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Gravity is one of the most widely used geophysical data types in subsurface exploration. In the recent developments of stochastic geologic modeling, gravity data serve as an additional constraint to the model construction. The gravity data can be included in the modeling process as the likelihood function in a probabilistic joint inversion framework and allow the quantification of uncertainty in geologic modeling directly. A fast but also precise forward gravity simulation is essential to the success of the probabilistic inversion. Hence, we have developed a gravity kernel method, which is based on the widely adopted analytical solution on a discretized grid. As opposed to a globally refined regular mesh, we construct local tensor grids for individual gravity receivers, respecting the gravimeter locations and the local sensitivities. The kernel method is efficient in terms of computing and memory use for mesh-free implicit geologic modeling approaches. This design makes the method well suited for many-query applications, such as Bayesian machine learning using gradient information calculated from automatic differentiation. Optimal grid design without knowing the underlying geometry is not straightforward before evaluating the model. Therefore, we further provide a novel perspective on a refinement strategy for the kernel method based on the sensitivity of the cell to the corresponding receiver. Numerical results are presented and found superior performance compared to the conventional spatial convolution method.

Geologic modeling has been widely adopted to investigate underground structures. However, modeling processes inevitably have uncertainties due to scarcity of data, measurement errors, and simplification of the modeling method. Recent developments in geomodeling methods have introduced a Bayesian framework to constrain the model uncertainties by considering the additional geophysical data in the modeling procedure. Markov chain Monte Carlo (MCMC) methods are normally used as tools to solve the Bayesian inference problem. To achieve a more efficient posterior exploration, advances in MCMC methods use derivative information. Hence, we introduce an approach to efficiently evaluate second-order derivatives in geologic modeling and adopt a Hessian-informed MCMC method, the generalized preconditioned Crank-Nicolson (gpCN), as a tool to solve the 3D model-based gravity Bayesian inversion problem. The result is compared with two other widely applied MCMC methods, random-walk Metropolis–Hastings and Hamiltonian Monte Carlo, on a synthetic geologic model and a realistic structural model of the Kevitsa deposit. Our experiment demonstrates that superior performance is achieved by the gpCN compared with the other two state-of-the-art sampling methods. This indicates the potential of the proposed method to be generalized to more complex models.

Methods from the field of computer graphics are the foundation for the representation of geological structures in the form of geological models. However, as many of these methods have been developed for other types of applications, some of the requirements for the representation of geological features may not be considered, and the capacities and limitations of different algorithms are not always evident. In this work, we therefore review surface-based geological modelling methods from both a geological and computer graphics perspective. Specifically, we investigate the use of NURBS (non-uniform rational B-splines) and subdivision surfaces, as two main parametric surface-based modelling methods, and compare the strengths and weaknesses of the two approaches. Although NURBS surfaces have been used in geological modelling, subdivision surfaces as a standard method in the animation and gaming industries have so far received little attention—even if subdivision surfaces support arbitrary topologies and watertight boundary representation, two aspects that make them an appealing choice for complex geological modelling. It is worth mentioning that watertight models are an important basis for subsequent process simulations. Many complex geological structures require a combination of smooth and sharp edges. Investigating subdivision schemes with semi-sharp creases is therefore an important part of this paper, as semi-sharp creases characterise the resistance of a mesh structure to the subdivision procedure. Moreover, non-manifold topologies, as a challenging concept in complex geological and reservoir modelling, are explored, and the subdivision surface method, which is compatible with non-manifold topology, is described. Finally, solving inverse problems by fitting the smooth surfaces to complex geological structures is investigated with a case study. The fitted surfaces are watertight, controllable with control points, and topologically similar to the main geological structure. Also, the fitted model can reduce the cost of modelling and simulation by using a reduced number of vertices in comparison with the complex geological structure. Graphical Abstract

Sealed geological models are commonly used as an input to process simulations, for example in hydrogeological or geomechanical studies. Creating these meshes often requires tedious manual work – and it is, therefore, difficult to adjust a once-created model. In this work, we propose a flexible framework to create and interact with geological models using explicit surface representations. The essence of the work lies in the determination of the control mesh and the definition of semi-sharp creases values which, in combination, enable the representation of complex structural settings with a low number of control points. We achieve this flexibility through the adaptation of recent algorithms from the field of computer graphics to the specific requirements of geological modelling, specifically the representation of non-manifold topologies and sharp features. We combine the method with a swarm optimization approach to enable the automatic determination of vertex position and crease sharpness values. The result of this work is implemented in an open-source software (PySubdiv) for reconstructing geological structures while resulting in a model which is (1) sealed/watertight, (2) controllable with a control mesh, and (3) topologically similar to the main geological structure. Also, the reconstructed model may include a fewer number of vertices compared to the input geological structure which results in reducing the cost of modelling and simulation. Furthermore, the proposed method resolves some problems known from spline surfaces. In addition to enabling a manual adjustment of sealed geological models, the algorithm also provides a method for the integration of explicit surface representations in inverse frameworks and the consideration of uncertainties.

Uncertainties in geological modeling have drawn increased attention in recent years. This is due to the fast-evolving computational power and more demanding evaluation of the modeling procedure.However, these uncertainty quantifications (UQ) often face high dimensionality. Advanced mathematical methods have been developed to significantly improve the efficiency of the UQ process. Still, many of the methods require not only the forward evaluation of the quantity of interest but also the partial derivative information to guide the posterior exploration (e.g., HMC, SVGD). Differentiable geological modeling methods have been introduced and have become an appealing tool to efficiently evaluate the partial derivatives w.r.t. the input parameters using Automatic Differentiation (AD) techniques.To successfully apply AD to geological modeling several challenges need to be addressed. One of these challenges is the aliased effect due to discretization. In this work, we will introduce a method to generate a trainable geological model under the framework of gravity inversion using the implicit geological modeling method. We present a smooth-step function in the scale value domain and adopt an order-reduction method to provide a visual evaluation of the trainability of the generated model. This work provides the fundamental step to the application of advanced derivative-informed UQ and optimization methods.

Thermochronological data and kinematic models are often combined to retrace exhumation, cooling or fault activity. However, structural uncertainty is often neglected in thermokinematic models, which can lead to bias when interpreting data. Here we aim to test the influence of structural uncertainty on the interpretation of low-temperature thermochronological data in Mount Rigi in the Subalpine Molasse, a key region in the Swiss foreland fold-thrust belt of the European Alps. The region has been incorporated into the Alpine orogenic wedge in the Miocene, which led to the development of a triangle zone at the leading edge of deformation. An extensive low-temperature thermochronological data set exists, including apatite fission track as well as apatite (U-Th)/He data, which contains outliers not easily to be explained with the existing kinematic models.To diminish bias in the thermokinematic model we first estimated the geological uncertainty by computing and comparing 1000 stochastically generated 3D implicit geometric models, varied within an assigned uncertainty range assigned to geological input parameters. Model generation is performed with the stochastic geological modeling engine implemented in the Python package GemPy. In a second step a kinematic model was created which was altered in areas of high uncertainty found in the first step. With this setup, minimum and maximum values of cooling associated with shortening within the fold-thrust belt can be determined. The remaining cooling signal (if present) must hence be associated with other drivers, such as erosion caused by drainage reorganization, or uplift and erosion due to deep seated processes. Additionally, hydrothermal fluids could be held responsible for explaining individual data points. With this research, we hope to give new insights into the temporal evolution of heat flow in foreland basins and show how including geological uncertainty can lead to better constrained time-temperature histories.

Coaxial Deep Borehole Heat Exchanger (DBHE) provides an alternative way to extract geothermal energy by circulating a working fluid without producing geofluids or performing injection processes. It can be used to avoid induced seismicity issues caused by injection operations in hydrothermal doublets or to repurpose damaged or non-productive wells. A detailed numerical model is required to accurately capture as well the thermo-hydraulic processes within the DBHE and the cooling effects in the surrounding reservoir. This numerical model is often high dimensional. For a real-time monitoring purpose and optimization study, a direct numerical simulation with this model is computationally intractable.In this study, we use a physics-based machine learning method to reduce the computational cost of the performed forward model run. The physics-based machine learning method here is based on the non-intrusive reduced-basis method which expresses a physical solution in a linear combination of basis functions and weights. It is a model-order reduction technique that is mathematically proven to produce physically consistent predictions. The structure of the physics is maintained in basis functions and a machine learning model is deployed to calculate the weight for each basis function.We show the advantages of using the physics-based machine learning method by applying it to the planned coaxial DBHE in Eden (Cornwall, UK). The forward simulation is performed using the open-source simulator GOLEM, a finite-element (FE) based simulator that is built within the MOOSE framework. In this study we provide a running time comparison between the FE simulations and the physics-based machine learning simulations. We will also evaluate the accuracy of the physics-based machine learning predictions towards the FE solutions. Here, we would like to emphasize the significant computational speed-up that allow us to obtain new temperature and pressure state predictions in real-time context and to perform optimization with numerous iterations.

Thermokinematic modeling often relies on prescribed geometric and kinematic models at depth without considering their uncertainty. This does not allow for the proper quantification of the relative contributions of different drivers to the exhumation signal. Considering uncertainty of structural data in thermokinematic models would help understanding how much shortening associated with the observed cooling signal occurred. The aim of this work is to combine probabilistic structural modeling with thermokinematic forward simulations to investigate the related uncertainties. For this purpose, the Bavarian Subalpine Molasse is particularly suited as a test case, as it connects the Alpine orogen with its foreland, and should shed light on the strain distributions during the latest stages of Alpine mountain building. 3D implicit geological modeling of the Bavarian Subalpine Molasse triangle zone was carried out and combined with a systematic random sampling approach to automatically generate an ensemble of geometric models in the range of assigned uncertainties. In addition, a probabilistic 3D kinematic forward model is constructed. A link can then be obtained between kinematic model parameters and present-day geometry in comparison with field observations at the surface and also in comparison to the range of geometric uncertainties in the 3D geological model. Results show that the uncertainty (quantified as information entropy) is distributed as a function of structural complexity, depth, and data density throughout the geometric model, and additionally where fault slip ranges are large in the kinematic model.In a next step, these models are combined with a thermokinematic forward model to integrate thermochronological measurements from previous campaigns, and eventually own measurements, to obtain an integrated picture of foreland evolution and associated uncertainties over space and time.

Deep geothermal aquifers can be a good source for industry applications with a high heat demand in the course of the energy transition. Industries such as paper production require heat sources that are independent of global market fluctuations, with predictable, reliable and green long-term supply. A number of clastic aquifers have been identified as potentially feasible for geothermal use within the North German Basin ranging from the lower Cretaceous to Middle Bunter Sandstone. Due to the extensive hydrocarbon exploration in the North German Basin, it is often possible to provide a local subsurface analysis of the geothermal potential at the location of the required use.In this work, we present a case study from the town of Hoya in the southern North German Basin, where an abandoned hydrocarbon exploration well is available, as well as numerous wells in the vicinity (20 km) and some 2D seismic lines. From well log analysis, well reports, seismic data and an available 3D model of Lower Saxony, we evaluate the quality of different clastic reservoirs in the study area. We found? that many of the reservoirs that are known for good quality sandstones in the North German Basin are presented as shales at the paper mill in Hoya. Potentially usable sandstone reservoirs include the lower Cretaceous Aptian (top ca. 1600 m) and Valendis Sandstone (top ca. 1700 m), the middle Jurassic Dogger Delta 2 (top ca. 1950 m) and the Lower Triassic Hardegsen Sandstone (top ca. 3850 m). Surrounding well data, however, show a large spatial heterogeneity for these lithologies over relatively short distances. We include these geological uncertainties, temperature measurements from wells, permeability measurements from well tests and the specific requirements of a paper production company to evaluate the geological and economic feasibility, based on geothermal energy estimations and levelized costs of heat for the individual reservoirs.

Characterizing the influence of geodynamical models is important to improve our understanding of the development and current state of subsurface properties. Which are, in turn, of great societal relevance, for questions such as renewable energy. However, enabling a quantifiable characterization is a major challenge in Geodynamics, due to the high computational cost associated with both the model and the analysis for characterizing the influential parameters. The high cost of the model is caused by a high dimensionality in space, and time and a large number of input parameters. The cost of the probabilistic analyses is related to the large number of individual model solves required for performing the characterization.To address this computational challenge, we employ the non-intrusive RB method, which combines advanced mathematical algorithms and novel machine learning methods. The method produces models that considerably reduce the dimensionality, yielding an acceleration of several orders of magnitude while maintaining the physical principles. In contrast, to other machine learning methods, the non-intrusive RB method produces explainable models, which is a crucial property for later analyses and predictions.In this work, we demonstrate how the methodology can be beneficially used for the construction of reliable surrogate models of large-scale geodynamical applications without impacting the underlying physics. Furthermore, we show the benefits of global variance-based sensitivity analysis to quantifiable characterize the influence of the densities and viscosities on both the topography and velocity for the designated case study of the Alpine Region. We employ a global sensitivity analysis to account for possible parameter correlations and nonlinearities.

The motivation behind this study is to repurpose idle wells from hydrocarbon exploration and production to provide heat for end users being located near the idle well. This is possible by prolonging the value-added chain of idle wells from the gas and oil industry by re-completion as geothermal closed loop wells. This is the most efficient way to produce green energy without drilling new wells by saving the carbon emission and costs of building a new geothermal well.With this feasibility study we quantify the concept of re-completing idle wells in the North German Basin (NGB) into deep coaxial borehole heat exchangers. With numerical models of two typical geological settings of the NGB and two different completion schemes it was possible to simulate the thermal performance over a lifetime of 30 years. The calculated heat extraction rates are in the range of 200 kW to 400 kW with maximum values of up to 600 kW. This is higher than from already installed deep borehole heat exchangers. Sensitivity analyses demonstrate that flow rate, injection temperature and the final depths of re-completion are the most impacting parameters of thermal output determination.In the final project stage, the heat demand around two exemplary boreholes was mapped and possible heating networks were simulated. The initial production costs for heat are comparable to other renewable energy resources like biomass and - depending on distance between source and user – well competitive against current gas prices. These calculations highlight not only the environmental valuable motivation behind the concept of repurposing idle wells but could also save capital expenditures for the geothermal industry.Using a vacuum isolated tubing characterized by very low thermal conductivity of 0.02 W/(m*K), would make it possible to use the geothermal resources even more efficiently from idle wells. This project highlights the major potential of usable geothermal resources in already installed deep wells. The application has almost no geological risk, as the concept is independent of reservoir uncertainties like permeability and reservoir fluid composition, drilling risks are skipped completely and it is realizable at any location.

Geological models can be constructed with a variety of mathematical methods. Generally, we can describe the modeling process in a formal way as a functional relationship between input parameters (geological observations, orientations, interpolation parameters) and an output in space (lithology, stratigraphy, rock property, etc.). We evaluate here the potential of using not only the output value (prediction) itself, but also its partial derivative with respect to the input parameters to gain insight into the interpolation process, to speed-up model calibration, and to enable high-dimensional uncertainty quantification.The calculation of this partial derivative through the complex modeling procedure requires additional work – however, this step has been greatly simplified due to progress in automatic differentiation approaches in recent years. Specifically, all modern machine learning frameworks enable a calculation of the derivatives, as this is an essential component of training in deep neural networks. We can benefit from these developments for specific geological modeling functions – and if we take specific care in the numerical implementation.In this presentation, we discuss the basic principles between differentiable geomodelling methods, show implementation methods, and discuss potential difficulties and open challenges. We exemplify the advantage of the additional work through efficient implementations of sensitivity analyses and gradient-based sampling methods for uncertainty quantification in geological models.

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.

An accurate assessment of the physical states of the Earth system is an essential component of many scientific, societal and economical considerations. These assessments are becoming an increasingly challenging computational task since we aim to resolve models with high resolutions in space and time, to consider complex coupled partial differential equations, and to estimate uncertainties, which often requires many realizations. Machine learning methods are becoming a very popular method for the construction of surrogate 5 models to address these computational issues. However, they also face major challenges in producing explainable, scalable, interpretable and robust models. In this manuscript, we evaluate the perspectives of geoscience applications of physics-based machine learning, which combines physics-based and data-driven methods to overcome the limitations of each approach taken alone. Through three designated examples (from the fields of geothermal energy, geodynamics, and hydrology), we show that the non-intrusive reduced basis method as a physics-based machine learning approach is able to 10 produce highly precise surrogate models that are explainable, scalable, interpretable, and robust.

Implicit neural representation (INR) networks are emerging as a powerful framework for learning three-dimensional shape representations of complex objects. These networks can be used effectively to implicitly model three-dimensional geological structures from scattered point data, sampling geological interfaces, units, and orientations of structural features, provided appropriate loss functions associated with data and model constraints are employed during training. The flexibility and scalability of these networks provide a potential framework for integrating new forms of related geological data and knowledge that classical implicit methods cannot easily incorporate. We present a methodology using an efficient INR network architecture, called GeoINR, consisting of multilayer perceptrons (MLP) that advance existing implicit methods for structural geological modelling. The developed methodology expands on the modelling capabilities of existing methods using these networks by: (1) including unconformities into the modelling, (2) introducing constraints on stratigraphic relations as well as global smoothness along with their associated loss functions, and (3) improving training dynamics through the geometrical initialization of learnable network variables. These three enhancements enable the modelling of more complex geology, improved data fitting characteristics, and reduction of modelling artifacts in these settings, as compared to existing INR frameworks for structural geological modelling. A provincial scale case study for the Lower Paleozoic portion of the Western Canadian Sedimentary Basin (WCSB) in Saskatchewan, Canada is presented to demonstrate the modelling capacity of the MLP architecture using the developed methodology. Modelling results illustrate the method’s capacity to fit noisy datasets, represent unconformities, and implicitly model large regional scale three-dimensional geological structures.

Uncertainty quantification is integral in geological modeling for mining, exploration and civil engineering projects. Workflows for uncertainty estimation for model input in implicit structural geological modeling and inversion are well-established when the parameter space is fixed. These approaches require a priori known or chosen number of parameters. In many cases, this may not be justified given the data, as the system’s number of layers or the number of phases is not known, resulting in overly complex or too simple models.Trans-dimensional Markov chain Monte Carlo methods can be crucial to model selection in pursuing parsimonious models in complex geological settings with large uncertainty. Reversible jump Monte Carlo (RJ-MCMC) algorithms yield excellent solutions for inverse problems with varying dimensionality. In this study, we evaluate how far RJ-MCMC concepts can be applied to parameters of geological interpolation functions. Using the RJ-MCMC algorithm, we can automatically determine the optimal number of parameters for a given problem, reducing the need for subjective user input.We simulate geophysical measurements for simple structural geological models built with GemPy to create a ground truth. In the second step, we create an ensemble of prior structural models, which are updated using geophysical inversion. We can address models’ likelihood in this probabilistic framework and enable further probabilistic inference given new data. We evaluate the performance of the trans-dimensional RJ-MCMC compared to standard, fixed-dimensional methods.

The provision of climate-neutral, sustainable, and independent heat sources is an essential part of the ongoing transformation of heating systems in Germany. The city of Minden, located at the junction of the river Weser and the Middleland Canal, with its strong industrial sector, faces a massive transition of how heat and energy will be provided for industrial processes as well as heating in the commercial and residential sectors. In our study, we evaluate the structural requirements for the exploitation and utilization of deep geothermal energy from regional Mesozoic rocks, which are known to source thermal springs in the greater Minden area, and geothermal projects in other parts of the North German Basin. The compilation of geological data, seismic data, and rock properties from wells is used to construct a regional structural model as well as temperature distributions based on depth uncertainties of the respective stratigraphic units. Our investigations indicate several stratigraphic units ranging from the Middle Jurassic, Keuper, and Muschelkalk to the Middle Bunter at depths greater than 4100 m below mean sea level with suitable temperatures greater than 150 ∘C. Seismic data reveal the presence of faults, which may act as a conduit for thermal waters in the northern part of Minden. Our study also provides a basis for further geothermal exploration and exploitation south of Minden, where an operating geothermal system has already been established in the city of Osnabrück and further north, where the potential reservoirs are located at greater depths as shown by hydrocarbon exploration data. First estimations of the geothermal power output for two selected reservoir horizons yield up to 11.3 and 14.3 MW (10 % probability to yield these or higher values), respectively. We conclude that the subsurface of the inverted part of the Lower Saxony Basin principally fulfills the requirements (formation temperatures) for deep geothermal production not only for residential and commercial use but also for industrial processes. However, future detailed reservoir analyses and thermo-hydraulic investigations on a regional scale require additional exploration work like newly acquired seismic surveys and deep exploration wells.

s & presentations / EGU General Assembly 2023 EGU General Assembly 2023, EGU 2023, Vienna, Austria, 23 Apr 2023 - 28 Apr 2023; Göttingen : Copernicus Gesellschaft mbH EGU23-11331, 1 Seite (2023). doi:10.5194/egusphere-egu23-11331, Published by Copernicus Gesellschaft mbH, Göttingen

We model hydrothermal convection using a partial differential equation formed by Darcy velocity and temperature—the velocity formulation. Using the Elder problem as a benchmark, we found that the velocity formulation is a valid model of hydrothermal convection. By performing simulations with Rayleigh numbers in the non-oscillatory regime, we show that multiple quasi-steady-state solutions can be one of the reasons that caused the Nusselt–Rayleigh discrepancy found in previous experiments. The results reveal more understandings about the nature of uncertainty of convection modes in porous media.

Methods from the field of Computer Graphics are the foundation for the representation of geological structures in the form of geological models. However, as many of these methods have been developed for other types of applications, some of the requirements for the representation of geological features may not be considered and the capacities and limitations of different algorithms are not always evident. In this work, we, therefore, review surface-based geological modelling methods from both a geological and computer graphics perspective. Specifically, we investigate the use of NURBS (Non-Uniform Rational B-Splines) and subdivision surfaces, as two main parametric surface-based modelling methods, and compare the strengths and weaknesses of both approaches. Although NURBS surfaces have been used in geological modelling, subdivision surfaces as a standard method in the animation and gaming industries have so far received little attention – even if subdivision surfaces support arbitrary topologies and watertight modelling, two aspects that make them an appealing choice for complex geological modelling. Watertight modelling is a type of modelling in which the surfaces of the model have sealed interactions with all surrounding surfaces, resulting in the generation of closed volumes. Watertight models are, therefore, an important basis for subsequent process simulations based on these models.Many complex geological structures require a combination of smooth and sharp edges. Investigating subdivision schemes with semi-sharp creases is therefore an important part of this paper, as semi-sharp creases characterize the resistance of a mesh structure to the subdivision procedure. Moreover, non-manifold topologies, as a challenging concept in complex geological and reservoir modelling, are explored, and the subdivision surface method, which is compatible with non-manifold topology is described. Finally, solving inverse problems by fitting the smooth surfaces to complex geological structures is investigated with a case study. The fitted surfaces are watertight, controllable with control points, and topologically similar to the main geological structure. Also, the fitted model can reduce the cost of modelling and simulation by using a reduced number of vertices in comparison to the complex geological structure.

Conventional exploration for oil and gas is based on the play concept, which puts a prospect into the regional geological context and allows selection of which areas or licences to choose and how to rank prospects. This is traditionally done by mapping the critical elements that constitute a play such as source, reservoir and seal, and then stacking them, which results in a mosaic of segments with similar common risk. However, this method does not distinguish between areas that have little data and therefore high uncertainties from areas with identified risks and therefore less uncertainty. This is the case because uncertainties are not appropriately taken into account in play element mapping. We propose an innovative method that quantifies and visualizes uncertainties of individual play elements using the principle of information entropy. The play fairway maps generated by this method can easily be updated by using principles of Bayesian inference, which in turn allow value of information calculations. With this approach, it can be estimated how much additional well or seismic information will impact play size, yet-to-find volumes, and prospect ranking.

For geothermal applications, we aim to obtain an understanding of the physical processes in the subsurface, to predict the geothermal potential reliable. Naturally, the reliability of the prediction of the physical processes directly relates to the reliability of the evaluation of the geothermal potential. Unfortunately, predicting the physical processes reliable is a non-trivial task because of high uncertainties related to these processes. These uncertainties may arise, for instance, from uncertainties of the rock properties (e.g. thermal conductivity, permeability), structural uncertainties, and not considered physical processes.Considering the various sources of uncertainties yields a high-dimensional and therefore computationally extremely demanding problem that is not solvable even with state-of-the-art high-performance software packages. To address this curse of dimensionality, we employ a methodology, namely the non-intrusive RB method, combining advanced mathematical algorithms and machine learning methods. This method aims to significantly reduce the dimensionality of the problem while maintaining the physical principles. In contrast to other machine learning methods, the method produces interpretable and scalable models, which are crucial to obtain reliable and robust predictions.In this work, we show how the method constructs surrogate models for coupled geothermal applications, which allow, in turn, the performance of global sensitivity analysis and uncertainty quantifications. Both methods are essential for improving our model understanding. Furthermore, we demonstrate how the method in combination with developments from the field of computer graphics (e.g. NURBS, and subdivision surfaces) can be used to quantify the influence of structural uncertainties on the temperature distribution.

Proper characterization of geologic structures that host geothermal systems is crucial for the efficiency and safety of their energy production. This includes estimating layer boundaries, complex geologic features, and lithology through means of inversion and its regularization. However, existing advanced regularization techniques (e.g., geostatistical regularization, minimum-gradient support, etc.) fail to capture the complexity of 3D geological models including fault networks, fault–surface interactions, unconformities, and dome structures. Förderer et al (2021) propose a solution by means of structure-based inversion, which implements implicit geological modeling and low-dimensional parametrization to produce sharp subsurface interfaces in 2D. This work aims to extend their approach to image realistic and complex geometries in 3D. We continue with the example of electrical resistivity tomography (ERT) and synthetic data; however, this approach is aimed towards independent and joint inversion of geophysical methods that are commonly used in geothermal exploration such as magnetotellurics, gravity, and seismic techniques.The 3D geological model is created using GemPy, an open-source Python library, which constructs a structural geological model from interface points and orientations using an implicit approach based on co-kriging (de la Varga et al., 2019). Subsequently, the 3D model is discretized, and physical parameters are assigned using minimal pilot points that are then interpolated. We use pyGIMLi (Rücker et al., 2017), another open-source multi-method library for geophysical modelling and inversion, to perform a structure-based inversion, where we include the interface points in the primary model vector of the inversion to update these points iteratively to estimate a geological model in agreement with the geophysical observations.In this work, special focus is placed on the sensitivity of each model parameter. To maintain low parametrization and account for the increase in computational power, the cumulative sensitivity is calculated and tested under criteria to optimize the model updates. This is relevant for geometries where the interface and pilot points are more influential in one dimension than others. The workflow has also been adapted to include more complex structures that can be defined in 3D, especially those that reflect geothermal systems. This work is part of the Innovative Training Network EASYGO (www.easygo-itn.eu), which aims to improve the efficiency and safety of geothermal operations but can be readily used in other applications. References:Förderer, A., Wellmann, F., and Wagner, F.: Geoelectrical imaging of subsurface discontinuities and heterogeneities using low-dimensional parameterizations, EGU General Assembly 2021, online, 19–30 Apr 2021, EGU21-10012, https://doi.org/10.5194/egusphere-egu21-10012, 2021.de la Varga, M., Schaaf, A., and Wellmann, F., 2019. GemPy 1.0: open-source stochastic geological modeling and inversion, Geosci. Model Dev., 12, 1–32, doi:10.5194/gmd-12-1-2019.Rücker, C., Günther, T., Wagner, F.M., 2017. pyGIMLi: An open-source library for modelling and inversion in geophysics, Computers and Geosciences, 109, 106-123, doi: 10.1016/j.cageo.2017.07.011.

Gravity is one of the most widely used geophysical data types in subsurface exploration. In the recent developments of stochastic geological modeling, gravity data serves as an additional constraint to the modeling construction and can be included in the modeling process as the likelihood function in a Bayesian workflow. A fast but also precise forward gravity simulation is key to the success of the geological modeling inverse problem. In this study, we present a gravity kernel method, which is based on the widely adopted analytical solution on a discretized grid. As opposed to a globally refined regular mesh, we construct local tensor grids for each sensor, respecting the gravimeter locations and the local sensitivities. The kernel method is efficient in terms of both computing and memory use for meshless implicit geological modeling approaches. This design makes the method well suited for many-query applications like Bayesian machine learning using gradient information calculated from Automatic Differentiation (AD). Optimal grid design without knowing the underlying geometry is not straightforward before evaluating the model. Therefore, we further provide a novel perspective on a refinement strategy for the kernel method based on the sensitivity of the cell to the corresponding receiver. Synthetic results are presented and show superior performance compared to the traditional spatial convolution method.

Marine geophysical research 43(2), 11 (2022). doi:10.1007/s11001-022-09475-z, Published by Springer Science + Business Media B.V., Dordrecht [u.a.]

Geological modeling has been widely adopted to investigate underground geometries. However, modeling processes inevitably have uncertainties due to scarcity of data, measurement errors, and simplification of modeling methods. Recent developments in geomodeling methods have introduced a Bayesian framework to constrain the model uncertainties by considering additional geophysical data into the modeling procedure. Markov chain Monte Carlo (MCMC) methods are normally used as tools to solve the Bayesian inference problem. To achieve a more efficient posterior exploration, advances inMCMC methods utilize derivative information. Hence, we introduce an approach to efficiently evaluate second-order derivatives in geological modeling and introduce a Hessian-informed MCMC method, the generalized preconditioned Crank-Nicolson (gpCN), as a tool to solve the 3D model-based gravity Bayesian inversion problem. The result is compared with two other widely applied MCMC methods, random walk Metropolis-Hasting and Hamiltonian Monte Carlo, on a synthetic three-layer geological model. Our experiment demonstrates that superior performance is achieved by the gpCN, which has the potential to be generalized to more complex models.

Geological models are commonly used to represent geological structures in 3D space. A wide range of methods exists to create these models, with much scientific work focusing recently on implicit representation methods, which perform an interpolation of a three-dimensional field where the relevant boundaries are then isosurfaces in this field. However, this method has well-known problems with inhomogeneous data distributions: if regions with densely sampled data points exist, modeling artifacts are common. We present here an approach to overcome this deficiency through a combination of an implicit interpolation algorithm with a local smoothing approach. The approach is based on the concepts of nugget effect and filtered kriging known from conventional geostatistics. It reduces the impact of regularly occurring modeling artifacts that result from data uncertainty and data configuration and additionally aims to improve model robustness for scale-dependent fit-for-purpose modeling. Local smoothing can either be manually adjusted, inferred from quantified uncertainties associated with input data or derived automatically from data configuration. The application for different datasets with varying configuration and noise is presented for a low complexity geologic model. The results show that the approach enables a reduction of artifacts, but may require a careful choice of parameter settings for very inhomogeneous data sets.

Calibrating geothermal simulations is a critical step, both in scientific and industrial contexts, with suitable model parameterizations being optimized to reduce discrepancies between simulated and measured temperatures. Here we present a methodology to identify model errors in the calibration and compensate for measurement sparsity. With an application to the Upper Rhine Graben, we demonstrate the essential need for global sensitivity studies to robustly calibrate geothermal models, showing that local studies overestimate the influence of some parameters. We ensure the feasibility of the study through a physics-based machine learning approach (reduced basis method), reducing computation time by several orders of magnitude.

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 with 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.

We present a probabilistic machine learning approach to determine lithologies for the wireline data from the Springbok Sandstone formation in the Surat Basin. Deterministic inversions of the data a...

This paper presents a novel perspective to understanding the spatial and statistical patterns of a cone penetration dataset and identifying soil stratification using these patterns. Both local consistency in physical space (i.e., along depth) and statistical similarity in feature space (i.e., logQt–logFrspace, where Qtis the normalized tip resistance and Fris the normalized friction ratio, or the Robertson chart) between data points are considered simultaneously. The proposed approach, in essence, consists of two parts: (i) a pattern detection approach using the Bayesian inferential framework and (ii) a pattern interpretation protocol using the Robertson chart. The first part is the mathematical core of the proposed approach, which infers both spatial pattern in physical space and statistical pattern in feature space from the input dataset; the second part converts the abstract patterns into intuitive spatial configurations of multiple soil layers having different soil behavior types. The advantages of the proposed approach include probabilistic soil classification and identification of soil stratification in an automatic and fully unsupervised manner. The proposed approach has been implemented in MATLAB R2015b and Python 3.6, and tested using various datasets including both synthetic and real-world cone penetration test soundings. The results show that the proposed approach can accurately and automatically detect soil layers with quantified uncertainty and reasonable computational cost.

Environmental earth sciences 81(3), 88 (2022). doi:10.1007/s12665-022-10202-5, Published by Springer, Berlin ; Heidelberg

Pragmatic and cost-effective representations of geological structures and features (e.g., heterogeneities, faults and folds) in full 3-D geological models are challenging. Implementations are highly dependent on the flexibility of the representation method. We investigate the use of parametric surface-based geological modelling methods for the purpose of low-dimensional model representations. Specifically, we focus on two grid-free and controllable parametric surfaced-based modelling methods: NURBS and subdivision surfaces. NURBS are the standard method in Computer-Aided Design (CAD) and have been used in geological reservoir modelling before. Subdivision surfaces are a common representation in the gaming and animation industry. They are very interesting as they can support watertight modelling and arbitrary topology (preserving the relationship between different parts of the model). However, this method is, to date, rarely used in geological modelling.Unlike implicit modelling, parametric surfaced-based modelling is a grid-free representation and exploits the boundary surfaces of the model. Also, the geological features (e.g., heterogeneities, faults, folds) can be represented by there bounding surfaces instead of grid-cells. Therefore, they do not suffer from the limitation of grid cells (e.g., Stair-stepping), which are often present in implicit representations.We discuss the advantages and shortcomings of both NURBS and subdivision surfaces for geological modelling. Furthermore, we investigate the approximation of geological structures by subdivision surfaces in this presentation. The approximated models are watertight (closed), controllable with few control points, smooth, and have less than 5% of the number of the vertices of the original model. Reducing the number of vertices of the model while preserving the topology can decrease the cost of both modelling and simulations. As the final step, we present the advantages of grid-free surface-based geological modelling for thermal finite element analyses by using a state-of-the-art finite-element solver, namely the MOOSE framework

Geophysical imaging is subject to inherent non-uniqueness due to the ill-posed nature of the inverse problem. To mitigate this, the solution is commonly subjected to regularization. Smoothing regularization is widely used in practice, but produces high-dimensional images without sharp contrasts between geological units. These tomograms stand in contrast to current implicit geological models, which are able to produce sharp subsurface interfaces with complex geometries using low-dimensional parametrizations. This work aims to bring together modelling concepts from geophysics and geology using the example of electrical resistivity tomography (ERT).An implicit geological model is used as the centerpiece of a 2D ERT inversion within a deterministic Gauss-Newton framework. The points that define the surfaces of the geological model are included into the model vector of the inverse problem along with a low-dimensional pilot point parametrization of the subsurface electrical resistivity. The point-based parameterization is translated to a triangular finite-element mesh to solve the geoelectrical forward problem. Sensitivities for the geological interfaces and resistivity parameters are efficiently calculated based on finite-differences and the reciprocity theorem, respectively. Each iteration step produces an update of both the geological interface as well as the parameter fields.The approach converges to an updated geological model and a distribution of subsurface resistivity, which are in accordance with the measured data. The tomograms show sharply localized and realistic subsurface interfaces that are described by only a few parameters. While the imaging of small-scale heterogeneities is challenging and would require a locally increased number of pilot points, the current approach allows for the estimation of smoothly distributed heterogeneities. Further advantages of the approach lie in the improved integration of a-priori geological knowledge, the straightforward extension to 3D, and the applicability to other geophysical methods as well as joint inversion.