After canister breach, the radionuclide inventory initially encapsulated in the waste form and contained in the canister is slowly released by the degradation of the waste form. The degradation of various waste forms, specifically spent nuclear fuel and amorphous waste forms such as vitrified, metallic, ceramic, and frozen halide salt waste forms, has been extensively studied theoretically, experimentally, numerically, and using natural analogues (Harper et al., 2024; Muñoz et al., 2024; McCloy et al., 2024; Deissmann et al., 2025; Marcial et al., 2025). This extensive body of work forms the empirical and theoretical foundation for the source term in PA models, offering rate constants, activation energies, and mechanistic insights that enable extrapolation over the geologic timescales relevant to repository safety. Focusing on vitrified waste, the dissolution and alteration of the glass matrix is recognized as a complex physicochemical process that includes several stages with changing degradation rates that depend on the glass composition, temperature, solution chemistry, self-irradiation, exposed surface area, and other factors (Gin et al., 2017; Curti, 2022). Furthermore, as the network of glass-forming oxide dissolves, the radionuclides are released either congruently or incongruently, i.e., radionuclides are released either faster or slower than the glass constituents. Different mechanisms with similar complexities arise in the degradation of spent nuclear fuel and other waste forms, as described in detail in, e.g., Thorpe et al. (2021), Curti (2022), and Nagra (2024) as well as the references cited in Table 3 below.

For application in a PA model, these complexities are typically simplified to different levels of abstraction, with uncertainties handled by the choice of effective parameters and conservative assumptions. The waste degradation and radionuclide release model described below attempts to capture the main factors and processes that affect the source term and to directly integrate them into a PA model.

The mass of radionuclides present in the waste form changes with time because of (a) radioactive decay and ingrowth, and (b) radionuclide release from the degrading waste form. Radioactive decay of the inventory encapsulated in the waste form—and, if it is a member of a decay chain, its potential generation by the decay of the parent radionuclide—is described by Equation 1: m t = m 0 · e − λ t + m p 0 λ p λ − λ p e − λ p t − e − λ t (1) where m 0 = m 0 (kg) is the initial radionuclide mass (referred to as inventory), λ = ln 2 / t 1 / 2 (s^-1) is the decay constant, t 1 / 2 (s) is the radionuclide’s half-life, and t is the elapsed time since the reference time, i.e., the time the inventory was determined. Note that the decay of a parent radionuclide (indicated by subscript   p in Equation 1) may increase the mass of a radioactive daughter product, which also needs to be tracked in the model, both within the waste form and in the geosphere. A relevant example is the ingrowth of ²⁴¹Am caused by the decay of ²⁴¹Pu. The production of short-lived daughter products is typically directly included in the dose conversion factor. The decay equation applies to radionuclides that are encapsulated in the solid waste matrix, dissolved in the pore fluid within the canister, or migrating through the engineered and natural barrier systems.

In addition to radioactive decay, the mass of radionuclides in the waste form declines as the waste form degrades and isotopes are released from the solid matrix. It is assumed here that the radionuclides are distributed homogeneously within the waste form, and that they are mobilized proportional to the rate with which the waste form dissolves; this is referred to as a congruent release of radionuclides. This assumption is generally conservative for vitrified wastes, because data indicate high retention for all nuclides in the glass alteration layer (Curti et al., 2006). Moreover, for sufficiently high degradation rates and limited removal of radionuclides away from the canister by diffusion and advection, the released radionuclides may reach their solubility limit in the water. This can be approximately simulated by assuming precipitation or sorption of radionuclides within the canister.

The waste degradation rate is typically described as a fraction of the remaining waste mass ω (s^-1) or by a specific degradation rate q 0 (kg m^-2 s^-1) acting on the exposed surfaces of the waste form. These normalized degradation rates q may be assumed constant in time or dependent on temperature and other factors (such as pH or an affinity term). The dependence of the degradation rate on temperature is described by the Arrhenius equation combined with adjustment factors to account for pH and other chemical, radiological, and geometrical impacts (see, e.g., Cassingham et al., 2015; Thorpe et al., 2021; Kienzler et al., 2012): q T , t = Υ pH · Υ SA · Υ α · Υ L · Υ Si t · q 0 · ⁡ exp − E a R · 1 T t − 1 T 0 (2)

In Equation 2, T (K) is temperature, q 0 (kg m^-2 s^-1) is the rate constant at reference temperature T 0 (K), E a (J mol^-1) is the activation energy, and R (J K⁻¹ mol^-1) is the universal gas constant. The dimensionless factors Υ i can be used to account for other effects, such as:

Υ pH : pH dependence with respect to neutral water, Υ pH = 10 η · p H − 7

Exponential decay model (Equation 3) Υ Si t = Υ r + 1 − Υ r · e − λ · t − t b (3)

Smoothstep model (Equation 4) Υ Si t = Υ r + 1 − Υ r · k 2 · 3 − 2 k (4)

where k is defined by Equation 5: k = 1 − t − t b t e − t b  for  t b < t < t e (5)

Potential rate resumption (Fournier et al., 2014) is currently not implemented in the model (see related discussion in Section 3.2).

Υ CCL Impact of chemical composition of the leachate (including salinity, dissolved clay, cement, sulfate, phosphate, corrosion products (Fe, Ni, Cr), etc.) on waste degradation rate.

Because the PA model does not explicitly simulate chemical reactions, Υ = Υ pH · Υ SA · Υ α · Υ Si · Υ CCL is a time-independent factor that can be adjusted in sensitivity or uncertainty propagation analyses. By contrast, the temperature dependence of the waste degradation rate (last term in Equation 2) is updated dynamically, as temperature is one of the solution variables of the coupled thermal-hydrological PA model.

Waste form degradation is often described by a fractional degradation rate, ω (s^-1), which is the rate with which the remaining waste mass degrades per year. The fractional waste degradation rate can be obtained from degradation experiments as the slope given by Equation 6 ω = − ln ⁡ ( m t / m t + Δ t Δ t (6) where m t is the mass of the waste form at a given time t . The fractional degradation rate ω can be considered temperature dependent based on the Arrhenius equation (Equation 2). The rate with which the mass of radionuclides in the waste form is reduced due to decay and congruent release is then given by Equation 7: d m d t = m t · λ + ω (7)

The radionuclide mass encapsulated in the degrading waste matrix at time t is given by Equation 8: m t = m 0 · 1 − I R F · e − λ t · e − ω t − t b (8) where m 0 (kg) is the initial inventory at time zero, and I R F is the instant release fraction, which is the fraction of certain semi-volatile radionuclides (specifically ¹²⁹I, ¹⁴C, and Cs isotopes) that are rapidly released, mainly from spent nuclear fuel with high in-reactor irradiation (Johnson et al., 2023). The time-dependent rate q R N (kg s^-1), with which the radionuclide mass remaining in the solid waste matrix is released to the pore fluid due to waste form degradation, is given by Equation 9: q R N t = m t · ω (9)

While the fractional waste degradation model is typically used to describe the degradation of spent fuel assemblies, other waste forms such as vitrified, metallic, or ceramic waste forms placed into a disposal canister have an approximately cylindrical shape. The initial volume of a waste cylinder of radius R (m) and length L (m) is given by Equation 10: V = π R 2 L (10)

The volume of unreacted waste changes as its radius and length are reduced by degradation, as given by Equation 11: d V = 2 π R L   d R + π R 2   d L (11)

The reduction in the radius (Equation 12) and the length (Equation 13) of the unreacted waste due to degradation is given by: d R = q ρ W F   d t (12) d L = 2 q ρ W F   d t (13) where q (kg m^-2 s^-1) is the adjusted waste degradation rate of Equation 2, and ρ W F (kg m^-3) is the density of the waste form. The volumetric waste degradation rate is therefore calculated as: q W F t = d V t d t = 2 π q ρ W F L t · R t + R t 2 (14)

If the surface rate q is constant with time, the mass degradation rate of the waste form is given by: q W F t = 2 π q 3 q ρ W F t − t b 2 − L + 4 R q ρ W F   t − t b + R L + R (15)

If L ≫ R , degradation from the two circular end faces of the cylinder can be ignored, and Equation 15 simplifies to Equation 16 q W F t = 2 π L r R − q ρ W F   · t − t b (16)

If q is time-dependent (e.g., due to the time-varying waste temperature), Equation 14 is evaluated numerically for each waste element, with the cylinder’s radius and length updated after each time step.

The waste form mass remaining in the canister is calculated by Equation 17: m W F t = m W F , 0 − ∫ t f t q W F τ d τ (17) where m W F , 0 (kg) is the initial mass of the waste form. The radionuclide mass fraction in the waste form is given by Equation 18: X W F R N t = m RN , 0 · 1 − I R F · e − λ t / m W F , 0 (18)

Finally, the congruent radionuclide release rate from the waste form is given by Equation 19: q R N t = q W F t · X W F R N t (19)

For constant q and a cylinder that is relatively long, i.e., L ≥ 2 R , the waste form is fully degraded at time t q W F = 0 = R · ρ W F / q . For a more disk-shaped waste form, i.e., L < 2 R , the waste form is fully degraded at time t q W F = 0 = L / 2 · ρ W F / q .

The instant release fraction ( I R F ) is implemented by specifying an initial radionuclide mass fraction dissolved in the water within the canister that yields the I R F ’s mass of the inventory, m I R F = m 0 · I R F . Furthermore, the initial inventory is reduced to m e = m 0 · 1 − I R F to properly reflect the fact that a smaller mass of radionuclides is encapsulated in the solid waste form. The mass m I R F will be mobilized instantaneously at the time of canister failure, t b , while mass m e will be mobilized by congruent release as the waste forms dissolves.

The waste degradation and radionuclide release models described above are implemented in the iTOUGH2 simulation-optimization software (Finsterle et al., 2017; Finsterle, 2025), allowing for a fully integrated analysis of the impact of waste-form related parameters on overall performance of the repository system, as demonstrated in Section 3.

To examine the importance of source-term model parameters on the safety of a radioactive waste repository, a PA model is developed that combines the source-term model with key components of the engineered and natural barrier systems and integrates them into a coupled thermo-hydrological numerical model that simulates fluid flow, heat transfer, and radionuclide transport from the canisters to the accessible environment. The activity of the main safety-relevant radionuclides in drinking water extracted from a near-surface aquifer is calculated and multiplied by the nuclide’s respective dose conversion factor (IAEA, 2003). Finally, the dose contributions from all tracked radionuclides are summed, and the maximum value over the entire performance period is determined and reported as the peak exposure dose, which is considered one of the main performance metrics of interest that represents long-term repository safety (e.g., Nagra, 2024). This general PA modeling approach is applicable to different repository types; in what follows, we use as an example the disposal of vitrified waste in a deep vertical borehole repository constructed in fractured, crystalline basement rock.

We use a simplified version of the model described in Finsterle et al. (2021b), which was developed to simulate the transport of radionuclides released from a deep vertical borehole repository. An array of parallel boreholes with a constant spacing of 50 m is modeled by taking advantage of vertical symmetry planes intersecting the borehole axis and the mid-plane between two neighboring boreholes. A total of 300 canisters are placed into a 1,500 m long disposal section, one canister every 5 m. Each canister holds three CSD-V ¹ containers, each with a cylinder of vitrified waste with a length of 1.4 m and a radius of 0.2 m, from which heat and radionuclides are released according to the source-term model described in Section 2.1.

A schematic of the considered repository system is shown in Figure 1. The repository is assumed to be constructed in crystalline bedrock, which may be encountered at relatively shallow depths, immediately below a 500 m thick sedimentary cover, which also includes a 200 m thick drinking water aquifer. The disposal section extends between a depth of 750 and 2,250 m. A well 200 m from the access hole of the repository extracts drinking water from the aquifer. Regional groundwater flow is driven by topographic elevation changes, here represented by two 500 m high, parallel mountain ridges that act as recharge zones. Groundwater flows preferentially laterally from the mountain ridges towards the discharge zone near a river that runs along the center of the valley. The flow field is affected by the drinking water well, which is assumed to pump at a constant rate of 1,000 m³ per year. Such a relatively small pumping rate leads to reduced dilution of the contaminated water and therefore a higher exposure dose. The narrow width of the symmetry cell ensures that essentially all radionuclides seeping into the aquifer will be captured by the well. In addition, a conservative, vertical upflow is introduced by specifying a constant pressure at the bottom of the model that is higher than the hydrostatic pressure. The hydrostatic pressure is affected by depth-dependent density changes due to the increase in pressure, salinity, and temperature, which is controlled by the mean annual surface temperature of 15 °C and an average geothermal gradient of 30 °C per kilometer.

At the depth of the repository, the granitic host rock is typically fractured, potentially providing pathways for advective radionuclide transport, which is retarded by diffusion into the essentially stagnant pore water of the rock mass between the connected fracture network. The degree of fracturing is related to the stress regime and is thus depth dependent. The discreteness of fractures, their connectivity, variability in aperture distribution, and gradient-dependent channeling effects lead to effective permeabilities that are heterogeneous and anisotropic. The key safety-relevant features of a crystalline geosphere—i.e., potentially fast fluid flow through a connected fracture network and radionuclide retention by matrix diffusion—are included in the model by a dual-continuum approach, with a depth-dependent trend of average fracture-network permeability (Achtziger-Zupančič et al., 2017) that is superimposed by geostatistically generated, spatially correlated and anisotropic variability. The exchange of fluids and radionuclides between the fracture and the matrix continua is controlled by the geometry of the fracture network, which is assumed to consist of two sets of planar, parallel, infinite fractures with an arbitrary angle between the two sets and a constant fracture spacing. The matrix is considered homogeneous. Global matrix-to-matrix fluid flow and radionuclide transport is allowed only in the vertical direction, consistent with the fact that the regional stress field at the depth of the repository likely generated subvertical fracture zones. For the current long-term analyses, some components of the engineered barrier system (i.e., the annulus between the canister and casing, the casing, and the cement between the casing and the borehole wall) are not individually discretized but represented by appropriate effective parameters that account for anisotropy in permeability between the axial and radial directions.

To summarize: The EBS is represented by an effective porous material with a strong axial-to-radial anisotropy; in the geosphere, fluid flow, heat transfer, and radionuclide transport through the fracture network is three-dimensional, with local exchange to the matrix continuum and global matrix-to-matrix flow and transport in the vertical direction.

The model domain is first discretized into a three-dimensional, structured Cartesian mesh with 69 × 8 × 104 = 57 , 408 grid blocks in X, Y, and Z direction, respectively. The smallest grid blocks have dimensions of d x = d y = 0.31 m, yielding the desired canister cross section of π · r c 2 = 0.096 m². Element sizes in X direction gradually increase and remain constant at 10 m up to a distance of 200 m from the borehole, after which they gradually increase again to a maximum d x of 500 m at the outer lateral boundaries of the model. In Y direction, the same increase in grid-block sizes away from the borehole is used up to the symmetry plane at y = 25 m. In vertical direction, grid spacing is constant at d z = 25 m from the land surface to a depth of 2,350 m (i.e., 100 m below the bottom of the borehole) after which layer thicknesses gradually increase to a maximum of 135 m at the bottom of the model. In a second mesh generation step, each grid block representing the crystalline bedrock is divided into two overlapping elements representing the fracture and matrix continuum, which are locally connected to each other, as described above. This results in a finite volume mesh with a total of 108,890 grid blocks and 216,724 connections between them. Five equations are solved for each grid block—one for pressure, temperature, salinity, and the mass fractions of ¹²⁹I and ¹³⁵Cs—for a total degree of freedom of 544,450. More details about processes, material properties, discretization in space and time, and the solution algorithm can be found in Muller et al. (2019) and Finsterle et al. (2020), Finsterle et al. (2021a), and Finsterle et al. (2021b).

The key processes considered in this study are non-isothermal flow of water and brine. Decaying radionuclides are transported by advection and diffusion, and they may be retarded by reversible sorption onto the solid phase. Heat is transported by conduction and convection. Porosity changes linearly as a function of pore pressure and temperature using volumetric pore compressibility and thermal expansivity. Details about the physical processes and the corresponding mathematical model and numerical scheme can be found in the documentation of the TOUGH2 code (Pruess et al., 2012), which is implemented in the iTOUGH2 simulation-optimization framework (Finsterle et al., 2017). iTOUGH2 is used in this study to take advantage of enhanced user features (Finsterle, 2025), specifically the capability to seamlessly integrate steady-state initialization runs with the transient simulation after repository construction and waste emplacement, which requires changes in properties and boundary conditions at discrete times. Moreover, the fractional and cylindrical waste degradation models and corresponding radionuclide releases described in Section 2.1 are implemented in iTOUGH2, where the factors influencing the source term are adjustable parameters for the calculation of composite sensitivity measures (Finsterle, 2015) as well as for inverse modeling, global sensitivity and uncertainty propagation analyses. In the current study, iTOUGH2 will be used for local sensitivity analyses and Monte Carlo simulations for examining prediction uncertainties, using Latin hypercube sampling to make sure parameter realizations cover the entire range and are in conformance with the assumed occurrence probabilities (Zhang and Pinder, 2003).

Properties of the main engineered and natural materials are listed in Table 1; they are not site specific but are considered representative of a generic borehole repository in a fractured, crystalline host rock. Note that this study focuses on the relative impact of changes in source-term parameters on peak exposure dose calculated for a generic reference case; the purpose is not to justify the absolute peak dose value, which depends on site-specific properties and conditions. While each material is associated with a full set of thermal and hydrological parameters, only a small subset of these parameters is provided, selected based on their significance for the storage and flow of fluids and heat as well as radionuclide transport within the respective material.

Only two radionuclides, ¹²⁹I and ¹³⁵Cs, are tracked in the PA model, down-selected from a comprehensive list of radionuclides present in the waste form according to their relative safety-relevance, which is estimated in a two-step approach based on the radionuclide’s inventory, half-life, specific activity, and dose coefficient, as well as their retardation in the geosphere, which is related to sorption and matrix diffusion. Table 2 lists key radiological and transport properties of the two safety-relevant radionuclides.

In the example discussed below, we consider the vitrification of high-level radioactive waste in a lanthanide borosilicate (LaBS) glass matrix. LaBS glasses are primarily composed of lanthanide oxides, silica, alumina, and boron oxide, as well as minor components such as barium, lead, or strontium oxide (Jantzen, 2011). Compared to traditional alkali aluminoborosilicate glasses, LaBS glasses are chemically more durable and exhibit higher glass transition temperatures, melting temperatures, and radionuclide solubilities (Jantzen, 2011). Compared to traditional alkali aluminoborosilicate glasses, LaBS glasses are chemically more durable and exhibit higher glass transition temperatures, melting temperatures, and radionuclide solubilities (Peeler et al., 1999). Other favorable qualities of these glasses include the presence of significant quantities of neutron absorbers (lanthanide oxides) that permit increased loading of fissile materials into the glass with negligible increases in the risk of criticality. Due to these characteristics, LaBS glasses have been explored as an effective waste form for the encapsulation of high-level radioactive wastes generated by existing and future facilities for the reprocessing and recycling of spent nuclear fuel.

Several properties affect the actual alteration rate of LaBS glass waste forms in a deep geological repository. These include (a) intrinsic waste form parameters such as rate constant and activation energy, (b) physical parameters such as surface area, (c) environmental factors such as temperature, leachate composition, and the precipitation of secondary phases, and (d) other factors such as radiation/radiolysis effects. The degradation parameters of LaBS glass along with some repository conditions that affect glass alteration are summarized in Table 3. The reference set is generic; some of the values are chosen to better demonstrate their potential impact on radionuclide release and repository performance during the sensitivity analyses, which are performed for a subset of the parameters by perturbing the parameter value one at a time within the range indicated in Column 3 of Table 3. A parameter scaling factor σ p is given in Column 4, which represents the approximate variation or uncertainty of the parameter. The same value is used as the parameter uncertainty to be propagated through the PA model to estimate the prediction uncertainty of the peak exposure dose and the time it occurs (see Section 3.3 below). Both the reference values of the waste degradation parameters and their expected uncertainty will likely be adjusted once waste-form- and site-specific characterization data become available.

We first present the prediction results obtained with the reference parameter set listed in Column 2 of Table 3. While the PA model calculates a comprehensive set of primary solution variables and derived quantities at each location within the model domain, we focus here on a small subset that highlights the system response to changes in the radionuclide source term and the exposure dose at the land surface, which is the performance metric of overall repository safety.

Figure 2A shows the total release rate of ¹²⁹I and ¹³⁵Cs from all canisters. The mass release rate, which depends on the parameters of Table 3 controlling waste degradation and congruent radionuclide mobilization according to the source-term model described in Section 2.1, is converted to Sieverts per year (Sv/yr) to allow for a direct comparison of the release rate at the underground disposal section of the repository with the eventual exposure dose at the land surface. The difference between the two rates reveals the effectiveness of the natural barrier system. Figure 2A also shows the fraction of the inventory released to the near field; the initial inventory of ¹²⁹I and ¹³⁵Cs in all 300 canisters amounts to a total dose of approximately 80,000 Sv.

Radionuclide release starts at the time the canisters are breached. The initial release rate is highest because (a) the exposed surface area of the waste form is at its maximum, (b) the surface-specific degradation rate q is not yet reduced by affinity effects, (c) reservoir temperature may still be elevated from decay heat release, and (d) no significant radioactive decay, which lowers the activity of ¹²⁹I and ¹³⁵Cs in the system, has occurred within the canister. During the first 10,000 years after canister breach, the waste degradation rate is reduced as the orthosilicic acid activity approaches chemical equilibrium, at which point the rate has been lowered by a factor of Υ r = 0.01 . Waste degradation and radionuclide release proceed at lower and continuously decreasing rates, as the surface area of the waste-form cylinder shrinks, temperatures approach ambient values, and radionuclides in the canister decay. For the reference parameter set, the waste form is never completely dissolved even after 10 million years, at which point a waste-form cylinder of length 1.14 m and radius 0.07 m remains at the locations of each of the inner CSD-V containers, highlighting the effectiveness of the LaBS waste form to retain radionuclides for very long times. As a result of containment and slow release, only about 60% of the inventory’s initial activity is released from the canisters. Finally, it is noted that ¹³⁵Cs has a higher contribution to the release rate than ¹²⁹I, initially by a factor of 11.3, which is the ratio of their respective inventories, specific activities, and dose coefficients (see Table 2). The dominance of ¹³⁵Cs release declines with time due to its shorter half-life.

¹³⁵Cs—as well as other radionuclides initially present in the waste form—may indeed have significantly higher radiological consequences than ¹²⁹I if it were released from a breached canister at the land surface, or if contaminated groundwater were extracted directly from the near field of the repository. However, it is important to recall that the safety of the repository is controlled by the performance of the natural barrier system with its various retention mechanisms that considerably delay exposure, thus reducing risks and changing the order of the radionuclides’ relative safety relevance. Many of the radionuclides in the inventory have an insignificant contribution to peak dose due to the effectiveness of the natural barrier system, while others (such as ¹²⁹I) that seem less problematic at the source become the most safety relevant. The screening process used to estimate each radionuclide’s relative safety-relevance is based on key metrics that account for characteristics of the radionuclide itself as well as its migration from the canister to the dose recipient. While it is essential to have a source-term model that accurately calculates the release of radionuclides into the geosphere, it must be integrated into a suitable PA model that simulates flow and transport processes and provides estimates of the ultimate exposure dose, which reflects repository safety.

Figure 2B shows the contributions of ¹²⁹I and ¹³⁵Cs to the annual exposure dose from the ingestion of drinking water extracted from the aquifer above the repository. The non-sorbing ¹²⁹I is highly mobile despite being somewhat retarded by matrix diffusion. It reaches its peak dose of 1.6 × 10^–5 mSv yr^-1 after about 130,000 years, a time too short for radioactive decay to reduce its concentration. Because the source term is not a single pulse, but radionuclides are released over an extended time with a declining rate, the breakthrough curve is relatively flat and exhibits a long tail.

By contrast, ¹³⁵Cs arrives at the land surface very late (after approximately 6 million years) as its migration is retarded by comparatively strong adsorption to the fracture surfaces and the grains of the rock matrix. Despite its long half-life of 2.3 million years, this prolonged travel time is sufficient for radioactive decay to reduce the ¹³⁵Cs mass by about a factor of six. The combination of radioactive decay and the lowering of the concentration peak due to adsorption leads to a considerable amplitude reduction of the ¹³⁵Cs breakthrough curve. As a result of different magnitudes and temporal separation of the ¹²⁹I and ¹³⁵Cs breakthrough curves, the peak dose of the sum of the two radionuclides is completely dominated by, and thus essentially identical to, that of ¹²⁹I. Note that the peak dose of 1.6 × 10^–5 mSv yr^-1 is almost four orders of magnitude below a typical dose standard of 0.1 mSv yr^-1.

The reference results shown in Figure 2 depend on numerous conceptual assumptions and simplifications, as well as on uncertainties in the model’s many input parameters. For the purpose of the current analysis, we focus on parameters of the source-term model and examine their impact on predictions of peak exposure dose through standard sensitivity analyses (Section 3.2) and a sampling-based uncertainty propagation analysis (Section 3.3).

The purpose of the sensitivity analyses is (a) to get physical insights into how the various processes implemented in the source-term model affect radionuclide release from the canisters, and (b) to identify the relative influence of source-term model parameters on the prediction of exposure dose, which helps formulate criteria for how accurately waste-form-related parameters need to be determined as not to induce unacceptably high prediction uncertainties that would render PA calculations less conclusive. Each sensitivity analysis presented below consists of perturbing one parameter at a time from its reference value and then calculating and visualizing the system response. The local parameter range covered by the perturbations is indicated in Table 3. Note that changing the waste loading factor leads to a proportional change in the release rates and peak dose. The impact of pH is also a simple proportionality factor as given by Equation 3. Similarly, the parameters modifying the degradation rates by a constant factor exhibit the same sensitivity behavior as changes in the intrinsic rate, scaled by the σ p ratio. Therefore, discrete sensitivity analyses are discussed for only six of the 11 parameters listed in Table 3. All 11 parameters will be varied during the uncertainty propagation analysis discussed in Section 3.3.

Figure 3 shows the influence of changes in the canister breach time on the sum of the ¹²⁹I and ¹³⁵Cs release rates and the cumulative release fraction of the inventory as well as on the individual and combined exposure doses. Changing the time of canister breach leads to a corresponding shift in the time when peak dose occurs. However, the peak dose itself remains essentially unchanged. Even if a very durable canister with a lifetime of 100,000 years is used, the peak dose is reduced only by the amount of radioactive decay occurring during the time shift, i.e., the period when the radionuclides were still contained in the canister, with some minor secondary effects related to changes in the temperature. Canister durability may be an important design criterion to prevent the early release of highly active radionuclides. However, highly active isotopes are typically short-lived, which means that they have decayed to insignificant levels once they arrive at the land surface, at least for the nominal case. Moreover, even in the very unlikely event that all canisters are breached immediately after repository closure, the radionuclides are still encapsulated in the waste form, which degrades very slowly even under the elevated temperatures prevalent during the first few decades of the thermal period. Temperatures encountered by the waste form range from 38 °C for shallow canisters after the thermal period, and 172 °C for the deepest canisters at the peak of the thermal period. Finally, the risks and consequences of severe failure of both the canister and waste form are partly addressed by pre-closure canister and waste-form performance requirements.

The criterion of a very long canister lifetime—with the considerable costs associated with the fabrication of such canisters—is sometimes driven by the attempt to demonstrate that the engineered barrier system by itself meets the long-term dose standard in the case of disruptive events that render the natural barrier system ineffective. Occurrence probabilities and consequences of such disruptive scenarios can be mitigated by proper site selection. Moreover, PA calculations for deep borehole repositories show considerable robustness and resilience to such adverse conditions and events (Finsterle et al., 2020; Finsterle et al., 2021a; Finsterle et al., 2021b).

Figure 4 reveals the sensitivities with respect to the intrinsic waste degradation rate q 0 , a parameter that can be inferred from leaching experiments, but may have relatively large estimation uncertainties due to its low value for materials (such as LaBS glasses) used as the matrix for the encapsulation of radioactive isotopes. Increasing or decreasing q 0 by one order of magnitude leads to a corresponding change in the initial radionuclide release rates immediately after canister breach. For the highest rate, about 85% of the radionuclides are released within the first 10,000 years after canister breach. However, for such a high intrinsic rate q 0 , the mass degradation rate of the waste form ( q F W ; see Equation 14) declines more quickly as the surface area of the cylindrical waste form shrinks faster compared to the reference case. Nevertheless, the entire waste form is consumed after about 1 million years. This rate profile can be described as a pulse release of most of the inventory, while for the two cases with a lower intrinsic rate, radionuclides are continually released over the entire performance period, as reflected by the long-tailed breakthrough curves shown in Figure 4B. The ratios of the peak exposure doses for the two low-rate cases are approximately proportional to the ratio of the intrinsic waste degradation rate. However, the dose ratio between the reference case and the high-rate case is less than a factor of ten because of the secondary effect of a faster reduction in the waste form’s surface area, as described above. Nevertheless, it is apparent that the intrinsic rate is one of the main parameters affecting peak dose.

The same impact is expected from the factors that directly increase the degradation rate by a constant, such as the factors accounting for the impact of waste-form fracturing ( Υ S A ), radiation ( Υ α ), leachate composition ( Υ C C L ), and pH effects ( Υ p H ). The relative impact of these factors depends on their expected uncertainty, which is expressed by σ p (see Table 3).

Figure 5 shows the influence of changing the maximum rate-reduction factor (or, equivalently, the residual rate q r ), which represents complex chemical affinity effects that lead to the formation of a passivating gel layer, moving the reaction front from the surface into the glass matrix, thus reducing the alteration rate as a function of time (see Υ S i t , Equations 3–5). In the reference case, it is assumed that the rate is reduced by a factor of 100 over a period of 10,000 years, which considerably slows waste degradation. If no affinity effects and thus no rate reduction occurs, i.e., if Υ S i = 1 (green curves), waste degradation remains high, and the entire waste form is consumed after about 200,000 years. This again results in an approximate pulse release of radionuclides, which leads to a higher peak dose compared to the case where affinity effects spread out the release of radionuclides over a very long time, which flattens the dose breakthrough curve. Interestingly, if the affinity effects are very strong and lead to a drastic reduction in the degradation rate (blue curves), the source term also has the characteristics of a pulse release, albeit only releasing about 20% of the inventory. Only in the intermediate case (the reference case, shown in red) is the reduced degradation rate still high enough to release about 40% of the inventory, creating the long-tailed dose breakthrough curve discussed previously. The peak dose, however, is controlled by the fast release of the initial 20% of the inventory, which is essentially the same amount for both the reference case (red) and the strong-affinity case (blue), leading to essentially the same peak dose.

It can be concluded that the presence or absence of affinity effects influences the peak dose. However, whether the rate reduction is on the order of a factor of 10 or much greater has no further influence on repository performance. This may also affect the conclusions about a possible rate resumption due to the precipitation of crystalline secondary phases (Thorpe et al., 2021). If that rate resumption is limited in magnitude or occurs after a prolonged period of waste degradation at the residual rate q r , rate resumption is unlikely to lead to a dose that exceeds the primary peak value. Even if rate resumption reverts to the initial forward rate, the volume of waste being dissolved is smaller because of a reduced waste form surface area, and the amount of congruently released radionuclides is further reduced because of inventory decay. Finally—unlike the pulse release immediately after canister breach—rate resumption occurs gradually, leading to a flattening of the breakthrough curve at the land surface.

While Figure 5 shows the impact of the strength of affinity effects, Figure 6 examines the influence of how fast the degradation rate is reduced from the intrinsic rate q 0 to the residual rate q r = q 0 · Υ S i t e . The duration of this rate-drop regime needed to transition from the forward to the residual rate regime (Thorpe et al., 2021) depends on many factors. If chemical equilibration progresses slowly, degradation rates are high for a longer period, leading to a higher peak dose. If chemical affinity processes are very fast, only a small fraction of the inventory is released at the initial high rate, while the rest is released with a low rate of q r over a long period, leading to a comparatively flat breakthrough curve with a lower peak dose that occurs at a later time (see green curve in Figure 6B).

The temperature effect on degradation rates—as described by the Arrhenius equation—is examined next. We first note that in all results discussed so far, temperature effects are included, with each canister having its own waste degradation rate in accordance with the local temperature conditions, which is depth-dependent given the assumed geothermal gradient of 30 °C km^-1. For example, in the reference case, waste degradation in the deepest canister is approximately seven times faster than in the shallowest canister due to an ambient temperature difference of 45 °C. The temperature increase caused by the waste’s decay heat may further accelerate glass degradation. However, the generated decay heat declines relatively fast, with the maximum temperature reached after just a few years, and temperatures approaching their pre-disposal ambient conditions within a few hundred years. If the canister is not breached until after the early portion of the thermal period, the additional impact of repository-induced temperature changes on waste-form degradation is likely negligible.

Figure 7 shows the sensitivity of peak dose to changes in the activation energy, which is the adjustable parameter that controls the temperature dependence of waste degradation. In the examined scenario, the temperature of the repository at the time of canister breach (i.e., 10,000 years) is lower than the reference temperature of T 0 = 90 ° C. As expected, when T t < T 0 , the lower the activation energy, the higher the waste degradation rate, with a corresponding sharpening of the initial release pulse and thus an increased peak dose. Note that if the canisters were breached during the early portion of the thermal period, where T t > T 0 , lower values of activation energy would correspond to lower waste degradation rates, and the opposite trend would be observed until the temperature of the system falls below that of the reference temperature.

Finally, having established that peak dose is sensitive to the kurtosis and amplitude of the source term, we consider the instant release of a fraction of the inventory at the time of canister breach, including the bounding case in which all radionuclides are immediately mobilized. Instant release is only assumed for radionuclides (specifically ¹²⁹I) that have the tendency to accumulate in gaps and on surfaces of pellets and cracks in spent nuclear fuel assemblies. However, no such accumulations are expected in vitrified waste. Nevertheless, we consider the unrealistic case with an I R F of 100%. While unrealistic, this scenario is considered useful as it provides an upper bound for the peak dose, which can also be interpreted as a case where repository performance relies almost exclusively on the natural barrier system, discounting long-term encapsulation of the radionuclides in the waste form, ignoring solubility limits, and assuming no adsorption occurs on corrosion products or other materials within the engineered barrier system. Note that none of the other parameters of the source-term model enter this extreme scenario, with the exception of the waste loading factor, which translates linearly into a corresponding change in the peak dose.

Figure 8A confirms that an I R F of 20% leads to the immediate release of that fraction of the inventory, verifying the correct implementation of the I R F feature in the PA model. Since the radionuclide mass encapsulated in the solid glass matrix is 20% less compared to the reference case, the release rates with a non-zero I R F are lower throughout the remainder of the waste form’s degradation. Instant release of 20% of the inventory creates a sharper release pulse, which increases peak dose by about 50%—i.e., more than the I R F itself—but reduces the radionuclide mass in the tail of the breakthrough curve (see Figure 8B).

The bounding case with I R F = 100 % leads to the highest peak dose of 6.6 × 10^–5 mSv yr^-1. This dose is only about four times higher than the reference case. While unrealistic (if not physically impossible), this result is reassuring. It suggests that (a) in the absence of any information about waste-form degradation, and (b) making the most conservative assumption about radionuclide releases from the canister and its further retention in the engineered barrier system, the peak dose increases by less than an order of magnitude. This systematic shift is likely less than the uncertainties associated with the estimation of peak dose as part of such performance calculations.

In the previous section, the source-term parameters were perturbed one at a time. These sensitivity analyses revealed considerable non-linearities in the dose response above a vertical borehole repository, with curves of radionuclide releases crossing each other and non-symmetric influences on peak dose. This calls for a sampling-based method to evaluate the impact of parameter uncertainties on the predicted performance metrics. Monte Carlo simulations with Latin hypercube sampling are used, drawing normally or log-normally distributed values around the reference parameter set with the standard deviation given in Column 4 of Table 3. The sample distribution is truncated at the bounds given in Column 3 of Table 3. It is further assumed that the uncertainties in the parameters are uncorrelated to each other. This is considered appropriate given that most of the source-term parameters are determined independently. An exception might be the experimentally determined intrinsic waste degradation rate, q 0 , which could be affected by errors in pH, temperature, and other variables measured to characterize the test conditions. Similarly, the estimates of the activation energy E a (Equation 2) or the pH exponent η (Eq. 2a) from leaching experiments may be weakly correlated to each other due to errors in the measured pH or temperature. However, the resulting correlations are expected to be minor and are unlikely to change the conclusions drawn from PA calculations. Finally, a small sample size of 500 realizations is simulated, which—in the context of this study—is considered sufficient to illustrate the degree of prediction uncertainty caused by assumptions about the radiological source-term model parameters. All other parameters, including thermal-hydrological properties and related flow and transport parameters, are fixed.

Figure 9 shows the dose breakthrough curves for the 500 random realizations along with the histograms of the peak dose and its time of occurrence. The reference case, which is close to the median peak dose, is highlighted in red; the realization leading to the highest peak dose value is shown in black. The 5th and 95th percentiles (green) are directly determined from the set of results ordered by their peak-dose and peak-dose-time values.

Figure 9 illustrates that various combinations of source-term parameters lead to a wide range of peak dose values; however, they are bounded by inherent, physical constraints. Only a few realizations exceed the peak dose value obtained by the I R F = 100 % scenario (see Figure 8B), mainly because that bounding case was run with the reference waste loading factor of 1.0, whereas the radionuclide inventory is considered an uncertain parameter during the Monte Carlo simulations, with about half of the sampled realizations having waste loading factors greater than 1.0. If waste loading and canister spacing is known, the corresponding simulation with an I R F of 100% still provides a robust estimate of the upper bound for peak dose. The fact that many breakthrough curves cross each other, and that the histograms are non-symmetric and likely multimodal is a clear indication that the exposure dose is a strongly nonlinear function of the source-term input parameters.

The individual peak dose values obtained by each of the 500 Monte Carlo realizations (shown as yellow diamonds in Figure 9) form two clusters: the first is vertically elongated around the time of the maximum peak dose (between approximately 100,000 and 200,000 years); the second forms a curved streak, which is bounded by the black breakthrough curve associated with the realization that generated the maximum peak dose. The separation of the two clusters is a consequence of the non-symmetric influence of the two factors that have an impact on the peak dose time, namely the duration of the affinity effects (see Figure 6B) and the canister breach time (see Figure 3B). The first cluster is associated with realizations in which the influential factors affecting the peak dose—i.e., intrinsic and residual waste degradation rate, activation energy, and all the Υ factors—are combined with factors that lead to early peak dose times, which are left-skewed. The curved cluster is associated with realizations that lead to late peak dose times. Since a late arrival of radionuclides at the land surface is correlated to higher spatial and temporal dispersion of the contaminant plume, the peak dose values also decline with time at a rate that is consistent with that of the tail of the breakthrough curve with the maximum peak dose (black line). This suggests that the breakthrough curve with an I R F of 100% (see Figure 8B) not only defines the upper limit for the maximum peak dose, but it also provides bounds on peak dose values for all combinations of source-term related parameter combinations.

Note that the uncertainty propagation analysis presented here is based on fairly large uncertainties and ranges in the input parameters (see Table 3), leading to a relatively large spread in the calculated peak dose values, which should be put in perspective to the spread caused by uncertainties in the conceptual models describing the thermal, hydrological, geochemical, mechanical, and biological processes and the large number of uncertain input parameters to the PA model. The contribution of the source-term model to prediction uncertainty is expected to be noticeable, but it may not dominate the conclusions of a comprehensive repository safety analysis.