Introduction

Safe geological disposal of radioactive waste relies on a combination of engineered and natural barriers representing a so called multiple barrier approach (Apted and Ahn 2017). Natural barriers, e.g., the host rocks, are chosen to provide stable hydro-chemical-geo-tectonic conditions and to slowdown a potential migration of radionuclides into the biosphere. Depending on the thermo-hydro-chemo-mechanical conditions provided by the host rocks, the system of engineered barriers containing the waste is optimised to ensure the mechanical integrity of waste packages, and to delay possible release of soluble radionuclides into the host rocks and biosphere (IAEA 2020).

The whats and whys of repository systems and coupled processes

Based on its physical-chemical properties and radiotoxicity, radioactive waste is grouped into Spent (nuclear) Fuel/High Level Waste (SF/HLW) and Low-/Intermediate Level Waste (L/ILW) (IAEA 2009). Accordingly, different concepts are used for the design of the either waste repository types.

The SF/HLW is also referred to as heat emitting waste. Depending on the inventory of spallation products and the predisposal history of the SF, its thermal output can have significant implications for the time evolution of the thermo-hydro-chemo-mechanical conditions in the repository. The combination of thermal pulse, mechanical strain, pore pressure build-up, solutes and moisture transport leads to complex transient conditions (Seyedi et al. 2017). These processes are coupled to chemical gradients and heterogeneous reaction fronts evolving in the engineered barrier system (EBS) and even in the adjacent host rocks (Bildstein et al., 2019b; Leupin et al. 2016a).

L/ILW is typically immobilised by using a cement based matrix (Ojovan et al. 2019). Other binders such as bitumen and geopolymers have also been used or are being considered. Cementitious materials are known for their durability, schielding capability, mechanical performance and therefore an important material in many repository designs. These materials have inherently high pH, which would be in chemical disequilibrium with common host rock types considered for the disposal of radioactive waste (Gaucher and Blanc 2006; Wilson et al. 2021). On a time-scale of several hundreds to thousand of years, a cementitious repository is expected to undergo complex chemical interaction with the surrounding host rocks (Blanc et al. 2024). In addition, the degradation of certain waste types and components of the EBS may lead to gas release which affects the saturation state of the repository and the pore fluid pressure (Levasseur et al. 2024; Moreno et al. 2001; Poller et al. 2016; Wendling et al. 2019). Gas release can therefore have a significant impact on the long-term hydro-chemo-mechanical evolution of the barrier system (Leupin et al. 2016b).

Considering the short- and long-term dynamics of the in situ conditions, the description of repository system is divided into near- and far-field domains. Near-field comprises the waste, the engineered barrier system and the adjacent host rocks affected by the repository. Recently, the term disposal cell is used to describe the waste package - barriers - adjacent host rock; whereas near-field may refer also to the combination of the engineered repository system and adjacent host rock (Jacques et al. 2024). The far-field comprises the distant part of host rocks and the biosphere, which, in essence, could be considered to be unaffected by the T-H-M-C phenomena in the repository even in the long term.

Why numerical models matter

The performance and safety assessments of the repositories rely, among various other aspects, on model-based descriptions and simulations of possible repository evolution scenarios. Due to the complexity of the repository systems and the long-time scales involved, modelling is the only way to evaluate the long-term evolution of the repository in situ conditions. For the same reasons reliable model-based predictions of repository evolution are challenging. The roots of the challenges are multi-fold and related to:

  • Intrinsic complexity and, for some phenomena, strong couplings of physical-chemical processes (Laviña et al. 2024; Noiriel and Soulaine 2021; Sellier et al. 2011; Xu et al. 2021).

  • Multi-scale nature of repository processes and the system anisotropy and heterogeneity in several spatial dimensions (Bary et al. 2014; Brough et al. 2017; Molins and Knabner 2019; Shao et al. 2019).

  • Combination of kinetically controlled processes (e.g. claystone oxidation, nuclear glass dissolution, radiolytic degradation) taking place at different temporal scales (Bleyen et al. 2023; De Craen et al. 2008; Vinsot et al. 2012).

  • Simulations of processes at highly reactive interfaces in presence of strong chemical gradients require high temporal/spatial model resolution projected to the long time-scales and large spatial domains controlling overall repository safety (Bildstein et al. 2019a; Damiani et al. 2020; Kiczka et al. 2021).

  • Conceptual differences in the interpretation and understanding of specific couplings (i.e. empirical aspects of chemo-mechanical couplings (Jenni et al. 2019), models variety used for mobility of adsorbed species in bentonite (Idiart et al. 2012; Tournassat and Appelo 2011).

  • Limited computational performance of the existing computer codes and their compatibility with the computer hardware (Zhang et al. 2023).

Why this opinion paper

Importance of numerical models for coupled processes have been recognised since the very first feasibility studies on concepts for geological disposal of radioactive waste (Bonano and Cranwell 1988b; Rechard 1999). The evaluation of repository systems concepts could strongly benefit form models and computer codes developed in the field of geochemistry (Steefel and Lasaga 1994). Still, even nowadays, complex models including several coupled phenomena quickly hit the limits of computational feasibility when the evolution of the system needs to be described in 3D geometries over geological time scales. To cope with such limitations, the models are reduced in dimensions and complexity sacrificing, in many cases, the model realism (e.g. 1D or 2D approximations are applied to describe 3D geometry, multi-component system chemistry is reduced to few basic species, etc.) thus limiting its predictive power. The problem of time scales and multi-scale heterogeneities can be dealt with upscaling and homogenization techniques (Bary et al. 2014). Recent review of the upscaling approaches and strategies for the model simplifications is provided in Govaerts et al. (2022). According to the taxonomy of model abstraction techniques (Frantz 1995), some are based on physical arguments and others merely use numerical approximations developed in computational and data sciences. As described in Govaerts et al. (2022), very often the approximations reported in literature are referred to either as physical or surrogate models.

Physical models describe T-H-M-C phenomena with partial differential equations (PDEs) derived from the corresponding conservation laws. The coupling is manifested either by cross dependencies of T-H-M-C-fluxes or via material properties. In practice, the PDEs are solved on a discrete set of grid elements which, together with the temporal discretization, determine the resolution and the computational performance.

Surrogate models-based simulation approaches have been initially developed by engineers for the sort of problems where the result of interest cannot be readily and directly assessed. Surrogate models are understood as approximate models that behave similarly to the reference ones but consuming considerably less computational resources. For many practical problems, running even a single simulation might take a considerable amount of time. Therefore, basic activities like design exploration, sensitivity analysis, and what-if analysis are not feasible since they need hundreds to millions of simulation realisations. The process is sometimes referred to as model abstraction based on physical arguments or on statistical approaches. The resulting models are interchangeably called surrogate models, metamodels, or emulators (Jiang et al. 2020).

Several national programs on the geological disposal of radioactive waste in Europe are entering their implementation phase. At this stage, the specific design of the multibarrier system and the repository layout will be selected and fixed for the realisation. Numerical modelling has been and will be the basis for the repository optimisation, safety, and performance assessment. Eventually, virtual repository prototypes, so called Digital Twins, will be built to support the construction, and implementation of a repository as well as it will be a tool for operation testing and addressing public awareness (see Editorial introducing the present Topical Collection (Kolditz et al. 2023)). High fidelity models describing the coupled processes will be embedded into digital twins to address the optimisation and safety related questions.

This paper presents the authors’ opinion towards the needs for future development of high fidelity numerical modelling tools for coupled processes in the field of the geological disposal of radioactive waste. High fidelity, in this context, refers to the numerical accuracy, computational efficiency and scalable application dependent level of realism or abstraction. In the following sections, key aspects of the state-of-the-art modelling in the field are discussed, seeking a balance between forward looking and traditional approaches. Based on our experience gained through the active participation in different national research and development programs and throughout collaboration at European level, we have prioritised and outlined some major challenges for the model-based description of coupled processes in repository systems. In Sect. “Description of repository relevant Feature-Events-Processes (FEPs) in argillaceous host rocks”, we summarise the current knowledge on the process coupling controlling the repository evolution. In Sect. “An overview of the state-of-the-art numerical models”, we review the state-of-the-art models developed within the Joint European program on radioactive waste disposal EURAD. In Sect. “Frontiers of realistic repository modelling, design and optimisation”, several applications of such models are discussed. Section “Author’s perspectives on coupled processes modelling” provides a brief summary of model development aspects which have potential to boost the application of coupled codes in the coming decades. As a disclaimer, we emphasise that some part of the manuscript conveys authors’ opinions supported by broad range of scientific publications and personnel experience.

Description of repository relevant feature-events-processes (FEPs) in argillaceous host rocks

The complex interplay of coupled processes in repository systems for SF/HLW in argillaceous host rocks and a cementitious repository for L/ILW is illustrated in Fig. 1.

Fig. 1
figure 1

Summary of coupled processes and their approximate temporal extent in a repository near-field located in a low permeability argillaceous host rock (middle). Semi qualitative consideration of main physical-chemical processes and in situ conditions based on a Swiss repository concept for SF/HLW (left) and cementitious repository for L/ILW (right) (Leupin et al. 2016a, b). EGTS stands for Engineered Gas Transport System. Whereas exact duration of the active phase or peak field values for specific processes (e.g. peak temperature or duration of re-saturation phase, etc. indicated by the colour intensity) depend on details of repository design and the properties of the waste, the detailed sequence of the physical process and their interdependence is characteristic for a wide range of current repository concepts

Full size image

In many countries, the SF/HLW is foreseen to be disposed in thick wall metal casks (e.g. carbon steel casks, copper or coated ones (Abdelouas et al. 2022). These casks, also referred to as canisters, are designed to sustain the corrosion and provide complete isolation of waste matrix from contact with formation water for 104 to even 106 years (in some repository concepts, waste isolation is considered at least for the period of elevated temperature in the near-field caused by the decay heat release). Bentonite-and cement-based materials are used as buffer material to backfill the disposal tunnel and to maintain homogeneous stresses around the canister (Sellin and Leupin 2013).

Decay heat from the waste governs the elevated temperatures in the repository near-field for a long period of time. The time evolution depends on the thermal power of the heat source, the selected repository layout and the thermal conductivity of the materials in the near-field and the host rock at repository depth (Ikonen 2009). The evolution of temperature in the near-field is a complex process depending on a number of factors as heat transport is back coupled to the saturation state of the medium and various source/sink terms related to latent heat of boiling/condensation or dissolution/precipitation reactions. Temperature increase causes transient increase of pore pressure and thermal expansion of the rocks (Seyedi et al. 2017; Xu et al. 2020). The re-saturation of the bentonite backfill is heavily impacted by the thermal transient in a coupled T-H-M-C framework. Depending on the saturation state of the repository and pore water chemistry, the waste canister is subject to corrosion processes which on the one hand releases gas and on the other hand, can lead to waste canister breaching (King and Kolár 2019). The release and migration of radionuclides starts once the formation water gets in contact with the waste matrix (Churakov et al. 2020).

The main physical-chemical processes that govern the evolution of a HLW cell system are thus:

  • Thermal pulse causing desaturation, pore pressure build-up in the near-field, and eventually pore scale mineral precipitation processes near the heat source.

  • Decrease of repository temperature and re-saturation of the barriers.

  • Thermo-hydro-mechanical evolution of the system, including convergence of the underground tunnels and heterogeneous stress field coupled to temperature distribution and saturation state of the near-field (i.e. development of swelling pressure, lateral and radial extent of the excavation damaged zone, pore fluid pressurization due to temperature increase and subsequent mechanical strains, etc.).

  • Corrosion of disposal casks, gas production and eventual access of formation water to the waste matrix.

  • Geochemical changes in buffer and structural materials, causing neoformation of new minerals and dissolution of primary minerals, potentially affecting the microstructure of the bulk material and changing the physical properties.

  • Geochemical interactions between the engineered barrier and the surrounding host rock, with interfacial processes leading to locally profound changes in mineralogy, porosity, or cracking.

  • Radionuclide release from the waste matrix and migration of radionuclides within the backfill and host rock.

L/ILWs disposed in deep geological repositories comprise a wide range of organic and inorganic materials. Their long-term degradation, reactions and mixing with groundwaters would lead to dynamic changes in the repository conditions and radionuclides release conditions (Leupin et al. 2016b). The main materials in a L/ILW repository are hydrated cements, aggregates, steel, and various radioactive waste forms. The evolution of the repository conditions is thus controlled by chemical gradients between waste forms, the cementitious materials used for repository construction, (clay-based) buffers, and host rocks, as well as the internal chemical degradation of cement and waste materials (Neeft 2022). Chemical gradients, in particular those associated with the strongly alkaline nature of cementitious materials, control the direction of diffusive chemical fluxes in the aqueous phase, whereas the principles of chemical thermodynamics govern the stability of the mineral phases that are in contact with pore water solution. These reactions depend on the water availability and are thus strongly coupled to the hydro-mechanical evolution of the system. Degradation of organic waste and metal corrosion are responsible for generation of gaseous species such as low molecular weight hydrocarbons, carbon oxides and, especially, hydrogen.

The major processes and couplings controlling the in situ conditions in the cementitious repository are:

  • Concrete degradation driven by exchange of hyperalkaline pore water solution with quasi neutral pH formation water, which leads to alkalis and calcium leaching from the cement matrix and to a potential formation of sulphate and carbonate-containing expansive products depending on formation water.

  • Carbonation of cement due to interaction of CO2(formation water or organic waste degradation) with the hydrated phases leading to alteration of the cementitious pore structure, accelerating also the corrosion of metals and hydrogen release .

  • Alkali-silica (ASR) and alkali-carbonate (ACR) reactions in concrete.

  • Waste degradation: Abiotic and radiolytic degradation of organic material leads to the release of small labile organic compounds such as CO2 and CH4.

  • De- and re-saturation phenomena: The saturation state of the repository is a delicate balance between water consumption and gas release reactions during the waste and concrete degradation, as well as supply of the formation water from host rocks.

  • Continuous recrystallization of waste matrix, radionuclide release, and waste-cement matrix interaction.

Depending on country specific legislation, if geological conditions and preferred host rocks satisfy the safety criteria for both SF/HLW and L/ILW repositories, a concept of a combined repository can be considered as preferred option (i.e., Belgium, France, Switzerland). To ensure that the repository safety is not compromised, a combined repository has to be designed to exclude mutual effect of FEP (Features, Events and Processes) on both repositories, resulting in even larger number of process couplings and their complexity. The key FEPs responsible for mutual repository influence to be considered in this context are the thermal and hydraulic pulse due to the SF/HLW and the chemical interactions, including the gas release and transport from the L/ILW repository.

An overview of the state-of-the-art numerical models

In this section, we examine the current state of process-based (Sect. “Physics and chemistry-based description of coupled phenomena and scales”) and data-driven (Sect. “Application of machine learning”) modelling approaches of T-H-M-C coupled processes in nuclear waste repository systems. We also elaborate on the benchmarking of coupled codes and opportunities towards collaborative platforms for collective code development in Sect. “Model validation and benchmarking”. The scope of Sect. “Physics and chemistry-based description of coupled phenomena and scales” is deliberately narrowed to the scale dependent processes. This focus is set based on authors’ observation that a significant improvement in the fidelity of T-H-M-C model predictions can be achieved by improving process and model couplings operating at different scales. The discussion of data driven models addressed in Sect. “Application of machine learning” is triggered by the exponential growth of the data volumes coming from monitoring or experimental studies both in terms of process coverage and sensor density. Use of Machine Learning and Artificial Intelligence (ML/AI) for such large datasets can support and improve both the direct and inverse T-H-M-C modelling approaches in future. Further, we provide several examples where ML based models are used to improve computational performance of the T-H-M-C models.

Physics and chemistry-based description of coupled phenomena and scales

Depending on the scientific question and physical-chemical-processes investigated, the system evolution can be modelled with continuum-, pore- hybrid and molecular scale models (Churakov and Prasianakis 2018; Lee et al. 2021; Plúa et al. 2021; Regenauer-Lieb et al. 2013; Soulaine 2024):

Scale dependent description of T-H-C phenomena

When describing the system evolution with partial differential equations at the continuum scale, the properties of the medium are approximated preferably with at least piecewise differentiable functions of space and time (possibly with the exception of the boundary conditions). This system of equations account for mass, energy and momentum conservation laws. The central quantity in this approximation is the Representative Elementary Volume (REV). The REV represents a generalised volume element large enough to provide homogenisation of possible heterogeneities at the smaller scale to ensure the applicability of the continuum scale approximation. REV would be representing surrogate of solids and fluid accessible pores with averaged macroscopic T-H-M-C properties and parameters such as diffusion coefficient and permeability.

In the pore-scale approximation, spatial distribution of individual phases is taken into account explicitly. The phase is a solid or fluid with distinct chemical and physical properties. Depending on the model, these phases could be pore space filled with multi-phase multi-component fluids and solids. The interaction between phases is directly defined at phase boundaries (interfaces) and may lead to the alteration of the pore space subject to dissolution/precipitation reactions. Considering the fact that any phase may have intrinsic heterogeneities, the pore scale approximation embarks on the “continuum scale” approach when describing properties of the individual phases in the system.

In the molecular scale models, the interaction between ions and molecules in fluid and solid phases is taken into account explicitly. Once again, depending on the level of abstraction and detail, these models range from the explicit consideration of electronic structure based quantum mechanical approach to coarse grained ones in which larger molecular segments are described as effective interaction sites (particles). Further simplifications are possible by employing effective media approach such as the dielectric continuum approach for solvent.

The choice of the optimal model and scale for the system description should be driven by the scientific question. Continuum scale modelling neglects spatial distribution of individual phases of the material, which can be critical if phase changes have strong effect on material properties. Pore scale modelling, on the other hand, is capable of tracking the interfaces between individual phases and provide the evolution of system parameters at REV scale. Molecular scale models can deliver the properties of individual phases and phase interfaces. Thus, further development of the computational tools and the models should be aimed at multiscale model coupling in which lager scale models define boundary conditions whereas the small-scale models provide material properties necessary for the accurate system description (Molins and Knabner 2019). Major limitations of multiscale modelling frameworks are related to computational effort and the cross-scale model coupling, which would not be solved by increase of computational resources alone. New efficient algorithms and the scale coupling-decoupling schemes should be in the first priorities of future developments.

Coupled reactive transport models for repository systems

The so-called reactive transport models include the coupling between fluid flow, mass transport and geochemistry. Such models are well established (Steefel 2019). Numerous codes exist at the Darcy or continuum scale (Steefel et al. 2015). Several computational benchmarks relevant to deep repositories have been published (Aguila et al. 2021; Idiart 2019; Marty et al. 2015; Poonoosamy et al. 2021). The corresponding models have been applied to simulate the evolution of barriers in deep geological repositories during the recent EU projects (e.g. EU CEBAMA (Duro et al. 2020) and work packages ACED (Jacques et al. 2024) and DONUT in EURAD (Claret et al. 2022). In EURAD WP-ACED, the components of repository systems are modelled at the scales of interfaces, waste packages and disposal cell, e.g. (Blanc et al. 2024; De Windt et al. 2024; Lemmens et al. 2023; Mon et al. 2023; Montenegro et al. 2023; Wittebroodt et al. 2024a; Wittebroodt et al. 2024b ).

More recently, coupled reactive transport models have been implemented at the pore-scale as well. Such models are able to relate pore-scale microstructural changes due to geochemical reactions to effective parameters that can be applied to the continuum scale. Examples of such approach, among many others, are given in Molins et al. (2017), Patel et al. (2018), Prasianakis et al.  (2017, 2018), Seigneur et al. (2017), and Varzina et al. (2020).

In situ conditions and material properties of the EBS system evolve due to the thermal, hydraulic, mechanical, biochemical and chemical gradients that exist within and among the different repository components. Within the EURAD project, two work packages have focussed on these changes, i.e. geochemistry induced changes (ACED WP) and coupled chemo-mechanical processes (MAGIC) (Claret et al. 2022). For the geochemical effects on the EBS properties, Neeft et al. (2022) and Deissmann et al. (2021) gave an overview of processes and available models at the interface scale. Several coupled reactive transport models have been used to evaluate the evolution at the interface of two materials (e.g. Idiart et al. (2020b), Marty et al. (2015), Savage et al. (2010) and also Bildstein et al. (2019a) for a recent overview). Recent coupled reactive transport models have been successful in coupling all relevant materials in a disposal cell containing high level vitrified waste: nuclear glass-steel-cement/bentonite-host rock (granite or clay) and reported in De Windt et al. (2024) and Montenegro et al. (2023). Also recently, hydro-chemo-mechanical modelling has been used to study the evolution of the Cigéo repository closure systems based on bentonite-based sealing components surrounded by cementitious materials (Idiart et al. 2020a; Laviña et al. 2023).

Chemo-mechanical couplings in massive concrete infrastructure have been addressed in WP MAGIC using multiscale simulations ranging from nano to cell scale models (Dauzères et al. 2022). Reactive transport codes and mechanical codes can be sequentially coupled to model cementitious material damage due to carbonation (Socié et al. 2023). The hydro-chemo-mechanical variational phase-field fracture approach, for example, is capable of handling chemical reactions, as well as the resulting material dissolution and/or precipitation caused by hydration or degradation (such as carbonation) of fractured cementitious materials (Zhang et al. 2018). Pore scale simulations are used to investigate microstructure evolution and to estimate the effective mechanical parameters of the media (Shen et al. 2020).

Application of machine learning

The computational time in a multi-physics modelling framework is often dominated by only a few or even just a single process. For example, it is well known that most of the computation time in coupled T-H-C reactive transport models is taken by the solution of the geochemical equations. Recently, several efforts have been made to replace these computationally expensive routines by cheap surrogate models often based on machine learning techniques or look-up tables. Huang et al. (2018) described the complex geochemistry of ageing cementitious waste resulting from carbonation and alkali-silica reactions via a look-up table approach to a multi-phase transport model of a concrete structure. Furthermore, emulating the geochemical models using machine learning techniques and by incorporating them into reactive transport models have been tested and implemented in some recent studies (De Lucia et al. 2017; Demirer et al. 2023; Huang et al. 2018; Jatnieks et al. 2016; Laloy and Jacques 2022; Leal et al. 2020; Prasianakis et al. 2020). Prasianakis et al. (2020) have shown several examples for successful use of machine learning for upscaling pore-scale models (lattice Boltzmann model) to a continuum (Darcy)-scale reactive transport model. Applications at the Darcy-scale show a gain in computational time of about an order of magnitude, while geochemical calculations can be accelerated between one and four orders of magnitude. Alternatively, a surrogate can be made to replace the complete model including all coupled physical processes. Recent applications in the field of coupled reactive transport are for uranium transport (Laloy and Jacques 2019) and electro-kinetic bioremediation (Sprocati and Rolle 2021). Machine learning techniques are also becoming popular for approximating the stress-strain constitutive behaviour of geomaterials, including artificial neural networks and genetic programming (Gao 2018). Graf et al. (2010) trained recurrent neural networks (RNN) with time-dependent data to assess the long-term behaviour of a reinforced concrete structure subject to mechanical and environmental loads. Capuano and Rimoli (2019) and Logarzo et al. (2021) also used RNN with time-history prediction capability to replace inelastic homogenization (i.e. multiscale) behaviour of materials consisting of a soft elastoplastic matrix with stiff elastic inclusions (e.g. concrete). Conti et al. (2018) applied concepts from data science to materials science with the so-called data-driven paradigm, consisting of reformulating the classical initial-boundary-value problems directly from material data.

Many numerical algorithms are mature enough to provide coupled description of multiphysics process, but are typically limited at a specific length- and time scale. The T-H-M-C processes that govern the repository evolution are intrinsically multiscale, comprising phenomena from the atomistic level (e.g. chemical sorption and ions mobility) to the macroscopic level. Furthermore, small-scale processes can have a strong effect on the system evolution at repository scale. In addition, the numerical algorithms use different programming languages (e.g. Fortran, C/C++, CUDA, Python, to name a few). Merging the different codes is a very challenging task. Machine learning techniques can be used to support the communication across the algorithms since the multi-dimensional complex output of the models may be represented by regressing a simple mathematical object, e.g. neural networks. At the same time, and for specific applications, it seems reasonable to create surrogate models, trained on the full physical algorithms, each at the respective length scale and to subsequently integrate them in order to accelerate the overall multiscale calculations.

The implementation of accelerated numerical methods and software engineering should go hand in hand and tuned to the future oriented development of High Performance Computing (HPC) infrastructure. This is not always the case for existing scientific software (Grannan et al. 2020). For example, Graphical Processor Units (GPU) based HPC systems have dominated the HPC landscape since nearly 10 years (www.top500.org). In contrast, rather few multi-physics codes are able to take full advantage of such architecture. Our assessment suggests that currently the development of coupled codes stays behind the progress in hardware development. Full exploitation of HPC infrastructure would enable faster and more complex calculations than currently feasible. This would provide a basis for development of digital twins of T-H-M-C coupled processes, and for integration of data collected prior and during the design, operational and/or the post closure phases of repository systems.

Model validation and benchmarking

Benchmarking plays an important role in the conceptual and numerical development of coupled T-H-M-C codes, where conservation laws (continuum mechanics), thermodynamics (e.g., equations of state), material behaviour based on highly non-linear constitutive laws, and chemistry (law of mass action, Gibbs energy) need to be considered simultaneously. Although the benchmarking of coupled process models has made great progress in recent decades, particularly in the analysis of laboratory experiments, a high degree of uncertainty remains from structural to practical applications (modelling of real repositories). The benchmarking of coupled processes started in the 1980s with the INTRACOIN, HYDROCOIN and INTRAVAL initiatives (Herbert et al. 1988; Konikow et al. 1997; Larsson 1992). Unlike previous benchmarking initiatives, DECOVALEX is an ongoing and growing project that has united a large community of modellers for more than 25 years for the development of coupled models and their validation against experiments (Birkholzer et al. 2018, 2019; Chen et al. 2009; Jing et al. 1995). DECOVALEX focuses mainly on near field thermo-hydro-mechanical processes. Recently a performance assessment task has been integrated into its portfolio. In this respect, the benchmarking initiatives MoMaS and SeS Bench for reactive transport processes have been launched in the past (Aguila et al. 2021; Bildstein et al. 2021; Carrayrou et al. 2010; Poonoosamy et al. 2021; Steefel et al. 2015). The combination of T-H-M and reactive transport initiatives is still not available, which also shows a lack of a truly multidisciplinary benchmarking initiative uniting geomechanics and geochemistry. Currently, this limitation still hinders the full potential of T-H-M-C code development (Kolditz et al. 2018). There is a large body of literature describing individual benchmarking efforts. A more systematic way of organising T-H-M/C has been developed in a book series linked to data repositories where input files and related code versions are stored (Kolditz et al. 2012). More recently, online versions of benchmark collections have become available, e.g. via the benchmarking galleries (https://www.opengeosys.org/docs/benchmarks/) and GeoML (https://geoml.eu/), allowing easier reproduction of benchmarks even offering collaborative interactive work environment via JupyterLabs (see also Future oriented collaborative platforms below). In authors’ opinion, support and development of such initiatives at European level (i.e. within EURAD research program) and its leverage by associated international research groups have high added value for the progress of scientific collaboration towards FAIR (Findable, Accessible, Interoperable, Reusable) research data principle (Wilkinson et al. 2016) and quality assured code development.

Besides benchmarking of numerical T-H-M-C coupled codes, equally important is the validation studies of the implemented models with available experimental data (Addassi et al. 2022; Pelegrí et al. 2023). For bentonite, which is an essential EBS component, recently compiled experimental databases are available in the literature and can be used as a means for collaborative T-H-M-C model validation (Cabrera et al. 2023; Thatcher et al. 2017).

Frontiers of realistic repository modelling, design and optimisation

In this section, we explore the present challenges of coupled process simulations, multi-physics, and chemistry relevant for the geological disposal of nuclear waste. Specifically, we discuss the development and application of advanced T-H-M-C models for inverse modelling, repository design optimisation and sensitivity analysis of model parameters. These are, in our opinion, the key areas for further development, driven by the needs of repository implementation. Aiming to exploit the full potential of the most powerful computational resources, we address the necessity of the coherence between the design of software packages and the HPC architecture. Finally, we elaborate on the advantages in the use of collaborative platforms for code development.

Optimisation, uncertainty analysis, and inverse modelling

Process-based numerical simulations are the basis for in-depth system understanding, analysis of experimental observations and their upscaling. Despite the continuous growth of the computational resources, the realism of the models applied in the simulations of repository systems remains limited in terms of dimensions, time-space resolution and process couplings. Interpretation of experimental data, safety and cost-driven design optimisation, and model uncertainty analysis belong to the class of inverse problems. Numerical solution of inverse problems implies iterative forward modelling until the solution converges to the optimal parameter set (e.g. satisfactory description of experimental data, cost-safety optimisation of repository design, or uncertainty analysis).

For both forward and inverse problems, orders of magnitude improvement in the computational efficiency can be obtained by replacing the physical based solvers or its components with high fidelity surrogate models (Sect. “Application of machine learning”). Particularly promising are the surrogate models based on machine learning for specific aspects of T-H-M-C coupled models, data exchange between models at different scales, reduction of big data and extraction of constitutive relations from large numerical, experimental, and monitoring datasets (Elodie et al. 2020; Hu et al. 2023, 2024; Ringel et al. 2024).

Abstraction and simplification methods

For a given numerical simulation problem with multiple processes, some processes or elements from the conceptual model might have only a limited effect on a model output (Frantz 1995). Model abstraction refers to a systematic method to reduce the complexity or the computational burden of the model while maintaining the validity of the simulation results with respect to the question that the simulation is being used to address. Model abstraction reduces the simulated system to its essential components and processes through a simplification of conceptual (sub-)models, selection of significant processes and appropriate time and spatial scales or more computationally efficient implementations (of specific model components and processes). In its most extreme form, the model is stripped down to a single component which just reproduces the desired output from the input in a computationally efficient way (Govaerts et al. 2022).

The classification of techniques described in Govaerts et al. (2022) are based on previous work by Pachepsky, (2006) and Razavi et al. (2012). A group of simplification methods is labelled lower fidelity numerical models and uses strategies as a pre-defined hierarchy of models, delimiting the input domain, the scale change and reducing numerical accuracy. A second group is based on statistically derived surrogate models which were discussed above. Hierarchy of models are developed for flow and transport in fractured porous media (Berre et al. 2019), T-H-M processes concerning mechanical barrier integrity (Pitz et al. 2023), diffusion in charged porous media (Hennig and Kühn 2021), or surface complexation modelling (Arora et al. 2018). Reducing the computational costs can be done by lowering the dimensionality of the problem (e.g. (Idiart 2019)) or sub-grid scale refinement (e.g. (Finsterle et al. 2020; Mariner, 2020)), but reduction of chemical system or coupled processes (e.g., the relevance of porosity feedback in Aguila et al. (2020)) are other options as well. A recent example of upscaling methods is to represent fractured nuclear glass as an effective medium (Repina et al. 2020).

Global sensitivity analyses

Treatment of uncertainties in the performance assessment of deep geological disposal has been recognized as an important topic for more than three decades (Bonano and Cranwell 1988a). In order to quantify the effect of parameter variation on predicted system performance, the local sensitivity analysis (LSA) and global sensitivity analysis (GSA) methods are relevant. While LSA determine the impact of small input perturbations around nominal values on the model output, GSA considers simultaneously the whole combinational variation range of the inputs. While both methods could provide relevant information for T-H-M-C coupling, GSA account for non-linearity and interactions among parameters in system responses in a more robust manner (Chaudhry et al. 2021; Delchini et al. 2021; Nguyen et al. 2009; Wainwright et al. 2013). Recently, GSA have also been used to tackle reactive transport problems and radionuclide migration (Ayoub et al. 2020). Surrogate models can also be used to decipher uncertainty propagation (Sochala et al. 2022) and for sensitivity analyses. The use of surrogate models can circumvent an issue related to GSA. Indeed, GSA requires many model evaluations to achieve satisfactory accuracy, which will lead to a great challenge in computational efforts for large models, which is of particular relevance for complex coupled T-H-M-C processes in repositories. Even running surrogate models can be remarkably challenging and even computationally prohibitive in the case of intensive simulations and large-dimensional systems and necessitate the use of reduced space surrogates (Vohra et al. 2019).

Software development and HPC infrastructure

Software engineering plays an increasingly important role in scientific projects. The main drivers are the complexity of the tasks to be tackled, e.g. multi-physics, multi-scale approaches for coupled processes, real-world application with complex geometries and the associated computational effort requiring the application of high-performance computing technologies (Park et al. 2021; Trinchero et al. 2017). We believe that, meeting these challenges, large international multidisciplinary development teams are needed for distributed development, so open source projects based on version control, continuous integration and code review have become a foundation of recent research software (Bilke et al. 2019; Bjorge et al. 2022; Fiorina et al. 2022). Quality-assurance of the software for safety assessment applications in nuclear waste management is particularly important, and require transparency, traceability and reproducibility of results. New software technologies are currently making their way into geoscience applications, such as container technologies for portability of complex software projects, automated workflows for solving complex tasks involving large application data, or automated benchmarking workflows (Lehmann et al. 2023). Modern software projects also make intensive use of ecosystems such as Python or Julia and integrate them into the entire workflow, i.e. pre- and post-processing as well as simulation kernels. This requires appropriate application interfaces (APIs) such as ogstools (e.g. (Buchwald et al. 2021). Professional software development and deploying technologies such as Virtual Reality are key to the successful implementation of digital twin concepts for nuclear waste management in the future (Kolditz et al. 2023).

The design and development of hardware for current and future HPC systems are shaped by several physical and economical challenges (Grannan et al. 2020). Modern HPC systems need significant amount of energy supply for operation and cooling, meaning that maximizing energy efficiency in their operation and usage is more important than ever. The miniaturized design of the hardware is reaching the physical limits of performance that can be achieved on each individual computer chip. The new generation of HPC (www.top500.org) increasingly combines Central Processing Unit (CPUs) with dedicated accelerators based on GPUs. Such hybrid CPU/GPU systems are energy efficient and provide, in theory, many more flops per watt of consumed energy (Ashraf et al. 2018). New programming models and software optimization are indispensable to be able to exploit the theoretical performance of such systems, considering the different hardware architecture between CPUs and GPUs; GPUs are composed of hundreds to thousands of cores focusing on parallel processing and high throughput at lower clock speed (Fiore et al. 2018). As a matter of fact, many widely used software packages and scientific applications developed for CPU-only systems are hardly able to benefit from the hardware potential offered by hybrid CPU/GPU systems, since dedicated GPU compatible source code has to be generated and compiled.

The use of artificial intelligence (AI), dynamic data processing and integration of data into the models are other changing aspects of scientific computing. Traditional physics driven modelling accept relatively small input datasets describing initial and boundary condition. Contrary, the AI and dynamic data integration strategies often require the management of massive data streams composed of millions of heterogeneous datasets, meaning that input and output algorithm for distributed file systems must be optimized for data processing. Whereas the high processing speed of data on CPU or GPU permits more complex ensemble simulations or multi-physics models, it also means generating ever-larger datasets for postprocessing and analysis.

Future oriented collaborative platforms

The landscape of computational algorithms relevant to machine learning is very broad and the respective software is updated at a very fast pace. This makes the consistent installation and use of these tools to be a very challenging task for scientists without scientific computing expertise, who however are interested to use and explore the machine learning potential. Moreover, in collaborative projects with many research partners there is no common working space which could allow to program and test algorithms in a collaborative way. Centralizing the efforts in providing an open access web-server based collaborative platform reduces duplication of work and reinvention of the wheel. As a response to these challenges, the www.geoml.eu open platform has been launched recently from the EURAD-DONUT and PREDIS work packages as a vehicle to enhance collaboration, education, joint code development and demonstration of results. A jupyter lab server, having pre-installed all necessary computational packages, allows to program, share and test numerical codes for typical classes of problems without the need of local computational resources, or of the installation of computational environments. Such platforms provide means to communicate results to the scientific community and the public in form of online interactive demonstrators significantly enhancing the outreach of scientific results. Similar platform has been recently launched for coupled process modelling as well (e.g. https://www.opengeosys.org/docs/benchmarks/).

Author’s perspectives on coupled processes modelling

Many highly reliable conceptual, mathematical and numerical models for physical and chemical processes at different scales are described in the literature. In authors’ opinion, the process coupling still remains challenging in some cases due to a combination of factors which include:

  1. 1)

    Different T-H-M-C processes which are often relevant at different spatial and temporal scales;

  2. 2)

    The computational effort rises exponentially with increasing number of coupled processes and the dimensionality of the system;

  3. 3)

    Increasing the level of coupling makes their conceptual modelling more difficult, sometimes with a clear lack of sufficient experimental data to parametrize and validate such coupling schemes;

  4. 4)

    Increasing level of coupling makes interpretation, visualization, and communication also much more challenging

The development of computer hardware is driven by energy efficiency and scalability of distributed systems. Efficient use of such a new hardware cannot be fully exploited without conceptual redesign of numerical code for specific HPC infrastructure. Machine learning and surrogate models development have been identified as powerful and promising approaches for acceleration of numerical simulations. Such models could provide computational speed-up for individual processes, improving the efficiency of optimisation involving numerically fast, albeit less accurate, surrogates for parameter sensitivity analysis and pre-optimisation in the inverse modelling workflows. Furthermore, the surrogate models, being more simple and robust parts of a code, can be more easily transferred between different hardware architectures and take advantage of dedicated accelerator devices and other future emerging hardware solutions.

It is now broadly accepted that the development of efficient numerical codes for simulation of coupled T-H-M-C processes in the repository near field is essential for the design and optimisation of repositories. These topics are heavily addressed in the context of EURAD Joint Program, specifically focusing on the development of high fidelity/high computation throughput models for multiscale simulations of coupled T-H-M-C processes in repository near-field and far-field. The future oriented modelling tools will be based on both physical and surrogate models with primary application for inverse modelling and large-scale simulations for repository design optimisation with respect to features, events, and processes (FEP catalogue) as well as performance and safety assessments as part of repository licence applications.