Scaleup of two-phase flow in porous media using phase permeability tensors

The effects of different levels of geological heterogeneity on a fluid displacement process may be captured at a larger scale using scaleup techniques. In the context of reservoir simulation, these are algorithms which should reproduce the results of fine grid calculations on a coarser grid. These techniques are referred to as pseudo-isation, the main objective being to produce pseudo-functions which can be used on the coarse grid. When carried out successfully, the pseudo-functions (e.g. pseudo relative permeabilities) incorporate the interaction between the fluid mechanics and the heterogeneity as well as correcting for numerical dispersion. These pseudo-functions also depend on the viscous/capillary and viscous/gravity ratios and are valid for the boundary conditions relevant to the particular flows. For single phase flow, the scaleup problem involves the derivation of effective permeability which, in general, is a tensor quantity. Multi-phase flow is more complex since scaled-up dynamic transport quantities must be calculated which depend on phase saturation, flow rate etc. In this paper, we present a method which extends the idea of tensor (absolute) permeability to tensor effective phase permeabilities. They are extensions of conventional functions which also include the off-diagonal phase crossflow terms which may be important in certain systems. Two phase tensor methods are presented which are valid (a) in the capillary equilibrium limit and (b) for arbitrary values of viscous/capillary and viscous/gravity ratios. Numerical examples of the application of these methods are presented for ripple-bedded systems where the two phase crossflow effects are significant, and where oil trapping within the lamina structure may occur under certain conditions. The results show that it is important to use phase tensors in upscaling where gravity effects are significant, in order to generate the correct vertical flows.