OECD/NEA Source Convergence Benchmark Program: Overview and Summary of Results

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1 23 OECD/NEA Source Convergence Benchmark Program: Overview and Summary of Results Roger BLOMQUIST *, Ali NOURI 2, Malcolm Armishaw 3, Olivier JACQUET 4, Yoshitaka NAITO 5, and Yoshinori MIYOSHI 6, Toshihiro YAMAMOTO 6 Argonne National Laboratory, 9700 S. Cass Ave., Argonne, IL USA 2 OECD Nuclear Energy Agency, Le Seine Saint-Germain, 2, boulevard des Iles, F-9230 Issy-les-Moulineaux FRANCE 3 Serco Assurance, Winfrith Technology Center, Dorchester, Dorset DT28DH, UNITED KINGDOM 4 IRSN/DPEA/SEC, B.P. 6, F Fontenay Aux Roses FRANCE 5 NAIS Co., 46 MuramatsuTokai-mura, Naka-gun, Ibaraki 39-2, JAPAN 6 JAERI, 2-4 Tokai-mura, Naka-gun, Ibaraki 39-95, JAPAN This paper describes the work of the OECD Nuclear Energy Agency Expert Group on Source Convergence in Criticality Safety Analysis. A set of test problems is presented, some computational results are given, and the effects of source convergence difficulties are described. KEYWORDS: Fission source convergence, Monte Carlo. Introduction Slow source convergence has long been a challenge in the analysis of some loosely coupled neutronics problems, ranging from the Whitesides criticality of the world problem identified in 97 to some very recent burnup credit benchmark calculation comparisons under the auspices of the OECD Nuclear Energy Agency (NEA). 2 Also, in the case of Monte Carlo calculations, slow convergence and statistical fluctuations can combine to produce unreliable source distributions and fission rates and underestimate both k eff and its uncertainty. Moreover, these calculations are characterized by a strong dependence of the results on the initial source distribution, an indication that source convergence is not achieved. This problem is especially important when no symptoms of non-convergence are apparent to the analyst. References 3, 4, and 5 describe some of the work on Monte Carlo source convergence methods. Obviously, the fundamental parameter of interest in criticality safety analysis is the system eigenvalue. It will be seen that the effect on k eff of incomplete convergence and other related computational difficulties is frequently small, but our premise is that an eigenvalue produced using an unconverged source is not reliably estimated and should not be used in a safety analysis. Consequently, in this work, both eigenvalues and fission distributions are examined to assess the state of convergence. Although results reported here were from Monte Carlo calculations, additional work has been done using deterministic methods. To explore these problems for the benefit of the international criticality safety community, the NEA Working Party on Nuclear Criticality Safety established an Expert Group on Source Convergence in Criticality Safety Analysis in the fall of 999. One of the group s objectives is to improve criticality safety analyses with respect to source convergence by comparing source convergence methods and tests, and by publishing and sharing information on source convergence problems relating to criticality safety analysis. The Expert Group is comprised of about ten participating laboratories in six countries, meeting annually to review progress and plan further work. The technical work is funded through programs at the participating institutions. 2. Source Convergence Test Problems A set of four idealized test problems that represent cases previously encountered in real criticality safety analyses has been developed. Their specifications have been modified to simplify input preparation and to reduce computation times, while retaining the source convergence properties of the original system. These are more test problems than benchmark problems in that they are used as a basis for comparison of source convergence performance, rather than comparison of physics results. Although all of these problems can be deemed to experience slow convergence, the dependence of the eigenvalue on the degree of source convergence is highly variable. Each exhibits slow convergence, undersampling effects, source-related statistical anomalies, or some combination of these effects. The effects of differences in nuclear data, transport equation approximations, and reference solutions have not been thoroughly examined because they are * Corresponding author, Tel. (630) , Fax. (630) , RNBlomquist@anl.gov

2 outside of the scope of the program. Instead, the focus has been on convergence and statistical behavior. 2. Checkerboard Fuel Storage Array Test problem, submitted by N. Smith (Answers Software), is a fuel storage facility surrounded by concrete on three sides and water on the fourth in which 36 identical fresh fuel assemblies with about 5 weight percent enriched UO 2 and Zr clad are stored in an alternating (checkerboard) pattern with empty locations, in a square grid composed of 0.5cm steel plate. The fueled locations consist of a square pin lattice, and all locations are flooded with water. Figure shows a plan view. 27 cm 40 cm thick concrete, 420 cm tall 30 cm thick water, 420 cm tall Fuel position (,3), 360 cm tall, 27 cm square 0.5 cm steel wall 2.5 cm water gap Fig. Checkerboard spent fuel layout (23,3) Square pin lattice (5x5,.4cm pitch), fuel Adjacent fuel elements are almost completely decoupled, so deterministic and Monte Carlo calculations suffer from extremely slow convergence. A simplified calculation using VIM established that the probability that a neutron born in one subassembly in one column will cause a fission in a particular neighboring column is about The converged fission rate peaks in the upper left corner location (,3) due to the superior reflecting properties of concrete, so a uniform guess, is quite unrealistic, although such a guess is suggested by the problem layout. For the fission sites in (23,3) to collect in unit (,3), many generations will be required. Monte Carlo calculations may also show the effects of undersampling if insufficient numbers of histories per generation are used. A set of 36 cases was defined across which several Monte Carlo parameters were varied. The number of neutrons per generation was set at 000, 2000, or The number of generations skipped was 20, 40, or 00. The starting source distributions were either uniform, all in location (,), all in location (2,2), or all in location (23,3). Some participants performed additional calculations skipping many more generations and using many more neutrons per generation, or simplifying the geometry further to study the convergence behavior or to quantify the effects of the asymmetric reflection in the x-y plane on the converged fission source and eigenvalue. Results were submitted by nine participants from four countries. The codes used were KENO, MCNP 4B and 4C, VIM, MORET4, and MONK8A. Because the source converges very slowly, its drift is not obvious from the drift in the eigenvalue during the course of the calculation. Figure 2 shows the influence of Monte Carlo computational strategy on estimated fission distributions when a straightforward generation-by-generation power iteration is used. Since the status of the fission source convergence is manifested mainly along the x-axis, we show only the fission fractions for row 3, which includes the location with the most reactive cell. The most conservative strategy of the specified cases (case 27: uniform initial source, 5000 histories per generation, skip 00 before tallying) does identify the most reactive array element, but badly underestimates its fission rate unless more than 500 generations are skipped. As commonly happens for very slowly converging problems, the uncertainties are also underestimated. Increasing the number of histories per generation to substantially improves both the cell (,3) fission rate and its uncertainty estimate. Skipping 500 generations instead of 00, and tallying over 000 generations brings the fission distribution close to convergence. Table shows the range of eigenvalues among the cases for each code, which can be viewed as a measure of the uncertainty due to the choice of Monte Carlo computational parameters. The table also shows the differences between the largest eigenvalue and case 27, which among the specified cases, is the closest case to a reference. The last digit(s) of the estimated uncertainties are shown in parentheses. Clearly, the range of eigenvalues substantially exceeds the estimated uncertainties, so these cases are deemed to suffer from insufficient source convergence and/or inaccurate statistical treatment. Statistical tests were unable to detect the eigenvalue drift (typically 0.002) during source iterations. At least 700 skip generations are required to converge the source using conventional power iterations. Fission Fraction H/G, Skip 00, Score H/G, Skip 00, Score H/G, Skip 500, Score Cell Column # Fig. 2 Checkerboard row 3 fission fraction profiles 2.2 Depleted Fuel Pin Array Test problem 2, submitted by Y. Naito (NAIS), is a flooded infinite lattice of depleted LWR pins. Six cases are considered which describe various symmetric and non-symmetric axial composition profiles due either to axial enrichment grading or

3 Table k eff ranges (and uncertainties) over all cases, and difference between largest eigenvalue and case 27. Participant/Code Max(k eff ) Min(k eff ) Max(k eff ) Case 27 k eff ANL/VIM a (6) (6) JAERI/MCNP4B a (9) (9) JNC/KENO-Va a (8) 0.00 (4) KFKI/MCNP4C a (4) (9) LANL/MCNP4C a (2) (6) ORNL/KENO-Va a (0) (0) Answers/MONK8A b (2) (8) EMS/KENO-Va a (8) (4) IRSN/MORET4 a (2) (9) IRSN/MORET4 b (4) (9) IRSN/MORET4 c (9) (7) a conventional power iteration, b superhistory method, c stratified sampling to an idealized burnup profile. This is a simplified version of the NEA Burnup Credit (Phase III-A) Benchmark Problem 2. Figure 3 shows the axial layout of the pin cell. The lattice pitch is.33 cm, the fuel radius is 0.42 cm and the cladding radius is cm. The fuel pins end with a.75 cm-long end plug, and there are 30 cm water reflectors beyond both ends of the pins. 5cm (2) 5cm (24) 0cm (30) 2 3 End plug (.75cm) A cm length of high-burnup, low-multiplication fuel in the center decouples the two reactive ends. Because there are only two reactive regions, undersampling is not experienced at either end. Nevertheless, a Monte Carlo calculation with a tractable number of histories will be effectively unable to generate fissions in the high-burnup section. Neutron transport between the ends will therefore not speed the redistribution of fission sites to the fundamental mode. Eight participants from five countries submitted results, using MORET4, MCNP 4B and 4C, KENO, and VIM. Figure 4 shows the slow evolution of the fission source distribution in case 2-3, the most nearly symmetric of the asymmetric spent fuel cases. The axial composition key is shown below the plot. For the cases with a larger asymmetry, source convergence is significantly faster, although still somewhat slow. The axially symmetric cases appeared to produce non-symmetric fission distributions, but were in fact exhibiting underestimates of fission rate uncertainties. The nearly symmetric cases require about 600 generations for source convergence, and the eigenvalue drift during convergence is about Source distribution 4.0E E E E E-02.5E-02.0E E-03 cycle 00 cycle 300 cycle cm (40) 4 5 Fuel regions E+00 Burnup GWD/MTU Axial distance (cm) B2G B24G B30G B40G B55G B40G B30G B30G B24G Fig. 4 Typical source evolution in the depleted pin cell problem case cm (55) 20cm (40or 55) water End plug (.75 cm) Fig. 3 Depleted pin cell, (burnups in MWD/MTU), not to scale 2.3 Loosely Uranyl Nitrate Slabs Problem 3 (Figure 5), submitted by T. Yamamoto (JAERI), describes two slabs of uranyl nitrate solution decoupled by an intervening moderating slab of water similar to the problem described in reference 4. The solution contains 300 gu/l at 0 wt.% enrichment and.0 mole/l nitric acid. The twelve cases consist of varying the left solution slab thickness to 2, 5, 8, and 20 cm, and the moderating slab to 0, 20, and 30 cm. This problem was used to test a fission matrix method 4, which acts as a restoring force to unduly fluctuating fission distributions. One computational difficulty discovered is that the homogenized compositions exacerbate the ill effects of using collisions to generate fission sites, a common Monte Carlo technique. When combined with the system s slow convergence, the result is a serious underestimate in reaction rate variances. This was most clearly evident in the moderately

4 decoupled symmetric case ( ) shown in Figure 6, when the fission fraction ratios reached 40. vacuum Reflective Vacuum Uranyl nitrate H 2 O Uranyl nitrate Vacuum variable variable 20 cm Source Ratio Fig. 5 Uranyl nitrate slabs VIM(collision) VIM(absorption) MCNP Reflective Generation Fig. 6 Fission source ratio for three source treatments Selecting fission sites on absorption produced a dramatic reduction in the fission distribution variation during the calculation. Even when 2,000 histories per generation are sampled, undersampling is possible, especially in the most decoupled of the symmetric systems. Increasing the generation size to 20,000 substantially reduced the fluctuations by reducing statistical variations. Seven participants in four countries submitted results using VIM, KENO-Va, MORET4 (both conventional power iterations and superhistory powering), and MCNP-4B and -4C. Due to relatively good coupling between the end slabs, the cases with the 0 cm water slab did not suffer from large fluctuations in fission distribution. When the water slab thickness was 30 cm, however, large fluctuations resulted, with fission fraction ratios approaching 0. In the most asymmetric cases, even with a 30 cm water slab, the source required only about 20 generations to converge. The eigenvalue drift during convergence is about x5x5 Array of Metal Spheres Problem 4 (Figure 7), submitted by O. Jacquet (IRSN), is a reduced-scale (5x5x) version 2 of the Whitesides problem, an array of metal spheres in vacuum. The initial source is chosen to be far from the converged source distribution, possibly resulting in a badly underestimated k eff. Fig. 7 5x5x array of spheres (not to scale) The center sphere is much larger than the other 24, so source sites will accumulate in it while the source is converging. Unlike the checkerboard test problem where a nearly one-dimensional array of cells acts to inhibit convergence, each sphere is directly coupled to two to four others in its row and column, and to roughly half of the spheres. The specified initial source contains only 25 neutrons per generation with a single neutron at the center of each sphere and 0 concentrated at the center of the lower left corner sphere. Figure 8 shows the evolution of a few fission fractions from a set of 0 statistically independent replica calculations. The randomness introduced by undersampling is dramatic, and is easily reduced by increasing the number of histories per generation. It is clear that this problem does not suffer from slow convergence per se; once the central sphere contains a few fission sites, its fission fraction rapidly approaches convergence. In fact, this problem converges easily when a uniform initial source is used. Center Sphere Fission Fraction Fig Batch # (5 Generations/Batch) 5x5x array source evolution in the central sphere Monte Carlo results were submitted by each participant for 00 statistically independent replica calculations. This allowed estimation of the probability that the source fails to converge. In 00 replicas, between 00 and 900 skip generations were

5 required to converge the fission source. The eigenvalue drift during convergence is The MORET4 results sere generated using three different methods, i.e., stratified sampling, superhistory powering, and conventional power iterations. The two enhanced methods produce superior performance. 4. Conclusion The test problem comparison program of the expert group has stimulated a useful set of investigations into the algorithm currently used to propagate neutron sources in Monte Carlo calculations. It is common knowledge that understanding the underlying physics of criticality problems is necessary to ensure satisfactory source convergence in criticality calculations These test problems demonstrate that reliance on statistical tests applied to eigenvalue estimates cannot be relied upon to detect inadequate source convergence. Furthermore, statistical tests commonly applied to detect non-convergence are being evaluated, and new ones are being proposed and tested. Acknowledgements Work supported by the U.S. Department of Energy, Nuclear Energy Program under W-3-09-ENG-38. References ) G. E. WHITESIDES, A Difficulty in Computing the k-effective of the World, Trans. Am. Nucl. Soc., 4, 680 (97) 2) OECD/NEA Expert Group on Burn-up Credit web site at 3) R. J. BRISSENDEN and A. R. GARLICK, Biases in the Estimation of k eff and its Error by Monte Carlo Methods, Ann. Nucl. Energy, Vol. 3, No. 2, pp (986). 4) T. YAMAMOTO, T NAKAMURA, and Y. MIYOSHI, Fission Source Convergence of Monte Carlo Criticality Calculations in Weakly Coupled Fissile Arrays, J. of Nucl. Sci. and Tech., Vol. 37, No., pp. 4-45, January, ) A. MOHAMED and E. GELBARD, Stratified Source-Sampling Techniques for Monte Carlo Eigenvalue Analysis, Proc. Int. Conf. On Physics of Nuclear Science and Technology, Hauppage, N.Y., Oct. 5-8, ) The OECD/NEA Expert Group on Source Convergence in Criticality-Safety Analysis web site at html. 7) SCALE: A Modular Code System for Performing Standardized Computer Analyses for Licensing Evaluation, NUREG/CR-0200, Rev. 6 (ORNL/NUREG/CSD-2/R6), Vols. I, II, and III (December, 999). 8) J. F. Briesmeister, Ed., MCNP A General Monte Carlo N-Particle Transport Code Version 4B, LA-2625 (997). 9) R. N. Blomquist, "VIM Continuous Energy Monte Carlo Transport Code," Proc. Int. Conf. On Mathematics, Computations, Reactor Physics, and Environmental Analyses, Portland, OR, April 30 - May 4, ) A. Nouri, et al, MORET4: A Versatile and Accurate Monte Carlo Code for Criticality Calculations, IPSN Report SEC/T/00.06 (France). ) N. SMITH, The Unification of MONK Extending the Monte Carlo Horizon, ICNC 99, Versailles, France (20-24 September, 999). 2) KADOTANI, Acceleration of Fission Distribution Convergence Using Eigenvectors from Matrix K Calculations in the KENO Code, ICNC 9, II-, Oxford, September, 99.