Can a Time Fractional‐Derivative Model Capture Scale‐Dependent Dispersion in Saturated Soils?

Time nonlocal transport models such as the time fractional advection‐dispersion equation (t‐fADE) were proposed to capture well‐documented non‐Fickian dynamics for conservative solutes transport in heterogeneous media, with the underlying assumption that the time nonlocality (which means that the current concentration change is affected by previous concentration load) embedded in the physical models can release the effective dispersion coefficient from scale dependency. This assumption, however, has never been systematically examined using real data. This study fills this historical knowledge gap by capturing non‐Fickian transport (likely due to solute retention) documented in the literature (Huang et al. 1995) and observed in our laboratory from small to intermediate spatial scale using the promising, tempered t‐fADE model. Fitting exercises show that the effective dispersion coefficient in the t‐fADE, although differing subtly from the dispersion coefficient in the standard advection‐dispersion equation, increases nonlinearly with the travel distance (varying from 0.5 to 12 m) for both heterogeneous and macroscopically homogeneous sand columns. Further analysis reveals that, while solute retention in relatively immobile zones can be efficiently captured by the time nonlocal parameters in the t‐fADE, the motion‐independent solute movement in the mobile zone is affected by the spatial evolution of local velocities in the host medium, resulting in a scale‐dependent dispersion coefficient. The same result may be found for the other standard time nonlocal transport models that separate solute retention and jumps (i.e., displacement). Therefore, the t‐fADE with a constant dispersion coefficient cannot capture scale‐dependent dispersion in saturated porous media, challenging the application for stochastic hydrogeology methods in quantifying real‐world, preasymptotic transport. Hence improvements on time nonlocal models using, for example, the novel subordination approach are necessary to incorporate the spatial evolution of local velocities without adding cumbersome parameters.     

Delineation of three-dimensional well capture zones for complex multi-aquifer systems

The delineation of well capture zones is a basic component of ground water protection. The conventional methodology for capture zone delineation is backward advective particle tracking, often applied under the assumption of a two-dimensional aquifer. The suitability of the conventional approach for complex heterogeneous multi-aquifer systems was investigated, using the Waterloo Moraine aquifer system as an example. It was found that the conventional approach produces irregular particle tracks that require judgment to interpret in a meaningful way, and it can raise questions that may affect the credibility of the capture zone delineation. As an alternative, the potentially powerful but little-used backward-in-time advective-dispersive transport approach was investigated. A key advantage of this approach is its capability to represent local heterogeneities through the dispersion term. The dispersion process has a natural smoothing effect that results in unambiguous capture zones without the need for interpretation, thus enhancing credibility. The question of capture zone validation is also addressed. The meaning of a three-dimensional capture zone is considered, and it is shown that a fully three-dimensional representation of the system is crucial for valid results. The distinction between the maximum extent capture zone and the surface capture zone is also explained. In the case of complex heterogeneous systems, advective particle tracking can be used as an initial screening tool, whereas the more realistic backward-transport modeling approach can be used for final capture-zone delineation.     

Magnetic resonance microscopy of biofouling induced scale dependent transport in porous media

Non-invasive magnetic resonance microscopy (MRM) methods are applied to study biofouling of a homogeneous model porous media. MRM of the biofilm biomass using magnetic relaxation weighting shows the heterogeneous nature of the spatial distribution of the biomass as a function of growth. Spatially resolved MRM velocity maps indicate a strong variation in the pore scale velocity as a function of biofilm growth. The hydrodynamic dispersion dynamics for flow through the porous media is quantitatively characterized using a pulsed gradient spin echo technique to measure the propagator of the motion. The propagator indicates a transition in transport dynamics from a Gaussian normal diffusion process following a normal advection diffusion equation to anomalous transport as a function of biofilm growth. Continuous time random walk models resulting in a time fractional advection diffusion equation are shown to model the transition from normal to anomalous transport in the context of a conceptual model for the biofouling. The initially homogeneous porous media is transformed into a more complex heterogeneous porous media by the biofilm growth.     

Lagrangian simulations of unstable gravity-driven flow of fluids with variable density in randomly heterogeneous porous media

A new Lagrangian particle model based on smoothed particle hydrodynamics (SPH) is developed and used to simulate Darcy scale flow and transport in porous media. The method has excellent conservation properties and treats advection exactly. The Lagrangian method is used in stochastic analysis of miscible density-driven fluid flows. Results show that heterogeneity significantly increases dispersion and slows development of Rayleigh–Taylor instability. The presented numerical examples illustrate the advantages of Lagrangian methods for stochastic transport simulations.     

Lessons Learned from 25 Years of Research at the MADE Site

Field studies at well‐instrumented research sites have provided extensive data sets and important insights essential for development and testing of transport theories and mathematical models. This paper provides an overview of over 25 years of research and lessons learned at one of such field research sites on the Columbus Air Force Base in Mississippi, commonly known as the Macrodispersion Experiment (MADE) site. Since the mid‐1980s, field data from the MADE site have been used extensively by researchers around the world to explore complex contaminant transport phenomena in highly heterogeneous porous media. Results from field investigations and modeling analyses suggested that connected networks of small‐scale preferential flow paths and relative flow barriers exert dominant control on solute transport processes. The classical advection‐dispersion model was shown to inadequately represent plume‐scale transport, while the dual‐domain mass transfer model was found to reproduce the primary observed plume characteristics. The MADE site has served as a valuable natural observatory for contaminant transport studies where new observations have led to better understanding and improved models have sprung out analysis of new data.     

A transport phase diagram for pore-level correlated porous media

Transport in porous media is often characterized by the advection–dispersion equation, with the dispersion coefficient as the most important parameter that links the hydrodynamics to the transport processes. Morphological properties of any porous medium, such as pore size distribution, network topology, and correlation length control transport. In this study we explore the impact of correlation length on transport regime using pore-network modelling. Earlier direct simulation studies of dispersion in carbonate and sandstone rocks showed larger dispersion compared to granular homogenous sandpacks. However, in these studies, isolation of the impact of correlation length on transport regime was not possible due to the fundamentally different pore morphologies and pore-size distributions. Against this limitation, we simulate advection–dispersion transport for a wide range of Péclet numbers in unstructured irregular networks with “different” correlation lengths but “identical” pore size distributions and pore morphologies. Our simulation results show an increase in the magnitudes of the estimated dispersion coefficients in correlated networks compared to uncorrelated ones in the advection-controlled regime. The range of the Péclet numbers which dictate mixed advection–diffusion regime considerably reduces in the correlated networks. The findings emphasize the critical role of correlation length which is depicted in a conceptual transport phase diagram and the importance of accounting for the micro-scale correlation lengths into predictive stochastic pore-scale modelling.     

Multicomponent competitive monovalent cation exchange in hierarchical porous media with multimodal reactive mineral facies

This paper presents a stochastic model for multicomponent competitive monovalent cation exchange in hierarchical porous media. Reactive transport in porous media is highly sensitive to heterogeneities in physical and chemical properties, such as hydraulic conductivity (K), and cation exchange capacity (CEC). We use a conceptual model for multimodal reactive mineral facies and develop a Eulerian-based stochastic theory to analyze the transport of multiple cations in heterogeneous media with a hierarchical organization of reactive minerals. Numerical examples investigate the retardation factors and dispersivities in a chemical system made of three monovalent cations (Na⁺, K⁺, and Cs⁺). The results demonstrate how heterogeneity influences the transport of competitive monovalent cations, and highlight the importance of correlations between K and CEC. Further sensitivity analyses are presented investigating how the dispersion and retardation of each cation are affected by the means, variances, and integral scales of K and CEC. The volume fraction of organic matter is shown to be another important parameter. The Eulerian stochastic framework presented in this work clarifies the importance of each system parameters on the migration of cation plumes in formations with hierarchical organization of facies types. Our stochastic approach could be used as an alternative to numerical simulations for 3D reactive transport in hierarchical porous media, which become prohibitively expensive for the multicomponent applications considered in this work.     

Incorporating Super‐Diffusion due to Sub‐Grid Heterogeneity to Capture Non‐Fickian Transport

Numerical transport models based on the advection‐dispersion equation (ADE) are built on the assumption that sub‐grid cell transport is Fickian such that dispersive spreading around the average velocity is symmetric and without significant tailing on the front edge of a solute plume. However, anomalous diffusion in the form of super‐diffusion due to preferential pathways in an aquifer has been observed in field data, challenging the assumption of Fickian dispersion at the local scale. This study develops a fully Lagrangian method to simulate sub‐grid super‐diffusion in a multidimensional regional‐scale transport model by using a recent mathematical model allowing super‐diffusion along the flow direction given by the regional model. Here, the time randomizing procedure known as subordination is applied to flow field output from MODFLOW simulations. Numerical tests check the applicability of the novel method in mapping regional‐scale super‐diffusive transport conditioned on local properties of multidimensional heterogeneous media.     

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.     

Simulation of less‐mobile porosity dynamics in contrasting sediment water interface porous media

Considering heterogeneity in porous media pore size and connectivity is essential to predicting reactive solute transport across interfaces. However, exchange with less‐mobile porosity is rarely considered in surface water/groundwater recharge studies. Previous research indicates that a combination of pore‐fluid sampling and geoelectrical measurements can be used to quantify less‐mobile porosity exchange dynamics using the time‐varying relation between fluid and bulk electrical conductivity. For this study, we use macro‐scale (10 s of cm) advection–dispersion solute transport models linked with electrical conduction in COMSOL Multiphysics to explore less‐mobile porosity dynamics in two different types of observed sediment water interface porous media. Modeled sediment textures contrast from strongly layered streambed deposits to poorly sorted lakebed sands and cobbles. During simulated ionic tracer perturbations, a lag between fluid and bulk electrical conductivity, and the resultant hysteresis, is observed for all simulations indicating differential loading of pore spaces with tracer. Less‐mobile exchange parameters are determined graphically from these tracer time series data without the need for inverse numerical model simulation. In both sediment types, effective less‐mobile porosity exchange parameters are variable in response to changes in flow direction and fluid flux. These observed flow‐dependent effects directly impact local less‐mobile residence times and associated contact time for biogeochemical reaction. The simulations indicate that for the sediment textures explored here, less‐mobile porosity exchange is dominated by variable rates of advection through the domain, rather than diffusion of solute, for typical low‐to‐moderate rate (approximately 3–40 cm/day) hyporheic fluid fluxes. Overall, our model‐based results show that less‐mobile porosity may be expected in a range of natural hyporheic sediments and that changes in flowpath orientation and magnitude will impact less‐mobile exchange parameters. These temporal dynamics can be assessed with the geoelectrical experimental tracer method applied at laboratory and field scales.     

Investigation of Small‐Scale Preferential Flow with a Forced‐Gradient Tracer Test

A new tracer experiment (referred to as MADE‐5) was conducted at the well‐known Macrodispersion Experiment (MADE) site to investigate the influence of small‐scale mass‐transfer and dispersion processes on well‐to‐well transport. The test was performed under dipole forced‐gradient flow conditions and concentrations were monitored in an extraction well and in two multilevel sampler (MLS) wells located at 6, 1.5, and 3.75 m from the source, respectively. The shape of the breakthrough curve (BTC) measured at the extraction well is strongly asymmetric showing a rapidly arriving peak and an extensive late‐time tail. The BTCs measured at seven different depths in the two MLSs are radically different from one another in terms of shape, arrival times, and magnitude of the concentration peaks. All of these characteristics indicate the presence of a complex network of preferential flow pathways controlling solute transport at the test site. Field‐experimental data were also used to evaluate two transport models: a stochastic advection‐dispersion model (ADM) based on conditional multivariate Gaussian realizations of the hydraulic conductivity field and a dual‐domain single‐rate (DDSR) mass‐transfer model based on a deterministic reconstruction of the aquifer heterogeneity. Unlike the stochastic ADM realizations, the DDSR accurately predicted the magnitude of the concentration peak and its arrival time (within a 1.5% error). For the multilevel BTCs between the injection and extraction wells, neither model reproduced the observed values, indicating that a high‐resolution characterization of the aquifer heterogeneity at the subdecimeter scale would be needed to fully capture 3D transport details.     

An ELLAM approximation for advective–dispersive transport with nonlinear sorption

We consider an Eulerian–Lagrangian localized adjoint method (ELLAM) applied to nonlinear model equations governing solute transport and sorption in porous media. Solute transport in the aqueous phase is modeled by standard advection and hydrodynamic dispersion processes, while sorption is modeled with a nonlinear local-equilibrium model. We present our implementation of finite volume ELLAM (FV-ELLAM) and finite element (FE-ELLAM) discretizations to the reactive transport model and evaluate their performance for several test problems containing self-sharpening fronts.     

Reactive transport in disordered media: Role of fluctuations in interpretation of laboratory experiments

We review the analysis of the dynamics of reactive transport in disordered media, emphasizing the nature of the chemical reactions and the role of small-scale fluctuations induced by the structure of the porous medium. We are motivated by results and interpretations of laboratory-scale experiments, for which detailed characterization of the system is possible. Modeling approaches based on continuum and particle tracking (PT) schemes are examined critically, highlighting how fluctuations are incorporated. The continuum approach spans a large literature. Traditional formats of reactive transport equations, such as the advection–dispersion–reaction equation (ADRE), are based on a series of assumptions related mainly to scale separation and relative magnitude of time scales involved in the reactive transport setting. These assumptions as well as further developments are assessed in depth. PT methods offer an alternative means of accounting for pore-scale dynamics, wherein space–time transitions are drawn from appropriate probability distributions that have been tested to account for anomalous transport. While PT methods have been employed for many years to describe conservative transport, their application to laboratory-scale reactive transport problems in the context of both Fickian and non-Fickian regimes is relatively recent. We concentrate on experimental observations of different types of reactions in disordered media: (1) the dynamics of a bimolecular reactive transport (A + B → C) in passive (non-reactive) media, and (2) a multi-step chemical reaction, as exemplified in the process of dedolomitization involving both dissolution and precipitation. The fluctuations in a number of the key variables controlling the processes prove to have a dominant role; elucidation of this role forms the basis of the present study and the comparison of methods.     

Application of a Bayesian approach to stochastic delineation of capture zones

This paper presents a Bayesian Monte Carlo method for evaluating the uncertainty in the delineation of well capture zones and its application to a wellfield in a heterogeneous, multiaquifer system. In the method presented, Bayes' rule is used to update prior distributions for the unknown parameters of the stochastic model for the hydraulic conductivity, and to calculate probability-based weights for parameter realizations using head residuals. These weights are then assigned to the corresponding capture zones obtained using forward particle tracking. Statistical analysis of the set of weighted protection zones results in a probability distribution for the capture zones. The suitability of the Bayesian stochastic method for a multilayered system is investigated, using the wellfield Het Rot at Nieuwrode, Belgium, located in a three-layered aquifer system, as an example. The hydraulic conductivity of the production aquifer is modeled as a spatially correlated random function with uncertain parameters. The aquitard and overlying unconfined aquifer are assigned random, homogeneous conductivities. The stochastic results are compared with deterministic capture zones obtained with a calibrated model for the area. The predictions of the stochastic approach are more conservative and indicate that parameter uncertainty should be taken into account in the delineation of well capture zones.     

Solute transport in sand and chalk: a probabilistic approach

A probabilistic approach is used to simulate particle tracking for two types of porous medium. The first is sand grains with a single intergranular porosity. Particle tracking is carried out by advection and dispersion. The second is chalk granulates with intergranular and matrix porosities. Sorption can occur with advection and dispersion during particle tracking. Particle tracking is modelled as the sum of elementary steps with independent random variables in the sand medium. An exponential distribution is obtained for each elementary step and shows that the whole process is Markovian. A Gamma distribution or probability density function is then deduced. The relationships between dispersivity and the elementary step are given using the central limit theorem. Particle tracking in the chalky medium is a non‐Markovian process. The probability density function depends on a power of the distance. Experimental simulations by dye tracer tests on a column have been performed for different distances and discharges. The probabilistic approach computations are in good agreement with the experimental data. The probabilistic computation seems an interesting and complementary approach to simulate transfer phenomena in porous media with respect to the traditional numerical methods. Copyright © 2006 John Wiley & Sons, Ltd.     

Analytical solutions of one-dimensional macrodispersion in stratified porous media by the quadrupole method: convergence to an equivalent homogeneous porous medium

In this article, the quadrupole method is implemented in order to simulate the effects of heterogeneities on one dimensional advective and diffusive transport of a passive solute in porous media. Theoretical studies of dispersion in heterogeneous stratified media can bring insight into transport artefacts linked to scale effects and apparent dispersion coefficients. The quadrupole method is an efficient method for the calculation of transient response of linear systems. It is based here on the Laplace transform technique. The analytical solutions that can be derived by this method assists understanding of upscaled parameters relevant to heterogeneous porous media.First, the method is developed for an infinite homogeneous porous medium. Then, it is adapted to a stratified medium where the fluid flow is perpendicular to the interfaces. The first heterogeneous medium studied is composed of two semi-infinite layers perpendicular to the flow direction each having different transport properties. The concentration response of the medium to a Dirac injection is evaluated. The case studied emphasises the importance in the choice of the boundary conditions.In the case of a periodic heterogeneous porous medium, the concentration response of the medium is evaluated for different numbers of unit-cells. When the number of unit cells is great enough, depending on the transport properties of each layer in the unit cell, an equivalent homogeneous behaviour is reached. An exact determination of the transport properties (equivalent dispersion coefficient) of the equivalent homogeneous porous medium is given.     

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.     

Non-Fickian mass transport in fractured porous media

The paper provides an introduction to fundamental concepts of mathematical modeling of mass transport in fractured porous heterogeneous rocks. Keeping aside many important factors that can affect mass transport in subsurface, our main concern is the multi-scale character of the rock formation, which is constituted by porous domains dissected by the network of fractures. Taking into account the well-documented fact that porous rocks can be considered as a fractal medium and assuming that sizes of pores vary significantly (i.e. have different characteristic scales), the fractional-order differential equations that model the anomalous diffusive mass transport in such type of domains are derived and justified analytically. Analytical solutions of some particular problems of anomalous diffusion in the fractal media of various geometries are obtained. Extending this approach to more complex situation when diffusion is accompanied by advection, solute transport in a fractured porous medium is modeled by the advection-dispersion equation with fractional time derivative. In the case of confined fractured porous aquifer, accounting for anomalous non-Fickian diffusion in the surrounding rock mass, the adopted approach leads to introduction of an additional fractional time derivative in the equation for solute transport. The closed-form solutions for concentrations in the aquifer and surrounding rocks are obtained for the arbitrary time-dependent source of contamination located in the inlet of the aquifer. Based on these solutions, different regimes of contamination of the aquifers with different physical properties can be readily modeled and analyzed.