This thesis explores the application of a Bayesian approach to hyperparameter optimization in surrogate modeling for geotechnical engineering problems. Surrogate modeling, particularly employing Gaussian Processes and Kriging, has become an essential tool for accelerating complex numerical simulations in geotechnical engineering. Traditional Maximum Likelihood Estimation (MLE) approaches to hyperparameter optimization, although straightforward, often overlook the inherent uncertainties in model hyperparameters, potentially leading to sub-optimal prediction accuracy and poor generalization.

The research presented in this thesis investigates the feasibility and benefits of applying a Bayesian inference approach to the tuning of hyperparameters in surrogate models. This approach allows for a probabilistic treatment of hyperparameters, providing a comprehensive quantification of uncertainty. The Markov Chain Monte Carlo sampling method, specifically the No-U-Turn Sampler, was employed to sample from the posterior distributions of hyperparameters, addressing the challenges posed by their non-Gaussian nature and non-linear relationship with model outputs.

Three case studies of varying complexity from geotechnical engineering practice are examined to compare the Bayesian approach against traditional MLE in terms of hyperparameter determination, prediction accuracy, uncertainty quantification, and computational efficiency. The findings suggest that the Bayesian approach, while computationally more intensive, could potentially offer more accurate predictions in terms of the mean-squared error and provide a deeper understanding of uncertainty, which is crucial for risk-informed decision-making in geotechnical engineering.

The study concludes that Bayesian hyperparameter optimization in surrogate modelling holds significant potential for improving the robustness and reliability of predictions in geotechnical engineering, particularly in applications involving complex dependencies and where a thorough understanding of uncertainty is crucial. Further research is recommended to enhance the computational efficiency of the Bayesian method and to explore its integration with multi-point enrichment strategies for practical engineering applications.

Several Geological Disposal Facilities rely on clay-rich formations as the host geological formation or engineered barrier. During the lifetime of a GDF, gasses may be generated primarily due to the anaerobic corrosion of metal, triggering gas migration throughout the natural or engineered barrier. If significant gas pressures build up, preferential dilatant pathways may develop. Laboratory gas injection tests have shown that these pathways significantly affect gas migration by locally increasing the porosity and decreasing the air entry value. Material heterogeneity is one method that has been used to replicate these pathways in numerical modelling. The effect of modelling material heterogeneity on gas transport properties of clay-rich materials has been studied qualitatively, but no fully stochastic analyses have been performed thus far. As a result, uncertainties remain about the quantitative effects of heterogeneity on transport properties of clay. Furthermore, the role of the scale of fluctuation is unclear.
In this thesis, spatially correlated random fields of porosity were generated, and the Finite Element software LAGAMINE was used to model water permeability, gas diffusion and gas injection tests. Results indicate that the time necessary to reach a steady-state water or gas outflow is unaffected by heterogeneity. Monte Carlo analyses of permeability and gas diffusion tests showed that the variability of the effective permeability is higher than that of the apparent diffusion coefficient due to the nonlinear relation between the porosity and the permeability. Additionally, the variability of the effective permeability and apparent diffusion coefficient depends strongly on the ratio between the sample size and the scale of fluctuation. With larger samples, the effects of the random field are averaged, narrowing the distribution of results.
In gas injection tests, water saturation profiles suggest that the onset of desaturation occurs in zones of high porosity, reinforcing the hypothesis that preferential pathway development is related to material heterogeneity. A Monte Carlo analysis of a gas injection test shows that variability in the gas outflow and the degree of saturation not only depends on the scale of fluctuation but also the injection pressure due to the non-linear dependency of the air entry pressure on the porosity.
These results suggest that heterogeneity and the scale thereof are important aspects of gas migration in clay-rich materials. Specifically for laboratory testing of clay-rich materials, it appears that an understanding of the approximate scale of the heterogeneity is essential for estimating the minimum number of samples and the minimum size of those samples. Too small or too few samples will not allow for a reliable estimate of parameters of interest.
Some limitations apply to the findings of this research. First, the sensitivity of the results to factors such as the standard deviation of the distribution underlying the random field and anisotropy has not been addressed yet. Second, the dependency of the effects of the random field on the injection pressure in gas injection tests is not fully understood yet. Ideally, additional simulations are performed to separate these effects. Finally, hydromechanical coupling should be introduced to replicate experimental findings more accurately.