Engineering Papers⌕ Search

SEARCH · Engineering Papers

Results for “unsaturated flow”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 19 records
All sourcesNASA NTRSDOE OSTI
DOE OSTI2022

Hydrobiological Mechanism Controlling the Synergistic Effects of Unsaturated Flow and Soil Organic Matter on the Degradation of Emerging Organic Contaminants in Soils

We report hydrology is a key factor influencing microbial degradation of emerging organic contaminants (EOCs) in soils, but the underlying mechanisms are not clear. In this study, biotic and abiotic column experiments were performed to investigate the removal and degradation of five EOCs in soils with different soil organic matter (SOM) contents under saturated and unsaturated flow conditions. In biotic experiments, 54–90% of bisphenol A (BPA) and 9–22% of ibuprofen (IBU) were removed from the aqueous phase of saturated columns due to adsorption and biodegradation. The biodegradation removed 26–65% of BPA and 1–22% of IBU. Decreasing soil pore water saturation from 100 to 80% increased BPA removal to 97–100% and IBU removal to 42–43% due to increased biodegradation (67–81% for BPA and 36–39% for IBU). No significant removal of BPA and IBU was observed in SOM-removed soils under saturated and unsaturated flow conditions. The desaturation did not influence sorptive losses of BPA (< 27%) and IBU (< 7%), suggesting their negligible adsorption at air–water interfaces but increased biodegradation of BPA and IBU sorbed at SOM–water interfaces. The study shows that soil drying and SOM can synergistically degrade BPA and IBU but have no effect on recalcitrant carbamazepine, tetracycline, and ciprofloxacin.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2017

Numerical convergence improvements for porflow unsaturated flow simulations

Section 3.6 of SRNL (2016) discusses various PORFLOW code improvements to increase modeling efficiency, in preparation for the next E-Area Performance Assessment (WSRC 2008) revision. This memorandum documents interaction with Analytic & Computational Research, Inc. (http://www.acricfd.com/default.htm) to improve numerical convergence efficiency using PORFLOW version 6.42 for unsaturated flow simulations.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2017

Effect of Unsaturated Flow Modes on Partitioning Dynamics of Gravity-Driven Flow at a Simple Fracture Intersection: Laboratory Study and Three-Dimensional Smoothed Particle Hydrodynamics Simulations

In this work, we study unsaturated gravity-driven free-surface flows on a synthetic surface intersected by a horizontal fracture and the importance of flow modes (droplet; rivulet) on the partitioning behavior at the fracture intersection. We present laboratory experiments, three-dimensional smoothed particle hydrodynamics (SPH) simulations using a heavily parallelized code, and an analytical solution. The flow-rate-dependent mode switching from droplets to rivulets is reproduced by the SPH model, and the transition ranges agree with the laboratory experiments. We show that flow modes heavily influence the bypass behavior. Flows favoring the formation of droplets exhibit a much stronger bypass capacity compared to rivulet flows, where nearly the whole fluid mass is initially stored within the horizontal fracture. This behavior is demonstrated for a multi-inlet laboratory setup where the inlet- specific flow rate is chosen so that either a droplet or rivulet flow persists. The effect of fluid buffering within the horizontal fracture is presented in terms of dimensionless fracture in- flow so that characteristic scaling regimes can be recovered. For both cases (rivulets and droplets), the flow within the horizontal fracture transitions into a Washburn regime until a critical threshold is reached and the bypass efficiency increases. For rivulet flows, the initial filling of the horizontal fracture is described by classical plug flow. Meanwhile, for droplet flows, a size-dependent partitioning behavior is observed, and the filling of the fracture takes longer. For the case of rivulet flow, we provide an analytical solution that demonstrates the existence of classical Washburn flow within the horizontal fracture.

unsaturated flow, fractured media, smoothed partic↗
DOE OSTI2017

Can Electrical Resistance Tomography be used for imaging unsaturated moisture flow in cement-based materials with discrete cracks?

Previously, it has been shown that Electrical Resistance Tomography (ERT) can be used for monitoring moisture flow in undamaged cement-based materials. In this work, we investigate whether ERT could be used for imaging three-dimensional (3D) unsaturated moisture flow in cement-based materials that contain discrete cracks. Novel computational methods based on the so-called absolute imaging framework are developed and used in ERT image reconstructions, aiming at a better tolerance of the reconstructed images with respect to the complexity of the conductivity distribution in cracked material. ERT is first tested using specimens with physically simulated cracks of known geometries, and corroborated with numerical simulations of unsaturated moisture flow. Next, specimens with loading-induced cracks are imaged; here, ERT reconstructions are evaluated qualitatively based on visual observations and known properties of unsaturated moisture flow. Results indicate that ERT is a viable method of visualizing 3D unsaturated moisture flow in cement-based materials with discrete cracks. - Highlights: • 3D EIT is developed to visualize water ingress in cracked mortar. • Mortar with different size discrete cracks are used. • The EIT results are corroborated with numerical simulations. • EIT results accurately show the temporal and spatial variation of water content. • EIT is shown to be a viable method to monitor flow in cracks and matrix.

36 MATERIALS SCIENCE↗
DOE OSTI2018

Uncertainty Quantification and Global Sensitivity Analysis of Subsurface Flow Parameters to Gravimetric Variations During Pumping Tests in Unconfined Aquifers

We study the contribution of typically uncertain subsurface flow parameters to gravity changes that can be recorded during pumping tests in unconfined aquifers. We do so in the framework of a Global Sensitivity Analysis and quantify the effects of uncertainty of such parameters on the first four statistical moments of the probability distribution of gravimetric variations induced by the operation of the well. System parameters are grouped into two main categories, respectively, governing groundwater flow in the unsaturated and saturated portions of the domain. We ground our work on the three-dimensional analytical model proposed by Mishra and Neuman (2011), which fully takes into account the richness of the physical process taking place across the unsaturated and saturated zones and storage effects in a finite radius pumping well. The relative influence of model parameter uncertainties on drawdown, moisture content, and gravity changes are quantified through (a) the Sobol' indices, derived from a classical decomposition of variance and (b) recently developed indices quantifying the relative contribution of each uncertain model parameter to the (ensemble) mean, skewness, and kurtosis of the model output. Our results document (i) the importance of the effects of the parameters governing the unsaturated flow dynamics on the mean and variance of local drawdown and gravity changes; (ii) the marked sensitivity (as expressed in terms of the statistical moments analyzed) of gravity changes to the employed water retention curve model parameter, specific yield, and storage, and (iii) the influential role of hydraulic conductivity of the unsaturated and saturated zones to the skewness and kurtosis of gravimetric variation distributions. The observed temporal dynamics of the strength of the relative contribution of system parameters to gravimetric variations suggest that gravity data have a clear potential to provide useful information for estimating the key hydraulic parameters of the system.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2016

Sequential ensemble-based optimal design for parameter estimation

The ensemble Kalman filter (EnKF) has been widely used in parameter estimation for hydrological models. The focus of most previous studies was to develop more efficient analysis (estimation) algorithms. On the other hand, it is intuitively understandable that a well-designed sampling (data-collection) strategy should provide more informative measurements and subsequently improve the parameter estimation. In this work, a Sequential Ensemble-based Optimal Design (SEOD) method, coupled with EnKF, information theory and sequential optimal design, is proposed to improve the performance of parameter estimation. Based on the first-order and second-order statistics, different information metrics including the Shannon entropy difference (SD), degrees of freedom for signal (DFS) and relative entropy (RE) are used to design the optimal sampling strategy, respectively. The effectiveness of the proposed method is illustrated by synthetic one-dimensional and two-dimensional unsaturated flow case studies. It is shown that the designed sampling strategies can provide more accurate parameter estimation and state prediction compared with conventional sampling strategies. Optimal sampling designs based on various information metrics perform similarly in our cases. Furthermore, the effect of ensemble size on the optimal design is also investigated. Overall, larger ensemble size improves the parameter estimation and convergence of optimal sampling strategy. Although the proposed method is applied to unsaturated flow problems in this study, it can be equally applied in any other hydrological problems.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2019

Numerical Evaluation of Unsaturated-Zone Flow and Transport Pathways at Rainier Mesa, Nevada

Sixty-one underground nuclear tests were conducted at Rainier Mesa at the Nevada National Security Site between 1957 and 1992. The mesa includes the highest-elevation areas at the Nevada National Security Site and experiences some of the highest estimated infiltration rates. Perched water is ubiquitous in many of the tunnels used for testing. The presence of sloping layers with strongly varying degrees of fracturing and overall permeability suggests that some infiltrating water may be diverted laterally along higher-permeability layers, thereby reducing overall percolation flux and radionuclide transport through the low-permeability testing horizons but also potentially transporting radionuclides laterally away from the test locations. A high-resolution, two-dimensional, cross-sectional model was created to simulate water and particle movement along potential lateral flow paths and to investigate flow and transport paths within and surrounding the perched water bodies. The model results indicate that particles with starting locations within the low-permeability perching layers will follow dominantly vertical trajectories through the perching horizons and reach the water table in the underlying carbonate aquifer only after many thousands of years. In contrast, particles starting near ground surface above the testing horizon are mostly diverted laterally before reaching the testing horizon, except where nearby faults through the perching layers are present. Where the rock above the perching horizon is fractured, particles originating near ground surface can move laterally with relatively high transport velocities. Transport calculations for 3 H show that transport extent is limited by slow matrix flow, radioactive decay, and matrix diffusion in the fractured units.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2023

Constraint energy minimizing generalized multiscale finite element method for multi-continuum Richards equations

In fluid flow simulation, the multi-continuum model is a useful strategy. When the heterogeneity and contrast of coefficients are high, the system becomes multiscale, and some kinds of reduced order methods are demanded. Combining these techniques with nonlinearity, we will consider in this paper a dual-continuum model which is generalized as a multi-continuum model for a coupled system of nonlinear Richards equations as unsaturated flows, in complex heterogeneous fractured porous media; and we will solve it by a novel multiscale approach utilizing the constraint energy minimizing generalized multiscale finite element method (CEM-GMsFEM). In particular, such a nonlinear system will be discretized in time and then linearized by Picard iteration (whose global convergence is proved theoretically). Subsequently, we tackle the resulting linearized equations by the CEM-GMsFEM and obtain proper offline multiscale basis functions to span the multiscale space (which contains the pressure solution). More specifically, we first introduce two new sources of samples, and the GMsFEM is used over each coarse block to build local auxiliary multiscale basis functions via solving local spectral problems, that are crucial for detecting high-contrast channels. Second, per oversampled coarse region, local multiscale basis functions are created through the CEM as constrainedly minimizing an energy functional. Various numerical tests for our approach reveal that the error converges with the coarse-grid size and that only few oversampling layers as well as basis functions are needed.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Enhanced Transport of TiO 2 in Unsaturated Sand and Soil after Release from Biodegradable Plastic during Composting

Biodegradable plastics can reach full degradation when disposed of appropriately and thus alleviate plastic pollution caused by conventional plastics. But additives can be released into the environment during degradation, and the fate of these additives can be affected by the degradation process. In this study, we characterized TiO 2 particles released from a biodegradable plastic mulch during composting and studied the transport of the released TiO 2 particles in inert sand and agricultural soil columns under unsaturated flow conditions. TiO 2 microparticles were released from the biodegradable plastic mulch in both single particle and cluster forms. The released TiO 2 microparticles were fully retained in unsaturated soil columns due to its ne texture, slow flow rate, and favorable attachment conditions. But in unsaturated sand columns, the released TiO 2 microparticles were highly mobile. Additionally, comparison with the pristine TiO 2 microparticles revealed that the enhanced mobility of the released TiO 2 microparticles was caused by the presence of compost residues which blocked attachment sites on the sand surface. This study demonstrates that TiO 2 microparticles can be released during composting of biodegradable plastics, and the transport potential of the released TiO 2 microparticles in the terrestrial environment can be enhanced by the presence of compost material.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2024

Development of inter-grid-cell lateral unsaturated and saturated flow model in the E3SM Land Model (v2.0)

Abstract. The lateral transport of water in the subsurface is important in modulating terrestrial water energy distribution. Although a few land surface models have recently included lateral saturated flow within and across grid cells, it is not a default configuration in the Climate Model Intercomparison Project version 6 experiments. In this work, we developed the lateral subsurface flow model within both unsaturated and saturated zones in the Energy Exascale Earth System Model (E3SM) Land Model version 2 (ELMv2.0). The new model, called ELMlat, was benchmarked against PFLOTRAN, a 3D subsurface flow and transport model, for three idealized hillslopes that included a convergent hillslope, divergent hillslope, and tilted V-shaped hillslope with variably saturated initial conditions. ELMlat showed comparable performance against PFLOTRAN in terms of capturing the dynamics of soil moisture and groundwater table for the three benchmark hillslope problems. Specifically, the mean absolute errors (MAEs) of the soil moisture in the top 10 layers between ELMlat and PFLOTRAN were within 1 %±3 %, and the MAEs of water table depth were within ±0.2 m. Next, ELMlat was applied to the Little Washita experimental watershed to assess its prediction of groundwater table, soil moisture, and soil temperature. The spatial pattern of simulated groundwater table depth agreed well with the global groundwater table benchmark dataset generated from a global model calibrated with long-term observations. The effects of lateral groundwater flow on the energy flux partitioning were more prominent in lowland areas with shallower groundwater tables, where the difference in simulated annual surface soil temperature could reach 0.3–0.4 ∘C between ELMv2.0 and ELMlat. Incorporating lateral subsurface flow in ELM improves the representation of the subsurface hydrology, which will provide a good basis for future large-scale applications.

58 GEOSCIENCES↗
DOE OSTI2020

Simulator for Hydrologic Unstructured Domains (SHUD v1.0): numerical modeling of watershed hydrology with the finite volume method

Abstract. Hydrologic modeling is an essential strategy for understanding and predicting natural flows, particularly where observations are lacking in either space or time or where complex terrain leads to a disconnect in the characteristic time and space scales of overland and groundwater flow. However, significant difficulties remain for the development of efficient and extensible modeling systems that operate robustly across complex regions. This paper introduces the Simulator for Hydrologic Unstructured Domains (SHUD), an integrated, multiprocess, multiscale, flexible-time-step model, in which hydrologic processes are fully coupled using the finite volume method. SHUD integrates overland flow, snow accumulation/melt, evapotranspiration, subsurface flow, groundwater flow, and river routing, thus allowing physical processes in general watersheds to be realistically captured. SHUD incorporates one-dimensional unsaturated flow, two-dimensional groundwater flow, and a fully connected river channel network with hillslopes supporting overland flow and baseflow. The paper introduces the design of SHUD, from the conceptual and mathematical description of hydrologic processes in a watershed to the model's computational structures. To demonstrate and validate the model performance, we employ three hydrologic experiments: the V-catchment experiment, Vauclin's experiment, and a model study of the Cache Creek Watershed in northern California. Ongoing applications of the SHUD model include hydrologic analyses of hillslope to regional scales (1 m2 to 106 km2), water resource and stormwater management, and interdisciplinary research for questions in limnology, agriculture, geochemistry, geomorphology, water quality, ecology, climate and land-use change. The strength of SHUD is its flexibility as a scientific and resource evaluation tool where modeling and simulation are required.

58 GEOSCIENCES↗
DOE OSTI2021

Multiscale simulations for multi-continuum Richards equations

In this paper, we study a multiscale method for simulating a dual-continuum unsaturated flow problem within complex heterogeneous fractured porous media. Mathematically, each of the dual continua is modeled by a multiscale Richards equation (for pressure head), and these equations are coupled to one another by transfer terms. On its own, Richards equation is already a nonlinear partial differential equation, and it is exceedingly difficult to solve numerically due to the extra nonlinear dependencies involving the soil water. To deal with multiple scales, our strategy is that starting from a microscopic scale, we upscale the coupled system of dual-continuum Richards equations via homogenization by the two-scale asymptotic expansion, to obtain a homogenized system, at an intermediate scale (level). Based on a hierarchical approach, the homogenization’s effective coefficients are computed through solving the arising cell problems. Furthermore, to tackle the nonlinearity, after time discretization, we use Picard iteration procedure for linearization of the homogenized Richards equations. At each Picard iteration, some degree of multiscale still remains from the intermediate level, so we utilize the generalized multiscale finite element method (GMsFEM) combining with a multi-continuum approach, to upscale the homogenized system to a macroscopic (coarse-grid) level. This scheme involves building uncoupled and coupled multiscale basis functions, which are used not only to construct coarse-grid solution approximation with high accuracy but also (with the coupled multiscale basis) to capture the interactions among continua. These prospects and convergence are demonstrated by several numerical results for the proposed method.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2017

Current Efforts in Immobilized Low-Activity Waste Glass Testing to Support the Hanford Site Integrated Disposal Facility - 17119

Cleanup plans for Hanford site high-level radioactive mixed tank waste calls for separating the waste into high-level waste (HLW) and low-activity waste (LAW) fractions. The fractions will be vitrified separately at the Hanford Tank Waste Treatment and Immobilization Plant (WTP). The HLW glass will be sent to an offsite geological disposal facility while the immobilized low-activity waste (ILAW) glass will be disposed at the onsite Integrated Disposal Facility (IDF). In order to inform models used to predict the performance of ILAW glass at IDF, it is necessary to understand the corrosion behavior of the glass. In this paper, we discuss tests that account for the effects of 1) solution chemistry, 2) the ion exchange reaction, and 3) secondary phase formation on the glass dissolution rate. To quantify the effects of solution chemistry, specifically changes in pH, T, and Si concentration, the single-pass flow-through (SPFT) test method is applied. A designed set of these tests results in a set of kinetic rate law parameters that are used in a glass dissolution rate law based on transition-state theory. Furthermore, for SPFT tests with relatively high concentrations of Si, an ion exchange rate is provided. Additionally, through tests such as the pressurized unsaturated flow (PUF) test method, a suite of secondary phases can be compiled. The collection of kinetic rate law parameters, the ion exchange term, and the suite of secondary phases are provided to IDF performance assessment (PA) modelers. Lastly, we present a description of the efforts made to verify and validate the use of these parameters and secondary phases through the use of a field lysimeter present at the Hanford site. (authors)

12 MANAGEMENT OF RADIOACTIVE AND NON-RADIOACTIVE W↗
DOE OSTI2017

Water Saturation Relations and Their Diffusion-Limited Equilibration in Gas Shale: Implications for Gas Flow in Unconventional Reservoirs

Large volumes of water are used for hydraulic fracturing of low permeability shale reservoirs to stimulate gas production, with most of the water remaining unrecovered and distributed in a poorly understood manner within stimulated regions. Because water partitioning into shale pores controls gas release, we measured the water saturation dependence on relative humidity (rh) and capillary pressure (P c ) for imbibition (adsorption) as well as drainage (desorption) on samples of Woodford Shale. Experiments and modeling of water vapor adsorption into shale laminae at rh = 0.31 demonstrated that long times are needed to characterize equilibrium in larger (5 mm thick) pieces of shales, and yielded effective diffusion coefficients from 9 × 10 -9 to 3 × 10 -8 m 2 s -1 , similar in magnitude to the literature values for typical low porosity and low permeability rocks. Most of the experiments, conducted at 50°C on crushed shale grains in order to facilitate rapid equilibration, showed significant saturation hysteresis, and that very large P c (~1 MPa) are required to drain the shales. These results quantify the severity of the water blocking problem, and suggest that gas production from unconventional reservoirs is largely associated with stimulated regions that have had little or no exposure to injected water. Finally, gravity drainage of water from fractures residing above horizontal wells reconciles gas production in the presence of largely unrecovered injected water, and is discussed in the broader context of unsaturated flow in fractures.

58 GEOSCIENCES↗
DOE OSTI2021

Verification of Coupled Hydraulic Fracturing Simulators Using Laboratory-Scale Experiments

In this work, we aim to verify the predictions of the numerical simulators, which are used for designing field-scale hydraulic stimulation experiments. Although a strong theoretical understanding of this process has been gained over the past few decades, numerical predictions of fracture propagation in low-permeability rocks still remains a challenge. Against this background, we performed controlled laboratory-scale hydraulic fracturing experiments in granite samples, which not only provides high-quality experimental data but also a well-characterized experimental set-up. Using the experimental pressure responses and the final fracture sizes as benchmark, we compared the numerical predictions of two coupled hydraulic fracturing simulators—CSMP and GEOS. Both the simulators reproduced the experimental pressure behavior by implementing the physics of Linear Elastic Fracture Mechanics (LEFM) and lubrication theory within a reasonable degree of accuracy. The simulation results indicate that even in the very low-porosity (1–2 %) and low-permeability (10 -18 m 2 - 10 -19 m 2 ) crystalline rocks, which are usually the target of EGS, fluid-loss into the matrix and unsaturated flow impacts the formation breakdown pressure and the post-breakdown pressure trends. Therefore, underestimation of such parameters in numerical modeling can lead to significant underestimation of breakdown pressure. The simulation results also indicate the importance of implementing wellbore solvers for considering the effect of system compressibility and pressure drop due to friction in the injection line. The varying injection rate as a result of decompression at the instant of fracture initiation affects the fracture size, while the entry friction at the connection between the well and the initial notch may cause an increase in the measured breakdown pressure.

15 GEOTHERMAL ENERGY↗
DOE OSTI2018

Toward an Understanding of Surface Layer Formation, Growth, and Transformation at the Glass–Fluid Interface

Silicate glass is a metastable and durable solid that has application to a number of energy and environmental challenges (e.g., microelectronics, fiber optics, and nuclear waste storage). If allowed to react with water over time silicate glass develops an altered layer at the solid-fluid interface. In this study, we used borosilicate glass (LAWB45) as model material to develop a robust understanding of altered layer formation (i.e., amorphous hydrated surface layer and crystalline reaction products). Experiments were conducted at high surface area-to-volume ratio (~200,000 m-1) and 90 °C in the pressurized unsaturated flow (PUF) apparatus for 1.5-years to facilitate the formation of thick altered layers and allow for the effluent solution chemistry to be monitored continuously. A variety of microscopy techniques were used to characterized reacted grains and suggest the average altered layer thickness is 13.2 ± 8.3 μm with the hydrated and clay layer representing 74.8% and 25.2% of the total altered layer, respectively. This estimate is within the experimental error of the value estimated from the B release rate data (~10 ±1 μm/yr) over the 1.5-year duration. PeakForce® quantitative nanomechanical mapping results suggest the hydrated layer has a modulus that ranges between ~20 to 40 GPa, which is in the range of porous silica that contains from ~20 to ~50% porosity, yet significantly lower than dense silica (~70 to 80 GPa). Scanning transmission electron microscopy (STEM) images confirm the presence of pores and an analysis of a higher resolution image of a region provides a qualitative estimate of ≥ 22% porosity in this layer with variations in the hydrated layer in void volume with increasing distance from the unaltered glass. Chemical composition analyses, based on a combination of time-of-flight secondary-ion mass spectrometry (ToF-SIMS) and scanning electron microscopy with X-ray energy dispersive spectroscopy (EDS) and STEM-EDS, clearly show that the altered layer is mainly composed of Al, H, Si, and O with the clay layer being enriched in Li, Zn, Fe, and Mg. The amorphous hydrated layer is enriched in Ca, H, and Zr with a minor amount of K. Furthermore, ToF-SIMS results also suggest the B profile is anti-correlated with the H profile in the hydrated layer. Our selected-area electron diffraction results suggest the structure of the hydrated layer closely resembles opal-AG (amorphous gel-like) with an average crystallite size of ~0.7 nm which is smaller than the critical nucleus for silica nanoparticles (i.e., 1.4 to 3 nm). These results suggest the hydrated layer is more consistent with a polymeric gel rather than a colloidal gel and is comprised of molecular units (<1 nm in size) that result from the difficult to hydrolyze bonds, such as Si—O—Zr units, during the glass corrosion process. The size of individual particles or molecular units is a function of formation conditions (e.g., pH, ionic strength, nano-confinement, solute composition) in the hydrated layer.

36 MATERIALS SCIENCE↗
DOE OSTI2019

A composite neural network that learns from multi-fidelity data: Application to function approximation and inverse PDE problems

Presently the training of neural networks relies on data of comparable accuracy but in real applications only a very small set of high-fidelity data is available while inexpensive lower fidelity data may be plentiful. We propose a new composite neural network (NN) that can be trained based on multi-fidelity data. It is comprised of three NNs, with the first NN trained using the low-fidelity data and coupled to two high-fidelity NNs, one with activation functions and another one without, in order to discover and exploit nonlinear and linear correlations, respectively, between the low-fidelity and the high-fidelity data. We first demonstrate the accuracy of the new multi-fidelity NN for approximating some standard benchmark functions but also a 20-dimensional function that is not easy to approximate with other methods, e.g. Gaussian process regression. Subsequently, we extend the recently developed physics-informed neural networks (PINNs) to be trained with multi-fidelity data sets (MPINNs). MPINNs contain four fully-connected neural networks, where the first one approximates the low-fidelity data, while the second and third construct the correlation between the low- and high-fidelity data and produce the multi-fidelity approximation, which is then used in the last NN that encodes the partial differential equations (PDEs). Specifically, by decomposing the correlation into a linear and nonlinear part, the present model is capable of learning both the linear and complex nonlinear correlations between the low- and high-fidelity data adaptively. By training the MPINNs, we can: (1) obtain the correlation between the low- and high-fidelity data, (2) infer the quantities of interest based on a few scattered data, and (3) identify the unknown parameters in the PDEs. In particular, we employ the MPINNs to learn the hydraulic conductivity field for unsaturated flows as well as the reactive models for reactive transport. The results demonstrate that MPINNs can achieve relatively high accuracy based on a very small set of high-fidelity data. Despite the relatively low dimension and limited number of fidelities (two-fidelity levels) for the benchmark problems in the present study, the proposed model can be readily extended to very high-dimensional regression and classification problems involving multi-fidelity data.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗