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AMS2005-14: Upscaling Tensorial Permeability Fields Based on Gaussian Markov Random Field Models and the Hybrid Mixed Finite Element Method

Zhuoxin Bi, John Trangenstein, Dave Higdon, and Herbert Lee
12/31/2005 09:00 AM
Applied Mathematics & Statistics
In this paper, a fine scale simulation-based upscaling procedure for full tensor permeability fields is presented based on the Hybrid Mixed Finite Element (HMFE) numerical scheme and synthetic permeability fields generated using a Gaussian Markov Random Field (GMRF) model. HMFE scheme is an accurate scheme that conveniently represents tensorial permeability field for forward flow/transport simulation. It is also one of the few choices that provides a natural technique for conservative upscaling, meaning that the coarse grid should produce the same flux and pressure as the upscaled fine values. As in any renormalization approach, we require that the coarse linear system to be in the same form as in the fine. This process produces good upsacling results if both the fine scale tensor and the upscaled tensor are assumed diagonal. However, results from a few examples show that it is not always possible to obtain a positive definite permeability tensor while conserving fluxes and pressures on upscaled cells. One explanation is that the discretized Darcy's law on the upscaled grid, based on conserving fine flow variables, may not be valid. Results from a generally used two-step upscaling procedure demonstrate that a symmetric, positive definite tensor field on coarser grids is obtainable. But it should be applied with caution because some of the off-diagonal values may be highly unreasonable.