Fractional Random Walk and Fractional Differential Equation Models of Transport by Time-Space Nonstationary Stochastic Fractional Flow

Starting from a time-space nonstationary general random walk formulation, first the general fractional random walk model of transport by time-space nonstationary flow in fractional time space is presented. From this general fractional random walk model, the development of the pure advection and advection-dispersion forms of the fractional ensemble average equations of solute transport by time-space nonstationary stochastic flow fields in fractional time-space are then presented. The purely advective form represents the Lagrangian form of the ensemble average mass conservation equation for solute transport in fractional time-space. In the case of the fractional ensemble average advection-dispersion transport equation, the moment and cumulant forms of the equation are quite different, and are both presented. Next, the developed fractional ensemble equations of transport, in their purely advective, and moment and cumulant advective-dispersive forms, are evaluated by numerical simulations against the corresponding Monte Carlo solutions. These comparisons show that the cumulant form of the advective-dispersive ensemble average fractional transport equation is generally superior in simulating the shape and mode of the ensemble average concentration of the contaminant. The derived fractional ensemble average equations of transport can accommodate both the non-Fickian and the Fickian behavior of transport.