The Upscaling of Miscible and Immiscible Displacement Processes in Porous Media

Abstract

SPE Members Abstract 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, the dynamic pseudofunctions 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. Introduction The detailed simulation of multi-dimensional flow in reservoirs in two- or three-dimensional models may require a large number of grid blocks. Such simulations can be very costly and may severely tax the capacity of currently available computers. It is often a convenience in reservoir modelling to reduce the number of grid cells in a system in order to reduce model run costs. However, spatial variability in absolute and relative permeability in reservoir rocks can have a great impact on the flow performance especially in enhanced oil recovery (EOR) processes. The ever-present heterogeneity occurs on many length scales in a reservoir down to the lamina scale (mm) where permeability contrasts greater than two orders of magnitude can be seen for certain sedimentary systems. It is clearly impossible to incorporate such detail into a simulation model since the lengthscale of these variations is generally much smaller than the typical grid block size in such reservoir simulations. For this reason, a scaleup method must be employed to translate the sub-grid system flow behaviour to the larger grid block system. Traditionally, flow within larger grid blocks is represented by saturation-dependent relative permeability pseudo- functions. The central purpose of the pseudo-curves is to reproduce the fluid and pressure distribution and displacement characteristics of the fine grid system on a coarse grid. The form of these pseudo-functions is usually derived from fine grid calculations which use estimated or experimentally measured curves for the relative permeability and capillary pressure. An alternative approach is to use a tensor-based formulation for coarse grid transmissibilities to model the upscaled fluid flow. P. 427