On the Impact of the Fuel Dissolution Rate Upon Near-Field Releases From Nuclear Waste Disposal

Calculations of the impact of the dissolution of spent nuclear fuel on the release from a damaged canister in a KBS-3 repository are presented. The dissolution of the fuel matrix is a complex process and the dissolution rate is known to be one of the most important parameters in performance assessment models of the near-field of a geological repository. A variability study has been made to estimate the uncertainties associated with the process of fuel dissolution. The model considered in this work is a 3D model of a KBS-3 copper canister. The nuclide used in the calculations is Cs-135. Our results confirm that the fuel degradation rate is an important parameter, however there are considerable uncertainties associated with the data and the conceptual models. Consequently, in the interests of safety one should reduce, as far as possible, the uncertainties coupled to fuel degradation.


INTRODUCTION
Estimation of the long-term impact of radionuclides on the biosphere and on man from a deep geological repository for spent nuclear fuel or high-level nuclear waste is a complicated task, owing to the uncertainties associated with many of the physical and chemical processes and to the huge time scales involved.Probabilistic calculations tend to be used to estimate the predictions of long-term releases that will impact on man and other biota.It is essential to support probabilistic modelling with deterministic calculations because the latter give us a deeper insight into the ongoing physical and chemical processes than do probabilistic simulations.In fact, probabilistic Monte Carlo calculations require the use of simplified models, fast enough to allow the thousands of simulations that are necessary to get good statistics for the analysis.Those simplified models (in general, 1D models) are not detailed enough to cope with coupled flow and physical-and chemical processes.
The aim of this work was to investigate the impact of the fuel degradation on the radionuclide release at the interface between the repository's near-field and the geosphere.The fuel dissolution is a complex chemical process and the laboratory data are affected by important uncertainties.We assumed that a fraction of the inventory in the interior of the canister is available for immediate migration, i.e. it is dissolved in water.This amount, the instant release fraction (IRF) is denoted by α.The IRF is nuclide specific and corresponds to gap-and grain boundary releases.The rest of the radionuclide inventory will be able to commence migration as a result of the gradual dissolution of the fuel matrix in which the nuclides are embedded (congruent dissolution).The fuel dissolution can be estimated by a simple release model as in Hedin [1], assuming a degradation rate D F that affects the (1 Ϫ α) fraction of the inventory.Following Hedin, the production rate P(t) is given by: (1) where: M 0 is the inventory at time t o (s) D F is the degradation rate for the fuel matrix (s Ϫ1 ) α is the fraction of the waste that is immediately available for migration (and is dimensionless) and t Delay is the time at which the canister starts leaking To assess the impact uncertainties of the dissolution rate have on the release from the nearfield, we use in this paper a near-field model which is an improved version of another threedimensional model developed in previous work Pereira [2,3].The main difference between the two-models is the way in which some of the boundary conditions are treated in addition to which there is the explicit inclusion of a fracture crossing the canister deposition hole and of a disturbed zone, as explained later on.

MODEL GEOMETRY OF A REPOSITORY SECTION 2.1 MODEL DESCRIPTION
The model geometry describes a fuel canister, approximately 4.8 m high with an inner diameter equal of 0.95 m.We assume that the welded copper lid had initial but undetected damage, so that after three hundred years a fully developed corrosion channel (a small pinhole) connects the interior of the canister to the outside.This pinhole is a pathway with an assumed cross-section of 10 Ϫ6 m 2 † .The pinhole is the pathway through which nuclides diffuse into the bentonite clay surrounding the canister.The diffusion gradient between the interior of the canister and the bentonite is controlled mainly by the fuel dissolution rate.
The repository is excavated in fractured granite (Figure 1).The nuclides that enter the bentonite buffer will be transported to the geosphere by groundwater circulating in the fractures adjacent to the buffer.Some short-lived nuclides will have time to decay completely and therefore will never reach the geosphere.Others will diffuse through the bentonite until they come into contact with the rock.In the model, we consider two horizontal rock fractures, one at the level of the corrosion pinhole and the other above it, at the level of the tunnel sole (Figure 1).These fractures are denominated Q1 and Q2 following the notation used by Hedin [1].The Q2 fracture was introduced to represent the higher conductivity of the disturbed zone formed at the bottom of the tunnel as a result of the excavation process when building the repository.In our model we "collect" the nuclides at the outlet of these two fractures.The flux at the outlet represents therefore the amount of nuclides leaving the near-field and entering in the geosphere.
Iodine, Caesium and Chlorine are between the first elements to be released when water comes in contact with the fuel matrix.The ions I Ϫ , Cs ϩ and Cl Ϫ form salts with high solubility in water and because a fraction of their inventory (the IRF fraction mentioned above) accumulates in the fuel-cladding gap and at grain boundaries, they are immediately 1 α λ released.Radionuclide concentrations are controlled by the solubilities of the different elements under the near-field conditions.The concentrations of Iodine, Caesium and Chlorine are not solubility limited.We can therefore expect that the impact of fuel degradation may be higher for these nuclides.Caesium is a weakly-sorbing nuclide and it is the one that we use for our calculations.

NUMERICAL MODEL AND INPUT DATA
The geometry of a repository section, shown in Figure 2, uses the symmetry of the problem to reduce the computational load of our three-dimensional finite element model.However symmetries are not fully exploited here (for instance a 2D symmetry axial model could be used).The motivation for keeping the geometry of the model and using the actual shape is that we intend to use it in other studies addressing more realistic situations, such as how bentonite erosion can impact on the release rates.Bentonite erosion has been identified as one of the most serious potential threats for the long-term safety of a repository SKB [4].
The model consists of several sub-domains: the interior of the canister or the "source term" is denoted as domain Ω 1 ; the pinhole, domain Ω 2 ; the bentonite, domain Ω 3 ; the fracture Q1, domain Ω 4 and the fracture Q2, domain Ω 5 .
The numerical model takes into account the main mass transport processes in the nearfield.These are advection, diffusion, sorption, and radionuclide decay.The length of the fractures considered here is short, so that the diffusion from the fracture to the adjacent rock (matrix diffusion) is ignored.Sorption on the walls of the fractures is also ignored.

THE SOURCE TERM
The mass transport within the canister (domain Ω 1 ) is described by a time-dependent diffusion equation that includes radionuclide decay and associated initial and boundary conditions: where c (r,t) is the nuclide concentration in the domain Ω 1 (mol m Ϫ3 ), c 0 is the initial nuclide concentration in the domain Ω 1 (mol m Ϫ3 ), D is the diffusivity of the water (m 2 s Ϫ1 ), λ is the radioactive decay constant (s Ϫ1 ) and α is the instant release fraction (dimensionless)

THE PINHOLE
The transport through the pinhole (Ω 2 domain) is also given by the time-dependent threedimensional diffusion equation and the relevant associated initial and boundary conditions: where c (r, t) is the nuclide concentration in the domain Ω 2 (mol m Ϫ3 ), c(0), the initial nuclide concentration in the domain Ω 2 , is nil (mol m Ϫ3 ), D is the water diffusivity (m 2 s Ϫ1 ) and λ is the radioactive decay constant (s Ϫ1 ).
Eqn. (2c) encapsulates the insulating boundary conditions at relevant surfaces of the domain.
The inlet and the outlet of the pinhole are interior boundaries.We assume that the pinhole is entirely filled with water.

THE BENTONITE
As long as the bentonite is intact (no occurrence of erosion), the advective water flow is extremely low and therefore we describe the mass transport in the bentonite (Ω 3 -domain) by a time-dependent diffusion-reaction equation that also includes radionuclide decay: where, c i (r,t) is the concentration of nuclide i, in the domain Ω 3 , (mol m Ϫ3 ), c i (0), the initial nuclide concentration in the domain Ω 3 , is nil (mol m Ϫ3 ), R i is the retardation coefficient of nuclide i and is dimensionless, D i is the diffusion coefficient of nuclide i (m 2 s Ϫ1 ) and λ i -is the radioactive decay constant of nuclide i (s Ϫ1 ).
The retardation coefficient R i is given by ε+ρ K d where ε is the porosity, ρ the density and K d the sorption coefficient.

THE FRACTURE Q1 AND THE DISTURBED ZONE Q2
The migration of radionuclides in the fracture Q1 and the disturbed zone Q2 (domains Ω 4 and Ω 5 , respectively) is described by coupling the advective flow in the fractures to the mass transport.To compute the advective velocity u of the groundwater in the fractures, we solve the three-dimensional steady state Navier-Stokes' equation under the assumption of fluid incompressibility, eq.(4a), together with the continuity equation, eq. ( 4b): where: u is the advective velocity in the fracture Q1 and the disturbed zone Q2 (m s Ϫ1 ), p is the pressure (N m Ϫ2 ), ρ is the water density (kg m Ϫ3 ) and η is the viscosity of water (N s m Ϫ2 ).
Eq. ( 4a) is valid for a Newtonian fluid in the steady state.The gradient ∇ p has been chosen so that it conforms with the value of the Darcy velocity given by Hedin for a fracture in the near-field, Hedin [1].The mass transport in the fracture Q1 and the disturbed zone Q2 is described by the time-dependent three-dimensional advection-diffusion-reaction equation, eq.(4c), to which the radionuclide decay has been added.The advective velocity u couples the Navier-Stokes' and continuity equations, eqs.(4a and 4b) to the mass transport equation, eq. ( 4c): where, c i (r,t) is the nuclide concentration in the domains Ω 4 and Ω 5 (mol m Ϫ3 ), R i is the retardation coefficient and is dimensionless, D i is the diffusion coefficient (m 2 s Ϫ1 ), u i is the advective velocity in the fracture Q1 and the disturbed zone Q2 (m s Ϫ1 ) and λ i is the radioactive decay constant (s Ϫ1 ).
The impact of the fuel dissolution rate has been studied using the caesium-135 nuclide as mentioned earlier.The nuclide-specific data is presented in Table1.It includes the half-life, the initial inventory M 0 , the instantaneous release fraction α, the effective diffusivity in bentonite and the sorption coefficient, D Bent eff and K Bent d respectively.Two values for the instantaneous release fraction α were used: the first one has been taken from Hedin [1] and the other is simply twice this value.The nuclide considered here has infinite solubility.
Table II presents the non-nuclide-specific data, namely, the effective diffusivity of water D water e , the bentonite density ρ buff , the bentonite porosity ε buff , the fracture apertures b Q1 and b Q2 , the pinhole cross-section A Hole , the pinhole length L Hole and the degradation rate of the fuel matrix D F .The degradation rate of the fuel matrix covers an uncertainty of two orders of magnitude.
The time at which the canister starts leaching, t delay is taken to be 300 years.
Part of the input data set is a subset of the data used by Hedin [1].Six case variations have been carried out for the two nuclides using three different rates for the fuel dissolution rate and two values for the IRF fraction.Table 3     Figure 4 shows the breakthrough curves for the six cases simulated.The input data in Case 3 correspond to the parameters that Hedin used in his analytic model.Hedin developed his model to speed up Monte Carlo calculations for risk analyses.Our 3D model and Hedin's model are conceptually close to one another although Hedin used the resistance approach to mass transfer introduced in the earlier work of Neretnieks [5].Given this similarity, we would expect to obtain a similar breakthrough curve for Case 3 to that of Hedin, which is in fact the case.

Time (yr)
It is observed that cases 2 and 3 (for α equal to 0.03) and cases 5 and 6 (for α equal to 0.06) are relatively tight at 10 000 years.There is a significant jump, however, when the dissolution rate D F increases by one order of magnitude from 10 Ϫ7 to 10 Ϫ6 for the same IRF (Cases 1 and 2, respectively Cases 4 and 5).At 10 000 years, the cases can be ordered by increasing order of importance as: Case 3 < Case 2 < Case 6 < Case 1 < Case 5 < Case 4 The breakthrough curve corresponding to Case 1 increases more rapidly than that of Case 6 curve, making Case 1 the third most important case at 10 000 years.However the continued increase for Cases 1 and 4 over the later time period is considerably greater than that of the other four cases, enabling one to conclude that the parameter D F is of paramount importance if the nuclide has a half-life that is not short and therefore we need to increase our understanding of uncertainties coupled to fuel degradation of nuclear fuels.

Figure 1
Figure 1 An overview of a section of the KBS-3 repository and its near field.The canister and the simulated disturbed zone at the bottom of the tunnel are indicated in the picture.

1 28Figure 2
Figure 2 The geometry of the near-field.

Figure 3
Figure 3 displays the hexahedral grid used in fracture Q1.Figure4shows the breakthrough curves for the six cases simulated.The input data in Case 3 correspond to the parameters that Hedin used in his analytic model.Hedin developed his model to speed up Monte Carlo calculations for risk analyses.Our 3D model and Hedin's model are conceptually close to one another although Hedin used the resistance approach to mass transfer introduced in the earlier work of Neretnieks[5].Given this similarity, we would expect to obtain a similar breakthrough curve for Case 3 to that of Hedin, which is in fact the case.

Figure 3 .
Figure 3.The hexahedral mesh taken for the Q1 fracture.

Table 1 :
Nuclide-specific data i

Table 2 :
Generic data

Table 4 :
Mesh statistics of the different domains