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DOE OSTI2024

Simulating water dynamics related to pedogenesis across space and time: Implications for four-dimensional digital soil mapping

Digital soil mapping (DSM) relies on machine-learning and geostatistics to represent soil property observations across space. DSM techniques are powerful but often empirical, being limited to the quality and density of point samples. Water dynamics are closely related to soil variability, and the physics that govern water movement are well known. Hydrological properties can hence be simulated by physical models through space and time, unveiling key characteristics about soils. We propose the use of hydrologic models to map soils across the surface (2D), depth (1D), and time (1D)–which provides a 4D approach to digital soil mapping (4DSM). The Distributed Hydrology Soil Vegetation Model (DHSVM) was applied to a watershed currently under pasture. Moisture sensors and wells were installed at different depths in the watershed on summit, sideslope and toeslope positions to validate the model. DHSVM simulations of soil moisture distribution and depth to saturation were performed during the hydrological year (October 2008-September 2009). Clusters of similar pixels based on soil moisture values were determined using Dynamic Time Warping (DTW) to align temporal data and K-means. Clustering was performed both seasonally and for the entire year. Temporal patterns simulated by DHSVM matched measurements given by moisture sensors and wells. Seasonal clusters differed from the annual cluster. Distinct clusters were observed for each season and with depth, showing that spatiotemporal soil variability is lost when statically assessing soils. Spatiotemporal clusters corroborated field observations of fragipan occurrence not explicitly spatially mapped by Soil Survey Geographic Database (SSURGO). If a connection can be made between water and soils, static and dynamic soil variability can be predicted using physically based hydrologic models. Hydrologic models can benefit soil mapping by enabling reliable 4D simulation of water dynamics, which are fundamental to soil variability and soil classification and directly relate to biological, physical and chemical soil processes not captured by typical soil sampling protocols.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2025

Thermal shock resistance of lightweight cements developed for geothermal conditions

Cements are a critical component in well construction, as they act to prevent well fluid and gas escape, prevent corrosion of the casing, and strengthen the wellbore to prevent deformation. Under the high temperature/pressure conditions common in geothermal systems, the injection of cold water for energy production is expected to induce cyclic damage to the borehole cement through the rapid temperature fluctuations. These “thermal shocks” are expected to cause casing shrinkage, annulus formation, and cement tensile stresses. To understand the effect of cold water injection on the wellbore environment, a set of rock-cement-steel samples were created to simulate the structure of a geothermal well. Lightweight thermally-insulating cement blends were tested under thermal shock conditions in this study. In each test, the samples were pressurized to an effective pressure of ~3.5 MPa and placed at high temperatures. Thermal shocks were performed by injecting cold water (~10-15 °C) through the samples at a constant rate while keeping the samples at high temperatures until the sample temperature stopped decreasing and deformation ceased. Eight thermal shock tests were conducted with each sample – two at 100 °C and six at 200 °C. Post-tests analysis was then conducted by cutting open each sample to examine the damage in each component of the simulated wellbore. Experimental results suggest that all samples experienced similar degrees of axial and lateral contraction during cold water injection, but for the most part this contraction is recoverable when injection halts. Post-test analysis revealed that fly ash cenosphere pre-treatment had the best effect on improving thermal shock resistance in the cement blends. Thermomechanical modeling of likely stress paths experienced by the cements during heating/cooling cycles shows that elasto-plastic cement constitutive behavior results in most plastic strain occurring during the initial heating steps, with mostly elastic strain occurring during the thermal shock cycles. In conclusion, this agrees with experimental evidence, suggesting that cement damage from shocking occurs via other mechanisms such as chemical alteration, corrosion, and fatigue.

36 MATERIALS SCIENCE↗

DOE OSTI2021

A missing layer in COVID-19 studies: Transmission of enveloped viruses in mucus-rich droplets

Here we evaluate the influence of mucus layers on the evaporation time and transport of enveloped viruses, including SARS-CoV-2. Enveloped viruses must remain moist to be fully infective. Yet, the Wells model based on water droplets divides respiratory droplets into either quickly evaporated aerosolized particles termed droplet nuclei (<10 s) or liquid droplets that fall to the nearest surface, leaving no physical mechanism for airborne transmission of fully infective enveloped viruses over large distances (greater than a few meters). Yet, the role of mucus layers on evaporation times has not been considered even though the formation of mucus shells around liquid cores of respiratory droplets has been shown experimentally. Here we show that mucus shells increase the drying time by orders of magnitude so that enveloped virions may remain well hydrated and, thus, fully infective at substantial distances. Further, this provides a mechanism by which infective enveloped virus particles can transmit as aerosols within buildings and between buildings over extended distances. This analysis is important because public health agencies typically follow the Wells model to establish health policies including social/physical distancing guidelines.

60 APPLIED LIFE SCIENCES↗

DOE OSTI2024

A tri-level optimization model for interdependent infrastructure network resilience against compound hazard events

Resilient operation of interdependent infrastructures against compound hazard events is essential for maintaining societal well-being. To address consequence assessment challenges in this problem space, we propose a novel policy-guided tri-level optimization model applied to a proof-of-concept case study with fuel distribution and transportation networks – encompassing one realistic network; one fictitious, yet realistic network; as well as networks drawn from three synthetic distributions. Mathematically, our approach takes the form of a defender-attacker-defender (DAD) model—a multi-agent tri-level optimization, comprised of a defender, attacker, and an operator acting in sequence. Here, in this study, our notional operator may choose proxy actions to operate an interdependent system comprised of fuel terminals and gas stations (functioning as supplies) and a transportation network with traffic flow (functioning as demand) to minimize unmet demand at gas stations. A notional attacker aims to hypothetically disrupt normal operations by reducing supply at the supply terminals, and the notional defender aims to identify best proxy defense policy options which include hardening supply terminals or allowing alternative distribution methods such as trucking reserve supplies. We solve our DAD formulation at a metropolitan scale and present practical defense policy insights against hypothetical compound hazards. We demonstrate the generalizability of our framework by presenting results for a realistic network; a fictitious, yet realistic network; as well as for three networks drawn from synthetic distributions. Additionally, we demonstrate the scalability of the framework by investigating runtime performance as a function of the network size. Steps for future research are also discussed.

11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗

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 OSTI2020

Geochemical narrowing of cement fracture aperture during multiphase flow of supercritical CO2 and brine

For carbon capture and storage operations, wells are used as a necessary conduit for injecting CO2 at depth, but they can also act as leakage conduits if the cement seals are compromised. The specific objective of this research was to investigate the nature of self-healing of a fracture within cement under multiphase flow of CO2 and brine, and to compare this to a newly developed model. This was accomplished through a laboratory investigation, flowing reacting supercritical CO2 and brine through a cement fracture with multiphase flow. The influent end of the fracture was degraded as evidenced by the formation of cracks across the surface. At the effluent end of the fracture, the initial aperture of 137 µm was reduced to 50 µm. This reduction by 87 µm was similar to results from a recently developed model tested in this study, which predicted an aperture reduction of 80 µm. Our results demonstrated that the model accurately predicted the reactions which occurred in this multiphase flow system.

wellbore, cement, fracture, multiphase-flow, super↗

DOE OSTI2021

A comprehensive analysis of transient pressure and rate data from CO 2 storage projects in a depleted pinnacle reef oil field complex, Michigan, USA

Pressure and rate data are commonly recorded as part of a basic monitoring program in CCS projects. This paper discusses the application of multiple analytical techniques to interpret pressure and rate transient data from CO 2 injection and storage operations. The techniques of interest, i.e., injection-falloff analysis, injectivity/productivity index analysis and pressure pulse arrival time analysis, are commonly used in the oil and gas industry to assess reservoir properties, but not well known in the CCS literature (especially the last two). Injection-falloff analysis involves log-log pressure derivative plotting for the falloff data and history-matching of the entire injection-falloff sequence to determine permeability. In the injectivity/productivity index analysis, rate-normalized pressure buildup is plotted against material balance time or ratio of cumulative injection to injection rate to determine the injectivity index (ratio of injection rate to stabilized pressure buildup) which can be related to the permeability-thickness product. The arrival time analysis identifies the arrival of a pressure disturbance (~0.1 psi change from ambient) to determine the hydraulic diffusivity from which permeability can be estimated. The applicability of these techniques is demonstrated via illustrative examples from multiple wells in different pinnacle carbonate reefs undergoing CO 2 -EOR in Northern Michigan. The paper ends with a discussion of the relative merits of each interpretive technique, as well as recommendations that could be useful for other field projects.

42 ENGINEERING↗

DOE OSTI2024

Impact of cement composition, brine concentration, diffusion rate, reaction rate and boundary condition on self-sealing predictions for cement-CO 2 systems

Geological CO 2 storage (GCS) plays an important role in curbing CO 2 emissions by reducing the carbon footprint of difficult to decarbonize operations and achieve negative CO 2 emissions through activities like Bioenergy with Carbon Capture and Storage (BECCS) and Direct Air Carbon Capture and Storage (DACCS). Leakage of CO 2 through wells is an important concern when it comes to deployment of large-scale GCS. Existing wells in sites that are otherwise suitable for GCS can act as conduits for stored CO 2 to escape the reservoir. There is broad consensus that the main risk of leakage through wellbores is via fractures/damaged pathways. The results from several studies evaluating the permeability evolution of cement fractures in wells upon leakage of CO 2 agree that smaller fracture apertures, slower brine velocities and higher brine residence times promote self-sealing of fractures by mineral precipitation. Quantitatively, however, the differences in sealing conditions are significant and are typically attributed to differences in experimental conditions or model assumptions. Here we examine the sensitivity of our model, describing CO 2 leakage through wellbores, to cement composition, brine concentration, diffusion rates, and reaction rates. We also evaluate the impact of the boundary condition to allow comparisons between observations from experiments performed at constant flow rate and model predictions made at constant pressure conditions. Our results show that diffusion and reactions rates have the most impact on the self-sealing criteria for cement-CO 2 systems. In addition, conditions associated with self-sealing of fractures at constant flow rate require longer fractures, smaller fracture apertures, and slower velocities than under constant pressure.

58 GEOSCIENCES↗

DOE OSTI2024

An assessment of controlled source EM for monitoring subsurface CO 2 injection at the wyoming carbonSAFE geologic carbon storage site

Here we evaluate if electromagnetic (EM) geophysical methods for monitoring geologic carbon storage (GCS) efforts at the Wyoming CarbonSAFE project adjacent to the Dry Fork Station power plant near Gillette, Wyoming. This first involved acquiring both electric and magnetic fields at eleven different locations ranging in distance from immediately adjacent to 4 km from the plant. Passive EM measurements were made to provide spectral EM noise measurements generated by electricity production at the plant and to determine if useful magnetotelluric (MT) data can be successfully collected in the region. The processed data indicate that useful MT data can be collected as long as the site is located more than 2km away from the power plant as well as active roads and rail lines. Controlled source EM data were collected using three different source configurations, two of which connected to steel casings used to complete the injection wells. Comparing the EM noise measurements to the CSEM data show measurable electric and magnetic field signals at all sites. Next a series of three-dimensional (3D) numerical models were built that simulate resistivity changes caused by the proposed CO2 injection at depths ranging from 2.4 to 3.0km. These models were used to simulate various EM measurement configurations. The modeling shows that casing-source CSEM monitoring can provide sensitivity to the injected CO 2 if source electrodes are connected to the bottom of one or both of the injection wells.

58 GEOSCIENCES↗

DOE OSTI2021

Study of the film boiling heat transfer and two-phase flow interface behavior using image processing

Here, in this study, based on a small-scale quench test facility, the two-phase flow interface behavior during quench transients is visualized and analyzed utilizing an image processing framework. The high-fidelity experimental results obtained for two-phase flow in the current framework can support various studies both in the time domain and in the frequency domain. In particular, visualization of the data obtained from different heating surfaces under different test conditions are used to perform a full-scale transient 2-D vapor film reconstruction. The liquid-vapor interface variations in various heat transfer regimes as well as at the initial film breakup point can be directly obtained through the processed data. Moreover, the temporal variation of the interfacial wave frequency approaching quench is investigated in detail. Based on the high-resolution data obtained for the liquid-vapor interface, the detailed phase velocity and temperature profiles are obtained through theoretic analysis, based on which the film boiling heat transfer coefficient (HTC) can be determined. In addition, an improved film boiling HTC model is developed considering the effects of wall superheat, liquid subcooling temperature, vapor film thickness as well as fluid properties. The model is found to predict film boiling HTC well within 15% error.

42 ENGINEERING↗

DOE OSTI2023

Detailed mixing measurements from single-hole converging ECN injectors using Rayleigh scattering

Time resolved liquid and vapor fields of dodecane and oxymethylene ethers are measured from Spray A-3 and Spray D using high speed Rayleigh scattering and diffuse back illumination at the Engine Combustion Network (ECN) Spray A condition of 900 K and 22.8 kg/m 3 . Global quantities including mixture fraction, vapor and liquid penetration, as well as spreading angle are measured. The mixture fraction fields and vapor penetration profiles are well predicted by the 1-D Musculus-Kattke model. The mixture fraction field and vapor penetration from Spray A-3 are similar to those measured from Spray A in previous works. Spray D exhibits higher mixture fraction fields and vapor penetration due to its larger nozzle diameter. The quasi-steady mixture fraction fields from these injectors scale well when distance from the injector is normalized by the nozzle diameter. The turbulent dissipation structures were also analyzed based on the orientation, thickness, and magnitude of the mixing layers. Furthermore, the orientation and thickness are similar to other measurements in atmospheric gas jets, while the magnitude is lower. The thickness and magnitude are subject to uncertainties due to limitations in the imaging resolution of the system but still provide an order of magnitude as a useful reference for comparison against computational fluid dynamic simulations.

33 ADVANCED PROPULSION SYSTEMS↗

DOE OSTI2026

Life cycle analysis of hydrogen production via methane pyrolysis using plasma arc

Steam methane reforming of natural gas is the primary method of producing hydrogen in the United States, accounting for 95% of all hydrogen produced there. Methane pyrolysis, an alternative production pathway that decomposes natural gas into solid carbon and hydrogen, both eliminates CO 2 emissions associated with methane reforming and allows for additional income from carbon black. A life-cycle inventory of this process has been developed using ASPEN Plus to model the methane pyrolysis (plasma arc) process. From well to gate, hydrogen production via methane pyrolysis produces 2.78 kg CO 2 e/kg H 2 of greenhouse gas emissions using mass allocation of emissions between hydrogen and carbon black coproducts. The well-to-gate emissions are mainly driven by electricity consumption (∼38 kW h/kg H 2 ), which accounts for 81% of the emissions; if renewable electricity is used, well-to-gate emissions can be reduced to −0.448 kg CO 2 e/kg H 2 .

08 HYDROGEN↗

DOE OSTI2020

A geospatial model for the analysis of time-dependent land subsidence induced by reservoir depletion

Land subsidence due to fluid depletion is an outcome of physical processes operating across a wide range of time and length scales. Although geomechanical models are crucial to simulate reservoir compaction and predict its long-term fate, their use across large regions often bears prohibitive computational costs. To overcome this obstacle, this paper proposes a simplified modelling framework consisting of (i) a near-field numerical solver simulating the coupling between fluid flow and rock deformation with reference to a simplified one-dimensional geometry and (ii) a far-field geospatial algorithm mapping ground settlements across a region through the superposition of poroelastic computations at multiple wells. The model computes the delay between depletion history and reservoir compaction in proximity of a producing well by assuming basal depletion of a fluid-saturated deformable disk, while the Geertsma solution of nucleus of strain is used to extrapolate the impact of such time-varying reservoir compaction around the well. This approach has been used to back-analyze the spatio-temporal progression of subsidence at the Groningen gas field. The results are compared against measurements collected over 50 year of production at 25 benchmark locations scattered over an area of 900 km 2 . It is shown that coupled simulations based on average values of rock compressibility and permeability lead to nonlinear trends of subsidence evolution in good agreement with field measurements, while uncoupled analyses overpredict settlements by more than 70%. Lastly, synthetic forecasts based on different rates of depletion were provided. The results suggest that slower depletion rates lead to lower subsidence at a given time, and that residual subsidence may continue to develop for several decades after interruption of production activities.

58 GEOSCIENCES↗

DOE OSTI2025

Inferring fracture dilation and shear slip from surface deformation utilising trained surrogate models

An important task in energy and CO 2 storage (sequestration) in the subsurface is to verify that the surrounding fractures and faults are not activated, acting as leakage pathways. This is achievable through effective and efficient Measurement, Monitoring and Verification (MMV) plans. In this work, two surrogate models are trained to captures dilation (opening) and shear deformation of fractures, and the associated surface deformation. The trained surrogate model, based on conditional Generative-Adversarial Networks (cGAN) receives fracture apertures from dilational fractures together with fracture slips from shear fractures and predicts the combined surface deformation. An inversion algorithm based on Bayesian framework is proposed to identify the geometry of both types of fractures, as well as volume of dilational fractures and deformation moment induced by shear fractures, all from the measured surface deformation data. The inversion algorithm utilises the Differential Evolution (DE) optimisation technique that has the superior performance in finding the global minimum of cost function. The proposed surrogate-assisted inversion successfully inferred the unknown dip, dip direction and the volume of the dilational fractures as well as the induced deformation moment in shear fractures. The model was further tested for the inversion of a field hydraulic fracturing tilt dataset applying different scenarios with varying unknowns to show the model's performance, as well as incorporating shear deformation for better match with the observed data.

Dilation and shear↗

DOE OSTI2025

Modeling supercritical CO2 injection induced rupture of a minor fault embedded in a poroelastic layered reservoir-caprock system

CO2 injection for geologic carbon sequestration involves hydromechanical processes that lead to changes in fluid pressure and stresses that can activate existing faults. This paper presents a new method and workflow of modeling fault activation considering more complex three-dimensional geometry of natural faults using the TOUGH-FLAC multiphase fluid flow and geomechanical simulator. In this method and workflow, FLAC3D mechanical interfaces and TOUGH3 finite volume elements are discretized using computer aided design and gridding software along with a tailored mesh translation routine. The method and workflow are demonstrated with a model of a curved minor fault embedded in a poro-elastic layered reservoir-caprock system. The model is used for a comprehensive sensitivity analysis of fault responses to fault length, injection mass rate, injection schedule, well-fault distance, and well locations versus fault location. Four metrics (CO2 plume, shear state of fault, pressure and stress path at fault monitoring points) are selected to assess CO2 migration, pressure change, and the reactivation of faults. The results reveal that CO2 can bypass around the tip of the minor impermeable fault, building up pressure and poro-elastic stress on both sides that tends to impede fault rupture. Our study shows the benefit of carefully designing the injection to achieve the targeted final storage volume, starting at a relatively low rate for considerable time, and then ramping up the injection rate to the full rate of injection. The initial low injection has two distinct benefits: (1) it allows for the formation of an extensive CO2 plume with a much higher mobility through a low viscosity that will result in a lower pressure for a given injection rate, and (2) it allows for gradual build-up of horizontal poro-elastic stress within the reservoir that will tend to impede activation of steeply dipping faults. The injection scenario starting at a low injection rate, denoted here as conservative injection, can significantly reduce the risk of fault activation as high fluid mobility and reservoir strengthening poro-elastic stress has been established long before reaching the peak injection rates. Moreover, simultaneous injection in two injection wells on both sides of fault can provide further reservoir strengthening through poro-elastic stress buildup acting on a fault under normal faulting stress regime. The findings presented in the paper can provide practical and effective guidance on long-term, safe, and reliable geological CO2 storage.

Cao, Meng↗

DOE OSTI2022

Facile polymerization in a bicellar template to produce polymer nano-rings

We report a well-defined discoidal bicelle composed of three lipids, specifically zwitterionic long-chain 1,2-dipalmitoyl phosphocholine (DPPC) and short-chain 1,2-dihexanoyl phosphocholine (DHPC) doped with anionic 1,2-dipalmitoyl phosphoglycerol (DPPG) provides a generalized template for the synthesis of hydrophobic polymer nano-rings. The lipid molar ratio of DPPC/DHPC/DPPG is 0.71/0.25/0.04. The detailed investigation and discussion were based on styrene but tested on three other vinyl monomers. The structure of nano-rings is identified through the detailed analysis of small angle X-ray/neutron scattering (SAXS and SANS) data and transmission electron micrographs (TEM), supported by the differential scanning calorimetric (DSC) data before and after polymerization. The investigation covers samples with a styrene-to-lipid ratio ranged varied from 1:50 to 1:10.The styrene monomers are initially located at both the discoidal planar (long-chain lipid rich) and rim (short-chain lipid rich) regions. During polymerization, they migrate to the more fluid rim region-section. The formation mechanism involves the interplay of hydrophobic interaction, mismatched miscibility of polystyrene between the ordered and disordered phases, and crystallinity of the long lipid acyl chains. This facile synthesis is proven applicable for several hydrophobic monomers. The well-defined nano-rings greatly enhance the interfacial area and have the potential to be the building blocks for functional materials, if monomers are incorporated with desirable functions, for future applications. Crosslinked polymer nano-rings are produced in well-defined discoidal bicelles. The enhanced immiscibility in the planar region drives the hydrophobic polymers to the fluidic rim. The platform can be generalized for synthesis of other hydrophobic polymer nano-rings.

36 MATERIALS SCIENCE↗

DOE OSTI2022

Biorefinery upgrading of herbaceous biomass to renewable hydrocarbon fuels, Part 1: Process modeling and mass balance analysis

Process simulation has long been a well-established tool to track key operational, design, and mass and energy balance metrics for pre-commercial technologies such as advanced lignocellulosic biofuels. While tools such as this are well-documented in the public literature around 2nd-generation cellulosic ethanol technologies (which have been scaled up to commercial deployment to some degree over the past decade), such models and analysis information remain more sparse for more complex biorefinery pathways focused on producing drop-in hydrocarbon fuels and blend-stocks, particularly regarding information required to support air emissions or other environmental analysis. In this work, we summarize key details for an established "design case" modeling the conversion of herbaceous lignocellulosic biomass into a renewable diesel hydrocarbon blend-stock based on a representative lipid pathway from oleaginous yeast. The process is based on a biochemical deconstruction and upgrading approach utilizing deacetylation and dilute acid pretreatment, followed by enzymatic hydrolysis, fermentation, and catalytic upgrading of hydrolysate sugars to fuels. We provide key mass and energy balance outputs from the process models, with accompanying stream tables and component-level flowrates. A total of 12 model scenarios are presented spanning two feedstocks, three biorefinery scales, and two processing approaches for the lignin/residual solids waste streams. This "Part 1" manuscript presents the resulting impacts across the 12 cases on fuel yields and key output streams, focused here on direct biorefinery air emissions for selected components including CO 2 as well as sulfur (SOx) and nitrogen oxides (NOx). In the context of cleaner production, the latter focus on selected biorefinery air emission outputs establishes an initial baseline estimate and accompanying framework of the model cases, upon which an accompanying "Part 2" study will build to refine the values for these and other air pollutants across these scenarios, also considering mitigation options to comply with applicable regulatory standards. We also highlight further optimization opportunities based on potential tradeoffs identified here between air emissions versus life-cycle greenhouse gas profiles attributed to the disposition of lignin/residual solids.

09 BIOMASS FUELS↗

DOE OSTI2020

NSFnets (Navier-Stokes flow nets): Physics-informed neural networks for the incompressible Navier-Stokes equations

In the last 50 years there has been a tremendous progress in solving numerically the Navier-Stokes equations using finite differences, finite elements, spectral, and even meshless methods. Yet, in many real cases, we still cannot incorporate seamlessly (multi-fidelity) data into existing algorithms, and for industrial-complexity applications the mesh generation is time consuming and still an art. Moreover, solving ill-posed problems (e.g., lacking boundary conditions) or inverse problems is often prohibitively expensive and requires different formulations and new computer codes. Here, we employ physics-informed neural networks (PINNs), encoding the governing equations directly into the deep neural network via automatic differentiation, to overcome some of the aforementioned limitations for simulating incompressible laminar and turbulent flows. We develop the Navier-Stokes flow nets (NSFnets) by considering two different mathematical formulations of the Navier-Stokes equations: the velocity-pressure (VP) formulation and the vorticity-velocity (VV) formulation. Since this is a new approach, we first select some standard benchmark problems to assess the accuracy, convergence rate, computational cost and flexibility of NSFnets; analytical solutions and direct numerical simulation (DNS) databases provide proper initial and boundary conditions for the NSFnet simulations. The spatial and temporal coordinates are the inputs of the NSFnets, while the instantaneous velocity and pressure fields are the outputs for the VP-NSFnet, and the instantaneous velocity and vorticity fields are the outputs for the VV-NSFnet. This is unsupervised learning and, hence, no labeled data are required beyond boundary and initial conditions and the fluid properties. The residuals of the VP or VV governing equations, together with the initial and boundary conditions, are embedded into the loss function of the NSFnets. No data is provided for the pressure to the VP-NSFnet, which is a hidden state and is obtained via the incompressibility constraint without extra computational cost. Unlike the traditional numerical methods, NSFnets inherit the properties of neural networks (NNs), hence the total error is composed of the approximation, the optimization, and the generalization errors. Here, we empirically attempt to quantify these errors by varying the sampling (“residual”) points, the iterative solvers, and the size of the NN architecture. For the laminar flow solutions, we show that both the VP and the VV formulations are comparable in accuracy but their best performance corresponds to different NN architectures. The initial convergence rate is fast but the error eventually saturates to a plateau due to the dominance of the optimization error. For the turbulent channel flow, we show that NSFnets can sustain turbulence at , but due to expensive training we only consider part of the channel domain and enforce velocity boundary conditions on the subdomain boundaries provided by the DNS data base. We also perform a systematic study on the weights used in the loss function for balancing the data and physics components, and investigate a new way of computing the weights dynamically to accelerate training and enhance accuracy. In the last part, we demonstrate how NSFnets should be used in practice, namely for ill-posed problems with incomplete or noisy boundary conditions as well as for inverse problems. We obtain reasonably accurate solutions for such cases as well without the need to change the NSFnets and at the same computational cost as in the forward well-posed problems. As a result, we also present a simple example of transfer learning that will aid in accelerating the training of NSFnets for different parameter settings.

97 MATHEMATICS AND COMPUTING↗