The fractional advection-dispersion equation for contaminant transport

Anomalous dispersion is observed throughout hydrology, yielding a contaminant plume with heavy leading tails. The fractional advection dispersion equation (FADE) captures this behavior by replacing the second-order spatial derivative with a Riemann-Liouville (RL) fractional derivative. The RL fractional derivative is a nonlocal operator and models large jumps of solute particles in heterogeneous media. This chapter reviews the FADE, including fundamental (point-source) solutions, which are expressed as stable probability density functions. The space FADE has been extended to space-dependent parameters (e.g., dispersivity) and multiple dimensions. Alternatively, the time FADE and fractional mobile immobile (FMIM) models, which utilize time-fractional derivatives to model long-waiting times (retention), are also used to model anomalous dispersion. Current applications of the FADE, including parameter estimation, source identification, space-time duality, and FADE models on bounded domains are discussed.