Treatment of the implicit effect in Shark-X

Shark-X is a set of Perl-based tools build around the deterministic neutron transport code CASMO-5, used to perform uncertainty quantification for lattice calculations at the Paul Scherrer Institut (PSI). While perturbing an input parameter, e.g. a cross section in a given energy range, the resulting perturbation to the effective cross section, e.g. multigroup cross section, can be expressed as the sum of an explicit and implicit component. The implicit term stems from changes in the neutron flux resulting from the cross section perturbation. This effect is neglected when using the standard perturbation theory methodology, even though its importance for uncertainty quantification is reported in the literature. The objectives of the paper are, in view of further enhancing Shark-X, to review a set of existing methods and to propose novel methodologies in order to accurately take the implicit effect into account in the resonance shielding treatment of CASMO-5; as well as to assess the importance of the new tool for typical LWR applications. Amongst the methods considered, those based on an equivalence in dilution allow perturbing the group-averaged cross sections without having to solve the slowing down problem in conjunction with the perturbed point-wise cross sections. However, such methods are found to show only limited improvements as compared to the standard perturbed self-shielded cross sections determined neglecting the implicit effect. As a result, a simpler approach based on the pre-determination of perturbation factors is adopted and implemented into Shark-X and CASMO-5. The new implementation is successfully tested and verified by comparing k-inf sensitivity coefficients computed with Shark-X with those from a Monte-Carlo reference solution, and this for a range of simple homogeneous mixtures as well as for LWR pin cells and lattices. Finally, the paper demonstrates through simple considerations, that the magnitude of the implicit effect is related to the energy group structure in the resolved resonance region, which might be specific to the code used to compute the sensitivity coefficients. The finer is the energy mesh; the smaller is also the impact of the implicit effect, and as such the necessity for a specific treatment. (C) 2019 Elsevier Ltd. All rights reserved.