Transport Solver Research Articles

Numerical dispersion of solute transport in an integrated surface–subsurface hydrological model

Integrated surface–subsurface hydrological models (ISSHMs) are increasingly being used for the assessment of contaminant transport in the environment, in addition to their more common use in water flow applications. However, the subsurface solute transport solvers in these models are prone to numerical dispersion errors. Numerical dispersion is a well-known issue in groundwater modeling, but its impacts on the results of ISSHM simulations are still poorly understood. In this study, the CATchment HYdrology (CATHY) model is used to assess the potential impacts of numerical dispersion on the simulation of coupled surface–subsurface solute transport. We first simulate the subsurface transport of a nonreactive tracer in two soil column test cases (1D and 3D) with known analytical solutions. The subsurface solute transport solver in CATHY adopts a computationally efficient time-splitting technique whereby the advection component of the governing equation is solved on elements and the hydrodynamic dispersion component is solved on nodes. Comparison between simulation results and analytical solutions with different mesh discretizations and different values for the hydrodynamic dispersion coefficients allows for accurate quantification of the numerical dispersion error and yields insights into the parameters and other factors that control it. It is shown that, taken alone, the advection and dispersion solvers are very robust, but their combination can result in significant numerical dispersion, stemming from the exchange of concentration information from elements to nodes and vice versa in the time-splitting procedure. The tests also show that these errors can be kept under control by ensuring that the grid Péclet number is in the range 0.5-1.0 or smaller. We then apply CATHY in a third test case involving two synthetic hillslopes (concave and convex) in fully coupled surface–subsurface mode, in order to examine the impact of this subsurface numerical dispersion on simulated streamflow hydrographs, in particular with reference to pre-event water contributions to runoff. Here as well the results show that the effect of numerical dispersion can be controlled by keeping the grid Péclet number sufficiently small. This work provides a new set of benchmark test cases for integrated surface–subsurface hydrological models, extending to solute transport the flow-only suite of benchmarks recently published in two intercomparison studies.

Development of a novel integrated model GOTRESS+ for predictions and assessment of JT-60SA operation scenarios including the pedestal

A novel integrated model GOTRESS+ has been developed, which consists of the iterative transport solver GOTRESS as a kernel of the integrated model, the equilibrium and current profile alignment code ACCOME and the neutral beam heating/current-drive code OFMC. GOTRESS is able to robustly find out an exact solution of the stationary-state transport equations even with a stiff turbulent transport model, taking advantage of global optimization techniques such as a genetic algorithm. GOTRESS+ is then suitable for self-consistently assessing the stationary-state plasma performance of JT-60SA as well as ITER and DEMO or validating their feasibility. Recently GOTRESS+ has been extended to incorporate the in-house EPED1 model exploiting the MHD stability code MARG2D and is now able to predict the plasma profiles even with the pedestal over the entire region from the magnetic axis to the plasma boundary in a self-consistent manner. The two JT-60SA operation scenarios including the ITER-like inductive scenario and the high β fully non-inductively current driven scenario have been assessed by GOTRESS+ with the CDBM turbulent transport model and then were found to be feasible with most of the target dimensionless parameters met.

Pore scale modeling and evaluation of clogging behavior of salt crystal aggregates in CO2-rich phase during carbon storage

The optimal CO2 storage operation requires high permeability in the near-well region in order to keep it safe and cost-efficient. Nucleation and growth of salt crystals driven by the evaporation of formation water into under-saturated (dry) super-critical CO2 streams result in the changes in porosity and permeability of the near well-bore area. Permeability reduction is one of the main reasons for injectivity losses in the context of CO2 storage in saline aquifers. According to recent studies, during CO2 storage, salt crystals grow in two different forms: 1) single, large crystals in the aqueous phase, and 2) aggregates of micro-meter size salt crystals in the CO2-rich vapor phase. All previous numerical studies at pore-scale have addressed the formation of single, large crystals in the aqueous phase. In this work we have developed a 3D pore-scale reactive transport solver based on a D3Q19 advection-diffusion Lattice-Boltzmann model. The model takes for the first time salt nucleation into consideration via a new probabilistic approach to simulate the formation of micro-meter size salt crystal aggregates in the CO2-rich phase and their effect on changes in pore morphology and permeability. Comparing the results of porosity-permeability relations with some of the well-known clogging models, confirms the need for a new clogging model to capture the permeability reduction caused by salt aggregates.

An Eddy‐Resolving Numerical Model to Study Turbulent Flow, Sediment, and Bed Evolution Using Detached Eddy Simulation in a Lateral Separation Zone at the Field‐Scale

AbstractTurbulence‐resolving simulations elucidate key elements of fluid dynamics and sediment transport in fluvial environments. This research presents a feasible strategy for applying state‐of‐the‐art computational fluid mechanics to the study of sediment transport and morphodynamic processes in lateral separation zones, which are common features in canyon rivers where massive lateral flow separation causes large‐scale turbulence that controls sediment erosion and deposition. An eddy‐resolving model was developed and tested at the field‐scale, coupling a viscous flow and sediment transport solver using Detached Eddy Simulation techniques. A morphodynamic model was applied to the viscous flow/sediment solver to calculate erosion and deposition. A simulation of turbulence was performed at the grid resolution for a straight channel to determine the relative contributions of modeled and resolved diffusivity. The time‐dependent, energetically important, correlative, non‐stationary signals of the simulated quantities were captured at the lateral separation zone. Strong periodic signals featured by high amplitude were found at the separation zone, while low frequency pulsations were observed at the reattachment zone of the lateral separation zone. Interactions between the eddies and the loose bed boundaries resulted in erosion of sediment at the main channel followed by deposition at the primary eddy and eddy bars.

Simulation of Pulsed-Laser-Induced Testing in Microelectronic Devices

A method to carry out a priori simulations of pulsed-laser-induced single-event effects experiments is reported. Nonlinear optical simulations are conducted to model the 3-D distributions of charge from a laser pulse. These pulsed-laser-induced charge distributions are then incorporated into a charge transport solver to model the movement of charge as the device returns to a steady state. The output of the simulation infrastructure is a current transient from which collected charge and the temporal characteristics of the transient can be determined. This complete approach to modeling, which includes full device structure information, nonlinear optical energy deposition, and the subsequent movement of optically generated charge in a device and measurement circuit, allows for direct comparison between experimental and simulated current transients. A large-area silicon diode was used as the test vehicle, and good agreement was found between simulations and experiments in both total injected charge as well as temporal characteristics. Importantly, the a priori simulation approach does not require knowledge from prior heavy-ion testing, making it a predictive tool with broad versatility and minimal reliance on approximations.

Eulerian solver sensitivity for computing long-duration blast drag loading on columns

‘Long-duration’ blast waves can exert damaging drag loads on finite structural columns that are complex to characterise. In the absence of suitable drag coefficients, computational fluid dynamics (CFD) can be the only satisfactory approach for calculating blast loading. Commercially available CFD software with shockwave modelling capabilities require solving the Euler equations that assume inviscid flow. Two Eulerian solver types are generally used: Godunov and flux-corrected transport, the latter of which is commonly perceived as more accurate for shockwave problems. This study investigates the capability of both solvers for calculating drag loading from long-duration blast interaction with an I-section, verified through comparison to experimental measurements. No solver sensitivity was observed for pressures on exposed geometry surfaces, although discrepancies were observed for shielded surfaces, particularly during later stage interaction. The Godunov scheme demonstrated increased computational efficiency and reliability in comparison to the flux-corrected transport solver type, generating results in better agreement to experimental data. The results provided new understanding of solver sensitivity for modelling long-duration blast drag loading on finite targets using Eulerian CFD analyses.

Recent progress in multi‐scale modeling and simulation of flow and solute transport in porous media

AbstractThe heterogeneity and multi‐scale characteristics of porous media (such as soil aquifers) bring a series of uncertainties in studying flow and solute transport. In particular, the mathematical representation of these physical processes at different temporal and spatial scales, that is, the constitutive equations and parameters, is largely varied, which creates grand challenges in multi‐scale modeling and numerical simulation of flow and solute transport in porous media. In recent years, we acknowledge that at‐scale models and simulations have been greatly progressed, which means that at each individual scale, the transport phenomena, numerical methods, and solvers have been developed and utilized at different levels. However, the long‐standing problem, scale coupling between the pore‐ and Darcy‐scale in porous media, has not been well resolved. In this review, we present an overview and classification of the current multi‐scale models, aiming to understand small‐scale flow and solute transport processes when/where needed in a larger system. This review begins with a brief introduction of the pore‐ and Darcy‐scale theories. Two groups of multi‐scale models that are state‐of‐the‐art, based on the Darcy–Brinkman–Stokes equation and the domain decomposition method, respectively, are explained in the context of research progress in model theory, numerical algorithms, and applications. Future research perspectives and challenges are also discussed from both modeling and application points of view. This review is expected to provide a fundamental understanding of flow and solute transport in porous media and concept of developing multi‐scale models to bridge the gaps between processes in adjacent scales.This article is categorized under: Science of Water > Water Quality Science of Water > Methods Science of Water > Hydrological Processes

An Adaptive Sequential Fully Implicit Domain-Decomposition Solver

SummaryModern reservoir simulation must handle complex compositional fluid behavior, orders-of-magnitude variations in rock properties, and large velocity contrasts. We investigate how one can use nonlinear domain-decomposition preconditioning to combine sequential and fully implicit (FI) solution strategies to devise robust and highly efficient nonlinear solvers. A full simulation model can be split into smaller subdomains that each can be solved independently, treating variables in all other subdomains as fixed. In subdomains with weaker coupling between flow and transport, we use a sequential fully implicit (SFI) solution strategy, whereas regions with stronger coupling are solved with an FI method. Convergence to the FI solution is ensured by a global update that efficiently resolves long-range interactions across subdomains. The result is a solution strategy that combines the efficiency of SFI and its ability to use specialized solvers for flow and transport with the robustness and correctness of FI. We demonstrate the efficacy of the proposed method through a range of test cases, including both contrived setups to test nonlinear solver performance and realistic field models with complex geology and fluid physics. For each case, we compare the results with those obtained using standard FI and SFI solvers.NOTE: This paper is also published as part of the 2021 SPE Reservoir Simulation Conference Special Issue.

PorousMedia4Foam: Multi-scale open-source platform for hydro-geochemical simulations with OpenFOAM®

porousMedia4Foam is a package for solving flow and transport in porous media using OpenFOAM® - a popular open-source numerical toolbox. We introduce and highlight the features of a new generation open-source hydro-geochemical module implemented within porousMedia4Foam, which relies on micro-continuum concept and which makes it possible to investigate hydro-geochemical processes occurring at multiple scales i.e. at the pore-scale, reservoir-scale and at the hybrid-scale. Geochemistry is handled by a third party package (e.g. PHREEQC) that is coupled to the flow and transport solver of OpenFOAM®. We conducted benchmarks across different scales to validate the accuracy of our simulator. We further looked at the evolution of mineral dissolution/precipitation in a fractured porous system. Application fields of this new package include the investigation of hydro-bio-geochemical processes in the critical zone, the modelling of contaminant transport in aquifers, as well as and the assessment of confinement performance for geological barriers.

GeoChemFoam: Direct Modelling of Multiphase Reactive Transport in Real Pore Geometries with Equilibrium Reactions

GeoChemFoam is an open-source OpenFOAM-based toolbox that includes a range of additional packages that solve various flow processes from multiphase transport with interface transfer, to single-phase flow in multiscale porous media, to reactive transport with mineral dissolution. In this paper, we present a novel multiphase reactive transport solver for simulations on complex pore geometries, including microfluidic devices and micro-CT images, and its implementation in GeoChemFoam. The geochemical model includes bulk and surface equilibrium reactions. Multiphase flow is solved using the Volume-Of-Fluid method, and the transport of species is solved using the continuous species transfer method. The reactive transport equations are solved using a sequential operator splitting method, with the transport step solved using GeoChemFoam, and the reaction step solved using Phreeqc, the US geological survey’s geochemical software. The model and its implementation are validated by comparison with analytical solutions in 1D and 2D geometries. We then simulate multiphase reactive transport in two test pore geometries: a 3D pore cavity and a 3D micro-CT image of Bentheimer sandstone. In each case, we show the pore-scale simulation results can be used to develop upscaled models that are significantly more accurate than standard macro-scale equilibrium models.

Development of integrated suite of codes and its validation on KSTAR

A newly developed integrated suite of codes coined as tokamak reactor integrated automated suite for simulation and computation (TRIASSIC) is reported. The suite comprises existing plasma simulation codes, including 1.5D/2D plasma transport solvers and neoclassical/anomalous transport, plasma heating/cooling, and cold neutral models. The components in TRIASSIC are fully modularized by adopting a generic data structure as its internal storage. Primary components such as the transport solver and the neutral beam or electron cyclotron wave actuator were verified to its standalone implementation. The interpretive and predictive simulations of 50 stationary plasma phases from 30 KSTAR discharges were conducted for validation, and a good agreement with experimental measurements on all simulation cases was found.

A Stable Method for Solving the Simplified Time-Dependent Neutron Transport Equation with Background Motion

The neutron transport equation is usually solved against a stationary background medium. When the background material is moving, the transport equation will need to be modified. Solving the transport equation with moving material is quite complicated, especially for the curved coordinate system because of the double angular redistributions. In this study, the discretization method of the simplified transport equation considering the moving-material effect is implemented in three-dimensional cylindrical geometry. Directly solving this modified transport equation with the standard solution technique is problematic since the advection term introduced by moving material may render the transport solver numerically unstable. The speed ratio λ is defined for stability analysis. A forced-stable method is proposed in this study to achieve good numerical stability for any material speeds and time-step sizes. The accuracy of this new method is verified using manufactured solutions. Steady numerical results demonstrate that the effects introduced by background motion cannot be neglected as the material speed starts to approach one-tenth of the neutron speed. Moreover, transient analysis indicates that the moving background has a considerable impact on the criticality of a system.

Verification of multiphysics coupling techniques for modeling of molten salt reactors

Crucial to the development of molten salt reactor (MSR) designs is the application of multiphysics codes to model the tightly coupled neutronics and thermal-hydraulics behaviour of the liquid fuel. However, the verification and validation of such codes is not a trivial task, in particular for fast reactor designs, where no experimental data are available. In absence of experimental data, a benchmark was developed by LPSC/CNRS-Grenoble for multiphysics codes dedicated to MSR studies. In this study we present two independent multiphysics approaches and apply them to this benchmark. The first approach utilizes the Serpent2 multiphysics interface, allowing for high fidelity coupling of the finite volume computational fluid dynamics code OpenFOAM and Serpent2. In this approach, Serpent2 serves as the neutronics solver and is coupled to an OpenFOAM based thermal-hydraulics solver and supplemented by a delayed neutron precursors transport solver implemented in OpenFOAM. The main advantage of this coupling approach is that it allows for using a high accuracy Monte-Carlo approach to solve the neutron transport equations. The second approach is a novel approach that utilizes the SEALION framework. The SEALION code employs a specialized thermal hydraulics solver based on OpenFOAM, coupled with a custom-made modified point kinetics neutronics solver, that explicitly accounts for the altered neutron importance due to the transport of delayed neutron precursors. The main advantage of this approach is that it allows for a pre-determination of the temperature feedback effects using Monte-Carlo codes, such as Serpent2. Both approaches are verified against results from the benchmark and the overall agreement between the results demonstrates the validity of both approaches.

Observation of synergy between lower hybrid waves at two frequencies in EAST

Synergistic effects between two frequencies of lower hybrid (LH) waves—operating at 2.45 and 4.6 GHz—were observed in experiment on EAST for the first time. At low density (ne,lin≈2.0×1019 m−3), simultaneous injection of a 65/35 mix of 2.45/4.6 GHz power achieved an lower hybrid current drive efficiency that was 25% higher than what should be expected from the linear combination of the two sources. The experiment was interpreted with time-dependent simulations, using the equilibrium and transport solver TRANSP, coupled with the ray-tracing code GENRAY and the Fokker–Planck solver CQL3D. For each discharge, profiles of current and hard x-rays from simulation and measurement agree within uncertainties. An examination of the electron distribution function indicates that the LH synergy is supported by the increased width of the LH resonance plateau in the simultaneous injection case compared to independent injection.

Self-Consistent Energy Normalization for Quasistatic Reactor Calculations

Use of the quasistatic (keff) approximation for producing steady-state solutions for non-critical fission systems is known to result in an imbalance between energy release and deposition within the system. In this paper, we formally quantify this imbalance and present a self-consistent energy normalization technique that preserves nuclear energy release per reaction, as well as enforces energy balance between release and deposition mechanisms, regardless of the criticality state of the system. The proposed technique is straightforward to implement in any type of transport solver through the use of a simple energy rebalance factor. Theoretical and numerical results are presented that demonstrate the energy deposition bias for non-critical systems and the effectiveness of the proposed energy normalization technique.

A multiscale mixed finite element method applied to the simulation of two-phase flows

The multiscale hybrid mixed finite element method (MHM-H(div)), previously developed for Darcy’s problems, is extended for coupled flow/pressure and transport system of two-phase flow equations on heterogeneous media under the effect of gravitational segregation. It is combined with an implicit transport solver in a sequential fully implicit (SFI) manner. The MHM-H(div) method is designed to cope with the complex geometry and inherent multiscale nature of the phenomena. The discretizations are based on a general domain partition formed by polyhedral subregions, where a hierarchy of meshes and approximation spaces are considered. The multiscale approach is applied to the flux/pressure kernel making use of coarse scale normal fluxes between subregions (trace variable). The fine-scale features inside each subregion are determined by resolving completely independent local Neumann problems, the boundary conditions being set by the trace variable, by the mixed finite element method using fine flux and pressure representations. These properties imply that the MHM-H(div) can be interpreted as a classical mixed formulation of the model problem in the whole domain, based on a H(div)-conforming space with normal components over the macro-partition interfaces constrained by the trace space, and showing divergence compatibility with the pressure space. Consequently, local mass conservation is observed at the micro-scale elements inside the subregions, an essential property for flows in heterogeneous media, and divergence-free constraint strongly enforced for incompressible flows. The efficient use of static condensation leads to a global system to be solved only in terms of primary degrees of freedom associated with the trace variable and of a piecewise constant pressure for each subregion. This procedure allows a substantial reduction of the dominant computational costs associated with the flux/pressure kernel embedded in the numerical model. An iterative coupling technique is adopted to solve the two-phase flow equations using a shared integration point memory implementation model. At each SFI time step, the efficiency iterative method for the transport equations is improved using a Quasi-Newton method with a simple but effective nonlinear acceleration. The numerical examples show that the proposed scheme is able to solve challenging coupled flow and transport problems.

The impact of self-heating and its implications on hot-carrier degradation – A modeling study

A combination of hot-carrier degradation (HCD) and self-heating (SH) was acknowledged to be the most detrimental reliability issue in ultra-scaled field-effect-transistors (FETs) with confined architectures, such as fin and nanowire (NW) FETs. Although the view on whether SH accelerates or inhibits HCD in n-channel devices is controversial, it is commonly accepted that in p-channel FETs HCD becomes more pronounced due to SH. Therefore, we develop a framework for modeling coupled HCD and SH and validate it against experimental data acquired in p-channel Si NWFETs. This framework incorporates a carrier transport solver linked to a solver for the lattice heat flow equation; the latter solver allows one to evaluate a non-uniform lattice temperature distribution in the device. Carrier energy distribution functions obtained considering coordinate-dependent lattice temperature are used to calculate the rates of the multiple- and single-carrier processes of SiH bond dissociation. In addition to perturbation of carrier transport by SH, elevated lattice temperature results in enhancement of the thermal contribution to the bond rupture rate and shortens vibrational lifetime of the SiH bond, thereby reducing the multiple-carrier process rate. All these three aspects are captured by our framework, importance of each of them is analyzed, and our approach was shown to accurately reproduce experimental data.

Canonical Implementation of Simplified Spherical Harmonics (SPL) in the Neutron Diffusion Code AZNHEX

Abstract The SPL approximation has been used to improve the accuracy of neutron transport solvers and has been widely used thanks to the diffusion like equations that can be solved with traditional diffusion codes. The objective of this work is to extend the AZtlan Nodal HEXagonal (AZNHEX) diffusion code capabilities to include SPL approximation. For achieving this goal, an independent preprocessor has been developed to determine new specific neutronic parameters based on the canonical form of the SPL approximation. The SPL neutronic parameters are used to conform an extended system of diffusion like differential equations that can be discretized and solved numerically. The degree of SPL approximation defines the number of equations. Since the original AZNHEX code solves one diffusion equation by means of an algebraic system (matrix) with size depending on energy groups, an artificial increase in the number of energy groups allows to increase the matrix order by adding additional rows and columns to the original matrix depending on the SPL order, SP3 increases the matrix order by two times, SP5 three times and SP7 four times. Two exercises are presented. The initial one for verification purposes in which the effects of mesh refinement and increase of SPL approximation degree are presented and discussed. It was shown that the percentage error between successive SPL approximations reduces in a consistent manner as the order of SPL approximation increases. A second case is presented for validation against a transport code. In this case it was shown that keff values approach asymptotically to the transport solver solution as the SPL degree of approximation increases and as mesh is finer.

An Analytic Benchmark for Neutron Boltzmann Transport with Downscattering—Part I: Flux and Eigenvalue Solutions

Computing in the energy dimension is one of the greatest challenges confronting present-day deterministic neutron transport solvers. Accurately resolving the neutron flux as neutrons downscatter across resonances in the nuclear cross sections currently requires considerable computing power and suffers from approximation errors. Flux uncertainty resulting from the uncertainty of the resonance structure is the single-largest cause of reactivity uncertainty. Any additional reference solution for the critical neutron downscattering problem with resonance phenomena would be a boon to verification and validation of neutronics codes. This paper establishes a benchmark to verify the accuracy of neutron transport criticality solvers along the energy dimension. For the first time, the analytic solution of the flux amplitude is derived in the particular case of an infinite homogeneous medium with isotropic scattering in the center of mass and an arbitrary number of no-threshold, neutral particle reaction resonances (e.g., radiative capture, fission, and resonance scattering). Original analytic expressions are established to quantify the discrepancy between the and flux amplitudes, respective solutions of the multiplication factor , or the exponential time-evolution frequency eigenproblems. The physical study of these relations led to analysis of their first-order relative difference near the criticality condition . Finally, numerical solutions are provided to a benchmark problem constituted of the first resonance of 239Pu, the 6.67-eV resonance of 238U, and a scattering isotope with a flat cross section, allowing for the computational verification of the energy resolution of current neutron transport criticality codes. Through these novel results, this analytic benchmark can serve as a reference to verify the energy resolution and sensitivity analysis of neutron transport criticality calculations.

The Effect of the Flux Separability Approximation on Multigroup Neutron Transport

The angular dependence of flux-weighted multigroup cross sections is commonly neglected when generating multigroup libraries. The error of this flux separability approximation is typically not isolated from other error sources due to a lack of availability of library generation and corresponding solvers that cannot relax this approximation. These errors can now be isolated and quantified with the availability of a multigroup Monte Carlo transport and multigroup library-generation capability in the OpenMC Monte Carlo transport code. This work will discuss relevant details of the OpenMC implementation, provide an example case useful for detailing the type of errors one can expect from making the flux separability approximation, and end with more realistic problems which show the impact of the approximation and highlight how it can strongly arise from an energy-dependent resonance absorption effect. Since the angle-dependence is intrinsically linked to the energy group structure, these examples also show that relaxing the flux separability approximation with angle-dependent cross sections could be used to reduce either the fine-tuning required to set a multigroup energy structure for a specific reactor type or the number of energy groups required to obtain a desired level of accuracy for a given problem. This trade-off could increase the costs of generating multigroup cross sections, and has the potential to require more memory for storing the multigroup library during the transport calculations, but it can significantly reduce the computational time required since the runtime of a discrete ordinates or method of characteristics neutron transport solver scales roughly linearly with the number of groups.