The fractional advection-dispersion equation: Development and application

The traditional 2{dollar}rmsp{lcub}nd{rcub}{dollar}-order advection-dispersion equation (ADE) does not adequately describe the movement of solute tracers in aquifers. This study examines and re-derives the governing equation. The analysis starts with a generalized notion of particle movements, since the second-order equation is trying to impart Brownian motion on a mathematical plume at any time. If particle motions with long-range spatial correlation are more favored, then the motion is described by Levy's family of {dollar}alpha{dollar}-stable densities. The new governing (Fokker-Planck) equation of these motions is similar to the ADE except that the order ({dollar}alpha{dollar}) of the highest derivative is fractional (e.g., the 1.65{dollar}rmsp{lcub}th{rcub}{dollar} derivative). Fundamental solutions resemble the Gaussian except that they spread proportional to time{dollar}sp{lcub}1/alpha{rcub}{dollar} and have heavier tails. The order of the fractional ADE (FADE) is shown to be related to the aquifer velocity auto-correlation function.; The FADE derived here is used to model three experiments with improved results over traditional methods. The first experiment is pure diffusion of high ionic strength CuSO{dollar}sb4{dollar} into distilled water. The second experiment is a one-dimensional tracer test in a 1-meter sandbox designed and constructed for minimum heterogeneity. The FADE, with a fractional derivative of order {dollar}alpha{dollar} = 1.55, nicely models the non-Fickian rate of spreading and the heavy tails often explained by reactions or multi-compartment complexity. The final experiment is the U.S.G.S. bromide tracer test in the Cape Cod aquifer. The order of the FADE is shown to be 1.6. Unlike theories based on the traditional ADE, the FADE is a "stand-alone" equation since the dispersion coefficient is a constant over all scales.; A numerical implementation is also developed to better handle the nonideal initial conditions of the Cape Cod test. The numerical method promises to reduce the number of elements in a typical numerical model by orders-of-magnitude while maintaining equivalent scale-dependent spreading that would normally be created by very fine realizations of the K field.