Marthe Faber

Accurate and efficient modelling of geothermal reservoirs is important for sustainable energy production and for the reliable assessment of operational risks. Predicting thermo-hydraulic (TH) system evolution under varying injection and production scenarios remains computationally challenging, particularly when physical knowledge of the subsurface system is incomplete and observational data are sparse. High-fidelity finite-element simulators are typically used to provide physics-based predictions of coupled flow and heat transport governed by complex partial-differential equations (PDEs). Such full-order simulations are, however, often prohibitively expensive for real-time forecasting, which is essential, for instance, in the context of digital twins.Physics-based machine-learning (PBML) approaches, such as the non-intrusive reduced basis (NIRB) method address this challenge by constructing physics-consistent surrogate models that project full-order simulation outputs onto a low-dimensional subspace learned from representative snapshots. By retaining only the dominantbasis functions, the NIRB surrogate enables orders-of-magnitude speedup in parametric predictions while staying consistent with the physical transport mechanisms and structural assumptions on fracture networks encoded in the full-order model. Despite these advantages, classical NIRB surrogates are intrinsically limited to the physical regimes represented by the governing PDEs, and consequently by the training simulations. If the surrogate does not fully capture the observed system behaviour, it is important to detect and adapt to missing or misrepresented local physics revealed by observational data, such as unmodeled convective heat transport or flow channelling arising from fracture activation.To address this need, we propose a complementary residual-learning framework that augments a baseline NIRB surrogate with parameter-to-state maps of residual temperature and pressure fields learned by Kolmogorov-Arnold Networks (KANs). The residual, defined as the difference between observed data (or a synthetic reference solution) and the NIRB model prediction, is interpreted as a proxy for missing or misrepresented physics not explicitly captured by the baseline model. KANs represent mappings as sums of learned univariate functions and provide explicit access to the functional structure of parameter dependence. Thereby, KANs could act as interpretable discrepancy models by learning the residual between observations and NIRB predictions. By analysing the dominant functional families emerging in the learned residual, such as linear dependence characteristic of conduction-dominated regimes or exponential dependence associated with convection, KANs can provide diagnostic insight into missing thermo-hydraulic processes and their relevance across parameter regimes.We validate the proposed approach synthetically by comparing a conduction-only NIRB surrogate against synthetic reference observations generated with an advection–diffusion model. We expect that KAN-based residual learning both improves predictive accuracy and reveals clear functional signatures of missing convective physics, even when only pointwise information is available. As an outlook, we aim to apply this workflow to real geothermal case studies, where sparse temperature and pressure measurements are available at well locations. In such settings, functional-family learning of residuals offers a promising pathway to improve surrogate predictions and to enhance the physical interpretability of geothermal systems, ultimately supporting more reliable assessments of reservoir behaviour.

The development of digital twins for subsurface applications faces several challenges, in this contribution we are focusing on the issue of providing near real-time predictions for numerical multi-physics applications describe by partial differential equations. Even when fronted against state-of-the-art high-performance computing infrastructures, conventional multi-physics simulations are not real-time compatible because of their huge computational demand. At the same time, they are subject to uncertainties from, for instance, the geometry, material properties, and boundary conditions.To address the computational demand, we introduce the usage of surrogate models. Surrogate models comprise data-driven and physics-based approaches. While data-driven techniques, such as neural-networks, well capture complex system responses, they typically lack interpretability, hindering the degree of reliability of the model outcomes. This, in turn, poses challenges for the integration into digital twins especially in applications where risks need to be assessed. In contrast, physics-based approaches are fully interpretable, but often limited to elliptic and parabolic partial differential equations. Hence, they cannot capture the full complexity of the systems dynamics. To overcome the limitations of both data-driven and physics-based techniques, we introduce a hybrid approach namely the non-intrusive reduced basis method within the class of projection-based model order reduction techniques.In this contribution, we demonstrate for a geothermal case study how this interpretable physics-based AI method can be used to reliably and efficiently accelerate the high-fidelity numerical multi-physics simulations. Furthermore, we illustrate their integration into a Bayesian uncertainty quantification framework, including hierarchical approaches. At last, we discuss possibilities to extend the aforementioned approaches to allow for a continuous integration of observational data.

SUMMARY: Material density remains poorly constrained in seismic imaging problems, yet knowledge of density would provide important insight into physical material properties for the interpretation of subsurface structures. We test the sensitivity to subsurface density contrasts of spatial and temporal gradients of seismic ambient noise wavefields, using wave equation inversion (WEI), a form of seismic gradiometry. Synthetic results for 3-D acoustic media suggest that it is possible to estimate relative density structure with WEI by using a full acoustic formulation for wave propagation and gradiometry. We show that imposing a constant density assumption on the medium can be detrimental to subsurface seismic velocity images. By contrast, the full acoustic formulation allows us to estimate density as an additional material parameter, as well as to improve phase velocity estimates. In 3-D elastic media, severe approximations in the governing wave physics are necessary in order to invert for density using only an array of receivers on the Earth's free surface. It is then not straightforward to isolate the comparatively weak density signal from the influence of phase velocity using gradiometric WEI. However, by using receivers both at the surface and in the shallow subsurface we show that it is possible to estimate density using fully elastic volumetric WEI.

It is of interest for environmental and resource applications to better characterise dynamic processes and properties of the near-surface critical zone of the solid Earth. Seismic wavefield gradiometry refers to a class of imaging techniques that estimate properties of the subsurface by calculating temporal and spatial gradients of incoming wavefields using dense array measurements, usually recorded at the Earth’s surface. One such method called wave equation inversion (WEI) has been shown to require only a few minutes of ambient seismic noise recordings to produce phase velocity maps, and shows promise for rapid field deployment.Previous applications of WEI are based on the assumption that the 2D scalar Helmholtz wave equation adequately describes the dynamics of recorded wavefields. This approximation is severe for seismic waves because the Helmholtz equation fails to describe elastic wave dynamics. Since ambient noise recordings contain all kinds of interfering elastic wave types, the accuracy of subsurface material property estimates is compromised.To investigate the potential to enhance the information available from WEI, we test the method synthetically using more sophisticated wave equations that represent wave propagation in the subsurface more accurately. Starting from a 3D seismic array geometry which provides wavefield gradient information both at the surface and at depth, WEI can be formulated in terms of the full elastic wave equation. From there we track approximations in both wave physics and field acquisition geometries that deplete information about the medium, eventually arriving at the conventional 2D scalar wave equation. These experiments highlight approximations that most deteriorate the solution, allowing us to target future effort to remove them.One approximation made in all previous WEI studies is to assume that density is constant across the local array. In reality, subsurface density varies both laterally and with depth, yet remains poorly constrained in seismic imaging problems. Accurate density estimates would provide important insight into subsurface properties. This prompts us to test wavefield sensitivities to subsurface density contrasts via WEI. Synthetic results for 3D acoustic media suggest that it is possible to estimate relative density structure with WEI by using a full acoustic formulation for wave propagation along the surface. We show that using a constant density assumption for the medium can be detrimental to subsurface images, whereas the full acoustic formulation of gradiometry improves our knowledge of material properties. It allows us to estimate density as an additional material parameter as well as to improve phase velocity estimates by incorporating approximations to the density structure. By expanding this methodology to the elastic case, we will discuss the feasibility of estimating density with gradiometric WEI in the solid Earth.