Uncertainty quantification with a physics-based machine learning method for geothermal-well targeting: A case study of The Hague, Netherlands

Drilling of production and injection wells for geothermal exploitation requires accurate knowledge of subsurface structures and processes related to fluid flow and heat transport, reactive transport, and mechanical processes. Since the number of exploration wells is limited, proper characterization of subsurface properties and structures is challenging. Therefore, quantifying uncertainties is essential for estimating the risk in selecting the location of suitable production and injection wells to increase the chance of profitable outcomes.

Uncertainty quantification is often conducted within a probabilistic framework which demands numerous forward model runs. It poses a computational challenge for geothermal applications since many geothermal subsurface models are high-dimensional due to covering a wide range of parameters and utilizing fine meshes to capture complex structures and address coupled processes. In this study, we introduce the use of the non-intrusive reduced-basis method (NI-RB), a physics-based machine learning method, to enable uncertainty quantification also for high-dimensional models. The NI-RB is a model-order reduction (MOR) technique that expresses a physical solution in a linear combination of basis functions and weights. The NI-RB preserves the structure of the physics in the basis functions and uses a machine-learning model to calculate the weight for each basis function. With this method, we can guarantee physical consistency with respect to a full Finite Element simulation in the prediction phase and gain significant speed-ups to enable probabilistic uncertainty quantification analyses.

As a test of feasibility, we use the model of The Hague case (in the Netherlands) to illustrate an uncertainty quantification with a physics-based machine learning method. With the use of this approach, we gain a significant computational speed-up for predicting a new temperature state, from 10 minutes in High-Performance Computing (HPC) infrastructures with 48 cores to 1 millisecond on a conventional laptop with a single core. It, therefore, enables performing a robust uncertainty quantification on such a high-dimensional model in only 30 minutes with 3 million realizations. The trade-off is on time required for preparing training samples for training the machine learning model. The selected training samples must be representative for the desired parameter ranges. We can, then, characterize the temperature distribution at any location with its uncertainty level. This information is valuable for a careful selection of the location of suitable production and injection wells and estimate the possible economic loss.