In geological characterization, the traditional methods that rely on the covariance matrix for continuous variable estimation often either neglect or oversimplify the challenge posed by subsurface non-stationarity. This study presents an innovative methodology using ancillary data such as geological insights and geophysical exploration to address this challenge directly, with the goal of accurately delineating the spatial distribution of subsurface petrophysical properties, especially, in large geological fields where non-stationarity is prevalent. This methodology is based on the geodesic distance on an embedded manifold and is complemented by the level-set curve as a key tool for relating the observed geological structures to intrinsic geological non-stationarity. During validation, parameters 𝜌 and 𝛽 were revealed to be the critical parameters that influenced the strength and dependence of the estimated spatial variables on secondary data, respectively. Comparative evaluations showed that our approach performed better than a traditional method (i.e., kriging), particularly, in accurately representing the complex and realistic subsurface structures. The proposed method offers improved accuracy, which is essential for high-stakes applications such as contaminant remediation and underground repository design. This study focused primarily on twodimensional models. There is a need for three-dimensional advancements and evaluations across diverse geological structures. Overall, this research presents novel strategies for estimating non-stationary geologic media, setting the stage for improved exploration of subsurface characterization in the future.

A document by Eungyu Park. Click on the document to view its contents.

Subsurface characterization remains a pivotal challenge in geology, hindered by the inherent complexity and nonstationarity of geological processes. Conventional geostatistical methods, relying on two-point statistics and kriging, often fall short in capturing the spatial heterogeneity and transitional dynamics of subsurface materials. This study introduces a novel three-dimensional lithofacies characterization method that incorporates manifold embedding and a transition probability-based spatial estimation technique, significantly deviating from conventional approaches. The proposed method addresses existing limitations by providing a robust framework for capturing the nonstationary nature of geological processes, utilizing data such as pole-to-plane orientations and lithological transitions for structural and juxtapositional information. Through hypothetical scenarios and simulations, the performance of the proposed method is demonstrated, showcasing its capability to model complex geological formations and accurately estimate subsurface characteristics. The study underscores the importance of considering nonstationarity in geological estimations and highlights the potential impact on hydrogeological modeling. Accurate lithological distribution estimation is crucial for reliable groundwater flow and solute transport modeling. This study advances the methodological toolkit for subsurface characterization and paves the way for future research, including empirical validation with real-world data and exploration of the method's applicability in early-stage site characterization, where data scarcity and quality are pressing concerns.

The concept of optimality in geostatistical inversion is traditionally rooted in a Euclidean framework, which often oversimplifies complex spatial relationships in geological structures, potentially leading to suboptimal representations of subsurface properties. This study challenges this conventional approach by introducing manifold embedding, a method that incorporates non-Euclidean geometries to better capture the complex and non-stationary nature of geological structures. Through the application of Kalman filtering (KF) and geostatistical principal component adaptation evolution strategy (GPCA-ES), we explore how different geometric frameworks influence the outcomes of hydraulic conductivity estimations in a synthetic aquifer. Our results demonstrate that while Euclidean-based methods may provide a single “optimal” solution, they do not necessarily yield the most geologically accurate models. By incorporating manifold geometries, we reveal a broader range of plausible subsurface interpretations, all of which produce similar hydraulic responses at observation points. This finding highlights the limitations of relying solely on Euclidean assumptions and challenges the conventional notion of a unique optimal solution in hydrogeological inverse problems. The study underscores the importance of adopting a more comprehensive geometric perspective in hydrogeological modeling, offering a pathway to more geologically meaningful and potentially more reliable subsurface characterizations. These findings advocate for a fundamental shift in the approach to geostatistical inversion, emphasizing the need to move beyond traditional optimality criteria and toward a more nuanced understanding of subsurface environments that acknowledges the inherent complexity and non-uniqueness of geological structures.

Constant head injection (CHI) tests are a standard method for estimating subsurface hydraulic properties, but traditional interpretations using analytical solutions like Moye’s equation are limited by their inability to capture transient dynamics and vertical heterogeneity. This study introduces a novel transient inverse method that allows for the estimation of layer-specific hydraulic conductivities (K) in multilayered aquifers, accommodating realistic boundary conditions. Using synthetic simulations with varying specific storage coefficients (S_s) and observation times (t_rep), the method consistently produced accurate K estimates and demonstrated robustness to S_s variations. In contrast, Moye’s equation significantly underestimated K values, highlighting the limitations of steady-state approaches. The proposed method offers improved flexibility and accuracy for CHI test interpretation, making it a valuable tool for more complex, real-world aquifer systems.

In this study, the combined application of geodesic kernel and Gaussian process regression was investigated to estimate nonstationary hydraulic conductivity fields in two-dimensional hydrogeological systems. Particularly, a semi-analytical form of the geodesic distance based on the intrinsic geometry of the manifold was derived and used to define positive definite geodesic covariance matrices that are employed for Gaussian process regression. Furthermore, the proposed approach was applied to a series of synthetic hydraulic conductivity estimation problems and the results show that the incorporation of secondary information, such as geophysical or geological interpretations, can considerably improve the estimation accuracy, especially in nonstationary fields. Moreover, groundwater flow and solute transport simulations based on the estimated hydraulic conductivity fields revealed that the accuracy of the simulations was strongly affected by the inclusion of secondary information. These results suggest that incorporating secondary information into manifold geometry can remarkably improve the estimation accuracy and provide new insights on the underlying structure of geological data. This proposed approach has crucial implications for hydrogeological applications, such as groundwater resource management, safety assessments, and risk management strategies related to groundwater contamination.