Simulation of dispersion in heterogeneous porous formations: statistics, first-order theories, convergence of computations

This paper discusses the results of numerical analysis of dispersion of passive solutes in two-dimensional heterogeneous porous formations. Statistics of flow and transport variables, the accuracy and the role of approximations implicit in existing first-order theories, and the convergence of computational results are investigated. The results suggest that quite different rates of convergence with Monte Carlo runs hold for different spatial moments and that over 1000 realizations are required to stabilize second moments even for relatively mild heterogeneity (σY2 < 1.6). This has implications for the extent of the spatial domain for single-realization numerical studies of the same type. A comparison of the variance of plumes with the results of linear theories (0.05 < σY2 < 1.6) shows an unexpectedly broad validity field for the theoretical solution obtained from a suitable linearization of flow and transport. Reformulation of the same problem linearizing in turn the flow or the transport equations shows opposite deviations from the linear theory. The interesting consequence is that the errors induced by linearizations in the flow or the transport equations have different signs, and their effects on the moments of dispersing plumes are compensating, thereby yielding consistent formulations. Unexpected features of the statistics of probability distributions of longitudinal and transverse velocities and travel times are also computed and discussed.