Analytical solution for one-dimensional advection–dispersion transport equation with distance-dependent coefficients

Nanoparticles transport in heterogeneous porous media using continuous time random walk approach

Knowledge of solute transport in heterogeneous porous media is crucial to monitor contaminant fate and transport in soil and groundwater systems. In this study, we present new findings from experimental and mathematical analysis to improve current understanding of solute transport in structured heterogeneous porous media. Three saturated columns packed with different sand combinations were used to examine the breakthrough behavior of bromide, a conservative tracer. Experimental results showed that bromide had different breakthrough responses in the three types of sand combinations, indicating that heterogeneity in hydraulic conductivity has a significant effect on the solute transport in structured heterogeneous porous media. Simulations from analytical solutions of a two-domain solute transport model matched experimental breakthrough data well for all the experimental conditions tested. Experimental and model results show that under saturated flow conditions, advection dominates solute transport in both fast-flow and slow-flow domains. The sand with larger hydraulic conductivity provided a preferential flow path for solute transport (fast-flow domain) that dominates the mass transfer in the heterogeneous porous media. Importantly, the transport in the slow-flow domain and mass exchange between the domains also contribute to the flow and solute transport processes and thus must be considered when investigating contaminant transport in heterogeneous porous media.

An analytical solution for describing the transport of dissolved substances in heterogeneous porous media with a distance‐dependent dispersion relationship has been developed. The analytical solution can be used to characterize differences in the transport process relative to the classical convection‐dispersion equation which assumes that the hydrodynamic dispersion in the porous medium remains constant. The form of the hydrodynamic dispersion function used in the analytical solution is , where α(x) is the distance‐dependent dispersivity and is the average pore water velocity. It is shown that for models which differ only in how the dispersion function is expressed, erroneous model parameters may result from parameter estimation techniques which assume a constant hydrodynamic dispersion coefficient if the porous medium is more accurately characterized by a distance‐dependent dispersion relationship. For such situations, the proposed model could be used to provide an alternate means for obtaining these parameters. The overall features of the solution are illustrated by several examples for constant concentration and flux boundary conditions.

Simulating solute transport in heterogeneous porous media poses computational challenges due to the high-resolution meshing required for traditional solvers. To overcome these challenges, this study explores a mesh-free method based on deep learning to accelerate solute transport simulation. We employ Physics-informed Neural Networks (PiNN) with a periodic activation function to solve solute transport problems in both homogeneous and heterogeneous porous media governed by the advection-dispersion equation. Unlike traditional neural networks that rely on large training datasets, PiNNs use strong-form mathematical models to constrain the network in the training phase and simultaneously solve for multiple dependent or independent field variables, such as pressure and solute concentration fields. To demonstrate the effectiveness of using PiNNs with a periodic activation function to resolve solute transport in porous media, we construct PiNNs using two activation functions, sin and tanh, for seven case studies, including 1D and 2D scenarios. The accuracy of the PiNNs’ predictions is then evaluated using absolute point error and mean square error metrics and compared to the ground truth solutions obtained analytically or numerically. Our results demonstrate that the PiNN with sin activation function, compared to tanh activation function, is up to two orders of magnitude more accurate and up to two times faster to train, especially in heterogeneous porous media. Moreover, PiNN’s simultaneous predictions of pressure and concentration fields can reduce computational expenses in terms of inference time by three orders of magnitude compared to FEM simulations for two-dimensional cases.

Analytical solution for the advection–dispersion transport equation in layered media

Three-dimensional modelling of flow and reactive transport in heterogeneous porous media : M. A. Cushey, A. Bellin & Y. Rubin, in: Groundwater quality: remediation and protection. Proc. conference, Prague, 1995, ed K. Kovar & J. Krasny, (IAHS; Publication, 225), 1995, pp 257–261

This paper presents a mixed finite element framework for coupled hydro-mechanical–chemical processes in heterogeneous porous media. The framework combines two types of locally conservative discretization schemes: (1) an enriched Galerkin method for reactive flow, and (2) a three-field mixed finite element method for coupled fluid flow and solid deformation. This combination ensures local mass conservation, which is critical to flow and transport in heterogeneous porous media, with a relatively affordable computational cost. A particular class of the framework is constructed for calcite precipitation/dissolution reactions, incorporating their nonlinear effects on the fluid viscosity and solid deformation. Linearization schemes and algorithms for solving the nonlinear algebraic system are also presented. Through numerical examples of various complexity, we demonstrate that the proposed framework is a robust and efficient computational method for simulation of reactive flow and transport in deformable porous media, even when the material properties are strongly heterogeneous and anisotropic.

Using constant coefficients is an inefficient way to explain the details of the problem of pollutant transport. This is due to factors such as nonuniform geometry, discharge changes, flow velocity and dispersion coefficient variations. Previous studies indicate that the ratio of studies on pollutant transport in rivers is lower than those on the porous media. However, the relative complexity of the solution of the pollutant transport equation in rivers (especially in the case of variable coefficients) in comparison with porous media highlights the importance of focusing on the solutions for rivers environment. In this study, the one-dimensional equation of pollutant transport in the river with location-dependent variables (velocity, dispersion coefficient and cross-section area) with arbitrary pattern is solved via the generalized integral transform technique (GITT) in a finite-length domain and general initial and boundary conditions. In this study, the results obtained from the presented analytical solution are compared to those obtained from the existing analytical solutions in previous studies for the two states of constant and variable coefficients. To further test the accuracy of this method, we have compared the results obtained from the presented analytical solution with the results obtained from numerical solution based on finite-difference method by applying the conditions of two real rivers. Comparison of the results of the GITT verification with available analytical solutions as well as numerical solutions shows the high accuracy and the applicability of the proposed solution considering arbitrary patterns of the ADRE coefficients and general initial and boundary conditions.

Powerful software tools for the sequential and parallel simulation of water and solute transport in strongly heterogeneous porous media are developed based on an extensive discussion of the involved processes, as well as of numerical and implementation aspects. The tools are applied to study problems involving parameter estimation in heterogeneous porous media, virtual soil systems and dynamic effects in heterogeneous soils. The excelent scalability of the numerical solver for both water as well as solute transport is investigated on BlueGene/P and /Q type supercomputers. As file transfer can be a bottleneck for large-scale simulations it is analysed carefully, obtaining near optimal data transfer rates with data sets of up to 17.5 Terabyte.

Transient Deposition of Colloidal Particles in Heterogeneous Porous Media

High-resolution characterization of nanoparticle transport in heterogeneous porous media via low-field nuclear magnetic resonance

We develop a probabilistic approach to map parametric uncertainty to output state uncertainty in first‐order hyperbolic conservation laws. We analyze this problem for nonlinear immiscible two‐phase transport in heterogeneous porous media in the presence of a stochastic velocity field. The uncertainty in the velocity field can arise from incomplete descriptions of either porosity field, injection flux, or both. This uncertainty leads to spatiotemporal uncertainty in the saturation field. Given information about spatial/temporal statistics of spatially correlated heterogeneity, we leverage the method of distributions to derive deterministic equations that govern the evolution of pointwise cumulative distribution functions (CDFs) of saturation for a vertical reservoir, while handling the manipulation of multiple shocks arising due to buoyancy forces. Unlike the Buckley‐Leverett equation, the equation describing the fine‐grained CDF is linear in space and time. Ensemble averaging of the fine‐grained CDF results in the CDF of saturation. Thus, we give routes to circumventing the computational cost of Monte Carlo simulations (MCS), while obtaining a pointwise description of the saturation field. We conduct a set of numerical experiments for one‐dimensional transport, and compare the obtained saturation CDFs, against those obtained using MCS as our reference solution, and the statistical moment equation method. This comparison demonstrates that the CDF equations remain accurate over a wide range of statistical properties, that is, standard deviation and correlation length of the underlying random fields, whereas the corresponding low‐order statistical moment equations significantly deviate from the MCS results, except for very small values of standard deviation and correlation length.

Trust-region based solver for nonlinear transport in heterogeneous porous media

Analytical solution for one-dimensional transport in heterogeneous porous media : Yates, S R Water Resour ResV26, N10, Oct 1990, P2331–2338

The present lecture summarizes some of the most recent joint research results from the cooperation between the Federal University of Rio de Janeiro, Brasil, and the University of Miami, USA, on the transient analysis of both fluid flow and heat transfer within microchannels. This collaborative link is a natural extension of a long term cooperation between the two groups, in the context of fundamental work on transient forced convection, aimed at the development of hybrid numerical-analytical techniques and the experimental validation of proposed models and methodologies [19]. The motivation of this new phase of the cooperation was thus to extend the previously developed hybrid tools to handle both transient flow and transient convection problems in microchannels within the slip flow regime. The analysis of internal flows in the slip-flow regime recently gained an important role in association with the fluid mechanics of various microelectromechanical systems (MEMS) applications, as well as in the thermal control of microelectronics, as reviewed in different sources [10-16]. For steady-state incompressible fully developed flow situations and laminar regime within simple geometries such as circular microtubes and parallel-plate microchannels, explicit expressions for the velocity field in terms of the Knudsen number are readily obtainable, and have been widely employed in the heat transfer analysis of microsystems, such as in [17-23]. Only quite recently, attention has been directed to the analysis of transient flow in microchannels [24-33]. Unsteady one-dimensional models have been extended from classical works, and analytical solutions have been sought for fully developed flows in simple geometries. These recent works are also concerned with situations in which a simple and well-defined functional form for the pressure gradient time variation is prescribed or for the time dependence of the wall imposed velocity, in the case of a Couette flow application. Research findings are yet to be further pursued in the analytical and robust solution of more generalized models, which will accommodate more general conditions and parameter specifications, and thus offer a wider validation range for the automatic general purpose numerical codes. Mikhailov and Ozisik [34] presented a unified solution for transient one-dimensional laminar flow models, with the usual no-slip boundary condition, based on the classical integral transform method. Their solution was then specialized to two situations: step change and periodically varying pressure gradient. The knowledge in regular size channels is therefore fairly well consolidated for models that use simple functional forms for the pressure gradient variation such as for the two cases cited above. One of the objectives of this paper is to illustrate the solution of a onedimensional mathematical model for transient laminar incompressible flow in microchannels such as circular tubes and parallel-plate channels, that accounts for a source term time variation in any functional form, including electrokinetic effects for liquid flows, by making use of the Generalized Integral Transform Technique (GITT) [35-40], and thus yielding analytical expressions for the time and space dependence of the velocity fields in the fully developed region. We then demonstrate this hybrid numerical-analytical solution for transient internal slip flow, obtained employing mixed symbolic-numerical computations with the Mathematica platform [41]. The goal here is to improve and complement existing analytical solution implementations to study laminar fully developed flows in micro-ducts subjected to arbitrary source term disturbances in space and time. On the other hand, the heat transfer literature of the last decade has demonstrated a vivid and growing interest in thermal analysis of flows in micro-channels, both through experimental and analytical approaches, in connection with cooling techniques of micro-electronics and with the development of micro-electromechanical sensors and actuators (MEMS), as also pointed out in recent reviews [12-16]. Since the available analytical information on heat transfer in ducts could not be directly extended to flows within microchannels with wall slip, a number of contributions have been recently directed towards the analysis of internal forced convection in the micro-scale. In the paper by Barron et al.