Contemporary Mathematics Volume 295, 2002 A New Approach to Upscaling for Two-Phase Flow in Heterogeneous Porous Media J0rg Aarnes & Magne S. Espedal ABSTRACT. We present a new dynamic scale-up technique for the coupled pressure-saturation two-phase porous media flow equations. This technique is motivated by the need for more robust coarse solvers for reservoir simu- lation applications where the heterogeneous porous media typically contain many spatial scales. The idea is to allow a fine mesh computation for the elliptic pressure equation based on initial reservoir data, ·and use information from this solution to construct data for the upscaled equations which reflect the important flow characteristics in the initial fine mesh solution. This is accomplished by generating intercell conductivities for the coarse scale pres- sure equation which mimic the effect of the reservoir data on the fine mesh generated velocity field, and coarse scale fractional flow curves which reflect the evolution of the saturation profile in a fine mesh steady state simulation neglecting capillary forces. The effect of capillary forces is integrated in a sim- ple fashion directly into the coarse scale equations. The method is applicable to general heterogeneous porous media flow without restrictive assumptions, and the numerical results show very good performance for flows with moderate capillary diffusion. 1. Introduction In heterogeneous porous media flow problems there are always multiple length scales in the physical coefficients for the governing equations. This implies that conventional coarse solvers might inherit very poor approximation properties, see eg [1]. On the other hand, the size of the geological model prohibits a full fine scale simulation over many time steps, even with the advent of modern computers and parallel computing technology. Therefore, a compromise have to be made between desired accuracy and available computer resources. The standard compromise is to up-scale the differential coefficients with the objective of improving the approxima- tion properties of the coarse grid operator, and thus obtain satisfactory accuracy with a pure coarse scale simulation. There has been published a vast number of papers on upscaling for porous media flow ranging from simple averaging techniques, eg. [13], to rigorous homogenization techniques, [3, 9]. For recent reviews on upscaling for both single-phase and two- phase flow, see [4, 5, 14, 15]. Unfortunately, upscaling has received a fair amount of criticism because few methods have proved to be robust and liable, and most scale- up techniques, if not all, lack satisfactory error estimates for general heterogeneous 1991 Mathematics Subject Classification. Primary 35R05,35B30 Secondary 65D99.' Key words and phrases. porous media, two-phase models, upscaling, elliptic and advection- diffusion equations, domain decomposition. The first author would like to acknowledge the support from the Norwegian Academy of Science and Letters and Statoil (The VISTA program). © 2002 American Mathematical Society 1
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