Use of groundwater lifetime expectancy for the performance assessment of a deep geologic waste repository: 1. Theory, illustrations, and implications

Long-term solutions for the disposal of toxic wastes usually involve isolation of the wastes in a deep subsurface geologic environment. In the case of spent nuclear fuel, if radionuclide leakage occurs from the engineered barrier, the geological medium represents the ultimate barrier that is relied upon to ensure safety. Consequently, an evaluation of radionuclide travel times from a repository to the biosphere is critically important in a performance assessment analysis. In this study, we develop a travel time framework based on the concept of groundwater lifetime expectancy as a safety indicator. Lifetime expectancy characterizes the time that radionuclides will spend in the subsurface after their release from the repository and prior to discharging into the biosphere. The probability density function of lifetime expectancy is computed throughout the host rock by solving the backward-in-time solute transport adjoint equation subject to a properly posed set of boundary conditions. It can then be used to define optimal repository locations. The risk associated with selected sites can be evaluated by simulating an appropriate contaminant release history. The utility of the method is illustrated by means of analytical and numerical examples, which focus on the effect of fracture networks on the uncertainty of evaluated lifetime expectancy.

💡 Research Summary

The paper presents a novel framework for assessing the safety of deep geological repositories for radioactive waste, particularly spent nuclear fuel, by introducing the concept of groundwater “lifetime expectancy” (LE). LE is defined as the probability distribution of the time required for water molecules (or released radionuclides carried by them) to travel from a potential repository location to the biosphere (i.e., any outlet of the groundwater system). The authors formulate LE as the solution of a backward‑in‑time advection‑dispersion (or Kolmogorov) equation, which is the formal adjoint of the conventional forward advection‑dispersion equation (ADE). By applying a unit flux pulse uniformly over all discharge boundaries, the LE probability density function g(x,τ) is obtained throughout the domain.

The first temporal moment of g, the mean lifetime expectancy E(x)=∫τ g(x,τ)dτ, is shown to be mathematically equivalent to the mean age of groundwater derived in earlier studies, but it is computed by reversing the velocity field and adding a source term equal to porosity. This makes the calculation straightforward in existing ADE solvers: one simply runs the model with reversed flow directions and a porosity source.

Risk assessment proceeds by convolving the LE PDF with a prescribed release history m(t) at any candidate repository location r. The resulting mass flux at the biosphere, b_j(t), is given by a single integral that uses the pre‑computed g(x,τ). Consequently, many “what‑if” scenarios (different repository sites, different release schedules, different radionuclides) can be evaluated without re‑solving the transport equations each time. Radioactive decay is incorporated by multiplying g with an exponential attenuation factor exp(−(ln2/ω)τ), where ω is the half‑life.

To illustrate the method, the authors derive analytical solutions for a semi‑infinite domain containing a set of equally spaced, vertical fractures embedded in a porous matrix. In the fractures, transport is assumed to be one‑dimensional advection‑dispersion; in the matrix, diffusion dominates perpendicular to the fractures. The backward‑in‑time equations for both domains are solved, yielding closed‑form expressions for the first and second moments of LE. Parametric studies show that increasing fracture aperture, spacing, or flow rate reduces the mean LE (i.e., faster migration), while also increasing the variance of LE, reflecting greater uncertainty in travel times.

Key insights and practical implications include:

1. Optimal Site Selection – By mapping the spatial distribution of mean LE, one can identify zones where radionuclides would take the longest to reach the biosphere, thereby selecting safer repository locations.

2. Computational Efficiency – Because g(x,τ) is computed once, subsequent evaluations of different release histories, source terms, or radionuclide half‑lives require only post‑processing, dramatically reducing computational cost.

3. Uncertainty Quantification – The framework naturally accommodates stochastic representations of hydraulic conductivity, fracture geometry, and dispersion coefficients. Monte‑Carlo sampling of these parameters yields probability distributions of LE, providing quantitative risk bounds.

4. Integration with Regulatory Metrics – Traditional safety criteria (maximum travel time, dilution, dose) can be expressed in terms of LE statistics, allowing regulators to set performance targets directly on a time‑based metric.

5. Extension to Real‑World Sites – Although the paper focuses on idealized fracture systems, the authors note that the same adjoint‑based approach can be implemented in full‑scale three‑dimensional groundwater models (e.g., MODFLOW‑MT3D) for actual candidate formations such as the Canadian Shield.

In summary, the study establishes groundwater lifetime expectancy as a robust, probabilistic safety indicator for deep geological waste repositories. By leveraging the backward‑in‑time transport equation, it provides a unified tool for optimal site identification, rapid scenario analysis, and rigorous uncertainty quantification, thereby enhancing the scientific basis for long‑term nuclear waste management decisions.