Fractional derivatives, continuous time random walks, and anomalous solute transport

This thesis describes the utility of fractional-in-time and space advection-dispersion equations (ADEs) for describing solute transport in heterogeneous aquifers. Fractional ADEs are generalizations of the classical ADE in which the integer-order derivatives can be replaced with fractional-order derivatives. The one-dimensional space fractional ADE has previously been applied in hydrogeology. This equation has a dispersion coefficient that is constant at all scales and can model the anomalous spreading of a conservative contaminant plume.; The probabilistic model underlying ADEs is the continuous time random walk (CTRW). While random walk models describe the path of a particle as the sum of instantaneous random jumps, CTRWs allow a random transition time between jumps. Allowing particle jump lengths to have infinite variance is akin to lifting the constraint that each jump is limited to an elementary size. Then dispersive flux is proportional to a fractional derivative, leading to a fractional Fick's Law. The space-fractional ADE can be derived by linking a conservation of mass equation with a flux term accounting for linear advection and fractional dispersion.; To apply a fractional ADE to anomalous solute plume growth in multiple dimensions, a multiscaling fractional ADE that allows unique scaling rates in different directions can be used. The relation between hydrologic parameters and the parameters of a spatially fractional ADE is obtained by linking CTRWs and compound Poisson processes. The multiscaling ADE allows non-orthogonal principal growth directions, suggesting that it may be ideal for modeling super-Fickian transport in fractured rock.; A CTRW with infinite-mean particle waiting times can be used as a model for solute transport with rate-limited mass transfer into the immobile zones of an aquifer. The CTRW can be broken into mobile and immobile components, leading to fractional-in-time ADEs that govern total, mobile, and immobile solute transport with retardation. The total solute transport equation is mass conservative, while the mobile solute transport equation predicts continual mass loss to the immobile zone.; Key features of plume evolution predicted by fractional-in-space and time ADEs include a power law rate of mass transfer from the mobile to immobile zone, power law late-time breakthrough curves, and concentration profiles with heavy-tailed leading edges.