Divergence of solutions to perturbation-based advection–dispersion moment equations

Moment equation methods are popular and powerful tools for modeling transport processes in randomly heterogeneous porous media, but the application of these methods to advection–dispersion equations often leads to erroneous oscillations. Perturbative methods, required to close systems of moment equations, become inaccurate for large perturbations; however, little quantitative theory exists for determining when this occurs for advection–dispersion equations. We consider three different methods (asymptotic approximation, Eulerian truncation, and iterative solution) for closing and solving advection–dispersion moment equations describing transport in stratified porous media with random permeability. We obtain approximate analytical expressions for time above which the asymptotic approximation to the mean diverges, in particular quantifying the impact that dispersion has on delaying—but not eliminating—divergence. We demonstrate that Eulerian truncation and iterative solution methods do not eliminate divergent behavior either. Our divergence criteria provide a priori estimates that signal a warning to the practitioner of stochastic advection–dispersion equations to carefully consider whether to apply perturbative approaches.