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Scientific reports 12(1), 17491 (2022). doi:10.1038/s41598-022-21739-7, Published by Macmillan Publishers Limited, part of Springer Nature, [London]

Geothermal energy plays an important role in the future energy mix by providing a renewable energy source with a low carbon footprint. This paper presents state of-the-art physics-based surrogate models for geothermal applications, providing predictions for the safe and efficient use of geothermal energy. A proper characterization of the subsurface is subject to high uncertainties, which are often estimated on the basis of probabilistic simulations. However, when dealing with complex, nonlinear physical simulations, a probabilistic approach is often infeasible due to long computing times. In this contribution, we discuss an alternative approach, based on physics-based and data driven machine learning methods in order to reduce computing times, hence, enabling probabilistic analysis including uncertainty quantification. The methods considered comprise the Physics Informed Neural Network (PINN) and the non-intrusive reduced-basis (NI-RB) method as physics-based machine learning approaches, while we rely on a Neural Network (NN) as the data-driven method. We make use of a hydro mechanical (HM) problem to compare their performances based on accuracy and efficiency at both the construction and prediction stage. We find that PINNs are less suitable than NI-RB methods for geothermal applications, especially in situations when only limited measurement data is available. NN requires a large number of training samples to achieve comparable accuracies in the models constructed by both PINN and NI-RB. The NI-RB method delivers the highest accuracy and physical consistency even for a small number of training samples. Therefore, we recommend the NI-RB method for geothermal simulations. With this study, we would like to raise the awareness of the geothermal communities about the careful deployment of machine learning.

Scientific reports 12(1), 4246 (2022). doi:10.1038/s41598-022-08017-2, Published by Macmillan Publishers Limited, part of Springer Nature, [London]

Hydro-mechanical (HM) simulations are essential aspects of geothermal reservoir studies to assess the heat production and the associated-environmental impacts, such as seismicity. HM simulations are numerically expensive (especially for large-scale simulations) since they require a relatively fine mesh to capture the complex interplay between the fluid-flow and geomechanical processes. This aspect makes it difficult to perform detailed studies on uncertainties in HM simulations. In this work, we present a comprehensive review and numerical demonstrations about critical elements in HM simulations for geothermal applications. We then discuss potential surrogate models to reduce the computational cost in performing the simulations, specifically for uncertainty quantification and optimization purposes. There are four important elements in HM simulations for geothermal applications: the equation of state, the porosity-permeability relationship for both the matrix and fracture, the stress-dependent porosity in the matrix and fracture, and lateral and vertical heterogeneities. We compile the discussion from more than 60 papers and numerically show the significance of these parameters using the MOOSE simulator. The incorporation of these parameters into HM simulations leads to realistic descriptions of geothermal applications. However, accommodating for these complex physics also elevates the computational cost. We compile surrogate-modeling approaches (dedicated for HM problems) from more than 40 papers. The approaches span from reduced-basis to polynomials chaos expansion methods and machine learning approaches. We found that the combination of reduced-basis methods and machine learning approaches enables to effectively deal with non-linearity in HM simulations, to preserve the physics, and to reduce computational cost for further uncertainty quantification.

Geothermal simulations are widely used in both scientific and applied industrial contexts. Typically, the temperature state is evaluated on the basis of the heat equation, with suitable parameterizations of the model domain and defined boundary conditions, which are calibrated to obtain a minimal misfit between measured and simulated temperature values. We demonstrate the essential need for global sensitivity studies for robust geothermal model calibrations since local studies overestimate the influence of the parameters. We ensure the feasibility of the study by using a physics-based machine learning approach, that reduces the computation time by several orders of magnitude.

Dissertation, Rheinisch-Westfälische Technische Hochschule Aachen, 2020; Aachen : RWTH Aachen University 1 Online-Ressource (lxxiv, 218 Seiten) : Illustrationen, Diagrammge (2020). = Dissertation, Rheinisch-Westfälische Technische Hochschule Aachen, 2020, Renewable and sustainable energy sources are becoming more and more important. To retrieve these earth's resources, we require reliable simulations. This is, however, a major challenge because of huge uncertainties related to geophysical simulations. Therefore, in this thesis, we focus on improving numerical simulations, to enable the usage of computationally demanding inverse processes, such as Markov Chain Monte Carlo, for complex sedimentary basins that are prohibitive with state-of-the-art finite element solvers. We demonstrate that the reduced basis method is a powerful tool, especially for parameter estimation studies, in the field of Geosciences. We use simple benchmark examples, of a three-layer and a fault model, to better illustrate the employed methods and then use the same methods for the real-case basin-scale models of the Upper Rhine Graben and Brandenburg. For both models, we obtain speed-ups of four to six orders of magnitude, while maintaining an approximation accuracy higher than the measurement accuracy. Hence, we generate approximations that can be equally well used for model predictions and parameter studies as the original full model. We illustrate how the obtained speed-up can be used to perform both deterministic and probabilistic inverse processes as, for instance, global sensitivity studies, model calibrations, and Markov Chain Monte Carlo studies. For these analyses, we are able to reduce the compute time from ten to hundreds of core years to a couple of core hours. The reduced basis method has the advantage that it provides the full state variable, and not only the quantities of interest, such as many other surrogate models. This allows the generation of 2D and 3D uncertainty quantification maps, providing us with interesting insights into the influence of the kind of boundary conditions on the overall spatial distribution of the uncertainties. Opposed to many machine learning approaches, we use a physics-based approach. This allows to incorporate our physical knowledge and to compensate for problems as data sparsity. For the here presented problems, we are able to use the certified reduced basis method. Hence, we obtain error bounds. These error bounds give us a guarantee of our approximation quality and allow for an effective basis construction. We focus in this thesis on geothermal conductive heat transfer problems. However, we also present and discuss the benefits of the reduced basis method for groundwater flow, and electrical resistivity tomography problems. For the transient simulations of the groundwater flow problems, we are able to show that the gained speed-ups are several orders of magnitude higher than those for the steady-state problems. In addition to the parameter estimation studies, we furthermore demonstrate under which conditions the reduced basis method can be used for data assimilation in groundwater studies., Published by RWTH Aachen University, Aachen

One of the biggest challenges in Computational Geosciences is finding ways of efficiently simulating high-dimensional problems. In this paper, we demonstrate how the RB method can be gainfully exploited to solve problems in the Geosciences. The reduced basis method constructs low-dimensional approximations to (high-dimensional) solutions of parametrized partial differential equations. In contrast to other widely used geoscientific reduction techniques, the reduced basis method reduces the Galerkin approximation space, and not the physical space and is consequently much less restrictive. Another advantage of the method is that for the problems considered in this paper, the method provides a bound to the error in the reduced order approximation, thus permitting an objective evaluation of the approximation quality. Using a geothermal conduction problem we demonstrate that depending on the model we obtain a maximum speed-up of three orders of magnitude with an approximation error that is very small in comparison to typical measurement errors. This significant reduction of the cost of the forward simulation allows performing uncertainty quantification, inversions, and parameter studies for larger and more complex models than currently possible.

Geophysical Research Abstracts. - 21 EGU General Assembly, EGU2019, Vienna, Austria, 7 Apr 2019 - 12 Apr 2019; Katlenburg-Lindau : European Geophysical Society EGU2019-16879, 1 Seite (2019)., Published by European Geophysical Society, Katlenburg-Lindau

[EGU General Assembly 2017, EGU2017, 2017-04-23 - 2017-04-28, Vienna, Austria] EGU General Assembly 2017, Vienna, Austria, 23 Apr 2017 - 28 Apr 2017; Geophysical Research Abstracts, 19, EGU2017-8356, 1 Seite (2017).