Fractional Ensemble Average Governing Equations of Transport by Time-Space Nonstationary Stochastic Fractional Advective Velocity and Fractional Dispersion. I: Theory

In this study, 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 were developed. 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 derived separately. 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. The fractional dispersion coefficient emerges as a $2\beta$-momentof the differential displacement per scaled differential time ${(dt)}^{\alpha }$, where $\beta$ and $\alpha$ denote, respectively, the fractional orders of the space and time derivatives. Since this fractional dispersion coefficient is in moment form, the corresponding ensemble average equation is a moment form of the fractional ensemble average advection-dispersion equation. The cumulant form of the fractional ensemble average advection-dispersion transport equation is derived for two different cases: (1) when the order of the space fractional derivative of the advective term is the same as that of the time derivative, while the order of the space fractional derivative of the dispersion term is twice that of the time derivative, and (2) when the orders of the time and space fractional derivatives are completely different. In case (1), the advection coefficient emerges as a combination of the fractional ensemble average advective flow velocity with an advective term that is due to dispersion, while the cumulant form of the dispersion coefficient emerges as a time-space-dependent variance of the $\alpha$-moment of the differential displacement per scaled time differential ${(dt)}^{\alpha }$, resulting in a fractional advection-dispersion equation. In case (2), when the orders of the time and space derivatives are completely different, the cumulant form of the fractional dispersion coefficient emerges as a combination of the moment form of the fractional dispersion coefficient and the square of a fractional ensemble average advective velocity. The derived fractional ensemble average governing equations of transport are rich in structure and can accommodate both the non-Fickian and the Fickian behavior of transport. The non-Fickian transport behavior can be modeled by the derived fractional ensemble average transport equations either by means of the long memory in the underlying stochastic flow, or by means of the time-space nonstationarity of the underlying stochastic flow, or by means of the time and space fractional derivatives of the transport equations.