The concept of pseudo-functions has been developed as a practical technique for representing the behaviour of small-scale multi-phase fluid mechanics and heterogeneity in coarse grid simulation models. Most of the existing methods for deriving pseudo-relative permeabilities make some specific assumptions and, for this reason, they do not apply under all conditions.

This paper explores a more general upscaling technique that may be applicable to both miscible and immiscible displacements and may be applied in arbitrary geometries and for all mobility ratios. In the new method, which is based on the approach of Stone,1  the dynamic pseudo-functions are produced from a fine grid simulation using a weighted fractional flow formulation with unit-mobility-ratio flow information. It avoids calculating the potential differences and has no limitation to flow rates. The flow rates out of the coarse grid block boundaries need to be calculated only once in producing the pseudo-relative permeabilities for "miscible" flow.

The method has been validated numerically by simulating certain immiscible and experimental miscible displacements for adverse mobility ratios within an areal quarter-five-spot porous medium on both fine and coarser two-dimensional grid meshes. Good agreement in effluent/recovery performance for the fine grid model, coarse grid model and the experiment has been obtained with a substantial saving in computing cost for the upscaled model. The new method is successful in capturing both the effects of interaction of the fluid displacement process with the small scale heterogeneity and also the numerical dispersion effects and the influence of the local boundary conditions.

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