The stability of the transient waterflood problem has been examined. The treatment includes both the effects of relative permeability and of capillary pressure. The analysis is formulated for a Buckley-Leverett type base state, although with capillary corrections. This base state is obtained for arbitrary water foil relative permeabilities and capillary pressure functions, using the method of matched asymptotic expansions.

Cases studies from three North Sea turbidite reservoirs will be presented, which together demonstrate our current understanding of permeability and relative permeability upscaling. The three formations, the Magnus, Magnus Sand Member (MSM), the Magnus, Lower Kimmeridge Clay Formation (LKCF), and the Andrew reservoir each provide distinct challenges for reservoir modelling, either because of reservoir complexity, the fluids in place, or the phase of field life. To meet these challenges, several novel upscaling approaches have been developed. Their use will be explored and current best practice delineated. This best practice differs significantly from previous definitions of "effective permeability" by placing more emphasis on extracting multiple properties from the fine scale geologic models. Distinct upscaling calculations are required to assessthe quality of sands,the quality of barriers, andthe tortuosity of flow around these barriers. Similarly, when constructing upscaled relative permeabilities, the "effective" curves are distinguished from the "pseudo" curves. The former describe the physical displacement of fluids, while the latter include the additional numerical dispersion corrections required when implementing the relative permeability functions within a coarsely gridded full field simulator. P. 133

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