| All petroleum reservoirs contain variations in permeability and porosity at length scales that are smaller than the smallest grid blocks used in reservoir simulations. Thus any simulation of displacements at the reservoir scale should systematically average local variations so that representative values of properties such as permeability and porosity may be assigned to the grid blocks. In most simulations, however, only the most rudimentary averaging techniques are applied.;The general tensor scaling procedure successfully modeled several two-phase displacements. The accuracy of the scaling procedure did not deteriorate when the viscosity ratio was increased, nor when moderate amounts of capillary pressure were included in the model.;The effects of truncation error and grid orientation are analyzed. Truncation error partially offsets the smoothing of the velocity field which results from the coarser discretization. Methods are suggested to correct the degree of smoothing to physically realistic levels. Using diagrams of the characteristics calculated from two-dimensional simulations, one-dimensional frontal advance pseudofunctions are shown to be inadequate for modeling two-dimensional flow through heterogeneous reservoirs. Coarse two-dimensional grids can reproduce many of the essential features of the two-dimensional fine-grid simulations.;Existing methods for averaging permeability are restricted by assumptions of low variance, erogodicity or isotropy. They are not suitable for computing the properties of finite-difference grids. This study formulated and tested new methods based upon fine-grid finite-difference solutions of the single-phase steady-state flow equations. A general tensor scaling algorithm which uses a global solution of the fine grid equations has been shown to be more accurate than either pairwise connection methods or power law averaging. For the degree of scaling considered, the coarse-grid solutions are hundreds of times faster than fine grid solutions. |
