Fractional ensemble average governing equations of transport by time-space nonstationary stochastic fractional flow

Starting from a time-space nonstationary general random walk formulation, the pure advection and advection-dispersion forms of the fractional ensemble average governing equations of solute transport by time-space nonstationary stochastic flow fields in fractional time-space will be reported. In the case of the purely advective fractional ensemble average equation of transport, the advection coefficient is a fractional ensemble average advective flow velocity in fractional time and space that is dependent on both space and time. As such, in this case the time space-nonstationarity of the stochastic advective flow velocity is directly reflected in terms of its mean behavior in the fractional ensemble average transport equation. In fact, the derived purely advective form represents the Lagrangian derivation 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. In the moment form of the fractional ensemble average advection-dispersion equation of transport, the advection coefficient emerges as a combination of the fractional ensemble average advective flow velocity in fractional time and space with an advective term that is due to dispersion.