Fractional advection-dispersion equation: A classical mass balance with convolution-Fickian flux

classical form of the partial differential equation governing advective-dis persive transport of a solute in idealized porous media is the advection-di spersion equation (ADE). This equation is built on a Fickian constitutive t heory that gives the local dispersive flux as the inner product of a consta nt dispersion tensor and the spatial gradient of the solute concentration. Over the past decade, investigators in subsurface transport have been incre asingly focused on anomalous (i.e., non-Fickian) dispersion in natural geol ogic formations. It has been recognized that this phenomenon involves spati al and possibly temporal nonlocality in the constitutive theory describing the dispersive flux. Most recently, several researchers have modeled anomal ous dispersion using a constitutive theory that relies on fractional, as op posed to integer, derivatives of the concentration field. The resulting ADE itself is expressed in terms of fractional derivatives, and they are descr ibed as "fractional ADEs." Here we show that the fractional ADE is obtained as a special case of the authors' convolution-Fickian nonlocal ADE [Cushma n and Ginn, 1.993]. To obtain the fractional ADE from the convolution-Ficki an model requires only a judicial choice of the wave vector and frequency-d ependent dispersion tensor.