Post on 12-Jan-2016
description
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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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Delft Center for Systems and Control
Contents
• Introduction • Estimating states and parameters - identification• Identifiability • Controllability and observability• Discussion
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Delft Center for Systems and Control
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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Delft Center for Systems and Control
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
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Delft Center for Systems and Control
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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Delft Center for Systems and Control
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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Delft Center for Systems and Control
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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Delft Center for Systems and Control
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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Delft Center for Systems and Control
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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Delft Center for Systems and Control
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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Delft Center for Systems and Control
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
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Delft Center for Systems and Control
Parameter estimation and system identification
Experimentdesign
Model SetData
ValidateModel
IdentificationCriterion
Construct model
prior knowledge /intended application
NOT OK
OK(Ljung, 1987)
Estimation
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Delft Center for Systems and Control
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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Delft Center for Systems and Control
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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Delft Center for Systems and Control
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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Delft Center for Systems and Control
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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Delft Center for Systems and Control
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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Delft Center for Systems and Control
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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Delft Center for Systems and Control
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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Delft Center for Systems and Control
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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Delft Center for Systems and Control
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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Delft Center for Systems and Control
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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Delft Center for Systems and Control
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
Controllability&Observability
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Delft Center for Systems and Control
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
Controllability&Observability
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Delft Center for Systems and Control
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.
Controllability&Observability
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Delft Center for Systems and Control
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)
Controllability&Observability
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Delft Center for Systems and Control
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
Controllability&Observability
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Delft Center for Systems and Control
Controllability/observability of saturations
time(k)
Pressure
Saturation
Controllable saturations
Observable saturations!!! Logarithmic scale
Calculated with empirical Gramians
Controllability&Observability
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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Delft Center for Systems and Control
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
Controllability&Observability
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Delft Center for Systems and Control
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)