Pseudo-relative permeabilities are usually generated to match fine grid models of typical cross-sections of the reservoir with coarse grid models and these are then used in full field models to reduce the number of cells. Various analytical formulae have been proposed to calculate these pseudos. In practice, the analytic formulae merely represent an educated first guess of the pseudo-relative permeabilities necessary to match the fine grid model. When used in the coarse grid equivalent of the fine grid model, these pseudo-relative permeability curves often require further trial and error adjustment in order to match the fine grid model results.
This paper describes an approach whereby these pseudos may be easily generated with non-linear regression analysis (automatic history matching). The relative permeability values of the saturation function tables are allowed to vary independently until a match of the observed saturations and production rates of the fine grid model is obtained with the coarse grid model. Calculated pseudo-functions are presented for a difficult previously published problem with non-communicating layers. A case study shows the sensitivity of the pseudos to coarse grid definition and production rates. Previously published formulae have been restricted to two-dimensional vertical cross-section pseudos. As a result, areal heterogeneity and sweep displacement efficiency have been ignored. In this paper, we extend the pseudo concept to three dimensions by generating a single pseudo-relative permeability curve to represent a fine three-dimensional model with a coarse three-dimensional model.