Transport In Heterogeneous Media Research Articles

Solute transport in highly heterogeneous media: The asymptotic signature of connectivity

The connectivity of the more conductive hydrofacies strongly determines flow and transport in heterogeneous media. Here we study solute transport in 3D binary isotropic samples with a proportion p of a high hydraulic conductivity facies (k+), and (1−p) of a low (k−) one. The k+ facies is characterized by two connectivity parameters: a connectivity structure type (no, low, intermediate and high), that controls how well the k+ facies is connected, and an integral scale Ib, that controls the heterogeneity characteristic lengthscale. Under ergodic conditions, and in the asymptotic Fickian regime that arises only very far from the injection plane, we analyze two transport quantities: the normalized mean solute arrival time 〈ta∗〉, and the longitudinal dispersivity αL. As p reaches the percolation threshold pc (pc depends on the connectivity parameters), k+ channels spanning the sample along the mean flow direction appear, giving rise to fast flow pathways . A sharp decrease of 〈ta∗〉, and a sharp increase of αL, occur when p→pc. As p exceeds pc, a subsequent minimum of 〈ta∗〉 and a maximum of αL are observed. This result is in contrast with previous ones by other authors that found a maximum of αL at p=pc. On the other hand, p kept fixed, αL decreases as the connectivity of the k+ facies increases. We conclude that the connectivity features sampled by the solute particles during their trajectories are retained in the transport quantities even after the asymptotic regime is attained. Also, that connectivity mainly affects αL through a shift or displacement of pc. Finally, the existence of a spatial connectivity structure may imply early, but also late, arrival times, compared with the absence of structure.

Chebyshev spectral method for the variable-order fractional mobile–immobile advection–dispersion equation arising from solute transport in heterogeneous media

Chebyshev spectral method for the variable-order fractional mobile–immobile advection–dispersion equation arising from solute transport in heterogeneous media

Evaluation of Dual Domain Mass Transfer in Porous Media at the Pore Scale.

Dual-porosity models are often used to describe solute transport in heterogeneous media, but the parameters within these models (e.g., immobile porosity and mobile/immobile exchange rate coefficients) are difficult to identify experimentally or relate to measurable quantities. Here, we performed synthetic, pore-scale millifluidics simulations that coupled fluid flow, solute transport, and electrical resistivity (ER). A conductive-tracer test and the associated geoelectrical signatures were simulated for four flow rates in two distinct pore-scale model scenarios: one with intergranular porosity, and a second with an intragranular porosity also defined. With these models, we explore how the effective characteristic-length scale estimated from a best-fit dual-domain mass transfer (DDMT) model compares to geometric aspects of the flow field. In both model scenarios we find that: (1) mobile domains and immobile domains develop even in a system that is explicitly defined with one domain; (2) the ratio of immobile to mobile porosity is larger at faster flow rates as is the mass-transfer rate; and (3) a comparison of length scales associated with the mass-transfer rate (Lα ) and those associated with calculation of the Peclet number (LPe ) show LPe is commonly larger than Lα . These results suggest that estimated immobile porosities from a DDMT model are not only a function of physically mobile or immobile pore space, but also are a function of the average linear pore-water velocity and physical obstructions to flow, which can drive the development of immobile porosity even in single-porosity domains.

A mobile-immobile model for contaminant transport through GCL/AL composite liner: analytical solutions.

The current study focuses on developing alternative formulations (mobile-immobile model) of transient analytical modelling for organic contaminant transport in a composite of geosynthetic clay (GCL) and attenuation layer (AL) system. The Laplace transform method is adopted to derive the solution. The proposed analytical frameworks are validated by two set of benchmarks. The varied examples of organic contaminant transport in the composite liner are evaluated by the proposed solution to discuss the effects of soil properties, flow conditions, adsorption, and mass transfer in mobile and immobile zones on the overall transport of contaminants. Finally, an example case is presented to show the application prospect of the proposed model. The result demonstrated that mass transfer between mobile and immobile may increase the breakthrough time by a factor of 2. This indicates that heterogeneities of clay are a non-negligible part for the performance assessment of the liner system. Péclet number in GCL ([Formula: see text]) is used to investigate the relative importance of advection and diffusion mechanisms. Increasing [Formula: see text] from 0.1 to 1 can lead to an increase of the breakthrough time by a factor of 15. The analytical solutions presented here may also serve as the basic benchmark test tool for alternative numerical studies of contaminants transport in heterogeneous media.

Tightly coupled hyperbolic treatment of buoyant two-phase flow and transport in porous media

Buoyant incompressible multiphase flow and transport in porous media typically is simulated by coupling an elliptic flow equation with a hyperbolic transport description. The tight coupling between flow and transport either requires a fully implicit solution algorithm or very small time steps, if solved sequentially. In any case, however, a large linear system has to be solved each time step.Here a new solution approach is devised, which relies on solving a coupled hyperbolic system of conservation laws with an explicit finite volume method. Consequently, only local operations have to be performed, which is a great advantage in case of massive parallel simulations and if GPUs are employed.The devised method is based on the isothermal Euler equations with momentum source terms accounting for resistance due to the porous medium and buoyancy. In this system the pressure is proportional to the density and in case of very small Mach numbers and relative density variation, the computed velocity field is divergence free and converges towards the mean total volumetric flow in the porous domain. To account for saturation transport, the system was augmented by an additional hyperbolic equation. In order to obtain the numerical fluxes, a characteristic based approximative Riemann solver was developed and 2nd order accuracy in space and time is achieved by piecewise linear reconstruction.Two-phase flow test cases with buoyant plumes and a lock exchange problem demonstrate that the devised hyperbolic approach recovers the correct incompressible solutions. Numerical experiments also demonstrate that the method is accurate and efficient.Thus the devised method is very promising, and conceptually it also can be applied for flow and transport in heterogeneous media; but it remains to be investigated how well it performs for such cases.

Physics-Informed Neural Network with Fourier Features for Radiation Transport in Heterogeneous Media

Typical machine learning (ML) methods are difficult to apply to radiation transport due to the large computational cost associated with simulating problems to create training data. Physics-informed Neural Networks (PiNNs) are a ML method that train a neural network with the residual of a governing equation as the loss function. This allows PiNNs to be trained in a low-data regime in the absence of (experimental or synthetic) data. PiNNs also are trained on points sampled within the phase-space volume of the problem, which means they are not required to be evaluated on a mesh, providing a distinct advantage in solving the linear Boltzmann transport equation, which is difficult to discretize. We have applied PiNNs to solve the streaming and interaction terms of the linear Boltzmann transport equation to create an accurate ML model that is wrapped inside a traditional source iteration process. We present an application of Fourier Features to PiNNs that yields good performance on heterogeneous problems. We also introduce a sampling method based on heuristics that improves the performance of PiNN simulations. The results are presented in a suite of one-dimensional radiation transport problems where PiNNs show very good agreement when compared to fine-mesh answers from traditional discretization techniques.

Data-driven discovery of the governing equationsfor transport in heterogeneous media by symbolic regression and stochastic optimization.

With advances in instrumentation and the tremendous increase in computational power, vast amounts of data are becoming available for many complex phenomena in macroscopically heterogeneous media, particularly those that involve flow and transport processes, which are problems of fundamental interest that occur in a wide variety of physical systems. The absence of a length scale beyond which such systems can be considered as homogeneous implies that the traditional volume or ensemble averaging of the equationsof continuum mechanics over the heterogeneity is no longer valid and, therefore, the issue of discovering the governing equationsfor flow and transport processes is an open question. We propose a data-driven approach that uses stochastic optimization and symbolic regression to discover the governing equationsfor flow and transport processes in heterogeneous media. The data could be experimental or obtained by microscopic simulation. As an example, we discover the governing equationfor anomalous diffusion on the critical percolation cluster at the percolation threshold, which is in the form of a fractional partial differential equation, and agrees with what has been proposed previously.

High-order asymptotic solutions for gas transport in heterogeneous media with multiple spatial scales

Subsurface structures generally exhibit strong heterogeneities at multiple spatial scales. In this study, two- and three-scale high-order models are developed to effectively predict nonlinear gas transport in heterogeneous porous media with multiscale configurations. The considered heterogeneous porous media are composed of the matrix and inclusions at the mesoscopic and microscopic scales, respectively. First, using the formal two-scale asymptotic analysis, the homogenized solutions, the two-scale first-order and higher-order solutions are derived, with the first-order and second-order cell functions defined at the mesoscopic cell. Second, by further expanding all the mesoscopic cell functions to the microscopic levels, the second-order expansions of the mesoscopic cell functions are established and the upscaled relationships for the permeability tensor from the microscale to the macroscale are developed accordingly. Finally, the three-scale low-order and high-order solutions are constructed by combining the multiscale expansions of the mesoscopic cell functions and the macro–meso two-scale solutions. Several representative cases are simulated to demonstrate the accuracy and reliability of the proposed multiscale solutions. The results show that the high-order solutions can perfectly capture the locally steep pressure fluctuations and non-equilibrium effects caused by the heterogeneities and large permeability contrast in porous media with two- or three-scale configurations. The strategies to obtain the multiscale high-order solutions follow the reverse thought process of the reiteration homogenization method, and can be easily extended to heterogeneous porous media with arbitrary multiple scales.

A Peridynamic Differential Operator-Based Model for Quantifying Spatial Non-Local Transport Behavior of Pollutants in Heterogeneous Media

Modeling pollutant transport in heterogeneous media is an important task of hydrology. Pollutant transport in a non-homogeneous environment typically exhibits non-local transport dynamics, whose efficient characterization requires a parsimonious model with the non-local feature. This study encapsulates the non-local transport characteristic of pollutants into the peridynamic differential operator (PDDO) and develops a PDDO-based model for quantifying the observed pollutant non-local transport behavior. The simulation results show that the proposed model can describe pollutant non-local transport behavior in various heterogeneous media. The non-local nature of pollutant transport can be adjusted by pre-defined weight function w(|ξ|) and horizon Hx. Applications show that the PDDO-based model can better capture pollutant non-local transport behavior than the classical advection–diffusion equation (ADE) model, especially for quantifying the tail of the experimental data late. Analyses further reveal that the PDDO-based model can characterize both normal (Fickian) and anomalous (Lévy) diffusion regimes.

Spatial Markov Model for the Prediction of Travel‐Time‐Based Solute Dispersion in Three‐Dimensional Heterogeneous Media

AbstractIn recent years, Spatial Markov Models have gained popularity in simulating solute transport in heterogeneous formations. They describe the transition times of particles between equidistant observation planes by statistical distributions, assuming correlation of the transit times of individual particles between subsequent steps. By this, the approach naturally captures preasymptotic solute dispersion. In this study, we analyze Spatial Markov Models assuming bivariate log‐normal distributions of the particle slowness (i.e., the inverse velocity) in subsequent transitions. The model is fully parameterized by the mean Eulerian velocity, the variance of the log‐slowness, and the correlation coefficient of log‐slowness in subsequent steps. We derive closed‐form expressions for distance‐dependent ensemble dispersion, which is defined in terms of the second‐central moments of the solute breakthrough curves. We relate the coefficients to the properties of the underlying log‐hydraulic conductivity field assuming second‐order stationarity. The results are consistent with linear stochastic theory in the limit of small log‐conductivity variances, while the approach naturally extends to high‐variance cases. We demonstrate the validity of the approach by comparison to three‐dimensional particle‐tracking simulations of advective transport in heterogeneous media with isotropic, exponential correlation structure for log‐conductivity variances up to five. This study contributes to relating solute dispersion to metrics of the porous‐medium structure in cases of strong heterogeneity.

Impact of velocity correlations on longitudinal dispersion in space-Lagrangian advective transport models

Space-Lagrangian random walk models conceptualize advective transport in heterogeneous media in terms of collections of particles undergoing fixed-length steps along flow streamlines. We study the impact of velocity correlation structure on longitudinal dispersion for different point velocity statistics. We find that asymptotic equivalence requires the step length of an uncorrelated continuous time random walk to be higher than the correlation length when velocity correlations decay exponentially with distance. We characterize the conditions for equivalence for broad classes of velocity distributions.

Stochastic parametric skeletal dosimetry model for humans: General approach and application to active marrow exposure from bone-seeking beta-particle emitters.

The objective of this study is to develop a skeleton model for assessing active marrow dose from bone-seeking beta-emitting radionuclides. This article explains the modeling methodology which accounts for individual variability of the macro- and microstructure of bone tissue. Bone sites with active hematopoiesis are assessed by dividing them into small segments described by simple geometric shapes. Spongiosa, which fills the segments, is modeled as an isotropic three-dimensional grid (framework) of rod-like trabeculae that "run through" the bone marrow. Randomized multiple framework deformations are simulated by changing the positions of the grid nodes and the thickness of the rods. Model grid parameters are selected in accordance with the parameters of spongiosa microstructures taken from the published papers. Stochastic modeling of radiation transport in heterogeneous media simulating the distribution of bone tissue and marrow in each of the segments is performed by Monte Carlo methods. Model output for the human femur at different ages is provided as an example. The uncertainty of dosimetric characteristics associated with individual variability of bone structure was evaluated. An advantage of this methodology for the calculation of doses absorbed in the marrow from bone-seeking radionuclides is that it does not require additional studies of autopsy material. The biokinetic model results will be used in the future to calculate individual doses to members of a cohort exposed to 89,90Sr from liquid radioactive waste discharged to the Techa River by the Mayak Production Association in 1949-1956. Further study of these unique cohorts provides an opportunity to gain more in-depth knowledge about the effects of chronic radiation on the hematopoietic system. In addition, the proposed model can be used to assess the doses to active marrow under any other scenarios of 90Sr and 89Sr intake to humans.

Estimation of Macrodispersivity in Bounded Formations by Circulant Embedding and Analysis of Variance

AbstractWe present a new method to estimate the displacement covariance and macrodispersivity for solute transport in bounded formations. Here we use circulant embedding, which is based on fast Fourier transform and is much more efficient than eigen‐decomposition, for the factorization of random spatial fields. We compute the displacement covariances by the analysis of variance approach, and introduce an interpolation process to significantly reduce the number of forward simulations. Once the effect of each eigenvector on the displacement covariance is obtained, it is unnecessary to rerun the simulator for different spatial covariance functions or anisotropy ratios, which saves a large amount of computational effort. The proposed method is validated in various tests in two‐dimensional and three‐dimensional examples and accurately matches the results from the Monte Carlo simulation. It is found that the longitudinal dispersivity is less sensitive to the impervious boundaries, while the transverse and vertical dispersivities are greatly affected. The method is applied to the Borden site and provides a better explanation of the observed data after considering the effect of vertical boundaries. These results show that our method could serve as a promising tool for studying and predicting the characteristics of solute transport in heterogeneous media.

A characteristic finite element method for the time-fractional mobile/immobile advection diffusion model

In this paper, we study the solute transport in heterogeneous media described by the time-fractional mobile/immobile advection diffusion model, where the integer and the fractional time derivatives are employed to characterize the movement of the particles in the mobile and immobile zone, respectively. We propose a fully discrete characteristic finite element scheme for the model, in which the modified method of characteristics is applied to handle the domination of advection. The optimal L2 error estimate is derived with first-order accuracy in time and second-order accuracy in space. Several practical numerical experiments are presented to validate the effectiveness and accuracy of the proposed method.

Interplay between percolation and glassiness in the random Lorentz gas

The random Lorentz gas (RLG) is a minimal model of transport in heterogeneous media that exhibits a continuous localization transition controlled by void space percolation. The RLG also provides a toy model of particle caging, which is known to be relevant for describing the discontinuous dynamical transition of glasses. In order to clarify the interplay between the seemingly incompatible percolation and caging descriptions of the RLG, we consider its exact mean-field solution in the infinite-dimensional d→∞ limit and perform numerics in d=2...20. We find that for sufficiently high d the mean-field caging transition precedes and prevents the percolation transition, which only happens on timescales diverging with d. We further show that activated processes related to rare cage escapes destroy the glass transition in finite dimensions, leading to a rich interplay between glassiness and percolation physics. This advance suggests that the RLG can be used as a toy model to develop a first-principle description of particle hopping in structural glasses.

Charge and Thermoelectric Transport in Polymer-Sorted Semiconducting Single-Walled Carbon Nanotube Networks.

Understanding the charge transport mechanisms in chirality-selected single-walled carbon nanotube (SWCNT) networks and the influence of network parameters is essential for further advances of their optoelectronic and thermoelectric applications. Here, we report on charge density and temperature-dependent field-effect mobility and on-chip field-effect-modulated Seebeck coefficient measurements of polymer-sorted monochiral small-diameter (6,5) (0.76 nm) and mixed large-diameter SWCNT (1.17-1.55 nm) networks (plasma torch nanotubes, RN) with different network densities and length distributions. All untreated networks display balanced ambipolar transport and electron-hole symmetric Seebeck coefficients. We show that charge and thermoelectric transport in SWCNT networks can be modeled by the Boltzmann transport formalism, incorporating transport in heterogeneous media and fluctuation-induced tunneling. Considering the diameter-dependent one-dimensional density of states (DoS) of the SWCNTs composing the network, we can simulate the charge density and temperature-dependent Seebeck coefficients. Our simulations suggest that scattering in these networks cannot be described as simple one-dimensional acoustic and optical phonon scattering as for single SWCNTs. Instead the relaxation time is inversely proportional to energy (τ ∝ (E - EC)s, s = -1, EC being the energy of the first van Hove singularity), presumably pointing toward the more two-dimensional character of scattering events and the necessity to include scattering at the SWCNT junctions. Finally, our observation of higher power factors in trap-free, 1,2,4,5-tetrakis(tetramethylguanidino)benzene-treated (6,5) networks than in the RN networks emphasizes the importance of chirality selection to tune the width of the DoS. To benefit from both higher intrinsic mobilities and a large thermally accessible DoS, we propose trap-free, narrow DoS distribution, large-diameter SWCNT networks for both electronic and thermoelectric applications.

Nonlocal transport models for capturing solute transport in one‐dimensional sand columns: Model review, applicability, limitations and improvement

AbstractModelling pollutant transport in water is one of the core tasks of computational hydrology, and various physical models including especially the widely used nonlocal transport models have been developed and applied in the last three decades. No studies, however, have been conducted to systematically assess the applicability, limitations and improvement of these nonlocal transport models. To fill this knowledge gap, this study reviewed, tested and improved the state‐of‐the‐art nonlocal transport models, including their physical background, mathematical formula and especially the capability to quantify conservative tracers moving in one‐dimensional sand columns, which represents perhaps the simplest real‐world application. Applications showed that, surprisingly, neither the popular time‐nonlocal transport models (including the multi‐rate mass transfer model, the continuous time random walk framework and the time fractional advection‐dispersion equation), nor the spatiotemporally nonlocal transport model (ST‐fADE) can accurately fit passive tracers moving through a 15‐m‐long heterogeneous sand column documented in literature, if a constant dispersion coefficient or dispersivity is used. This is because pollutant transport in heterogeneous media can be scale‐dependent (represented by a dispersion coefficient or dispersivity increasing with spatiotemporal scales), non‐Fickian (where plume variance increases nonlinearly in time) and/or pre‐asymptotic (with transition between non‐Fickian and Fickian transport). These different properties cannot be simultaneously and accurately modelled by any of the transport models reviewed by this study. To bypass this limitation, five possible corrections were proposed, and two of them were tested successfully, including a time fractional and space Hausdorff fractal model which minimizes the scale‐dependency of the dispersion coefficient in the non‐Euclidean space, and a two‐region time fractional advection‐dispersion equation which accounts for the spatial mixing of solute particles from different mobile domains. Therefore, more efforts are still needed to accurately model transport in non‐ideal porous media, and the five model corrections proposed by this study may shed light on these indispensable modelling efforts.

Fast Mixing in Heterogeneous Media Characterized by Fractional Derivative Model

This study aims at investigating non-Fickian temporal scaling of fast mixing processes using fractional advection dispersion equation (FADE) model in Indiana carbonate, multi-lognormal hydraulic conductivity field, self-affine fractures and cemented porous media, in which the fundamental solution of the FADE model is a standard symmetric Levy stable distribution. The temporal scaling of the scalar dissipation rate (SDR) induced by the FADE model is a function of fractional derivative order $$\alpha$$ , $$\chi (t) \sim t^{{ - \frac{\alpha + 1}{\alpha }}}$$ ( $$1 \le \alpha \le 2$$ ), and it reduces to the Fickian scaling $$t^{{ - \frac{3}{2}}}$$ when $$\alpha = 2$$ . Smaller values of $$\alpha$$ reflect more efficient and a better mixing state at early time. The fitted results show that the FADE model is much more accurate than the traditional model, which can also well interpret the fast mixing scaling from clearer physical mechanism than the empirical power law fitting line. The fitted values of $$\alpha$$ capture the complexity of the heterogeneous media, which are consistent with the existing empirical results. Thus, the SDR of the FADE model is feasible to describe the temporal scaling of the fast mixing for the tracer transport in heterogeneous media.

Upscaling non-linear reactive transport in correlated velocity fields

While commonly non-local transport models have been shown to reproduce breakthrough curves resulting from transport in heterogeneous aquifers successfully, open questions include the formal link between the upscaled governing equations and the sub-scale heterogeneity, and the ability to account for the effect of heterogeneity on effective chemical reaction rates in the presence of non-linear multi-component reactions. Time-domain random walk approaches based on velocity Markov models provide a framework to resolve these issues by incorporating the spatial correlation of velocity of a solute particle in consecutive locations along its trajectory. These approaches often rely on particle tracking approaches, which, however, can be computationally burdensome especially when non-linear reactions are sought to be modeled. In this paper, an integro-differential equation is proposed for upscaling multi-component reactive transport in heterogeneous media that relies on copulas for representing the velocity correlation structure. For this purpose, we express concentration or flux of solutes as a distribution over their velocity. We then derive the integro-differential equation that governs the evolution of concentration distribution over a quantity defined as velocity-rank. In this way, the spatial evolution of breakthrough curves away from the source is predicted based on ergodic cross-sectional velocity distributions and a parameterized copula function which expresses the correlation between velocity-ranks of a solute particle along its trajectory. We demonstrate the validity of the proposed model by comparing breakthrough curves for conservative and non-linearly reacting solutes based on realizations of hydraulic conductivity fields to the results of the upscaled model.

Classical Advection-Diffusion in Heterogeneous Media

A method is proposed for solving the problem of impurity transport in heterogeneous medium due to the classical diffusion and advection. The case when advection is absent was analyzed separately. It focuses on distances from the impurity source, which are much larger than the main body of its localization, and the asymptotic approach developed by one of the authors (P.S.K.) is used. The problem is reduced to solving a differential equation of the first order, which determines the linear trajectory of the concentration signal from the source that arises with this approach to the point of observation. The result for concentration is expressed in terms of one-dimensional integrals along the concentration signal line. The solution of the transport problem in the presence of advection is obtained by transition into the coordinate system accompanying advection. The key elements entering the resulting concentration expression are the effective time and impurity displacement, both are caused by advection.