## Abstract

This work illustrates an integrated model for (i) predicting the evolution of radionuclide concentrations due to releases from a spent fuel storage pool and (ii) providing a preliminary assessment of the radiological risk for the local population. The model integrates (i) a particle tracking algorithm for radionuclide transport based on the Kolmogorov–Dmitriev theory of branching stochastic processes and (ii) a semi-empirical model for representing the effects of seasonal variations in the water table level on the radionuclide concentrations. A real case study is considered concerning the release of ^{90}Sr from the spent fuel storage pool at the EUREX site in Italy. The integrated model is calibrated by a genetic algorithm optimization based on observation data of water table levels and ^{90}Sr concentrations available from the local monitoring network. The proposed model offers a satisfactory trade-off between a detailed physical and geometrical description of the transport processes and computational efficiency.

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## Acknowledgments

The authors are indebted to Dr. Laura Porzio and Dr. Luca Albertone from ARPA Piemonte for their technical support in the elaboration of the case study.

## Author information

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### Corresponding author

## Appendices

### Appendix 1

In what follows, the main features of the KDPT are illustrated with reference to a one-dimensional domain representing the unsaturated zone, although the method can be easily extended to deal with two- or three-dimensional problems, as shown in Sect. 2.1.3 and Appendix 2. The domain is subdivided in *N*
_{
z
} discrete zones *i*, *i* = 1, 2,…,*N*
_{
z
}. Several categories of particles can be introduced according to the different states in which the solute particle under analysis can be found. Thus, the system is made up in general of *m* = *N*
_{
c
} × *N*
_{
z
} different kinds of particles, representing the *N*
_{
c
} solute particle types at their location in one of the zones *i* = 1, 2,…,*N*
_{
z
}. Each particle may disappear or give rise to one particle of the *m* − 1 remaining kinds according to given transition probability laws. It is assumed that: (i) the stochastic process is Markovian, i.e. a particle of the *l*th kind, l = 1,…,m, gives rise to a branching process independently of its past history; (ii) the process is linear, i.e. the particles do not interact among each other; (iii) within a generic time interval (*t*,* t* + *dt*), with *dt* sufficiently small, only one particle transition may occur.

Within this framework, we consider *N*
_{
c
} = 1 particle type, i.e. the *solutions*, representing the solute particles migrating within the partially filled pores of the unsaturated zone. For simplicity, but with no loss of generality, we further assume that the solute transport occurs under the linear isotherm equilibrium hypothesis. Since we deal with a partially saturated medium, the corresponding retardation factor, *R*, depends on the spatial discrete coordinate *i* = 1, 2,…,*N*
_{
z
} through the saturation degree *θ*(*i*) as follows (Giacobbo and Patelli 2007):

where *ρ*
_{
b
} [cm^{3}/g] is the dry bulk density and *k*
_{
d
} [cm^{3}/g] is the partition coefficient.

The mass transport process is, then, described as follows. In the time interval *dt* the solution can only travel to one of the adjacent zones, *i* + 1 or *i* − 1, with transition rates \( f_{u}^{ret} \left( {i \to i + 1,t} \right) \) (forward), and \( b_{u}^{ret} \left( {i \to i - 1,t} \right) \) (backward), respectively. To describe the space and time evolution of the system of particles, it is possible to write the following system of *N*
_{
z
} FKEs for the evolution of the expected values of the numbers of solutions at time *t* in zone *i* of the unsaturated medium (Cadini et al. 2010):

where *S*
_{
u
}(*i*, *t*) is the expected value of the number of solutions in the *i*th cell of the spatial domain representing the unsaturated zone at time *t*.

The effectiveness of such description is conditioned by the capability of identifying the values of the transitions rates. To this aim, in analogy to the approach followed in (Cadini and Zio 2013; Cadini et al. 2013, 2012; Giacobbo and Patelli 2007; Ferrara et al. 1999), a term-by-term correspondence is set up between (8) and the classical advection-dispersion equation of one-dimensional reactive solute transport in an unsaturated, homogeneous and isotropic porous medium (Simunek and Van Genuchten 2006):

where \( G\left( {z,t} \right) \) [Bq/l] is the contaminant concentration in the pore water and *D*
_{
u
} [m^{2}/h] is the hydrodynamical dispersion coefficient in the unsaturated zone, which, being *z* the principal axis for the contaminant transport, can be written as (Bear 1979; Bear and Cheng 2010) \( D_{u} = D_{u}^{D} + \alpha_{L}^{u} \times v_{u}, \) where \( D_{u}^{D} \) is the molecular diffusion coefficient and \( \alpha_{L}^{u} \) [m] is the longitudinal dispersivity, in the unsaturated zone.

The entire procedure for the identification of the transition rates values is not repeated here for brevity’s sake; let it suffice to recall that upon spatial discretization of (9) by a centered Euler finite difference method with spatial cell (compartment) length ∆*z* (Giacobbo and Patelli 2007) and (i) assuming that \( D_{u}^{D} \) is constant all over the unsaturated zone (ii) and recalling that all the hydrogeological parameters are stationary, one finally obtains:

Note that the transition rates are stationary but not homogeneous in space, and that the spatial cell length ∆*z* must be chosen so that the rate \( b_{u}^{ret} \) is positive in the whole domain. In our application, the domain length is \( L_{u} = 7.30 \times 10^{ - 1} \) m, corresponding to the distance between the mean depth of the irregular pool bottom, i.e. 3.26 m (ARPA Piemonte 2006–2013), and the stationary average water table depth, and we assume ∆*z* = 10^{−2} m.

Given proper boundary and initial conditions, the solution of (8), with the parameters (10) and (11) identified by analogy with the discretized form of (9), can be found by a Monte Carlo procedure derived from the underlying Markovian description of the stochastic evolution of the system of particles (Cadini et al. 2010a, b, 2012, 2013; Cadini and Zio 2013). The stochastic migration (random walk) of a large number \( N_{s}^{u} \) of solute particles is simulated by repeatedly sampling their births from the release sources and their transitions across the medium compartments. During each simulation, the position of the particle is tracked, so that, at the end of the \( N_{s}^{u} \) simulations, expected values can be estimated by computing the sample means of the number of visits at the positions recorded at the different time instants. For further details about the functioning of the KDPT algorithm, the interested reader is referred to (Cadini et al. 2010a, b, 2012, 2013; Cadini and Zio 2013).

Operatively, the first cell of the array (*i* = 1) representing the unsaturated domain is set as the location of the source of ^{90}Sr radionuclides released from the pool, whose associated release rate is \( \mu ' \). A reflecting barrier is imposed at this cell, so that the particles injected in the domain are not allowed to diffuse back in the pool.

The random walk of the individual radionuclide particle is simulated either until it exits the domain to the *N*
_{
z
} + 1 cell representing the aquifer, that is assumed to be absorbing (i.e. the particle cannot come back into the migration domain), or until its lifetime exceeds the time horizon \( T_{miss}^{u} \) of the analysis.

In order to account for the continuous injection of contaminant from the source cell, we first use the KDPT algorithm to find the solution for an instantaneous injection and, then, exploiting the assumption of process linearity, we resort to a discrete-time convolution for estimating the concentrations resulting from continuous sources [see (Cadini et al. 2010a, b, 2012) for further details].

Operatively, we discretize the time horizon \( T_{miss}^{u} = 100,000 \) h in \( N_{t}^{u} = 10,000 \) equally spaced time instants, with a time step ∆*t*
_{
u
} = 10 h, and compute the effects of an instantaneous injection of ^{90}Sr radionuclides at *t* = 0 of magnitude \( \varLambda_{u} = \mu ' \times \Delta t_{u} \) [Bq/m^{2}]. To this aim, during the KDPT simulation, the times at which each of the \( \left( {i,j} \right) \) particles exit the unsaturated zone domain are recorded in a counter \( Count\left( {k_{u} } \right) \), \( k_{u} = 1,2,\ldots,N_{t}^{u} \).

The probability \( \hat{P}_{u} \left( {k_{u} } \right) \) that a particle enters the aquifer within the time interval *k*
_{
u
} can be estimated as:

The ^{90}Sr activity \( R_{aq}^{\varLambda } \left( {k_{u} } \right) \) [Bq/m^{2}] released into the aquifer within the time interval *k*
_{
u
} as a consequence of the instantaneous release at *t* = 0 can be computed as:

As anticipated above, the total release \( R_{aq}^{{}} (k_{u} ) \) into the aquifer resulting from the continuous injection into the unsaturated zone (i.e. the source of the aquifer transport model described in the next Section) can, then, be obtained by discrete-time convolution.

### Appendix 2

Under the same assumptions made for the case of the unsaturated zone, the mass transport process in the two-dimensional aquifer is mimicked again by a KDPT algorithm. Since the domain is two-dimensional, in a time interval *dt*, besides moving forward or backward with respect to the water flow direction, with rates \( f_{aq}^{ret} \left( {i \to i + 1,j,t} \right) \) and \( b_{aq}^{ret} \left( {i \to i - 1,j,t} \right) \), respectively, the solution can also travel to one of the adjacent lateral cells \( \left( {i,j + 1} \right) \) or \( \left( {i,j - 1} \right) \), with rates \( r_{aq}^{ret} \left( {i,j \to j + 1,t} \right) \) (rightward), and \( l_{aq}^{ret} \left( {i,j \to j - 1,t} \right) \) (leftward), respectively.

Let us now further assume that the aquifer is a homogeneous medium, so that all the rates are constant over the spatial domain, i.e. \( f_{aq}^{ret} \left( {i \to i + 1,j,t} \right) = f_{aq}^{ret} \left( {i - 1 \to i,j,t} \right) = f_{aq}^{ret} \left( t \right) \), and similarly for the remaining transition rates; by a derivation similar to that leading to the FKEs in one-dimension (Marseguerra and Zio 1997), we obtain the following coupled equations for the expected value of the number of solutions in the cell (*i*, *j*) at time *t*, \( S_{aq} \left( {i,j,t} \right) \):

In this case, to identify the values of the transition rates, the term-by-term correspondence is made with the two-dimensional advection-dispersion equation describing the reactive solute transport in a saturated, homogeneous and isotropic porous medium, i.e. (Bear 1979; Bear and Cheng 2010):

where \( \bar{C}\left( {x,y,t} \right) \) [Bq/l] is the contaminant concentration in (*x*, *y*) at time *t*, \( D_{aq}^{x} \) [m^{2}/h] and \( D_{aq}^{y} \) [m^{2}/h] are the hydrodynamical dispersion coefficients in the aquifer in the directions *x* and *y*, respectively. Since *x* and *y* are principal axes, the hydrodynamical dispersion coefficients can be expressed as \( D_{aq}^{x} = D_{aq}^{D} + \alpha_{L}^{aq} \times v_{aq} \) and \( D_{aq}^{y} = D_{aq}^{D} + \alpha_{T}^{aq} \times v_{aq} \), where \( D_{aq}^{D} \) [m^{2}/h] is the molecular diffusion coefficient in the aquifer, and \( \alpha_{L}^{aq} \) [m] and \( \alpha_{T}^{aq} \) [m] are the longitudinal and the transversal dispersivities in the aquifer, respectively.

Again, the procedure of parameter identification is not detailed here for brevity’s sake and we only mention that upon spatial discretization of the Eq. (15) by a centered Euler finite difference method with spatial cell dimension \( \Delta x \times \Delta y \) [m^{2}] (Marseguerra and Zio 1997; Cadini et al. 2010a, b, 2013, 2012; Cadini and Zio 2013; Ferrara et al. 1999), assuming that \( D_{aq}^{D} \) is constant over the aquifer and recalling that all the hydrogeological parameters are stationary, one finally obtains:

As expected, the transition rates turn out to be stationary and homogeneous in space. Again, note that ∆*x* must be chosen so that the rate \( b_{aq}^{ret} \) is positive. In our application we assume ∆*x* × ∆*y* = 1 × 1 m^{2}. The domain size considered is *L*
_{
x
} × *L*
_{
y
} = 300 × 200 m^{2}, with the origin of the coordinate system placed such that \( x \in \left[ { - 100,\,200} \right] \) m and \( y \in \left[ { - 100,\,100} \right] \) m.

For convenience of reference, we assume that the radionuclide release into the aquifer occurs in correspondence of the cell of the two-dimensional grid located in the origin (0, 0); the ^{90}Sr activity injected within the time step *k*
_{
u
} as a result of the transport through the unsaturated zone is \( R_{aq} \left( {k_{u} } \right) \times A \) [Bq], where *A* is the area of the leaking crack in the pool pavement (in fact, no lateral spreading was considered in the one-dimensional unsaturated zone analysis).

Note that in order for the assumption of zero-dimensionality of the aquifer source to be satisfied, we impose that the leaking area *A* be smaller than, at least, a single cell of the two-dimensional array representing the aquifer domain.

The KDPT algorithm, then, proceeds as for the case of the unsaturated zone, the only difference being that the time-dependent ^{90}Sr concentration is computed at a fixed point, i.e. the SPB well.

Similarly to the approach followed for the unsaturated zone, we first compute the solution for an instantaneous injection of radionuclides of magnitude \( R_{aq} \left( {k_{u} = 1} \right) \times A \) at time *t* = 0 in the source cell.

The random walk of the individual radionuclide particle is simulated from the initial time *t* = 0 until it exits the domain or its lifetime exceeds the time horizon \( T_{miss}^{aq} = 100,000 \) h of the analysis. The time horizon \( T_{miss}^{aq} \) is, again, discretized in \( N_{t}^{aq} = 10,000 \) equally spaced time instants, with a time step ∆*t*
_{
aq
} = 10 h, which is here set equal to ∆*t*
_{
u
} for computational simplicity but could be taken different in general. A counter \( Count_{s} \left( {\left( {i,j} \right)_{SPB} ,k_{aq} } \right) \) is associated to the cell \( \left( {i,j} \right)_{SPB} \) corresponding to the location of the SPB well and to each discrete time \( k_{aq} = 1,2,\ldots,N_{t}^{aq} .\) During the simulation, a one is accumulated in the counter \( Count_{s} \left( {\left( {i,j} \right)_{SPB} ,k_{aq} } \right) \) if a solution particle resides, during the random walk, in the cell \( \left( {i,j} \right)_{SPB} \) at time step *k*
_{
aq
}. At the end of the \( N_{s}^{aq} \) simulated random walks of the contaminant particles, the value accumulated in the counter allows computing the estimate of the time-dependent probabilities of cell occupation by a solution at the discrete times *k*
_{
aq
}, \( \hat{P}_{{\left( {i,j} \right)_{SPB} }} (k_{aq} ) \):

The ^{90}Sr concentration in the pore water at \( \left( {i,j} \right)_{SPB} \) at time step *k*
_{
aq
}, corresponding to a release of an amount of radionuclides \( R_{aq} \left( {k_{u} = 1} \right) \times A \) [Bq] from the unsaturated zone is, then, computed as:

where \( \Delta V = \Delta x \times \Delta y \times \Delta z_{aq} = 3.01\,m^{3} \) is the volume of an aquifer domain cell.

In order to account for the fact that the ^{90}Sr radionuclides are continuously released from the unsaturated zone, we exploit again the process linearity, so that the time-dependent ^{90}Sr concentration at the SPB well, \( \tilde{C}\left( {\left( {i,j} \right)_{SPB} ,k_{aq} } \right) \), can be obtained by convolution of the \( N_{t}^{u} = 10,000 \)
^{90}Sr releases of magnitude \( R_{aq} \left( {k_{u} } \right) \times A \) in correspondence of each time step *k*
_{
u
}.

Finally, as explained at the beginning of Sect. 2, the effects of the radioactive decay are considered at the end of the computation, so that the output of the average model at a generic time step *k*
_{
aq
} can be written as:

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### Cite this article

Cadini, F., Tosoni, E. & Zio, E. Modeling the release and transport of ^{90}Sr radionuclides from a superficial nuclear storage facility.
*Stoch Environ Res Risk Assess* **30, **693–712 (2016). https://doi.org/10.1007/s00477-015-1112-7

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### Keywords

- Radioactive waste
- Near-surface disposal
- Radionuclide transport modeling
- Uncertainty analysis
- Risk assessment