Risk Assessment for CO 2 Sequestration- Uncertainty Quantification based on Surrogate Models of Detailed Simulations

Transcription

1 Risk Assessment for CO 2 Sequestration- Uncertainty Quantification based on Surrogate Models of Detailed Simulations Yan Zhang Advisor: Nick Sahinidis Department of Chemical Engineering Carnegie Mellon University 3/11/2012

2 Modeling CO 2 Underground Plume Injection well Aquitard/caprock CO 2 plume Aquifer/reservoir Mass balance Darcy s law

3 Uncertainty Analysis Options v Numerical model + Monte Carlo simulation v Stochastic response surface methods Surrogate model Approximate the output of the original model by polynomial functions of the uncertain parameters Monte Carlo simulation with surrogate model The primary goal of this work is to provide an assessment of surrogate-based uncertainty analysis for CO 2 injection into a saline aquifer

4 Polynomial Chaos Expansion (PCE) v A physical model v Random è random model response v Assuming finite variance, chaos representation

5 Orthogonal Polynomial Basis v Orthogonal polynomials for a single random v Multivariate polynomial basis for Independent Dependent: Nataf Transformation correlated variables correlated normals correlated standard normals uncorrelated standard normals

6 Coefficient Estimation v Regression Number of points to choose: M d = 1 d = 2 d =

7 Stepwise PCE Approximation Select X and collect y Start: degree d = 0, set of basis A = {0} Q 2 =Q 2 tgt? No d = d+1 Yes Forward selection: for all degree-d basis functions, add one term to A and retain the term if Q 2 increases Backward elimination: for all lower degree (< d) basis functions, remove one term from A if Q 2 does not decrease much No d = d max? Yes S. Weisberg, Applied Linear Regression, 3 rd edition,

8 Assessment of PCE approximation v Empirical error R 2 1 as N t increases è overfitting v Leave-one-out cross validation (N p -1) points Metamodel N p points 1 point Predicted residual

9 Assessment of PCE approximation v Empirical error R 2 1 as N t increases è overfitting v Leave-one-out cross validation N p points (N p -1) points 1 point 8

10 Case Study v CO 2 injection into a deep saline aquifer v Simulated using TOUGH2 developed by LBNL since m Upper Shale Cap (impermeable) Pressure at well Temperature 110 bar m Symmetry Plane (no flux) 30 m 30 m 30 m 30 m 22 m Shale layers Horizontal injection well Lower Shale (impermeabile) Hydrostatic pressure Salinity 3.2 wt.-% NaCl Injection rate kg/s Output: Gas saturation (space, time) Pressure (space, time) CO 2 mass distribution 6000 m ECO2N manual,

11 Specify Input Random Variables v National Petroleum Council Public Database v Non-standard marginal distributions NPC database v Strong correlation v Transform to normal variables 10

12 Derive Orthogonal Polynomial Basis v Hermite polynomials for a standard normal random variable v Basis for the two-dimensional independent input vector (ξ 1, ξ 2 ) 11

13 Build PCE Model v Select an experimental design ξ (Latin Hypercube Sampling) ξ 1 ξ 2 12

14 Build PCE Model v Select an experimental design ξ (Latin Hypercube Sampling) ξ 1 ξ 2 v Reverse of Nataf transformation v Model output evaluations v Stepwise regression for coefficients and degree 12

15 PCE Model Example simulations Q 2 = nd order expansion 13

16 PCE Model Example simulations Q 2 = th order expansion 14

17 Use PCEs for Uncertainty Analysis Input: Randomly sampled from probability distribution function of independent parameters Monte Carlo simulation Output: Statistical analysis Surrogate Model Sample # Generated random samples x 1 x 2 x n Permeability Porosity Outputs evaluated using PCEs y 1 y 2 y m Mass Pressure 15

18 Correlated Sampling Using Copulas 16

19 MC Simulation Result 1 17

20 MC Simulation Result 2 18

21 Pressure Map with TOUGH2 19

22 Pressure Map with PCE Model Caprock 20

23 Maximum Caprock Pressure p frac Probability of overpressure = 0.01 Pa 21

24 Gas Saturation Map with TOUGH2 22

25 Gas Saturation Map with PCE Model 23

26 Optimal Injection under Uncertainty Nonconvex stochastic NLP solved with BARON using scenario-based approach Optimal injection rates under various realizations of uncertain parameters 24

27 Conclusions v Polynomial chaos expansion model was iteratively built with stepwise regression for correlated uncertain parameters v Results obtained with PCE-based surrogate model match well the results obtained with TOUGH2 v Developed a nonconvex stochastic NLP model for finding optimal injection rate under model uncertainty 25

28 Disclaimer