Upscaling, Computational Aspects, and Statistics of the Velocity Field

Abstract

This chapter deals with computing the velocity fields in heterogeneous media. This is a broad area, and we shall concentrate here on upscaling, on the spatial correlation pattern of the velocity, and on accuracy measures for techniques that compute velocity fields. Numerical simulations of velocity fields in heterogeneous media (Ababou et al., 1988, 1989; Bellin et al., 1992, 1994; Bellin and Rubin, 1996; Dykaar and Kitandis, 1992a,b; Hassan et al., 1998a,b; Salandin and Fiorotto, 1998) indicate that to capture accurately the effects of the spatial variability of the conductivity on the velocity field, the conductivity field should be modeled with high resolution. Techniques for generating highly detailed realizations of rock properties were reviewed earlier. Because of the huge level of detail included in these realizations, large-scale flow simulations can become computationally intensive. However, the need for fine detail varies over the aquifer. For example, a high level of detail is needed where the velocity field may vary rapidly, such as near wells, or over areas traversed by a contaminant plume, or for describing small-scale features which dominate the flow, such as high-conductivity channels. Coarsening the grid over areas where high resolution is unnecessary can reduce the computational effort. To be able to do that, a procedure is needed for assigning properties such as conductivity on a coarser scale which is more appropriate for simulation, while avoiding the loss of important details. Such a procedure is called upscaling (also scale-up). Upscaling assigns properties to blocks based on subgrid-scale heterogeneity. Upscaling leads to block-effective properties. Unlike effective properties, block-effective properties depend on the size of the block. In the limit of block dimensions much larger than the integral scale of the heterogeneity, the block-effective properties become equal to the media's effective properties. Unlike the case of effective conductivities, there is no consensus about the definition of block conductivity. For example, Rubin and Gomez-Hernandez (1990) defined the block conductivity as the coefficient of proportionality between the block-averaged flux and the gradient. Indelman and Dagan (1993a, b) stipulated that the block-effective conductivity should dissipate energy at a rate equal to the dissipation due to the small-scale heterogeneity.