摘要
Perturbation methods are of interest to hydrologists because they provide a way to incorporate upscaling and accuracy assessment capabilities into practical groundwater models. In particular, these methods may be used to obtain approximate solutions for the ensemble moments of solute concentration and for related effective properties in randomly heterogeneous porous media. Although the many perturbation methods proposed in the groundwater literature seem to be rather different they are actually closely related. For example, we show here through a systematic analysis that the second-order asymptotic expansion approach [Flow and Transport in Porous Formations, Springer-Verlag, Berlin, 1989], random Green’s function methods [Dispersion in Heterogeneous Geological Formations, Kluwer Academic Publishers, the Netherlands, 2001], and Taylor series methods [Water Resour. Res. 32(1) (1996) 85; Water Resour. Res. 28(12) (1992) 3269; Water Resour. Res. 21(3) (1985) 359] all give identical expressions for the ensemble mean, for random fluctuations about this mean, and for related closure covariances. The Eulerian truncation approach [Water Resour. Res. 14(2) (1978) 263; Water Resour. Res. 19(1) (1983) 161; Phys. Fluids 31 (1988) 965; Water Resour. Res. 27(7) (1991) 1598] gives expressions similar to the other methods except that its closure covariances depend on the unknown ensemble mean rather than the known deterministic solution (the solution obtained when random velocities are replaced by their means). This reflects the fact that Eulerian truncation works with perturbations taken about the mean while the other methods work with perturbations taken about the deterministic solution. The ensemble mean accounts for the large-scale effects of small-scale variability while the deterministic solution does not. Eulerian truncation solutions can be more difficult to compute than other alternatives but they can also be more accurate. This is demonstrated for the case of Gaussian velocities with a general truncation analysis and with a specific numerical example. The results provided here suggest that Eulerian truncation is superior to alternative perturbation methods of comparable computational complexity.