Role of reactive transport in the alteration of vitrified waste packages: the MOS model

France vitrifies high-level waste from spent fuel reprocessing and plans deep geological disposal (CIGEO). Operational performance assessments have used a highly conservative operational model (MOP Vv) in which the glass source term dominates and alteration proceeds at an initial maximum rate until environmental reactivity is exhausted, then drops to a residual rate. A more mechanistic alternative, the GRAAL model, has been implemented in reactive transport codes to capture glass–environment interactions (iron, clay). This paper introduces MOS (Modélisation Simplifiée), a simplified intermediate tool designed to quantify how the diffusive, reactive environment near the glass—specifically silicon consumption by iron/corrosion products/argillite and diffusion through successive layers—controls silicon concentration at the glass interface and thus the glass alteration rate. The research question is how silicon reactive diffusion and environmental reactivity across multilayer barriers impact the time evolution of glass alteration rate and package lifetime in repository conditions, and how to capture this with a tractable model that bridges conservative operational models and full reactive transport simulations.

The paper situates MOS between two modeling approaches: (1) the Operational Model MOP Vv (CEA, 2004), which assumes alteration proceeds at the initial rate until environmental reactivity is exhausted, then immediately drops to a residual rate, and (2) the GRAAL geochemical model coupled to reactive transport codes (e.g., HYTEC), which accounts for detailed chemistry, transport, passivation, and glass–environment interactions. Table 2 contrasts mechanisms included in MOS versus GRAAL+transport: MOS neglects detailed aqueous chemistry, convection, temperature gradients, and internal transport within the fractured block, and considers diffusion only for silicon with environmental reactivity represented by a partition coefficient R. Prior studies (e.g., GRAAL applications, HYTEC simulations) support identifying dominant long-term mechanisms and justify simplifying assumptions used by MOS.

Theory and equations: MOS focuses on silicon as the controlling species for glass alteration, since silicon is central to gel formation and rate decline. A mass balance at the glass/environment interface equates silicon released by glass dissolution (considering retention in alteration film) to silicon removed by reactive diffusion into the environment. For a spherical, homogeneous, semi-infinite medium, this yields an analytical time-dependent expression for the alteration rate V(t) that depends on diffusion coefficient D, porosity, the environmental silicon reactivity expressed as a partition coefficient R (solid/liquid), the silicon saturation concentration at the interface Csat, the far-field concentration C0x, the package radius R0, the glass silicon mass fraction xsg, silicon molar mass Msi, retention ratio Td, and an effective fracturing ratio under initial-rate conditions (Tx0 or TD). Spherical 1D geometry is adopted to capture radial removal in three dimensions and keep equations tractable. For multilayer heterogeneous environments (successive layers with distinct porosity, D, and R), there is no general analytical solution. MOS introduces a simplified numerical resolution by decoupling fast reaction from slower diffusion: (1) compute the time to saturate a mesh cell’s reactive capacity (silicon dissolved in porewater plus silicon sequestered in newly formed solids parameterized by R), (2) once saturated, treat that cell as a purely diffusive barrier and compute the diffusive flux across it using a steady-state spherical wall expression. The approach iteratively advances the diffusion front outward across adaptively sized spherical shells. A reference mesh thickness ΔL0 at radius R1 is defined such that the shell can remove the same silicon flux as the glass dissolving at the initial rate; mesh sizes are adapted to local parameters to avoid over- or under-estimating fluxes. The equivalent diffusion coefficient across already saturated layers is computed using a series-law combination adapted to spherical geometry. The method is validated against the analytical solution for a homogeneous medium and against the HYTEC reactive transport code for both homogeneous and heterogeneous cases. Water arrival and phase partitioning: MOS represents progressive wetting via a piecewise-linear function h(t) for the submerged fraction of package height between times t0 and t3. During partial saturation, above-water surfaces alter at a hydration rate Vhydr(T) and submerged surfaces alter at V(t) bounded between V0 and Vf. The total altered glass fraction is the sum of vapor-phase and submerged contributions, accounting for evolving submerged area. Rates and parameters: V0(T,pH) is computed from an Arrhenius-type dissolution constant corrected for pH and solution composition; the final (closed-system residual) rate Vf(T) and vapor-phase hydration rate Vhydr(T) also follow Arrhenius forms with given activation energies. The silicon diffusion coefficient at temperature is obtained from D(20 °C) via a polynomial fit to Stokes–Einstein behavior between 0–90 °C. Environmental reactivity R is defined as a partition-like coefficient: the ratio of maximum silicon in solids after reaction to the maximum silicon in solution in contact with glass. Illustrative upper bounds are derived for iron and magnetite transforming to iron silicates, and for clay consumption capacity. Inputs include package geometry and fracturing ratios, porosities and diffusion coefficients per layer, R per layer, temperature (often 50 °C), pH, Csat, silicon retention ratios in alteration products, and timing/extent of wetting. Implementation is provided in a spreadsheet (mesh-by-mesh computation) and a dedicated C/C++ code suitable for uncertainty propagation (URANIE platform). Model assumptions (Table 1): MOS neglects long-term glass crystallization, self-irradiation, and fracturing evolution; ignores convection; considers diffusion only for silicon; assumes constant porosity and D; omits aqueous speciation/acid-base/redox; uses constant temperature; adopts spherical 1D geometry; treats environmental reactivity via fast partition coefficient R; and uses preset vapor-phase alteration parameters.

- The numerical resolution method is validated against the analytical homogeneous solution and the HYTEC reactive transport code for both homogeneous and heterogeneous media; mesh refinement near the glass is important to capture early high fluxes. - Time to reduce the glass alteration rate to V0/10 is generally very short relative to geological times under repository-relevant ranges of D and R. With slightly reactive materials (R < 10^3) and clay-like diffusion coefficients D(20 °C) < 1e-11 m^2/s, the drop from the initial rate is very rapid, indicating that using V0 is strongly overconservative for repository conditions. - Time to reach the final rate Vf depends strongly on the interplay between environmental reactivity and diffusion across layers, with two regimes: (1) when R is modest and/or D is low, the metallic envelopes alone control transport and the diffusion front does not extend beyond them; increasing D increases time to reach Vf; (2) when R is high and D is high, the diffusion front extends beyond metallic envelopes and the exterior medium controls reaching Vf; increasing D in metallic envelopes then decreases the time to reach Vf by more quickly feeding the outer, rate-controlling zone. - In highly diffusive and reactive 3D media, the diffusive removal flux tends asymptotically to a non-decreasing value due to the expanding 3D diffusion volume; the corresponding asymptotic alteration rate is several orders of magnitude lower than Vf for the parameters tested (homogeneous-medium limit given by the large-time limit of the analytical expression). - Package lifetime: For parameter sets tested (including 50 °C), even with metallic envelope reactivity up to R ≈ 10^6, calculated lifetimes are almost always ≥ 2 × 10^5 years and only weakly dependent on D. The rapid transition from V0 to Vf means the final-rate regime dominates lifetime. - Practical implication: The immediate vicinity materials’ ability to consume silicon (often via secondary silicate precipitation) has a strong impact on early-time rates, but clogging accompanying mineral transformation may limit this consumption over time. A simple stainless steel thickness or its corrosion products suffice to induce at least a one-order-of-magnitude rate drop from V0.

MOS quantifies how reactive diffusion of silicon in multilayer environments around vitrified waste reduces the glass alteration rate from the initial hydrolysis rate toward the final closed-system rate. The model demonstrates that maintaining V0 for significant durations is highly unlikely under repository conditions because diffusion barriers and finite reactivity rapidly elevate interfacial silicon concentrations, triggering rate decline. The finding that time to V0/10 is short supports safety assessments that do not rely on sustained initial-rate alteration. The sensitivity analysis clarifies when metallic envelopes versus exterior clayey media control the transition to Vf, helping interpret design choices (e.g., envelope thickness, material reactivity). The asymptotic 3D diffusion behavior highlights a fundamental geometric constraint on long-term fluxes. The discussion emphasizes that while environmental silicon consumption can be substantial, it may be self-limiting due to clogging by newly formed phases. The authors outline two avenues: (1) strengthen robustness by validating key assumptions (e.g., silicon-only diffusion, fast reaction embodied in R, chemistry simplifications, spherical geometry, retention ratios) with coupled geochemical reactive transport models (e.g., GRAAL/HYTEC), and (2) exploit MOS for sensitivity and uncertainty propagation studies (via C++/URANIE) to derive parameter constraints meeting performance goals (source term, altered fraction, lifetime).

The study introduces MOS, a simplified yet mechanistically grounded model that explicitly incorporates environmental reactive diffusion into predictions of vitrified waste glass alteration in deep geological disposal. By framing a silicon mass balance in spherical geometry and representing environmental reactivity with a partition coefficient, MOS provides an analytical solution for homogeneous media and a tractable iterative resolution for multilayer heterogeneous systems. Validation against analytical limits and HYTEC simulations supports its credibility for scoping and sensitivity analyses. Results show rapid decline from the initial rate and lifetimes largely governed by the final-rate regime for representative parameters, implying that operational models based on sustained V0 are conservative. Beyond nuclear glass, the modeling framework and multilayer resolution approach may be applicable to other corrosion and alteration problems where surface reactions in diffusive media control removal. Future work should validate and refine assumptions (chemistry, transport, geometry), reduce conservative margins, better quantify input parameters and their uncertainties, and propagate these uncertainties to performance metrics.

- Strong simplifying hypotheses (Table 1): no long-term glass crystallization, self-irradiation, or evolving fracturing; no convection; silicon-only diffusion considered; constant porosity and diffusion coefficients; no aqueous chemistry (acid-base, complexation, redox); constant temperature; spherical 1D geometry; environmental reactivity represented by a fast partition coefficient R. - Mesh-based decoupling of reaction and diffusion intentionally overestimates flux under certain conditions and depends on appropriate adaptive meshing; spherical geometry slightly overestimates volumes and reactive inventories near the package. - Reactivity R is a lumped partition-like parameter; real processes (e.g., specific precipitation pathways, kinetics, multi-component interactions) may deviate from this simplification. - Input parameters (e.g., D in metallic envelopes, R values, retention ratios, fracturing ratios) carry significant uncertainties that affect quantitative predictions; comprehensive validation against reactive transport simulations and experiments is still needed. - Vapor-phase alteration and the availability of vapor-altered material to transport are simplified; internal transport within the fractured block is not explicitly modeled beyond effective fracturing ratios.