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

classical form of the partial differential equation governing advective-dispersive transport of a solute in idealized porous media is the advection-dispersion equation (ADE). This equation is built on a Fickian constitutive theory that gives the local dispersive flux as the inner product of a constant dispersion tensor and the spatial gradient of the solute concentration. Over the past decade, investigators in subsurface transport have been increasingly focused on anomalous (i.e., non-Fickian) dispersion in natural geologic formations. It has been recognized that this phenomenon involves spatial and possibly temporal nonlocality in the constitutive theory describing the dispersive flux. Most recently, several researchers have modeled anomalous dispersion using a constitutive theory that relies on fractional, as opposed to integer, derivatives of the concentration field. The resulting ADE itself is expressed in terms of fractional derivatives, and they are described as fractional ADEs. Here we show that the fractional ADE is obtained as a special case of the authors' convolution-Fickian nonlocal ADE [Cushman and Ginn, 1.993]. To obtain the fractional ADE from the convolution-Fickian model requires only a judicial choice of the wave vector and frequency-dependent dispersion tensor.