Application of the Fractional Advection‐Dispersion Equation in Porous Media

According to the classical advection‐dispersion equation (ADE), the variance of travel distance increases linearly with the time elapsed after the release of a solute tracer. However, several field studies showed nonlinear relationships exist between the variance of travel distance for solute tracers and time. Therefore, the transport at the field scale often shows non‐Gaussion property. To describe the non‐Gaussian transport process, a fractional advection‐dispersion equation (FADE) is a possibility. Fractional ADE has been applied to the transport process of non‐reactive solutes in a laboratory sandbox as well as that for the Cape Cod field site. In this paper, we discuss several issues regarding the application of FADE. Because of the complexity of the FADE, its application often needs justification. We propose an F‐test to judge whether a FADE is necessary. For the Cape Cod experiment, the FADE of order 1.82 is necessary according to our F‐test results. In addition, we present a method to estimate the dispersion coefficient and fractional order simultaneously. The obtained parameters from joint estimation will recover the variance‐time relationship. Finally, we pointed out that the dispersivity‐time or dispersivity‐mean travel distance relationship is needed to estimate the fractional order α. Use of dispersivity‐distance relationship for estimating the fractional order is theoretically unjustified and may result in unreasonable α values.