NUREG/CR-6655,Vol.2 

ORNL/TM- 13692/V2

Sensitivity and Uncertainty

Analyses Applied to 

Criticality Safety 

Validation 

Illustrative Applications and Initial Guidance 

Oak Ridge National Laboratory 

U.S. Nuclear Regulatory Commission 

Office of Nuclear Regulatory Research 

Washington, DC 20555-0001 J.
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NUREG/CR-6655, Vol. 2 

ORNL/TM-13692/V2 

Sensitivity and Uncertainty 

Analyses Applied to 

Criticality Safety 

Validation 

Illustrative Applications and Initial Guidance 

Manuscript Completed: October 1999 

Date Published: November 1999 

Prepared by 

B.L. Broadhead, C.M. Hopper, C.V. Parks 

Oak Ridge National Laboratory 

Managed by Lockheed Martin Energy Research Corporation 

Oak Ridge, TN 37831-6370 

C.W. Nilsen, NRC Project Manager 

Prepared for 

Division of Systems Analysis and Regulatory Effectiveness 

Office of Nuclear Regulatory Research 

U.S. Nuclear Regulatory Commission 

Washington, DC 20555-0001 

NRC Job Code W6479
ABSTRACT 

This report presents the application of sensitivity and uncertainty (S/U) analysis methodologies developed in 

Volume 1 to the code/data validation tasks of a criticality safety computational study. Sensitivity and uncertainty 

analysis methods were first developed for application to fast reactor studies in the 1970s. This work has 

revitalized and updated the existing S/U computational capabilities such that they can be used as prototypic 

modules of the SCALE code system, which contains criticality analysis tools currently in use by criticality safety 

practitioners. After complete development, simplified tools are expected to be released for general use. 

The methods for application of S/U and generalized linear-least-squares methodology (GLLSM) tools to the 

criticality safety validation procedures were described in Volume 1 of this report. Volume 2 of this report 

presents the application of these procedures to the validation of criticality safety analyses supporting uranium 

operations where the enrichments are greater than 5 wt %. 

Specifically, the traditional kftrending analyses are compared with newly developed kff trending procedures, 

utilizing the D and ck coefficients described in Volume 1. These newly developed procedures are applied to a 

family of postulated systems involving U( 1)02 fuel, with H/X values ranging from 0 to 1000. These analyses 

produced a series of guidance and recommendations for the general usage of these various techniques. 

Recommendations for future work are also detailed. 

ii NUREG/CR-6655, Vol. 2
CONTENTS 

A B STRA CT .................................................................... i 

LIST OF H GURES .............................................................. vii 

LIST OF TABLES ............................................................... ix 

EXECUTIVE SUMMARY ......................................................... xi 

ACKNOW LEDGMENTS ......................................................... xv 

1 INTRODUCTION ........................................................... 1 

1.1 BACKGROUND ........................................................ 2 

2 DEVELOPMENT OF CRITICAL EXPERIMENT DATABASE ........................ 5 

2.1 LEU System s ........................................................... 5 

2.2 CSEW G Benchmarks ..................................................... 5 

2.3 PCTR Experiments ....................................................... 9 

2.4 HEUM ET Systems ...................................................... 11 

2.5 Rocky Flats Experiments ................................................. 12 

2.6 Russian Critical Experiments .............................................. 13 

2.7 Other System s ......................................................... 13 

2.8 Tabulation of Calculated-versus-Experimental Results and Uncertainties ............. 14 

3 USE OF METHODOLOGY FOR GREATER THAN 5-WT % ENRICHED URANIUM 

APPLICATIONS ........................................................... 19 

3.1 Description of U(l 1)02 Systems ............................................ 19 

3.2 Sensitivity and Uncertainty Analyses of the U(1 1)02 Systems ...................... 19 

3.3 Traditional Trending Analysis .............................................. 25 

3.4 Trending Analysis Using D Values .......................................... 34 

3.5 Trending Analysis Using ck Values .......................................... 38 

3.6 GLLSM Analysis of 11 -wt % Systems ....................................... 43 

3.7 Summary of Trending Analyses ............................................ 45 

4 GUIDANCE FOR DETERMINATION OF APPLICABILITY AND ADVANCED 

TRENDING PROCEDURES .................................................. 47 

4.1 Recommendations on the Use of Trending Procedures ........................... 47 

4.2 Trending Procedures for ck and D Parameters .................................. 48 

4.3 Guidance on Estimating Experimental Uncertainties ............................. 48 

4.4 Future Developments .................................................... 49 

5 SU M M ARY ............................................................... 51 

6 REFERENCES ............................................................. 53

V NUREG/CR-6655, Vol. 2
LIST OF FIGURES 

Figure Page 

1 Correlation coefficients, ck, for U(1 1)02, H/X = 0 system with 102 benchmark experiments ......... 21 

2 Correlation coefficients, cý, for U(1 1)02, H/X = 3 system with 102 benchmark experiments ......... 22 

3 Correlation coefficients, ck, for U(1 1)02, H/X = 40 system with 102 benchmark experiments ........ 23 

4 Correlation coefficients, ck, for U(11)0 2, H/X = 500 system with 102 benchmark experiments ....... 24 

5 D coefficients for U(1 1)02, HIX = 0 system with 102 benchmark experiments .................... 26 

6 D coefficients for U(1 1)02, H/X = 3 system with 102 benchmark experiments .................... 27 

7 D coefficients for U(1 1)02, H/X = 40 system with 102 benchmark experiments ................... 28 

8 D coefficients for U(l 1)02,, H/X = 500 system with 102 benchmark experiments .................. 29 

9 Trend plot for kffversus energy of average lethargy causing fission (EALF) ..................... 30 

10 Trend plot for kffversus energy of average lethargy causing capture (EALC) ..................... 31 

11 Trend plot for kf versus hydrogen-to-• 5U ratio (H/X) ...................................... 32 

12 Trend plot for k versus enrichment .................................................... 33 

13 Trend plot for k.versus D. value for the U(11)0 2 H/X = 3 system ........................... 35 

14 Trend plot for kfversus D. value for the U(1 1)02 H/X = 40 system .......................... 36 

15 Trend plot for kf versus D. value for the U(1 1)02 H/X = 500 system ......................... 37 

16 Trend plot for kff versus ck value for the U(11)O 2 H/X = 0 system ............................. 39 

17 Trend plot for kfversus ck value for the U(1 1)02 HIX = 3 system ............................. 40 

18 Trend plot for ke versus ck value for the U(1 1)02 H/X = 40 system ............................ 41 

19 Trend plot for kefversus ck value for the U(1 1)02 HIX = 500 system ........................... 42 

20 Plot of convergence in predicted k. bias, [(e-a)/c], for U(1 1)02 systems as a function of number of 

experimental groups considered ....................................................... 44

vii NUREG/CR-6655, Vol. 2
LIST OF TABLES 

Table Page 

1 Description of 102 benchmark experiments ............................................... 6 

2 PCTR central region experiments ...................................................... 10 

3 Dimensions for Tuballoy- and nickel-reflected spheres ...................................... 11 

4 Spherical models for Rocky Flats H/U = 0.77 experiments ................................... 12 

5 Spherical models for Rocky Flats H/U = 2.03 experiments ................................... 13 

6 Benchmark descriptions for ORNL L-8 to L-1 1 experiments ................................. 14 

7 Measured-versus-calculated kff values for 102 critical benchmarks ............................. 15 

8 Correlation coefficients for the U(1 1)02 systems .......................................... 20 

9 Predicted Ak bias and its standard deviation based on traditional trending procedure ............... 25 

10 Predicted Ak bias and its standard deviation based on D.m trending procedure .................... 34 

11 Predicted Ak bias and its standard deviation based on ck trending procedure ...................... 38 

12 Predicted Ak bias and its standard deviation based on GLLSM procedure ....................... 45 

13 Comparison of predicted Ak bias and the standard deviation of the estimated k. for various procedures. 46 

14 Comparison of predicted USL1 and USL2 values for the various procedures ..................... 46

ix NUREG/CR-6655, Vol. 2
EXECUTIVE SUMMARY 

Sensitivity and uncertainty (S/U) analysis and the subsequent application of generalized linear-least-squares 

methods (GLLSM) were developed in the 1970s primarily for application to fast reactor studies. In the 1980s 

these techniques were expanded in the area of reactor pressure vessel damage analyses. Recently, interest in the 

United States has increased in the area of cross-section evaluation, with the recognition that these methods can be 

used to improve the consistency of the nuclear data with respect to integral nuclear criticality experiments. 

Volume 1 of this report presented the methodology for application of the S/U and GLLSM techniques to the 

validation of data used in a criticality safety analysis. 

Current generation techniques in criticality safety validation studies typically use trending analyses where the 

calculated effective neutron multiplication factors (k.) for sets of benchmark critical experiments are trended 

against physical and spectral indices, like hydrogen-to-fissile ratio (H/X), energy of average lethargy causing 

fission (EALF), enrichment, solution concentration, etc., to observe their patterns of variation with measured ky 

values (typically 1.00). This trending allows for visual estimates of measured- versus-calculated discrepancies 

(i.e., biases and associated uncertainties) as a function of these spectral/physical parameters. Under this 

approach, the set of critical benchmarks must be shown via an expert judgement determination to be "applicable" 

to the particular application area intended for criticality safety studies. 

Thus, a great deal of judgement is often needed in the current validation techniques in order to establish the area of 

applicability. Each of these traditional trending parameters like H/X and EALF can be useful in establishing 

possible areas of applicability; however, most systems have multiple variables and their simultaneous variation can 

make a definite determination of applicability difficult. The combined variations in HIX, soluble poison 

concentrations, reflected/unreflected geometry, enrichment, and impurity concentrations are treated poorly by using 

single- or even multiple-parameter trend curves. Similarly, no method is available to determine if there is sufficient 

coverage for the entire range of the parameter trend curves. For example, given systems with HIX values of 200, 

500, 1500, is there coverage of the entire range from 200-1500? If experiments with H/X of 0 and 50 are added, is 

there adequate coverage from 0-1500? If two of these systems have soluble poison levels of 200 ppm and 

500 ppm, is there sufficient coverage for 0-500 ppm levels of poison over the full range of H/X? These are 

difficult questions, with answers typically derived from expert judgement. 

Conversely, the application of S/U techniques allows for a formal estimation of the applicability of each of the 

critical benchmarks in the set to the application area under consideration. Volume 1 of this report has presented 

techniques that can be utilized in a formal estimation of applicability using integral parameters based on 

differences in sensitivity profiles between systems (D) and the correlation coefficients between systems (ck). The 

D values are defined as the following: 

g 

D = I Si- Si [ 

i=1 

g Do I ~-s~i I 

i=1 

g 

D =. I Sai- sei 

i=l 

where S is the sensitivity of ke for the safety application, a, or experimental configuration, e, to T (n) for group i or 

to the capture (c) or scattering (s), cross sections. These parameters are useful, not only as formal estimations of 

critical benchmark data applicability, but additionally as trending parameters in the traditional criticality safety 

data validation approach.

xi NUREG/CR-6655, Vol. 2
Executive Summary

A criterion for determining the appropriate cut-off values of D and q, for adequate "similarity" has been 

established. The parameters developed for use in this process are calculated by performing a sensitivity and 

uncertainty analysis on both the selected experiments and the applications of interest. The complete S/U analysis capability is presently limited to one-dimensional (1-D) and two-dimensional (2-D) models of the systems of 

interest. Future development work will hopefully eliminate this limitation. 

A further development under this current work was the investigation of GLLSM. This methodology is an 

alternative approach to the traditional trending analysis for the determination of bias and the associated uncertainty. Physically the GLLSM is designed to minimize differences between the measured and calculated values of k.f by predicting data changes based on the entire set of criticals used in the data validation process. The inputs needed 

for such an analysis are almost identical to the concepts presented thus far: the sensitivity coefficients, the cross

section uncertainties, and the actual calculated and measured k, values, with the addition of an estimate of the 

uncertainty in the measured kff values. The "data changes" that result from the application of the GLLSM can then 

be used to predict (via interpolation or extrapolation) the biases for any application determined to be similar to the 

benchmark area of applicability. 

One of the benefits of the GLLSM approach is that, not only can the bias for a given application be estimated, the 

cumulative "combination" of critical benchmarks can be used to determine the convergence of the procedure. Questions that can be addressed include: how many experiments are needed to verify an application, and how much 

correlation to the application is necessary in order to validate the application area? 

In Volume 1, the GLLSM is used to predict how many experiments and how much correlation between experiments 

are necessary to validate an application area. In Volume 2, the GLLSM is applied as a data validation tool for 

comparison with trending analyses using typical characteristic parameters as well as the D and cý parameters. 

This current report presents an illustrative application of both the S/U and GLLSM procedures to the validation 

of criticality safety studies for facilities processing uranium fuels with enrichments greater than 5 wt % for use in 

commercial power reactors. In the past, these processing facilities have been limited to enrichments at or below 5 

wt %. Hence, much of the critical experiment data correspond to these lower enrichments. As a part of this 

study, a number of critical experiments in the 5-20-wt % range were identified as having been performed in 

Russia. A number of these experiments were obtained and documented as a result of this work. 

As with any criticality data validation, the goal is to estimate the bias trends for ranges over which the criticality safety calculational studies are to be performed. The usefulness of S/U and GLLSM methods in validation 

studies is demonstrated by performing a validation of a hypothetical set of application scenarios, which consist of 

14 systems, each having U(l 1)02 fuel with H/X values varying from 0 to 1000. The 11 -wt % enrichment is 

chosen so the entire range of moderations, including dry, is able to achieve criticality. The data validation 

included traditional trending analyses, trending analyses with the D and ck parameters developed in Volume 1, and finally the full GLLSM approach. Volume 2 explores the advantages and disadvantages of each approach 

and presents guidance for general use of these techniques. 

The comparisons among the various trending techniques are quite interesting in that they give very different 

answers, depending on the particular system being analyzed. The predicted biases for various systems are in 

some cases up to a factor of 15 different between the various trending techniques. The primary reason for these 

differences is that various systems "look" very similar from the standpoint of certain parameters, but very different with respect to other parameters. In particular, the HIX and EALF parameters, frequently used in 

current validation efforts, predict similarity between dry systems with high enrichments (93 wt %) and dry 

systems with intermediate enrichments (10 wt %), while the ck and D parameters indicate that these systems are

NUREG/CR-6655, Vol. 2 xii
Executive Summary

quite different. The net effect of trending with traditional HIX and EALF parameters is a cancellation of effects 

(the high-enriched systems underpredict by about 1%, the intermediate-enriched systems overpredict by about 

2%), which produce a predicted bias of about 0.5% overprediction for a dry U(1 1)02 system. The trending with 

cq and D parameters produces an estimated bias of between 1 and 2% overprediction because only the 

intermediate-enriched systems are predicted to be similar to the dry U(1 1)02 system. Although the predicted bias 

from these applications are all positive, the differences in magnitude are a concern, since the prudent application 

of trending procedures is very important in criticality validation exercises. 

As a result of these observations, the following guidance was developed for the usage of the various trending 

techniques. 

"* Traditional trending procedures are acceptable for estimation of the bias and associated uncertainty if about 

20 benchmarks with a Cq value of 0.8 or higher are included in the benchmark database. It is further 

recommended that in using the traditional ktrending procedures that systems with very different physical 

characteristics, such as high- versus-low enrichment, poisoned-versus-nonpoisoned, wet-versus-dry, etc., 

not be included together unless that specific effect is being analyzed. 

"* The advanced trending analyses with the D and ck coefficients are recommended if fewer experiments than 

those recommended above are available. Also the primary motivation behind their use is that widely 

varying systems can be included in the analysis, hence making the effective extrapolation of the existing 

data more meaningful. The procedure will allow for automatic collecting of experiments that have similar 

D and ck coefficients such that the predicted kffbias should be valid. 

"* The GLLSM procedure is recommended if existing experiments with ck values of 0.8 or higher with respect 

to the area of application are not available. These experiments, possibly having Ck values 0.2-0.6, are not 

similar to the application; but they are still relevant. A suite of such experiments can be expected to 

perform quite well using a GLLSM technique if they are suitably chosen to highlight different aspects of 

the given application system. Under these conditions, the use of GLLSM for validation is possible, 

although it may still be desirable to initiate an experimental program to obtain additional data. This 

procedure also allows for the inclusion of noncritical experiments and non-k responses in the validation 

exercises. Further development is required before these GLLSM techniques can become fully functional in 

a routine criticality safety validation study. 

Additional work is needed to enable full use of these methodologies by the criticality safety practitioner. Of 

immediate importance is the need to extend the sensitivity analysis methodologies to the realm of three-dimensional 

Monte Carlo codes typically used in criticality safety analysis. Improved sensitivity information can also be 

obtained by including the effects of the problem-specific cross-section processing. Additional applications need to 

be explored to gain further confidence in the guidance developed herein and aid in the development of standard 

processes that can be more easily implemented and applied by criticality safety analysts. Methods to enable full 

implementation of GLLSM need particular attention. Finally, cross-section uncertainty information is currently 

rather limited and the need exists to obtain uncertainty data for a number of nuclides important to criticality safety.

°°.i NUREG/CR-6655, Vol. 2
ACKNOWLEDGMENTS 

This work was supported by the U.S. Nuclear Regulatory Commission under Project JCN W6479, "Development 

and Applicability of Criticality Safety Software for Licensing Review." The direction of C. W. Nilsen, the NRC 

project manager, along with D. R. Damon, S. A. Parra and K. J. Hardin, the NRC technical monitors, is 

acknowledged. 

The technical work described in the report has been presented to several audiences and the reports have been 

reviewed by individuals with a wide spectrum of interests and experience. This process has been valuable to the 

authors and appreciation to those willing to provide opinions and feedback is gratefully appreciated. In particular 

the authors would like to recognize the efforts of D. R. Damon and S. A. Parra who provided review comments for 

the NRC. Useful discussions and technical reviews were also received from R. M. Westfall, W. C. Jordan, and J. 

F. Mincey of ORNL. The initial suggestions from J. J. Wagschal of the Hebrew University, Jerusalem, relative to 

presenting the methodology as well as his careful review of the final report were of great benefit to the authors. 

Similarly the comments and suggestions provided by R. L. Murray of N. C. State University, M. L. Williams of 

Louisiana State University, R. E. Wilson of Safe Sites of Colorado, Keyes Niemer of Duke Engineering and 

Services, and F. Trumble of Westinghouse Safety Management Systems helped the authors identify areas where 

improved clarity and explanation were needed.

XV NUREG/CR-6655, Vol. 2
1 INTRODUCTION 

Sensitivity and uncertainty (S/U) analysis and the subsequent application of generalized linear-least-squares 

methods (GLLSM) were developed in the 1970s, primarily for application to fast reactor studies. "2 In the 1980s 

they were expanded in the area of reactor pressure vessel damage analyses.3 Recently interest in the 

United States has increased in the area of cross-section evaluation with the recognition that these methods can be 

used to improve the consistency of the nuclear data with respect to some non-criticality integral experiments.4 

Current-generation techniques in criticality safety validation studies5 use trending analyses where the calculated 

effective neutron multiplication factors (ke) values for sets of benchmark critical experiments are trended against 

physical and spectral indices, like hydrogen-to-fissile ratio (H/X), energy of average lethargy causing fission 

(EALF), enrichment, or solution concentration. This trending allows for visual estimates of measured-versus

calculated discrepancies (i.e., biases) as a function of these spectral/physical parameters. Under this approach, a 

set of critical benchmarks must be shown via an expert judgement determination to be "applicable" to the 

particular application area intended for criticality safety studies. 

Conversely, the application of S/U techniques allows for a formal determination of the applicability of each of the 

critical benchmarks in the set to the application area under consideration. Volume 1 of this report has presented 

techniques that can be utilized in a formal determination of applicability using integral parameters based on 

differences in sensitivity profiles between systems (D) and the correlation coefficients between systems (ci). 

These D values are defined as the following: 

g 

D.= Si- Snie 

i=1 

g 

i=1 

g 

Ds= Ssa - S 

i=1 

where S is the sensitivity of kff for the safety application, a, or experimental configuration, e, to ' (n) for group i or 

to the capture (c) or scattering (s) cross sections. These parameters are useful, not only as formal determinations 

of critical benchmark data applicability, but additionally as trending parameters in the traditional criticality safety 

data validation approach. 

The results from such a S/U analysis can also be directly incorporated into a full GLLSM procedure quite readily, 

in that only additional information on the measurement uncertainties need be added to perform the analyses. The 

GLLSM approach described in Volume 1 of this report 6 was used as a research tool to understand the systematics 

of the D and ck parameters previously mentioned. These GLLSM results of Volume 1 supported definition of 

general ranges in the D and cý parameters which indicate applicability. Based on these studies, systems with ck 

values of 0.80 and higher, the sum of the D values less than 1.2, and individual D values of 0.40 and lower, can 

be used to designate similar systems. The use of the GLLSM procedures in a criticality safety data validation 

exercise has the advantage of reducing the underlying uncertainties in the criticality safety limits, which can 

allow for enhanced operational flexibility with little or no safety implications. Additionally, application of the 

GLLSM procedure should allow for the inclusion of non-k. responses (e.g., reaction rates) and even noncritical 

(e.g., subcritical and dosimetry) experimental data into the data validation procedure.

1 NUREG/CR-6655, Vol. 2
Introduction

This current report presents an application of both the S/U and GLLSM procedures to the validation of criticality 

safety studies for facilities processing uranium fuels with enrichments greater than 5 wt % for use in commercial 

power reactors. In the past, these processing facilities have been limited to enrichments at or below 5 wt %. 

Hence, much of the critical experiment data correspond to these lower enrichments. As a part of this study, a 

number of critical experiments in the 5-20-wt % range were identified as having been performed in Russia.7 

Several of these experiments were obtained and documented as a result of this work.8 The advantage of having 

these data available is so that the effect of inclusion or exclusion of these data from the benchmark database can 

be studied. 

As with any criticality data validation, the goal is to estimate the bias and associated uncertainty for ranges over 

which the criticality safety calculational studies are to be performed. The use of S/U and GLLSM methods in 

these validations studies will be demonstrated herein by performing a validation for a hypothetical set of 

application scenarios, which consist of 14 systems, each having U( 1)02 fuel with H/X values varying from 0 to 

1000. The 11 -wt % enrichment is chosen so the entire range of moderations, including dry, is able to achieve 

criticality. The data validation will be performed using both the traditional trending analyses, trending analysis 

with the D and ck parameters, and finally the full GLLSM approach. Advantages and disadvantages of each 

approach will be explored. Guidance for general use of these techniques will then be given. 

The database of 102 critical benchmark experiments used in this study will be described in Section 2. The 

application of the S/U and GLLSM techniques to the data validation of the hypothetical 11 -wt % systems will be 

discussed in Section 3. Finally, guidance on the general application of these techniques to other situations will be 

presented in Section 4. 

1.1 BACKGROUND 

This volume applies a series of newly developed data validation techniques to criticality safety applications. 

Before launching into the specific analyses, a summary of a current approach is needed. Certainly there are many 

possible approaches, but the approach briefly described herein is representative of those currently in use at Oak 

Ridge National Laboratory. 5 

The data validation exercises are most appropriately named trending analyses, since the calculated values of ke 

for a set of benchmark critical experiments are plotted against a single parameter, x, typically H/X, EALF, 

leakage, concentration, etc., to determine a linear fit of k. with respect to that parameter. The entire set of 

calculated k~values is then averaged, and a "pooled" standard deviation, ap, (the standard deviation of the entire 

data set about the average) determined. A bias, [(x), is then determined as the difference between the linear fit 

and the measured kffvalue. The bias is a function of the x parameter since the linear fit to kf.can and typically 

does have a non-zero slope. An uncertainty in this predicted bias is assumed to be the uncertainty in the pooled 

data times a statistical multiplier, F. In terms of these quantities two upper subcritical limit (USL) parameters are 

defined: 

USLI(x) = 1 - Akm - F1 oa + P3(x), or alternately 

USL2(x) = 1 - F2 ap + P() 

where Akm is an additional margin to ensure subcriticality, and F2 is a more stringent multiplier than F1. If the 

analysis finds P(x) to be positive at any point in the trending, the methodology of Ref. 5 sets P(x) = 0 such that 

no credit is provided for a positive bias (over prediction of kf).

NUREG/CR-6655, Vol. 2

Section 1

2
This work has applied these currently used validation methods, but replaced the standard parameters with the D 

and ck values developed in Volume 1 of this report. These techniques combine the validation and determination 

of areas of applicability tasks, since the D and cý parameters are utilized for both operations. Similarly, the 

results from an application of the GLLSM tools to a data validation problem is simply a prediction of the bias 

and its uncertainty, which can also be utilized in the safety analysis procedures described above.

NUREG/CR-6655, Vol. 2

Section 1 Introduction

3
2 DEVELOPMENT OF CRITICAL EXPERIMENT DATABASE 

The key to any criticality safety data validation procedure is the generation of a database of critical benchmark 

experiments that typically covers a broad range of systems that are in some way similar to the particular 

application(s) of interest Several studies5' 7-10 have reported the generation of large databases of critical 

experiments and corresponding code input. The inputs for the benchmark experiments are typically generated 

corresponding to a three-dimensional (3-D) geometry model. Currently, only 1-D and 2-D sensitivity tools are 

available, which necessitates the conversion of many of these inputs to 1-D or 2-D models. The impact of the 

revised models on the sensitivities is expected to be small, based on selected studies of this effect. 

This section will briefly describe each of the 102 critical benchmark systems included in this analysis. In each 

case a summary of the problem materials and dimensions will be given, along with a description of the 

calculation procedures followed. These problems can be broken into several groups, which include: 14 

low-enriched uranium (LEU) oxide or fluoride systems with 2-5-wt % uranium fuel and paraffin or stereotex 

moderators, 14 Cross-Section Evaluation Working Group (CSEWG) data testing benchmarks (a full range of 

enrichments from about 1 to 93 wt %, dry to fully moderated), the 11 Physical Constants Testing Reactor 

(PCTR) k. experiments on 2-wt % uranium fuel, 9 high-enriched uranium metal experiments (HEUME'T), 

8 low-HNX Rocky Flats experiments (RF low H/X) with uranium enrichments of 4.5 wt %, 36 Russian 

experiments, which consist of various configurations of solution tanks and fuel rod arrays with 25U enrichments 

ranging from 5 to 89 wt %, and 10 various experiments that include HISS, UH3, and LXX configurations. The 

complete list of experiments, their enrichments, NIX values, and a brief description are given in Table 1. The 

following sections give more detailed information on each system. 

2.1 LEU Systems 

This group of experiments consists of 2-5-wt % uranium oxide and fluoride systems. The UF4-paraffln systems 

(experiments Nos. 1-8 in Table 1) are 2- and 3-wt % uranium systems. The critical dimensions for the 

unreflected homogeneous spherical systems are radii of 44.91, 38.50, 36.38 36.36, 37.67, 49.65, 36.64 and 

28.78 cm, respectively.'1 Even though these systems were cuboid systems, the critical spherical dimensions were 

obtained from Ref. 11 and were converted from measured critical bucklings and extrapolation distances. A total 

of six experiments with U(5) 30 and stereotex were also analyzed (experiment Nos. 9-14 in Table 1). These six 

criticals were modeled as homogeneous spheres with radii of 35.57, 33.82, 32.17, 32.94, 33.55, and 41.05 cm, 

respectively.12 These radii differ slightly from the estimates reported in Ref. 12; however, they were obtained by 

buckling conversions based on the reported actual critical dimensions and extrapolation distances. 

2.2 CSEWG Benchmarks 

This group of experiments (15-19, 30-34,46-48, and 51-52 of Table 1) consists of a number of benchmarks 

developed over a number of years by CSEWG. These benchmarks are documented in the CSEWG benchmark 

book;13 however, they will be described briefly here. GODIVA is a 93-wt %-enriched bare metal sphere with 

radius of 8.741 cm, and a measured eigenvalue of 1.00 ± 0.003. Big-10 is a cylindrical system consisting of a 

uranium-metal core, averaging 10 wt % 235U, reflected by depleted-uranium metal. The core has an homogeneous 

axial region surrounded by interleaved plates of high-enriched uranium and natural uranium such that the average 

1 5

U content is uniform. An equivalent spherical system was modeled with a core radius of 30.48 cm and a 

reflector radius of 45.72 cm. The uncertainty in the measured value of kf.is approximately ±0.003. The ZPR 

3.11 assembly is a U(14) metal and steel assembly with a depleted uranium (DU) metal and steel reflector.

5 NUREG/CR-6655, Vol. 2
Development of Critical Experiment Database

Table 1 Description of 102 benchmark experiments

Enrichment 

Exp. No. (wt %) H/X Application description Ref. 

1 2 195 u(2)f4-ch2 h/x = 195 11 

2 2 294 u(2)f4-ch2 h/x = 294 11 

3 2 406 u(2)f4-ch2 h/x = 406 11 

4 2 496 u(2)f4-ch2 h/x = 496 11 

5 2 614 u(2)f4-ch2 h/x = 614 11 

6 2 972 u(2)f4-ch2 h/x = 972 11 

7 3 133 u(3)f4-ch2 h/x = 133 11 

8 3 277 u(3)f4-ch2 h/x = 277 11 

9 5 147 u(5)3o8-h2o h/x = 147 12 

10 5 245 u(5)3o8-h2o h/x = 245 12 

11 5 320 u(5)3o8-h2o hlx = 320 12 

12 5 396 u(5)3o8-h2o h/x = 396 12 

13 5 503 u(5)3o8-h2o h/x = 503 12 

14 5 757 u(5)3o8-h2o h/x = 757 12 

15 93 0 godiva-u(93) 13 

16 1.3 306 bapl-1 u(1.3) lattice h/x = 306 13 

17 1.3 382 bapl-2 u(1.3) lattice h/x = 382 13 

18 1.3 515 bapl-3 u(1.3) lattice h/x = 515 13 

19 10 0 big-10 u(n)-refl 13 

20 93 0 hiss(hug) u(93)-c-b c/x 303 19 

21 98 0 u(98) h2o-refl 15 

22 93 0 u(93) 2-u(n)-refl 15 

23 93 0 u(93) 3-u(n)-refl 15 

24 93 0 u(93) 4-u(n)-refl 15 

25 93 0 u(93) 5-u(n)-refl 15 

26 93 0 u(93) 7-u(n)-refl 15 

27 93 0 u(93) 8-u(n)-refl 15 

28 93 0 u(93) 11-u(n)-refl 15 

29 93 0 u(93) 8-ni-refl 15 

30 93 1378 oral-1 u(93)o2(no3)2-h2o h/x = 1378 13 

31 93 1177 ornl-2 u(93)o2(no3)2-h2o-b h/x = 1177 13 

32 93 1033 ornl-3 u(93)o2(no3)2-h2o-b h/x = 1033 13 

33 93 972 oral-4 u(93)o2(no3)2-h2o-b h/x = 972 13

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Development of Critical Experiment Database

Table 1 (continued) 

Enrichment 

Exp. No. (wt %) H/X Application description Ref. 

34 93 1835 ornl-10 u(93)o2(no3)2-h2o h/x = 1835 13 

35 1.157 322 pctr 1.157% 3.73 h/x = 322 14 

36 1.071 353 pctr 1.071% 3.78 h/x = 353 14 

37 1.006 381 pctr 1.006% 3.83 h/x = 381 14 

38 1.071 545 pctr 1.071% 5.84 h/x = 545 14 

39 1.157 518 pctr 1.157% 5.99 h/x = 518 14 

40 1.006 619 pctr 1.006% 6.23 h/x = 619 14 

41 1.157 619 pctr 1.157% 6.90 h/x = 619 14 

42 1.006 596 pctr 1.006% 6.95 h/x = 596 14 

43 1.071 667 pctr 1.071% 7.14 h/x = 667 14 

44 1.006 748 pctr 1.006% 7.52 h/x = 748 14 

45 1.157 650 pctr 1.157% 7.52 h/x = 650 14 

46 14 0 zpr-3/11 u(14)-fe-ni-cr-mn u(d)-refl 13 

47 21 0 zpr-3/12 u(21)-c u(d)-refl 13 

48 16 0 zpr-6/6a u(16)-na-fe u(d)-refl 13 

49 93 0 uh3 ni-refl 18 

50 93 0 ua3 u(n)-refl 18 

51 1.3 251 trx-1 u(1.3) lattice h/x = 251 13 

52 1.3 429 trx-2 u(1.3) lattice h/x = 429 13 

53 93 76 17 u(93)o2f2-h2o h2o-refl h/x = 76 20 

54 93 1110 18 u(93)o2f2-h2o h/x = 1110 20 

55 93 1390 19 u(93)o2f2-h2o h/x = 1390 20 

56 93 126 110 u(93)o2f2-h2o h2o-refl h/x = 126 20 

57 93 1270 111 u(93)o2f2-h2o h2o-refl h/x = 1270 20 

58 5 405 sheba 2 u(5)o2f2-h2o h/x = 405 15 

59 4.5 17 rfc 4.5% h/u=.77 met h/x = 17 16 

60 4.5 17 rfc 4.5% h/u=.77 met h/x = 17 16 

61 4.5 17 rfc 4.5% h/u=.77 met h/x = 17 16 

62 4.5 17 rfc 4.5% h/u=.77 met h/x = 17 16 

63 4.5 70.7 rf 4.5% h/u=2.03 h/x = 70.7 17 

64 4.5 120.1 rf 4.5% h/u=2.03 h/x = 120.1 17 

65 4.5 45 rf 4.5% h/u=2.03 h/x = 45 17 

66 4.5 45 rf 4.5% h/u=2.03 h/x = 45 17 

67 89 92 U(89)o2(no3)2 abrod p = 0 7,8 

68 89 92 U(89)o2(no3)2 abrod p = 6 7, 8 

69 89 92 U(89)o2(no3)2 abrod p = 7 7, 8

NUREG/CR-6655, Vol. 2

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Table 1 (continued) 

Enrichment 

Exp. No. (wt %) HIX Application description Ref. 

70 89 92 U(89)o2(no3)2 abrod p = 10.57 7,8 

71 89 92 U(89)o2(no3)2 abrod p = 12.3 7, 8 

72 89 92 U(89)o2(no3)2 abrod p = 14 7, 8 

73 89 92 U(89)o2(no3)2 abrod p = 16 7, 8 

74 89 375 U(89)o2(no3)2 76 g/l 0 abrods 7, 8 

75 89 375 U(89)o2(no3)2 76 g/l 3 abrods 7, 8 

76 89 375 U(89)o2(no3)2 76 g/l 4 abrods 7, 8 

77 89 92 U(89)o2(no3)2 289 g/l 0 abrods 7, 8 

78 89 92 U(89)o2(no3)2 289 g1l 3 abrods 7, 8 

79 89 92 U(89)o2(no3)2 289 g/l 4 abrods 7, 8 

80 89 92 U(89)o2(no3)2 289 g/l 6 abrods 7, 8 

81 89 92 U(89)o2(no3)2 289 g/i 18 p = 4 rods 7, 8 

82 89 92 U(89)o2(no3)2 289 g/i l6 p = 4 rods 7, 8 

83 89 92 U(89)o2(no3)2 289 g/i 18 p = 6 rods 7, 8 

84 89 92 U(89)o2(no3)2 289 gil 36 p = 6 rods 7, 8 

85 17 628 U(17)o2 34 rods 22.7 C 7, 8 

86 17 628 U(17)o2 34 rods 218 C 7, 8 

87 17 611 U(17)o2 74 rods 16 C 7,8 

88 17 611 U(17)o2 74 rods 151 C 7,8 

89 17 565 U(17)o2 68 rods 15 C 7, 8 

90 17 565 U(17)o2 68 rods 151 C 7,8 

91 10 50 U(10)o2 p = 0.7 20 C 7,8 

92 10 50 U(10)o2 p = 0.7 166 C 7,8 

93 10 50 U(10)o2 p = 0.7 263 C 7, 8 

94 10 340 U(10)o2 p = 1.4 20 C 7,8 

95 10 340 U(l0)o2 p = 1.4 206 C 7,8 

96 10 340 U(10)o2 p = 1.4 274 C 7,8 

97 10 629 U(10)o2 p = 1.852 20C 7,8 

98 10 629 U(10)o2 p = 1.852 193 C 7,8 

99 10 629 U(10)o2 p = 1.852 263 C 7,8 

100 5.64 973 U(5.64)o2(no3)2 400 g/l 0 rods 7, 8 

101 5.64 973 U(5.64)o2(no3)2 400 g/I 1 rod 7, 8 

102 5.64 973 U(5.64)o2(no3)2 400 g/l 7 rods 7, 8

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Development of Critical Experiment Database

The specifications in the CSEWG benchmark book gave a homogeneous spherical model with core radius of 

31.61 cm and a 30-cm-thick reflector, with a reported eigenvalue of 1.0000 ± 0.0025. ZPR 3.12 is a U(25) metal, 

graphite, and steel assembly with a DU metal and steel blanket. The system was modeled as specified in Ref. 13 as 

homogeneous concentric spheres with radii of 28.76 and 59.26 cm, respectively. The measured value of k'f was 

reported as 1.0000 with no uncertainty quoted. An uncertainty in kff due to the use of a 1 -D model was specified to 

be 0.0023. The final ZPR-type assembly included in this analysis was the ZPR 6.6a, which is a large 

16 wt %-enriched uranium oxide, sodium and steel core with a DU and steel blanket. A 1-D homogeneous 

spherical model is specified that consists of core radius and blanket thicknesses of 95.67 and 33.81 cm, 

respectively. The measured eigenvalue was 1.0000 ± 0.0005, with an estimated 0.05% uncertainty due to the 

spherical model. 

The experiments labeled BAPL-l, -2 and -3 (experiments 16-18 in Table 1) are a series of experiments consisting 

of a lattice of water-moderated U(1.3)02 fuel pins in a triangular pitch. The fuel pins for each experiment are 

identical and have pellet, gap and clad radii of 0.4864, 0.5042, and 0.5753 cm, respectively. The respective pin 

pitches are 1.5578, 1.6523 and 1.8057 cm for BAPL-1, -2, and -3. These specifications were used to perform cell

weighting calculations with the resulting cell-weighted cross sections input into a 1 -D cylindrical model with core 

and reflector dimensions of 38.1269 and 59.0 cm for BAPL-1, 36.3428 and 56.0 cmn for BAPL-2, and 37.6250 and 

58.0 cmn for BAPL-3. The respective heights for buckling were 137.245, 129.697, and 125.535 cm. No 

uncertainties on the measured eigenvalues were reported. 

The next family of CSEWG critical experiment benchmarks corresponds to the ORNL spheres: ORNL-1, -2, -3, 

4, and -10 (experiments 30-34 of Table 1). These unreflected spheres consisted of U(93)0 2(NO3)2 and H20 at 

various concentrations and with/without boron. ORNL-1, -2, and -3 have a radius of 34.595 cm, and ORNL-10 

has a radius of 61.011 cm. The reported eigenvalues have an uncertainty of 0.25%. 

The TRX experiments (51-52 of Table 1) are water-moderated lattices of slightly enriched (1.3%) uranium metal 

rods with diameters of 0.4915 cm in a triangular pitch. The pitches were set to 1.8060 and 2.1740 cm to define the 

TRX-1 and TRX-2 configurations, respectively. Both experiments were modeled using a cell-weighting step, which 

characterized the pin lattice, followed by a 1-D cylindrical model consisting of core radii of 26.2093 and 

27.4419 cm, followed by a 20-cm-thick water reflector. In both cases an axial buckling height of 136.98 cm was 

assumed. 

2.3 PCTR Experiments 

This group of 11 experiments (35-45 of Table 1) were performed at the Physical Constants Test Reactor (PCTR)"4 

and consisted of the measurement of k, for three different enrichments near 1.0 wt %, with variable H/U values. 

The experiments are identified by their respective H/U values: 3.83, 6.23, 6.95, and 7.52, which have an 

enrichment of 1.006 wt %; 3.78, 5.84, and 7.14, which have enrichments of 1.071 wt %; and 3.73, 5.99, 6.90, and 

7.52, which have enrichments of 1.157 wt %. The experimentally inferred values of k. are given in Table 2, along 

with their uncertainties.

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Development of Critical Experiment Database

Table 2 PCTR central region experiments" 

Case No. 13U wt % H/U Exp. k 

1 1.006 3.83 0.986 

2 1.006 6.23 0.986 

3 1.006 6.95 0.974 

4 1.006 7.52 0.960 

5 1.071 3.78 1.005 

6 1.071 5.84 1.005 

7 1.071 7.14 0.992 

8 1.157 3.73 1.031 

9 1.157 5.99 1.031 

10 1.157 6.90 1.030 

11 1.157 7.52 1.019 

aExperimental uncertainty is ± 0.006 Ak (varies from 0.005 to 0.007).

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Development of Critical Experiment Database

2.4 HEUMET Systems 

This group of nine experiments (21-29 of Table 1) include two separate measurement sets.'5 The first is a single 

experiment consisting of a water-reflected U(98) metal sphere with radii corresponding to 6.5537 cm for the core 

and 33.4717 cm for the reflector. The remaining set consists of eight experiments with a U(93.5) metal (Oralloy) 

core, along with variable thickness reflectors of natural uranium (Tuballoy) or nickel. The dimensions are given in 

Table 3 for each of these situations. 

Table 3 Dimensions for Tuballoy- and nickel-reflected spheresa 

Reflector thickness Outer radius 

(in) Region (cm) Material 

2.0 1 6.7820 Oralloy 

2 11.8620 Tuballoy 

3.0 1 6.4423 Oralloy 

2 14.0623 Tuballoy 

4.0 1 6.2851 Oralloy 

2 16.4451 Tuballoy 

5.0 1 6.1535 Oralloy 

2 18.8535 Tuballoy 

7.0 1 6.0740 Oralloy 

2 23.8540 Tuballoy 

8.0 1 6.0509 Oralloy 

2 26.3709 Tuballoy 

11.0 1 6.0276 Oralloy 

2 33.9676 Tuballoy 

8.0 1 6.4627 Oralloy 

2 26.7827 Ni 

"aThe experimental values of kfr for all cases is 1.000, with uncertainties in the experiments of ±0.003 for 

reflector thicknesses of 5, 7, 8, and 11 in. and ±0.005 for all others.

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Development of Critical Experiment Database

2.5 Rocky Flats Experiments 

These experiments 6

", 7 consist of a series of measurements (59-66 in Table 1) corresponding to low-enrichment and 

low-moderation conditions. The fuel in all cases was U(4.5)0 2, which was packed into 1.6-mm-thick aluminum 

cans forming a 152-mm cube. Water was added to the oxide to achieve HIU ratios of 0.77 and 2.03. The 

experimental setup consisted of reflected 5 x 5 x 5 arrays of these cans typically driven by a high-enriched uranium 

driver to achieve criticality. The drivers were a high-enriched uranium metal sphere, and high- and 

low-concentration uranyl nitrate solution boxes. 

Eight configurations were included in this work, four each for the H/U ratios of 0.77 and 2.03. The 1-D spherical 

models shown in Tables 4 and 5 were generated based on conservation of the actual experimental volumes. 

Table 4 Spherical models for Rocky Flats H/U = 0.77 experiments 

Configuration Region Outer radius Material 

(cM) 

RF H/U = 0.77 A 1 7.334 U metal 

2 9.454 Air 

3 49.627 Hon. U02 + HI0 2 + A1 

4 75.227 Plastic reflector 

RF HIU = 0.77 B 1 11.958 High conc. solution 

2 14.769 Void 

3 15.015 Steel 

4 49.600 Hom. U0 2 + H20 + Al 

5 75.200 Concrete reflector 

RF H/U = 0.77 C 1 12.248 High conc. solution 

2 14.769 Void 

3 15.015 Steel 

4 49.656 Hom. U02 + H20 + Al 

5 75.256 Plastic reflector 

RF H/U =0.77 D 1 13.041 Low conc. solution 

2 14.746 Void 

3 14.983 Steel 

4 49.650 Hon. UO2 + H 20 + Al 

5 75.250 Plastic reflector 

'The experimental values are 1.000, the uncertainties in the measured values of kf are not 

explicitly quoted, but are estimated to be ±0.001.

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Development of Critical Experiment Database

Table 5 Spherical models for Rocky Flats H/U = 2.03 experimentsa 

Configuration Region Oute~ rdius Material 

RF H/U = 2.03 A 1 34.857 Horn. U0 2 + H-20 + Al + Poly (0.93 cm) 

2 60.457 Plastic reflector 

RF H/U = 2.03 B 1 32.628 Horn. U02 + H20 + Al + Poly (2.44 cm) 

2 58.228 Plastic reflector 

RF H/U = 2.03 C 1 5.669 U metal 

2 9.454 Void 

3 45.919 Hom. U02 + H 20 + Al 

4 71.519 Plastic reflector 

RF H/U =2.03 D 1 2.340 Void 

2 5.670 U metal 

3 9.454 Void 

4 45.746 Hom. U02 + H20 + Al 

5 71.346 Plastic reflector 

"The experimental values are 1.000, the uncertainties in the measured values of kfare not explicitly quoted, 

but are estimated to be ±0.001. 

2.6 Russian Critical Experiments 

The Russian experiments included in this work (67-102 of Table 1) have been documented in the Benchmark 

Handbook7 and separately described and analyzed in another phase of this work-8 These experiments include HEU

SOL-THERM-029 (7 configurations), HEU-SOL-THERM-030 (7 configurations), HEU-SOL-THERM-031 

(4 configurations), IEU-COMP-THERM-02 (6 configurations), LEU-COMP-THERM-32 (9 configurations), and 

LEU-SOL-THERM-005 (3 configurations). Each of these configurations was modeled in 1-D. The models and 

results from these 1-D studies are given in Ref. 8. 

2.7 Other Systems 

The first two systems (49-50 of Table 1) included in this group are the UH3-UR and UH3-NI critical assemblies. 8 

These assemblies consist of approximate spheres of enriched-uranium hydride within a 20.32-cm thick natural 

uranium and nickel reflectors. The average empirical formula of the hydride composition is U(93.15)H2 97C.11 00 .

25. 

The benchmarks are modeled as a sphere within a 20.32 cm natural uranium reflector (UH3-UR) and a 20.32 cm 

nickel reflector (UH3-NI). The UH3-UR core has a radius of 7.7902 cm, and the UH3-NI has a core radius of 

7.7978 cnm The reported value of k,- is 1.000 with no uncertainties quoted.

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Development of Critical Experiment Database

The HISS benchmarks are measurements of kf and are based on central region experiments."9 The single 

experiment considered in this work (20 of Table 1) is the homogeneous uranium graphite (HUG) measurement. 

The HISS(HUG) contains HEU plus graphite with a graphite/U-235 ratio of 302, and a boron/U-235 ratio of 1.16. 

The measured value of kf for this system is 1.000 ± 0.004. 

The ORNL L-7 to L-11 experiments20 are a family of criticals (53-57 of Table 1) that consist of reflected and 

unreflected spheres of uranyl fluoride in water. Details of the models for each of these systems are given in 

Table 6. The vessel wall is made of aluminum. The measured kff values for each experiment are also given in the 

table. No uncertainty on these values is quoted. A 20-cm-thick region of water at room temperature was assumed 

for the reflector. 

The SHEBA experiment15 is an unreflected cylinder of U(5)0 2F2 + H20. The sensitivity calculations for this 

experiment were performed with a 2-dimensional model and the SEN2 code. 

Table 6 Benchmark descriptions for ORNL L-8 to L-1 I experiments 

Inner radius Vessel wall Water 

Assembly Measured kff (cm) thickness (cm) H/X reflector 

L-7 1.0000 11.5176 0.16 76.1 Yes 

L-10 1.0000 11.8442 0.16 126.5 Yes 

L-8 1.0004 27.9132 0.20 1112 No 

L-11 0.9999 27.9132 0.20 1276 Yes 

L-9 1.0000 34.6327 0.32 1393 No 

2.8 Tabulation of Calculated-versus-Experimental Results and Uncertainties 

The calculated values of kf and associated sensitivity profiles for the set of 102 benchmark experiments described 

above were evaluated using the sensitivity analysis modules SEN1 (1-D models) and SEN2 (2-D models). 2

' The 

cross-section library used in the analyses was the SCALE 44-group library' which is derived from a 238-group 

library processed from ENDF/B-V.' The necessary cross-section covariances were processed from ENDF/B-V 

into the 44-group structure. In most cases, the value of kff was based on the 1-D models also described above. In 

certain cases, like the Rocky Flats experiments, the values of ke are based on 3-D Monte Carlo (KENO V.a) 

models, although the sensitivity results are based on 1-D models. This "substitution" is necessary because the 1-D 

models for these cases contain significant model bias, which should be accounted for in order for the GLLSM 

techniques to be valid. 

In Table 7, the calculated kf values of each of the 102 benchmark experiments are presented, along with their 

uncertainty due to cross-section uncertainties. The uncertainties in the system specification are included in the 

measurement uncertainties by convention. The uncertainties in the calculated kff values were estimated using the 

SEN1 or SEN2 modules, which incorporates the S/U methodology presented in Vol. 1 of this report. The measured 

kf values are also presented, along with their estimated uncertainties. The uncertainties in the measured values are 

incomplete, since a great deal of work is needed in most cases to quantify these values.

NUREG/CR-6655, Vol. 2

Section 2

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Development of Critical Experiment Database

Table 7 Measured-versus-calculated kff values for 102 critical benchmarks

15 NUREG/CR-6655, Vol. 2

System No. 

1 U(2)F4 - 195 

2 U(2)F4 - 294 

3 U(2)F4 - 406 

4 U(2)F4 - 496 

5 U(2)F4 - 614 

6 U(2)F4 - 972 

7 U(3)F4 - 133 

8 U(3)F4 - 277 

9 U(5)308 - 147 

10 U(5)308 - 245 

11 U(5)308 - 320 

12 U(5)308 - 396 

13 U(5)308 - 503 

14 U(5)308 - 757 

15 Godiva 

16 Bapl-1 

17 Bapl-2 

18 Bapl-3 

19 Big-10 

20 HISS (HUG) 

21 U(98) H20 refl. 

22 HEUMET A 

23 HEUMET B 

24 HEUMET C 

25 HEUMET D 

26 HEUMET E 

27 HEUMET F 

28 HEUMET G 

29 HEUMET H 

30 ORNL-1 

31 ORNL-2 

32 ORNL-3 

33 ORNL-4 

34 ORNL-10

Calculated 

k., value 

1.0065 

1.0079 

1.0049 

1.0029 

1.0020 

0.9954 

1.0219 

1.0193 

0.9964 

0.9839 

1.0077 

1.0021 

1.0013 

1.0038 

1.0041 

1.0017 

1.0016 

1.0023 

1.0173 

1.0119 

0.9986 

0.9881 

0.9865 

0.9911 

0.9897 

0.9956 

0.9966 

0.9988 

1.0048 

0.9990 

0.9987 

0.9957 

0.9972 

0.9991

Uncertainty in 

calculated k6 

% std. dev. 

1.6 

1.4 

1.3 

1.2 

1.2 

1.1 

1.6 

1.3 

1.3 

1.1 

1.1 

1.0 

1.0 

0.9 

1.6 

1.2 

1.1 

1.0 

1.7 

2.1 

1.4 

1.6 

1.6 

1.6 

1.6 

1.6 

1.6 

1.6 

1.8 

0.8 

0.8 

0.8 

0.8 

0.9

Measured kff 

value 

1.000 

1.000 

1.000 

1.000 

1.000 

1.000 

1.000 

1.000 

1.000 

1.000 

1.000 

1.000 

1.000 

1.000 

1.000 

1.000 

1.000 

1.000 

1.000 

1.000 

1.000 

1.000 

1.000 

1.000 

1.000 

1.000 

1.000 

1.000 

1.000 

1.000 

1.000 

1.000 

1.000 

1.000

Uncertainty in 

measured k, 

% std. dev. 

0.5 

0.4 

0.5 

0.4 

0.4 

0.5 

0.5 

0.5 

0.4 

0.4 

0.4 

0.3 

0.3 

0.3 

0.1 

NR* 

NR 

NR 

0.3 

0.4 

0.5 

0.5 

0.5 

0.5 

0.3 

0.3 

0.3 

0.3 

0.5 

0.3 

0.3 

0.3 

0.3 

0.3

(Measured/ 

calculated) - 1, 

-0.65 

-0.79 

-0.49 

-0.29 

-0.20 

0.46 

-2.15 

-1.90 

0.36 

1.63 

-0.76 

-0.21 

-0.13 

-0.38 

-0.41 

-0.17 

-0.16 

-0.23 

-1.70 

-1.18 

0.14 

1.21 

1.37 

0.90 

1.04 

0.45 

0.34 

0.12 

-0.48 

0.10 

0.13 

0.43 

0.29 

0.09

Section 2
Development of Critical Experiment Database

Table 7 (continued)

System No. 

35 PCTR 3.73 

36 PCTR 3.78 

37 PCTR 3.83 

38 PCTR 5.84 

39 PCTR 5.99 

40 PCTR 6.23 

41 PCTR 6.90 

42 PCTR 6.95 

43 PCTR 7.14 

44 PCTR 7.52 

45 PCTR 7.52a 

46 ZPR 3/11 

47 ZPR 3/12 

48 ZPR 6/6a 

49 UH3 NI 

50 UH3 UR 

51 TRX-1 

52 TRX-2 

53 ORNL L7 

54 ORNL L8 

55 ORNL L9 

56 ORNL L10 

57 ORNL L11 

58 SHEBA 

59 RF 0.77 A 

60 RF 0.77 B 

61 RF 0.77 C 

62 RF 0.77 D 

63 RF 2.03 A 

64 RF 2.03 B 

65 RF 2.03 C 

66 RF 2.03 D 

67 HST29-1 

68 HST29-2 

69 HST29-3 

NUREG/CR-6655, Vol. 2

Calculated 

k, value 

1.0342 

1.0099 

0.9897 

1.0097 

1.0370 

0O.9829 

1.0293 

0.9750 

0.9971 

0.9678 

1.0225 

1.0201 

1.0138 

1.0247 

1.0108 

1.0072 

0.9947 

0.9975 

1.0053 

1.0063 

1.0034 

1.0029 

1.0016 

1.0089 

1.0037 

1.0116 

1.0061 

1.0107 

1.0062 

1.0105 

1.0065 

1.0001 

1.0032 

1.0062 

1.0002

Table 7 (continued)

16

Uncertainty in 

calculated k, 

% std. dev. 

1.5 

1.5 

1.5 

1.3 

1.3 

1.3 

1.3 

1.3 

1.3 

1.2 

1.2 

1.7 

1.6 

1.9 

1.9 

1.6 

1.1 

0.9 

0.9 

0.8 

0.8 

0.8 

0.8 

1.0 

1.3 

1.1 

0.9 

1.2 

1.3 

1.0 

1.5 

1.5 

1.1 

1.1 

1.1

Measured k. 

value 

1.031 

1.005 

0.986 

1.005 

1.031 

0.986 

1.030 

0.974 

0.992 

0.960 

1.019 

1.000 

1.000 

1.000 

1.000 

1.000 

1.000 

1.000 

1.000 

1.0004 

1.000 

1.000 

0.9999 

1.000 

1.000 

1.000 

1.000 

1.000 

1.000 

1.000 

1.000 

1.000 

1.000 

1.000 

1.000

Uncertainty in 

measured k, 

% std. dev. 

0.6 

0.6 

0.6 

0.6 

0.6 

0.6 

0.6 

0.6 

0.6 

0.6 

0.6 

0.3 

0.2 

0.1 

NR 

NR 

NR 

NR 

NR 

NR 

NR 

NR 

NR 

NR 

0.1 

0.1 

0.1 

0.1 

0.1 

0.1 

0.1 

0.1 

0.7 

0.6 

0.7

(Measured/ 

calculated) - 1, 

-0.31 

-0.49 

-0.37 

-0.47 

-0.58 

0.32 

0.07 

-0.10 

-0.51 

-0.80 

-0.34 

-1.97 

-1.36 

-2.41 

-1.07 

-0.71 

0.53 

0.25 

-0.53 

-0.63 

-0.34 

-0.29 

-0.16 

-0.88 

-0.37 

-1.15 

-0.61 

-1.06 

-0.62 

-1.04 

-0.65 

-0.01 

-0.32 

-0.62 

-0.02

Section 2
Development of Critical Experiment Database

Table 7 (continued)

System No. 

70 HST29-4 

71 HST29-5 

72 HST29-6 

73 HST29-7

Calculated 

k value 

0.9964 

1.0015 

1.0050 

1.0031

Uncertainty in 

calculated kff, 

% std. dev. 

1.1 

1.1 

1.1 

1.1

Measured kf 

value 

1.000 

1.000 

1.000 

1.000

Uncertainty in 

measured k, 

% std. dev. 

0.7 

0.7 

0.7 

0.6

(Measured/ 

calculated) - 1, 

0.36 

-0.15 

-0.50 

-0.31

74 HST30-1 

75 HST30-2 

76 HST30-3 

77 HST30-4 

78 HST30-5 

79 HST30-6 

80 HST30-7 

81 HST31-1 

82 HST31-2 

83 HST31-3 

84 HST31-4 

85 ICT02-1 

86 ICT02-2 

87 ICT02-3 

88 ICT02-4 

89 ICT02-5 

90 ICT02-6 

91 LCT32-1 

92 LCT32-2 

93 LCT32-3 

94 LCT32-4 

95 LCT32-5 

96 LCT32-6 

97 LCT32-7 

98 LCT32-8 

99 LCT32-9 

100 LST05-1 

101 LST05-2 

102 LST05-3 

*None reported (NR)

NUREG/CR-6655, Vol. 2

1.0002 

1.0006 

0.9995 

1.0046 

1.0006 

1.0024 

1.0007 

1.0012 

1.0035 

1.0027 

1.0000 

0.9944 

0.9927 

0.9997 

0.9953 

0.9927 

0.9915 

0.9977 

0.9912 

0.9881 

1.0095 

0.9977 

0.9982 

1.0079 

1.0020 

1.0026 

0.9988 

0.9989 

0.9989

1.0 

1.0 

1.0 

1.1 

1.1 

1.1 

1.3 

1.1 

1.1 

1.1 

1.1 

1.0 

1.0 

1.0 

1.0 

1.0 

1.0 

1.2 

1.3 

1.3 

1.0 

1.0 

1.0 

1.0 

1.0 

1.0 

1.1 

1.1 

1.1

1.000 

1.000 

1.000 

1.000 

1.000 

1.000 

1.000 

1.000 

1.000 

1.000 

1.000 

1.000 

1.000 

1.000 

1.000 

1.000 

1.000 

1.000 

1.000 

1.000 

1.000 

1.000 

1.000 

1.000 

1.000 

1.000 

1.000 

1.000 

1.000

0.4 

0.3 

0.3 

0.6 

0.6 

0.6 

0.6 

0.5 

0.6 

0.6 

0.7 

0.4 

0.4 

0.4 

0.4 

0.4 

0.4 

0.4 

0.4 

0.4 

0.4 

0.3 

0.3 

0.5 

0.4 

0.4 

0.4 

0.5 

0.6

-0.02 

-0.06 

0.05 

-0.46 

-0.06 

-0.24 

-0.07 

-0.12 

-0.35 

-0.27 

0.00 

0.56 

0.74 

0.03 

0.47 

0.74 

0.86 

0.23 

0.89 

1.20 

-0.94 

0.23 

0.18 

-0.78 

-0.20 

-0.26 

0.12 

0.11 

0.11

17

Section 2
Development of Critical Experiment Database

A complete analysis of the measurement uncertainties was carried out on the first 14 experiments. The details of 

this analysis are described in Section 4 to serve as guidance on the generation of similar results for all systems 

included in a GLLSM analysis. 

For the purposes of this work, a range of measurement uncertainties corresponding to 0.1 to 0.4% was assumed. 

Various scenarios were assumed to determine the sensitivity of the final results to the explicit values assumed for 

the measurement uncertainties. This procedure should be useful in establishing a standard approach toward the 

estimation of measurement uncertainties.

NUREG/CR-6655, Vol. 2

Section 2

18
3 USE OF METHODOLOGY FOR GREATER THAN 5-WT % 

ENRICHED URANIUM APPLICATIONS 

The GLLSM methodology, as described in Volume 1 of this document,6 is quite complex and may be quite daunting 

to the criticality safety practitioner. For this reason, this section is designed to proceed through a criticality 

validation exercise using'standard techniques, S/U techniques, and GLLSM techniques as demonstration exercises. 

Each of these techniques will use the same benchmark database which consists of the 102 experiments described in 

the previous section. These experiments are not specifically "tailored" for the greater than 5 wt % applications; 

they represent a generic set which could be used for a number of different applications using these new validation 

techniques. The S/U and GLLSM approaches have several options, which will be examined; conclusions will be 

drawn from the various approaches. 

3.1 Description of U(11)0 2 Systems 

The illustrative application for which validation is sought is a series of U(l 1)02 systems with H/X values of 0, 3, 5, 

10, 20, 40, 80, 200, 300, 400, 500, 600, 800, and 1000. These U(11)0 2 systems are "artificial" critical bare 

spheres generated for calculational comparison purposes. The object of this demonstration problem is the 

validation of the underlying cross sections and criticality code for these 14 systems. 

The dimensions for the U(1 1)02 systems were generated via a critical radius search calculation using the 

SCALE/XSDRNPM module. The critical radii corresponding to the 14 unreflected systems for H/X of 0 to 1000 

were 50.55, 40.00, 36.85, 31.77, 26.48, 22.34, 19.87, 18.99, 19.53, 20.39, 21.42, 22.51, 25.41, and 29.17 cm. 

3.2 Sensitivity and Uncertainty Analyses of the U(11)0 2 Systems 

These 14 systems were analyzed with the SEN1 sensitivity module, which generates detailed sensitivity 

information, as well as the uncertainty in the system kff value due to cross-section uncertainties obtained from the 

Evaluated Nuclear Data Files (ENDF/B). The same cross-section library (SCALE 44-group) and associated 

covariances used in the analysis of the critical experiments were used for the analysis of these 14 applications. 

Using this sensitivity and uncertainty information, the uncertainty in the calculated value of kff for each system can 

be generated. Also, the covariance of any two systems can easily be determined. To illustrate, Table 8 provides the 

uncertainty matrix for the 14 U(1 1)02 systems. This matrix gives the system uncertainty for the calculated ke on 

the diagonal and the correlation coefficient on the off-diagonal element. These off-diagonal elements are the ck 

parameters, and can be used for defining similar systems. Such a matrix can also be generated for the entire set of 

benchmark experiments plus applications. In this case, an 116 x 116 matrix was generated, corresponding to the 

102 criticals and 14 applications. For this study, only a 14 x 102 matrix was necessary (i.e., each row 

corresponding to each of the 14 applications and 102 columns corresponding to each of the benchmarks). Shown in 

Figures 1-4 are rows of the full matrix corresponding to the U(1 1)02 systems for H/X values of 0, 3, 40 and 500. 

The values plotted correspond to the ck coefficients for all 102 systems. (Note that labels are only given to every 

third system. The system order is the same as shown in Tables 1 and 7.) These plots allow, at a glance, the 

determination of the number of systems in the benchmark dataset that are similar to the particular application. 

From Figure 1, it is noted that 4 systems have c, values at or greater than 0.9 compared to the H/X = 0 system. 

These values indicate that, based on the conclusions from Volume 1 where five systems with ck of 0.9 or higher are 

suggested to ensure validation, the U(1 1)02 system with an H/X value of 0 is marginally covered by this benchmark 

set. However, it is clear from Figure 2 that the U(1 1)02 system with an H/X of 3 is not adequately covered by this

19 NUREG/CR-6655, Vol. 2
Application of Methodology Section 3

benchmark set. The remaining two graphs, shown in Figures 3-4, indicate that the U(1 1)02 systems with HI/X 

values of 40 and 500 are adequately covered by this benchmark series. 

Table 8 Correlation coefficientsa for the U(l 1)02 systems 

Critical system 11%-0 110/&-3 11%-5 11%-10 110/c-20 11%-40 11%0-80 11%-200 11%-300 11%-400 11%0-500 11%-600 11%-800 11%-1000 

ll%H/X=0 0.0191 

11% H/X= 3 0.8328 0.0185 

11%H/X=5 0.7379 0.9818 0.0188 

11% H/X= 10 0.6011 0.9205 0.9725 0.0188 

11%H/X=20 0.4887 0.8409 0.9161 0.9784 0.0176 

11% I/X = 40 0.4067 0.7562 0.8403 0.9253 0.9763 0.0151 

11%HIX=80 0.3428 0.6585 0.7392 0.8327 0.9094 0.9698 0.0128 

11% HI-X=200 0.2800 0.5240 0.5888 0.6760 0.7696 0.8705 0.9526 0.0106 

11%HIX=300 0.2633 0.4751 0.5315 0.6115 0.7058 0.8157 0.9148 0.9832 0.0099 

11% I/X = 400 0.2557 0.4452 0.4953 0.5687 0.6604 0.7727 0.8798 0.9668 0.9846 0.0095 

11% HIX=500 0.2517 0.4329 0.4688 0.5359 0.6235 0.7349 0.8453 0.9448 0.9717 0.9845 0.0091 

11% HIX= 600 0.2490 0.4076 0.4482 0.5097 0.5927 0.7014 0.8123 0.9200 0.9543 0.9742 0.9847 0.0089 

11%H-X= 800 0.2432 0.3755 0.4076 0.4567 0.5278 0.6265 0.7331 0.8514 0.8991 0.9330 0.9576 0.9734 0.0087 

11% H/X= 1000 0.2353 0.3452 0.3697 0.4071 0.4652 0.5509 0.6484 0.7702 0.8277 0.8731 0.9097 0.9367 0.9752 0.0087 aNote the diagonal element contains the fractional standard deviation instead of the correlation coefficient.

NUREG/CR-6655, Vol. 2 20
W 

0.9 

0.8 

0.7 

0.6

0EU(11)02 H/=O 

0 0.5 

0.4 

0.3 

0.2 

0.1 

to, -t CV) 0 C0 C~ ) 0 -l 0 (3 CO 14 ~ I'- 0 Cl) C~N C%4 

0) m C 0 0 ) '0 r M 9 0 

0 0 

N CQ CQ 

Cl CL CL' CL0 

C)D 

Figure 1 Correlation coefficients, ck, for U(1 1)02, H/X =0 system with 102 benchmark experiments 

0
V,) 0 V, 1 V ) ) 0 C a) ) M' .0 90 o 0- M 

C C O) CO)*

E 

0 OR N 1' f-v 999 (z 

CO (0 f0 MgM.l ) M 0 :

o V o o

Figure 2 Correlation coefficients, ck, for U(1 1)02, HLX = 3 system with 102 benchmark experiments

z 

0 

0\ 

(it 

(.ji 

C 

IN)

1 

0.9 

0.8 

0.7 

0.6 

0.5 

0.4 

0.3 

0.2 

0.1 

0

0 

0 

0 

OfQ 

o--

CO 

0
En 

U-) 

1* 

Ml U(l1 )02 H/X=401 

0.9 

0.8 

0.7 

0.6 

0.5 

0.4 

0.3 

0.2 

0.1 

O - 0 E C! Da CF ) 0 0. 

a tor - P- C) C 

0. 0L a- 0 

(ON 

LA Figure 3 Correlation coefficients, ck, for U(1 1)02, H/X = 40 system with 102 benchmark experiments C 

0 

o 00 cI 

t'J
v7z Z'IOA '9999-?ID/Da3fIfN

Ck 

& w IN, C~n 0) -4 00 (a crc 

0 

0 

0 

0 

0 

0 

0 

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II 

Lu 

ci, 

0 

0 

0 

0 -t 

0 

ci,

u(2) R h(= 195 

u(2)f4 h/x=614 

u(5)3o8 h/x=147 

u(5)3a8 hlx=503 

bapl-2 lattice 

U(98) h20 refi 

heumet d 

heumet h 

om"

pctr 1.006% 32.83 

pctr 1.157% 6.90 

pctr 1.157% 7.52 

uh3-ni 

ornl 17 

orn!1 II 

rf 0.77 c 

rf 2.03 c 

hst-029-03 

hst-029-07 

hst-030-04 

hst-031-01 

ict-002 core 1 

ict-002 case 5 

lct-032-3

ctd-032-7

Ist-05-02.

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a, 

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ý, UOTIOOS
Application of Methodology

Similarly, a matrix of values corresponding to the D,, D,, and D, values previously defined can also be generated 

based only on the sensitivity coefficients generated using the SENI module. In Figures 5-8, the sum of the D•, Dn, 

and Ds values are plotted for the U(l1)0 2 systems with HI/X = 0, 3, 40, 500, respectively, to each of the 

102 benchmark systems. Using this technique, the criterion for similar systems is that the sum of the D values 

should be less than about 1.2 (3 times the 0.4 criterion). In Figure 5, similarity is indicated for the U(l1l)0 2 

H/X = 0 system and the same four systems previously determined to be similar. Figure 6 predicts similarity 

between the U( 1)02 II/X = 3 system and only one other system. In Figures 7 and 8, the systems with H/X values 

40 and 500 indicate about 10 and 40 systems, respectively, meet the less than 1.2 criterion. Hence, conclusions 

relative to system similarity based on these D coefficients agree with those of the preceding analysis using the ck 

values. 

3.3 Traditional Trending Analysis 

In order to clearly show the relationship between the GLLSM techniques and the more traditional techniques for 

criticality safety validation, a traditional trending analysis using the same 102 benchmark experiments is presented. 

In Figure 9, k. is trended versus the energy of average lethargy causing fission (EALF). The prediction from this 

analysis would be a nearly constant positive bias of about 0.3%. Results shown in Figure 10 correspond to a 

similar parameter, energy of average lethargy causing capture (EALC). These results predict essentially the same 

biases as the EALF parameter and will not be further discussed. 

Trend plots are also shown in Figures 11 and 12 for HIX and enrichment parameters. The H/X trend plot shows a 

slight trend, with the predicted Ak bias near zero for high H/X values and about +0.005 for low H/X values. The 

trend with enrichment is similar, but not enough data are present for the intermediate enrichments to confirm the 

trend. 

Note that the largest variations (±2%) about the trend lines are seen for fast systems (i.e., the upper portion of the 

EALF, EALC trend plots and the lower portion of the H/X trend plot). Upon examination, it was observed for H/X 

= 0 systems that the predicted ky values less than unity were from the IIEUMET set of criticals, while the systems 

with predicted eigenvalues greater than unity were from the Big-10 and ZPR sets. The combined effects of the 

high-versus-low enrichments and the processing of the SCALE 44-group library (collapsed using a thermal reactor 

spectrum) are believed to be responsible for this variation. 

As a result of the trending analysis, a prediction of the Ak bias and its uncertainty can be obtained for each of the 

U(l1)0 2 systems. Predictions using the USLSTATS procedure (Ref. 5) for systems with HIX values of 0, 3, 40, 

and 500 are given in Table 9. 

Table 9 Predicted Ak bias and its standard deviationa based on traditional trending procedure 

H/X = 0 System H/X = 3 System H/X = 40 System H/X = 500 System 

Trending % bias % std. dev. % bias % std. dev. % bias % std. dev. % bias % std. dev. of 

Parameters of biased k• of biased kff, of biased kf biased k• 

EALF 0.17 0.70 0.26 0.70 0.26 0.70 0.26 0.70 

H/X 0.32 0.71 0.32 0.71 0.31 0.71 0.17 0.71 

"aStandard deviations correspond to the "pooled standard deviation" as specified in Ref. 5.

NUREG/CR-6655, Vol. 2

Section 3

25
Application of Methodology Section 3

-90-ISI 

ir aseo zoo 

L 9-100 zoo-P! 

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CO-OCD-Isq 

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6 C-ý C1 1-: 6

NUREG/CR-6655, Vol. 2 26
0 

3.5 

H DC 

2.5

2 

D 

1.5 

0.5 

t• (D C, U, O -0 M -0 M '- - IN o 'T 

0 0 0 0 0 0 "7E M7 W t- i 99990 

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V, vl. 0 

0 

("5 (C) :5 CL CL. ON• a. a a.• 

0 

0 (10 

Figure 6 D coefficients for U(1 1)02, H/X = 3 system with 102 benchmark experiments
Figure 7 D coefficients for U(1 1)02, H/X = 40 system with 102 benchmark experiments

z 

C) C-, tll 

(J 

CI

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29

Application of Methodology

NUREG/CR-6655, Vol. 2
z 

0 

ON 

ON 

Ut 

C,' 

0 

I,J

1.OOE-02 11.OOE+00 1.OOE+02 1.OOE+04 1.OOE+06

EALF (ev) 

Figure 9 Trend plot for kf versus energy of average lethargy causing fission (EALF)

1.03 

1.02 

1.01 

1 

I0.99 

0,0.98 

0.97 

0.96 

0.95 

0.94 

0.93

C) 

0

0 

0

0

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1.03 

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0.995 " 

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0.985 

9 

0.98 , 

1.00E-02 1.OOE+00 1.OOE+02 1.OOE+04 I.OOE+06 

o• ~EALC (ev) ° t.p. 

Figure 10 Trend plot for kff versus energy of average lethargy causing capture (EALC) 0 0 On
x 

0 ON 1.03 0 

S~~1.02" 

0 0 

C0 

• S0.99 

' 0.98 

0.97 -- ---------------------------- us 

0.96 

0.95 

0 9 ........... "................ :.......... ........................................ i............................................... ..... 0.93 

0 500 1000 1500 2000 

HIX 

Figure I 1 Trend plot for keff versus hydrogen-to-25U ratio (H/X) 

0 t)
1.03 

1.025

1.02 : 

1.015 

I 1.01 S1.005 

0.995

0.985

0.98,, 

0 20 40 60 80 100 

0 Enrichment (wt %) 

Figure 12 Trend plot for kf versus enrichment t.• . 

0 0
Application of Methodology

3.4 Trending Analysis Using D Values 

This section will discuss trending analyses using the same set of 102 benchmarks as the traditional analyses shown 

above; however, the trending parameters are now the D coefficients, described in Volume 1. Even though it is 

possible to perform the trending on each of the D coefficients independently, it was decided to trend kff versus the 

sum of these coefficients (i.e., D.. = D, + Dn + D8). This combination reduces the number of trends plots to be 

examined. 

The trend plot of kff versus Dý, is given in Figure 13 for the U(11)0 2 H/X=3 system.* These plots are analyzed in 

quite a different method from the traditional approach. Since the D coefficients reflect the deviations of the 

sensitivities of benchmark systems from the respective sensitivities of the application system, D equal to zero is our 

goal, i.e., the system does not deviate from itself. A D, value of zero corresponds to the U(1 1)02 H/X = 3 system 

for Figure 13. The trend line must therefore be extrapolated to zero in order to estimate the Ak bias. Therefore, the 

slope of the trend line is not nearly as important as where it crosses zero and how many systems are in the region of 

D..m less than 1.2 (i.e., 3 times 0.4 as postulated earlier). Extrapolation to D = 0 should not be misinterpreted as an 

extrapolation to systems that are not bracketed between similar systems. From this plot it is clear that perhaps only 

one other system could be considered similar to the U(l1)0 2 HIX = 3 system (i.e., Duless than 1.2). Hence, the 

predicted bias will have somewhat large uncertainties associated with it. The trend plot for the U(l 1)02 H/X = 40 

system is shown in Figure 14. Here the coverage near a D, value of zero is much better than the value shown in 

Figure 13. In this case, there are at least 8 systems with D, values of 1.2 or less. The trend plot for the last 

U(l 1)0, system (i.e., H/X = 500) is shown in Figure 15. Here the conclusions are very similar to those in 

Figure 14. A large number of systems are within a Dm value of 1.2, with a resulting good prediction of the Ak 

bias for this system. 

These trending analysis results are generated using the same software that was used in the traditional trending 

approach in Section 3.3. Therefore, the same type of estimates for the Ak bias and its uncertainty can be obtained 

from these analyses. These bias predictions are given in Table 10. Note that the uncertainties in the predicted bias 

shown in Table 10 all have about the same value. This is due to the use of all benchmarks in the trend for each of 

these four application systems. An optional choice of using only similar experiments in the trending would have 

produced biases corresponding closely to the similar experiments, but would have very large (and perhaps invalid) 

uncertainties for the HIX = 0 and 3 cases due to the very limited number of applicable experiments. 

Table 10 Predicted Ak bias and its standard deviation0 based on D,8 m trending procedure 

H/X = 0 System HIX = 3 System HX =40 System H/X = 500 System 

Trending % bias % std. dev. % bias % std. dev. % bias % std. dev. % bias % std. dev. 

Parameters of biased kff of biased ke of biased keg of biased kff 

Dsum 0.74 0.71 1.20 0.70 0.19 0.72 0.09 0.71 

0

Standard deviations correspond to the "pooled standard deviation" as specified in Ref. 5. 

The U(1 1)02 H/X = 0 system was analyzed, but the trend plot is not shown. See Table 10 for a 

summary of the complete results.

NUREG/CR-6655, Vol. 2

Section 3

34
o• 0 

"iuo s =s X/H -iO(i I)fl ofl JoJ onfA '"'(I snSJOA gff.IOJ iold puoju I oJn•Id 

(, Il ,JI , 6o 

96"0 9 .6"0tt 

•1sn -96'0 c$ 96* 

9660 

" " ". ............................ ....1 

ZO' .•, 

Co 

0
z> 

1.03o 

1.02 

1.01 "" * * 0 

0.99 

0.98 - USL, 

0 .9 7 - . .. . . . . . . . . . . . . . . . . . . . . . . . .I.. . . .. .--.. . . .. 

0.96 

0.95 USLI 

0 .9 4 . ................................................................................................................................................... ,....... ......... ................ 0.94 

0.93 

0 1 2 3 4 

Dn+Dc+Ds 

Figure 14 Trend plot for kff versus D,,m value for the U(I 1)02 HIX =40 system 

0
C~c 

1.03 

1.02 

1.01 

0.99 " " . • 4 

c, 0.98 USL2 

0.96 

0.95 -USL 

0 .9........................................................................................................... ................................. ......................... ..... 0.93 , 

0 1 2 3 4 

Dn+Dc+Ds 0 

LA Figure 15 Trend plot for kfversus Dum value for the U(l 1)02 HX = 500 system 

0 C C
Application of Methodology

3.5 Trending Analysis Using Ck Values 

The trending analyses using the cý values follow very closely to the analyses using the D coefficients discussed in 

Section 3.4. Here the trend curves are interpreted as an extrapolation to a ck value of unity, which corresponds to 

the particular application system of interest; i.e., correlation coefficient with an experiment identical to the 

application. The slope of the trend curve is again of secondary importance. The items of primary importance are 

the number of systems with a c. value greater than 0.8 and the value of the predicted Ak bias at a ck value of unity. 

The kf trend plot for ck of a U(l 1)02 H/X = 0 system is shown in Figure 16. This trend plot is interesting when 

compared with the traditional trend plot shown in Figure 9. The four data points in the upper-right-hand portion of 

both plots correspond to the same four systems (three ZPR and the Big-10). In Figure 9, the predicted Ak bias is 

about 0.4% because the overprediction of ky for these four systems is counteracted by the underprediction of the 

HEUMET systems, which all have very similar values of EALF. However, the trend seen for kff in Figure 16 is 

caused by the lack of similarity between the U(1 1)02 H/X = 0 and HEUMET systems. These HEUMET systems 

have a c, value of about 0.5-0.6, indicating only minor correlations with the U(1 1)02 H/X = 0 system This 

example shows the potential improvement from the use of a trending analysis with these new parameters, since 

trends can be observed as a function of systems that are expressly determined to be similar. It is clear from the 

preceding analyses that sometimes the traditional parameters indicate that systems should be similar, but are not. 

The trend plots for the remaining U(1 1)02 systems with H/X values of 3, 40, and 500 are given in Figures 17 to 19. 

For the system with an HIX value of 3, the predicted biases are higher than those predicted by the standard 

techniques. The specific reasons for these differences were not explored in depth as with the HIX of 0 cases, but 

are believed to be caused by the separation of effects that tended to cancel each other. In any case it should be 

noted that all points with ck Ž 0.8, i.e., similar systems, have kff values that are greater or equal to one. The Ak 

bias predicted for the H/X = 40 and H/X = 500 systems are in line with those of the standard techniques since there 

are a large number of experiments considered to be similar, and no cancellation of effects is seen. 

These trending analysis results are generated using the same software that was used previously in Sections 3.3 and 

3.4. Estimates of the Ak bias and its uncertainty from this trending approach are given in Table 11. Again, note 

the similarities of the uncertainties in the bias predictions. This is due to the use of all benchmarks in the trending, 

regardless of the magnitude of their ck values. 

Table 11 Predicted Ak bias and its standard deviationa based on ck trending procedure 

H/X = 0 System H/X = 3 System H/X = 40 System HIX = 500 System 

Trending % bias % std. dev. % bias % std. dev. % bias % std. dev. % bias % std. dev. 

Parameters of biased k, of biased k. of biased k. of biased kf 

ck 1.34 0.66 1.15 0.66 0.26 0.72 0.39 0.79 

"aStandard deviations correspond to the "pooled standard deviation" as specified in Ref. 5.

NUREG/CR-6655, Vol. 2

Section 3

38
0 

L~) 

1.03 

1.02 

1.01 . ........................................ 

0.99 . 0.98 USL 

- -- - - - - -- - - - - - ------ - -- - -

0.97 -

0.96 

0.95 - USU 

0.94*. 

0.93 

0 0.2 0.4 0.6 0.8 

Ck 

0 Lh Figure 16 Trend plot for kff versus ck value for the U(1 1)0O2 H/X =0 system 0 

0 0 

0. 2
0 

1.03 

i 1.02 

101 

-99. USL2 

--- -- " --- - - - -

0.-97 

0 0.96 

0.95 USL. 

0 .9 4 - .......... ........................................ . . .................... -........ . :........... '......................... '..................... ....., ... 

0.93 , 1 

0 0.2 0.4 0.6 0.8 1 

Ck 

Figure 17 Trend plot for kf versus ck value for the U(I 1)02 H/X =3 system 

0
4

4

G)

1.03 

1.02 

1.01 

1 

0.99 

0.98 

0.97 

0.96 

0.95 

0.94 

0.93

0

* 4 8 

8 4 4

U..., 4

4

4 

* 4 

4-p.,.,, 4 # 4

- p

* *4 

*' 

4$

USL2

USLI

0.2 0.4 0.6

Ck 

Figure 18 Trend plot for keff versus cq value for the U(l 1)02 HIX = 40 system

Lto 

10

(n

0.8 1

0 

0 

0 

0 

o 

0"e
z> 

1.03 0 

C - O1I'-'-h O 0 

1.02 

0 

101 .0 .

'0.99 

S0.98 

USL2 0.97 ---

0.96 

0.95 0.94 USLI 

0.93 

0 0.2 0.4 0.6 0.8 1 

Ck 

Figure 19 Trend plot for kff versus ck value for the U(1 1)0-2 H/X =500 system 0• 

0•
Application of Methodology

3.6 GLLSM Analysis of 11-wt % Systems 

The GLLSM technique is a procedure completely different from the preceding trending techniques, although it does 

have the same intended endpoint (i.e., the determination of the predicted Ak bias and associated uncertainty for the 

set of U(l 1)02 systems) and GLLSM uses the same sensitivities and correlations as the other techniques. As was 

shown in Volume 1 of this report, the GLLSM procedure is capable of indicating not only the predicted Ak bias, 

but also demonstrates its convergence to a single value as the number of experiments is increased. The (e-a)/c 

quantity used in Figure 20 is the same output value discussed in Volume 1; however, it has a different meaning with 

regard to an application area as used here. The difference in meaning is due to the lack of an experimental value, e, 

for an unmeasured application. By convention the calculated and measured values are set to 1.0 for the application, 

thus (e-a)/c is directly related to ka(a) - m,' shown in Eq. (20) in Volume 1, which is the prediction of Ak bias. 

Shown in Figure 20 is the convergence of the predicted Ak bias as a function of experiment groups. The 

experimental groups are defined based on similar system characteristics as follows: 

Group No. Experiment descriptions 

1 Godiva, HISS, HEUMET 

2 ORNL spheres 

3 UH3, ORNL L7-L11 

4 U(5) 30 8, Big-10, ZPR, 

5 SHEBA, Rocky Flats 

6 U(2)F4, U(3)F 4 

7 BAPL, PCTR, TRX 

8 Russian experiments 

The moderated systems (H/X > 20) appear to converge quicker than the fast systems. In this example, the fast 

systems require four of the groups to be included for convergence, while the moderated systems only required three 

groups. This difference is likely due to the larger number of moderated systems, since the procedure does not favor 

one group of systems over the other. The predicted Ak bias and uncertainties for all 14 of these systems are shown 

in Table 12.

NUREG/CR-6655, Vol. 2

Section 3

43
z0 

6- a 0 

5-U- H/X=3 

4 -N- ./X=5 

x-- H/X=10 

-- w3 H/X=20. 

"-.-- H/X=40 

-- H/X=80 " 2 

(U - H/X=200 

) - H/X=300 

o H/X=400 

--- H/X=500 

-- H/X=600 

-1 --- H/X=800 

-'- H/X=1000 

-2 i.. 

0 1 2 3 4 5 6 7 8 

experiment groups considered 

Figure 20 Plot of convergence in predicted kff bias, [(e-a)/c], for U(1 1)02 systems as a function of number of 

experimental groups considered 

0
Application of Methodology

Table 12 Predicted Ak bias and its standard deviationa based on GLLSM procedure 

H/X = 0 System H/X = 3 System H/X 5 System H/X = 10 System 

% bias % std. dev. % bias % std. dev. % bias % std. dev. % bias % std. dev. 

of biased kf of biased kf of biased k. of biased k. 

2.59 0.46 1.90 0.39 1.50 0.35 0.93 0.35 

H/X =20 System H/X =40 System HIX = 80 System H/X = 200 System 

% bias % std. dev. % bias % std. dev. % bias % std. dev. % bias % std. dev. 

of biased ke of biased kf of biased kgf of biased kff 

0.51 0.39 0.35 0.43 0.36 0.46 0.42 0.46 

H/X = 300 System H/X = 400 System H/X = 500 System H/X = 600 System 

% bias % std. dev. % bias % std. dev. % bias % std. dev. % bias % std. dev. 

of biased kf of biased kf of biased kf of biased ke 

0.41 0.42 0.40 0.39 0.37 0.35 0.35 0.33 

H/X = 800 System H/X = 1000 System 

% bias % std. dev. of biased kff % bias % std. dev. of biased kff 

0.30 0.29 0.24 0.28 

,Standard deviations correspond to the value output from the GLLSM process. See Appendix B of Volume 1 for details. 

3.7 Summary of Trending Analyses 

In the preceding sections, results from a number of approaches to criticality safety data validation were presented. 

Quite interestingly, they give very different answers for the set of application problems chosen for study. The 

predicted bias are in some cases up to a factor of 15 different. The primary reason for these differences seems to be 

the inclusion of systems that may "look" very similar from the standpoint of certain parameters, but vary with 

respect to other parameters. In particular, according to both the H/X and EALF parameters, the HEUMET and 

ZPRfBig-10 problems are similar. However, with respect to the sensitivities and uncertainties, they appear to be 

quite different. Cancellation of effects due to systems that "appear" to be similar causes the traditional trending 

approaches to underpredict the actual bias for low-moderation systems with intermediate enrichments. This 

situation is evident in Table 13, where each of the previously reported results are presented in summary form. 

Note also that the differences between the ck, D, and GLLSM results for low H/X systems would have been 

minimized if only the similar experiments had been included in the trending studies. The validity for inclusion and 

the best approach options for inclusion of dissimilar (ck < 0.8 or D > 1.2) experiments in the trending analyses will 

be studied in future workNUREG/CR-6655, Vol. 2

Section 3

45
Application of Methodology

Table 13 Comparison of predicted Ak bias and the standard deviationa of the estimated kff for various procedures 

H/X = 0 System HIX = 3 System HfX = 40 System H/X = 500 System 

Trending % bias % std. dev. % bias % std. dev. % bias % std. dev. % bias % std. dev. 

Procedures of biased ky of biased k, of biased kf of biased kf 

EALF 0.17 0.70 0.26 0.70 0.26 0.70 0.26 0.70 

H/X 0.32 0.71 0.32 0.71 0.31 0.71 0.17 0.71 

Dsum 0.74 0.71 1.20 0.70 0.19 0.72 0.09 0.79 

ck 1.34 0.66 1.15 0.66 0.26 0.72 0.39 0.79 

GLLSM 2.59 0.46 1.90 0.39 0.35 0.43 0.37 0.35 

aStandard deviations correspond to the "pooled standard deviation" as specified in Ref. 5. 

The predicted bias from these applications are all positive. Since the USLSTATS program5 sets a positive bias to 

zero for the purposes of determining USL1 and USL2, the overprediction found in these applications does not 

present a safety concern. However, a similar situation can be easily postulated where a predicted positive bias is 

actually a negative bias. With the inclusion of strict confidence levels, along with an additional margin of 

subcriticality, the cumulative effect of these factors should still be conservative. However, prudent application of 

trending procedures is very important in criticality safety validation exercises. 

It can be seen in Table 14 that the predicted USL1 (with an assumed 5% administrative margin of subcriticality for 

safety) and USL2 values are nearly identical for all the procedures, with the exception of the GLLSM, which 

predicts USL values some 0.5-1.5% higher due to the lower uncertainty estimates. Caution is advised with the 

direct use of the USL values from the GLLSM since these results are estimated based on a comparison of the 

standard deviations and USL values from the ck trending analysis. The development of the statistical basis for the 

USL values using the GLLSM approach is currently under way in another related project. 

Table 14 Comparison of predicted USL1 and USL2 values for the various procedures 

Trending H/X = 0 System H/X = 3 System H/X = 40 System HI/X = 500 System 

Procedures USL1 USL2 USLI USL2 USL1 USL2 USL1 USL2 

EALF 0.938 0.970 0.938 0.970 0.938 0.970 0.938 0.970 

HI/X 0.937 0.969 0.937 0.969 0.937 0.969 0.937 0.969 

DSUM 0.937 0.968 0.937 0.967 0.938 0.970 0.938 0.971 

ck 0.938 0.971 0.938 0.971 0.937 0.969 0.936 0.966 

GLLSM 0.942 0.979 0.943 0.982 0.942 0.982 0.944 0.985

NUREG/CR-6655, Vol. 2

Section 3

46
4 GUIDANCE FOR DETERMINATION OF APPLICABILITY AND 

ADVANCED TRENDING PROCEDURES 

The proposed use of multiple criticality safety data validation techniques is not to make the task of validation more 

difficult, neither is it to make the task less difficult. The goal of this work is to allow for criticality safety validation 

under conditions not possible previously, primarily the extrapolation beyond the range of the experimental 

benchmarks. Techniques developed in this work provide a more rigorous and formal definition of what 

applicability really connotes. 

4.1 Recommendations on the Use of Trending Procedures 

As stated in the Volume 1 summary of this document,' this work has recommended that two systems with a Ck value 

of 0.8 or higher can be considered similar enough that one can serve as a benchmark in the validation of the other. 

This conclusion does not imply that a formal S/U analysis needs to be redone for all previous criticality validation 

exercises. Clearly, a series of systems with the nearly identical characteristics to the application of interest (i.e., 

same enrichments, same moderators, a very tightly spaced sequence of moderator/fuel ratio values, similar 

geometries, etc.,) should be acceptable for use as a validation set. Indeed, in Volume 1 such a series of experiments 

with constant enrichment was shown to have ck values greater than 0.9 for H/X values within a factor of 2 of each 

other. 

Thus, the recommendation is that for benchmark data sets that contain more than 20 benchmarks with a ck value of 

0.8 or higher to the corresponding application area, a traditional criticality safety data validation via the standard 

k. trending analysis method is acceptable. This recommendation doubles the Volume 1 predicted numbers of 

benchmarks needed for a reasonable estimate of the kf bias or the resulting upper subcritical limit. It is further 

recommended that in using the traditional kff trending approach that systems with very different characteristics such 

as high-versus-low enrichment, poisoned-versus-nonpoisoned, wet-versus-dry, etc., not be included together unless 

that specific effect is being analyzed. For example, do not include three or four random systems with boron poisons 

in an analysis of nonpoisoned systems. As was seen in the comparison of the traditional-versus-advanced validation 

procedures in Table 12, the inclusion of widely varying benchmarks can cause cancellation of trends, resulting in an 

under prediction of the ke bias. Similar situations could also result in an over prediction of the kf bias. 

The advanced trending analyses, using the D and ck coefficients as trending parameters, are recommended under 

conditions that are similar but complimentary to the above recommendations for the traditional trending analyses. 

The conditions on the advanced trending analyses parameters are as stated in Volume 1 for a meaningful estimate 

of the kff bias (i.e., 5-10 experiments with a ck value of 0.9 or higher or 10-20 experiments with a c, value of 0.8 

or higher). These recommended parameter ranges are only marginally different from those above; however, the 

motivation for use of these advanced trending analyses is that the inclusion of widely varying system conditions like 

HJX, enrichment, poisons, etc., are actually encouragecd These advanced techniques are very adept at 

differentiating between dissimilar systems, while allowing for a wide range of system types to be validated using the 

single validation data set. To date, the practice has been to include all systems in the benchmark set in the trending 

analyses; however, future work is planned to explore using only those experiments that are determined to be similar. 

Applications for which the advanced validation techniques should be applicable are extrapolation scenarios, which 

are necessary when the desired experiments meeting the strict requirements of the standard techniques are not 

available. This set of recommended application areas is very well represented by the examples shown in Section 3 

for the greater-than-5-wt % enrichment cases.

47 NUREG/CR-6655, Vol. 2
Guidance for Determination of Applicability

The final data validation technique presented in this work is the GLLSM methodology. This technique has been 

extremely useful in the prediction of values of ck and D that are necessary to define an area of applicability, as well 

as the determination of the number of experiments needed to obtain a valid estimate of the kef bias and associated 

uncertainty. Additionally, this technique can be a useful data validation procedure under practically any set of 

conditions. However, since the technique is currently much more complex to implement than the other procedures, 

it is recommended that it only be used in situations where the other approaches are not expected to proceed 

satisfactorily. The GLLSM technique corresponds to a general validation, where the values of ck and D fall outside 

the ranges specified above, or when responses other than kff are needed for inclusion into the validation. Using the 

full GLLSM techniques for criticality safety validation can allow for noncritical experiments to be included into the 

validation procedures. Further development is required before the full GLLSM procedure can be used in these 

applications, since the "similarity" criteria for the ck and D parameters may be different when using noncritical 

systems. A means for ensuring that all important material/reactions and energy regions of a given application 

problem are tested by the validation procedure must be developed. This procedure would then allow for validation 

to occur even with a very diverse set of benchmark data (e.g., noncritical measurements). 

4.2 Trending Procedures for Ck and D Parameters 

The techniques for doing trending analyses with the ck and D parameters are quite different from those of traditional 

trending analyses. The traditional analyses allow for multiple systems to be validated from a single trending curve, 

while the trends with ck and D parameters only allow a single system per trend plot to be validated. This limitation 

results from the fact that the trend parameters are only valid between pairs of systems. Thus the procedure for 

doing a trend analysis, with either the ck or D parameters, is to first generate sensitivity coefficients for all the 

benchmark problems and all the application scenarios. The application scenarios should cover the range of 

materials, moderation, enrichment, impurities, etc., that are expected to be used in the criticality safety analyses. If 

a broad range is needed, it is recommended that the range be broken up into discrete values (like the H/X values in 

the U(1 1)02 example above). A simple postprocessing code should then be able to generate either the ck or D 

parameters for all systems. These parameters can then be trended using standard data-fitting codes with the 

resulting bias and uncertainties of fit and future predictions being extrapolated to either the zero parameter value 

for D or the unity parameter value for ck. This trending must be done for each of the application systems that span 

the range of scenarios for the criticality safety analyses. 

The above techniques specify that the sensitivities for all of the benchmark and application systems be quantified. 

Currently, only 1-D and 2-D geometry capabilities exist for the generation of this sensitivity information; therefore, 

it is necessary to convert the critical systems into either of these two geometrical systems. Standard procedures are 

available that make this conversion process easier (cell-weighting, equivalent buckling, etc.), but the conversion can 

be a time-consuming process that requires some degree of expertise.8 The development of 3-D sensitivity tools 

should help alleviate this limitation. 

4.3 Guidance on Estimating Experimental Uncertainties 

The estimate of an uncertainty matrix can be quite difficult to construct, depending on the availability of a complete 

description of the experiment. Experimentalists routinely give estimates of the uncertainty in a single experiment, 

but do not always give enough information to estimate the common uncertainties between a series of critical 

measurements. If the number of experiments is relatively large, the detailed uncertainty information is typically not

NUREG/CR-6655, Vol. 2

Section 4

48
Guidance for Determination of Applicability

needed. Under these circumstances, it is often desirable to assume that the uncertainties in the respective 

experiments are uncorrelated and allow for the experimental uncertainties to vary between maximum and minimum 

quantities. This variation in uncertainties allows a test of the variation in the bias results to be determined. The 

remainder of this section describes how a full matrix of uncertainties can be generated if sufficient experimental 

detail is available. 

The report describing the U(2)F4 and U(3)F 4 experiments" estimates the uncertainties in the individual material 

number densities. The report gives standard deviations in the UF4 weight percent of 0.7%, paraffin weight percent 

of 1.2%, enrichment of 1.0% (0.7% for the 3 wt % cases), and density correction of 0.3%. These uncertainty 

contributions can be used to define a covariance matrix (x104) corresponding to the individual isotopes for the 

2 wt % cases as follows: 

Material 235U 238

U H C F 

235u 0.72 + 1.02 + 0.32 0.72 + 0.32 0.32 0.32 0.72 + 0.32 

238u 0.72 + 0.32 0.72 + 0.32 0.32 0.32 0.72 + 0.32 

H 0.32 0.32 1.22 + 0.32 1.22 + 0.32 0.32 

C 0.32 0.32 1.22 + 0.32 1.22 + 0.32 0.32 

F 0.72+ 0.32 0.72 + 0.32 0.32 0.32 0.72 + 0.32 

These covariances can then be combined with the respective sensitivities of kfto the number densities for each 

isotope to obtain the uncertainty in kf due to these effects. The resulting k, standard deviations for each of the 

eight experiments are 0.49, 0.45, 0.46, 0.44, 0.42, 0.45, 0.53, 0.52%, respectively. The remaining six experiments 

for U(5) 30 8 are constructed using the same methods. 

4.4 Future Developments 

Volume 1 of this report presented the development of S/U and GLLSM techniques for application to criticality 

safety data validation studies. This volume has applied those methods to the analysis of uranium systems with 

enrichments greater than 5 wt %. Specifically, a family of U(1 1)02 systems with H/X values of 0-1000 have 

been analyzed. The guidelines given in the previous sections explain when usage of traditional procedures is 

recommended as well as the situations when the use of these newly developed procedures is recommended. 

The procedures for using the 1-D sensitivity tool, SEN1, and for trending bias with the resulting D and ck 

parameters are established as a result of this development work, however, additional experience with other 

application areas is required with these new procedures before complete guidance on their use is possible. Work 

is planned to address the trending with D and ck values which are outside the proposed similarity criterion of this 

report. Comparison of trending procedures which exclude experiments which are not similar, include them as 

shown in this work, and weight themn based on their ck and D values are exercises planned for the future. This 

experience should allow for development of an upper subcritical limit (USL) procedure that completely takes 

advantage of these new validation methods. For example, the possibility of a lower administrative margin due to 

the inclusion of cross section uncertainty effects could be explored.

NUREG/CR-6655, Vol. 2

Section 4

49
Guidance for Determination of Applicability Section 4

There are a number of development areas that need further study regarding the use of these techniques. The first 

area is the need for flexible tools for applying the sensitivity methods to 2-D and 3-D problems. A 2-D capability is currently available, 2

' but further work on implementation of a more flexible package is planned. The use of 

existing inputs for a 3-D Monte Carlo code is considered a must for the routine application of these procedures. Currently work is underway on this task,24 but further work is needed before this tool will be generally available 

for use. 

The sensitivity methodology applied in this work obtains the sensitivity of kff to the problem-specific cross sections 

used in the analysis. Future work is also needed to explore extension of the sensitivity capability to obtain the 

sensitivity of kf to the code and data parameters that impact the problem-specific cross section processing. 

Reference 20 indicates via direct re-calculations how omission of these effects can cause errors in the sensitivity 

coefficients that are obtained. 

The procedures that govern the use of the GLLSM techniques require some additional development before they 

can be routinely applied in a criticality safety data validation process. These additional developments involve a 

concept called "completeness," the determination of the rigor involved for data validation versus data evaluation 

(i.e., self-shielding effects, inclusion of resonance parameter uncertainties, etc), and the addition of non-critical 

experiment and/or non-ky response methods. 

The completeness concept is needed to ensure that in the general application of GLLSM techniques, where there 

are no limits for the values of ck and D, the benchmark data set is complete in the sense that all important reactions and energies are validated. There is a recognized need to investigate and address, as appropriate, the 

influence and magnitude of shielded and unshielded effects of resonance reactions on the sensitivity and 

uncertainty analyses relating the GLLSM procedure. The resonance effects include the option of incorporating uncertainties due to self-shielding into the analysis, and also the generation of and processing of resonance 

parameter uncertainties. The use of non-critical and/or non-kg responses in the GLLSM analysis is currently 

available, however, it is not in the SEN1 package that was developed for this project. Either the capability could 

be added to the SENI package or currently available programs that have more complex input requirements could 

be used. The full use of these capabilities in a 3-D sensitivity generation procedure would simplify the criticality safety validation efforts. Additional measures are also expected to be needed in order for the full GLLSM 

techniques to be readily used by criticality safety practitioners. 

The trending procedures with ck and the GLLSM applications require the availability of cross section covariance 

files for processing to a desired group structure. There are currently 27 materials/isotopes with covariance data in 

ENDF/B-V, and 43 materials/isotopes with covariance data in ENDF/B-VI. However, there are still many important nuclides without covariance information. This information needs to be generated for many isotopes 

that are of importance to the various criticality application areas. 

While the methods and procedures generated as a result of this project are felt to be quite general, they have 

currently only been applied to uranium systems. Application to other fissile material (e.g., plutonium or mixed 

oxide) systems is desired to ensure the generality of these procedures. For systems containing primarily uranium 

fissile materials, the D. and ck parameters appear to work equally well. Application to other fissile materials 

should allow for confirmation of these trends.

NUREG/CR-6655, Vol. 2 50
5 SUMMARY 

This report presents the application of sensitivity and uncertainty (S/U) analysis methodologies developed in 

Volume 1 to the code/data validation tasks of a criticality safety computational study. Sensitivity and uncertainty 

analysis methods were first developed for application to fast reactor studies in the 1970s. This work has 

revitalized and updated the existing S/U computational capabilities such that they can be used as prototypic 

modules of the SCALE code system, which contains criticality analysis tools currently in use by criticality safety 

practitioners. After complete development, simplified tools are expected to be released for general use. 

The methods for application of S/U and generalized linear-least-squares methodology (GLLSM) tools to the 

criticality safety validation procedures were described in Volume 1 of this report. Volume 2 of this report presented 

the application of these procedures to the validation of criticality safety analyses, supporting uranium operations for 

use in commercial power reactors where the enrichments are greater than 5 wt %. 

As a part of this study, a benchmark database of 102 experiments was developed which included not only 

calculational models and measured-versus-calculational kffresults but also sensitivity and uncertainty results 

generated by the S/U methodology described in Volume 1. The same tools were then applied to a set of 

14 application scenarios corresponding to U(11)02 systems with H/X values ranging from 0 to 1000. A range of 

applicability determination was performed using the D and ck coefficients generated by the S/U analysis. The 

analysis indicated that the benchmark dataset had good coverage of the 11 -wt % systems with HIX values greater 

than 40, while the systems with HIX less than 40 had only marginal-to-inadequate coverage. The sparsity of 

low-HIX systems is a concern. Additional experiments in this area would be useful. 

Additionally, analyses were performed to predict biases for systems with enrichments greater than 5 wt %. 

Specifically, the traditional kf trending analyses were compared with newly developed kf trending procedures, 

utilizing the D and ck coefficients described in Volume 1. Application of these newly developed procedures was 

made to the family of applications involving U( 1)02 systems with H/X values ranging from 0 to 1000. 

The comparisons among the various trending techniques were quite interesting in that they give very different 

answers, depending on the particular system being analyzed. The predicted biases for various systems were in 

some cases up to a factor of 5 different between the various trending techniques. The primary reason for these 

differences was that various systems "look" very similar from the standpoint of certain parameters, but are very 

different with respect to other parameters. In particular, the H/X and EALF parameters predicted similarity 

between dry systems with high enrichments (93 wt %) and dry systems with intermediate enrichments (10 wt %), 

while the ck and D parameters indicated that these systems were quite different. The net effect of trending with H/X 

and EALF was a cancellation of effects (the HEU systems were underpredicted by about 1%, the IEU systems were 

overpredicted by about 2%), which produced a predicted bias of about 0.5% overprediction for a dry U(1l)0 2 

system. The trending with ck and D parameters produced an estimated bias of between 1 and 2% overprediction 

since only the IEU systems were predicted to be similar to the dry U(1 1)02 system. Although the predicted biases 

from these applications are all positive, the differences in magnitude are a concern, since the prudent application of 

trending procedures is very important in criticality validation exercises.

51 NUREG/CR-6655, Vol. 2
Summary Section 5

As a result of these observations, the following guidance was developed for usage of the various trending 

techniques. 

Traditional trending procedures are acceptable for bias estimation if about 10-20 benchmarks with a ck value 

of 0.8 or higher are included in the benchmark database. It is further recommended that in using the 

traditional ke trending procedures that systems with very different physical characteristics, such as high versus-low enrichment, poisoned-versus-nonpoisoned, wet-versus-dry, etc., not to be included together unless 

that specific effect is being analyzed. 

" The advanced trending analyses with the D and ck coefficients are recommended if fewer experiments than 

those recommended above are available. Also the primary motivation behind their use is that widely varying 

systems can be included in the analysis, hence making the effective extrapolation of the existing data more 

meaningful. The procedure will allow for automatic collection of experiments that have similar D and c, 

coefficients such that the predicted kff bias should be valid. 

" The GLLSM procedure is recommended if existing experiments with ck values of 0.8 or higher with respect to 

the area of application are not available. These experiments, possibly having ck values 0.2-0.6, are not 

similar to the application; but they are still relevant. A suite of such experiments can be expected to perform quite well using a GLLSM technique if they are suitably chosen to highlight different aspects of the given application system. Under these conditions, the use of GLLSM for validation is possible, although it may still 

be desirable to initiate an experimental program to obtain additional data. This procedure also allows for the 

inclusion of noncritical experiments and non-kf responses in the validation exercises. Further development is 

required before these GLLSM techniques can be fully functional in a routine criticality safety validation study.

NUREG/CR-6655, Vol. 2 52
6 REFERENCES 

1. C. R. Weisbin, J. H. Marable, J. L. Lucius, E. M. Oblow, F. R. Mynatt, R. W. Peelle, and F. G. Perey, 

Application of FORSS Sensitivity and Uncertainty Methodology to Fast Reactor Benchmark Analysis, 

ORNL/TM-5563, Union Carbide Corp., Oak Ridge National Lab., December 1976. See also NucI. Sci. 

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2. J. L. Lucius, C. R. Weisbin, J. H. Marable, J. D. Drischler, R. Q. Wright, and J. E. White, A Users Manual 

for the FORSS Sensitivity and Uncertainty Analysis Code System, ORNL-5316, Martin Marietta Energy 

Systems, Oak Ridge National Lab., January 1981. 

3. R. E. Maerker, B. L. Broadhead, and J. J. Wagschal, "Theory of an New Unfolding Procedure in Pressurized 

Water Reactor Pressure Vessel Dosimetry and Development of an Associated Benchmark Data Base," 

Nucl. Sci. Eng. 91, 369 (1985). See also Development and Demonstration of an Advanced Methodology for 

LWR Dosimetry Applications, EPRI NP-2188, Electric Power Research Institute, Palo Alto, Calif. 

(December 1981). 

4. N. M. Larson, L. C. Leal, and H. Derrien, Integral Data Analysis for Resonance Parameters 

Determination, ORNL/TM-13495, Lockheed Martin Energy Systems, Inc., Oak Ridge National Lab., 

September 1997. 

5. J. J. Lichtenwalter, S. M. Bowman, M. D. DeHart, and C. M. Hopper, Criticality Benchmark Guide for 

Light-Water-Reactor Fuel in Transportation and Storage Packages, Appendix C, NUREG/CR-6361 

(ORNL/TM-1321 1), U.S. Nuclear Regulatory Commission, Oak Ridge National Lab., March 1997. 

6. B. L. Broadhead, C. M. Hopper, R. L. Childs, and C. V. Parks, "Sensitivity and Uncertainty Analyses 

Applied to Criticality Validation: Volume 1: Methods Development," NUREG/CR-6655, Vol. 1, 

ORNL/TM-13692/Vl, U.S. Nuclear Regulatory Commission, Oak Ridge National Lab., July 1999. 

7. International Handbook of Evaluated Criticality Safety Benchmark Experiments, NEA/NSC/DOC(95)03/lI, 

Nuclear Energy Agency, Organization for Economic Co-operation and Development, Paris, France (1995). 

8. B. T. Rearden, J. J. Lichtenwalter, and C. M. Hopper, Comparative Results of Reported Critical Experiment 

Parameters and Uncertainties with First Order Multigroup Perturbation Theory, NUREG/CR-5 624 

(ORNL/TM-13718), U.S. Nuclear Regulatory Commission, Oak Ridge National Lab., December 1999. 

9. W. C. Jordan, N. F. Landers, and L. M. Petrie, Validation of KENO V.a Comparison with Critical 

Experiments, ORNL/CSD/TN-238, Martin Marietta Energy Systems, Oak Ridge National Lab., December 

1986. 

10. G. R. Smolen, Validation Studies Based on Data from Low Concentration Mixed Pu+ U Aqueous Critical 

Experiments, ORNL-6449, Martin Marietta Energy Systems, Oak Ridge National Lab., August 1988.

53 NUREG/CR-6655, Vol. 2
References

11. S. J. Raffety and J. T. Mihalczo, Homogenized Critical Assemblies of 2 and 3% Enriched Uranium in 

Paraffin, Y-DR-14, Union Carbide Corp., Nuclear Division, Oak Ridge Y-12 Plant (1969). 

12. D. F. Cronin, Critical Mass Studies, Part X, ORNL-2968, Union Carbide Nuclear Corp., Oak Ridge 

National Lab., 1960. 

13. ENDF-202 Cross-Section Evaluation Working Group Benchmark Specifications, BNL-19302, Brookhaven 

National Lab., November 1974. 

14. W. C. Jordan and J. C. Turner, Minimum Mass of Moderator Required for Criticality of Homogeneous 

Low-Enriched Uranium Systems, ORNL/CSD/TM-284, Martin Marietta Energy Systems, Inc., Oak Ridge 

National Lab., December, 1992. 

15. Experiments HEU-MET-FAST-003, HEU-MET-FAST-004, and LEU-SOL-THERM-001 in International 

Handbook of Evaluated Criticality Safety Benchmark Experiments, NEA/NSC/DOC(95)03/II, Nuclear 

Energy Agency, Organization for Economic Co-operation and Development, Paris, France (1995). 

16. G. Tuck and I. Oh, Benchmark Critical Experiments on Low-Enriched Uranium Oxide Systems with 

HJU = 0. 77, NUREG/CR-0674, U.S. Nuclear Regulatory Commission (1979). 

17. R. E. Rothe and G. R Goebel, Critical Experiments on Low-Enriched Uranium Oxide Systems with 

H/U = 2.03, NUREG/CR-2500, RFP-3277, U.S. Nuclear Regulatory Commission (1982). 

18. G. A. Linenberger et al., "Enriched-Uranium Hydride Critical Assemblies," Nucl. Sci. Eng. 7,44 (1960). 

19. W. N. Fox et al., "Reactor Physics Measurements on 1 5 U and "Pu Fuels in an Intermediate Spectrum 

Assembly," Journal British Energy Society, 9 (1970). 

20. J. K. Fox et al., Critical Parameters of Uranium Solutions in Simple Geometry, ORNL-2609, Oak Ridge 

National Laboratory, Oak Ridge, Tennessee (1958). 

21. R. L. Childs, SENJ: A One-Dimensional Cross-Section Sensitivity and Uncertainty Module for Criticality 

Safety Analysis, NUREG/CR-5719 (ORNL/TM-13738), U.S. Nuclear Regulatory Commission, Oak Ridge 

National Lab., July 1999. 

22. SCALE: A Modular Code System for Performing Standardized Computer Analyses for Licensing 

Evaluations, NUREG/CR-0200, Rev. 5 (ORNL/NUREG/CSD-2/R5), Vols. I-rII, March 1997. Available 

from Radiation Safety Information Computational Center at Oak Ridge National Laboratory as CCC-545.

NUREG/CR-6655, Vol. 2

Section 6

54
References

23. M. D. DeHart and S. M. Bowman, Validation of the SCALE Broad Structure 44-Group ENDF/B-V 

Cross-Section Library for Use in Criticality Safety Analyses, NUREG/CR-6102 (ORNL/TM-12460), 

U.S. Nuclear Regulatory Commission, Oak Ridge National Lab., September 1994. 

24. B. T. Rearden, "Development of SAMS: A Three Dimensional Sensitivity Analysis Module for the SCALE 

Code System," report of dissertation work, Texas A&M University, December 1999.

NUREG/CR-6655, Vol. 2

Section 6

55
NRC FORM 336 U.& NUCLEAR REGULATORY COMMISSION 1. REPORT NUMBER 

(2-M (Assigned by NRC, Add Vol.. Supp.. Rev., 

NRCM 1102 and Addendum Numbers, If any.) 

3201,3= BIBUGRAPHIC DATA SHEET 

(See , ictions on ft. reve) NUREG/CR-6655, Vol. 2 ORNLITM-1 3692NV2 

2. TITLE AND SUBTITLE 

Sensitivity and Uncertainty Analyses Applied to Criticality 3. DATE REPORT PUBLISHED 

Safety Validation 3. MONTH DTE RED YEAR 

Illustrative Applications and Initial Guidance November 1999 

4. FIN OR GRANT NUMBER 

W6479 

5. AUTHOR(S) 6. TYPE OF REPORT 

B.L. Broadhead, C.M. Hopper, C.V. Parks Technical 

7. PERIOD COVERED ttnclusive Dates) 

8. PERFORMING ORGANIZATION - NAME AND ADDRESS (lf NRC, provide Division, Office or Region, U.S. Nucer Regulatoy Commision, and mailing addmss; dfconb'acta, 

provide name and mailing address.) 

Oak Ridge National Laboratory 

Managed by Lockheed Martin Energy Resarch Corporation 

Oak Ridge, TN 37831-6370

9. SPONSORING ORGANIZATION - NAME AND ADDRESS (IffNRC, type"Samne as above;, ifwcn~acfW, provide NRC Division, Office or Region, U.S. NuceAr Reguslatoy Commission, 

and mailing ad&wes.) 

Division of Systems Analysis and Regulatory Effectiveness 

Office of Nuclear Regulatory Research 

U.S. Nuclear Regulatory Commission 

Washington, DC 20555-0001

10. SUPPLEMENTARY NOTES

C.W. Nilsen, NRC Project Manager

11. ABSTRACT r2Oo words orless) 

This report presents the application of sensitivity and uncertainty (S/U) analysis methodologies developed in Volume 1 to the 

code/data validation tasks of a criticality safety computational study. Sensitivity and uncertainty analysis methods were first 

developed for application to fast reactor studies in the 1970s. This work has revitalized and updated the existing S/U computational 

capabilities such that they can be used as prototypic modules of the SCALE code system, which contains criticality analysis tools 

currently in use by criticality safety practitioners. After complete development, simplified tools are expected to be released for 

general use. 

The methods for application of S/U and generalized linear-least squares methodology (GLLSM) tools to the criticality safety 

validation procedures were described in Volume I of this report Volume 2 of this report presents the application of these 

procedures to the validation of criticality safety analyses supporting uranium operations where the enrichments are greater than 5 

wt%.

12. KEY WORDS/DESCRIPTORS (List words orphrasw that will assist reseembrs in /oaning the report.) 13. AVAILABILITY STATEMENT 

unlimited 

criticality safety, data validation, senst analysis, uncertainty analysis unlimited 

14. SECURITY CLASSIFICATION 

(This Page) 

unclassified 

(Tis Report) 

unclassified 

15. NUMBER OF PAGES 

16. PRICE

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