Numerical Subgrid Upscaling for Waterflood Simulations

Abstract

Abstract We present a subgrid-scale numerical technique for upscaling waterflood simulations. We scale up the usual parameters porosity and relative and absolute permeabilities, and also the location of wells and capillary pressure curves. Some of these are critical nonlinear terms that need to be resolved on the fine scale, or serious errors will result. Upscaling is achieved by explicitly decomposing the differential system into a coarse-grid-scale operator coupled to a subgrid-scale operator. The subgrid-scale operator is approximated as an operator localized in space to a coarse-grid element. An influence function (numerical Greens function) technique allows us to solve these subgrid-scale problems independently of the coarse-grid approximation. The coarse-grid problem is modified to take into account the subgrid-scale solution and solved as a large linear system of equations. Finally, the coarse scale solution is corrected on the subgrid-scale, providing a fine grid scale representation of the solution. In this approach, no explicit macroscopic coefficients nor pseudo-functions result. The method is easily seen to be optimally convergent in the case of a single linear parabolic equation. The method is several times faster than solving the fine-scale problem directly, generally more robust, and yet achieves good results as it requires no ad hoc assumptions at the coarse scale and retains all the physics of the original multiphase flow equations. Introduction There is a large and growing literature on upscaling techniques. We will not attempt a literature review here, but merely mention a few of the main techniques. The first and basic techniques developed involved in an essential way averaging or homogenization of physical parameters such as permeability (see, e.g., [1], [2], [3]). While such upscaling techniques can be very effective for purely linear problems, they are less satisfactory for nonlinear problems. They suffer from the elementary observation that a nonlinear function of an average is not the average of the nonlinear function. For example, over a coarse grid-block, the value of capillary pressure evaluated at the average saturation is not at all the same as the average over the grid-block of the capillary pressure. More sophisticated techniques have been developed to circumvent the inadequacies of simple averaging (see, e.g., [4], [5], [6], [7]), including the development of renormalization techniques to successively upscale to coarse levels, pseudo- functions, modified finite element basis functions, and explicit subgrid techniques that seek to improve the resolution of the coarse solution after it has been computed. These techniques all attempt in some way to represent fine-scale information on coarse scales in an indirect way, and some times require at least some information about the nature of the flow that is expected under field management conditions. Although most upscaling techniques are dynamic in that they respond to the changing state of the reservoir, many do so through anticipation of the possibilities. Often one needs some kind of closure assumption such as the imposition of local boundary conditions, the expected primary flow direction, or expected limits on certain parameters such as flow rates.