Report

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 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 www.simtech.uni-stuttgart.de 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 ▪ ▪ ▪ ▪ ▪ ▪ 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): (, ; ) ≈ orthon. basis: Ω =0 , () ≡ (, ; ) () = projection: , = analysis: E () ≈ E () Ω (, ; ) () www.simtech.uni-stuttgart.de www.simtech.uni-stuttgart.de ▪ See real-time PCE demo by SimTech www.simtech.uni-stuttgart.de 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 www.simtech.uni-stuttgart.de 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! www.simtech.uni-stuttgart.de 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 www.simtech.uni-stuttgart.de 1 10 5 10 10 Size of data sample 6 10 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 ▪ Bayes Theorem: ∝ () ▪ …where… : : : probability density function parameters data : prior PDF (the initial belief) : likelihood (data fit for given ) : 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 www.simtech.uni-stuttgart.de ▪ Bayes Theorem: ∝ () ▪ …where… = 0, 2 = − () ▪ Use the PCE for residuals : () ≈ () ▪ Use in bootstrap filter to avoid direct model calls www.simtech.uni-stuttgart.de ▪ 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? www.simtech.uni-stuttgart.de 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 www.simtech.uni-stuttgart.de ▪ 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” www.simtech.uni-stuttgart.de ▪ Simulation problem implemented in ▪ One single run of this model requires 1-2 days of CPU time using a computational cluster with 40 CPU. www.simtech.uni-stuttgart.de 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 www.simtech.uni-stuttgart.de Strong offset of Prior Prior is simply wrong Expansion is outside of area of interest, resulting posterior would be inaccurate www.simtech.uni-stuttgart.de 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 www.simtech.uni-stuttgart.de