In order to evaluate the uncertainties in prediction of lattice-averaged parameters, input data of core neutronics codes, Exercise I-2 of the OECD benchmark for uncertainty anal-ysis in modeling (UAM) was proposed. This work aims to perform a sensitivity/uncertainty analysis of the BWR configurations defined in the benchmark for the purpose of Exercise I-2. Criticality calculations are done for a 7x7 BWR fresh fuel assembly at HFP in four configurations: single unrodded fuel assembly, rodded fuel assembly, assembly/reflector and assembly in a color-set. The SCALE6.1 code package is used to propagate cross section covariance data through lattice physics calculations to both k-effective and two-group assembly-homogenized cross sec-tions uncertainties. Computed sensitivities and uncertainties for all configurations are analyzed and compared. It was found that uncertainties are very similar for the four test-problems, showing that the influence of the assembly environment on uncertainty prediction is very small

Cabellos de Francisco, Oscar Luis

García Herranz, Nuria

Garrido, J.

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Sensitivity/Uncertainty Analysis for BWR Configurations of Exercise I-2 of UAM Benchmark

Servicio de Coordinación de Bibliotecas de la Universidad Politécnica de Madrid

PHYSOR 2014 – The Role of Reactor Physics Toward a Sustainable Future 
The Westin Miyako, Kyoto, Japan, September 28 – October 3, 2014, on CD-ROM (2014) 
 
SENSITIVITY/UNCERTAINTY ANALYSIS FOR BWR CONFIGURA-
TIONS OF EXERCISE I-2 OF UAM BENCHMARK 
 
N. García-Herranz*
nuria.garcia.herranz@upm.es
, J. Garrido, O. Cabellos  
Department of Nuclear Engineering 
Universidad Politécnica de Madrid, Spain 
 
oscar.cabellos@upm.es 
 
 
ABSTRACT 
 
In order to evaluate the uncertainties in prediction of lattice-averaged parameters, input 
data of core neutronics codes, Exercise I-2 of the OECD benchmark for uncertainty anal-
ysis in modeling (UAM) was proposed.  
 
This work aims to perform a sensitivity/uncertainty analysis of the BWR configurations 
defined in the benchmark for the purpose of Exercise I-2. Criticality calculations are done 
for a 7x7 BWR fresh fuel assembly at HFP in four configurations: single unrodded fuel 
assembly, rodded fuel assembly, assembly/reflector and assembly in a color-set. The 
SCALE6.1 code package is used to propagate cross section covariance data through lattice 
physics calculations to both k-effective and two-group assembly-homogenized cross sec-
tions uncertainties. Computed sensitivities and uncertainties for all configurations are 
analyzed and compared. It was found that uncertainties are very similar for the four 
test-problems, showing that the influence of the assembly environment on uncertainty 
prediction is very small.     
 
Key Words: lattice physics uncertainty propagation, sensitivity/uncertainty analysis, 
few-group homogenized cross section uncertainties  
 
 
1. INTRODUCTION 
 
The OECD benchmark for Uncertainty Analysis in Modeling (UAM) for design, operation and 
safety analysis of LWRs [1] was established in order to quantify the uncertainties at all stages of 
coupled reactor physics/thermal-hydraulics LWR calculations. It is based on the introduction of 9 
steps or Exercises, carried out in 3 phases: neutronics phase, core phase and system phase, fol-
lowing the calculation scheme for LWR design and safety analysis established in the nuclear 
industry.  
 
Phase I is focused on understanding uncertainties in prediction of core parameters associated with 
stand-alone core neutronics simulations. In that phase, the main source of uncertainty considered 
is the cross section uncertainties. It consists of 3 Exercises: 
 
• Exercise I-1 (Cell Physics) propagates the uncertainties starting from evaluated nuclear data 
libraries to multi-group microscopic cross-section libraries used as input by lattice physics 
codes. 
• Exercise I-2 (Lattice Physics) propagates mainly uncertainties coming from Exercise I-1 
                                                   
* Corresponding author. E-mail address: nuria.garcia.herranz@upm.es 
García-Herranz N., Garrido J., Cabellos O. 
2 / 8 PHYSOR 2014 – The Role of Reactor Physics Toward a Sustainable Future 
Kyoto, Japan, September 28 – October 3, 2014 
 
through lattice physics calculations to k-inf and few-group homogenized parameters, used as 
input data by core neutronics codes.  
• Exercise I-3 (Core Physics) propagates uncertainties coming from Exercise I-2 to core 
stand-alone neutronics parameters.  
 
Three representative designs, corresponding to a PWR, a BWR and a VVER-1000, are defined for 
the purposes of those Exercises. 
 
In the present work, the uncertainty analysis corresponding to Exercise I-2 for the BWR design is 
performed using SCALE6.1 code system [2]. The calculation methodology, the models developed 
for the different BWR configurations proposed in the benchmark and the modeling assumptions 
are presented. Then, a sensitivity/uncertainty analysis is performed. Computed sensitivities and 
uncertainties in both k-effective and homogenized two-group cross sections due to cross section 
covariance data are compared and the influence of the assembly environment is assessed.  
 
 
2. DESCRIPTION OF BWR TEST PROBLEMS 
 
Cases under study are four different configurations of a BWR fuel assembly representative of the 
initial loading of Peach Bottom-2 (PB-2) plant: single fuel assembly, rodded fuel assembly, 1-D 
assembly/reflector and 2-D assembly color set or mini-core. Specifications can be found in [1, 
Section 3.2].  
 
The fresh fuel assembly is a 7x7 lattice array with 45 UO2 rods with enrichments ranging from 
1.33 w/o to 2.93 w/o and 4 UO2-Gd2O3 rods. The assembly has diagonal symmetry. Criticality 
calculations are carried out at Hot Full Power (HFP) conditions, as well as uncertainty quantifi-
cation due to cross section uncertainties. 
 
 
3. METHODOLOGY 
 
The calculation methodology consisted of employing the TRITON multipurpose sequence of 
SCALE, version 6.1.2, to perform coupled cross-section processing and transport calculations 
using the 2-D discrete-ordinates NEWT code, and sensitivity/uncertainty analysis using TSU-
NAMI-2D.  
 
• All calculations were performed using the SCALE 238-group cross section library from 
ENDF/B-VII.0 (v7-238). The covariance library was 44GROUPCOV.  
• For cross-section processing, the latticecell treatment in CENTRM was used for the UO2 fuel 
rods. Accurate Dancoff factors were calculated using de MCDancoff module. Rods with the 
same enrichment and similar Dancoff factors were lumped to reduce the self-shielding 
processing effort. For Gadolinium-bearing fuel rods, a multiregion treatment with five 
equal-area rings to capture radial depletion of gadolinium was used.  
• For the NEWT transport calculations, the order of SN was set to 10. All fuel mixtures and 
structural materials used P1 scattering, while all moderator mixtures used P3 scattering. The 
recommended 4x4 grid mesh was used for the square-pitched units. Convergence criteria for 
the eigenvalue were set at 1.0E-5. Coarse-mesh finite-difference acceleration (CMFD) option 
was employed on the global grid, in particular the original method for rectangular-domain 
S/U ANALYSIS FOR BWR CONFIGURATIONS OF EXERCISE I-2 OF UAM BENCHMARK 
PHYSOR 2014 – The Role of Reactor Physics Toward a Sustainable Future 
Kyoto, Japan, September 28 – October 3, 2014 3 / 8 
 
configurations (cmfd=rect). Moreover, the second-level two-group CMFD accelerator was 
enabled (cmfd2g=yes).  
• For S/U calculations, TSUNAMI-2D sequence solved forward and adjoint transport problems 
using NEWT. Cross-section processing used the modules by default (BONAMIST and 
CENTRM with ENDF/B-VII.0). Then, the SAMS module was invoked to calculate the sen-
sitivity coefficients in 238 energy groups. Since the covariance matrix in SCALE is given in 
44 energy groups, the sensitivity profiles were collapsed inside SAMS and uncertainty quan-
tification carried out using “sandwich formula”. 
 
 
4. SCALE MODELING OF EXERCISE I-2/ BWR 
 
4.1. Unrodded Fuel Assembly (FA) Model 
 
The 2-D single assembly with reflective boundary conditions (standard case used for cross-section 
generation in LWR analysis) was modeled. Inputs were prepared using the geometry, materials and 
temperatures at HFP conditions included in the specifications [1] (40% void fraction for the 
moderator in-channel). For the analysis using SCALE, the whole fuel assembly was modeled, as 
plotted in Figure 1.a, where different colors identify different materials modeled in the configu-
ration.  
Figure 1.b presents the Dancoff factors computed with MCDancoff as well as the SCALE-default 
value. Wrong Dancoff factors were obtained when the origin of the global unit was placed in a 
corner of the assembly whereas right values were computed placing the origin in the center, as 
shown in Figure 1.b.  
 
   
 
Figure 1. TRITON BWR assembly model and Dancoff factors map 
 
4.2. Rodded Fuel Assembly Model 
 
In order to model the controlled assembly (Figure 2), a cruciform control rod blade was included. 
Dancoff factors were not re-computed, and the ones calculated for the unrodded assembly were 
used instead.  
García-Herranz N., Garrido J., Cabellos O. 
4 / 8 PHYSOR 2014 – The Role of Reactor Physics Toward a Sustainable Future 
Kyoto, Japan, September 28 – October 3, 2014 
 
     
 
Figure 2. TRITON rodded BWR assembly model and detail of the control rod blade. 
 
 
4.3. 1-D Assembly/Reflector Model   
 
A 1-D assembly/reflector configuration was modeled, with reflective boundary conditions on the 
left boundary and vacuum boundary condition on the right boundary. This is the standard model 
used for reflector cross-section generation in LWR analysis.  
 
 
 
Figure 3. TRITON BWR radial reflector model 
 
4.4. 2-D Color-set (mini-core) Model  
 
A mini-core problem involving unrodded and rodded assemblies was model, with reflective 
boundary conditions.  
 
S/U ANALYSIS FOR BWR CONFIGURATIONS OF EXERCISE I-2 OF UAM BENCHMARK 
PHYSOR 2014 – The Role of Reactor Physics Toward a Sustainable Future 
Kyoto, Japan, September 28 – October 3, 2014 5 / 8 
 
 
 
Figure 4. TRITON 2-D BWR color-set model 
 
 
5. SENSITIVITY/ UNCERTAINTY ANALYSIS 
 
5.1. Comparison of k-effective uncertainties  
 
For all test problems, k-effective values and their associated uncertainties due to cross section 
uncertainty propagation (taking into account all the possible reactions for which there is uncer-
tainty information) are shown in Table 1. In addition, the reactivity values computed in both the 
forward and adjoint calculations are included, verifying an excellent agreement.  
 
Table 1. Multiplication factors and associated uncertainties (relative standard deviation of 
k-effective, %∆k/k) for all configurations due to cross-section covariance data 
 
Test problems k-effective % ∆k/k  
Unrodded Fuel Assembly Forward Calculation:  
Adjoint Calculation:  
1.088974 
1.088965 0.5474 
Rodded Fuel Assembly Forward Calculation:  
Adjoint Calculation:  
0.778953 
0.778951 0.5950 
1-D Assembly/Reflector Forward Calculation:   
Adjoint Calculation:  
0.620948 
0.620942 0.5364 
2-D Color-set Forward Calculation:  
Adjoint Calculation:  
0.943464 
0.943460 0.5497 
 
It can be seen that the total uncertainty in k-eff for the unrodded fuel assembly is very similar to the 
cases where control rod is inserted, or where the environment of the single fuel assembly is 
modified. 
 
García-Herranz N., Garrido J., Cabellos O. 
6 / 8 PHYSOR 2014 – The Role of Reactor Physics Toward a Sustainable Future 
Kyoto, Japan, September 28 – October 3, 2014 
 
The top eight reaction-nuclide contributors to the uncertainty in k-eff are given in Table 2 in order 
to identify reactions that contribute most to the uncertainties. The contributions are given in rela-
tive std. dev. (%). Only relative contributions larger than 0.05 are included, sorted in descending 
order only for the unrodded fuel assembly. The square root of the sum of the square rel.std.dev. 
would provide the total uncertainty in k-eff as rel.std.dev.(%) shown in Table 1. 
 
Table 2. Top eight nuclide-reaction contributors to the uncertainty in k-effective for all test 
problems 
 
Nuclide-Reaction Contributions to Uncertainty in keff (% Δk/k) 
Unrodded FA Rodded FA FA/Reflector Color-set 2D 
238U (n,γ) 0.30457 0.27556 0.31037 0.29392 
235U ν 0.26782 0.25007 0.26660 0.26124 
238U (n,n’) 0.20541 0.31075 0.14342 0.22530 
235U χ 0.14316 0.20507  0.15604 
235U (n,γ) 0.14202 0.12390 0.13531 0.13470 
235U (n,f) 0.12643 0.14358 0.14192 0.13973 
235U (n,f) (n,γ) 0.12018 0.11255 0.12105 0.11756 
238U ν 0.93636 0.13109 0.10197 0.10723 
1H (n,n)   0.14275  
 
It can be seen that the same cross-sections are the main responsible of k-eff uncertainties for all 
configurations except the (n,n) reaction of 1H, which becomes one of the main contributors in the 
assembly/reflector case, whereas is not as important for the other configurations.  
 
5.2. Comparison of sensitivity profiles of k-effective 
 
It is interesting to compare the sensitivity profiles among all configurations. As an example, the 
sensitivity profiles to 238U (n,γ) reaction cross section are presented in Figure 5 for the four con-
figurations. Differences in sensitivity profiles are found, and the same trend is observed for other 
isotope-reactions, but they do not lead to large differences in the k-effective uncertainty.  
 
Figure 5. Sensitivity profiles to 238U (n,γ) for all test problems at HFP 
S/U ANALYSIS FOR BWR CONFIGURATIONS OF EXERCISE I-2 OF UAM BENCHMARK 
PHYSOR 2014 – The Role of Reactor Physics Toward a Sustainable Future 
Kyoto, Japan, September 28 – October 3, 2014 7 / 8 
 
 
5.3. Comparison of homogenized macroscopic cross sections uncertainties 
 
Two-group assembly homogenized macroscopic cross-sections and associated uncertainties are 
given in Tables 3 and 4, using 0.625 eV as cut-off point. Uncertainties in other parameters of 
interest for core calculations such us discontinuity factors have not been computed.  
 
Table 3 contains values for the unrodded and rodded fuel assembly homogenized over the as-
sembly. Table 4 contains values for the assembly/reflector and 2-D color-set, homogenized over 
the fuel assembly and over the reflector in the first configuration, and over the unrodded and 
rodded assembly in the second one.  
 
It can be seen that uncertainties in the analyzed few-group cross-sections are lower than 1.5% for 
all configurations and homogenization regions. In addition, uncertainties in the cross-sections 
homogenized over the fuel assembly exhibit very slight differences among the four configurations, 
that is, contributions are very similar independently on the assembly environment.   
 
Table 3. Two-group homogenized macroscopic cross-sections and uncertainties for the unrodded 
and rodded fuel assemblies 
 
Reaction Unrodded FA Rodded FA 
 Σ (cm-1) ∆Σ/Σ (%) Σ (cm-1) ∆Σ/Σ (%) 
Absorption 1  6.7846E-3 0.85 9.2300E-3 0.87 
Absorption 2 5.2333E-2 0.22 7.4123E-2 0.20 
Nu-fission 1 4.7086E-3 1.08 4.6012E-3 1.06 
Nu-fission 2 6.6870E-2 0.45 7.4958E-2 0.45 
Scattering 1→2 1.4396E-2 1.24 1.1322E-2 1.29 
Scattering 2→1 9.3617E-4 0.33 1.3409E-3 0.33 
 
Table 4. Two-group homogenized macroscopic cross-sections and uncertainties for the assem-
bly/reflector and for the 2-D color-set 
 
 1-D Assembly/Reflector 2-D Color-set 
Reaction Unrodded FA  Reflector Unrodded FA Rodded FA 
 Σ (cm-1) ∆Σ/Σ (%) Σ (cm-1) ∆Σ/Σ (%) Σ (cm-1) ∆Σ/Σ (%) Σ (cm-1) ∆Σ/Σ (%) 
Absorption 1  6.3897E-3 0.81 3.7851E-4 1.19 6.6981E-3 0.81 9.3502E-3 0.92 
Absorption 2 5.2396E-2 0.21 1.0452E-2 0.49 5.2293E-2 0.21 7.3132E-2 0.19 
Nu-fission 1 4.6890E-3 1.19 - - 4.6966E-3 1.10 4.6134E-3 1.02 
Nu-fission 2 6.7195E-2 0.45 - - 6.6826E-2 0.45 7.4326E-2 0.45 
Scattering 1→2 1.2955E-2 1.11 4.4891E-2 1.37 1.3982E-2 1.17 1.1814E-2 1.39 
Scattering 2→1 7.8688E-4 0.34 2.6621E-4 0.23 9.6142E-4 0.32 1.2740E-3 0.33 
  
García-Herranz N., Garrido J., Cabellos O. 
8 / 8 PHYSOR 2014 – The Role of Reactor Physics Toward a Sustainable Future 
Kyoto, Japan, September 28 – October 3, 2014 
 
 
6. CONCLUSIONS 
 
The propagation of cross section uncertainties in criticality calculations for the different BWR 
configurations at HFP proposed in Exercise I-2 of UAM benchmark has been performed using 
SCALE6.1 code package. Those configurations are unrodded fuel assembly, rodded fuel assembly, 
1-D assembly/reflector and 2-D color set.  
 
First, the uncertainties in k-effective have been calculated, obtaining very similar uncertainty 
values, lower than 0.6%, for all test problems. The most important contributors are the same in the 
four cases: 238U(n,γ), 235U-ν  and 238U (n,n’). Additionally, the sensitivity profiles have been compared 
and although some differences are found, they do not lead to important differences in the k-effective un-
certainty.  
 
Second, the uncertainties in some few-group homogenized cross sections have been computed. 
Very similar uncertainties for the assembly-homogenized cross sections can be found, indepen-
dently on the assembly environment. That means that the assembly environment does not play a 
major role in the evaluation of few-group homogenized cross sections uncertainties. Consequently, 
the standard model for fuel assembly cross-section generation in LWR analysis (single assembly 
calculations with reflective boundary conditions) can be used to compute few-group cross-sections 
libraries and their uncertainties, which are to be propagated to core calculations in Exercise I-3 of 
UAM.  
 
Further calculations are in progress to assess the uncertainties in discontinuity factors and other 
parameters for core simulators. Also, in order to address the uncertainties in the depletion calcu-
lation, extension to Exercise II-2a of UAM will be performed. 
 
ACKNOWLEDGMENTS 
 
This work is partially funded by the Spanish Nuclear Safety Council (CSN), within the agreement 
with the Universidad Politécnica de Madrid in the area of Propagation of Uncertainties for Neu-
tronic Calculations (P110530207). 
 
REFERENCES 
 
[1] K. Ivanova, M. Avramova, S. Kamerow et al., Benchmark For Uncertainty Analysis in Mod-
eling (UAM) For Design, Operation and Safety Analysis of LWRs, NEA/NSC/DOC (2012) 
 
[2] Oak Ridge National Laboratory, “SCALE: a comprehensive modeling and simulation suite for 
nuclear safety analysis and design,” ORNL/TM-2005/39, version 6.1, Radiation Safety 


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Sensitivity/Uncertainty Analysis for BWR Configurations of Exercise I-2 of UAM Benchmark

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