Upscaling of flow in heterogeneous porous formations: Critical examination and issues of principle |
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| Institution: | 1. Université de Tunis El Manar, Institut Supérieur d’Informatique, Research Team on Intelligent Systems in Imaging and Artificial Vision (SIIVA), LR16ES06 Laboratoire de recherche en Informatique, Modélisation et Traitement de l’Information et de la Connaissance (LIMTIC), 2 Rue Abou Rayhane Bayrouni, Ariana 2080, Tunisia;2. Université de Carthage, Ecole Nationale d’Ingénieurs de Carthage, 45 Rue des Entrepreneurs, Tunis-Carthage 2035, Tunisia;1. School of Chemical Engineering and Analytical Science, University of Manchester, Manchester, UK;2. School of Minerals and Energy Resources Engineering, The University of New South Wales, Sydney, Australia;1. Intelligent Energy Laboratory, Peng Cheng Laboratory, Guangdong, PR China;2. College of Engineering, Peking University, Beijing, PR China;3. School of Environmental Science and Engineering, Southern University of Science and Technology, Guangdong, PR China |
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| Abstract: | We consider heterogeneous media whose properties vary in space and particularly aquifers whose hydraulic conductivity K may change by orders of magnitude in the same formation. Upscaling of conductivity in models of aquifer flow is needed in order to reduce the numerical burden, especially when modeling flow in heterogeneous aquifers of 3D random structure. Also, in many applications the interest is in average values of the dependent variables over scales larger or comparable to the conductivity length scales. Assigning values of the conductivity Kb to averaging domains, or computational blocks, is the topic of a large body of literature, the problem being of wide interest in various branches of physics and engineering. It is clear that upscaling causes loss of information and at best it can render a good approximation of the fine scale solution after averaging it over the blocks.The present article focuses on upscaling approaches dealing with random media. It is not meant to be a review paper, its main scope being to elucidate a few issues of principle and to briefly discuss open questions. We show that upscaling can be usually achieved only approximately, and the result may depend on the particular upscaling scheme adopted. The typically scarce information on the statistical structure of the fine-scale conductivity imposes a strong limitation to the upscaling problem. Also, local upscaling is not possible in nonuniform mean flows, for which the upscaled conductivity tensor is generally nonlocal and it depends on the domain geometry and the boundary conditions. These and other limitations are discussed, as well as other open topics deserving further investigation. |
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