Solution Of Transport Equation Research Articles

Simulating groundwater flow and contaminant transport

This study presents a comprehensive simulation of groundwater flow and contaminant transport using advanced numerical modeling techniques. The research focuses on understanding the complex interactions between groundwater movement and pollutant dispersion across various geological settings. By applying the governing equations of fluid dynamics and solute transport, the model incorporates essential parameters such as hydraulic conductivity, porosity, and dispersivity, enabling the accurate prediction of contaminant behavior over time. Simulations were carried out under a range of scenarios to assess the effects of differing recharge rates and contaminant source configurations on groundwater quality. The results provide critical insights into the mechanisms governing contaminant migration, highlighting the importance of soil properties, boundary conditions, and flow dynamics in influencing transport behavior. Moreover, the study demonstrates how variations in geological features and pollutant sources can significantly alter contaminant dispersal patterns. This work contributes to the development of robust groundwater management strategies and emphasizes the role of simulation tools in assessing environmental risks tied to groundwater contamination.

Mathematical model of solute transport in channels with variable geometry and permeable walls

This paper investigates solute transport dynamics within a steady, viscous, Newtonian fluid flow through channels of varying geometry (convergent–divergent boundaries) and permeable walls. The significant impact of osmotic pressure on solute concentration is particularly highlighted by modeling osmotic pressure as a cubic function of solute concentration. Approximate solutions to the Navier–Stokes equations and solute transport equations, along with the corresponding boundary conditions, have been derived. The effects of varying geometry and various emerging parameters on hydrostatic and osmotic pressures, solute concentration, and solute clearance are illustrated through graphs. Results indicate a linear decline in hydrostatic pressure and a nonlinear rise in osmotic pressure along the channel length, with divergent channels producing higher solute and wall concentrations than uniform or convergent channels. Increased ultrafiltration and absence of osmotic pressure yield greater wall concentrations. Solute clearance rates increase with the wall slope, transmittance coefficient, permeability, and ultrafiltration parameters, underscoring intricate parameter relationships within filtration. A dataset is used to visually present and analyze the influence of different physiological factors on solute concentration, with graphs corresponding to real physiological conditions. For channels with the uniform width, the findings closely align with previous results, demonstrating its applicability to solute exchange in glomerular capillaries as well as to other blood vessels or capillaries.

An Improved Scheme for the Finite Difference Approximation of the Advective Term in the Heat or Solute Transport Equations

Transport equations are widely used to describe the evolution of scalar quantities subject to advection, dispersion and, possibly, reactions. Numerical methods are required to solve these equations in applications, adopting either the advective or conservative formulations. Conservative formulations are usually preferred in practice because they conserve mass. Advective formulations do not, but have received more mathematical attention and are required for Lagrangian solution methods. To obtain an advective formulation that conserves mass, we subtract the discretized fluid flow equation, multiplied by concentration, from the conservative form of the transport equation. The resulting scheme not only conserves mass, but is also elegant in that it can be interpreted as averaging the advective term at cell interfaces, instead of approximating it at cell centers as in traditional centered schemes. The two schemes are identical when fluid velocity is constant, and both have second-order convergence, but the truncation errors are slightly different. We argue that the error terms appearing in the proposed scheme actually imply an improved representation of subgrid spreading/contraction and acceleration/deceleration caused by variable velocity. We compare the proposed and traditional schemes on several problems with variable velocity caused by recharge, discharge or evaporation, including two newly developed analytical solutions. The proposed method yields results that are slightly, but consistently, better than the traditional scheme, while always conserving mass (i.e., mass at the end equals mass at the beginning plus inputs minus outputs), which the traditional centered finite differences scheme does not. We conclude that this scheme should be preferred in finite difference solutions of transport.

First-principles studies on the electronic and thermal transport properties of twin graphene and its analogues

The novel two-dimensional (2D) carbon allotropes with various carbon networks have provided an unprecedented platform to explore fascinating device applications beyond graphene. In this work, the electronic and thermal transport properties of the twin graphene and its structural analogues, i.e., γ-graphyne, twin T-graphene, and twin 4–8 graphene, have been systematically revealed through first-principles calculations. Our results confirm the energetic and dynamical stability of the twin graphene family, and the intrinsic semiconducting nature of these 2D carbon sheets superior to graphene. Based on the solution of the phonon Boltzmann transport equation, the evaluated thermal conductivity of the considered 2D carbon sheets indicates that the absence of acetylenic linkages in carbon networks leads to a relatively enhanced heat transfer capacity, i.e., a higher thermal conductivity in the twin graphene family than the γ-graphyne case. More interestingly, a way to effectively tune thermal transport properties in the twin graphene family has been proposed via the utilization of atom-embedded carbon nanocages. Our results indicate that a notable 63.8% reduction in thermal conductivity can be achieved for twin graphene through the embedding of Ti atoms into the nanocages, exhibiting great potential for robust thermal management in low-dimensional carbon networks.

Numerical solution of static and spatial kinetics self-adjoint angular flux neutron transport equation

This paper elucidates a comprehensive derivation of the variational formulation pertaining to the static and spatial kinetics self-adjoint angular flux (SAAF) neutron transport equation. The methodology employed for discretization of the spatial variable is the finite element method, while the energy group discretization is executed via the group method, and the directional discretization is conducted using the discrete ordinates method. Analytic expressions for the discretization to variable separation under both vacuum and reflective boundary conditions are furnished. Constructed upon the MOOSE framework, a code designated as SAAFCGSN has been developed for the resolution of the SAAF neutron transport equation. The zero-order scattering matrix within this computational framework is managed through an innovative "decoupling" method, thereby enhancing the computational efficiency significantly. The functionality and robustness of the SAAFCGSN code are corroborated through meticulous evaluation involving seven distinct steady-state scenarios as well as two transient states. Empirical outcomes verify the compatibility of the SAAFCGSN code with both structured and unstructured mesh, inclusive of their amalgamations, thus facilitating maintenance and ensuring elevated computational accuracy. In addition, a performance analysis benchmarked against IAEA standards reveals that the adoption of the scattering matrix "decoupling" method propels a computational speed increase exceeding 30 %, signifying a notable advancement in calculation efficiency.

Conjugate mass transfer from a spherical moving droplet: Direct numerical simulations of enhancement by a chemical reaction

Chemical reactions can have a significant impact on the overall rate of mass transfer in fluid–fluid applications (such as gas absorption in a liquid or liquid–liquid extraction). However, as for non reacting systems, the rate at which a solute is exchanged between the two fluids is difficult to predict accurately. Yet from a theoretical point of view, most studies address uniform diffusion transport or creeping flow, i.e. conditions enabling the derivation of analytical or simple numerical solutions of the solute transport equation, therefore neglecting the effect of convection in the case of coupled mass transfer problems. On the other hand, experimental investigations available in the literature provided correlations corresponding to flow conditions, shape and contamination of the interface which are extremely difficult to control precisely.Direct numerical simulation (DNS) is therefore a powerful tool to investigate the coupling of hydrodynamics with transfer and reaction. It was recently applied to model the internal and external resistances (or Sherwood numbers, Shi and She) prevailing during the transfer of a non reacting solute, in the general case of conjugate mass transfer from a moving spherical droplet (Godé et al., 2023; 2024). This study is complemented by considering a first order reaction that consumes the transferred solute in the continuous phase. The results indicate that the double film model, based on these last correlations for Shi and She, with She corrected by an enhancement factor allows obtaining accurate predictions of the overall mass transfer coefficient in the case of a reactive conjugate problem, thus extending the work of Juncu (2002) to flows at moderate or even high Re, as long as the droplet remains spherical and the flow axisymmetric.

Tensor networks for solving the time-independent Boltzmann neutron transport equation

Tensor network techniques, known for their low-rank approximation ability that breaks the curse of dimensionality, are emerging as a foundation of new mathematical methods for ultra-fast numerical solutions of high-dimensional Partial Differential Equations (PDEs). Here, we present a mixed Tensor Train (TT)/Quantized Tensor Train (QTT) approach for the numerical solution of time-independent Boltzmann Neutron Transport equations (BNTEs) in Cartesian geometry. Discretizing a realistic three-dimensional (3D) BNTE by (i) diamond differencing in space, (ii) multigroup-in-energy, and (iii) discrete ordinates collocation in angle leads to large generalized eigenvalue problems that generally require a matrix-free approach and large computer clusters. Starting from this discretization, we construct a TT representation of the PDE fields and discrete operators, followed by a QTT representation of the TT cores. We then solve the tensorized generalized eigenvalue problem using a fixed-point scheme with tensor network optimization techniques. We validate our approach by applying the method to two examples of 3D neutron transport problems, currently solved by the Los Alamos National Laboratory PARallel TIme-dependent SN (PARTISN) solver.1 We demonstrate that our TT/QTT method, executed on a standard desktop computer, leads to large compression. This allows for the storage of terrabyte-sized neutron angular flux eigenvectors in megabytes. Additionally, we create megabyte-sized full access TT representations of yottabyte-sized transport matrix operators. By leveraging the TT operators and solution methods, we obtain a 7500 times speedup when compared to the PARTISN solution time with an error of less than 10−5.

Prescribing transport equation solution's decay via multiplicity manifold and autoregressive boundary control

AbstractThis article addresses the boundary control problem of the transport equation. Namely, we propose a control method, which is merely a delayed output feedback relying on a partial pole placement idea, that consists in assigning an appropriate exponential decay rate to the closed‐loop system's solution. The proposed control structure appearing in the transport boundary, which has proven its effectiveness in controlling finite dimensional systems, consists of an autoregressive relation linking the transport equation's input and output. The obtained result provides an analytical lower bound for the solution's exponential decay. The contribution is concluded with a numeric discussion on the concrete implementation of the method with a special emphasis on the delay seen as a control parameter.

Phase-field modeling and effective simulation of non-isothermal reactive transport

We consider single-phase flow with solute transport where ions in the fluid can precipitate and form a mineral, and where the mineral can dissolve and release solute into the fluid. Such a setting includes an evolving interface between fluid and mineral. We approximate the evolving interface with a diffuse interface, which is modeled with an Allen–Cahn equation. We also include effects from temperature such that the reaction rate can depend on temperature, and allow heat conduction through fluid and mineral. As Allen–Cahn is generally not conservative due to curvature-driven motion, we include a reformulation that is conservative. This reformulation includes a non-local term that makes the use of standard Newton iterations for solving the resulting non-linear system of equations very slow. We instead apply L-scheme iterations, which can be proven to converge for any starting guess, although giving only linear convergence. The three coupled equations for diffuse interface, solute transport and heat transport are solved via an iterative coupling scheme. This allows the three equations to be solved more efficiently compared to a monolithic scheme, and only few iterations are needed for high accuracy. Through numerical experiments we highlight the usefulness and efficiency of the suggested numerical scheme and the applicability of the resulting model.

B-Spline finite element solution of 1D contaminant transport equation along unsteady flow in saturated contaminant free porous media with general boundary conditions

B-Spline finite element solution of 1D contaminant transport equation along unsteady flow in saturated contaminant free porous media with general boundary conditions

On modeling radiation-induced charge effects in objects of complex geometry

Algorithms for supercomputer modeling of the radiation-induced charge effects in materials of a complex multimodal structure are constructed. A geometric model of an object is created using Stilinger-Lubachevsky algorithms for objects of complex geometry. The model includes a system of detectors for statistical evaluation of functionals on the space of solutions of the photon-electron cascade transport equations. Visualization of simulation results is carried out using the three-dimensional approximation based on the neural network technology. The thechnique of mathematical simulating the generation and development of a photon-electronic cascade in a medium is used. The examples of results of calculations of the charge effects in a fragment of a bimodal material are presented. The results of the calculations show that the spatial distribution of the radiation-induced charges has a sharply inhomogeneous structure caused by the presence of boundaries of materials with different radiation properties.

High-Resolution Sensor System to Investigate Solute Transport in Saturated Sediments

Highlights Innovative sensor system was developed to detect solute transport at very high time resolutions. The sensors measure electrical conductivity as voltage drops at a frequency of up to 10 Hz, allowing for detailed solute breakthrough curves (BTCs). One-dimensional solute transport equations were modeled numerically using the finite differences technique, and the results show good agreement between the modeled BTCs and those obtained in the laboratory for all materials. Abstract. Hyporheic exchange flow, the movement of river water into streambed sediments and its subsequent return to the surface water after suffering biogeochemical transformations, impacts a stream’s water quality and ecology. These processes, which are the basis for the self-purification capacity of river systems, depend on the quantity and travel time of the flow within the sediments. Different tracer-based approaches have been deployed for studying flow in saturated sediments, but few of these experimental methodologies are able to attain the high temporal resolutions that are needed to adequately capture the passage of a solute plume in a porous medium. To allow for a better description of such flows, we designed and fabricated an innovative, low-cost sensor system able to detect solute transport at multiple locations simultaneously at very high time resolutions. The sensors measure electrical conductivity as voltage drops at a frequency of up to 10 Hz, allowing for detailed solute breakthrough curves (BTCs). The fabricated sensors were tested in a laboratory experimental setup designed to examine solute transport under conditions that reproduce 1D flow in saturated sediments. The sensor system was able to successfully detect electrical conductivity (expressed as voltage drop) in real-time, at high temporal resolution, and at multiple locations. Solute BTCs collected with the system were consistent and highly repeatable under the same conditions. One-dimensional solute transport equations were modeled numerically using the finite differences technique, and the results showed good agreement between the measured and modeled BTCs for all materials. The proposed system is inexpensive compared to conventional EC probes, easy to operate, and sensitive to low solute concentrations. Keywords: Hyporheic exchange flow, Saturated sediments, Sensor, Solute transport.

A Mechanistic Model for Simulation of Carbendazim and Chlorothalonil Transport through a Two-Stage Vertical Flow Constructed Wetland

A mechanistic model was developed to simulate one-dimensional pesticide transport in two-stage vertical flow constructed wetland. The two pesticides taken under study were carbendazim and chlorothalonil. The water flow patterns within the constructed wetland were simulated using the Richards equation. Water content and vertical flux, which are the outputs of the substrate water flow model, were used to calculate the substrate moisture-related parameters and advection term in the solute transport model. The governing solute transport equation took into account a total of six processes: advection, molecular diffusion, dispersion, adsorption to the solid surface, degradation and volatilization. A total of 14 simulation cases, corresponding with available experimental data, were used to calibrate the model, followed by further simulations with standardized influent pesticide concentrations. The simulations indicated that the constructed wetland reached a steady state of pesticide removal after 7 days of operation. Two distinct water flow patterns emerged under saturated and unsaturated conditions. The patterns observed while varying the hydraulic loading rates were similar for each individual saturation condition. Two-factor ANOVA of the simulated data further revealed that the carbendazim and chlorothalonil removal was dependent on the hydraulic loading rates, but it was independent of the influent pesticide concentration. Analysis of the simulated pesticide removal showed that degradation emerged as the predominant removal process over time for both the pesticides. The model developed in this study can be an important tool for the design and construction of treatment wetlands for pesticide removal from wastewater.

Explicit K-Means Clustering Assisted Direct Model for Identification of Groundwater Pollution Sources

A computationally inexpensive and data parsimonious direct model for identification of groundwater pollution sources that captures their location, spatial extent and flux intensities at a chosen reference time is presented in this study. The model is based upon instantaneous micro-level mass balance that is depicted by a pseudo steady state solute transport equation. Clusters of polluting locations were identified by finite differencing the solute transport equation and applying K-means clustering algorithm. The model is validated by the problem taken from Singh, Datta, and Jain (2004) and is also illustrated by applying it to an area lying between river Krishni and Hindon. Validation results showed that the model captured closely the pollution source locations; their corresponding spatial extents and flux intensities. Model captured the sources even when the concentration plumes are far away from the source locations due to advection dominated contaminant transport. Illustration results showed that the model identified the leachate flux intensity from irrigation in the form of a regional source. Further, it established a few local sources arising possibly out of localized industrial and/or municipal waste water disposal.

2D numerical modeling of solute transport Na-K from a geothermal reservoir: The case of Las Tres Vírgenes, BCS

Las Tres Vírgenes geothermal field (LTVGF) is located within the volcanic complex of the same name with temperatures between 250 and 275 °C and a total power generation capacity of 10 MW. For geothermal systems, understanding the thermodynamic-chemical equilibrium of the fluid-rock interaction allows us to study the reactions involved in the precipitation or dissolution of minerals. In this work, we use the hydrothermal alteration mineralogical phases albite, microcline, and zoisite that are in equilibrium with the geothermal fluid as a source of cations (Na+, K+, and Ca2+) to model the chemical composition of the geothermal fluids in the last 20 years. The heat transfer (conductive-convective) and solute transport (advective-diffusive) equations in 2D, were discretized using the Finite Volume Method (FVM) and were coupled to simulate the temperature profile and spatial distribution of the main cations in the LTVGF reservoir. The code evaluates the thermal and chemical evolution of the reservoir from an initial geothermal gradient and an initial chemical composition of the fluid to reach the present concentrations at reservoir conditions. Based on the model results, we consider a reservoir between 1000 and 2000 m in depth and initial concentrations of 100 ppm for sodium to get the average concentrations at reservoir conditions of wells LV-13, LV-11, LV-6, LV-3, LV-4, and LV-1 corresponding to values between 3138 and 3661 ppm and temperatures between 225 and 249 °C. For potassium, the initial concentration was 10 ppm to reproduce the average concentrations between 604 and 663 ppm for wells LV-13, LV-11, LV-4, and LV-5 obtaining temperatures between 223 °C and 257 °C. For calcium, the initial concentration was 2 ppm and reproduced the concentrations of wells LV-5 and LV-6 (251 and 260 ppm) which are at depths of 1737 and 2162 m and temperatures of 225 and 260 °C. This work presents perhaps the first approach to solute transport of the LTVGF reservoir and it was determined that the alteration minerals considered influence the chemical composition of the current discharge fluids of the field. These results promote using Na/K geothermometers based on ionic activities proposed in the literature to estimate the reservoir temperature according to the equilibrium conditions and the chemical composition of the fluids.

Semi-empirical model of galactic cosmic ray spectrum

The galactic cosmic rays are high-energy particles that reach us from all directions of the vast universe. Their transport in the interplanetary medium is well described by the transport equation, first derived by Parker in 1965. An approximate solution of the cosmic ray transport equation for 1 AU is used to construct a simplified, semi-empirical model of the cosmic ray spectrum. The model parameters are obtained by fitting the proton spectra of the PAMELA experiment. The values of the model parameters are presented for the period 2006-2014, which includes the solar minimum between cycles 23 and 24 and the maximum phase of cycle 24. Our analysis shows that the proposed semi-empirical model agrees approximately well with the force-field approximation to the cosmic ray transport equation.

Coupling water, solute, and sediment transport into a new computationally efficient hydrologic model

While surface–subsurface interactions strongly impact water quality and quantity in ecosystems, only relatively little attention has been given to solute mass transfers at this boundary. The KINEROS2 (K2) model solves the 1D kinematic wave equation for water flow and the advection–dispersion equation for non-absorbing solute (or sediment) transport at the surface, whereas the HYDRUS-1D (H1D) model solves the Richards equation for flow and the advection–dispersion equation for solute transport in the subsurface. We previously coupled water flow in the surface and subsurface environments using the K2 and H1D models (H1D-K2). Herein, an innovative framework was developed to additionally couple solute transport at the surface–subsurface boundary in the simple, accurate, and computationally efficient H1D-K2 model. To test the H1D-K2 model’s accuracy and robustness, simulation results are compared to analytical solutions, experimental data, and results from the two-dimensional HYDRUS-2D (H2D) code. The relative differences in water and solute balance components between the H2D and H1D-K2 models are less than 1% for homogeneous cases and 4% for heterogeneous cases. The simulated hydrograph and chemograph are almost identical for the two models. Furthermore, simulated Bromide (Br) concentration in runoff is in good agreement with observations. The proposed model also accurately reproduces soil erosion under different rainfall patterns and intensities.

Non‐monotonic salt concentration dependence of inverted electrokinetic flow

AbstractModeling of electrokinetic flows is crucial to understand numerous phenomena associated with electrochemistry, biophysics, and colloidal science. Here, we incorporate the modified Gaussian renormalized fluctuation theory into transport equations for electrolyte solutions to study the ion‐correlation‐induced inversion of electrokinetic flows, also known as charge inversion. We are able to capture the non‐monotonic dependence of inverted streaming current and reversed electrophoretic mobility on salt concentration. By analyzing the double‐layer structure, we elucidate that this non‐monotonicity is a consequence of the competition between spatially varying ion correlations and the translational entropy of the ions. We find that for practical values of surface charge densities, the excluded volume effect does not play any significant role. In a significant improvement over existing theories, our theoretical predictions are in quantitative agreement with experimental measurements for charge inversion in trivalent salts.

Analytical and numerical solutions of radially symmetric aquifer thermal energy storage problems

Abstract Aquifer thermal energy storage (ATES) systems offer reduced energy costs, lower carbon emissions, and increased energy resilience. The feasibility, however, depends on several factors and usually require optimization. We study an ATES system with injection and extraction wells (cf. graphical abstract). The purpose of the investigation was to calculate the recovery factor of an ATES system with a cyclic repetition of injection and pumping. In the paper, we discuss analytical and numerical radial solutions of differential equations for heat transport in water-saturated porous media. A similar solution was obtained for a 2-D-horizontal confined aquifer with a constant radial flow. Numerical solutions were derived by using a high-resolution Lagrangian approach suppressing spurious oscillations and artificial dispersion. The numerical solution and the analytical solutions give consistent results and match each other well. The solutions describe instantaneous and delayed heat transfer between fluid and solid, as well as time-varying water flow. In hydrological terms, these solutions are relevant for a wide range of problems where groundwater reservoirs are utilized for extraction and storage (namely, irrigation; water supply; geothermal extraction).

Auxiliary physics-informed neural networks for forward, inverse, and coupled radiative transfer problems

In this paper, we develop and employ auxiliary physics-informed neural networks (APINNs) to solve forward, inverse, and coupled integrodifferential problems of radiative transfer theory. Specifically, by focusing on the relevant slab geometry and scattering media described by different types of phase functions, we show how the proposed APINN framework enables the efficient solution of Boltzmann-type transport equations through multi-output neural networks with multiple auxiliary variables associated with the Legendre expansion terms of the considered phase functions. Furthermore, we demonstrate the application of APINN to the coupled radiation-conduction problem of a participating medium and find distinctive temperature profiles beyond the Fourier thermal conduction limit. Finally, we solve the inverse problem for the Schwarzschild–Milne integral equation and retrieve the single scattering albedo based solely on the knowledge of boundary data, similar to what is often available in experimental settings. The present work significantly expands the current capabilities of physics-informed neural networks for radiative transfer problems that are relevant to the design and understanding of complex scattering media and photonic structures with applications to metamaterials, biomedical imaging, thermal transport, and semiconductor device modeling.