David A. Benson

Application of a fractional advection-dispersion equation

Abstract. A transport equation that uses fractional-order dispersion derivatives hasfundamental solutions that are Le´vy’s a-stable densities. These densities represent plumesthat spread proportional to time 1/a , have heavy tails, and incorporate any degree ofskewness. The equation is parsimonious since the dispersion parameter is not a functionof time or distance. The scaling behavior of plumes that undergo Le´vy motion isaccounted for by the fractional derivative. A laboratory tracer test is described by adispersion term of order 1.55, while the Cape Cod bromide plume is modeled by anequation of order 1.65 to 1.8. 1. Introduction Anomalous, or non-Fickian, dispersion has been an activearea of research in the physics community since the introduc-tion of continuous time random walks (CTRW) by Montrolland Weiss [1965]. These random walks extended the predictivecapability of models built on the stochastic process of Brown-ian motion, which is the basis for the classical advection-dispersion equation (ADE). The CTRW assign a joint space-time distribution, called the transition density, to individualparticle motions. When the tails are heavy enough (i.e., powerlaw), non-Fickian dispersion results for all time scales andspace scales.

The fractional‐order governing equation of Lévy Motion

A governing equation of stable random walks is developed in one dimension. This Fokker-Planck equation is similar to, and contains as a subset, the second-order advection dispersion equation (ADE) except that the order (a) of the highest derivative is fractional (e.g., the 1.65th derivative). Fundamental solutions are Levys a-stable densities that resemble the Gaussian except that they spread proportional to time 1/a , have heavier tails, and incorporate any degree of skewness. The measured variance of a plume undergoing Levy motion would grow faster than Fickian plume, at a rate of time 2/a , where 0 , a # 2. The equation is parsimonious since the parameters are not functions of time or distance. The scaling behavior of plumes that undergo Levy motion is accounted for by the fractional derivatives, which are appropriate measures of fractal functions. In real space the fractional derivatives are integrodifferential operators, so the fractional ADE describes a spatially nonlocal process that is ergodic and has analytic solutions for all time and space.

Fractional Dispersion, Lévy Motion, and the MADE Tracer Tests

The macrodispersion experiments (MADE) at the Columbus Air Force Base in Mississippi were conducted in a highly heterogeneous aquifer that violates the basic assumptions of local second-order theories. A governing equation that describes particles that undergo Levy motion, rather than Brownian motion, readily describes the highly skewed and heavy-tailed plume development at the MADE site. The new governing equation is based on a fractional, rather than integer, order of differentiation. This order (α), based on MADE plume measurements, is approximately 1.1. The hydraulic conductivity (K) increments also follow a power law of order α=1.1. We conjecture that the heavy-tailed K distribution gives rise to a heavy-tailed velocity field that directly implies the fractional-order governing equation derived herein. Simple arguments lead to accurate estimates of the velocity and dispersion constants based only on the aquifer hydraulic properties. This supports the idea that the correct governing equation can be accurately determined before, or after, a contamination event. While the traditional ADE fails to model a conservative tracer in the MADE aquifer, the fractional equation predicts tritium concentration profiles with remarkable accuracy over all spatial and temporal scales.

Eulerian derivation of the fractional advection–dispersion equation

A fractional advection-dispersion equation (ADE) is a generalization of the classical ADE in which the second-order derivative is replaced with a fractional-order derivative. In contrast to the classical ADE, the fractional ADE has solutions that resemble the highly skewed and heavy-tailed breakthrough curves observed in field and laboratory studies. These solutions, known as alpha-stable distributions, are the result of a generalized central limit theorem which describes the behavior of sums of finite or infinite-variance random variables. We use this limit theorem in a model which sums the length of particle jumps during their random walk through a heterogeneous porous medium. If the length of solute particle jumps is not constrained to a representative elementary volume (REV), dispersive flux is proportional to a fractional derivative. The nature of fractional derivatives is readily visualized and their parameters are based on physical properties that are measurable. When a fractional Ficks law replaces the classical Ficks law in an Eulerian evaluation of solute transport in a porous medium, the result is a fractional ADE. Fractional ADEs are ergodic equations since they occur when a generalized central limit theorem is employed.

Subordinated advection‐dispersion equation for contaminant transport

A mathematical method called subordination broadens the applicability of the classical advection-dispersion equation for contaminant transport. In this method the time variable is randomized to represent the operational time experienced by different particles. In a highly heterogeneous aquifer the operational time captures the fractal properties of the medium. This leads to a simple, parsimonious model of contaminant transport that exhibits many of the features (heavy tails, skewness, and non-Fickian growth rate) typically seen in real aquifers. We employ a stable subordinator that derives from physical models of anomalous diffusion involving fractional derivatives. Applied to a one- dimensional approximation of the MADE-2 data set, the model shows excellent agreement.

Transport of conservative solutes in simulated fracture networks: 1. Synthetic data generation

[1] This paper investigates whether particle ensembles in a fractured rock domain may be adequately modeled as an operator-stable plume. If this statistical model applies to transport in fractured media, then an ensemble plume in a fractured rock domain may be modeled using the novel Fokker-Planck evolution equation of the operator-stable plume. These plumes (which include the classical multi-Gaussian as a subset) are typically characterized by power law leading-edge concentration profiles and super-Fickian growth rates. To investigate the possible correspondence of ensemble plumes to operator-stable densities, we use numerical simulations of fluid flow and solute transport through largescale (2.5 km by 2.5 km), randomly generated fracture networks. These two-dimensional networks are generated according to fracture statistics obtained from field studies that describe fracture length, transmissivity, density, and orientation. A fracture continuum approach using MODFLOW is developed for the solution of fluid flow within the fracture network and low-permeability rock matrix, while a particle-tracking code, random walk particle method for simulating transport in heterogeneous permeable media (RWHet), is used to simulate the advective motion of conservative solutes through the model domain. By deterministically mapping individual fractures onto a highly discretized finite difference grid (1 m � 1m � 1 m here), the MODFLOW ‘‘continuum’’ simulations can faithfully preserve details of the generated network and can approximate fluid flow in a discrete fracture network model. An advantage of the MODFLOW approach is that matrix permeability can be made nonzero to account for any degree of matrix flow and/or transport.

Apparent directional mass-transfer capacity coefficients in three-dimensional anisotropic heterogeneous aquifers under radial convergent transport

Aquifer hydraulic properties such as hydraulic conductivity (K) are ubiquitously heterogeneous and typically only a statistical characterization can be sought. Additionally, statistical anisotropy at typical characterization scales is the rule. Thus, regardless of the processes governing solute transport at the local (pore) scale, transport becomes non-Fickian. Mass-transfer models provide an efficient tool that reproduces observed anomalous transport; in some cases though, these models lack predictability as model parameters cannot readily be connected to the physical properties of aquifers. In this study, we focus on a multirate mass-transfer model (MRMT), and in particular the apparent capacity coefficient (β), which is a strong indicator of the potential of immobile zones to capture moving solute. We aim to find if the choice of an apparent β can be phenomenologically related to measures of statistical anisotropy. We analyzed an ensemble of random simulations of three-dimensional log-transformed multi-Gaussian permeability fields with stationary anisotropic correlation under convergent flow conditions. It was found that apparent β also displays an anisotropic behavior, physically controlled by the aquifer directional connectivity, which in turn is controlled by the anisotropic correlation model. A high hydraulic connectivity results in large β values. These results provide new insights into the practical use of mass-transfer models for predictive purposes.

A Microarray Biosensor for Multiplexed Detection of Microbes Using Grating-Coupled Surface Plasmon Resonance Imaging

Grating-coupled surface plasmon resonance imaging (GCSPRI) utilizes an optical diffraction grating embossed on a gold-coated sensor chip to couple collimated incident light into surface plasmons. The angle at which this coupling occurs is sensitive to the capture of analyte at the chip surface. This approach permits the use of disposable biosensor chips that can be mass-produced at low cost and spotted in microarray format to greatly increase multiplexing capabilities. The current GCSPRI instrument has the capacity to simultaneously measure binding at over 1000 unique, discrete regions of interest (ROIs) by utilizing a compact microarray of antibodies or other specific capture molecules immobilized on the sensor chip. In this report, we describe the use of GCSPRI to directly detect multiple analytes over a large dynamic range, including soluble protein toxins, bacterial cells, and viruses, in near real-time. GCSPRI was used to detect a variety of agents that would be useful for diagnostic and environmental sensing purposes, including macromolecular antigens, a nontoxic form of Pseudomonas aeruginosa exotoxin A (ntPE), Bacillus globigii, Mycoplasma hyopneumoniae, Listeria monocytogenes, Escherichia coli, and M13 bacteriophage. These studies indicate that GCSPRI can be used to simultaneously assess the presence of toxins and pathogens, as well as quantify specific antibodies to environmental agents, in a rapid, label-free, and highly multiplexed assay requiring nanoliter amounts of capture reagents.

A particle number conserving Lagrangian method for mixing‐driven reactive transport

The purely Lagrangian algorithm for chemical reactions introduced by Benson and Meerschaert (2008) suffers from a low-concentration resolution problem. We alleviate the problem by redefining the probabilistic collision/reaction (birth/death) stochastic process as a mass-reduction operation. Theoretically, this corresponds to replacing an on/off particle with a large number of “subparticles” and tracking the number fraction. The new particle reaction process maintains the original particle numbers but adjusts each particles mass upon reaction. Several simulations show the veracity as well as the gains in low-concentration resolution offered by the algorithm. We also compare the results to those obtained by a traditional finite difference model with suitably defined initial condition, demonstrating that the Lagrangian models match these.

Arbitrarily complex chemical reactions on particles

Previous particle-tracking (PT) algorithms for chemical reaction conceptualize each particle being composed of one species. Reactions occur by either complete or partial birth/death processes between interacting particles. Here we extend the method by placing any number of chemical species on each particle. The particle/particle interaction is limited to mass exchange. After exchange, reactions of any sort are carried out independently on each particle. The novel components of the algorithms are verified against analytic solutions where possible. This article is protected by copyright. All rights reserved.