Divergence of solutions to solute transport moment equations

We provide explicit solutions to one‐dimensional moment equations for solute transport in random porous media using asymptotic perturbation expansions up through fourth order in standard deviation of log hydraulic conductivity. From these solutions, we demonstrate the source of multi‐modal behavior in this special case; namely, oscillatory terms that increase with the variance of velocity (or of log conductivity) and time. We show that over time higher‐order moments become less accurate than second‐order moments. Moreover, we show that the complete asymptotic series solution diverges for any value of log conductivity variance after sufficient time, using an analytical bound and assuming Gaussian‐distributed velocity. This bound depends on the zero‐order mean velocity, correlation length, and properties of the initial data. We find that the bound is also a good approximation when applied to our solutions of moment equations for a non‐Gaussian velocity distribution.