Analytical Upscaling of Permeability for 3D Gridblocks

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Abstract
With modern conditional simulation methods, detailed reservoir descriptions can be constructed on a very small scale. Many methods also allow detailed descriptions which include basic geologic features such as cross beds and laminations. Unfortunately, present flow simulators are not capable of incorporating these small scale descriptions due to computational limitations. The small scale values, especially, the permeabilities, need to be upscaled appropriately to the size suitable for flow simulations. Many methods, some numerical and some analytical, have been reported in the literature to upscale the permeability values. Numerical methods are computationally expensive, whereas, many analytical methods are restricted to two dimensions. This paper presents an analytical method to estimate the upscaled permeability tensor in three dimensions using small scale permeability values. The method is based on dividing the upscaled block into eight smaller blocks and using cross flow and non-cross flow averaged permeabilities based on Darcy's law. The method is validated by comparing the results from a numerical simulator.