Joint Investigations of Structural and Process Related Variabilities using Physics-Based Machine Learning

Geothermal applications, like other Solid Earth studies, suffer from different sources of uncertainty. These may arise, among others, from structural variations, and material properties. Joint considerations of these different sources are challenging since taking structural changes for process simulations into account requires a mesh for the given structural configuration. For the geological model generation, either implicit or explicit techniques are available. Implicit models would allow for an easy adaptation of the structural features but pose challenges in constructing water-tight unstructured meshes, as required for the process simulations. For explicit approaches, already the initial mesh construction is a labor-intensive procedure, potentially generating a couple of hundred thousand to millions of meshes, which are needed for probabilistic analyses, exceeds the typically available resources. Furthermore, fully automatized meshing procedures for complex explicit subsurface models remain an open challenge. In this contribution, we present methods from the field of computer vision, such as subdivision surfaces, to leverage some of these issues.

However, we face another computational challenge: Even if we are able to generate the desired amount of meshes, this does not address the computational burden of the process simulations themselves. Even for simple physical principles, large-scale geothermal models easily require a couple of hours per simulation using state-of-the-art solvers and high-performance computing infrastructures. This makes a probabilistic consideration unfeasible. Therefore, we illustrate in this study the construction of reliable and physically consistent surrogate models via physics-based machine learning methods that capture both the impact of structural variations and material properties on both conductive and convective temperature distributions. The obtained surrogate models typically reduced the computation time for a single simulation to a couple of milliseconds, reducing the computational burden by several orders of magnitude. Nonetheless, we require about a hundred simulations for the construction of the surrogate models. This entails the generation of a hundred meshes and the execution of a hundred simulations. However, this computational cost is significantly lower than the cost for the later analyses. Furthermore, the surrogate generates a continuous representation for the geometry. Consequently, we can represent, for instance, interface positions or dip angles for the fault for which no mesh has been generated, as long as the values are within the pre-defined training ranges.

We want to highlight, especially the convective aspect of the study since most approaches that have been presented so far are applicable to linear problems only. Hence, the transferability of these approaches to nonlinear hyperbolic partial differential equations, as required for hydrothermal studies, is a major challenge. By using the here proposed methodology this challenge is overcome and demonstrates great potential for future applications.