Eulerian‐Lagrangian Analysis of Transport Conditioned on Hydraulic Data: 3. Spatial Moments, Travel Time Distribution, Mass Flow Rate, and Cumulative Release Across a Compliance Surface

Abstract

In paper 1 of this series we described an analytical‐numerical approach to predict deterministically solute transport under uncertainty. The approach allows conditioning such predictions on hydraulic measurements and assessing the corresponding reduction in uncertainty. In paper 2 we examined the effects of log transmissivity and hydraulic head data on conditional predictions of concentration due to instantaneous point and nonpoint sources. In this paper we show how the same approach can be used directly to estimate mass flow rate across a “compliance surface,” cumulative mass release, and the probability distribution of travel times across this surface and the associated uncertainty. Contrary to some other methods in the literature, our approach requires neither a special theory for travel times nor a prior assumption about their probability distribution. We also show how one can compute explicitly the second spatial moment of the conditional mean plume about its center of mass, the conditional mean second spatial moment of the actual plume about its center of mass, and the conditional covariance of the plume center of mass. We illustrate these quantities and the effect of conditioning on some of them by considering instantaneous point, line, and area sources in a two‐dimensional, statistically homogeneous, isotropic, mildly varying log transmissivity field under uniform prior (unconditional) mean flow.