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NASA NTRS2018

Outer Solar System Material in Inner Solar System Regolith Breccias

There is excellent evidence that a dynamical instability in the early solar system led to gravitational interactions between the giant planets and trans-Neptunian objects (TNOs). Giant planetary migration triggered by the instability dispersed a disk of primordial TNOs and created a number of small body reservoirs (e.g. the Kuiper Belt, scattered disk, irregular satellites, and the Jupiter/Neptune Trojan populations). It also injected numerous bodies into the main asteroid belt, where modeling shows they can successfully reproduce the observed P and D-type asteroid populations. During the injection era and after implantation, some of these “main belt TNOs” would have collided with S-class asteroids. Some of this material may have survived as a component of asteroid regolith breccias. Thus, we have been searching for evidence of these impact events in the form of carbonaceous xenoliths in brecciated ordinary chondrites. These xenoliths would have experienced a wide range of impact velocities, and therefore we should expect to see everything between relatively unaltered material to completely shock-melted lithologies. This material might also be different from the carbonaceous chondrites that represent standard C-complex asteroids. A goal of this research is to define useful criteria for distinguishing between these two classes of materials, including O, Cr, N and C isotopes, petrographic characteristics, and chronology.

Zolensky, M.↗

DOE OSTI2021

A Mixed Fracture-Matrix Model for Evaluating Well Orientation and Completion Options for the Utah FORGE Site

Orientation and completion for well pairs that have been subjected to multi-zonal stimulation play a critical role in the long-term performance of an Enhanced Geothermal Reservoir. Here we present the development of a methodology to rapidly and efficiently numerically simulate mixed fracture-matrix flow systems for evaluation of well design and completion options. An example evaluation based on a small fracture network representative of FORGE Well 16(A and B)-78(32) follows a discussion of the theory and model validation.

58 GEOSCIENCES↗

DOE OSTI2021

A Mixed Fracture-Matrix Model for Evaluating Well Orientation and Completion Options for the Utah FORGE Site

15 GEOTHERMAL ENERGY↗

DOE OSTI2019

Framework for Characterizing Strain in the Vicinity of Leaky Reservoirs

The identification of leaks in the vicinity of reservoirs is critical to determining the feasibility of long-term geologic storage of CO2, methane, wastewater and other fluids. We have established a series of basic reservoir leakage scenarios and performed numerical analysis to characterize the associated pressure and strain fields created from injection into these leaky reservoir settings, as well from ambient loads, such as barometric pressure and earth tides. The end-member conceptual models evaluated include injection and ambient loads applied to: 1) a thin, horizontal layer with no leakage pathways (baseline), 2) a thin layer with a vertical conduit to the surface (e.g., a leaky fault or abandoned well) and 3) a thin layer with a conduit to an overlying aquifer. The pressure and strain fields are sensitive to the reservoir and confining unit geometry, material properties, the distribution, composition and compressibility of the pore fluid and the type of applied load (e.g., constant vs. cyclical injection, barometric pressure fluctuations, earth tides, etc.). Initial results from the numerical analyses indicate that an induced pressure change of 1 MPa in a formation at 1 km depth with a Young’s modulus of 10 GPa is sufficient to create vertical and horizontal strain signals as high as 2 με at a depth of 100 m. The signals predicted from leaky conditions compared to an intact reservoir differ by as much as 0.2 με in magnitude and evolve at different rates. We have developed instruments capable of measuring vertical and horizontal strain at a resolution of better than 5 nε and have recorded repeatable strain signals at 30 m depth in response to wastewater injection into the Avant field in Oklahoma. Based on these findings, strain signals predicted in the shallow subsurface (< 100 m) with numerical analysis are expected to be measurable, and this signal may be inverted to characterize the location and magnitude of leaks, as well as to characterize the geometry and hydromechanical properties of the formation and overlying confining units. This suggests that shallow strain measurements could be a cost effective method for characterizing and monitoring gas storage projects.

Robert, Moak↗

NASA NTRS1992

Thermal and dynamical regimes of single- and two-phase magmatic flow in dikes

The coupling between thermal and dynamical regimes of single- and two-phase magmatic flow in dikes, due to temperature-dependent viscosity and dissipation, was investigated using finite element calculations of magma flow in dikelike channels with length-to-width ratios of 1000:1 or more. Solutions of the steady state equations governing magma flow are obtained for a variety of conditions ranging from idealized plane-parallel models to cases involving nonparallel geometry and two-phase flows. The implications of the numerical simulations for the dynamics of flow in a dike-reservoir system and the consequences of dike entrance conditions on magmatic storage are discussed. Consideration is also given to an unmixing/self-lubrication mechanism which may be important for the lubrication of silicic magmas rising to the earth's surface in mixed magma ascent scenarios, which naturally segregates magma mixtures of two components with differing viscosities to minimize the driving pressure gradient.

Carrigan, Charles R.↗

DOE OSTI2024

3D modeling of deep borehole electromagnetic measurements with energized casing source for fracture mapping at the Utah Frontier Observatory for Research in Geothermal Energy

Here, we present a 3D numerical modelling analysis evaluating the deployment of a borehole electromagnetic measurement tool to detect and image a stimulated zone at the Utah Frontier Observatory for Research in Geothermal Energy geothermal site. As the depth to the geothermal reservoir is several kilometres and the size of the stimulated zone is limited to several 100 m, surface-based controlled-source electromagnetic measurements lack the sensitivity for detecting changes in electrical resistivity caused by the stimulation. To overcome the limitation, the study evaluates the feasibility of using a three-component borehole magnetic receiver system at the Frontier Observatory for Research in Geothermal Energy site. To provide sufficient currents inside and around the enhanced geothermal reservoir, we use an injection well as an energized casing source. To efficiently simulate energizing the injection well in a realistic 3D resistivity model, we introduce a novel modelling workflow that leverages the strengths of both 3D cylindrical-mesh-based electromagnetic modelling code and 3D tetrahedral-mesh-based electromagnetic modelling code. The former is particularly well-suited for modelling hollow cylindrical objects like casings, whereas the latter excels at representing more complex 3D geological structures. In this workflow, our initial step involves computing current densities along a vertical steel-cased well using a 3D cylindrical electromagnetic modelling code. Subsequently, we distribute a series of equivalent current sources along the well's trajectory within a complex 3D resistivity model. We then discretize this model using a tetrahedral mesh and simulate the borehole electromagnetic responses excited by the casing source using a 3D finite-element electromagnetic code. This multi-step approach enables us to simulate 3D casing source electromagnetic responses within a complex 3D resistivity model, without the need for explicit discretization of the well using an excessive number of fine cells. We discuss the applicability and limitations of this proposed workflow within an electromagnetic modelling scenario where an energized well is deviated, such as at the Frontier Observatory for Research in Geothermal Energy site. Using the workflow, we demonstrate that the combined use of the energized casing source and the borehole electromagnetic receiver system offer measurable magnetic field amplitudes and sensitivity to the deep localized stimulated zone. The measurements can also distinguish between parallel-fracture anisotropic reservoirs and isotropic cases, providing valuable insights into the fracture system of the stimulated zone. Besides the magnetic field measurements, vertical electric field measurements in the open well sections are also highly sensitive to the stimulated zone and can be used as additional data for detecting and imaging the target. We can also acquire additional multiple-source data by grounding the surface electrode at various locations and repeating borehole electromagnetic measurements. This approach can increase the number of monitoring data by several factors, providing a more comprehensive dataset for analysing the deep-localized stimulated zone. The numerical analysis indicates that it is feasible to use the combination of the energized casing and downhole electromagnetic measurements in monitoring localized stimulated zone at large depths.

58 GEOSCIENCES↗

DOE OSTI2023

Investigation of the Effect of Injected CO 2 on the Morrow B Sandstone through Laboratory Batch Reaction Experiments: Implications for CO 2 Sequestration in the Farnsworth Unit, Northern Texas, USA

About one million tons of CO 2 have been injected into the Farnsworth unit to date. The target reservoir for CO 2 injection is the Morrow B Sandstone, which is primarily made of quartz with lesser amounts of albite, calcite, chlorite, and clay minerals. The impact of CO 2 injection on the mineralogy, porosity, and pore water composition of the Morrow B Sandstone is a major concern. Although numerical modeling studies suggest that porosity changes will be minimal, significant alterations to mineralogy and pore water composition are expected. Given the implications for CO 2 storage effectiveness and risk assessment, it is crucial to verify the accuracy of theoretical model predictions through laboratory experiments. To this end, batch reaction experiments were conducted to model conditions near an injection well in the Morrow B Sandstone and at locations further away, where the CO 2 has been diluted by formation water. The laboratory experiments involved submerging thin sections of both coarse- and fine-grained facies of the Morrow B Sandstone in formation water samples with varying levels of CO 2 . The experiments were conducted at the reservoir temperature of 75 °C. Two experimental runs were conducted, one lasting for 61 days and the other for 72 days. The initial fluid composition used in the second run was the same as in the first. The mineralogy changes in the thin sections were analyzed using SEM and the Tescan Integrated Mineral Analyzer (TIMA), while changes in the composition of the formation water were determined using ICP-AES. During each experiment, a thin layer of white fine-grained particles consisting mainly of dolomite and silica formed on the surface of the thin sections, leading to significant reductions in Ca, Mg, and Sr in the formation water. This outcome is consistent with numerical model predictions that dolomite would be the primary mineral that would react with injected CO 2 and that silica would be oversaturated in the formation water. Changes in mineral abundance in the thin sections themselves were much less systematic than in the theoretical modeling experiments, perhaps reflecting heterogeneities in the mineral grain size surface area to volume ratios and mineral distributions in the thin sections not considered in the numerical models.

58 GEOSCIENCES↗

DOE OSTI2021

Connect the Dots: In Situ 4-D Seismic Monitoring of CO 2 Storage With Spatio-Temporal CNNs

4-D seismic imaging has been widely used in CO 2 sequestration projects to monitor the fluid flow in the volumetric subsurface region that is not sampled by wells. Ideally, real-time monitoring and near-future forecasting would provide site operators with great insights to understand the dynamics of the subsurface reservoir and assess any potential risks. However, due to obstacles such as high deployment cost, availability of acquisition equipment, exclusion zones around surface structures, only very sparse seismic imaging data can be obtained during monitoring. That leads to an unavoidable and growing knowledge gap over time. The operator needs to understand the fluid flow throughout the project lifetime and the seismic data are only available at a limited number of times. This is insufficient for understanding reservoir behavior. To overcome those challenges, we have developed spatio-temporal neural-network-based models that can produce high-fidelity interpolated or extrapolated images effectively and efficiently. Specifically, our models are built on an autoencoder, and incorporate the long short-term memory (LSTM) structure with a new loss function regularized by optical flow. We validate the performance of our models using real 4-D post-stack seismic imaging data acquired at the Sleipner CO 2 sequestration field. We employ two different strategies in evaluating our models. Numerically, we compare our models with different baseline approaches using classic pixel-based metrics. We also conduct a blind survey and collect a total of 20 responses from domain experts to evaluate the quality of data generated by our models. Finally, via both numerical and expert evaluation, we conclude that our models can produce high-quality 2-D/3-D seismic imaging data at a reasonable cost, offering the possibility of real-time monitoring or even near-future forecasting of the CO 2 storage reservoir.

4-D seismic imaging↗

DOE OSTI2019

Testing the efficacy of active pressure management as a tool to mitigate induced seismicity

Manipulation of subsurface stresses through fluid injection/extraction may induce seismicity through modulation of overpressure along faults. Globally, most industrial operations do not induce significant earthquakes. However, the possibility of inducing moderate to large earthquakes is recognized as a significant risk for basin-scale carbon dioxide (CO 2 ) storage operations (Niass et al., 2017). One possible method to reduce the occurrence of induced seismicity is to maintain only small (or zero) fluid overpressure changes in the subsurface through extraction of formation brine. For active pressure management (APM), operators may employ different procedures including extraction of various brine volumes and different configurations of injection/extraction well locations. However, the impact each these operational procedures on induced seismicity must be understood in order to develop management strategies. Here, we describe a numerical modeling workflow to evaluate the efficacy of APM techniques, that when applied to a specific site may be used to inform operational management strategies. We couple the multiphase reservoir model, NUFT, with the earthquake simulator, RSQSim and simulate seismicity induced by CO 2 injection into a hypothetical brine saturated reservoir. Reservoir overpressure is modified by co-extraction of brine at various fractions of the injected CO 2 volume from five potential wells. Our goal is to understand how the well-pair configuration and the extracted brine volume impact the total number and the maximum event magnitude. Additionally, the performance of each scenario is also evaluated in terms of the volume of CO 2 that can be stored before an earthquake above a tolerance threshold is induced.

58 GEOSCIENCES↗

DOE OSTI2021

Thermal and Mechanical Energy Performance Analysis of Closed-loop Systems in Hot-Dry-Rock and Hot-Wet-Rock Reservoirs

To understand the potential and limitations for recovering thermal and mechanical energy from closed-loop geothermal systems a collaborative study is underway that will investigate an array of system configurations, working fluids, geothermal reservoir characteristics, operational periods, and heat transfer enhancements. Closed-loop geothermal systems are distinguished from hydrothermal or enhanced geothermal systems (EGS) in that the working fluid only circulates through drilled boreholes. Principal objectives of this study are to determine upper limits for thermal and mechanical energy recovery and optimal operational and configuration parameters for each scenario. Teams of scientists and engineers are applying a suite of numerical simulation and analytical tools to model the heat recovery from closed-loop geothermal systems, and then optimizing operational and configuration parameters to maximize the thermal and mechanical energy recovery. Results from the suite of numerical simulators and analytical tools, such as outlet and inlet states and temperature profiles in the geothermal reservoir over time are intercompared to increase confidence in the analysis. This paper documents the study findings for closed-loop systems in hot-dry-rock and hot-wet-rock reservoirs, where water is the working fluid. The characteristics of the hot-dry-rock reservoir were based on the U.S. Department of Energy's Utah Frontier Observatory for Research in Geothermal Energy (FORGE) site, near Milford Utah. Two objective functions are defined to optimize the operational and configuration parameters of the system, one each for the recovery of mechanical and thermal energy over the period of operation. For both objective functions, a surface plant thermal to mechanical energy conversion factor and an energy drilling cost is required. In keeping with the study objectives the surface plant conversion factor is determined from a second-law of thermodynamics analysis of a generic binary plant, and drilling costs are based on those from the Utah FORGE site and current national electrical costs.

closed-loop geothermal systems↗

DOE OSTI2021

Thermal and Mechanical Energy Performance Analysis of Closed-loop Systems in Hot-Dry-Rock and Hot-Wet-Rock Reservoirs

To understand the potential and limitations for recovering thermal and mechanical energy from closed-loop geothermal systems a collaborative study is underway that will investigate an array of system configurations, working fluids, geothermal reservoir characteristics, operational periods, and heat transfer enhancements. Closed-loop geothermal systems are distinguished from hydrothermal or enhanced geothermal systems (EGS) in that the working fluid only circulates through drilled boreholes. Principal objectives of this study are to determine upper limits for thermal and mechanical energy recovery and optimal operational and configuration parameters for each scenario. Teams of scientists and engineers are applying a suite of numerical simulation and analytical tools to model the heat recovery from closed-loop geothermal systems, and then optimizing operational and configuration parameters to maximize the thermal and mechanical energy recovery. Results from the suite of numerical simulators and analytical tools, such as outlet and inlet states and temperature profiles in the geothermal reservoir over time are intercompared to increase confidence in the analysis. This paper documents the study findings for closed-loop systems in hot-dry-rock and hot-wet-rock reservoirs, where water is the working fluid. The characteristics of the hot-dry-rock reservoir were based on the U.S. Department of Energy’s Utah Frontier Observatory for Research in Geothermal Energy (FORGE) site, near Milford Utah. Two objective functions are defined to optimize the operational and configuration parameters of the system, one each for the recovery of mechanical and thermal energy over the period of operation. For both objective functions, a surface plant thermal to mechanical energy conversion factor and an energy drilling cost is required. In keeping with the study objectives the surface plant conversion factor is determined from a second-law of thermodynamics analysis of a generic binary plant, and drilling costs are based on those from the Utah FORGE site and current national electrical costs.

Closed-loop geothermal systems, hot-dry-rock, hot-↗

DOE OSTI2022

Voronoi Meshing to Accurately Capture Geological Structure in Subsurface Simulations

Mesh generation lies at the interface of geological modeling and reservoir simulation. Highly skewed or very small grid cells may be necessary to accurately capture the geometry of geological features, but the resulting poorly scaled or small grid cells can have a substantial negative impact on simulator accuracy and speed. One way to minimize numerical errors caused by gridding complex structures is to simulate on high-quality Voronoi meshes, which reduce grid orientation effects in fluid flow. This work presents a complete methodology to create Voronoi simulation grids, model fluid flow in complex geological systems, and visualize the results. A recently developed Voronoi meshing method that can automatically generate provably good unstructured meshes that conform to input surfaces creating closed volumes is used. Initially an analytical benchmark simulation is presented to validate the quality of the meshes and simulation results and demonstrate the superiority of simulation results using Voronoi meshes over flexed-hexahedral meshes on a domain with internal features. Next, meshes are created for test structures representing four of the most common geological features in the subsurface: layering, pinch-out, an interior lens that tapers to zero thickness on all sides and a fault with offset. Two benchmark flow simulations are run for each test structure. Finally, a realistic geological example for CO 2 injection into an anticline is simulated. Three realizations of the Voronoi mesh at the same resolution are generated for the simulations. Each mesh is highly refined near the injection wells and coarse in areas of less interest. These three meshes are used to model the CO 2 plume in the subsurface as it migrates to the top of the structure and then fills downward. Simulations on the meshes with randomly generated elements inside the input volumes each give slightly different fingering patterns for the viscous-unstable buoyant gas flow. The results presented in this work show a promising step towards utilizing fully automated Voronoi meshing for subsurface flow simulations in complex geology.

58 GEOSCIENCES↗

DOE OSTI2024

Predicting Geologic Behavior in Carbon Storage Projects Using Graph Neural Network

This study was invited to presented at NVIDIA's GTC conference to highlight the potential of Graph Neural Network as a novel and promising methodology for predicting pressure and saturation evolution in carbon storage projects. Carbon capture and storage (CCS) technology plays a pivotal role in mitigating greenhouse gas emissions, facilitating the transition to a low-carbon future. Effective management of subsurface reservoirs is essential to ensure the safe and efficient storage of captured carbon dioxide (CO₂). Accurate predictions of pressure and saturation over time are critical for evaluating the long-term performance and integrity of CCS projects. In recent years, Graph Neural Network (GNN) has emerged as a powerful framework for analyzing complex data in graph-structured domains. This abstract explores the application of GNN to forecast pressure and saturation evolution in carbon storage projects. Traditional numerical simulations of subsurface reservoirs have proven successful in providing pressure and saturation forecasts. However, these simulations involve massive amounts of computational effort and require extensive domain expertise for proper model calibration and validation. Graph Neural Operator offers an alternative approach that harnesses the inherent graph structure of reservoirs, where nodes represent reservoir grid cells and edges represent the geological connectivity between them.

Shih, Chung Yan↗

DOE OSTI2023

Controls on Barite (BaSO 4 ) Precipitation in Unconventional Reservoirs

Barite (BaSO 4 ) precipitation is one of the most ubiquitous examples of secondary sulfate mineral scaling in shale oil and gas reservoirs. Often, a suite of chemical additives is used during fracturing operations to inhibit the accumulation of mineral scales, though their efficacy is widely varied and poorly understood. This study combines experimental data and multi-component numerical reactive transport modeling to offer a more comprehensive understanding of the geochemical behavior of barite accumulation in shale matrices under conditions typical of fracturing operations. A variety of additives and conditions are individually tested in batch reactor experiments to identify the factors controlling barite precipitation. Our experimental results demonstrate a pH dependence in the rate of barite precipitation, which we use to develop a predictive model including a pH-dependent term that satisfactorily reproduces our observations. Further, this model is then extended to consider the behavior of three major shale samples of highly variable mineralogy (Eagle Ford, Marcellus, and Barnett). This data-validated model offers a reliable tool to predict and ultimately mitigate against secondary mineral accumulation in unconventional shale reservoirs.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2024

Stochastic reconstruction of fracture network pattern using spatial point processes

Fracture spatial patterns can strongly affect fluid flow in the subsurface. Proximity and distribution of fractures control reservoir flow behavior over various length scales. In many studies, however, simplified geometrical patterns are generated for fractures which may lead to unrealistic subsurface models. Here we introduce a new method for characterization and modeling of fracture spatial patterns based on outcrop observations. We use Ripley's K-function to characterize the arrangement of fracture barycenters and intersection points over various length scales. In addition, we use semivariograms to quantify spatial correlation in fracture intensity maps. Using this information, we develop a stochastic algorithm that generates two-dimensional fracture network realizations with spatial properties similar to those of a real fracture network measured in the field. Numerical simulation models indicate that the generated fracture realizations exhibit similar flow behaviors as that of the original fracture network. Furthermore, such modeling tools expand and improve our capability in building representative fracture models and in quantification of uncertainty in naturally and hydraulically fractured reservoirs.

58 GEOSCIENCES↗

DOE OSTI2020

Deterministic and Statistical Modeling of a New Thermal Breakout Technology for Measuring the Maximum Horizontal In-Situ Stress

The current state-of-the-art technology for in-situ stress measurements involves an integrated approach that combines borehole breakout observations, drilling-induced tensile fractures, and hydraulic fracturing tests (i.e., “mini-fracs”). This methodology has achieved wide application in the oil and gas industry but has several limitations that often prevent successful in-situ stress measurements. One major limitation is that breakouts do not appear in all boreholes and are generally only a natural occurrence that cannot easily be controlled. Because borehole breakouts are used to directly measure the maximum horizontal in-situ stress magnitude, the absence of borehole breakouts presents a major data gap for in-situ stress measurements. In response to this data gap, a new US Department of Energy (US DOE)-sponsored thermal breakout technology that will provide a method for thermally inducing borehole breakouts and allow the consistent measurement of the maximum horizontal stress magnitude is currently in development. This thermal breakout technology involves heating the borehole and increasing the thermoelastic compressive stress in the rock until a breakout develops, which can be directly correlated to the maximum horizontal stress magnitude. The first step in this project was an analytical modeling study of the thermal breakout process. Based on the Kirsch solution (Kirsch 1898), a deterministic and statistical analysis was performed on the pertinent parameters that influence the maximum horizontal stress calculation. As a result of the analysis, the findings indicate that the thermal breakout technology is feasible and provides improved accuracy and/or an enhanced ability to measure the maximum horizontal stress. Future work as part of this US DOE-sponsored project includes additional validation through more detailed numerical modeling, laboratory testing, and field testing of the thermal breakout technology.

58 GEOSCIENCES↗

DOE OSTI2025

GEOS-DEV/FORGE

This dataset is a repository that provides the input deck for numerical models associated with two Utah FORGE research projects: "Closing the Loop Between In situ Stress Complexity and EGS Fracture Compexity" (Project Number: 2-2446) and "Coupled Investigation of Fracture Permeability Impact on Reservoir Stress and Seismic Slip Behavior" (Project Number: 5-2428). Three kinds of simulation decks are included: (i) hydraulic fracturing simulation (HydroFrac folder), (ii) phase-field simulation (PhaseField folder), and (iii) thermo-hydro-mechanical simulation (ThermoHydroMech folder).

Cusini, Matteo [Lawrence Livermore National Labora↗

DOE OSTI2020

Fast Computational Algorithms for Partial Differential Equations and Uncertainty Quantifications

This project concerned the construction, testing and analysis of computational algorithms for solving parameterized and stochastic partial differential equations. The study and understanding of equations of this type is of fundamental importance in numerous engineering and scientific applications. Examples include simulation of plasma dynamics in models of electric propulsion and nuclear fusion, simulation of multiphase flows, such as the flow of water, gas and oil in reservoirs, and structural analysis of the dependence of structures on materials. Parametrization is used in such settings when properties of the models such as viscosity of fluids or electric resistivity of materials are not precisely understood and instead are treated as random variables. The resulting solutions are themselves random, and having such solutions will enable engineers to use probabilistic methods to assess the likelihood of events, for example, whether a pollutant in a liquid will exceed a limit, and to use such analyses to develop ways to ensure positive outcomes. Construction of accurate (high resolution) computational solutions is expensive, requiring significant computer time and computational resources, and there is need to reduce computational cost to make simulation useful and effective. The aim of the project was to construct algorithms to efficiently compute surrogate solutions to parameterized problems to allow for efficient and accurate simulation. The technical approach used focused on two related strategies, based on rank-reduction methods and reduced-order models. These methods construct surrogate solutions of parameter-dependent models by projection or interpolation into low-dimensional approximation spaces. Cost savings are achieved if the low-dimensional spaces can be identified and constructed efficiently and if the resulting low-dimensional algebraic systems can be solved cheaply. Accomplishments include: Theoretical and empirical demonstration of the effectiveness of fast multigrid solution strategies for computing low-rank representations of parameter-dependent solutions to discrete partial differential equations, including the first proof establishing so-called textbook convergence properties for low-rank methods. Development of efficient solution algorithms for solving nonlinear parameter-dependent partial differential equations used in models of fluid dynamics. Developent of efficient algorithms for low-rank representation of solutions of time-dependent simulations of fluid dynamics using multi-dimensional tensor representations of solutions.

97 MATHEMATICS AND COMPUTING↗