Continuous time random walks for non-local radial solute transport

This study formulates and analyzes continuous time random walk (CTRW) models in radial flow geometries for the quantification of non-local solute transport induced by heterogeneous flow distributions and by mobile–immobile mass transfer processes. To this end we derive a general CTRW framework in radial coordinates starting from the random walk equations for radial particle positions and times. The particle density, or solute concentration is governed by a non-local radial advection–dispersion equation (ADE). Unlike in CTRWs for uniform flow scenarios, particle transition times here depend on the radial particle position, which renders the CTRW non-stationary. As a consequence, the memory kernel characterizing the non-local ADE, is radially dependent. Based on this general formulation, we derive radial CTRW implementations that (i) emulate non-local radial transport due to heterogeneous advection, (ii) model multirate mass transfer (MRMT) between mobile and immobile continua, and (iii) quantify both heterogeneous advection in a mobile region and mass transfer between mobile and immobile regions. The expected solute breakthrough behavior is studied using numerical random walk particle tracking simulations. This behavior is analyzed by explicit analytical expressions for the asymptotic solute breakthrough curves. We observe clear power-law tails of the solute breakthrough for broad (power-law) distributions of particle transit times (heterogeneous advection) and particle trapping times (MRMT model). The combined model displays two distinct time regimes. An intermediate regime, in which the solute breakthrough is dominated by the particle transit times in the mobile zones, and a late time regime that is governed by the distribution of particle trapping times in immobile zones. These radial CTRW formulations allow for the identification of heterogeneous advection and mobile-immobile processes as drivers of anomalous transport, under conditions relevant for field tracer tests.     

Statistical scale-up of 3D particle-tracking simulation for non-Fickian dispersive solute transport modeling

Numerical techniques for subsurface flow and transport modeling are often limited by computational limitations including fine mesh and small time steps to control artificial dispersion. Particle-tracking simulation offers a robust alternative for modeling solute transport in subsurface formations. However, the modeling scale usually differs substantially from the rock measurement scale, and the scale-up of measurements have to be made accounting for the pattern of spatial heterogeneity exhibited at different scales. Therefore, it is important to construct accurate coarse-scale simulations that are capable of capturing the uncertainties in reservoir and transport attributes due to scale-up. A statistical scale-up procedure developed in our previous work is extended by considering the effects of unresolved (residual) heterogeneity below the resolution of the finest modeling scale in 3D. First, a scale-up procedure based on the concept of volume variance is employed to construct realizations of permeability and porosity at the (coarse) transport modeling scale, at which flow or transport simulation is performed. Next, to compute various effective transport parameters, a series of realizations exhibiting detailed heterogeneities at the fine scale, whose domain size is the same as the transport modeling scale, are generated. These realizations are subjected to a hybrid particle-tracking simulation. Probabilistic transition time is considered, borrowing the idea from the continuous time random walk (CTRW) technique to account for any sub-scale heterogeneity at the fine scale level. The approach is validated against analytical solutions and general CTRW formulation. Finally, coarse-scale transport variables (i.e., dispersivities and parameterization of transition time distribution) are calibrated by minimizing the mismatch in effluent history with the equivalent averaged models. Construction of conditional probability distributions of effective parameters is facilitated by integrating the results over the entire suite of realizations. The proposed method is flexible, as it does not invoke any explicit assumption regarding the multivariate distribution of the heterogeneity. In contrast to other hierarchical CTRW formulation for modeling multi-scale heterogeneities, the proposed approach does not impose any length scale requirement regarding sub-grid heterogeneities. In fact, it aims to capture the uncertainty in effective reservoir and transport properties due to the presence of heterogeneity at the intermediate scale, which is larger than the finest resolution of heterogeneity but smaller than the representative elementary volume, but it is often comparable to the transport modeling scale.     

Perspective on theories of non-Fickian transport in heterogeneous media

Subsurface fluid flow and solute transport take place in a multiscale heterogeneous environment. Neither these phenomena nor their host environment can be observed or described with certainty at all scales and locations of relevance. The resulting ambiguity has led to alternative conceptualizations of flow and transport and multiple ways of addressing their scale and space–time dependencies. We focus our attention on four approaches that give rise to nonlocal representations of advective and dispersive transport of nonreactive tracers in randomly heterogeneous porous or fractured continua. We compare these approaches theoretically on the basis of their underlying premises and the mathematical forms of the corresponding nonlocal advective–dispersive terms. One of the four approaches describes transport at some reference support scale by a classical (Fickian) advection–dispersion equation (ADE) in which velocity is a spatially (and possibly temporally) correlated random field. The randomness of the velocity, which is given by Darcy’s law, stems from random fluctuations in hydraulic conductivity (and advective porosity though this is often disregarded). Averaging the stochastic ADE over an ensemble of velocity fields results in a space–time-nonlocal representation of mean advective–dispersive flux, an approach we designate as stnADE. A closely related space–time-nonlocal representation of ensemble mean transport is obtained upon averaging the motion of solute particles through a random velocity field within a Lagrangian framework, an approach we designate stnL. The concept of continuous time random walk (CTRW) yields a representation of advective–dispersive flux that is nonlocal in time but local in space. Closely related to the latter are forms of ADE entailing fractional derivatives (fADE) which leads to representations of advective–dispersive flux that are nonlocal in space but local in time; nonlocality in time arises in the context of multirate mass transfer models, which we exclude from consideration in this paper. We describe briefly each of these four nonlocal approaches and offer a perspective on their differences, commonalities, and relative merits as analytical and predictive tools.     

Influence of small-scale heterogeneities on contaminant transport in fractured crystalline rock

We present a sequence of purely advective transport models that demonstrate the influence of small-scale geometric inhomogeneities on contaminant transport in fractured crystalline rock. Special weight is placed on the role of statistically generated variable fracture apertures. The fracture network geometry and the aperture distribution are based on information from an in situ radionuclide retardation experiment performed at Grimsel test site (Swiss Alps). The obtained breakthrough curves are fitted with the advection dispersion equation and continuous-time random walks (CTRW). CTRW is found to provide superior fits to the late-arrival tailing and is also found to show a good correlation with the velocity distributions obtained from the hydraulic models. The impact of small-scale heterogeneities, both in fracture geometry and aperture, on transport is shown to be considerable.     

Computational issues in the determination of solute discharge moments and implications for comparison to analytical solutions

Solute discharge moments (mean and variance) are computed using numerical modeling of flow and advective transport in two-dimensional heterogeneous aquifers and are compared to theoretical results. The solute discharge quantifies the temporal evolution of the total contaminant mass crossing a certain compliance boundary. In addition to analyzing the solute discharge moments within a classical absolute dispersion framework, we also analyze relative dispersion formulation, whereby plume meandering (deviation from mean flow path caused by velocity variations at scales larger than plume size) is removed. This study addresses some important issues related to the computation of solute discharge moments from random walk particle tracking experiments, and highlights some of the important differences between absolute and relative dispersion frameworks. Relative dispersion formulation produces maximum uncertainty that coincides with the peak mean discharge. Absolute dispersion, however, results in earlier arrival of the uncertainty peak as compared to the first moment peak. Simulations show that the standard deviation of solute discharge in a relative dispersion framework requires increasingly large temporal sampling windows to smooth out some of the large fluctuations in breakthrough curves associated with advective transport. Using smoothing techniques in particle tracking to distribute the particle mass over a volume rather than at a point significantly reduces the noise in the numerical simulations and removes the need to use large temporal windows. Same effect can be obtained by adding a local dispersion process to the particle tracking experiments used to model advective transport. The effect of the temporal sampling window bears some relevance and important consequences for evaluating risk-related parameters. The expected value of peak solute discharge and its standard deviation are very sensitive to this sampling window and so will be the risk distribution relying on such numerical models.     

Experiment and Simulation of Non-Reactive Solute Transport in Porous Media

To more accurately predict the migration behavior of pollutants in porous media, we conduct laboratory scale experiments and model simulation. Aniline (AN) is used in one-dimensional soil column experiments designed under various media and hydrodynamic conditions. The advection-dispersion equation (ADE) and the continuous-time random walk (CTRW) were used to simulate the breakthrough curves (BTCs) of the solute transport. The results show that the media and hydrodynamic conditions are two important factors affecting solute transport and are related to the degree of non-Fickian transport. The simulation results show that CTRW can more effectively describe the non-Fickian phenomenon in the solute transport process than ADE. The sensitive parameter in the CTRW simulation process is , which can reflect the degree of non-Fickian diffusion in the solute transport. Understanding the relationship of with velocity and media particle size is conducive to improving the reactive solute transport model. The results of this study provide a theoretical basis for better prediction of pollutant transport in groundwater.     

Characterization of mixing and spreading in a bounded stratified medium

Matheron and de Marsily [Matheron M, de Marsily G. Is the transport in porous media always diffusive? A counter-example. Water Resour Res 1980;16:901–17] studied transport in a perfectly stratified infinite medium as an idealized aquifer model. They observed superdiffusive solute spreading quantified by anomalous increase of the apparent longitudinal dispersion coefficient with the square root of time. Here, we investigate solute transport in a vertically bounded stratified random medium. Unlike for the infinite medium at asymptotically long times, disorder-induced mixing and spreading is uniquely quantified by a constant Taylor dispersion coefficient. Using a stochastic modeling approach we study the effective mixing and spreading dynamics at pre-asymptotic times in terms of effective average transport coefficients. The latter are defined on the basis of local moments, i.e., moments of the transport Green function. We investigate the impact of the position of the initial plume and the initial plume size on the (highly anomalous) pre-asymptotic effective spreading and mixing dynamics for single realizations and in average. Effectively, the system “remembers” its initial state, the effective transport coefficients show so-called memory effects, which disappear after the solute has sampled the full vertical extent of the medium. We study the impact of the intrinsic non-ergodicity of the confined medium on the validity of the stochastic modeling approach and study in this context the transition from the finite to the infinite medium.     

An integrated approach for modeling solute transport in streams and canals with applications

A new modeling approach for solute transport in streams and canals was developed to simulate solute dissolution, transport, and decay with continuously migrating sources. The new approach can efficiently handle complicated solute source feeding schemes and initial conditions. Incorporating the finite volume method (FVM) and the ULTIMATE QUICKEST numerical scheme, the new approach is capable of predicting fate and transport of solute that is added to small streams or canals, typically in a continuous fashion. The approach was tested successfully using a hypothetical case, and then applied to an actual field experiment, where linear anionic polyacrylamide (LA-PAM) was applied to an earthen canal. The field experiment was simulated first as a fixed boundary problem using measured concentration data as the boundary condition to test model parameters and sensitivities. The approach was then applied to a moving boundary problem, which included subsequent LA-PAM dissolution, settling to the canal bottom and transport with the flowing canal water. Simulation results showed that the modeling approach developed in this study performed satisfactorily and can be used to simulate a variety of transport problems in streams and canals.     

Transport of nonsorbing solutes in a streambed with periodic bedforms

Previous studies of hyporheic zone focused largely on the net mass transfer of solutes between stream and streambed. Solute transport within the bed has attracted less attention. In this study, we combined flume experiments and numerical simulations to examine solute transport processes in a streambed with periodic bedforms. Solute originating from the stream was subjected to advective transport driven by pore water circulation due to current–bedform interactions as well as hydrodynamic dispersion in the porous bed. The experimental and numerical results showed that advection played a dominant role at the early stage of solute transport, which took place in the hyporheic zone. Downward solute transfer to the deep ambient flow zone was controlled by transverse dispersion at the later stage when the elapsed time exceeded the advective transport characteristic time t_c (= L/u_c with L being the bedform length and u_c the characteristic pore water velocity). The advection-based pumping exchange model was found to predict reasonably well solute transfer between the overlying water and streambed at the early stage but its performance deteriorated at the later stage. With dispersion neglected, the pumping exchange model underestimated the long-term rate and total mass of solute transfer from the overlying water to the bed. Therefore both advective and dispersive transport components are essential for quantification of hyporheic exchange processes.     

Effects of uncertainty of lithofacies, conductivity and porosity distributions on stochastic interpretations of a field scale tracer test

We investigate the importance of selecting two different methodologies for the determination of hydraulic conductivity from available grain-size distributions on the stochastic modeling of the depth-averaged breakthrough curve observed during a forced-gradient tracer test experiment. The latter was performed in the Lauswiesen alluvial aquifer, located near the city of Tübingen, Germany, by injecting NaBr into a well at a distance of about 50 m from a pumping well. We also examine the joint effect of the choice of the transport model adopted to describe solute transport at the site and the way the spatial distribution of porosity is assessed. In the absence of direct measurements of porosity, we consider: (a) the model used by Riva et al. (J Contam Hydrol 88:92–118, 2006; J Contam Hydrol 101:1–13, 2008), which relates the natural logarithms of effective porosity and conductivity through an empirical, experimentally-based, linear relationship derived for a nearby experimental site; and (b) a model based on a commonly used relationship linking the total porosity to the coefficient of uniformity of grain size distributions. Transport is described in terms of a purely advective process and/or by including mass exchange processes between mobile and immobile regions. Modeling of flow and transport is performed within a Monte Carlo framework, upon conceptualizing the aquifer as a random composite medium. Our results indicate that the model adopted to describe the correlation between conductivity and porosity and the way grain-sieve information are incorporated to depict the heterogeneous distribution of hydraulic conductivity can have relevant effects in the interpretation of the data at the site. All the conceptual models employed to describe the structural heterogeneity of the system and transport features can reasonably reproduce the global characteristics of the experimental depth-averaged breakthrough curve. Specific details, such as the peak concentration and the time of first arrival, can be better reproduced by a double porosity transport model when a correlation between conductivity and porosity based on grain size information at the site is considered. The best prediction of the late-time behavior of the measured breakthrough curves, in terms of the observed heavy tailing, is offered by directly linking porosity distribution to the spatial variability of particle size information.     

An examination of the steady‐state assumption in soil development models with application to landscape evolution

Soil depth and soil production are highly complicated phenomena, generated from a complex interaction of physical, biological and chemical processes. It has, nevertheless, become increasingly clear that soil formation rates are closely related to chemical weathering rates. Somewhat paradoxically, it is likewise becoming apparent that such biogeochemical reactions as slowly transform rock to soil are limited by physical processes, such as flowing water and the formation of fractures. We have formulated a theoretical approach that relates soil formation rates to chemical weathering rates, and those, likewise, to solute transport rates. For such a theoretical framework to be relevant, the solute transport rates cannot equal those of the flowing water, as is the case in Gaussian solute transport. Rather, solute transport must be slowed in accordance with heavy‐tailed solute arrival time distributions. The inference is that the traditional advection–dispersion equation formulation for solute transport is inadequate in the typically heterogeneous geological media that weather to form soils. Here we examine the implications of this soil production model on the assumption of the approach to steady state. Particularly at slow erosion rates we find that many soil columns are not in equilibrium. This tendency may be accentuated in dry climates. Copyright © 2017 John Wiley & Sons, Ltd.     

Effect of permeability variations on solute transport in highly heterogeneous porous media

The effect of aquifer heterogeneity on flow and solute transport in two-dimensional isotropic porous media was analyzed using the Monte Carlo method. The two-dimensional logarithmic permeability (ln K) was assumed to be a non-stationary random field with its increments being a truncated fractional Lévy motion (fLm). The permeability fields were generated using the modified successive random additions (SRA) algorithm code SRA3DC [http://www.iamg.org/CGEditor/index.htm]. The velocity and concentration fields were computed respectively for two-dimensional flow and transport with a pulse input using the finite difference codes of MODFLOW 2000 and MT3DMS. Two fLm control parameters, namely the width parameter (C) and the Lévy index (α), were varied systematically to examine their effect on the resulting permeability, flow velocity and concentration fields. We also computed the first- and second-spatial moments, the dilution index, as well as the breakthrough curves at different control planes with the corresponding concentration fields. In addition, the derived breakthrough curves were fitted using the continuous time random walk (CTRW) and the traditional advection-dispersion equation (ADE). Results indicated that larger C and smaller α both led to more heterogeneous permeability and velocity fields. The Lévy-stable distribution of increments in ln K resulted in a Lévy-stable distribution of increments in logarithm of the velocity (ln v). Both larger C and smaller α created sharper leading edges and wider tailing edges of solute plumes. Furthermore, a relatively larger amount of solute still remained in the domain after a relatively longer time transport for smaller α values. The dilution indices were smaller than unity and increased as C increased and α decreased. The solute plume and its second-spatial moments increased as C increased and α decreased, while the first-spatial moments of the solute plume were independent of C and α values. The longitudinal macrodispersivity was scale-dependent and increased as a power law function of time. Increasing C and decreasing α both resulted in an increase in longitudinal macrodispersivity. The transport in such highly heterogeneous media was slightly non-Gaussian with its derived breakthrough curves being slightly better fitted by the CTRW than the ADE, especially in the early arrivals and late-time tails.     

Effects of compound-specific dilution on transient transport and solute breakthrough: A pore-scale analysis

This pore-scale modeling study in saturated porous media shows that compound-specific effects are important not only at steady-state and for the lateral displacement of solutes with different diffusivities but also for transient transport and solute breakthrough. We performed flow and transport simulations in two-dimensional pore-scale domains with different arrangement of the solid grains leading to distinct characteristics of flow variability and connectivity, representing mildly and highly heterogeneous porous media, respectively. The results obtained for a range of average velocities representative of groundwater flow (0.1–10 m/day), show significant effects of aqueous diffusion on solute breakthrough curves. However, the magnitude of such effects can be masked by the flux-averaging approach used to measure solute breakthrough and can hinder the correct interpretation of the true dilution of different solutes. We propose, as a metric of mixing, a transient flux-related dilution index that allows quantifying the evolution of solute dilution at a given position along the main flow direction. For the different solute transport scenarios we obtained dilution breakthrough curves that complement and add important information to traditional solute breakthrough curves. Such dilution breakthrough curves allow capturing the compound-specific mixing of the different solutes and provide useful insights on the interplay between advective and diffusive processes, mass transfer limitations, and incomplete mixing in the heterogeneous pore-scale domains. The quantification of dilution for conservative solutes is in good agreement with the outcomes of mixing-controlled reactive transport simulations, in which the mass and concentration breakthrough curves of the product of an instantaneous transformation of two initially segregated reactants were used as measures of reactive mixing.     

Chaotic advection at the pore scale: Mechanisms,upscaling and implications for macroscopic transport

The macroscopic spreading and mixing of solute plumes in saturated porous media is ultimately controlled by processes operating at the pore scale. Whilst the conventional picture of pore-scale mechanical dispersion and molecular diffusion leading to persistent hydrodynamic dispersion is well accepted, this paradigm is inherently two-dimensional (2D) in nature and neglects important three-dimensional (3D) phenomena. We discuss how the kinematics of steady 3D flow at the pore scale generate chaotic advection—involving exponential stretching and folding of fluid elements—the mechanisms by which it arises and implications of microscopic chaos for macroscopic dispersion and mixing. Prohibited in steady 2D flow due to topological constraints, these phenomena are ubiquitous due to the topological complexity inherent to all 3D porous media. Consequently 3D porous media flows generate profoundly different fluid deformation and mixing processes to those of 2D flow. The interplay of chaotic advection and broad transit time distributions can be incorporated into a continuous-time random walk (CTRW) framework to predict macroscopic solute mixing and spreading. We show how these results may be generalised to real porous architectures via a CTRW model of fluid deformation, leading to stochastic models of macroscopic dispersion and mixing which both honour the pore-scale kinematics and are directly conditioned on the pore-scale architecture.     

Inverse sequential simulation: Performance and implementation details

For good groundwater flow and solute transport numerical modeling, it is important to characterize the formation properties. In this paper, we analyze the performance and important implementation details of a new approach for stochastic inverse modeling called inverse sequential simulation (iSS). This approach is capable of characterizing conductivity fields with heterogeneity patterns difficult to capture by standard multiGaussian-based inverse approaches. The method is based on the multivariate sequential simulation principle, but the covariances and cross-covariances used to compute the local conditional probability distributions are computed by simple co-kriging which are derived from an ensemble of conductivity and piezometric head fields, in a similar manner as the experimental covariances are computed in an ensemble Kalman filtering. A sensitivity analysis is performed on a synthetic aquifer regarding the number of members of the ensemble of realizations, the number of conditioning data, the number of piezometers at which piezometric heads are observed, and the number of nodes retained within the search neighborhood at the moment of computing the local conditional probabilities. The results show the importance of having a sufficiently large number of all of the mentioned parameters for the algorithm to characterize properly hydraulic conductivity fields with clear non-multiGaussian features.     

Application of a data assimilation method via an ensemble Kalman filter to reactive urea hydrolysis transport modeling

With growing importance of water resources in the world, remediations of anthropogenic contaminations due to reactive solute transport become even more important. A good understanding of reactive rate parameters such as kinetic parameters is the key to accurately predicting reactive solute transport processes and designing corresponding remediation schemes. For modeling reactive solute transport, it is very difficult to estimate chemical reaction rate parameters due to complex processes of chemical reactions and limited available data. To find a method to get the reactive rate parameters for the reactive urea hydrolysis transport modeling and obtain more accurate prediction for the chemical concentrations, we developed a data assimilation method based on an ensemble Kalman filter (EnKF) method to calibrate reactive rate parameters for modeling urea hydrolysis transport in a synthetic one-dimensional column at laboratory scale and to update modeling prediction. We applied a constrained EnKF method to pose constraints to the updated reactive rate parameters and the predicted solute concentrations based on their physical meanings after the data assimilation calibration. From the study results we concluded that we could efficiently improve the chemical reactive rate parameters with the data assimilation method via the EnKF, and at the same time we could improve solute concentration prediction. The more data we assimilated, the more accurate the reactive rate parameters and concentration prediction. The filter divergence problem was also solved in this study.     

Solute transport scales in an unsaturated stony soil

Solute transport parameters are known to be scale-dependent due mainly to the increasing scale of heterogeneities with transport distance and with the lateral extent of the transport field examined. Based on a transect solute transport experiment, in this paper we studied this scale dependence by distinguishing three different scales with different homogeneity degrees of the porous medium: the observation scale, transport scale and transect scale. The main objective was to extend the approach proposed by van Wesenbeeck and Kachanoski to evaluating the role of textural heterogeneities on the transition from the observation scale to the transport scale. The approach is based on the scale dependence of transport moments estimated from solute concentrations distributions. In our study, these moments were calculated starting from time normalized resident concentrations measured by time domain reflectometry (TDR) probes at three depths in 37 soil sites 1 m apart along a transect during a steady state transport experiment. The Generalized Transfer Function (GTF) was used to describe the evolution of apparent solute spreading along the soil profile at each observation site by analyzing the propagation of the moments of the concentration distributions. Spectral analysis was used to quantify the relationship between the solid phase heterogeneities (namely, texture and stones) and the scale dependence of the solute transport parameters. Coupling the two approaches allowed us to identify two different transport scales (around 4-5 m and 20 m, respectively) mainly induced by the spatial pattern of soil textural properties. The analysis showed that the larger transport scale is mainly determined by the skeleton pattern of variability. Our analysis showed that the organization in hierarchical levels of soil variability may have major effects on the differences between solute transport behavior at transport scale and transect scale, as the transect scale parameters will include information from different scales of heterogeneities.     

Influence of conduit network geometry on solute transport in karst aquifers with a permeable matrix

In karst aquifers with significant matrix permeability, water and solutes are exchanged between the conduits and carbonate matrix. Transport through the matrix increases the spread of solutes and increases travel times. This study numerically evaluates advective solute transport in synthetic karst systems that contain 3D branching conduit networks. Particle tracking is performed to analyze the spatial and temporal transport history of solute that arrives at the conduit outlet. Three measures of transport connectivity are used to quantify the solute migration behavior: the skewness of the particle arrival time distribution, the normalized fifth percentile of arrival times, and the fraction of the total travel time that occurs within conduits. All three of these metrics capture the influence of conduit network geometry on solute transport. A more tortuous network leads to enhanced conduit-matrix mixing, which reduces the transport connectivity and yields a broader distribution of solute arrival times. These results demonstrate that the conduit network geometry is an important control on solute transport in karst systems with a permeable matrix.