# A methodology for the design of non-uniform core configurations in the modified STACY facility

B. Dechenaux, Mariya Brovchenko, S. Araki, S. Gunji, K. Suyama

► To cite this version:

B. Dechenaux, Mariya Brovchenko, S. Araki, S. Gunji, K. Suyama. A methodology for the design of non-uniform core configurations in the modified STACY facility. Annals of Nuclear Energy, 2025, 223, pp.111555. 10.1016/j.anucene.2025.111555. hal-05175569

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# A methodology for the design of non-uniform core configurations in the modified STACY facility

B. Dechenaux<sup>a, ID,*</sup>, M. Brovchenko<sup>a</sup>, S. Araki<sup>b</sup>, S. Gunji<sup>b</sup>, K. Suyama<sup>b</sup>

## ARTICLE INFO

**Keywords:**

Critical experiment

Modified STACY facility

Missing fuel rods

Heterogeneous configurations

Fuel debris

## ABSTRACT

The safe retrieval of the fuel debris generated during the Fukushima Daiichi nuclear accident poses a number of challenges, among which an important one is to ensure the criticality safety of the recovery operations. At the heart of the problem lies the intrinsically heterogeneous nature of the problem, and the appearance of complex and disordered media whose neutronic properties are difficult to accurately characterize and reproduce in neutron transport simulations. Typically, a similar, simpler, problem is encountered in the modeling of assemblies with missing fuel rods. In both problems, the availability of experimental facilities capable of validating both the nuclear data and the simulation schemes in heterogeneous configurations is crucial. The modified STACY installation has been specially designed to provide and carry out critical experiments, allowing for highly non-uniform configurations, that will directly support fuel debris recovery operations, but can also be used for other experimental programs. The present work describes a method to consistently and orderly sample non-uniform core configurations in the modified STACY facility, and proposes two critical heterogeneous core configurations. They have the advantage to present a high sensitivity to the water thermal scattering law, whose importance was found to be more significant for more heterogeneous configurations. The proposed experiments will contribute to the exploration and validation of heterogeneous critical systems.

## 1. Introduction

The fuel debris produced during a nuclear accident are characterized by highly disordered and heterogeneous configurations, both in terms of geometry and composition. This situation presents significant challenges for ensuring criticality safety during retrieval operations, as these configurations differ greatly from those typically encountered in the fuel cycle. Three main consequences can be shown to arise from this issue.

First, since criticality safety ultimately depends on the accurate simulation of the neutron population in fissile materials, it becomes necessary to develop specialized methods capable of handling neutron transport in disordered media. In recent years, significant efforts have been directed toward the development of such methods (Yasunobu et al., 2023; Monange, 2022; Larmier et al., 2017).

Along the same lines, the accurate simulation of neutron interactions in fuel debris configurations requires extended validation of the underlying nuclear data, which are a crucial input for all simulation codes. The nuclear data provided in general-purpose libraries such as JEFF (NEA, 2009; Plompen et al., 2020), JENDL (Iwamoto et al.,

2023), or ENDF (Chadwick et al., 2011; Brown et al., 2018) must be tested against experimental data to ensure their accuracy. Although the ICSBEP Handbook (NEA, 2022) contains a wide range of critical experiments, new benchmarks are required to expand the domain of validity of nuclear data to meet new, specific objectives, such as the modeling of heterogeneous configurations representative of fuel debris.

The final and perhaps greatest challenge is addressing the vast diversity of situations encompassed by the generic term “fuel debris”. This diversity results from both the intricate history of the accident sequence and the complex physico-chemical processes involved. Consequently, a wide range of possible situations may be encountered during the retrieval operations. It is therefore essential to ensure that a sufficient variety of scenarios have been considered, so that one can confidently trust that the simulation methods and nuclear data are representative of the actual operating conditions.

To overcome these challenges, one must rely on solid experimental foundations, and one must make sure to dispose of experimental capabilities to validate the codes and nuclear data in a vast range of

<footnote><sup>a</sup> Autorité de Sûreté Nucléaire et de Radioprotection (ASNR), PSN-RES/SNC/LN, Fontenay-aux-Roses, F-92260, France</footnote>

<footnote><sup>b</sup> Japan Atomic Energy Agency (JAEA), Nuclear Safety Research Center, Tokai-mura, Ibaraki, Japan</footnote>

<footnote>* Corresponding author.</footnote>

<footnote>E-mail address: benjamin.dechenaux@asn.fr (B. Dechenaux).</footnote>
possibles conditions. The modified STACY facility can address, at least partially, these unprecedented needs.

The experimental setup primarily consists of a regular rectangular array, that can accommodate fuel rods, concrete rods, steel rods, or water (i.e. empty positions). By varying the placement of these different materials, one can easily design non-uniform core configurations that can then be adjusted to emulate fuel debris configurations (Gunji et al., 2023). This adjustment can be achieved, for example, by resorting to similarity studies (Shigeki et al., 2023) if the physical properties of the debris fuels are known (or can be inferred). It can also be based on comparisons with dedicated simulation codes (Watanabe et al., 2024). It should be noted, however, that this comparison is necessarily and ultimately limited by experimental constraints and the representation of a complex medium in 3D by disordered arrangements on a 2D lattice.

Even in this simplified, 2D setting, the question immediately arises as to how to navigate the vast number of possible configurations. Specifically, the question is how to identify, within the high combinatorial complexity of the problem, which configurations are physically truly distinct and/or representative.

These combinatorial aspects can be compared to another, seemingly unrelated issue, occasionally encountered in criticality safety assessments. This issue involves fuel assemblies with missing fuel rods and is associated with rare cases where some fuel rods are randomly withdrawn from a fuel assembly (due to random failures, for instance). This situation may lead to highly heterogeneous moderation conditions, which must be properly taken into account in criticality safety assessments. To this end, methods must be devised to account for the full range of possibilities in order to effectively ensure that the reactivity of the fuel assembly remains subcritical in all cases. In this context, recent results have suggested a method for consistently exploring the full configuration space and describing the associated reactivity variations, Dechenaux et al. (2024), Dechenaux (2022) and Dechenaux (2024). The proposed approach relies on the introduction of a set of descriptive variables, which enables the mapping and adequate covering of possible configurations, while minimizing the use of computational resources.

One of the main goals of the present work is to transpose the methods developed in the context of criticality safety assessments of assemblies with missing fuel rods to the design of critical experiments in the modified STACY facility. To this end, the problem will be assigned appropriate “coordinates” and the latter will be used to efficiently pave and explore the whole configuration space associated to the possible lattice arrangements. It is worth mentioning that the bridge established between the two subjects will ultimately prove beneficial for both. By doting STACY with adapted methods to cover the space of all possible configurations, one will be left better equipped to address the issues associated to the representativeness and completeness of critical experiments related to fuel debris retrieval operations. Regarding the problem of the missing fuel rods, the approach that is proposed paves the way for conducting critical experiments under heterogeneous moderation conditions.

The experimental facility is introduced in Section 2, along with a set of core configurations with different moderator to fuel ratios, that will serve as references throughout. In Section 3, we aim to adapt the methods developed for the missing fuel rods problem to the design of critical experiments in a regular array (using a simplified approach with two components: fuel rods and water). It is demonstrated that this approach works well in this context, and the use of a set of simple variables effectively describes the possible configurations in the STACY facility. In Section 4, the methodology is used to propose possible critical configurations. The choice is based on the importance of the thermal scattering law and the difference to existing experiments documented in the ICSBEP handbook.

## 2. The STACY experimental facility

The STACY installation was completed in December 2023. The modified STACY core consists of an open-top core tank, grid plates, guide pin, and fuel rods, as illustrated in Fig. 1. The tank is approximately 180 cm in diameter and 190 cm high, with grid plates installed in it. Two types of grid plates with 1.27 cm and 1.50 cm lattice pitches are currently available. The 1.27 cm lattice plate has 41 × 71 and the 1.50 cm lattice plate has 39 × 59 holes for loading of fuel rods. The fuel rods have a length of approximately 150 cm and an outer diameter of 0.95 cm. They are constituted of $UO_2$ pellets, with an enrichment of about 5%, encapsulated in a zirconium alloy cladding. The reactivity is mainly controlled by the water level of the light water moderator in the core tank. In addition, to simulate the composition of fuel debris, experimental equipments of the same size as the fuel rods can be loaded into the core to simulate the composition of structure materials such as steel and concrete.

The modified STACY has begun executing experiments since August 2024. In order to understand the performance of the core, experiments are being conducted with a core consisting only of fuel and water. Preliminary analysis of the experiments is being conducted with the benchmark model described in the Gunji (2023).

Three lattice pitch configurations (1.80 cm, 1.50 cm, and 1.27 cm) that could be achieved with existing fuels were analyzed at three water levels (50 cm, 90 cm, and 130 cm) as shown in Fig. 2, and will be designated as “reference” STACY configurations in the frame of this study. The 1.80 cm and 1.27 cm lattice pitch configurations are both based on the 1.27 cm grid plate. The 1.80 cm lattice pitch configuration is loaded with the fuel rods in a checkerboard pattern, and the corresponding volume ratio of moderator to fuel (Vm/Vf) is 4.77. The 1.27 cm lattice pitch configuration contains an external zone with 1.80 cm lattice pitch with the fuel rods uniformly loaded in the center providing a Vm/Vf of 1.71. This is because the existing number of the fuel rods is not enough to achieve criticality even if they are uniformly loaded on the 1.27 grid plate. The 1.50 cm lattice pitch configuration, uniformly loaded on the 1.50 cm grid plate, has moderator to fuel ratio of 2.92 and is close to an optimum moderation pitch.

## 3. The sampling of non-uniform configurations

The modified STACY facility is an experimental installation that allows critical integral experiments to be performed on non-uniform configurations of fuel, concrete and stainless steel rods, arranged on a regular lattice and immersed in water. Among all of the possible configurations, that is, every arrangement of *N* fuel pins, *M* concrete rods and *K* stainless steel rods, over a rectangular lattice of size $H \times L$, the crux is to identify those that best cover and represent the various needs for validation of both the simulation schemes and the nuclear data.

This appears to be quite a daunting task, and for the purposes of the present work, a more limited framework has been adopted. First, the focus is placed on the configurations using only fuel pins and the lattice grids as used in the so-called “first criticality” experimental campaign, detailed in Gunji (2023). Concrete or stainless steel rods are thus not considered in this work. Second, for both simplicity and definiteness, the lattice used in the following is fixed to a square lattice of size $31 \times 31$ (both pitches of 1.27 cm and 1.50 cm were considered).

With these additional constraints, the experimental setup can be viewed as a $31 \times 31$ isolated fuel assembly immersed in water. An interesting feature of the system is that the fuel assembly is not completely filled with fuel pins. Some of the 961 slots in the square lattice are left vacant, and these empty positions are filled with interstitial water. Thus, even if this simplified setting does not really embrace the full complexity associated with fuel debris modeling, it nevertheless manages to capture an interesting aspect of the overall problem.
<figure><img src="image_4.png" /><figcaption>Fig. 1. Photo of the modified STACY taken from the top of the core tank. The fuel rods, guide pins, safety plates are loaded in the core.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Fig. 2. The three reference core configurations analyzed in the preliminary analysis in Gunji (2023).</figcaption></figure>

The presence of water holes in the fuel assembly generates highly heterogeneous local moderation conditions. This is a first step in the right direction, as this heterogeneous nature of the moderation will be naturally present in fuel debris. The subject is all the more interesting as international databases of integral experiments (e.g. ICSBEP, NEA, 2022) mainly feature experiments with rather homogeneous moderation conditions. STACY is therefore a good opportunity to enrich these databases and to challenge the adequacy of the simulation schemes and associated nuclear data in this relatively less explored context. The simplified setting chosen for this work presents yet another advantage. It addresses at the same time the experimental need for validation of configurations of assemblies with missing fuel rods problems. Recent results have shown that for an isolated fuel assembly with missing fuel rods, a set of different methods can be devised and have been proposed to address the problem (Dechenaux et al., 2024; Hilbert, 2004; Hoeffer et al., 2019; Albert et al., 2019).

A promising approach consists in assigning a simple coordinate system to the problem, which can be used to describe the reactivity variations observed for different lattice arrangements (Dechenaux et al., 2024). This method has proven useful in studying the sensitivities of multiplication coefficient $k_{\text{eff}}$ to the various cross-sections involved in the system (Dechenaux, 2024). In this section, the methodology is adapted to the STACY experimental setup, and to the design of critical experiments.

### 3.1. The configuration space and coordinate system of the problem

The “configuration space” of the problem is defined as the set of all possible configurations, meaning all the ways in which the fuel rods can be arranged on the regular lattice of the experiment. Currently, the facility is limited to a maximum of 400 fuel rods, though this number may increase to 900 in future experimental campaigns. Therefore, a maximum of 900 fuel rods was considered in this study. To provide a clear and definite boundary of the configuration space, a lower limit on the number of fuel rods was also set, and configurations with fewer than 200 rods were excluded from the analysis. This cutoff on the number of fuel rods might be considered to be somewhat too restrictive, as critical configurations can be found in this region. However, these configurations correspond to very specific patterns and are therefore better studied separately from the bulk of possible configurations.

As previously mentioned, it was decided, for the sake of simplicity, to limit this study to a 31 × 31 square lattice. The method developed in this section could easily be adapted to accommodate smaller, larger, or rectangular lattices. However, as a first approach, we opted to keep the problem simple by using a square lattice of size 31 × 31. This size was chosen to ensure it could accommodate both available grid lattices (with 1.27 cm and 1.50 cm pitches).

With all these elements, the total configuration space of the problem consists of all the possible arrangements of $N$ identical fuel pins (with $N \in [200; 900]$) on a lattice of size 31 × 31. Recent work (Dechenaux
<figure><img src="image_6.png" /><figcaption>Fig. 3. The configuration space of the problem, represented in the 2-dimensional plane (N, S). Blue points correspond to the coordinate of configurations that were either drawn at random (left) or selected through an optimization procedure, using a genetic algorithm (right).</figcaption></figure>

et al., 2024) suggests that it is possible to equip the space with a simple coordinate system (N, S), where,

* N is either the number of fuel pins or the number of vacant positions on the lattice (in the present work, it will always represent the number of fuel pins);

* S is related (but not simply equal) to the number of adjacent pairs (fuel rod, empty positions), see Dechenaux et al. (2024).

The precise definition of the coordinate *S* in the literature was tailored to fit the needs of a fuel lattice with very few empty positions (at least compared to the present situation). In particular, the coordinate was normalized so that, in this case, the variable *S* took values within the bounded interval [0; 1]. The same definition and normalization are used here, but the increased dimensionality of the problem (mainly due to the greater number of possible vacant positions) requires careful attention to identifying the boundaries of the configuration space. The lower bound of the coordinate can be shown to be slightly negative but identical for all *N* (a property that has already been identified and discussed in the literature (Dechenaux et al., 2024)). In the present context, negative coordinates can be shown to indicate highly ordered configurations, where all fuel pins are tightly packed together some- where on the lattice. Although these configurations cannot entirely be dismissed, they can always be analyzed separately. For this reason, and since heterogeneous configurations are ultimately sought, it will be assumed in the following that one can safely set min(*S*) = 0. The upper bound of the variable *S* is a function of the number of fuel pins, max(*S*) = *f*(*N*). From this, it can be concluded that the total, two-dimensional configuration space associated with the problem has a non-trivial shape. Fortunately, its boundaries can still be readily estimated. The overall shape of the configuration space is illustrated by the light purple shaded areas in both plots of Fig. 3. In these, it is observed that the maximum value of *S* significantly deviates from unity for fuel rod numbers below 500.

It can easily be shown that to each possible lattice arrangement, there corresponds a unique coordinate (N, S) (the converse is not true). Each and every possible configuration thus finds a place in the configuration space. The next step would now consist in devising a method to sample the configuration space, in such a way that no region is left unexplored. As the left plot of Fig. 3 shows, a simple random sampling of the lattice arrangements proves a very inefficient approach. If one aims to sample the most diverse set of possible arrangements, relying on random sampling is insufficient. A dedicated method must be implemented to systematically and consistently explore the problem. The simple coordinate system assigned to the configuration space makes this task relatively straightforward. In the right plot of Fig. 3, a genetic algorithm (McCall, 2005) optimization method was used to

generate configurations for a total of 580 target coordinate values. Although the sampling is still not perfect (primarily because the algorithm was not run for a sufficiently long time), it proves to be significantly more effective in exploring the diversity of configurations that can be encountered.

## 3.2. Unraveling a relationship between the coordinate system and the reactivity variations

The key aspect of the introduced coordinate system is its connection to the reactivity of the associated lattice configurations. Our primary interest lies in examining configurations that reflect different physical behaviors related to neutron interactions, often characterized by *k*<sub>eff</sub> sensitivities. The similarity in *k*<sub>eff</sub> values serves as an initial indicator that the interactions occurring within these configurations may be comparable. Furthermore, the predictive capability of *k*<sub>eff</sub> is essential for assessing the feasibility of the proposed configurations. To understand this relationship, 300 samples were randomly selected among those generated with the genetic algorithm. For a fixed water height level of *h* = 70 cm, and using the lattice with a 1.27 cm pitch, their reactivity was estimated through Monte Carlo simulations in criticality mode, employing the Serpent 2 code (version 2.1.32) (Leppanen et al., 2015) and the ENDF/B-VII.1 nuclear data library (Chadwick et al., 2011). To perform the simulations, a simplified model of the modified STACY facility was employed. A description of this simplified model and of the underlying assumptions is presented in Gunji (2023). For each simulation, the statistical uncertainty on the estimated multiplication coefficient was systematically maintained within a range of 100 pcm. The results are presented in Fig. 4, which shows the estimated *k*<sub>eff</sub> variations as a function of the number of fuel pins and the *S* coordinate (represented on a color scale).

The overall shape observed in the figure closely resembles results previously obtained in the literature (Dechenaux et al., 2024), leading to the conclusion that *k*<sub>eff</sub> appears to depend smoothly on the variables *N* and *S*. It seems that for any fixed value of these coordinates, one can refer to Fig. 4 to infer the corresponding reactivity.

If this were entirely true, it would drastically simplify the overall problem, as all configurations sharing the same coordinate values, namely, the thousands of possible lattice arrangements with the same (or similar) values of (N, S), would also share the same reactivity. This property could then be put to good use as one could group together all of the configurations sharing similar coordinates and use one configuration in each group to serve as a representative element of the ensemble. It is, of course, not possible to draw a definitive conclusion based solely on the results of Fig. 4. A more comprehensive and in-depth analysis of the problem would be required to confidently assert the existence of such a smooth relationship between reactivity and the
<figure><img src="image_7.png" /><figcaption>Fig. 4. Relation $k_{\text{eff}}$ vs number of fuel pins for 300 lattice arrangements chosen in the sampled configuration space. The color scale encodes for the coordinate $S$ of each configuration.</figcaption></figure>

coordinates of the configuration space and quantify the remaining reactivity spread within each ensemble. One might argue, however, that in the present context, exhaustive precision is not strictly necessary. It may suffice to rely on approximately accurate relationships and expand on them as needed for deeper analysis. Simply put, the ultimate goal is to identify a few interesting configurations that can be performed, rather than exhaustively exploring all possibilities, like it would be the case in criticality safety assessments. For this purpose, postulating the existence of a smooth map $k_{\text{eff}} = f(N, S)$ will suffice to efficiently explore the configuration space and pinpoint the subset corresponding to potential critical experiments.

## 4. Identifying and exploring critical configurations in STACY

Equipped with an appropriate coordinate system and robust sampling methods, it is now relatively straightforward to explore the full range of possibilities and extract candidate configurations for potential experimental realization. To do so, it is however necessary to ensure that the proposed configurations meet the constraints of the facility. As was detailed in Section 2, in the STACY facility, the criticality is approached by adjusting the water height in the cylindrical tank surrounding the fuel assembly. As such, the most important constraint that a given configuration must satisfy is that it must achieve criticality for an admissible water height $h_{\text{crit.}}$, which, in STACY, must fall within the interval [40; 140] cm. The relationship between reactivity and the variables (N, S) is of crucial importance here. It allows us to postulate that all of the configurations that have been grouped together, as they are expected to have the same $k_{\text{eff}}$, will also share similar critical heights.

This problem is specific to the design of critical experiments, but the tools that has been devised make it relatively easy to define a strategy to tackle this new problem. Starting from the samples provided by the genetic algorithm in Section 3, we can perform a set of Monte Carlo simulations for each configuration at different water heights and, from these, interpolate the critical water height (i.e. the height $h_{\text{crit.}}$ leading to $k_{\text{eff}} = 1$) for each one of them. In this way, one obtains a preliminary idea of the regions of space containing potentially critical configurations, that meet the experimental constraint. The sampling of configurations in these regions of interest can be further refined, and the critical height again extracted, etc.

These regions of interest will evidently differ depending on the lattice pitch that is used. Fig. 5 shows the results obtained for the lattice with a 1.27 cm pitch and the region that satisfies the experimental constraint related to the critical water height. The figure is

the result of the iterative process described above. For each individual configuration, 10 Monte Carlo simulations were performed using the Serpent 2 code (with a low precision on the multiplication coefficient, of the order of several hundreds of pcm), at water heights spaced every 10 cm between 40 and 140 cm. An interpolation method is then put in place to infer the critical height of the configuration. It has been observed that the critical height obtained in this way typically does not differ significantly from the actual critical height. The results in Fig. 5 show a smooth variation of the critical height with respect to the coordinates. A gradual increase is noted, primarily driven by the number of fuel pins present on the lattice. This trend aligns with expectations, as the critical water height decreases when the number of fuel pins increases. Furthermore, an increase in $S$ corresponds to more effective moderation of the configurations. In under-moderated pitch configurations, $S$ tends to increase the $k_{\text{eff}}$ or equivalently, lower the critical height. When the critical height falls below the acceptable criteria, the configuration is rejected, allowing only those with fewer fuel pins to reach higher $S$ values. This explains the observed shape of the admissible configurations and highlights that the large area in the upper right section of the figure is actually over-critical. This reactivity reservoir could be explored in the future, for example, by introducing absorbing materials.

The edges of the region of interest are not clearly defined, and additional efforts could be made to better delineate this subspace. However, for the practical realization of experiments, it proves more convenient to avoid configurations near the upper and lower bounds, as they are too close to the operating constraints for the water level. By focusing on the interior region, there are already plenty of configurations to explore, so further investigation of configurations on the boundaries will not be pursued. Along the same line, it should also be noted that the region identified is not claimed to contain all possible configurations; for example, it does not include the reference configurations displayed in Section 2. The region includes most configurations, and this is more than sufficient for the purpose of the present work.

The same process was applied to the lattice with a 1.50 cm pitch. The results are presented in Fig. 6, and it is now observed that the set of admissible configurations is split into two almost disconnected regions in the (N, S) space. A detailed inspection reveal that this separation arises from the fact that the upper region appears to be associated with configurations regularly arranged along the diagonal of the lattice. This is partly illustrated in Fig. 7, which shows two configurations, each extracted from the two distinct regions identified in the configuration space. The configuration on the left side of Fig. 7 is associated with a dense array of fuel pins arranged along the standard axes of the lattice
<figure><img src="image_8.png" /><figcaption>Fig. 5. Identification of the configurations whose corresponding critical water height meets the experimental constraint ($h_{crit.}$ ∈ [40; 140] cm). The figure shows the results obtained with the lattice with a pitch of 1.27 cm.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 6. Identification of the configurations whose corresponding critical water height meets the experimental constraint ($h_{crit.}$ ∈ [40; 140] cm). The figure shows the results obtained with the lattice with a pitch of 1.50 cm.</figcaption></figure>

with a homogeneous moderation distribution in this region. It can be shown to have the coordinates ($N, S$) = (360, 0.15), and thus belongs to the lower cluster in the ($N, S$) space.

The configuration on the right side of the figure belongs to the other cluster, and one can easily convince oneself that the fuel pins are in this case arranged along the diagonal axes of the lattice. A visual inspection of a sufficient number of configurations confirms that this is also the case for the other configurations. This analysis provides valuable insights into the physical understanding of the potential space available for investigating critical configurations.

Note that in the case of the 1.27 cm pitch lattice, it is very likely that a second component, corresponding to configurations arranged along the diagonal of the lattice, could eventually be observed. In this scenario, and because this lattice pitch is associated to a global under-moderation of the configurations, it is expected that this component corresponds to lower values of $N$. The lower bound of 200 fuel pins likely prevents its observation.

## 5. The selection of critical experiments

So far, the configuration space has been effectively explored, and several configurations have been generated that can be considered as

proposals for experiments. The most delicate question remains: among the candidate configurations, which experiments will best meet the needs for validating codes and nuclear data? Traditionally, the selection process for experiments is guided by well-established criteria (e.g., typical sensitivity profiles, etc.). However, in the case of fuel debris, as with assemblies containing missing fuel rods, the problem is more complex, as there is no single reference configuration to rely on. Rather than focusing on specific criteria, attention can be shifted toward identifying gaps in existing databases. The reasoning is that, given the need to cover a wide range of cases, it is crucial to ensure validation data is available for all relevant regions. This approach aligns well with the method developed, as a large set of possible configurations is available for STACY. It can therefore be assessed how this set integrates into the existing experimental data landscape.

### 5.1. Sensitivity

As demonstrated in the literature, Dechenaux (2024), the coordinates that have been introduced also have the property to parameterize the $k_{eff}$ sensitivities to nuclear data of the problem. This is illustrated, for example, in Fig. 8, which shows the sensitivity spectra for the fission cross-section of $^{235}$U for all of the 110 critical configurations
<figure><img src="image_10.png" /><figcaption>Fig. 7. Configurations extracted from the two disconnected regions found for the lattice with a pitch of 1.50 cm.</figcaption></figure>

found for the 1.27 cm lattice pitch. In the figure, each individual sensitivity profile is the result of a dedicated simulation, performed at the extrapolated critical height and using the 252 groups energy grid associated with the SCALE-6.3 covariance data (and pre-defined in Serpent). The sensitivity calculations were performed using 100 000 neutrons per cycle, 200 active and 70 passive cycles. These simulation parameters, as well as the usage of 10 latent generations, were chosen to ensure that a sufficient precision (around few percent) is obtained in the most important energy bins of the sensitivity profiles of the capture and fission cross sections of the main isotopes present in the system. This level of precision was set in order to be able to discriminate between the different configurations. It should be noted, however, that obtaining a similar level of precision for the scattering cross sections and the associated thermal scattering law would require significantly higher computational costs. The sensitivity spectra were reconstructed and analyzed with the SERPENTTOOLS Python module (Andrew E. Johnson and Ridley, 2020).

The Fig. 8 shows that the sensitivity profiles for the ²³⁵U cross section are ordered according to the coordinate *S* associated with each configuration. Sensitivity increases with *S* for energies below around 0.1 eV, but this trend reverses for higher energies. Similar variations were also observed for ²³⁸U capture, ¹H capture and ²³⁵U capture. A possible explanation for this behavior can be understood quite simply: *S* ultimately measures the total amount of water accessible around each fuel pin. The greater this quantity (i.e., the larger *S*), the higher the proportion of thermal fissions.

A more detailed analysis of the dependence of the sensitivity profiles on the coordinates would be interesting, but it would divert us from the main objective pursued in the present work. Relying on a simplified description, in terms of energy-integrated sensitivities, was found satisfying enough to discriminate between the different configurations. Fig. 9 shows the energy-integrated sensitivities to the capture cross section on ¹H and the fission cross section on ²³⁵U respectively, as a function of both the number of fuel rods and of the variable *S* (on a color scale) for all critical configurations of the 1.27 cm lattice pitch. The similar behavior obtained on the two plots suggests that there exists a strong correlation between those two sensitivities for STACY configurations.

To determine which of the configurations are most suitable for the realization of critical experiments, a comparison was performed with the existing benchmarks documented in the ICSBEP Handbook, with a focus on the benchmarks that closely align with STACY's experimental setup: low-enrichment UOx fuel rods immersed in water, with varying moderation ratios and, whenever possible, a heterogeneous distribution of the moderation. The sensitivity profiles of the benchmarks used

were partially extracted from the DICE database (Hill et al., 2015), primarily calculated using KENO code. For some experimental series, sensitivities were calculated with MORET (Cochet et al., 2015), using either the JEFF-3.1.1 or ENDF/B-VII.1 libraries. While there is an inconsistency in the nuclear data libraries and codes employed for the sensitivity calculations, their impact is regarded as second-order effects within the context of the present study.

Here again, one could very well proceed with a comparison of the full, energy-dependent sensitivity profiles, but the integrated sensitivities were found to be sufficient to discriminate between the different experiments. The Fig. 10 shows the data extracted for the selected Low Enrichment Uranium (LEU) benchmarks, in the two dimensional space of the, previously analyzed, integral sensitivities to ¹H capture and ²³⁵U fission cross sections. The figure also contains the integrated sensitivities of the reference STACY configurations, described in Section 2, shown as red crosses of the figure. The critical configurations resulting from the selection process described in the previous section are also displayed, they appear as red (1.27 cm pitch) or green (1.50 cm pitch) diamond on the figure. As was already noted in Fig. 9, it appears that all of the identified critical configurations fall more or less along a unique diagonal line in the bi-dimensional space of the integrated sensitivities. Additionally, this diagonal line overlaps only very partially with the set of pre-existing experiments, and there therefore seem to be plenty room to propose new and original experiments with the STACY facility. The region in the lower right corner of the figure, with an integrated sensitivity to the fission cross section on ²³⁵U greater than 0.34, appears to exhibit physical properties not found in any pre-existing experiments or in the reference STACY configurations. As a result, it was decided to focus on this subspace and extract candidate configurations from within this region. It is worthwhile to mention that the inspection of different planes, associated with other energy-integrated sensitivities, ultimately led to the selection of a similar subset of candidate configurations.

In the selected sub-space the configurations can employ either of the studied pitches. To further refine the selection of configuration within this sub-space, one should proceed with a careful inspection of the sensitivity profiles of each configuration, but also consider any other relevant metrics that could be used to better distinguish and delineate possibly different physical regions. Because the focus is set on proposing a limited set of experiments to be performed, working at the level of energy-integrated sensitivity profiles was deemed sufficient. A further exploration of the problem is therefore left for future work. Although a lattice pitch of 1.5 cm allows for a slightly greater distance from existing experiments compared to the pitch 1.27 cm, the choice was ultimately made to use the under-moderated, 1.27 cm pitch. A careful study of the configurations associated to the 1.50 cm lattice pitch is
<figure><img src="image_11.png" /><figcaption>**Fig. 8.** $k_{\text{eff}}$ sensitivity profiles for the $^{235}\text{U}$ fission cross section, for each and every critical configuration sampled for the 1.27 cm lattice pitch.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>**Fig. 9.** Critical configurations on the lattice pitch 1.27 cm in the (N, S) space, sharing the same ordinate, colored according to the integral sensitivity to $^{1}\text{H}$ capture (left) and $^{235}\text{U}$ fission (right) cross sections.</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>**Fig. 10.** ICSBEP experiments, reference STACY configurations and the possible STACY configurations in the two dimensional space of energy-integrated $k_{\text{eff}}$ sensitivities to $^{1}\text{H}$ capture and $^{235}\text{U}$ fission cross sections.</figcaption></figure>
<figure><img src="image_14.png" /><figcaption>Fig. 11. Two proposed core configurations: ordered on the left, and unordered on the right.</figcaption></figure>

left for future study. Finally, from the subset of the configurations with 1.27 cm lattice pitch, a configuration was chosen based on its reactivity worth of water thermal scattering law, that will be discussed on more details below. All of the aforementioned considerations ultimately led to the selection of the configuration illustrated in the right-hand plot of Fig. 11.

This heterogeneous configuration can be paired with another, highly ordered, configuration that shares very similar global physical properties. This associated configuration, arranged along the diagonal of the lattice, is illustrated on the left-hand plot of Fig. 11. For simplicity, these two candidate configurations will be respectively referred to as ordered and unordered in the following. The joint analysis of both experiments would represent an almost unprecedented opportunity to probe the impact of the heterogeneity in STACY-like configurations, offering new and interesting experimental support for the validation of neutron transport methods and nuclear data in disordered media.

Some important global physical parameters of the two candidate configurations are listed in Table 1. Both configurations are in average over-moderated, as can be seen from the global moderation ratio ($V_m/V_f$), that is within the specifications of STACY operation, as described in Izawa et al. (2023). Other parameters, for instance, the energy of average lethargy causing fission (EALF), the resonance escape probability $P$ and other studied physical parameters remain comparable between the two proposed configurations. The most notable difference lies in the reactivity worth of the thermal scattering law data, which significantly varies between them. Although the $k_{eff}$ sensitivity to $S(\alpha, \beta)$ is also expected to differ, the challenges of accurately calculating this sensitivity using Monte Carlo methods led us to focus on the total reactivity worth, evaluated through two independent calculations, as the primary quantity of interest.

## 5.2. Thermal scattering law reactivity worth

The overall approach for proposing new experiments aimed to identify configurations that differ significantly from any pre-existing experiment in terms of their physical characteristics. The selection process relied on an analysis of the integral sensitivities to ¹H capture and ²³⁵U fission cross-sections. The analysis of the selected unordered configuration highlighted that its distinguishing feature, compared to more ordered configurations, was its $S(\alpha, \beta)$ reactivity worth, reaching a value of around 3700 pcm. To further investigate this specific criterion, an additional study was conducted.

For each of the 580 sampled configurations displayed in Fig. 3, two Serpent simulations were successively performed, at a fixed water level of 70 cm, with or without the activation of the water thermal scattering

<table><caption>Table 1
Parameters describing the final configurations.</caption><thead><tr><th></th><th>Ordered</th><th>Unordered</th></tr></thead><tbody><tr><td>Number of fuel rods</td><td>184</td><td>329</td></tr><tr><td>S variable</td><td>0.2</td><td>0.26</td></tr><tr><td>Fuel rod array size</td><td>23 × 23</td><td>31 × 31</td></tr><tr><td>Approx. critical height (cm)</td><td>110</td><td>82</td></tr><tr><td>k<sub>eff</sub></td><td>0.99952 ± 23</td><td>0.99999 ± 20</td></tr><tr><td>V<sub>m</sub>/V<sub>f</sub></td><td>7.8</td><td>7.9</td></tr><tr><td>EALF (eV)</td><td>0.079</td><td>0.082</td></tr><tr><td>eta factor from 6 factor formula</td><td>1.93</td><td>1.93</td></tr><tr><td>F factor from 6 factor formula</td><td>0.532</td><td>0.536</td></tr><tr><td>P factor from 6 factor formula</td><td>0.914</td><td>0.913</td></tr><tr><td>e factor from 6 factor formula</td><td>1.079</td><td>1.081</td></tr><tr><td>LF factor from 6 factor formula</td><td>0.990</td><td>0.982</td></tr><tr><td>LT factor from 6 factor formula</td><td>0.998</td><td>0.997</td></tr><tr><td>Fission rate in fuel (a.u.)</td><td>0.0726</td><td>0.0717</td></tr><tr><td>Capture rate in water (a.u.)</td><td>0.451</td><td>0.446</td></tr><tr><td>Sensitivity to ¹H capture</td><td>-0.241</td><td>-0.289</td></tr><tr><td>Sensitivity to ²³⁵U fission</td><td>0.344</td><td>0.347</td></tr><tr><td>S(a, b) worth: Δk<sub>eff</sub> = <sup>k<sub>eff</sub>S<sub>eff</sub>ab - k<sub>eff</sub>S<sub>eff</sub>ab</sup>&frasl;<sub>k<sub>eff</sub>ab</sub></td><td>1260 pcm</td><td>3700 pcm</td></tr></tbody></table>

law data in the simulation. In both cases, the simulation parameters were set so that the uncertainty on the final multiplication factor was of the order of 100 pcm. The relative differences $\Delta k_{eff}$ (see the definition in Table 1) observed for both lattice pitches are displayed in Fig. 12. In the figure, the regions corresponding to configurations accessible within the experimental constraints are delineated by red dashed lines.

It should be noted that since $k_{eff}$ may deviate significantly from unity, directly comparing the evaluated reactivity worths might be misleading in some cases. Nevertheless, some preliminary but insightful observations can be drawn from these plots. First, the variation in $S(\alpha, \beta)$ reactivity worth in the $(N, S)$ space appears to be less smooth compared to the variation observed in the multiplication coefficient, as illustrated in Figs. 5 and 6. This suggests that for a given $(N, S)$ value, the variability in $S(\alpha, \beta)$ reactivity between different configurations with the same $(N, S)$ value might be too high to be represented by a single configuration. To confirm this analysis, further statistical studies are required.

Nevertheless, an overall trend can be identified, indicating that the $(N, S)$ region with a higher $S(\alpha, \beta)$ reactivity worth typically corresponds to a lower number of fuel pins. Another interesting observation is that, across this extensive exploration of configurations, the highest reactivity worth within the region of feasible configurations is around 5000 pcm. Given that the explored configurations are not necessarily critical, this value can be compared to the 3700 pcm observed for
<figure><img src="image_15.png" /><figcaption>Fig. 12. Exploration of the thermal scattering law reactivity worth in the coordinate system of S value and the number of rods, for the lattice pitches 1.27 cm on the left and 1.5 cm on the right.</figcaption></figure>

the selected unordered configuration. These preliminary results offer valuable insights that could guide future efforts to potentially identify configurations with even higher S(α, β) reactivity worth.

## 6. Conclusion

This paper has demonstrated that designing critical experiments to represent the diverse heterogeneous conditions encountered in both fuel debris retrieval operations and the criticality safety assessment of fuel assemblies with missing fuel rods is a particularly challenging task. The inherent complexity of the problem, stemming ultimately from the wide range of possible configurations, means that no single application case can be defined, whose sensitivity profiles could be set as reference and used to optimize the design of a critical experiment.

To overcome this problem, the approach that has been taken in the present work proposed to provide the whole configuration space of the problem with an adapted coordinate system, which could be used to pave and systematically ensure that every physically distinct arrangements of the fuel pins on the lattice grid of the experimental setup were considered. Relying on the methods recently developed in the context of the criticality safety of fuel assemblies with missing fuel rods, and adapting these to the design of experiment in the modified STACY facility, revealed an invaluable tool to systematically explore the space of possible configurations and precisely engineer a specific, highly heterogeneous configuration, to fill the gap existing in the current ICSBEP database.

The two configurations that were ultimately proposed are interesting in two respects. First, although these two configurations appear very dissimilar at first glance, they share similar global physical properties (spatially averaged observables, so to speak) but differ greatly in the local arrangement of the fuel rods. There conjoint evaluation as critical experiments will therefore constitute an interesting opportunity for the validation of the simulation codes and nuclear data in the context of disordered geometries. The second notable aspect of the proposed candidate configurations lies in their differing sensitivities to the thermal scattering laws of hydrogen atoms in water. Here also, the evaluation of the pair of configurations reveal beneficial, as a joint analysis might help to isolate the specific impact of the water S(α, β), which is crucial for the experimental validation of this specific nuclear data.

Note finally that the present work opens several promising new avenues for further study. Only one highly disordered STACY core design was selected from a vast range of possibilities. It may be beneficial to select additional configurations from the studied space to achieve a broader coverage of potential setups. Another natural extension of these studies is to consider incorporating other types of rods into the

core configuration, such as concrete and stainless steel rods, which will be used in STACY to mimic fuel debris. Preliminary studies suggest that the same coordinates (N, S) could still be efficiently used, with only some minor adjustments, to integrate other materials into this approach (Dechenaux et al., 2025). In more complex situations, an extension of the present work could be carried out by resorting to a more heuristic approach, trying to make emerge a set of representative variables, extracted from a careful data driven approach of the problem (Dechenaux, 2022).

## Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

## Data availability

The authors do not have permission to share data.

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