<figure><img src="image_19.png" /><figcaption>FIGURE 15
Colored fracture aperture distribution at artificial granite block fracture (Jankovec et al., 2019).</figcaption></figure>

heterogeneous dual porosity systems. They concluded that both discrete and continuous adjoint states provide similar solutions. They claimed that the continuous adjoint state is the most versatile form as its non-intrusive character allows for plugging an inversion toolbox quasi-independent from the code employed for solving the forward problem.

Samper et al. (2021, 2022) presented the AS formulation for multicomponent solute transport systems involving mineral phases and cation exchange reactions. Contrary to the partial differential equations (PDE) of multicomponent reactive solute transport for mineral phases and cation exchange reactions, which are highly nonlinear, the PDE's of the adjoint state variables are always linear. The vector of adjoint states, $\bar{\tau}$, for a multicomponent reactive system with kinetically-controlled mineral phases satisfies similar PDE's and algebraic equations as those of the original problem (formulated in terms of solute and mineral concentrations). Sink/source terms of the adjoint states are zero- and first-order linear terms. The

coefficients of the sink/source terms involve the derivatives of the objective function with respect to the concentrations of aqueous primary species and a rectangular matrix containing the first-order derivatives of the precipitation/dissolution kinetic rates with respect to the concentrations of the aqueous primary species (Samper et al., 2021).

The adjoint states, $\bar{\tau}$ and $\bar{\epsilon}$ of the concentrations of dissolved species and minerals, respectively for a chemical system at chemical equilibrium have the following properties: 1) The adjoint states $\bar{\tau}$ satisfy a system of coupled linear PDE's. After solving these equations one can solve for $\bar{\epsilon}$; 2) The time derivative of the vector $\dot{S}_p^T \bar{\tau}$ (where $S_p$ is the stoichiometric matrix of the minerals) is equal to $t$ the derivative of the objective function with respect to mineral concentrations. If such derivative is zero, then $\dot{S}_p^T \bar{\tau}$ is constant in time and varies only in space.

The adjoint states of dissolved and exchanged concentrations for a multicomponent reactive system with multicomponent cation
exchange reactions can be calculated in a decoupled manner. First, the adjoint states of the dissolved concentrations, $\bar{\tau}$, are calculated by solving a linear system of PDE's with constant sink/source terms. These PDE's are solved backwards in time starting with zero adjoint state values at the first time. The adjoint states of the exchanged concentrations satisfy linear equations which are solved in terms of known values of the adjoint states of dissolved concentrations, $\bar{\tau}$ (Samper et al., 2021).

Samper et al. (2009) reported a semi-analytical solution for linearized multicomponent cation exchange reactive transport in groundwater. The resulting linearized cation exchange reactions can be solved analytically. Samper et al. (2021) reported also the AS equations for linearized cation exchange reactions. The adjoint states of a reactive system with multicomponent cation exchange transport satisfy similar PDE's as those of the original problem with constant sink/source terms which depend on the derivatives of the objective function with respect to the dissolved concentrations. The retardation matrix of the adjoint states $\bar{\tau}$ is the transpose of the retardation matrix of the dissolved concentrations.

## 4.2.2 Statistical metamodeling

### 4.2.2.1 State of the art compilation of efforts regarding emulation with machine learning techniques for reactive transport modelling

Although the resulting benefits are potentially huge in terms of computational savings, emulation of reactive transport models has been scarcely addressed so far. Jatnieks et al. (2016) presented a short comparison of a range of nonlinear regression methods for the emulation of the geochemical component of a relatively simple RTM. Leal et al. (2020) demonstrated for a simple reactive transport problem how the geochemical calculations can be efficiently replaced by an online first-order Taylor approximation around the closest previously simulated point by the original geochemical solver. In addition, this approach has been further applied on two reactive transport problems with more complex chemical and geologic conditions in the simulations such as dolomitization process or H2S-scavenging process (Kyas et al., 2022). Huang et al. (2018) proposed to use a lookup table to also replace the geochemical solver of an RTM for a 1D reactive transport problem with two parameters. The studies by Jatnieks et al. (2016), Leal et al. (2020), Huang et al. (2018) are thus concerned with replacing the geochemical component of an RTM by a computationally cheap approximation while assuming the geochemical model parameters to be fixed. Since geochemical calculations commonly take most of the running time of an RTM, this can be very useful when the goal is to perform one or more otherwise CPU-intensive simulation(s) with fixed model parameters. Noticeably, the approach by Leal et al. (2020) includes a check on surrogate prediction quality and automatic replacement by a full physics geochemical simulation of those surrogate predictions that are considered to be inaccurate. Guérillot and Bruyelle (2020) replaced the geochemical solver by a trained neural network (NN) for a 3D case study of CO2 storage in geological formation. Prasianakis et al. (2020) used NNs to either replace the porosity-permeability relationship or predict saturation index from the concentration of two chemical components in a reactive transport code. In a similar spirit of Leal et al. (2020), De Lucia and Kühn (2021)

presented a strategy in which at runtime, full physics geochemical simulations are performed only if surrogate predictions are deemed implausible.

In the framework of the DONUT project, Laloy and Jacques (2022) further advanced the topic by investigating the replacement of the geochemical solver by a machine learning (or data-driven) surrogate for a series of 2D reactive transport problems involving cement degradation. Laloy and Jacques (2022) considered two types of transport conditions, advection and diffusion, and replaced the PHREEQC-3 geochemical simulator (Parkhurst and Appelo, 2013) in the HPx reactive transport code (Jacques et al., 2018) either by a kNN nearest neighbor algorithm (e.g., Hastie et al., 2009) or by a deep neural network (DNN, see, e.g., Goodfellow et al., 2016) trained beforehand. In terms of geochemistry and associated dimensionality of the surrogate model, Laloy and Jacques (2022) considered non-trivial cement systems with C-S-H solid solutions that involve both a 2-input (Ca, Si) - 4-output (Ca, Si, O, H) setup and a 5-input (Ca, Si, Al, C, S) - 7-output (Ca, Si, Al, C, S, O, H) setup. To overcome the observed drawbacks associated with using a purely random training set to build the considered emulators, an approach was proposed where geochemical knowledge obtained from a small (and thus computationally cheap) physics-based RT simulation is used to guide the construction of the training set. This process ranges from simply extracting lower and upper bounds for the input space to identify which portions of the input space should be preferentially represented in the training set. Overall, Laloy and Jacques (2022) found that after training, both our DNN-based and kNN-based codes, HPx-DNN and HPx-kNN, can make satisfactorily to very accurate predictions while providing either a 3-9 speedup factor compared to HPx with parallelized geochemical calculations over 4 brendlers. Benchmarking against single-threaded HPx, these speedup factors become 8-33. Overall, HPx-DNN and HPx-kNN are found to achieve a close to optimal speedup when DNN regression and kNN search are performed on a GPU. Lastly, within the ACED project, Laloy and coworkers (chapter 5 in Samper et al., 2024) further researched the topic by considering better ways of creating the DNN training sets and considering other complex cement systems.

Nevertheless, the above strategy is not suited to tasks involving varying the model parameter values such as sensitivity analysis (SA), uncertainty quantification (UQ), and model calibration (inversion). To address these situations, Laloy and Jacques (2019) compared different machine learning methods, namely, polynomial chaos expansion (PCE), Gaussian processes (GP) and deep neural networks (DNN), for the emulation of the full RTM response to (some) model parameter and boundary condition changes. Here the considered problem was the 1D reactive transport of Uranium VI through a soil column with 8-13 uncertain model parameters. The strategy of emulating the full RTM response permits emulation-based SA, UQ, and inversion (model calibration), and has the advantage of being non-intrusive: no modifications to the RTM need to be made. Note that in principle, one could plug an emulator of the geochemical solver within an RTM and try to build a new emulator on the fly for each evaluated model parameter set within a SA, UQ, or inversion procedure. The technical feasibility and tractability of such approach remain to be investigated though. Moreover, depending on the ML method used to create the surrogate model, the fine tuning or re-training of the model
might require significant computational power, which has to be
taken into account. Turunen and Lipping (2023) revisited the
Hanford case (Saiers and Hornberger, 1996) to study the
feasibility of Convolutional Neural Network (CNN) metamodels
in sensitivity analysis of ¹³⁷Cs propagation in sand. The case is based
on experimental results and corresponding 1-D advection-
dispersion (ADE) model. It was found that in the case of
sufficiently large training set (over 40,000 samples in this case)
the metamodel was capable of emulating the ADE model and the
experimental results with high accuracy. The sensitivity of the model
output to four ADE parameters was studied using Monte Carlo
analysis and the Sobol method. The sensitivity indices obtained
using the first and total order analysis differed significantly
indicating the interaction between the parameters. Also, it was
found that altering the parameter tolerances in Monte Carlo
simulation affected the sensitivity analysis results of the ADE
model but did not affect significantly those of the CNN
metamodel. The work by Sun et al. (2012) reports on the
calibration of 4 first-order reaction rates using a rather simple
RTM that does not account for thermodynamic equilibrium,
inter-species interactions, and sorption processes. In addition,
Sun and coworkers do not mention which emulation method
they use exactly. Keating et al. (2016) resort to multivariate
adaptive regression splines to emulate a model of the multi-phase
transport of CO₂ with non-reactive transport of associated solutes.
Keating et al. (2016) use their emulator for an UP application that
consists of predicting the ensemble behavior of some given model
outputs across the considered parameter space.

## 4.2.3 Experiences on metamodelling based on machine learning

Uncertainty is a common denominator in modelling real-world
phenomena. Uncertainty arises from the variability of the input and
internal parameters of the model. Also, there may be uncertainties
related to the structure of the model. Various frameworks exist to
quantify these uncertainties (see overview by Borgonovo and
Plischke (2016), for example,). In the following, some of the
most common sensitivity analysis frameworks are briefly presented.

### 4.2.3.1 FAST and EFAST

Fourier Amplitude Sensitivity Test (FAST) is a variance based sensitivity analysis method by Cukier et al. (1973). Outputs of the model are converted into coefficients using multiple Fourier Transforms that represent conditional variance. After that, the multidimensional Fourier coefficient integral is converted into one-dimensional integral using the theory developed by Weyl (1938). The frequencies are usually irregular non-integer frequencies and to reduce computation, a set of integer frequencies (multiples of base frequency) are selected. Integer frequencies are selected in a way that enables the error to be controlled. By using integers, the resulting function of one-dimensional integral is periodic and only a single period is needed for evaluation. The continuous integral function can be sampled using Nyquist-Shannon sampling theorem and the resulting sampled function is evaluated from the summation of function values. FAST method is so-called "first-order sensitivity index" method.

Extended Fourier Amplitude Sensitivity Test (FAST) is a
modification of FAST proposed by Saltelli et al. (1999). The

EFAST method estimates total effect indices by computing the
variance of a complementary set to that introduced in FAST.
Also, more flexible sampling scheme is introduced when
compared to the FAST algorithm. EFAST algorithm can produce
both "first-order" and "total-order" sensitivity indices.

### 4.2.3.2 RBD

Random Balance Design (RBD) was first proposed for regression problems by Satterthwaite (1959). Tarantola et al. (2006) modified the RBD method by combining it with the FAST method. While FAST algorithm uses integer multiples of frequencies, RBD uses random permutations of the selected frequencies to generate scrambled set that covers the input space. RBD is a "first-order" sensitivity method.

### 4.2.3.3 EASI

Effective Algorithm for Sensitivity Indices (EASI) was developed by Plischke (2010). EASI algorithm is basically similar to FAST and RBD but the frequencies are obtained by sorting and shuffling the input samples and not the frequencies of the Fourier transform. Input samples are sorted and output samples are re-arranged based on sorted input samples. Sorted data is then analyzed using power spectrum. EASI is a "first-order" sensitivity method, but it can be extended to higher order method by implementing multidimensional search curve to the search algorithm.

### 4.2.3.4 SOBOL

In Sobol method, the output variance of the model is decomposed into summands of input parameter variances for increasing dimensionality (Sobol, 1993; Zhang et al., 2015). Sobol sensitivity analysis determines the contribution of each input parameter and their interactions related to the output parameter variance. While the FAST method uses sinusoidal components of the Fourier method for pattern search, Sobol uses Monte Carlo simulation for that purpose. Sobol algorithm can produce both "first-order" and "total-order" sensitivity indices.

We have recently compared the described sensitivity analysis
methods for evaluating a radionuclide transport compartment-type
model. The model considers a lake-farm scenario at a site of nuclear
waste repository. There were altogether 34 probabilistic variables
describing distribution coefficients ($K_d$) of the model compartments,
features of the modelled lake (area, volume), nutrient root uptake of
plants etc. We found that for ³⁶Cl, for example, when considering the
first-order sensitivity, all the methods ranked the same
parameter-$K_d$ of the soil-as the feature contributing most to the
output variability (i.e., the variability of the dose to humans living in
the modelled site). Also, the three next contributing parameters were
the same for all methods. However, the contribution of the
respective input variables varied significantly for different
methods (from 54.8% to 83.8% for $K_d$ of soil). When applying
total-order sensitivity analysis (EFAST and SOBOL methods), the
difference was even higher (the most contributing variable, $K_d$ of
soil, contributed 28.3% in EFAST and 71.1% in SOBOL analysis).
Recently SOBOL analysis has been used for the uncertainty and
sensitivity analysis of the chemistry of cesium sorption in deep
geological repositories (Ayoub et al., 2020).

Sensitivity analysis using the frameworks described above,
requires the model to be executed repeatedly with parameter
<figure><img src="image_20.png" /><figcaption>FIGURE 16
Using metamodeling in sensitivity analysis.</figcaption></figure>

valued drawn from the corresponding distributions (i.e., Monte Carlo simulation). This is often not feasible as the implementation of the physical models may require excessive computational power. The physical model can, however, be replaced by a metamodel, emulating the input-output relationship of the true model. In case the metamodel is capable of approximating the model function sufficiently well, it can be used to further explore the behaviour of the modelled phenomenon. Also, the metamodel can be used for sensitivity analysis through Monte Carlo simulation (Figure 16). Neural Network based metamodel has been used in Chan and Elsheikh (2018), for example, to emulate a geological Multiscale Finite Volume model.

Recently, a methodology has been developed to trace back the input variables contributing most to the output of a trained deep neural network (Montavon et al., 2018). For example, in the case of applying a Convolutional Neural Network to a 2-D input data matrix, the method enables to determine the significance of each input pixel to the network output. The methodology is generally applicable to all neural network type structures.

A step forward is:

* to compare different sensitivity analysis frameworks in the evaluation of THM(C) models used for modelling the phenomena related to geological repositories of nuclear waste

* to develop and evaluate Deep Neural Network based metamodels in emulating physical THM(C) models for sensitivity analysis through Monte Carlo simulation

* to apply the Relevance Propagation technique proposed by Montavon and Müller (2018) to the sensitivity analysis of THM(C) models using metamodeling.

#### 4.2.3.5 VARS

Sophisticated sensitivity analysis methods have been developed for numerical models based on the analysis of model results. VARS (Variogram Analysis of Response Surfaces) is a variance-based tool to analyse the structure of a response function Y (Razavi and Gupta, 2016). It is linked to the High Dimensional Model Reduction (HDMR) proposed by Sobol and has connections with the Morris elementary effects. Samper et al. (2024) reported the application of VARS and HDMR to the evaluation of the global sensitivities of the predictions of a reactive transport model for a HLW repository in granite. The output geochemical variables Yi considered in the analysis include pH, redox potential and volume of precipitated corrosion products at $t = 10^4$, 2. 10⁴ y 5. 10⁴ years. The input parameters considered include: the corrosion rate of the metallic canister, the bentonite porewater diffusion coefficient, the flow of groundwater through the granitic host rock, the cation exchange selectivity of Fe and the solubility of magnetite. The most influential input parameters for pH are the corrosion rate, the flow in the granite and the cationic selectivity of Fe. The sensitivity indexes of the corrosion rate and the Fe selectivity for pH increase with time while that of the log K magnetite decreases. The sensitivity index of the bentonite diffusion coefficient, however, remains constant in time. The main interactions between parameters for pH occur between the corrosion rate and the groundwater flow through granite. Similar to pH, the most influential parameters for Eh are the corrosion rate and the flow in the granite. The sensitivity indexes of the corrosion rate and the log K of magnetite for Eh decrease with time while that of groundwater flow increases with time. Similar to pH and Eh, the most influential parameter for the volume of corrosion products is the corrosion rate while the second is the diffusion coefficient of the
bentonite. All the sensitivity indexes for the corrosion products increase with time except for the corrosion rate. Sobol and VARS lead to similar results in terms of rankings and significance. However, they show some differences. Morris indexes tend to reach values larger than those of Sobol and VARS the less influential parameters.

#### 4.2.3.6 Surrogate model construction

Uncertainty quantification (UQ) techniques have been extensively developed in the last decade in order to estimate the uncertainties in a Quantity of interest (QoI) caused by uncertainties in model inputs. A cornerstone in UQ is an accurate estimation of the changes in the uncertain QoI caused by changes in the uncertain inputs. The goal is generally to estimate the statistics of the QoI, such as its mean and its variance, and this can be achieved using Monte Carlo technique (and its variants) whereby an ensemble of simulations is performed with randomly selected input data. The convergence rate of the Monte Carlo method is independent of the dimension of the uncertain space but is rather slow (the error in estimating the mean, for example, decreases as the square root of the number of realizations). As a consequence, the number of ensemble members needed to estimate the statistical moments with high accuracy increases dramatically with the number of uncertain parameters and becomes nearly impractical when the forward model is computationally expensive and can only be run hundreds of times (as opposed to millions of times).

So, for a computationally expensive model, the use of a surrogate can be the only way to do an uncertainty propagation problem. The principle of a surrogate (a.k.a. metamodel, emulator, proxy or response surface) is to build, from a finite (relatively slow) number of samples, an approximation of the original forward model able to estimate the response of the QoI to changes in the input. The main advantages of a surrogate are (i) to efficiently propagate the uncertainties in the model inputs in order to compute the uncertainties in the model output, (ii) to identify the dominant contributors to the output uncertainties through the sensitivity analysis, and (iii) to perform parameter calibrations efficiently (the so-called backward propagation) if observational data is available. From a practical point of view, the surrogate is built unintrusively (i.e., without changes to the forward model that is treated as a "black box") at the cost of performing ensemble calculations. The input-output relationships of the ensemble members are in fact exploited to obtain a simplified representation of the model output whose computational cost is negligible in comparison with forward model evaluation.

Two of the most popular and commonly used surrogate construction techniques are polynomial chaos (PC) expansions and Gaussian process (GP) regressions. On the one hand, PC expansion (Ghanem and Spanos, 1991; Le Maître and Knio, 2010) represents the QoI by a finite sum of orthogonal polynomials. Many works have been devoted to improve the computation of the PC expansions including for instance (Blatman and Sudret, 2011; Doostan and Owhadi, 2011; Conrad and Marzouk, 2013; Alemazkoor and Meidani, 2017). On the other hand, in GP regression (Rasmussen and Williams, 2005), a.k.a. kriging (Matheron, 1973) in geostatistics, the QoI is modeled as

the realization of a stochastic process. As well, an abundant literature exists for the GP (Williams and Seeger, 2001; Lawrence et al., 2003; Conti and O'Hagan, 2010; Hensman et al., 2013; Gramacy and Apley, 2015). Some recent studies (Le Gratiet L, 2016; Owen et al., 2017; Roy et al., 2017; Sochala and Iskandarani, 2019), have showed that the two approaches lead to a similar level of accuracy, the differences being on the computational complexity and flexibility.

Suitable methods for constructing surrogate models include polynomial approximation via the stochastic collocation method (Babuška et al., 2007; Nobile et al., 2008) and radial basis functions interpolation (Powell, 2001; Chen et al., 2005). These methods have proven valuable in the context of Markov chain Monte Carlo (MCMC) methods to accelerate the process, as demonstrated by Bliznyuk et al. (2008), Hwang and Martins (2018). further explored challenges in constructing surrogate models with high-dimensional data, a common situation in advanced computational models. One of the motivations for the development of surrogate models is the acceleration of sampling processes in Bayesian inversion. Blaheta et al. (2018a) utilizes polynomial surrogate models derived from the stochastic Galerkin approach. The works Domesová et al. (2020), Domesová (2020) and Blaheta et al. (2020) present practical applications of surrogate models and show applications of polynomial and radial basis functions surrogate models to complex nonlinear models, where the number of observations is limited due to their computational complexity. Furthermore, Domesová (2020) thoroughly investigates the radial basis function model type. These results, along with further analysis, are detailed in the PhD thesis by Bérešová (2022a), which delves into both surrogate model construction and Bayesian inverse analysis, with applications to geoscience problems. The results of employing the stochastic Galerkin approach for constructing polynomial models are discussed in Béreš (2020) and Béreš (2024). These papers present efficient approaches to solving underlying systems, including the very recent Tensor-Train decomposition, with applications to geoscience problems. All these findings are compiled in the PhD thesis by Béreš (2023). Still related to high dimension problem and coupling of different models sparse polynomial chaos expansion has been used by Sochala et al. (2022, 2024) for uncertainty quantification in both pore water composition calculation and model sorption. Doing so it was possible not only to quantify the uncertainties of the quantities of interest for future performance evaluation calculations, but also to identify the main influential input parameters in the pore water models. This latter plays a minor role into the caesium adsorption process.

## 4.3 Uncertainty treatment and sensitivity analysis

### 4.3.1 Compilation of the recent efforts taken for developing sensitivity analysis methods

Sensitivity analysis is a means of treating uncertainties that is generally considered essential in repository performance assessment. By analysing the effects of input variation to the output of performance assessment (PA) models, it allows the
identification of the most relevant uncertainties or, from the reverse
point of view, the exclusion of practically irrelevant uncertainties
from further investigation. It has to be kept in mind, however, that
sensitivity analysis is performed on the model, not on the reality.
Therefore, its results describe the sensitivities of the model and must
not be misinterpreted as statements about the actual sensitivities of
the repository system.

Sensitivity analysis can be performed in different ways. While
local sensitivity analysis investigates the behaviour of the model
system under variation of one specific model condition (typically
a parameter value) keeping all others constant, in a global
sensitivity analysis one varies all conditions or model
parameters that are considered uncertain simultaneously,
executing a higher number of model runs, and analyses the
outcome using appropriate methods.

Local sensitivity analysis is normally performed by executing a
few model runs with deterministically selected conditions. This is
often done if these conditions are not describable by a simple
parameter but represent more complex situations, such as
scenario variants, but also for dedicated investigation of the
influence of a specific parameter. Global sensitivity analysis,
however, is technically limited to parameter uncertainties. The
usual approach to global sensitivity analysis is the probabilistic
approach. It is assumed that the uncertainties of the model input
parameter values can be represented by statistical distributions.
According to these distributions, sample values are drawn and
combined in such a way that the total parameter space is covered
as well as possible. The model has then to be calculated with each of
the sampled parameter combinations. The collected outcome of
these calculations can be mathematically analysed in view of model
sensitivity to individual parameters.

Coming along with the increase of computer capabilities,
probabilistic sensitivity analysis has been a subject of specific
interest in mathematical research during the recent two or three
decades. A variety of more or less sophisticated mathematical
methods have been developed, which produce numerical
sensitivity measures or graphically coded information (Saltelli
et al., 2008; Borgonovo and Plischke, 2016). Different types of
methods act differently to models with different properties.
Repository PA models often exhibit a distinctly nonlinear
behaviour or even discontinuities and the model output is
typically distributed over several orders of magnitude. As a
consequence, such models should not just be analysed using
some standard method as the results might be misleading or do
not reveal the actually interesting details of sensitivity.

The methods of sensitivity analysis applied so far to repository
PA models can be grouped in the following categories:

*   Correlation or regression methods. Such methods linearize the model input-output relation and calculate the parameter sensitivities from correlation or regression coefficients, which are normalised to a range between -1 and 1, negative values indicating inverse sensitivity (higher input -> lower output). For models with a close-to-linear behaviour, these methods produce clear and unique sensitivity statements. The degree of linearity can be assessed via the coefficient of model determination, rsup2;, which is always between 0 and 1. While 1 means true linearity,

a low rsup2; indicates that the model can only poorly be
approximated by a linear approach. This does not necessarily
mean, however, that the calculated sensitivities are
meaningless or of low relevance, as they may represent just
the interesting aspects of the model. Application of such
methods to nonlinear models can be useful, but requires
thorough model understanding.

*   Rank-based correlation or regression methods. For these methods, a rank transformation is performed on the model output, replacing each value by its rank in an ordered list. For nonlinear but monotonic models, this normally leads to significant increase of rsup2; and, apparently, to more significant sensitivity results. It should be kept in mind, however, that rank transformation tends to undervalue high and overvalue low model output values, which might influence the sensitivity statements. Moreover, the rank transformation is non-reversible and causes loss of information.

*   Non-parametrical sensitivity tests. Such tests work by dividing the input sample into two or more subsamples according to some criteria and comparing the corresponding model output. The resulting sensitivity statements are not easy to interpret. These methods should be applied with care and only for specific reasons.

*   Variance-based sensitivity analysis. The variance of the model output that occurs under variation of the input parameters can-under certain circumstances-be uniquely decomposed in contributions caused by the individual parameters and their combinations. The relative contributions to the total variance, the Sobol indices, provide a measure for sensitivity that does not require model linearity or monotonicity. 1st-order indices quantify the influences of the individual parameters alone, higher-order indices describe parameter interactions. The sum of all indices of any order that contain a specific parameter is called the total-order index of that parameter and quantifies its total influence on the model output in interaction with all other parameters. A number of computational methods are available for calculating these indices.

*   Graphical methods of sensitivity analysis. A number of methods have been invented to represent the sensitivities graphically instead of compressing them to a single numerical value. Such figures provide a clear optical impression of the sensitivities and are often more meaningful than a single value. A typical drawback is that many separate figures are needed to visualise the sensitivities of time-dependent models.

*   Density-based sensitivity analysis. This kind of method is based on the idea to compare the conditional and the unconditional probability density of the model output. Such methods work with any model and can improve model understanding but are not easy to apply and interpret.

Since for probabilistic sensitivity analysis with several
uncertain parameters it is necessary to perform a rather high
number of model runs, such investigations can become very
time-consuming and occupy considerable computational
resources. This problem can possibly be mitigated by using metamodels. Metamodels are analytical functions of the parameters, such as polynomials, designed to simulate the behaviour of the actual model under investigation, including its sensitivities. Where the original model might run several hours, a metamodel can be evaluated within milliseconds. So, it is possible to perform a high number of metamodel runs for calculating sensitivity measures. In exchange, however, it is necessary to adjust the metamodel accurately by performing a sufficient number of runs of the original model. Experiments showed that defining and adjusting a metamodel and using it for sensitivity analysis with standard methods in separate steps does not provide actual advantages compared to direct sensitivity analysis of the original model. Specific methods, however, have been developed that combine metamodeling with calculation of sensitivity indices. The main strength of such methods seems to be the ability to effectively compute higher-order sensitivity indices, which is a task difficult to perform with direct methods. This is a field of current and future research.

Already in the 1990s, but specifically in the past decade, some research has been done or initiated aiming at applying different sensitivity analysis methods to different repository PA models and investigating their specific strengths and weaknesses under specific consideration of the typical properties of such models (Sinclair, 1993; Becker et al., 2009; Kuhlmann et al., 2013; Spiessl, 2017). Currently, an exercise is being carried out with a number of integrated PA models and one process model from United States, Germany and Belgium. Teams from France, Switzerland, Finland and Russia recently joined the group either for observation or for active contribution (Röhlig et al., 2018).

## 4.3.2 Uncertainty treatment methods

To sample from the Bayesian posterior distribution, MCMC methods based on the Metropolis-Hastings algorithm are commonly utilized. These methods, while effective, often require numerous evaluations of the underlying numerical model, resulting in substantial computational costs. A strategy to mitigate these costs involves the use of surrogate models in place of the numerical model. However, this substitution can lead to sampling from a modified distribution. To preserve the exact posterior distribution, it is preferable to use the surrogate model within the Delayed Acceptance Metropolis-Hastings (DAMH) algorithm, as highlighted by Christen (2005). Recent literature has introduced DAMH-based sampling processes that iteratively update the surrogate model. For example, Zhang et al. (2013) employed adaptive sparse grid interpolation, while Cui et al. (2011), Cui et al. (2015) developed methods for constructing stochastic models of the error in the approximated posterior distribution and for creating reduced-order models using forward model evaluations.

The papers Blaheta et al. (2018a), Domesová and Béreš (2018) demonstrated that computation costs can be reduced by iterative updating of surrogate models during the sampling process. Recent extensions of this work (Domesová et al. (2020); Blaheta et al. (2020) performed Bayesian inverse analysis on multiple material parameters for complex and costly non-linear models using

DAMH and adaptive surrogate refinement. These outcomes have been incorporated into the PhD thesis by Bérešová (2022a). There is also open-source software available, which accompanies the numerical examples in the aforementioned papers. SurrDAMH (Bérešová, 2022b) is a parallel tool for scheduling sampling procedures for multiple chains in DAMH. Its link with Flow123d (Březina et al., 2024), a library for hydromechanical modelling in fractured porous media, also resulted in software ENDORSE (Březina et al., 2022) for UQ in safety calculations for the excavation disturbed zone of a deep repository of radioactive waste.

Additionally, error-controlled polynomial surrogates derived from the stochastic Galerkin approach were used to fully replace the costly forward model. Geo-inspired examples of Bayesian inverse analysis in this setting were presented in the PhD thesis by Béreš (2023).

We also explored inverse problems outside the Bayesian framework as well. Haslinger et al. (2021) addressed uncertainties and sensitivities in optimal control problems with flux cost functionals using Tikhonov regularization. Malík and Kolcun (2021, 2023) treated the uncertainty in measurements for determining the initial stress tensor by employing spectral analysis of the least square matrix.

# 5 Conclusion and perspectives

DONUT, by providing numerical means for assessment, contributes to: increase the multi-physical description of processes within radioactive waste disposal, and thus, to the ability to manage integrated scientific knowledge in the scope of safety assessment abstraction by being more accurate in quantifying margins, - Understand the consequence of coupled processes and qualify long-term expected safety functions of components, - Improve tools that could be integrated in safety/ performance assessment. Moreover, digitalization becomes a pillar of nuclear waste management (Kolditz et al., 2023). While the frontiers between model and digital twin is a debate (Wright and Davidson, 2020), DONUT has contributed to many scientific bricks in the scope of digital twin. Indeed, a digital twin is developed by, as far as possible, reproducing its multi-physical twin's behavior. As demonstrated the better understanding and modeling of coupled multi-physics phenomena was one of the main goals of DONUT. The digital twin requires the use of high-performance computing, and this was tackled by developing the relevant numerical methods within the project. The digital twin needs model calibration, uncertainty analysis and model reduction, and all these topics have been addressed and advanced. Surrogate models were developed and tested for strong non-linear calculations such as those involving chemistry and reactive transport. At last, but not least digital twin needs to integrate multiscale representation that can be supported by upscaling methods developed within DONUT. The best framework to support the development of digital twins will be investigated in a strategic study entitled "Next-generation Digital Twins to support Optimization, Construction and Operation of surface and subsurface radioactive waste
management facilities” within EURAD 2. Still in this latter context, a R&D work package entiled “High fidElity numeRical siMulations of strongly coupled processes for rEpository syStems and design optimisation with physical models and machine learning” will also be launched.

## Author contributions

FC: Writing-review and editing, Writing-original draft. NP: Writing-review and editing, Writing-original draft. ABa: Writing-review and editing, Writing-original draft. DL: Writing-review and editing, Writing-original draft. GP: Writing-original draft. EA: Writing-original draft. BA: Writing-original draft. GB: Writing-original draft. DB: Writing-original draft. ABe: Writing-original draft. MB: Writing-original draft. SB: Writing-original draft. ZB: Writing-original draft. VB: Writing-original draft. KB: Writing-original draft. JB: Writing-original draft. FC: Writing-original draft. SC: Writing-original draft. MH: Writing-original draft. DH: Writing-original draft. DJ: Writing-original draft. FJ: Writing-original draft. CK: Writing-original draft. TK: Writing-original draft. TKo: Writing-original draft. TKr: Writing-original draft. JK: Writing-original draft. EL: Writing-original draft. JL: Writing-original draft. TL: Writing-original draft. TLI: Writing-original draft. CL-V: Writing-original draft. RM: Writing-original draft. JM: Writing-original draft. MM: Writing-original draft. AM: Writing-original draft. LM: Writing-original draft. VP: Writing-original draft. JP: Writing-original draft. SP: Writing-original draft. ZS: Writing-original draft. JS: Writing-original draft. A-CS-P: Writing-original draft. GS: Writing-original draft. SS: Writing-original draft. KY: Writing-original draft. YY: Writing-original draft. MZ: Writing-original draft. OK: Writing-review and editing, Writing-original draft.

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## Funding

The author(s) declare that financial support was received for the research, authorship, and/or publication of this article. The project leading to this application has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement No. 847593. The work of UDC was funded by ENRESA and the Galician Regional Government (Grant ED431C2021/54). The work of IGN, TUL, ÚJV, CU, and CTU was partly funded by SÚRAO (contract No. SO2020-017).

## Acknowledgments

<figure><img src="image_21.png" /></figure>

## Conflict of interest

Author ABa was employed by TS Enercon.

The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

## Publisher's note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

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