Direct Use Geothermal Systems (DUGS) are rapidly and densely deployed to meet the growing demand for renewable energy with less carbon emissions globally. The simulation of DUGS can provide a reservoir-scale understanding of geothermal resource assessment, where the geothermal system’s lifetime and the injection well Bottom Hole Pressure (BHP) are used as performance indicators. However, there are inherent errors from numerical simulations of any engineering problems, due to approximating continuous partial differential equations by their discretized approximation in time and space. In this work, we establish an optimal numerical setup with reduced errors across the homogeneous, stratified and heterogeneous models for the simulation of a geothermal system. Next, we develop a standardized method for calculating recoverable Heat In Place (HIP) and an analytical solution for evaluating the HIP recovery factor across various geological models using a single forward simulation. We present reference examples on the design of DUGS simulations using the open-source software Delft Advanced Research Terra Simulator (open-DARTS). The open-DARTS platform enables accurate and efficient sensitivity and uncertainty analysis. Using Distance-Based Generalized Sensitivity Analysis (DGSA), we identify reservoir depth and discharge rate as the most influential parameters for geothermal projects across all three types of geological models.@en

Direct Use Geothermal Systems (DUGS) are rapidly and densely deployed to meet the growing demand for renewable energy with less carbon emissions globally. The simulation of DUGS can provide a reservoir-scale understanding of geothermal resource assessment, where the geothermal system's lifetime and the injection well Bottom Hole Pressure (BHP) are used as performance indicators. However, there are inherent errors from numerical simulations of any engineering problems, due to approximating continuous partial differential equations by their discretized approximation in time and space. In this work, we establish an optimal numerical setup with reduced errors across the homogeneous, stratified and heterogeneous models for the simulation of a geothermal system. Next, we develop a standardized method for calculating recoverable Heat In Place (HIP) and an analytical solution for evaluating the HIP recovery factor across various geological models using a single forward simulation. We present reference examples on the design of DUGS simulations using the open-source software Delft Advanced Research Terra Simulator (open-DARTS). The open-DARTS platform enables accurate and efficient sensitivity and uncertainty analysis. Using Distance-Based Generalized Sensitivity Analysis (DGSA), we identify reservoir depth and discharge rate as the most influential parameters for geothermal projects across all three types of geological models.

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This study presents a method to address the significant uncertainties in subsurface modeling that impact the efficiency of energy transition applications such as geothermal energy extraction and CO₂ geological sequetsration. The approach combines a physics-based geomechanical proxy model with an ensemble smoother with multiple data assimilation (ES-MDA), aimed at enhancing uncertainty quantification through the integration of vertical displacement measurements from fluid production and injection. The data from wells is limited in spatial coverage, while these measurements offer extensive spatial information, improving the understanding of subsurface behavior by reflecting changes in reservoir pressure and temperature. The open-DARTS simulator for fluid flow and a geomechanical proxy are used to perform data assimilation with ES-MDA. By generating an ensemble of model realizations with varied permeability, calculating vertical displacements at the surface, and applying ES-MDA, we effectively identify the probability distribution of the vertical displacement of the model conditioned to observed subsidence data. Entropy is used as a statistical measure to quantify the reduction of uncertainty of subsurface models based on observations. Our approach was tested on a 2D conceptual and 3D realistic datasets, demonstrating its capability to provide data assimilation. This workflow represents an advancement in subsurface modeling, supporting informed decision-making in geothermal energy production and CO₂ sequestration by offering an improved alternative for data assimilation and enhancing tools for uncertainty quantification.

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Pore pressure fluctuation in subsurface reservoirs and its resulting mechanical response can cause fault reactivation. Numerical simulation of such induced seismicity is important to develop reliable seismic hazard and risk assessments. However, modeling of fault reactivation is quite challenging, especially in the case of displaced faults, i.e., faults with non-zero offset. In this paper, we perform a systematic benchmarking study to validate two recently developed numerical methods for fault slip simulation. Reference solutions are based on a semi-analytical approach that makes use of inclusion theory and Cauchy-type singular integral equations. The two numerical methods both use finite volume discretizations, but they employ different approaches to represent faults. One of them uses a conformal discrete fault model (DFM) while the other employs an embedded (non-conformal) fault model. The semi-analytical test cases cover a vertical frictionless fault, and inclined displaced faults with constant friction and slip-weakening friction. It was found that both numerical methods accurately represent pre-slip stress fields caused by pore pressure changes. Moreover, they also successfully cope with a vertical frictionless fault. However, for the case with an inclined displaced fault with a constant friction coefficient, the embedded method can not converge for the post-slip phase, whereas the DFM successfully coped with both constant and slip-weakening friction coefficients. In its current implementation, the DFM is therefore the model of choice when accurate simulation of local faulted systems is required.@en

Featured Application: This study emphasizes the importance of comprehensive monitoring, calibration, and optimization of storage strategies in a saline aquifer. It also highlights the need to manage geomechanical risks and uncertainties. By understanding these risks and employing suitable monitoring techniques, the integrity and safety of GCS can be ensured, contributing to the reduction of CO ₂ emissions. Geological Carbon Storage (GCS) involves storing CO ₂ emissions in geological formations, where safe containment is challenged by structural and stratigraphic trapping and caprock integrity. This study investigates flow and geomechanical responses to CO ₂ injection based on a Brazilian offshore reservoir model, highlighting the critical interplay between rock properties, injection rates, pressure changes, and ground displacements. The findings indicate centimeter-scale ground uplift and question the conventional selection of the wellhead as a monitoring site, as it might not be optimal due to the reservoir’s complexity and the nature of the injection process. This study addresses the importance of comprehensive sensitivity analyses on geomechanical properties and injection rates for advancing GCS by improving monitoring strategies and risk management. Furthermore, this study explores the geomechanical effects of modeling flow in the caprock, highlighting the role of pressure dissipation within the caprock. These insights are vital for advancing the design of monitoring strategies, enhancing the predictive accuracy of models, and effectively managing geomechanical risks, thus ensuring the success of GCS initiatives.

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Injection of high-pressure CO2 into depleted gas reservoirs can lead to low temperatures promoting formation of hydrate in the near wellbore area resulting in reduced injection rates. The design of effective mitigation methods requires an understanding of the impact of crucial parameters on the formation and dissociation of CO2 hydrate within the porous medium under flowing conditions. This study investigates the influence of water saturation (ranging from 20 % to 40 %) on the saturation and kinetics of CO2 hydrate during continuous CO2 injection. The experiments were conducted under a medical X-ray computed tomography (CT) to monitor the dynamics of hydrate growth inside the core and to calculate the hydrate saturation profile. The experimental data reveal increase in CO2 hydrate saturation with increasing water saturation levels. The extent of permeability reduction is strongly dependent on the initial water saturation: beyond a certain water saturation the core is fully blocked. For water saturations representative of the depleted gas fields, although the amount of generated hydrate is not sufficient to fully block the CO2 flow path, a significant reduction in permeability (approximately 80 %) is measured. It is also observed that the volume of water + hydrate phases increases during hydrate formation, indicating a lower-than-water density for CO2 hydrate. Having a history of hydrate at the same water saturation leads to an increase in CO2 consumption compared to the primary formation of hydrate, confirming the existence of the water memory effect in porous media.@en

The reduction of temperature caused by Joule-Thomson effect during injection of CO2 in low pressure reservoirs combined with presence of water can lead to formation of hydrates, which in turn reduces rock permeability and hence CO2 injectivity. This paper introduces an empirical model to evaluate impact of hydrate formation on injectivity of CO2 injection wells. Experiments were also conducted to validate the model. The model was then used to simulate injection of CO2 into a multi-layered depleted gas field. The results indicate that operational parameters, particularly CO2 injection rate and temperature, have a large influence on hydrate formation. This is because a higher CO2 injection rate leads to a greater pressure drop within the injection well, potentially triggering conditions conducive to hydrate formation. It is also shown that the dynamics of the competition between the dry-out and temperature fronts play an important role in the final saturation of the hydrate within porous media. For large evaporation rates, the evaporation of water reduces water saturation near wellbore and hence formation of hydrates is limited.@en

CO2 storage in deep saline aquifers is an effective strategy for reducing greenhouse gas emission. However, salt precipitation triggered by evaporation of water into injected dry CO2 causes injectivity reduction. Predicting the distribution of precipitated salts and their impact on near-well permeability remains challenging. Therefore, a detailed investigation of the interactions between salt precipitation and porous domain is essential for of revealing the mechanisms of pore blockage due to salt crystallization. Through series of microfluidic experiments, direct observations, coupled with detailed imaging processing, form the basis for explaining these phenomena and provide a relationship between water and salt saturations, highlighting the critical roles played by local capillary-driven flow and water film along grains in influencing water relocation. The results reveal two distinct types of salt crystallization: occurring inside the brine with smooth edges and at the CO2-brine interface with rough edges. Furthermore, the impact of local heterogeneity and surface wettability on salt precipitation patterns is discussed. The transition region between the porous domains and inlet/outlet channels exhibits brine backflow and a larger amount of salt accumulation. This paper presents a comprehensive analysis of the dynamic process of salt dry-out occurring during CO2 injection at the pore scale.@en

A geothermal doublet has been installed in a sedimentary reservoir for direct-use heating on the TU Delft campus, targeted to supply around 25 MW of thermal energy at peak conditions. This contribution presents the implementation and initial data collection from the doublet, including an initial evaluation of the logging and coring campaign. Nearly half of Netherlands natural gas consumption is allocated to heating, and the on-campus CO2 emissions from heating exceed 50%. The doublet has been designed with two primary aims of research and commercial heat supply, with the wells being completed in December 2023. The project will be operated by a commercial entity, and built into a larger thermal energy system including a high temperature underground storage system, with the first energy production planned in 2025. The research questions relate to field-scale geothermal operations, e.g. how reliable is the long-term energy production?, how do materials perform in the long-term? and how can geothermal projects be best monitored? The research programme involves the installation of a wide range of instruments alongside an extensive logging and coring program and monitoring network. The doublet has been cored, with substantial continuous samples from the heterogenous reservoir, alongside a large suite of open hole well logs in the reservoir and through casing logs in overlying geological units. A fiber-optic cable will monitor distributed pressure throughout the producer reservoir section, at approximately 2300m depth, which will be installed during commissioning. A local seismic monitoring network has been installed in the surrounding area with the aim of monitoring very low-magnitude natural or induced seismicity. The project is a key national research infrastructure and is being incorporated into the European EPOS (European Plate Observing System, https://www.epos-eu.org/), such that accessibility and data availability will be as wide as possible. All observations will be included in a digital-twin framework that will allow to make better decisions in future geothermal projects.@en

In this study, we conduct a comprehensive history matching study for the FluidFlower benchmark model. This benchmark was prepared and organized by the University of Bergen, the University of Stuttgart, and Massachusetts Institute of Technology, for promoting understanding of the complex physics of geological carbon storage (GCS) through in-house experiments and numerical simulations. This paper synthesizes the experiences of history matching the benchmark data encountered by the Delft-DARTS and CSIRO participants. History matching is first performed based on a low-dimensional-zonated structured model using a simple Poisson-like solver. The permeability of six facies and two faults is inferred in this stage to match the digitized concentration data. The history matching is then further enhanced to richer spatial and physical models to capture the spatial variation of permeability and buoyancy effects, using an unstructured grid. Efficient adjoint methods are used to evaluate the gradient used in the optimization of data misfits or equivalent Bayesian log-likelihoods. With efficient optimization methods available for both maximum a posteriori model inference and Randomized Maximum Likelihood methods for model uncertainty, we perform history matching using both binary and continuous concentration observations. The results show that the tracer plumes in the enriched model match the experimental plumes more accurately compared with the results from the parsimonious-zonated model. The history matching results based on the concentration observations provide more similar plume shapes compared with the case based on the binary observations. The permeability difference between the model before and after history matching reveals that the tracer plume zone and the high permeable zone are the regions of high sensitivity in terms of data misfit between the model response and observations. Surprisingly, CO ₂ concentration plume forecasts based on these history-matched models were not especially sensitive to the improvements observed in the enhanced model.

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Hydrogen plays a crucial role in the transition to low-carbon energy systems, especially when integrated into energy storage applications. In this study, the concept of exergy-return on exergy-investment (ERoEI) is applied to investigate the exergetic efficiency (defined as the ratio of useful exergy output to invested exergy input) and CO2 equivalent intensity of the hydrogen supply chain, with a specific focus on the underground hydrogen storage process. Our findings reveal that the overall exergetic efficiency of the electricity-to-hydrogen-to-electricity conversion process can reach up to 25 %. Among the hydrogen production methods, green hydrogen, produced via electrolysis powered by renewable energy, exhibits the lowest CO2 equivalent intensity. Blue hydrogen, produced from natural gas with carbon capture, can significantly reduce the carbon footprint of electricity generation, though this advantage comes at the expense of decreased exergetic efficiency. Analysis further indicates that the exergetic efficiency of underground storage components ranges from 72 % to 92 %. A substantial fraction of the exergy is lost during compression and injection of the stored hydrogen. Nevertheless, the subsurface operations contribute a minimal CO2 emission, between 1.46–4.56 grams of equivalent CO2 per megajoule (gr-CO2eq/MJ) when powered by low-carbon energy sources. Furthermore, it is found that hydrogen loss in the reservoir, along with methane and hydrogen leak during surface operations, notably affects the overall efficiency of the storage process.@en

Carbon dioxide capture and storage in subsurface geological formations is a potential solution to limit anthropogenic CO₂ emissions and combat global warming. Depleted gas fields offer significant CO₂ storage volumes; however, injection of CO₂ into these reservoirs poses some potential challenges for the injectivity, containment and well/facility integrity due to low temperatures caused by isenthalpic expansion of CO₂. A key injectivity risk is due to possible formation of hydrates at the low expected temperatures. This study aims to address main causes of CO₂ hydrate formation and its impact on permeability of porous media. This review highlights the current state of knowledge in the literature while emphasizing the need to bridge existing gaps in derisking CO₂ injection into (depleted) low-pressure gas reservoirs. In summary, according to the existing literature, the potential for hydrate formation is assessed to be credible. Current industry solutions exist to manage this risk; however, they are costly and energy intensive. Future research will be needed to provide capabilities to manage this risk more efficiently.

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This study investigates the integration of machine learning (ML) and data assimilation (DA) techniques, focusing on implementing surrogate models for Geological Carbon Storage (GCS) projects while maintaining the high fidelity physical results in posterior states. Initially, we evaluate the surrogate modeling capability of two distinct machine learning models, Fourier Neural Operators (FNOs) and Transformer UNet (T-UNet), in the context of CO₂ injection simulations within channelized reservoirs. We introduce the Surrogate-based hybrid ESMDA (SH-ESMDA), an adaptation of the traditional Ensemble Smoother with Multiple Data Assimilation (ESMDA). This method uses FNOs and T-UNet as surrogate models and has the potential to make the standard ESMDA process at least 50% faster or more, depending on the number of assimilation steps. Additionally, we introduce Surrogate-based Hybrid RML (SH-RML), a variational data assimilation approach that relies on the randomized maximum likelihood (RML) where both the FNO and the T-UNet enable the computation of gradients for the optimization of the objective function, and a high-fidelity model is employed for the computation of the posterior states. Our comparative analyses show that SH-RML offers a better uncertainty quantification when compared to the conventional ESMDA for the case study.

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In this study, we present an efficient and flexible adjoint-based framework for history matching and forecasting geothermal energy extraction at a large scale. In this framework, we applied the Principal Component Analysis to reduce the parameter space for representing the complex geological model. The adjoint method is implemented for gradient calculation to speed up the history-matching iteration process. Operator-based linearization (OBL) used in this framework makes the calculation of the physical state and its derivatives very efficient and facilitates the matrix assembly in the adjoint method. This study primarily focuses on history matching based on combined observation of well production and in-situ electromagnetic measurements to predict the temperature front. However, different types of misfit terms can be added to the objective function based on practical considerations. For example, our history-matching case studies include model misfit terms applied for regularization purposes. The measurement data is extracted from the true model, and realistic measurement errors are considered. Also, in this work, we propose an optimal weighting strategy for the terms of the objective function to balance their sensitivity with respect to the model control variables. The high efficiency of the framework is demonstrated for the geothermal doublet model implemented at the heterogeneous reservoir with multiple realizations. The framework allows for generating posterior Randomized Maximum Likelihood (RML) estimates of the entire ensemble of the realizations with a reasonable computational cost. Results show that the framework can achieve reliable history-matching results based on the doublets production data and the reservoir electromagnetic measurement.

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Nearly half of the Netherlands’ natural gas consump tion is allocated to heating, with direct -use geothermal heating being one of the available low-carbon energy solutions. A geothermal well doublet, designed with the two primary aims of research and commercial heat supply, is currently being installed on the campus of Delft University of Technology. The project is a key national research infrastructure and is being incorporated into the European sustainable and distributed infrastructure (EPOS: European Plate Observing System, https://www.epos-eu.org/), such that accessibility and data availability will be as wide as possible. All observations will be included in a digital-twin framework, which will allow us to make better decisions in future geothermal projects. The project includes a comprehens ive research program, involving the installation of a wide range of instruments alongside an extensive logging and coring program and monitoring network. The doublet has been cored, with substantial continuous samples from the heterogeneous reservoir, alongside a large suite of well logs in both the reservoir and overlying geological units. Such investigation is rarely undertaken in geothermal projects. A fiber-optic cable will monitor the producer well all the way down to the reservoir section, at approximately 2300m depth, in the Lower Cretaceous Delft Sandstone that is used as a geothermal reservoir in a series of existing and planned doublets in the West Netherlands Basin. A local seismic monitoring network has been installed in the surrounding area with the aim of monitoring very low-magnitude natural or induced seismicity. A vertical observation well with electromagnetic sensors will be drilled in the near future between the injector and producer to monitor cold-front propagation. This paper presents the initial modeling for the project and steps towards the production of a digital twin. Two modeling examples in the paper will emp hasize current operational challenges relevant to the project.@en

A geothermal well doublet, designed with two primary aims; one of research and the second of commercial thermal energy supply, is currently being installed on the campus of Delft University of Technology, with the wells being drilled in the second half of 2023. The project includes a comprehensive research program, involving the installation of a wide range of instruments alongside an extensive logging and coring program and monitoring network. The doublet has been cored, with continuous samples from the heterogenous reservoir being complimented with more distributed side-wall cores, alongside a large suite of open-hole well logs in the reservoir section of both wells. Such investigation is rarely undertaken in geothermal projects. A fiber optic cable will monitor the production well, and will be installed all-the-way down to the reservoir section when the well completion is installed, at approximately 2300m depth. The reservoir is the fluvial Lower Cretaceous Delft Sandstone that is used as a geothermal reservoir in a series of existing and planned doublets in the West Netherlands Basin. A local seismic monitoring network has been installed in the surrounding area with the aim of monitoring very low-magnitude natural or induced seismicity. A vertical observation well with electromagnetic sensors will be drilled in a few y ears’ time between the injector and producer to monitor cold-front propagation. The total project is targeted to supply around 25 MW of thermal energy at peak conditions, next to this project a thermal energy storage system is planned to provide a seasonal buffer. The project is a key national research infrastructure and is being incorporated into the European infrastructure EPOS (European Plate Observing System, https://www.epos-eu.org/), such that accessibility and data availability will be as wide as possible. All observations will be included in a digital-twin framework that will allow better decisions to be made in future geothermal projects. This paper presents the implementation and initial data collection from the project, including an initial evaluation of the logging and coring campaigns.@en

The data assimilation process for geothermal reservoirs often relies on well data, which primarily offer insights into the immediate vicinity of the borehole. However, integrating geophysical methods can provide valuable information beyond well proximity, possibly enhancing reservoir predictions. Current methods of monitoring geothermal reservoirs struggle to maintain a good signal-to-noise ratio for deep reservoirs. Diffusive electromagnetic (EM) methods can be sensitive to the decreasing conductivity from heat extraction in geothermal reservoirs and offer promising additional value. To test their potential effectiveness, numerical examples are simulated. A scheme to incorporate diffusive EM observations into a data assimilation process for geothermal reservoirs is presented and implemented in this study. First, an ensemble of prior models representing the reservoir uncertainty is used to determine the moments of the resulting temperature field using a forward geothermal simulation. Subsequently, a conductivity model is calculated from the temperature field using an empirical relationship. The expected electric field response can then be simulated using an EM forward model. EM sources are placed on the surface around the expected cold plume location. The receiver is placed at reservoir depth. To assimilate the data, the ensemble smoother with multiple data assimilation method is used. The findings demonstrate that the incorporation of EM data provides valuable information regarding the temperature field. This improves the accuracy of the temperature forecast of the entire reservoir when combined with the localized data from the production well and, therefore, helps to resolve the complex migration of the cold front. These results highlight the monitoring potential of EM observations for geothermal reservoirs.@en

In this paper, we present an efficient inverse modeling framework for energy transition applications. The key feature of this framework is a combination of adjoint gradients and Operator-based Linearization (OBL) technique to achieve high efficiency in inverse modeling based on forward simulations. This framework allows conducting the history matching of practical industrial applications using the gradient descent method with considerable model control variables in a reasonable time. Generally, the inverse modeling of industrial applications involves large amounts of gradient calculations in algorithms based on gradient descent. In this study, we analytically compute the gradient using the adjoint gradient method as an alternative to the widely used numerical gradient method where many time-consuming forward simulation runs are needed. In the adjoint gradient approach, the objective function is linearly combined with the governing equation by introducing a Lagrange multiplier. That allows for finding the analytical gradient in a backward manner. The developed adjoint gradient method takes full advantage of the OBL efficiency and flexibility when assembling the Jacobian and some relevant derivatives. We demonstrate the applications of the proposed inverse modeling framework to different energy transition applications, including petroleum production, extraction of geothermal energy, and CO₂ storage. We demonstrate various treatments of objective function definitions, well controls, and measurement errors for these industrial applications. For petroleum production, the proposed framework is tested on the multiphase multi-component flow problem, which is illustrated by an example of data-driven Discrete Well Affinity model. For this application, only production data is considered. The geothermal problem involves an additional energy balance equation and various property calculations for water and steam. In this application, together with the production data, additional electromagnetic monitoring is used in the history matching process. The results show that electromagnetic monitoring significantly improves the inversion process. We conclude the description of our framework with an application relevant to CO₂ sequestration process. The CO₂ storage modeling is complicated due to the complex physical phenomena to be considered. In this application, tracer data are used as an additional observation, which allows considering uncertainties in the dynamics of CO₂. In this study, the adjoint gradient method is specially designed and customized for OBL infrastructure of the Delft Advanced Research Terra Simulator (DARTS). This allows us to design the general-purpose inversion module with efficient gradient computation, while most existing simulation platforms lack this capability. Based on the multiphysics simulation engine in DARTS, the various observation information can be combined in the proposed framework. This allows us to solve the general-purpose inverse modeling problems for most energy transition applications.

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In this work, we present a unified framework for the simulation of CO₂ sequestration problems at various time and space scales. The parametrization technique utilizes thermodynamic state-dependent operators expressing the governing equations for the thermal-compositional-reactive system to solve the nonlinear problem. This approach provides flexibility in the assembly of the Jacobian, which allows straightforward implementation of advanced thermodynamics. We validate our simulation framework through several simulation studies including complex physical phenomena relevant to CCUS. The proposed simulation framework is validated against a set of numerical and experimental benchmark tests, demonstrating the efficiency and accuracy of the modeling framework for CCUS-related subsurface applications. Important physical phenomena resulting from the complex thermodynamic interactions of CO₂ and impurities with reservoir fluids can be accurately captured now in detailed dynamic simulation. The investigated simulation scenarios include a reproduction of lab experiments at the core scale, investigation of macro-scale analog model and simulation of large-scale industrial application. The simulation time can also span from hours to years among various applications. Complex thermal-compositional-reactive phenomena can be addressed at each of these space and time scales. The unified thermodynamic description allows us to perform all these simulations for a reasonable CPU time due to advanced parametrization techniques and efficient GPU capabilities in our in-house reservoir simulator DARTS.

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Focus is on comparing stochastic, process-based and deterministic methods for modelling heterogeneity in hydraulic properties of fluvial geothermal reservoirs. Models are considered a generalized representation of a fluvial sequence in the upper part of the Gassum Formation in northern Denmark. Two ensemble realizations of process-based and stochastic heterogeneity were simulated using the state-of-the-art numerical modelling software, Delft Advanced Research Terra Simulator (DARTS), to assess differences on a statistically relevant sample size. Simulator settings were optimized to achieve two orders of magnitude improvement in simulation time. Our general findings show that the stochastic and process-based methods produce nearly identical results in terms of predicted breakthrough time and production temperature decline for high net-to-gross ratios (N/G). Simple homogenous and layered models overestimate breakthrough and underestimate temperature decline. More complex representation of facies in process-based models show smaller variance in results and stay within the limits of ensemble runs compared to simpler facies representation. Results indicate that a multivariate Gaussian based stochastic representation of heterogeneity provides comparable thermal response to a process-based model in fluvial systems of similar quality.

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