P R E S E N T E D B Y
Sandia National Laboratories is a multimissionlaboratory managed and operated by National Technology & Engineering Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell International Inc., for the U.S. Department of
Energy’s National Nuclear Security Administration under contract DE-NA0003525.
Comparison of Probabilistic Sensitivity Analysis Methods Applied to a 3-D Reference Case Performance Assessment Dataset
from GDSA Framework
E m i l y S t e in , Lau r a S wi l e r , an d D av id S e v oug ian
SAND2018-12331 C
Dakota
•Flexible, extensible interface between simulation software and variety of system analysis methods• Uncertainty quantification (aleatory and epistemic)
• Sensitivity/variance analysis
• Deterministic/stochastic parameter estimation
• Design optimization
•C++ Software Toolkit
•Open source GNU LGPL
•Supports parallel computation
https://dakota.sandia.gov
Sensitivity Analysis
•Factor Prioritization: Identify the model inputs (factors) in which a reduction of uncertainty would most reduce the uncertainty in the model output; prioritize research.
•Factor Fixing: Identify model inputs that could be fixed or simplified without affecting model output; create a parsimonious, less computationally expensive, and possibly more defensible model.
•Variance Cutting: Reduce the variance (a measure of uncertainty) in simulation output to below a specified tolerance by identifying the smallest possible set of factors to fix.
•Factor Mapping: Identify regions of the input parameter space that lead to extreme or interesting results.
Saltelli et al. 2008
Sensitivity Analysis – Correlation and Regression
• SCC – Simple correlation coefficients measurelinear correlation.
• PCC – Partial correlation coefficients measure linear correlation with the effects of other variables removed.
• SRC – Standardized regression coefficients are calculated from stepwise multiple linear regression.
• Rank transformation improves correlation for monotonic but nonlinear relationships.
Sensitivity Analysis – Variance Decomposition
• Fraction of variance in the output due to the variance in an input.
• Sj – Main sensitivity index measures the main effect of an input variable (without variable interactions).
• Sij – Higher order sensitivity indices measure the effects of variable interactions (multiplicative).
• ST – Total sensitivity index measures the total effect of an input variable (including interactions).
• Model independent (theoretically).
• Computationally expensive.
Gaussian Process
•Predicts the most likely value of output variable 𝑦 at points on a response surface.
•Possible values for 𝑦 at any point are part of a normal (Gaussian) distribution.
•The variance of that distribution is smaller close to the training points (zero at the training points) and larger further away.
•𝑚(𝑑 + 2) evaluations gives Sj and ST
Polynomial Chaos Expansion
•Approximates values of the output variable using a series expansion comprised of orthogonal polynomials.
•Obtain Sj, Sij, …, ST from analytic functions of the coefficients.
•Number of terms: 𝑃 =𝑑+𝑝 !
𝑑!𝑝!
•Use compressive sensing to fit the coefficients.
Repository Performance Assessment
Natural Barrier SystemEngineered Barrier SystemWaste Source Term
•Coupled heat and fluid flow
•Waste package degradation
•Waste form dissolution
•Decay & ingrowth
•Solubility, sorption
•Advection, dispersion,
diffusion
•Biosphere
Biosphere
Natural Barrier System
Host Rock
Shales
Sandstone Aquiferoverburden
Limestone Aquifer
Lower Shales & Sandstones
Depth (m)
90
675
Perm. k
~500
930
1200
1e-19
1e-13
1e-13
1e-14
1e-20
1e-17
1e-15
repository
1e-20
siltstone
marine shale
marine shale 1e-19
Perry and Kelley 2017Low permeability ◆ High sorption ◆ Self-healing
Engineered Barrier System
12-PWR waste package
In-drift axial emplacement
30-m drift spacing
20-m center-to-center WP spacing
Bentonite/sand buffer
WP degradation
Temperature-dependent
Log-normal distribution
50% breached by 31 ky
Base Rate, R
Pro
babilit
y𝑅𝑒𝑓𝑓 = 𝑅 ∙ 𝑒
𝐶1
60℃−
1
𝑇(𝑥,𝑡)
Reff = canister degradation rate
Heat and Radionuclide Source Terms
Assumptions ➤ Source Terms
60 GWD/MT burn-up
4.73 wt% 235U enrichment
100 yrs Out of Reactor
18 Radionuclides241Am, 243Am, 238Pu, 240Pu, 242Pu, 237Np, 233U, 234U, 236U, 238U, 229Th, 230Th, 226Ra, 99Tc, 129I, 135Cs, 36Cl
Normalized
Canister
Thickness
101 102 103 105 106 107
Time [yr]
104
× 5.22 𝑀𝑇𝐻𝑀𝑊𝑃
1
10
102
103
104
10 102 103 104 105 106 107
Time [yr]
0.1Deca
y H
eat
[W/M
TH
M]
108
Initial WP heat = 3084 W
Model Domain
~6800 m
Pressure gradient -12 Pa/m
Half-symmetry domain
• 42 1035-m drifts
• 50 WPs / drift
• 2100 WPs
~7 million hexes ◆ 512 cores ◆ ~3 hours
129I Concentration – Deterministic
100,000 y
1,000,000 y
Direction of forced flow
Sampled Parameters
Parameter Distribution
Lower
Bound
Upper
Bound
Upper sandstone k (m2) Log uniform 10-15 10-13
Limestone k (m2) Log uniform 10-17 10-14
Lower sandstone k (m2) Log uniform 10-14 10-12
Buffer k (m2) Log uniform 10-20 10-16
DRZ k (m2) Log uniform 10-18 10-16
Mean waste package
degradation rate (1/yr)Log uniform 10-5.5 10-4.5
Shale porosity Uniform 0.1 0.25
Buffer Np Kd (m3/kg) Log uniform 0.1 702
Shale Np Kd (m3/kg) Log uniform 0.047 20
Parameter Description Range Units Distribution
rateSNF SNF Dissolution Rate 10-8 – 10-6 yr-1 log uniform
rateWPMean Waste Package
Degradation Rate 10-5.5 – 10-4.5 yr-1 log uniform
kSand Upper Sandstone k 10-15 – 10-13 m2 log uniform
kLime Limestone k 10-17 – 10-14 m2 log uniform
kLSand Lower Sandstone k 10-14 – 10-12 m2 log uniform
kBuffer Buffer k 10-20 – 10-16 m2 log uniform
kDRZ DRZ k 10-18 – 10-16 m2 log uniform
pShale Host Rock (Shale) 𝜙 0.1 – 0.25 - uniform
bNpKd Np Kd Buffer 0.1 – 702 m3kg-1 log uniform
sNpKd Np Kd Shale 0.047 – 20 m3kg-1 log uniform
129I Concentration – Uncertainty
0 m 1777 m 4267 m 6772 m
Scatter Plotssa
nd_obs1
sand_obs2
sand_obs3
Max 1
29I Concentr
ati
on (
mol/
L)
rateWP pShale rateSNF kSand kLime kLSand kBuffer kDRZ
Sensitivity Measures
Conclusions
•Shale Reference Case
• All SA methods identify pShale and kSand as the input variables having the greatest effect on max [129I].
• Given the current conceptual/numerical model, reduction of uncertainty in these input variables wouldmost reduce the uncertainty in the output variable.
• For pShale, kSand, and rateWP, the total sensitivity index is several times larger than the main sensitivityindex, indicating the importance of parameter interactions.
• The finding of near zero ST for kDRZ, kBuffer, kLime, and kLSand indicates that values of these inputvariables could be fixed without affecting uncertainty in max [129I].
•Next Steps
• Investigate methods and work flow that enable effective use of variance decomposition in repositoryperformance assessment where the computational expense of the simulations is high and the number ofuncertain inputs is potentially large.
References
Adams, B. M., M. S. Ebeida, M. S. Eldred, G. Geraci, J. D. Jakeman, K. A. Maupin, J. A. Monschke, J. A. Stephens, L. P. Swiler, D. M. Vigil, T.M. Wildey, et al. 2018. Dakota, A Multilevel Parallel Object-Oriented Framework for Design Optimization, Parameter Estimation, Uncertainty Quantification,and Sensitivity Analysis: Version 6.8 User’s Manual. SAND2014-4253. Sandia National Laboratories, Albuquerque, NM.
Helton, J. C. and F. J. Davis. (2000). Sampling-Based Methods. In A. Saltelli, K. Chan, & E. M. Scott (Eds.), Sensitivity Analysis (pp. 102-153).Chichester, U.K.: Wiley.
Lichtner, P. C., G. E. Hammond, C. Lu, S. Karra, G. Bisht, B. Andre, R. T. Mills, J. Kumar, and J. M. Frederick, PFLOTRAN User Manual,http://www.documentation.pflotran.org, 2018.
Mariner, P. E., E. R. Stein, S. D. Sevougian, L. J. Cunningham, J. M. Frederick, G. E. Hammond, T. S. Lowry, S. Jordan, and E. Basurto 2018. Advances in Geologic Disposal Safety Assessment and an Unsaturated Alluvium Reference Case. SFWD-SFWST-2018-000509; SAND2018-11858R. Sandia National Laboratories, Albuquerque, NM.
Saltelli, A., P. Annoni, I. Azzini, F. Campolongo, M. Ratto, and S. Tarantola 2010. "Variance based sensitivity analysis of model output. Design and estimator for the total sensitivity index". Computer Physics Communications, 181(2), 259-270. doi: 10.1016/j.cpc.2009.09.018
Saltelli, A., M. Ratto, T. Andres, F. Campolongo, J. Cariboni, D. Gatelli, M. Saisana, and S. Tarantola 2008. Global Sensitivity Analysis The Primer. Chichester, U.K.: Wiley.
Sudret, B. 2008. "Global sensitivity analysis using polynomial chaos expansions". Reliability Engineering & System Safety, 93(7), 964-979. doi: 10.1016/i.ress.2007.04.002
Weirs, V. G., J. R. Kamm, L. P. Swiler, S. Tarantola, M. Ratto, B. M. Adams, W. J. Rider, and M. S. Eldred 2012. "Sensitivity analysis techniques applied to a system of hyperbolic conservation laws". Reliability Engineering & System Safety, 107, 157-170. doi: 10.1016/j.ress.2011.12.008