The French nuclear industry produces high-level waste (HLW), primarily from spent fuel reprocessing. Vitrification is employed for HLW containment, and deep geological disposal is the chosen method for long-term management. ANDRA (French National Nuclear Waste Management Agency) is responsible for the design and safety assessment of this repository. Initial release models, like the Operational Model (MOP) Vv, oversimplify the process by focusing on the glass source term, assuming a constant initial alteration rate until the surrounding medium's reactivity is exhausted. More sophisticated models like GRAAL, which incorporate geochemical and reactive transport codes, offer greater accuracy but increased complexity. The MOS model aims to provide an intermediate level of simplification, incorporating the crucial factors of silicon consumption and diffusive transport limitations without the computational burden of the GRAAL model. The design of the CIGEO repository involves several layers of materials surrounding the vitrified waste (stainless steel container, overpack, liner, MREA, and Callovo-Oxfordian clay). These materials interact and transform over time, affecting silicon transport and consumption. The MOS model aims to quantify these effects.

The paper references previous models such as the Operational Model (MOP) Vv which provides a simplified approach, focusing primarily on the glass source term, and the more complex geochemical model GRAAL, which incorporates reactive transport and environmental material behavior. The literature review highlights the need for a model that balances simplicity and accuracy. Existing models such as GRAAL are mentioned for their ability to validate simplifying assumptions made in the MOS model. The role of silicon in glass alteration and the limitations of existing models in accurately representing long-term behavior are discussed.

The MOS model simplifies the alteration process by focusing on the diffusion of silicon and its consumption by surrounding materials. Key assumptions include the dominance of silicon diffusion and consumption, the neglect of other chemical reactions (acid-base, complexation, redox), constant temperature, and a simplified spherical geometry for calculations. For a homogeneous semi-infinite medium, a simple analytical equation is derived (Equation 1). This equation accounts for the alteration rate (V(t)), silicon mass fraction in glass, silicon retention in the alteration film, diffusion coefficient, porosity, silicon saturation concentration, package radius, and a reactivity coefficient (R). The reactivity coefficient (R) represents the ability of the surrounding materials to consume silicon. For heterogeneous media, a numerical approach is used, decoupling transport and chemistry. The method iteratively calculates the time required to saturate the reactivity of each mesh node, then considers the diffusional limitations imposed by saturated nodes before proceeding to the next node. This methodology involves the calculation of the reference mesh size (ΔLo) to ensure accuracy. The model considers water arrival as liquid or vapor, influencing the alteration rate. Equations 8 and 9 describe the total altered fraction of the package. Glass alteration rates (Vo, Vf, Vhydr) are calculated as functions of temperature and pH (Equations 10-12). The model includes parameters for the package, water arrival, glass alteration, and the surrounding environment (Tables 3-6). A C++ code and a spreadsheet implementation are used for calculations.

The MOS model demonstrates that the reactivity and diffusion properties of materials surrounding the glass package significantly impact the alteration rate. The initial alteration rate (Vo) is shown to be an overestimation of the long-term rate, even in highly reactive environments. The model predicts that a relatively short time is needed for the alteration rate to drop to one-tenth of the initial rate (Vo/10). This is due to the rapid accumulation of silicon in the vicinity of the glass. The time to reach the final alteration rate (Vf) is significantly longer. Sensitivity analyses show the importance of the silicon diffusion coefficient (D) and reactivity (R) in determining the time to reach both Vo/10 and Vf. A higher reactivity results in a longer time to reach Vf. The package lifetime is largely determined by the final rate regime and is predicted to be very long (greater than 2 x 10^5 years, even with high reactivity). The validation of the numerical resolution method is demonstrated by comparison with analytical calculations and results from the HYTEC reactive transport code. The model highlights the importance of silicon consumption by surrounding materials in controlling the long-term alteration rate. Results are presented in the form of graphs and tables showing the evolution of alteration rates, quantities of altered glass, and the position of the diffusion front over time.

The MOS model successfully integrates the influence of reactive diffusion on the long-term alteration of vitrified waste packages. By explicitly considering silicon consumption and transport limitations, it provides a more realistic estimate of the alteration rate than simpler models. The model's sensitivity to diffusion coefficient and reactivity highlights the importance of characterizing these parameters accurately for reliable predictions. The overestimation inherent in the model, stemming from simplifying assumptions, provides a conservative safety margin. Future work should focus on validating the model's assumptions using more detailed simulations, improving the accuracy of input parameters, and analyzing the propagation of uncertainties. The model's flexibility allows for exploration of various scenarios and parameter values, offering valuable insights into the long-term behavior of vitrified waste packages.

The MOS model offers a practical and relatively accurate way to estimate the long-term alteration of vitrified nuclear waste packages. It bridges the gap between simpler operational models and computationally intensive geochemical transport models. Future work should focus on validating the model hypotheses, refining input parameters, and utilizing the model for uncertainty analyses to improve predictive capabilities. The MOS methodology provides a flexible framework adaptable to various mineral and material alteration studies.

The MOS model employs several simplifying assumptions, including a simplified spherical geometry, constant temperature, and neglect of certain chemical reactions. These simplifications may lead to overestimations of the alteration rate. Additionally, the accuracy of the model is dependent on the reliability of the input parameters, some of which are associated with uncertainty. The model's predictions should be interpreted considering these limitations.