Flow and transport in highly heterogeneous formations: 3. Numerical simulations and comparison with theoretical results

Abstract

In parts 1 [Dagan et al., 2003] and 2 [Fiori et al., 2003] a multi‐indicator model of heterogeneous formations is devised in order to solve flow and transport in highly heterogeneous formations. The isotropic medium is made up from circular (2‐D) or spherical (3‐D) inclusions of different conductivities K, submerged in a matrix of effective conductivity. This structure is different from the multi‐Gaussian one, even for equal log conductivity distribution and integral scale. A snapshot of a two‐dimensional plume in a highly heterogeneous medium of lognormal conductivity distribution shows that the model leads to a complex transport picture. The present study was limited, however, to investigating the statistical moments of ergodic plumes. Two approximate semianalytical solutions, based on a self‐consistent model (SC) and on a first‐order perturbation in the log conductivity variance (FO), are used in parts 1 and 2 in order to compute the statistical moments of flow and transport variables for a lognormal conductivity pdf. In this paper an efficient and accurate numerical procedure, based on the analytic‐element method [Strack, 1989], is used in order to validate the approximate results. The solution satisfies exactly the continuity equation and at high‐accuracy the continuity of heads at inclusion boundaries. The dimensionless dependent variables depend on two parameters: the volume fraction n of inclusions in the medium and the log conductivity variance σY2. For inclusions of uniform radius, the largest n was 0.9 (2‐D) and 0.7 (3‐D), whereas the largest σY2 was equal to 10. The SC approximation underestimates the longitudinal Eulerian velocity variance for increasing n and increasing σY2 in 2‐D and, to a lesser extent, in 3‐D, as compared to numerical results. The FO approximation overestimates these variances, and these effects are larger in the transverse direction. The longitudinal velocity pdf is highly skewed and negative velocities are present at high σY2, especially in 2‐D. The main results are in the comparison of the macrodispersivities, computed with the aid of the Lagrangian velocity covariances, as functions of travel time. For the longitudinal macrodispersivity, the SC approximation yields results close to the numerical ones in 2‐D for n = 0.4 but underestimates them for n = 0.9. The asymptotic, large travel time values of macrodispersivities in the SC and FO approximations are close for low to moderate σY2, as shown and explained in part 1. However, while the slow tendency to Fickian behavior is well reproduced by SC, it is much quicker in the FO approximation. In 3‐D the SC approximation is closer to numerical one for the highest n = 0.7 and the different σY2 = 2, 4, 8, and the comparison improves if molecular diffusion is taken into account. Transverse macrodispersivity for small travel times is underestimated by SC in 2‐D and is closer to numerical results in 3‐D, whereas FO overestimates them. Transverse macrodispersivity asymptotically tends to zero in 2‐D for large travel times. In 3‐D the numerical simulations lead to a small but persistent transverse macrodispersivity for large travel times, whereas it tends to zero in the approximate solutions. The results suggest that the self‐consistent semianalytical approximation provides a valuable tool to model transport in highly heterogeneous isotropic formations of a 3‐D structure in terms of trajectories statistical moments. It captures effects like slow transition to Fickian behavior and to Gaussian trajectory distribution, which are neglected by the first‐order approximation.