Ensemble-averaged equations for reactive transport in porous media under unsteady flow conditions

We present a method for deriving the ensemble-averaged reactive solute tran sport equation for unsteady, non-divergence-free flow field conditions. Our approach uses a cumulant expansion, Lie group theory, and time-ordered exp onentials to develop the ensemble-averaged transport equation. The cumulant expansion is in powers of a alpha tau(c), where alpha measures the magnitu de of the perturbations of the transport and reaction operators and tau(c) is the correlation time of these perturbations. Because the cumulant expans ion avoids secular terms (terms in powers of time), the problem can be clos ed by rationally truncating the expansion. The truncated terms can be shown to be of lower order than those terms that are kept, provided that a parti cular constraint (in terms of the Kubo number) is met. The use of Lie group theory allows one to automatically combine the Eulerian and Lagrangian app roaches. 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 trajector y formed by the ensemble-averaged velocity field. This approach does not re quire neglecting the local dispersion tensor and has the advantage that no integral transformations are conducted; therefore all results are expressed in terms of real space variables.