# Ex-vessel stabilization of corium: An analysis of corium-concrete interaction with top flooding for siliceous concrete

Florian Fichot

► To cite this version:

Florian Fichot. Ex-vessel stabilization of corium: An analysis of corium-concrete interaction with top flooding for siliceous concrete. Annals of Nuclear Energy, 2025, 223, pp.111584. 10.1016/j.anucene.2025.111584. hal-05318086

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# Ex-vessel stabilization of corium: An analysis of corium-concrete interaction with top flooding for siliceous concrete

Florian Fichot

<figure><img src="image_3.png" /></figure>

Autorité de Sûreté Nucléaire et de Radioprotection (ASNR), PSN-RES/SAM/LEPC, F13115 Saint Paul lès Durance, France

## ARTICLE INFO

**Keywords:**

Corium

Concrete

Water ingression

Ex-vessel stabilization

## ABSTRACT

In case of severe accident without possibility of in-vessel retention, the corium must be stabilized outside the vessel, either in a designed core-catcher or directly on the concrete basemat of the reactor building, after spreading. In this last case, the stabilization strategy must be efficient enough to avoid the progression of corium through the concrete basemat and to the environment.

In order to ensure corium stabilization, the physical processes leading to quench some part of the liquid corium must be fast enough, compared to the erosion of concrete. One of the possible strategies consists in spreading the corium over a rather large area (i.e. the reactor pit plus one or two adjacent rooms) and flood it with water on top. Two processes have been identified experimentally for a potential quenching of corium: melt eruption and water ingression. Melt eruption is a process that is driven by the flow of gases (CO₂ and H₂O) generated by concrete erosion: the gas flow is likely to entrain droplets of corium into the overlying water, quenching the droplets quickly. Water ingression is a process where cracks are generated by the contraction of the top crust due to cooling: this process enables the flow of water into the cracks and the propagation of the cracking front deep into the corium crust. When the basemat is made of siliceous concrete, stabilization depends mostly on water ingression because there is little gas content in such concrete and melt eruption may only have a limited effect.

This paper presents the analysis of a situation where corium is spread over a rather large area (around 80 m²) in order to reduce the corium thickness after spreading. Of course, this thickness depends on the reactor design (total volume of corium divided by total spreading area). After spreading, corium is flooded with water on top. First, the melt eruption process is investigated: it is shown that it has a limited efficiency in case of siliceous concrete but it plays an important role in triggering the water ingression earlier. Then, the process of water ingression is examined, as it is the process leading to complete stabilization. The maximum heat flux extracted by water ingression (CHF) is the most important parameter in this process and needs to be evaluated accurately. Therefore, a theoretical determination of the CHF is made, as a function of the concrete content in the corium, based on the existing Lister-Epstein model, on data obtained earlier at Argonne National Laboratory and on data measured in volcanic rocks. In addition, a simple modeling of transient heat conduction through the concrete basemat is proposed, to determine a criterion of stop of erosion that can be used in a lumped-parameter code. As a conclusion, it is shown that the quenching of corium may be divided in two main stages. The first stage is characterized by a fast erosion of concrete and a significant eruption and quenching of debris. This stage lasts from half a day to a few days, depending on the type of concrete and on the model parameters. The second stage is characterized by a slower erosion of concrete and the start of water ingression. This stage may last from half a day up to one week depending on the type of concrete and on the model parameters. Finally, a sensitivity analysis is made to determine the effects on the erosion depth and on the time of stabilization. It appears that there is a non-trivial dependence of water ingression efficiency on the occurrence of melt eruption at the very beginning of corium concrete interaction. Melt eruption, even if limited, is likely to significantly enhance the triggering of water ingression and favor a fast stabilization of corium.

<footnote>E-mail address: florian.fichot@asnr.fr</footnote>
<figure><img src="image_4.png" /><figcaption>Fig. 1. Schematic view of corium after spreading and flooding with water.</figcaption></figure>

# 1. Introduction

In case of severe accident without possibility of in-vessel retention, the corium must be stabilized outside the vessel, either in a designed core-catcher or directly on the concrete basemat of the reactor building, after spreading. In this last case, the stabilization strategy must be efficient enough to avoid the progression of corium through the concrete basemat and to the environment.

One of the strategies consists in pre-flooding the reactor cavity where the corium is expected to flow and break up into a particle bed which may be coolable: this strategy is implemented is several BWRs where the height of water below the vessel is high enough to predict a complete break-up of the corium jet into a particle bed (Allelein and Bürger, 2006; OECD/NEA, 2017). An alternative strategy consists in spreading the corium over a rather large area including the reactor pit plus one or two adjacent rooms and flood it with water on top, as illustrated in Fig. 1.

Before introducing the phenomenology, it is useful to mention that the objective of this paper is clearly not to discuss about the “validity” or “physical consistency” of the existing models for corium coolability: those models have been developed and used for quite a long time now, they have been justified and discussed as much as possible in the past by specialists and they can be considered as the “state-of-the-art” (OECD/NEA, 2017). It is probably still possible to improve those models, but this was not the objective of this paper. The objective is to select the models which are commonly used in the reference codes and to make a thorough analysis of a mitigation strategy which has rarely been investigated (flooding after spreading). Therefore, the description of the phenomenology is taken from what is considered as the “state-of-the-art” understanding. Some improvements are provided in this paper, for the purpose of a more accurate evaluation of the safety margin of implementation of the “spreading and flooding” strategy, in the specific case of siliceous concrete.

Two processes have been identified experimentally for a potential quenching of corium by top cooling: melt eruption (Farmer, 2006; Robb, 2011; Robb and Corradini, 2011) and water ingression (Lom-perski and Farmer, 2007; Nayak et al., 2009; Villarreal-Larrauri et al., 2021). Melt eruption is a process that is driven by the flow of gases ($CO_2$ and $H_2O$) generated by concrete erosion: the gas flow is likely to entrain droplets of corium into the overlying water, quenching the droplets quickly. The resulting debris layer is assumed to be permeable enough to avoid remelting. Water ingression is a process where cracks are generated by the contraction of the top crust due to cooling: this process enables the flow of water into the cracks and the propagation of the cracking front may progress deeper into the corium crust. When the basemat is made of siliceous concrete, stabilization depends mostly on water ingression because there is little gas content in such concrete and melt eruption may only have a limited effect. The stabilization depends mainly on the competition between the ablation of concrete and the quenching of corium (either by formation of fuel-bearing debris or cracks in the crust). In this paper, the quenching rate is estimated for each of the two processes and compared to the ablation rate. A rather simple model is used, where the main governing parameters appear. First, the role of melt eruption is explained and discussed. Then water ingression is examined. Because water ingression is the

most important cooling mechanism for siliceous concrete basemat, the maximum heat flux extracted by water ingression (CHF) appears as one of the most important parameters. A theoretical determination of the CHF is proposed as an extension of the existing Lister-Epstein correlation (Epstein, 1999; Lister, 1974). A criterion of end of ablation is proposed too, to complete the modeling. Finally, a simple application is presented to illustrate the interactions between the two cooling processes. Before concluding, a sensitivity analysis is performed and shows some of the non linear effects coming out of the equations of the model.

# 2. A basic model to analyze the phases of corium cooling and concrete erosion

In this paper, a rather simple approach is chosen in order to make a clear distinction between the two cooling mechanisms and show their respective impact. Considering that corium is spread over a total surface of around 80 m², the corium layer has an initial thickness close to 20 cm. Therefore, it is reasonable to consider that the problem is one-dimensional. The main variables of the problem are the masses (or heights) and compositions of each layer. The layers are represented in Fig. 2. From bottom to top, they are:

1. the concrete layer (ablated by the heat-generating corium located just above)

2. the corium layer (which is a mixture of fuel and concrete elements, cooled from the top)

3. the compact crust layer (above the corium, resulting from solidification)

4. the fractured crust layer (resulting from crack formation and water ingression into the crust)

5. the debris layer (resulting from melt eruption)

An important simplification is made in the thermal analysis: it is assumed that the partitioning of residual power between the bottom and the top of the corium layer is given by a constant parameter. In general, this is not absolutely true (it evolves with time and ranges typically between 0.5 and 0.9) but this assumption makes the analysis easier to understand and is sufficient to get relevant orders of magnitude. With this assumption, the corium temperature plays no direct role, and it is not necessary to calculate it (of course, in the complete formulation, the corium temperature and the power partitioning coefficient are implicitly linked through the heat transfer coefficients which depend on temperature (Allelein and Bürger, 2006; OECD/NEA, 2017)). The concrete temperature is fixed, and the debris and fractured crust layers are at the saturation temperature of water (quenched corium). The remaining unknowns are essentially the masses of fuel in the layers and the masses of concrete in the layers. The masses of fuel in the corium, compact crust, cracked crust and debris are respectively denoted $m_f$, $m_s$, $m_{ic}$, $m_d$. The mass of concrete in the corium layer is denoted $m_c$ and the total mass of ablated concrete is denoted $m_{tot}$. Molten steel is also present in the initial corium but it is assumed not to play any role in the stabilization process because no published information was found on the effect of molten steel on water ingression or on melt eruption (this might need some revision if new data are obtained in the future). A more detailed nomenclature is provided in B. The fuel in the corium
## Layers involved in the model

<figure><img src="image_5.png" /><figcaption>Fig. 2. The five layers considered in the problem (represented here as a function of time).</figcaption></figure>

layer is depleted by debris formation and (compact and cracked) crust formation. This mass balance is written as:

$$ \frac{dm_f}{dt} = -\frac{dm_d}{dt} - \frac{dm_w}{dt} - \frac{dm_s}{dt} \quad (1) $$

where the debris formation is proportional to the ablation rate and to the content of gas in concrete:

$$ \frac{dm_d}{dt} = X_f k_e X_g Q_c \quad (2) $$

And the fractured crust formation is given by the expression proposed by Epstein (1999):

$$ \frac{dm_w}{dt} = \rho_f S X_f \frac{\phi_{CHF}^* - \phi_{up}}{\frac{\Delta h}{C_p(T_{sol} - T_{sat})}} \quad (3) $$

where $\phi_{CHF}^*$ is the actual $\phi_{CHF}$ reduced by the equivalent heat flux resulting from the volumetric heat dissipated in the cracked crust. Indeed, in the original Lister-Epstein model, steam is generated only at the bottom of the cracks whereas in the real situation, the volumetric power in the cracked crust also produces steam which reduces the effective flow of water which is able to reach the bottom of the cracks. Therefore we have:

$$ \phi_{CHF}^* = \phi_{CHF} - \frac{m_w E_f}{S} \quad (4) $$

At any time, according to Epstein model (Epstein, 1999), the mass of fuel in the compact crust is:

$$ m_s = X_f \rho S \frac{k(T_{sol} - T_{sat})}{\phi_{CHF}^* - \phi_{up}} \quad (5) $$

The ablation rate is proportional to the heat flux to the bottom:

$$ Q_c = \frac{dm_c^{tot}}{dt} = \rho_c S U_{abl} = \frac{\phi_{dn} S}{\Delta h_{abl}} \quad (6) $$

And the partitioning of the residual power is simply written as:

$$ m_f E_f = (\phi_{dn} + \phi_{up}) S \quad (7) $$

where:

$$ \phi_{up} S = k_{up} m_f E_f \quad (8) $$

Similar equations apply for the masses of concrete and steel in the layers. In the model, steel is transported with the other elements and rebars may be considered in the model by providing the mass fraction of steel in the concrete composition. However, in the experiments on melt eruption or water ingression performed up to now, the effect of

<table><caption>Table 1
Reference case parameters.</caption><thead><tr><td>Parameter</td><td>Reference value</td><td>Description</td></tr></thead><tbody><tr><td>$m_c^{tot} E_f$</td><td>20 MW</td><td>Residual power at initial time</td></tr><tr><td>$S$</td><td>80 m²</td><td>Spreading surface</td></tr><tr><td>$\phi_{CHF}$</td><td>80 kW/m²</td><td>Critical heat flux (water ingression)</td></tr><tr><td>$k_e$</td><td>0.06</td><td>Melt eruption coefficient</td></tr><tr><td>$k_{up}$</td><td>0.9</td><td>Power fraction to the top</td></tr><tr><td>$k_b$</td><td>0.03</td><td>Initial fraction of corium solidified (bulk cooling)</td></tr></tbody></table>

the presence of molten steel was not analyzed nor quantified: melt eruption depends mainly on the amount of gases in the concrete and water ingression depends mainly on the mass fraction of concrete in the corium. Therefore, in this paper, we do not pay more attention to the presence of steel since it has no impact on the conclusions (but we recall that this is due to the lack of experimental data on the effect of steel).

The reference values of the model parameters are provided in Table 1.

### 2.1. Evolution during the eruption-ablation phase

At first, erosion is intense, generating a strong flow rate of gas from the decomposition of concrete. The formation of quenched fuel-bearing debris leads to a decrease of the power in the corium layer. If we look at the system of equations involving only the melt eruption process (Eqs. (2), (6) and (7)), we have:

$$ \frac{dm_f}{dt} = -k_e X_g \frac{dm_c^{tot}}{dt} \quad (9) $$

$$ \frac{dm_c}{dt} = (1 - k_{up}) m_f \frac{E_f}{\Delta h_{abl}} - k_e X_g \frac{m_c}{(m_f + m_c)} \frac{dm_c^{tot}}{dt} \quad (10) $$

This set of equations may be solved analytically when $m_f$ is larger than $m_c$, which corresponds to the beginning of concrete erosion (approximately the first few hours or even up to the first day) and with an exponential decay of the residual power. The approximate solution is:

$$ m_f(t) = m_f(t=0) \exp \left( \frac{\tau}{t_{me}} (\exp(-t/\tau) - 1) \right) \quad (11) $$

where $\frac{1}{t_{me}} = k_e X_g (1 - k_{up}) E_0 / \Delta h_{abl}$. From this equation, we can see that the mass of corium cannot decrease down to zero, but, instead, it would reach an asymptotic value, after a time equal to a few $\tau$. This means that melt eruption alone is not sufficient to quench all the corium: melt eruption has an efficiency inversely proportional to the constant $t_{me}$ but it loses its efficiency with time, proportionally to $\exp(-\frac{\tau}{t_{me}})$. It is also interesting to notice that the ratio of the melt eruption rate to the ablation rate is proportional to the mass of fuel in the corium layer: hence, it decreases with time. This means that melt eruption decrease is faster than erosion decrease: the corium thickness is always growing if melt eruption is the only process considered and complete stabilization cannot be reached (but the residual power in liquid corium can be significantly reduced in the case of strong melt eruption, which occurs with limestone concrete). This is illustrated in Fig. 3 where the corium layer (yellow) grows faster than the debris layer (brown) and where only a little part of corium is quenched after 1 day, in spite of choosing the value $k_e = 0.08$, which is the maximum recommended in Farmer (2006).

To conclude, the melt eruption process should be considered as a precursor to the beginning of water ingression which is the only process able to totally quench the corium. This role of precursor is very important in determining the total time for stabilization: in the first hours, melt eruption reduces the mass of fuel in the corium and therefore significantly reduces the heat flux to the top, which is a necessary condition for the onset of water ingression into the crust.
<figure><img src="image_6.png" /><figcaption>Fig. 3. Example of evolution during the first two days without water ingression.</figcaption></figure>

## 2.2. Evolution during the water ingression phase

When the heat flux $\phi_{up}$ from the corium layer to the top crust reaches the value $\phi_{CHF}^*$, the crust thickness becomes thick enough for the generation of cracks (by excessive stress due to thermal contraction) at the top and the penetration of water, according to the theoretical approach of Lister (1974): this corresponds to the start of the water ingression.

Afterwards, the cracks progress downwards, separated from the liquid corium layer by a rather thin and compact crust, as illustrated in Fig. 4. After this initial trigger, the cracked front progresses steadily and water is able to flow into the cracks and quench the corium. This phenomenon is not specific to corium materials and it has been observed in particular in the solidification of lava lakes (Hardee (1980)) where the velocity of the solidification front was measured and appeared to be constant.

An example of evolution of corium quenching is presented in Fig. 5, where quench is examined during a few days, with water ingression only (the melt eruption is not considered). For this application, the value of $\phi_{CHF}$ is chosen equal to 80 kW/m² and does not decrease with time: it is not a realistic case but it is useful for understanding the dynamics of the process.

We see that there is an acceleration of the quench front (i.e. the limit of the cracked zone moves faster downwards), followed by a progression at a more linear rate. The initial acceleration is due to the decrease of $\phi_{up}$ caused by the decay of residual power and by the decrease of the mass of fuel in the corium layer: the driving force for solidification ($\phi_{CHF}^* - \phi_{up}$) increases. The quasi-linear progression corresponds to a period when ($\phi_{CHF}^* - \phi_{up}$) varies more slowly. This time-dependent variation of velocity was not addressed by Lister because he considered only lava without internal heat generation ($\phi_{CHF}$ and $\phi_{up}$ were constant in his study). It is interesting to examine in more details the time dependent behavior of the crack front velocity. The derivative of the velocity is expressed as:

$$ \frac{dU_{wi}}{dt} \propto \frac{d\phi_{CHF}^*}{dt} - \frac{\phi_{up}}{dt} = -\frac{d(m_f^{wi} E)}{dt} - \frac{d(k_{up} m_f E)}{dt} \quad (12) $$

Which may be developed as:

$$ \frac{dU_{wi}}{dt} \propto -\left(\frac{dm_w}{dt} + k_{up} \frac{dm_f}{dt}\right)E - (m_w + k_{up} m_f) \frac{dE}{dt} \quad (13) $$

If we neglect the formation of debris (melt eruption), we can write that $m_f^{wi} + m_f \approx m_f^{tot}$ because the compact crust is rather small. Therefore, we have:

$$ m_w \approx m_f^{tot} - m_f \quad (14) $$

So we may re-write:

$$ \frac{dU_{wi}}{dt} \propto (1 - k_{up}) \frac{dm_f}{dt} E - (m_w + k_{up} m_f) \frac{dE}{dt} \quad (15) $$

Numerical evaluations show that the first term (which is negative) is quickly smaller than the second term (which is positive): therefore the cracked front velocity is always increasing at the end of the process. This is illustrated in Fig. 8.

It is interesting to note that, to completely quench in one day a corium layer spread over a surface of 80 m² and generating 20 MW, it is necessary to have $\phi_{CHF}$ larger than 200 kW/m². In most cases, the CHF will be lower, which has two implications: the onset of water ingression will occur later and the quench front velocity will be lower. As a result, the complete stabilization will be reached after a few days (or even several days in some cases). This will be illustrated in Section 2.3.

## 2.3. Discussion on the combined effects of both cooling processes

In this section, we investigate the combined effects of both quenching processes and the feedback between them. A simple case is defined, with all the parameters set to the reference values indicated in Table 1. The results are presented in Fig. 6. During the first two days, melt eruption is the only active process because the heat flux to the top of the corium layer is too high to allow the crack formation and water ingression. As a consequence, the behavior is comparable to the case shown in Fig. 3: a small layer of debris is formed and fast erosion leads to the ablation of approximately 50 cm of concrete. Then, water ingression starts because of the decrease of $\phi_{up}$. This leads to a rather fast decrease of the height of the corium layer which is progressively quenched. As a consequence, the erosion of concrete strongly decreases. Another consequence is the decrease of the melt eruption. As it was explained, melt eruption has an asymptotic behavior with time: this reduction of efficiency with time is amplified by the onset of water ingression and the thickness of the debris layer remains almost constant after that. In this example, the stabilization is reached in a little more than 4 days.

An analysis of the powers, in Fig. 7, reveals that the debris layer, although rather thin, bears a significant part of the total power: approximately 30% of the total residual power is generated in the 15 cm of debris located on top. It means that, after the first 24 h, 30% of the power was already removed from the corium layer. After quenching, 70% of the power is generated in the cracked crust which is fully quenched. The onset of water ingression is clearly seen to accelerate the decrease of power in the corium layer by progressive solidification. Looking at the velocities of the interfaces between layers shown in Fig. 8, the two phases are also obvious. Up to 50 h, the ablation velocity is larger than the velocity of growth of debris bed. After 60 h and the start of water ingression, the solidification (by water ingression) is quickly faster than the erosion velocity, leading to the complete stabilization. It is interesting to notice the quasi linear growth of the crack front velocity, as discussed in Section 2.2. It is important to remind that this is only an example of application and that a variety of results may be obtained, depending on the scenario (residual power), on the spreading surface (reactor design), on the type of concrete (gas content) and on model parameters. This will be illustrated in the sensitivity analysis in Section 3. More detailed models would also add more parameters and other dependencies (for example in the power partitioning coefficient).

## 3. Sensitivity analysis

In order to determine the effects of the different parameters governing the problem, a sensitivity analysis is made. The range of variation of each parameter is provided in Table 2. For some parameters, the range of variation was deliberately chosen to exceed the likely range: this was done to better observe the trends and the non linear behaviors.

The concrete conductivity could have been included in the variable parameters (in particular because of the presence of rebars which may
<figure><img src="image_7.png" /><figcaption>Fig. 4. Schematic view of water ingression (partially cracked crust with water penetration).</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Fig. 5. Example of evolution without melt eruption.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 6. Example of application with both quenching processes acting simultaneously.</figcaption></figure>
<table><caption>**Table 2**
Uncertain parameters and variability.</caption><thead><tr><td>Parameter</td><td>Range of variation</td><td>Description</td></tr></thead><tbody><tr><td>m<sub>f</sub> E<sub>f</sub></td><td>15–25 MW</td><td>Residual power</td></tr><tr><td>S</td><td>80–120 m<sup>2</sup></td><td>Spreading surface (range corresponds to area achievable in some existing designs)</td></tr><tr><td>φ<sub>CHF</sub></td><td>40–100 kW/m<sup>2</sup></td><td>Critical heat flux (range does not include some high values observed with low concrete content)</td></tr><tr><td>k<sub>e</sub></td><td>0.02–0.10</td><td>Melt eruption coefficient (range is deliberately wider than experimental observations)</td></tr><tr><td>k<sub>up</sub></td><td>0.5–0.95</td><td>Power fraction to the top</td></tr><tr><td>k<sub>b</sub></td><td>0.01–0.10</td><td>Initial fraction of corium solidified by bulk cooling</td></tr></tbody></table>

<figure><img src="image_10.png" /><figcaption>**Fig. 7.** Distribution of power between the layers.</figcaption></figure>

increase the equivalent conductivity). However, it was not considered at this stage because it may play a role only in the end of the ablation process. It will be taken into account in the next steps of this work.

Four variables are examined: the time of onset of water ingression, the time to achieve complete stabilization, the erosion depth at complete stabilization and the erosion depth at the onset of water ingression.

### 3.1. Effect of the water ingression heat flux $\phi_{CHF}$

The results are shown in Fig. 9. The effect of $\phi_{CHF}$ on the erosion depth (Fig. 9(a)) looks quite linear, if a constant value is assumed, with a decreasing trend, as expected. Of course, it would be non linear if we considered a decrease with the increase of mass fraction of concrete, as in the correlation proposed in 4.1. The effect of $\phi_{CHF}$ on the erosion time (Fig. 9(b)) is more non linear, with a stronger increase of the stabilization time when $\phi_{CHF}$ decreases. A similar trend is observed for the time of onset of water ingression. When $\phi_{CHF}$ is low, most of the erosion occurs before the start of water ingression. This means that the main impact of reducing $\phi_{CHF}$ is to delay the onset of cooling by water ingression.

### 3.2. Effect of the melt eruption coefficient $k_e$

The results are shown in Fig. 10. The effect of $k_e$ on the erosion depth (Fig. 10(a)) is slightly non linear, with a rather strong impact when approaching the minimum value of $k_e$: the total erosion depth is more than doubled compared to the value obtained with the maximum value. However, it is reminded that the chosen range of variation deliberately exceeds the values recommended in the literature: this was done to study the impact of a possible limitation of melt eruption in the reactor situation, compared to the experiments. The effect of $k_e$ on the erosion time (Fig. 10(b)) is slightly non linear too. In the range of variability chosen, this parameter has a rather strong impact, with a total time for stabilization varying by a factor of 2.

### 3.3. Effect of the power partitioning coefficient $k_{up}$

The results are shown in Fig. 11. The effect of $k_{up}$ on the erosion depth (Fig. 11(a)) is very non linear: increasing this factor up to 1 leads to a strong decrease of the erosion depth, which tends asymptotically to 0 (because no power is transferred downwards to the concrete in that case). Overall, $k_{up}$ has a strong impact on the erosion depth which varies almost by a factor 10 over the chosen range of variability. The impact on the erosion depth at the onset of water ingression is non monotonic: increasing at first, then decreasing. This shows the complex balance between the decay of power and the reduction of power in corium by melt eruption. The effect of $k_{up}$ on the total stabilization time (Fig. 11(b)) is also non linear, especially for high values of $k_{up}$ but, overall, it does not have a strong impact. The trend is stronger and more non linear for the time of onset of water ingression. Again, this shows the effect of the power decay, which is more and more important as $k_{up}$ tends towards 1.

### 3.4. Effect of the total spreading surface $S$

The results are shown in Fig. 12. The effect of the total spreading surface $S$ on the erosion depth (Fig. 12(a)) is non linear, with a strong decrease (almost a factor 5) over the chosen range of variability. The erosion depth at the onset of water ingression has a comparable behavior and appears to be simply shifted by 0.15 m in this example. This seems to indicate that most of the erosion occurs before the onset of water ingression. The effect of the total spreading surface $S$ on the total stabilization time (Fig. 12(b)) is rather linear: it decreases by a factor around 2 in the range of variability chosen. Overall, it may be seen that for the maximum chosen value, $S = 120$ m$^2$, the stabilization occurs very quickly (half a day), with a very limited erosion depth (10 cm). This shows that $S$ is a very important parameter to consider in the “design” of that ex-vessel stabilization strategy.

## 4. Specific features of water ingression with siliceous concrete

### 4.1. A correlation for the water ingression CHF as a function of silica content

The detailed process of crack formation and water ingression has been investigated mostly in geology. A model was proposed by Lister (1974) for the quenching of hot lava. It was adapted to corium with residual power by Epstein (1999). Basically, the model consists in determining the width of the cracks and the distance between cracks, resulting from the thermal contraction due to water penetrating into the cracks: from the geometry of the cracks, a permeability is deduced, from which the maximum heat flux (by evaporation inside the cracks) is determined.

In the Lister-Epstein correlation, three parameters have the most important impact: the solidification temperature of the corium/concrete mixture $T_{sol}$, the expansion coefficient $\alpha_p$, and the heat conductivity of the crust $k$. The other parameters may be considered as almost constant or having a small influence.

$$ \phi_{CHF} = C_{CHF} \left( \Delta h_{vap} \frac{(\rho_l - \rho_g)g}{v_g} \right)^{\frac{5}{13}} (\alpha_p(T_{cr} - T_{sat}))^{\frac{15}{13}} $$
<figure><img src="image_11.png" /><figcaption>Fig. 8. Evolution of front velocities.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>Fig. 9. Sensitivity to $\phi_{CHF}$ (W/m²).</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>Fig. 10. Sensitivity to $k_c$ (-).</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>Fig. 11. Sensitivity to $k_{up}$ (-).</figcaption></figure>

$$ \times \left( \frac{k^2 (\Delta h + C_p(T_{sol} - T_{sat}))^2}{C_p \Delta h + C_p(T_{sol} - T_{cr})} \right)^{\frac{4}{13}} \quad (16) $$

$T_{cr}$ is called the cracking temperature: it is a little less than 100 degrees below the solidification temperature $T_{sol}$, as it was estimated in Lomperski and Farmer (2009) and measured in Lamur et al. (2018).

The determination of $T_{sol}$ is one of the key points because it strongly decreases with the concrete mass fraction $X_c$ when it is low, leading to a strong decrease of $\phi_{CHF}$ up to a value of $X_c$ where the solidification temperature becomes constant. This value is denoted $T_{eul}$. In this work, we used the thermochemical database NUCLEA-ToolBox (developed
<figure><img src="image_15.png" /><figcaption>Fig. 12. Sensitivity to S (m²).</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>Fig. 13. Comparison of the CHF correlation with experimental data (SSWICS) and proposed extension.</figcaption></figure>

by IRSN, Bakardjieva et al. (2014)) to estimate the solidification temperature of various mixtures of silicious concrete with corium. The composition of those mixtures corresponds to SSWICS tests (Lomperski and Farmer, 2007). Details about the results of the thermochemical calculations are not provided here, as they are not directly useful with respect to the conclusions of the paper. The conclusions of that thermochemical study is that a linear correlation was found between the temperature for which the solid fraction is about 90% (almost complete solidification) and the molar ratio Y between the major atoms from concrete (Si and Ca) and the major atoms from the fuel (U and Zr). This correlation is:

$$T_{sol} = T_{eut} + Max[0; Y_{eut} - X](1700 - T_{eut}) \quad (17)$$

$$Y = (n_{Ca} + n_{Si})/(n_{U} + n_{Zr})(10) \quad (18)$$

where $Y_{eut}$ is near the value 1.5 (which is equivalent to a mass fraction $X_{eut} = 0.3$) and $T_{eut} = 1200$ °C. As a result, the solidification temperature decreases linearly down to the minimum value of 1200 °C, then it is constant. This means that after this threshold value, $\phi_{CHF}$ decreases much more slowly. The conductivity and the expansion coefficient both decrease with the increase of the concrete fraction because silica has a lower conductivity than fuel materials (1 W/K/m compared to 2.5 W/K/m) and a much lower expansion coefficient ($1.10^{-6}$ K⁻¹ compared to $8.10^{-6}$ K⁻¹). If we calculate the conductivity of the mixture with a weighted average, and do the same for the expansion coefficient, we get the variation of $\phi_{CHF}$ as presented in Fig. 13. It shows that there is a strong decrease down to a threshold mass fraction (here a value near 0.2 gives the best fit to experimental data, instead of the theoretical value 0.3 proposed before) and then a slower decrease. Here the best

<figure><img src="image_17.png" /><figcaption>Fig. 14. Comparison of eroded concrete (gray) and cracked crust (red) thicknesses when using a constant CHF and a quasi-piecewise linear fit of the CHF.</figcaption></figure>

fit is obtained with the constant $C_{CHF} = 2$. This curve differs mostly from the curve proposed in Lomperski and Farmer (2007) after the mass fraction 0.2. In Lomperski and Farmer (2007), a constant extrapolation was proposed, which is not consistent with the lower conductivity and lower expansion coefficient of silica. The small number of experimental data make it impossible to determine accurately the value of $X_{eut}$. Therefore, it is necessary to take into account an uncertainty in the range [0.15;0.25]. This sensitivity is studied in section Section 4.3.

From a practical point of view, the curve in Fig. 13 shows that, except for cases where the stabilization is reached very quickly (because of a low residual power and/or a large spreading surface) and the concrete content remains low, the CHF will be more likely around 50 kW/m² or even lower when stabilization is reached after several days. Lister (1974) mentions that the heat flux could be estimated by Björnsson et al. (1982) during the cooling of a lava flow: it was roughly estimated at 40 kW/m². The model presented here would give a value between 20 and 30 kW/m² for a concrete mass fraction of 1, which may be considered as quite satisfactory considering all the simplifying assumptions made to derive the model. Unfortunately, there are no data for pure corium (concrete mass fraction equal to 0.) because such experiments are quite difficult to perform due to the very high temperature required to completely melt a mixture of UO₂ and ZrO₂. Therefore, the left part of the curve presented in Fig. 13 cannot be assessed properly. But it is not so important from a practical point of view as, in the mitigation strategy considered here, water is poured over the corium after some time of concrete erosion and the mass fraction of concrete is already larger than 10% when top cooling by water starts.

## 4.2. Comparison between constant and decreasing CHF evaluations

The comparison between a constant CHF law and a quasi-piecewise linear law (as shown in Fig. 13) is presented in Fig. 14. It can be
<figure><img src="image_18.png" /><figcaption>Fig. 15. Sensitivity to $X_{eut}$ (-).</figcaption></figure>

observed that, in the chosen example, the use of the quasi-piecewise linear law induces a faster start of water ingression (because of the initial high value of CHF) but the decrease of CHF due to addition of silica slows down the progression of water ingression and, eventually, the stabilization time is about 20% longer (120 h instead of 100 h). Looking at the thickness of eroded concrete, It can be observed that, in the chosen example, the use of the quasi-piecewise linear law leads to a smaller erosion depth (by approximately 15 cm) in spite of the longer erosion time. This shows the non intuitive and integral character of the whole process and emphasizes the importance of describing the CHF evolution as accurately as possible.

### 4.3. Sensitivity to the value of $X_{eut}$

As explained above, a sensitivity to $X_{eut}$ must be considered, in the range [0.15;0.25] because this parameter cannot be determined accurately. The results of this study are presented in Fig. 15. It shows a rather significant sensitivity of approximately 20 cm in the erosion depth and 20 h in the stabilization time. This sensitivity is comparable to the sensitivity to the other parameters.

## 5. Conclusions

The objective of this paper is to provide an insight and orders of magnitude about the quenching of molten corium spread over a siliceous concrete basemat, in the containment of a nuclear reactor, since it is one of the possible strategies to mitigate the consequences of a severe accident in a GenII/III reactor. The case of siliceous concrete is less favorable (compared to limestone concrete) and this is why it deserves a careful evaluation. It is reminded that a generic case was considered (in particular, the decay power evolution was simplified and the initial corium composition may be different). This paper does not aim at providing quantitative evaluations for specific reactor designs or specific cases. Instead, the aim is to draw the attention to the important parameters and elements of modeling that are necessary for the evaluation of a specific case. A simple parametric model was used to investigate the problem. There are several characteristic times in the problem: the decay time of residual power, the duration of efficient melt eruption and the time of cracks progression into the crust. Those different times are not independent. The evolutions of quenching and concrete erosion may be divided in two main stages. The first stage is characterized by a fast erosion of concrete and a significant eruption and quenching of debris. During this stage, erosion is faster than solidification. This stage lasts from half a day to a few days, depending on the type of concrete and on the model parameters.

The second stage is characterized by the start of water ingression and slowing down of erosion of concrete. During this stage, solidification is faster than erosion. This stage may last from half a day up to several days depending on the type of concrete and on the model parameters. Among the parameters governing the process, the maximum heat flux (CHF) that can be extracted (through the network of cracks that are formed in the top crust) is of crucial importance. A revised correlation is proposed for this maximum heat flux, considering the negative effect of adding silica into the corium mixture, which results in smaller cracks and a reduced CHF. From the correlation, it appears that the CHF may be significantly reduced if the concrete content in the corium is large: values lower than 50 kW/m² are possible, which would lead to a deeper erosion of concrete and a longer time for total quenching.

The problem is strongly non-linear with respect to several of the model parameters, including the CHF and the power partitioning coefficient. An early triggering of water ingression results in a fast quenching whereas a late triggering may lead to quite long stabilization and deeper erosion of the concrete basemat (up to a couple of meters and a few weeks until complete quenching). Because of the non-linearities and because of the existence of different characteristic times, there is a non-trivial dependence of water ingression efficiency on the potential occurrence of melt eruption at the very beginning of corium concrete interaction. Melt eruption, even if limited, is likely to significantly accelerate the triggering of water ingression and favor a fast stabilization of corium. This is the reason why adding a rather thin layer of limestone concrete on top of the siliceous concrete could be a very efficient option to speed up the stabilization and limit the erosion of the basemat: with the high gas content in the limestone concrete, it would significantly enhance the debris formation in the beginning and favor a quicker onset of water ingression. This is an option to consider when the total spreading surface is not large enough to ensure a fast stabilization with limited erosion depth.

## 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.

## Appendix A. Determination of a criterion for the complete stabilization

In many lumped-parameter models, the concrete basemat is represented as a layer with uniform temperature $T_\infty$. However, in reality,
Heat flux criterion for stop of erosion

<figure><img src="image_19.png" /><figcaption>Fig. 16. Bottom heat flux corresponding to the stop of concrete erosion.</figcaption></figure>

there is a conductive heat transfer from the corium to the concrete and a temperature gradient near the interface. In the end of corium quenching, the downwards heat flux $\phi_{dn}$ decreases and eventually becomes balanced by the conductive heat flux through the concrete, which leads to stop the erosion. For an ablation front at temperature $T_{abl}$ moving at the velocity $U_{abl}$ in a medium initially at temperature $T_{\infty}$, the evolution of temperature is solved in Carslaw and Jaeger (1959):

$$T(z) = T_{\infty} + \frac{1}{2}(T_{abl} - T_{\infty}) \left[ erfc \frac{z - tU_{abl}}{2\sqrt{\alpha}} + \exp\left(\frac{zU_{abl}}{\alpha}\right) erfc \frac{z + tU_{abl}}{2\sqrt{\alpha}} \right] \quad (19)$$

The heat flux into the concrete layer is given by estimating the derivative at position $z = 0$ and after a time corresponding to a quasi-steady progression

$$\phi_c = -k_c(dT/dz)_{(z=0)} = -k_c(T_{abl} - T_{\infty})\sqrt{\pi/(\alpha_c t)} \quad (20)$$

This equation is valid only when the ablation velocity is close to zero and decreases faster than $1/\sqrt{t}$. Typically, this gives a heat flux of a few kW/m², which decreases with time. With this result, we can write the time for which concrete erosion stops, as a function of the downwards heat flux $\phi_{dn}$:

$$t_{end} = \left[ \frac{(k_c(T_{abl} - T_{\infty}))}{\phi_{dn}} \right]^2 \pi / \alpha_c \quad (21)$$

This evaluation is of practical interest for an implementation in a lumped-parameter model but it must be used carefully because it is only valid with the two conditions written just above. It must also be kept in mind that we used an analytical solution derived for a constant velocity of the front whereas this velocity slowly decreases in our case. To give an order of magnitude, the heat flux corresponding to the stop of erosion is represented in Fig. 16, as a function of time. However, it was shown in Section 2.2 that the solidification front accelerates in case of water ingression: therefore, it is not expected that the criterion of Eq. (21) would change the total stabilization time in most of the cases.

## Appendix B. Nomenclature

<table><thead><tr><th colspan="2">Main variables</th></tr></thead><tbody><tr><td>E<sub>0</sub></td><td>Power per unit mass of fuel materials at t = 0</td></tr><tr><td>E<sub>f</sub></td><td>Power per unit mass of fuel materials (= E<sub>0</sub> exp(-t/τ))</td></tr><tr><td>m<sub>c</sub></td><td>Mass of concrete in the corium</td></tr><tr><td>m<sub>f</sub></td><td>Mass of fuel materials in the corium</td></tr><tr><td>m<sub>s</sub></td><td>Mass of fuel in the crust</td></tr><tr><td>m<sub>w</sub></td><td>Mass of fuel in fractured crust (due to water ingression)</td></tr><tr><td>m<sub>d</sub></td><td>Mass of fuel in debris (generated by melt eruption)</td></tr><tr><td>Q<sub>c</sub></td><td>Concrete erosion rate (= dm<sub>c</sub><sup>tot</sup> / dt)</td></tr><tr><td>U<sub>abl</sub></td><td>Velocity of concrete erosion</td></tr><tr><td>U<sub>wi</sub></td><td>Velocity of water ingression front</td></tr><tr><td>X<sub>f</sub></td><td>Mass fraction of fuel materials in a layer X<sub>f</sub> = m<sub>f</sub> / (m<sub>f</sub> + m<sub>c</sub>)</td></tr><tr><td>φ</td><td>Heat flux</td></tr><tr><td>φ<sub>up</sub></td><td>Heat flux at the upper surface of corium layer (towards the crust)</td></tr><tr><td>φ<sub>dn</sub></td><td>Heat flux at the bottom surface of corium layer (towards concrete)</td></tr><tr><td>φ<sub>CHF</sub></td><td>Maximum heat flux extracted in the cracked crust (by water ingression)</td></tr><tr><td>τ</td><td>Decay time of residual power</td></tr><tr><th colspan="2">Physical properties</th></tr><tr><td>ρ</td><td>Volumetric mass</td></tr><tr><td>α</td><td>Thermal diffusivity (α = k/(ρCp))</td></tr><tr><td>α<sub>ρ</sub></td><td>Thermal expansion coefficient</td></tr><tr><td>k</td><td>Thermal conductivity</td></tr><tr><td>Cp</td><td>Specific heat per unit mass</td></tr><tr><td>Δh<sub>sol</sub></td><td>Enthalpy of melting of the corium mixture (fuel and concrete elements)</td></tr><tr><td>Δh<sub>abl</sub></td><td>Enthalpy of melting of concrete</td></tr><tr><td>T<sub>abl</sub></td><td>Ablation temperature of concrete</td></tr><tr><td>T<sub>sol</sub></td><td>Solidification temperature of the corium/concrete mixture</td></tr><tr><td>T<sub>sat</sub></td><td>Saturation temperature of water</td></tr></tbody></table>

## Data availability

The data that has been used is confidential.

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