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

Geologic Disposal Safety Assessment (GDSA) Framework

https://pa.sandia.gov

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