Abstract. Fast upscaling of hydraulic conductivity is a recurrent problem in modeling flow through heterogeneous porous media. We propose a new renormalization technique. It is based on the iterative application of the Cardwell and Parsons [1945] bounds on elementary groups of cells. The combination of the bounds with a heuristic formula allows anisotropy to be taken into account. The new technique is tested and compared with other fast techniques. Among the tested techniques the two most reliable ones are the tensorial renormalization and the new simplified renormalization. The numerical efficiency of the simplified renormalization leads us to recommend it when a diagonal tensor of equivalent conductivity is sufficient.
IntroductionModeling groundwater flow and transport requires a realistic description of the spatial distribution of hydraulic conductivity to capture flow paths and to make realistic forecasts of groundwater behavior. As discussed by de Marsily et al. [1998] and Koltermann and Gorelick [1996], statistical and genetic tools can be used to describe the underground distribution of hydraulic conductivity. Geological models generated by these tools often produce a very high spatial resolution which cannot be used directly in groundwater flow and transport models given currently available computer resources. It is necessary to adopt a much coarser description. This upscaling comprises the calculation of spatial averages of the hydraulic conductivity over blocks of the geological model that can be used directly in the groundwater model. This problem is different from the problem of finding a unique effective conductivity for the whole aquifer. Renard and de Marsily [1997] and Wen and G6mez-Herngndez [1996] present extensive reviews of upscaling theories, analytical results, and numerical techniques.In this paper, we develop a fast real-space renormalization algorithm to calculate block conductivities. The essence of renormalization is to apply composition rules to groups of cells at the local scale to produce "composite" cells and then to successively reapply the same rules to the resulting composite cells until the desired coarse-scale resolution is achieved. The basic assumption is that the upscaling rules are scale invariant. When the probability distribution of the local-scale conductivity is known, it is possible to follow the evolution of the prob-
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