bruges, Belgium, 23-26 June 2008

Control+

-

objectives production

Contributors:Okko Bosgra, Jan Dirk Jansen, Maarten Zandvliet, Jorn Van Doren, Gijs van Essen, Sippe Douma

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Contents

• Introduction • Estimating states and parameters - identification• Identifiability • Controllability and observability• Discussion

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Systems and Control

• Successes of advanced control are widespread - from aerospace to vehicles, robots, and chemical plants

Product

Fines

FeedHX

Malvern

Helos Hot water

Dissolution vessel

Opus

Condensedvapor

Dilution

HXCooling water

Annular zone

Draft tube

Skirt baffle

Product

Fines

FeedHX

Malvern

Helos Hot water

Dissolution vessel

Opus

Condensedvapor

Dilution

HXCooling water

Annular zone

Draft tube

Skirt baffle

Product

Fines

FeedHX

Malvern

Helos Hot water

Dissolution vessel

Opus

Condensedvapor

Dilution

HXCooling water

Annular zone

Draft tube

Skirt baffle

• Effective use of dynamic models (and their limitations) is of central importance

Introduction

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The reservoir problem:

• Challenging and attractive!• Poorly known models• Highly nonlinear behaviour• One-shot (batch) type of process• High levels of uncertainty in information• Large scale (manipulated/measured variables and more)• High computational load• Slow / low sampling rates• Options for learning/adaptation

Introduction

5


Introduction

Here we will focus on issues around

model construction and estimation /

data assimilation /history matchingwith reference to tools from systems

and control theory

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Estimating states and parameters

reservoir

disturbances

valvesettings

actualflow rates,seismics...

management,storage,

transport

economicperformancecriteria

optimization

reservoirmodel

reservoirmodel

gain

+-

update

stateestimation

Estimation

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Two roles of reservoir models

• Reservoir model used for two distinct tasks: state estimation and prediction.

• Distinct role of parameters: essential model properties states: initial conditions for predictions

past

Estimation

present future

Predictionreservoir

disturbances

valvesettings

actualflow rates,seismics...

management,storage,

transport

economicperformancecriteria

optimization

reservoirmodel

reservoirmodel

gain

+-

update

disturbance + stateestimation

Estimation

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Parameter and state estimation in data assimilation

Model-based state estimation:

past data

initial state

state update

saturations, pressurese.g. permeabilities

Options: Ensemble Kalman Filter (EnKF) (Evensen, 2006)

Estimation

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Parameter and state estimation in data assimilation

If parameters are unknown, they can be estimated byincorporating them into the state vector:

past data

initial state/parameter

state/parameter update

Can everything that you do not know be estimated?

Estimation

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With respect to large-scale parameter vector:

• Singular parameter-update matrix (data not sufficiently informative) • Parameters are updated only in directions where data contains information (in the best case)

Result and reliability is crucially dependent on initial (prior) model

Matching the history may add/contribute little to the priors

Problem of identifiability

Estimation

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With respect to large-scale state vector:

• Similar mechanisms • States are updated only in directions where data

contains information Only that part of the state space that can be appropriatelyobserved and controlled is relevant for the optimization

Reservoir models typically live in low-dimensional spaces

Problems of controllability and observability

Estimation

12


Parameter estimation and system identification

Experimentdesign

Model SetData

ValidateModel

IdentificationCriterion

Construct model

prior knowledge /intended application

NOT OK

OK(Ljung, 1987)

Estimation

13


Parameter estimation in identification

Parameter estimation by applying LS/ML criterion to (linearized) model prediction errors

e.g. areparameters that describepermeabilities

G0 (q)+++u yv

H0 (q)

e

G(q,)+-

H(q,)-1

(t)

presumed datagenerating system

predictor model

Estimation

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Parameter estimation in identification

Particular issues/tools:

G0 (q)+++u yv

H0 (q)

e

G(q,)+-

H(q,)-1

(t)

presumed datagenerating system

predictor model

• Experiment design (u has to excite dynamics)• Best model fit is not the goal• Validation should prevent overfit of parameters• Mature theory and tools for linear models

Estimation

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Starting from (linearized) state space form:

the model dynamics is represented in its i/o transfer function form:

with the shift operator:

Structural identifiability

Identifiability

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Principle problem of physical model structures

Different might lead to the same dynamic models

This points to a lack of structural identifiability

There does not exist experimental data that can solve this!

Solutions:• Apply regularization (additional penalty term on criterion) to enforce a unique solution (does not guarantee a sensible solution for )• Find (identifiable) parametrization of reduced dimension

Identifiability

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Structural identifiability (cont’d)

A model structure is locally (i/o) identifiable at if for anytwo parameters in the neighbourhood of it holds that

At a particular point the identifiable subspace of can be computed! This leads to a map

with

Van Doren et al., Proc. IFAC World Congress, 2008, to appear.

Identifiability

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Analyse (svd) the matrix

with

Limitation: only local linearized situation can be handled

Tool:

and Markov parameters

The svd gives the directions in parameter space that havethe greatest influence on the i/o dynamics (columns of )

Identifiability

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Identifiable directions

• Consider a single-phase, 21x21 grid block model with 5 wells.

• Directions in the permeability space that are best identifiable from pressure measurements:

Identifiability

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Analysis extends to nonlinear (two-phase) situation,by calculating the svd of the Hessian of the cost function

Result:Insight into information content of data, with respect to parameters to be estimated

Even if structural identifiability is OK, the input has to excite the dynamics in order to make parameter visible in the data

Analysis can be applied to geological parametrizationVan Doren et al., ECMOR, 2008

Identifiability

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Physical parameters (permeabilities) determine predictive quality but one parameter per grid block leads to excessive over-parametrization (hard to validate)

Parameter estimation:

Identifiability

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Controllability and observability

• Controllability:• Can we (dynamically) steer all pressures and

saturations by manipulating the inputs?• Observability:

• Do all states (dynamically) appear in the observed output?

Controllability&Observability

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In the linear(ized) case:

Controllability:

Observability:

The controllable and observable part of the state spacedetermines that part of the states that affects the i/o mapping of the system


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Controllability and observabilityBesides a yes/no answer, the notions can be quantified:

Minimum input energy to reach a state is

Maximum output energy obtained from state is

Small s.v.’s of Grammians refer to state directionsthat are poorly controllable/observable


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Controllability and observability

Eigenvalues of Grammian product determine the minimumnumber of states required to describe the i/o dynamics

Reservoir models live in low-dimensional state spaceZandvliet et al., Comput. Geosciences, submitted, 2008.


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Quantifying controllability and observability

• In which area’s in the reservoir are the states more controllable and observable than others?

• Notions can be extended to nonlinear situation• Methodology:

1. Linearize along current state2. Calculate LTV controllability and

observability matrices 3. Visualize dominant directions with SVD

(for pressures and saturations separately)


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Controllability/observability of pressures

time(k)

Pressure(k)

Saturation(k)

Controllable pressures!!! Logarithmic scale

Observable pressures!!! Logarithmic scale

Calculated with LTV controllability and observability matrices


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Controllability/observability of saturations

time(k)

Pressure

Saturation

Controllable saturations

Observable saturations!!! Logarithmic scale

Calculated with empirical Gramians


p(1)

s(1)

Cemp,s

Oemp,s

p(61)

s(61)

Cemp,s

Oemp,s

p(121)

s(121)

Cemp,s

Oemp,s

p(181)

s(181)

Cemp,s

Oemp,s

p(227)

s(227)

Cemp,s

Oemp,s

0

0.5

1

-0.4

-0.2

0

-30

-20

-100

9.2

9.4

x 106

0

0.5

1

-0.2

-0.1

0

-30

-20

-10

0

9.2

9.4

x 106

0

0.5

1

-0.15

-0.1-0.050

-30

-20

-100

9.2

9.4

x 106

0

0.5

1

-30

-20

-100

1.1

1.15x 10

7

0

0.5

1

-30

-20

-10

0

-0.15-0.1-0.05

-0.2-0.15-0.1-0.05

1

1

1x 10

7

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Observations

• Most observable phenomena occur in the direct vicinity of the wells

• Saturations are most controllable around the oil-water front

• This would motivate a representation of the state space (e.g. in terms of basis functions), with emphasis on the oil-water front….

• Notions can be instrumental in (a) determining the control-relevant model aspects(b) optimal well placement

Yortsos et al., 2006


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Discussion

Discussion

• Basic methods and tools have been set, but there remain important and challenging questions

• Complexity reduction of the physical models: limit attention to (a) what is known or verifiable by data(b) what is relevant for the ultimate optimization.... (control-relevant modelling)

• Can we increase information content in the data?(learning / dual control)

model-based control and optimization in reservoir engineering

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