The German site selection procedure for a high-level nuclear waste repository is entering a stage in which preliminary safety assessments have to be conducted and the release of radionuclides has to be estimated for a large number of potential sites.

Here, we present TransPyREnd, a 1D finite-differences code for modeling the transport of radionuclides in the subsurface at geological timescales. The code simulates the processes advection, diffusion, equilibrium sorption, decay of radionuclides, and the build-up of daughter nuclides. We summarize the modeled physical processes, their mathematical description and our numerical approach to solve the governing equations. Finally, two simple tests are shown, one considering diffusion, sorption, and radioactive decay, the other involving diffusion and a radioactive decay chain. In both tests, the code shows good agreement with the reference solutions. Caveats of the model and future additions are discussed.

In the German site selection procedure for a high-level nuclear waste repository preliminary safety assessments have to be evaluated in three consecutive phases.
In the first phase (starting 2020) representative preliminary safety assessments have to be carried out for a large number of potential sites

In general, the large number of radionuclides to consider (several hundreds up to a thousand) make such evaluations difficult.
Apart from the complexity of considering three different types of host rock, the particular challenge in the German site selection procedure is two-fold: First, the number (and area) of potential sites is large, with 54 % of the area of Germany being a potential site after the initial stage of the site selection procedure

At present, a workflow for estimating the transport parameters is set up, which is inspired by the one developed by Nagra for the Swiss site selection procedure

In the literature, several models and codes for assessing the migration of radionuclides out of a deposit exist, e.g.

Our transport model for the representative preliminary safety assessments and its implementation is named TransPyREnd (“

A number of choices have to be made to develop a model for the transport of radionuclides.
Here only one spatial dimension (1D) is considered, that is, we model transport along a line, e.g. in the vertical, horizontal, or any other meaningful direction.
Given the limited available data and the large number of potential sites, this is a justified approximation also applied in other safety assessments, e.g.

Furthermore, the scope of the model is limited to the transport through the geological barriers far from the emplacement area (also termed as the “far-field” in contrast to the “near-field” that would include the technical barriers near the emplacement area). The transport along e.g. mining shafts or other technical elements is currently not included in the model.

Transport is included in the host rock layer and in geological layers above and below (if transport is vertical) or next (in case of horizontal transport) to the host rock layer.
A sketch of an example model domain for vertical transport is shown in Fig.

Regarding the modeled processes, diffusion, advection, mechanical dispersion, and sorption are considered.
Diffusion is assumed to be Fick'ian

Additionally, we include radioactive decay and the formation of daughter nuclides, as required by the regulatory stipulations. The inventory of nuclear waste that has to be stored contains hundreds of different radionuclides: however, many of them can be shown to be not relevant for the effective release of radionclides. For example, a radionuclide with a half-life of hours will not be transported significantly in the designated host rock before it decays, even in a pessimistic scenario. However, its daughter nuclide might be relevant, if it has a long half-life. At present, a simplified nuclide scheme following this line of argument is employed, taking into account 47 radionuclides

The nuclides included in the simplified nuclide scheme of

Several additional physical processes and their coupling are not taken into account which is discussed together with other simplifications at the end of Sect.

Schematic depiction of the model domain. Note that the direction of transport is not necessarily vertical as shown here. Note that while the zoom-in only shows the host rock, all geological layer are discretized.

The model computes the evolution of concentration

The aforementioned processes can be included into a set of coupled differential equations, namely one equation for each of the

Note that we consider transport in various geological layers, so the retardation factor

In the following, we do not consider steady state solutions to the system in Eq. (

In order to solve Eq. (

Illustration of the grid structure.

Accordingly, the transient problem in the time domain is split into specific time-steps.

The solution of transient partial differential equations requires initial and boundary conditions.

We begin by writing down discretized approximations of the right hand side of Eq. (

For the diffusive term, we use the following central difference scheme

The advection term is also approximated using central differences:

Evaluation of the decay and source term in Eq. (

To solve the equation, we need to choose a time-evolution scheme. We use a semi-implicit scheme governed by a meta parameter

The

Inserting our discretizations from Eqs. (

As can be seen from Eq. (

Identifying the right hand side with a vector

Again, it becomes evident here why we treat the decay terms as an external source term evaluated at time step

Dirichlet boundary conditions are used to fix the concentration at the outer edges of the domain (here termed left and right) to a constant value, i.e.

Neumann boundary conditions can be considered as well. For the left boundary at

We can use Eq. (

We assume instantaneous release of the radionuclides from canisters and the emplacement area and neglect solubility limits. Thus, the initial conditions are simply given by the total amount of each radionuclide in the repository and the volume of the repository. Future development will be focused on overcoming this simplification, see Sect.

The matrix Eq. (

Given the sparse structure of the matrix, we employ the

Since the nuclide scheme with its 47 nuclides contains a few linear decay chains and many activation or fission products that directly decay to a stable nuclide, many of the nuclides can be solved for in parallel. For example, all of the activation and fission products (see Table

To exploit this, we make use of the pathos

Predefined time steps can be used with an initial “ramping up” of the time step size from a small initial value to a specified size

In the given geological context, the timescale associated with radioactive decay or advection is usually the shortest. The selection of

As the time step is usually determined by the radioactive decay of the shortest-lived radionuclide in question, a large number of time steps might be required to reach the designated accuracy. For some nuclides, this can become very expensive in terms of computational effort. It is possible to increase both accuracy and performance by applying operator splitting (i.e.

Consider an ordered, linear decay chain with

For illustration, consider a chain with three members,

To employ this operator splitting scheme in the transport model, we drop the original decay terms in Eq. (

Testing TransPyREnd is an on-going effort that will be continuously documented. Here, we briefly show first tests on analytic or semi-numeric solutions.

First, we show a comparison of TransPyREnd with an analytic solution for the transport of nuclides in porous media given by

The parameters used in the test

Comparison between the analytic solution by

We find a good agreement with the analytic solution: The root mean square error (RMSE) is

In Fig.

Convergence of the

Secondly, we show a test involving both the radioactive decay in a decay chain and diffusion to test the mass conservation of our scheme. We arbitrarily place 1 mol of Cm-245 in the center of a domain with a total length of 1000 m and let the system evolve under decay and diffusion for

The simplified Neptunium chain from

Figure

Comparison of a simplified nuclide chain (Actinium chain) between radioactivedecay and TransPyREnd. While radioactivedecay

Naturally, the performance gain grows as the minimum half-life in the problem goes down. For example, for the Actinium decay chain, the allowed time step is of the order of 10 years, resulting in a runtime of 36 s without operator splitting, and 4 s with operator splitting. Enforcing the same time step as before will render the system unstable in this case. Thus, depending on the details of the decay chains in question, the operator splitting scheme can provide a relevant improvement in performance.

In this paper, we have presented the radionuclide transport code TransPyREnd, developed specifically for application in the representative preliminary safety assessments of the German site selection procedure for high-level radioactive waste.
The 1D model includes the processes diffusion, advection, sorption, and radioactive decay in the geological barrier.
For the model we use a nuclide scheme

As noted in Sect.

Our current approach of neglecting these effects is justified by the degree of knowledge about the parameters governing these processes at the current stage of the site selection procedure. However, it is clear that in later stages, these processes need to be carefully addressed to assess their importance to radionuclide migration.

Also, the one-dimensional approach might become less appropriate at later stages, in particular to assess effects arising from complex geometries. While the latter is generally not expected to lead to an increased discharge of radionuclides in the model, the former might have effects in both directions, lowering or enhancing the release of radionuclides in the geosphere/biosphere. At the current stage, the arising uncertainties (i.e.

In the next phases of the site-selection procedure more complex 3D numerical models will be computed.
For this aim the OpenWorkFlow project was initiated by BGE (e.g. see

TransPyREnd will be made public under an Open-Source license.

CB and EL developed the model code, contributed to writing, conceptualizing the manuscript, ran the calculations and created the plots. EL reviewed the manuscript. SM contributed to writing, calculations, and review. MG contributed to writing, review, and created Fig.

The contact author has declared that none of the authors has any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This article is part of the special issue “European Geosciences Union General Assembly 2022, EGU Division Energy, Resources & Environment (ERE)”. It is a result of the EGU General Assembly 2022, Vienna, Austria, 23–27 May 2022.

We would like to thank Olaf Kolditz, Haibing Shao and Renchao Lu for fruitful discussions of the validation strategy. TransPyREnd makes use of Python

This paper was edited by Michael Kühn and reviewed by two anonymous referees.