First presented at the SPE Annual Technical Conference & Exhibition held in New Orleans, 25-28 September, 1994, as " Development and Application of a New Two-Phase Scaleup Method Based on Tensor Permeabilities".
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 maybe 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.
The modelling of two-phase flow through porous media is important for forecasting hydrocarbon recovery in oil and gas reservoirs. This process is complicated by the fact that rocks may be very heterogeneous over a wide range of length scales. If the effects of these heterogeneities are to be taken into account in full-scale simulations, a method is required for averaging flow properties from small-scale models, for incorporation in larger-scale ones. The calculation of effective flow properties (sometimes known as pseudoproperties)has been studied, from different view points, by a number of authors. Generally speaking, scaleup can only be carried out successfully when there is a separation of length-scales. In fact, many sedimentary rocks do exhibit heterogeneities at distinct length scales, e.g. laminae (em), beds (m), formations (10m). Corbett et al have shown that it is feasible to scaleup two-phase flow using such geologically based length scales.