Reconstructing age and reactivity distributions from the moments carried by central-moment transport models

Shahram Asgari 

A central-moment transport model follows a property that varies across a population, such as the age of water parcels or the decay rate of organic-matter particles. Instead of storing the whole distribution of that property, it keeps only a few summary numbers, the moments, and moves those through the model. But the distribution itself is usually what a scientist wants, so it has to be rebuilt from the moments afterward. This rebuilding is an old problem, because many different distributions share the same few moments. It is well understood in aerosol and population-balance modeling, but it has not been tested for age and reactivity distributions. Here we bring those methods to the age and reactivity setting and measure how well they work against fully resolved references. We test five ways to rebuild a distribution from its moments, the gamma, log-normal, translated Weibull, and maximum-entropy shapes and a triangle, and we score each by the area between the rebuilt curve and the true one. The result is simple. Accuracy depends on how well the assumed shape matches the true shape, not on how many moments are carried. The log-normal shape wins on a log-normal reference, with an error of 0.03 against 0.04 for the gamma; the gamma wins on a gamma reference at 0.02; and the three-moment Weibull wins on the skewed age distribution at 0.06, while the triangle cannot fit these skewed shapes at all. The error grows as pore-scale mixing increases. Storing the maximum-entropy shape in a lookup table makes it about 700 times faster to use, but the gamma and Weibull shapes are already far cheaper, so the table helps only when maximum entropy is needed.