Asymptotic properties of the Eulerian truncation approximation: Analysis of the perfectly stratified transport problem

It is well known that stochastic groundwater problems are difficult to solve without making approximations of one kind or another. A popular approximation used in both subsurface flow and transport applications is the so-called "Eulerian truncation." This approximation, which relies on a perturbation expansion of the governing equation, neglects certain terms involving products of small fluctuations. Several recent publications [e.g., Dagan and Neuman, 1991; Neuman and Orr, 1993; Neuman, 1993] question the validity of Eulerian truncation, arguing that it is inconsistent because neglected terms are of the same order as those retained. In this paper we analyze the asymptotic properties of the Eulerian truncation solution, using a Taylor series approach which clearly reveals the properties of different order approximations. We show that the Eulerian truncation does, in fact, generate an asymptotic ensemble mean expansion for the perfectly stratified example considered by Dagan and Neuman [1991]. Moreover, we show that the Eulerian truncation solution to this problem is identical to the well-known cumulant expansion approximation, which is also based on an asymptotic expansion.

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