Upscaling of the Geological Models of Large-Scale Porous Media Using Multiresolution Wavelet Transformations

To model fluid flow and energy transport in a large-scale porous medium, such as an oil or a geothermal reservoir, one must first develop the porous medium’s geological model (GM) that contains all the relevant data at all the important length scales. Such a model, represented by a computational grid, usually contains several million grid blocks. As a result, simulation of fluid flow and energy transport with the GM, particularly over large time scales (for example, a few years), is impractical. Thus, an important problem is upscaling of the GM. That is, starting from the GM, one attempts to generate an upscaled or coarsened computational grid with only a few thousands grid blocks, which describes fluid flow and transport in the medium as accurately as the GM. We describe a powerful upscaling method, which is based on the wavelet transformation of the spatial distribution of any static property of the porous medium, such as its permeability, or a dynamic property, such as the spatial distribution of the local fluid velocities in the medium. The method is a multiscale approach that takes into account the effect of the heterogeneities at all the length scales that can be incorporated in the GM. It generates a nonuniform computational grid with a low level of upscaling in the high permeability sectors but utilizes high levels of upscaling in the rest of the GM. After generating the upscaled computational grid, a critical step is to calculate the equivalent permeability of the upscaled blocks. In this paper, six permeability upscaling techniques are examined. The techniques are either analytical or numerical methods. The results of computer simulations of displacement of oil by water, obtained with each of the six methods, are then compared with those obtained by the GM.