Stephen W. Wheatcraft

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.

Gravity-driven infiltration instability in initially dry nonhorizontal fractures

Experimental evidence demonstrating gravity-driven wetting front instability in an initially dry natural fracture is presented. An experimental approach is developed using a transparent analog rough-walled fracture to explore gravity-driven instability. Three different boundary conditions were observed to produce unstable fronts in the analog fracture: application of fluid at less than the imbibition capacity, inversion of a density-stratified system, and redistribution of flow at the cessation of stable infiltration. The redistribution boundary condition (analogous to the cessation of ponded infiltration) is considered in a series of systematic experiments. Gravitational gradient and magnitude of the fluid input were varied during experimentation. Qualitative observations imply that finger development is strongly correlated to the structure of the imbibition front at the onset of flow redistribution. Measurements of fingertip velocity are used to develop a first-order relationship with fingertip length. Measured finger width is compared to theoretical predictions based on linear stability theory.

Macrodispersivity Tensor for Nonreactive Solute Transport in Isotropic and Anisotropic Fractal Porous Media: Analytical Solutions

Using spectral stochastic theory, macrodispersivity tensors are developed for one-, two-, and three-dimensional isotropic and anisotropic fractal porous media. Since natural geologic aquifers are always bounded, we introduce the concept of a maximum length scale ((Lmax). In the one-dimensional case, the asymptotic (long-time) longitudinal macrodispersivity (A∞) is proportional to Lmax2, and in the multidimensional cases, A∞ ∝ Lmax. In the multidimensional cases, as the fractal dimension increases, A∞ decreases, because a larger fractal dimension means the tortuosity of solute particles is larger, leading to more mixing in the transverse direction, which in turn reduces longitudinal spreading. The transverse macrodispersivities are shown to be analogous to traditional spectral stochastic results. In the multidimensional cases, Lmax is shown to be controlled by the aquifer thickness. As a result, ergodic conditions may be reached in relatively thin aquifers, allowing the use of the asymptotic macrodispersivity expressions obtained here for single realizations.

Is Old Faithful a strange attractor

Geysers are systems that exhibit hydrothermal eruptive behavior at irregular and, occasionally, relatively regular intervals. Because geyser dynamics are governed by strongly nonlinear differential equations, it is reasonable to expect that these systems are capable of displaying chaotic behavior. Geyser dynamics are shown to be analogous to a simple conceptual model known to exhibit chaotic behavior. State space reconstructions of eruption interval data then show that the geyser Old Faithful erupts at chaotic intervals. Establishing the system as chaotic explains the inability to predict long-term eruptive behavior of Old Faithful.

Non-Fickian ionic diffusion across high-concentration gradients

A non-Fickian physico-chemical model for electrolyte transport in high-ionic strength systems is developed and tested with laboratory experiments with copper sulfate as an example electrolyte. The new model is based on irreversible thermodynamics and uses measured mutual diffusion coefficients, varying with concentration. Compared to a traditional Fickian model, the new model predicts less diffusion and asymmetric diffusion profiles. Laboratory experiments show diffusion rates even smaller than those predicted by our non-Fickian model, suggesting that there are additional, unaccounted for processes retarding diffusion. Ionic diffusion rates may be a limiting factor in transporting salts whose effect on fluid density will in turn significantly affect the flow regime. These findings have important implications for understanding and predicting solute transport in geologic settings where dense, saline solutions occur.

Including Multi-Scale Information in the Characterization of Hydraulic Conductivity Distributions

Abstract Transport in heterogeneous porous media is highly dependent on the spatial variability of aquifer properties, particularly hydraulic conductivity. It is impractical, however, when modeling a real situation, to obtain aquifer properties at the scale of each grid block. Wavelet transforms are investigated as a tool to assess the movement of information between scales, and to incorporate the scale and location of hydraulic conductivity test data when interpolating to a fine grid. Wavelet transforms are particularly useful as they include parameters that define a spatial localization center that can be positioned in space, and changed in size. As a result, one can incorporate multiple measured hydraulic conductivity values in the wavelet transformation at a resolution matched to each measurement scale. A multi-scale reconstruction method is developed here which uses forward and inverse wavelet transforms in conjunction with a pseudo-fractal distribution to fill in missing information around sparse data. This wavelet reconstruction method is compared to several more traditional interpolation schemes with respect to accuracy of solute transport prediction. The wavelet reconstruction algorithm is then used to examine the issue of optimum sample size and density for stationary and fractal random fields.