# Ensembled–Averaged Solute Transport

We present a method for determining the differential equations that describe the ensemble-averaged moments for reactive solute transport in an unsteady, non-divergence-free flow field. Our approach uses a renormalized cumulant expansion, Lie group theory, and time-ordered exponentials to develop the ensemble-average transport equation. The approach results in an expansion in powers of a α τ_{c}, where α measures the magnitude of the perturbations of the transport and reaction operators, and τ_{c} is the correlation time of these perturbations. The cumulant expansion avoids secular terms, and allows one to close the problem rationally by truncating the expansion; the truncated terms can be shown to be of lower order than those terms that are kept, provided that the Kubo number restriction is met. The use of Lie group theory allows one to automatically combine the Eulerian and Lagrangian approaches, and begins with an Eulerian conservation equation for the solute. A particular time-ordered exponential that arises in the analysis can be interpreted as a translation operator that possesses a well defined algebra. These translation operators appear in the second-order (covariance) terms of the cumulant expansion, and their effect is to shift one of the terms of the covariance functions relative to the other along the trajectory formed by the ensemble-average velocity field. The Lagrangian trajectories can be described in terms of a Lie series that involves only the ensemble-average Eulerian velocity field and its spatial derivatives, and leads to exact closed-form representations for the trajectories when the velocity field is spatially stationary. When the velocity field is nonstationary, the Lie series may still provide adequate representation of the trajectories when truncated to a finite number of terms. This approach has the advantages that no integral transformations are conducted, so all results are in real space.

## References

Wood, B.D., M.L. Kavvas. 1999. "*Ensemble-averaged Equations for Reactive Transport in Porous Media Under Unsteady Flow Conditions.*" **Water Resources Research.** Vol. 35 , No. 7 , p. 2053 and Vol. 35 , No. 9 , p. 2887.