www.simtech.uni-stuttgart.de
W. Nowak, S. Oladyhskin, M. Sinsbeck
Stochastic Modelling of Hydrosystems
Chair for Hydromechanics and Modelling of Hydrosystems
Institute for Modelling Hydraulic and Environmental Systems
University of Stuttgart
www.simtech.uni-stuttgart.de
▪ Engineering & mangement in the subsurface


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Nuclear waste
Geothermal energy
Groundwater
Remediation…
▪ Uncertainty!
 UQ, PRA,…
 RD, OD,…
 BU, DA,…
nuclear
waste
deposits?
contaminations,
legacies
CO2 injection
& storage?
geothermal
energy
CO2
injection
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www.simtech.uni-stuttgart.de
▪
▪
▪
▪
?
nuclear
?
?
waste
deposits?
open
dynamic
complex
heterogeneous,
unique
▪ limited
observability
▪ Stochastic
contaminations,
legacies description!
▪ Uncertainty
CO injection
is a nuisance!
& storage?
2
geothermal
energy
?
www.simtech.uni-stuttgart.de
▪ “belief about system parameters” encoded as PDF
▪ When new data become available:
}
Bayesian
updating
▪ Result: updated belief with less uncertainty
▪ Special cases: Least squares, EnKF, Particle Filter,…
www.simtech.uni-stuttgart.de
▪ Goal: Bayesian updating for large CO2 models!
▪ Problem 1: this will be way too expensive!
 Idea: replace model by faster surrogate
 Here: will “Polynomial chaos expansion” (PCE) help?
▪ Problem 2: data far off from expectations!
 How can a model surrogate be constructed,
if we do not even know in what parameter range?
 Here: can we make the PCE work in this situation?
www.simtech.uni-stuttgart.de
▪
▪
▪
▪
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Goals, problems & questions
PCE as a surrogate model
Bayesian updating with PCE
Iterative PCE/BU
Application to CO2 pilot injection site
Conclusions
www.simtech.uni-stuttgart.de
Idea:
▪ Project model‘s dependence on parameters
onto a polynomial basis in the parameter space
▪ Use resulting polynomial as surrogate model
PCE (order d): 𝑀(𝐱, 𝑡; 𝛏) ≈
orthon. basis:
Ω
𝑃
𝑘
𝑑
𝑗=0 𝑐𝑗
𝐱, 𝑡 𝑃
𝑗
(𝛏) ≡ 𝑀(𝐱, 𝑡; 𝛏)
𝑃 𝑙 𝑓(𝛏)𝑑𝛏 = 𝛿𝑘𝑙
projection:
𝑐𝑗 𝐱, 𝑡 =
analysis:
E 𝑔(𝑀) ≈ E 𝑔(𝑀)
Ω
𝑀(𝐱, 𝑡; 𝛏)𝑃
𝑗
𝑓(𝛏)𝑑𝛏
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▪ See real-time PCE demo by SimTech
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Two relevant properties:
1. PDF-weighted fitting
 poor response surface in low-probability regions
(compare: extrapolation)
2. Polynomial approximations can oscillate
(Gibbs and Runge phenomena)
 poor can mean REALLY poor
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5 different experts:
▪ log-normal PDF
 (moment matching)
▪ beta PDF
 (ML fitting)
▪ Log-transform, Gauss PDF
 (visual fit)
▪ Box-Cox, log-normal PDF
 (visual fit)
▪ log-normal PDF
 (visual fit)
THIS is subjectivity!
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numerics
physics
parameters
subjectivity
Oladyshkin, Class, Helmig, Nowak:
A concept for data-driven uncertainty quantification and its
application to carbon dioxide
storage in geological formations.
Advances in Water Resources
34, 1508–1518, 2011
(confidence in prior PDF → )
0
10
-1
10
-2
-4
10
2
10
3
10
4
d=4
d=3
d=4
d=5
d=3
-3
10
d=2
d=1
d=2
10
d=1
Error of
of Variance
Variance
Error
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1
10
5
10
10
Size of data sample
6
10
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▪
▪
▪
▪
▪
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Goals, problems & questions
PCE as a surrogate model
Bayesian updating with PCE
Iterative PCE/BU
Application to CO2 pilot injection site
Conclusions
www.simtech.uni-stuttgart.de
▪ Bayes Theorem:
𝑓 𝑝 𝑑 ∝ 𝑓 𝑑 𝑝 𝑓(𝑝)
▪ …where…
𝑓
𝑝:
𝑑:
𝑓 𝑝
𝑓 𝑑
𝑓 𝑝
:
probability density function
parameters
data
: prior PDF (the initial belief)
𝑝 : likelihood (data fit for given 𝑝)
𝑑 : posterior PDF (the updated belief)
www.simtech.uni-stuttgart.de
▪ Idea: represent posterior by a MC sample
▪ Approach:
 Sample at first from prior
 Reflect likelihood through sample weights
 All statistics become weighted statistics
▪ Advantage: direct implementation of Bayes’ law
▪ Problem: high-weight samples by blind trial and error
(curse of dimension, filter collapse)
 Alternatives such as MCMC,… are available
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▪ Bayes Theorem:
𝑓 𝑝 𝑑 ∝ 𝑓 𝑑 𝑝 𝑓(𝑝)
▪ …where…
𝑓 𝑑 𝑝 = 𝑁𝜀 0, 𝜎𝜀 2
𝜀 = 𝑑𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑 − 𝑑𝑠𝑖𝑚 (𝑝)
▪ Use the PCE for residuals 𝜀:
𝑑𝑠𝑖𝑚 (𝑝) ≈ 𝑑𝑠𝑖𝑚 (𝑝)
▪ Use in bootstrap filter to avoid direct model calls
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▪ What if data fall out of the prior?
 Monitored pressures are outside the
uncertainty intervals predicted by the prior?
 Formation porosity/permeability was poorly estimated?
▪ Then we use PCE outside area of good match:
 Extrapolation,
 Oscillation…
▪ Approximate posterior is simply wrong!
www.simtech.uni-stuttgart.de
▪
▪
▪
▪
▪
▪
Goals, problems & questions
PCE as a surrogate model
Bayesian updating with PCE
Iterative PCE/BU
Application to CO2 pilot injection site
Conclusions
www.simtech.uni-stuttgart.de
▪ Task: solve a least-squares fitting problem
▪ Principle:
 use pseudo-2nd-order Taylor expansion
 Go to optimum of corresponding paraboloid
 Iterate until no improvement can be found
▪ Difference to our problem:
 Taylor expansion (local) instead of PCE (global)
 Searching for optimum instead of searching a PDF
▪ Similarity:
 Use a low-order approximation to guide some search
 So what about successive PCE?
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Parameter 2
POSTERIOR
STATE 1
PRIOR
STATE
POSTERIOR
STATE N
Parameter 1
▪ Constantly update
the PDF for „fitting“
the PCE
▪ Add new integration
points for PCE
▪ Adequately increase
approximation order
of the PCE on the go
[Oladyshkin, S., Schroeder P., Class, H., Nowak, W., Chaos expansion based Bootstrap filter
to calibrate CO2 injection models. Energy Procedia, Elsevier, N.40, P. 398-407, 2013]
www.simtech.uni-stuttgart.de
▪ Nested integration
(recycling points between
several iterations, optimal
extension of old point cloud
under current PDF estimate)
▪ Arbitrary PDF shapes
and dependent variables
(PDF shapes become arbitrary
and non-linear dependence
emerges during Bayesian
updating!)
Sinsbeck and Nowak: An optimal sampling rule for non-intrusive polynomial chaos expansions of expensive models.
Submitted to Int. J. Unc. Quant. (2013)
www.simtech.uni-stuttgart.de
▪
▪
▪
▪
▪
▪
Goals, problems & questions
PCE as a surrogate model
Bayesian updating with PCE
Iterative PCE/BU
Application to CO2 pilot injection site
Conclusions
www.simtech.uni-stuttgart.de
Well-known CO2
pilot site in Europe
▪ Task: fit model to reproduce data time series
(pressure at monitoring well during 1-year injection)
and provide uncertainty estimates after fitting
▪ Problem: no-one obtained acceptable fit up to now
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▪ Geological model: from geophysical investigation
▪ Parameterization: permeability multipliers for three layers
(keeping internal contrasts from geophysics)
▪ Subjective prior: each multiplier: mean 1, lognormal,…
region “sand”
region “flood”
region “rest”
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▪ Simulation problem implemented in
▪ One single run of this model requires 1-2 days of CPU
time using a computational cluster with 40 CPU.
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Option A
Run DuMuX ~1.000.000 times
and relax for next 5.000 years
Option B
Try the proposed ideas…
Feasibility
Bayesian
Updating
+
=
Chaos
Expansion
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Strong offset of Prior
Prior is simply wrong
Expansion is outside of area of interest,
resulting posterior would be inaccurate
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Iterative runs
Initial runs
Total number
of runs: 15
[Oladyshkin, S., Schroeder P., Class, H., Nowak, W., Chaos expansion based Bootstrap filter
to calibrate CO2 injection models. Energy Procedia, Elsevier, N.40, P. 398-407, 2013]
www.simtech.uni-stuttgart.de
www.simtech.uni-stuttgart.de
region “rest”
region “flood”
region “sand”
www.simtech.uni-stuttgart.de
▪
▪
▪
▪
▪
▪
Goals, problems & questions
PCE as a surrogate model
Bayesian updating with PCE
Iterative PCE/BU
Application to CO2 pilot injection site
Conclusions
www.simtech.uni-stuttgart.de
www.simtech.uni-stuttgart.de
Funding sources
▪ DFG
▪ EXC 310/1 (SimTech)
▪ IRTG 1398 (NUPUS)
▪ IRTG 1829 (Hydromod)
▪ Volkswagen Foundation
▪ State of Baden-Württemberg
▪ Zweckverband
Landeswasserversorgung / DVWG
3
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